Atmospheric Chemistry and Physics: From Air Pollution to Climate Change [ebook ed.] 1119221161, 9781119221166

Table of contents :
Table of Contents

Preface to the First Edition xxiii
Preface to the Third Edition xxv

PART I The Atmosphere and Its Constituents
Chapter 1 | The Atmosphere 3
Chapter 2 | Atmospheric Trace Constituents 18

PART II Atmospheric Chemistry
Chapter 3 | Chemical Kinetics 69
Chapter 4 | Atmospheric Radiation and Photochemistry 88
Chapter 5 | Chemistry of the Stratosphere 119
Chapter 6 | Chemistry of the Troposphere 175
Chapter 7 | Chemistry of the Atmospheric Aqueous Phase 265

PART III Aerosols
Chapter 8 | Properties of the Atmospheric Aerosol 325
Chapter 9 | Dynamics of Single Aerosol Particles 362
Chapter 10 | Thermodynamics of Aerosols 396
Chapter 11 | Nucleation 448
Chapter 12 | Mass Transfer Aspects of Atmospheric Chemistry 493
Chapter 13 | Dynamics of Aerosol Populations 537
Chapter 14 | Atmospheric Organic Aerosols 573

PART IV Physical and Dynamic Meteorology, Cloud Physics, and Atmospheric Diffusion
Chapter 16 | Physical and Dynamic Meteorology 661
Chapter 17 | Cloud Physics 708
Chapter 18 | Atmospheric Diffusion 763

PART V Dry and Wet Deposition
Chapter 19 | Dry Deposition 829

PART VI The Global Atmosphere, Biogeochemical Cycles, and Climate
Chapter 21 | General Circulation of the Atmosphere 891
Chapter 22 | Global Cycles: Sulfur and Carbon 908
Chapter 23 | Global Climate 931
Chapter 24 | Aerosols and Climate 970

PART VII Chemical Transport Models and Statistical Models
Chapter 25 | Atmospheric Chemical Transport Models 1011
Chapter 26 | Statistical Models 1051

Citation preview

ATMOSPHERIC CHEMISTRY AND PHYSICS

ATMOSPHERIC CHEMISTRY AND PHYSICS From Air Pollution to Climate Change Third Edition John H. Seinfeld Spyros N. Pandis

Copyright  2016 by John Wiley & Sons, Inc. All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Names: Seinfeld, John H. | Pandis, Spyros N., 1963­ Title: Atmospheric chemistry and physics : from air pollution to climate change / John H. Seinfeld, Spyros N. Pandis. Description: Third edition. | Hoboken, New Jersey : John Wiley & Sons, Inc., [2016] | “A Wiley-Interscience publication.” | Includes index. Identifiers: LCCN 2015043236 (print) | LCCN 2015045406 (ebook) | ISBN 9781118947401 (cloth) | ISBN 9781119221166 (pdf) | ISBN 9781119221173 (epub) Subjects: LCSH: Atmospheric chemistry. | Air–Pollution–Environmental aspects. | Environmental chemistry. Classification: LCC QC879.6 .S45 2016 (print) | LCC QC879.6 (ebook) | DDC 551.51/1–dc23 LC record available at http://lccn.loc.gov/2015043236

Printed in the United States of America 10 9 8

7 6 5 4

3 2 1

To Benjamin and Elizabeth and Angeliki and Nikos

Contents

Preface to the First Edition Preface to the Third Edition

PART I Chapter 1

|

xxiii xxv

The Atmosphere and Its Constituents | The Atmosphere

3

1.1 1.2 1.3 1.4

History and Evolution of Earth’s Atmosphere 3 Climate 5 Layers of the Atmosphere 5 Pressure in the Atmosphere 7 1.4.1 Units of Pressure 7 1.4.2 Variation of Pressure with Height in the Atmosphere 7 1.5 Temperature in the Atmosphere 10 1.6 Expressing the Amount of a Substance in the Atmosphere 10 1.7 Airborne Particles 14 1.8 Spatial and Temporal Scales of Atmospheric Processes 14 Problems 16 References 17 Chapter 2

| Atmospheric Trace Constituents 2.1 2.2

2.3

2.4

Atmospheric Lifetime 19 Sulfur-Containing Compounds 23 2.2.1 Dimethyl Sulfide (CH3SCH3) 26 2.2.2 Carbonyl Sulfide (OCS) 26 2.2.3 Sulfur Dioxide (SO2) 27 Nitrogen-Containing Compounds 27 2.3.1 Nitrous Oxide (N2O) 28 2.3.2 Nitrogen Oxides (NOx = NO + NO2) 2.3.3 Reactive Odd Nitrogen (NOy) 30 2.3.4 Ammonia (NH3) 31 2.3.5 Amines 32 Carbon-Containing Compounds 32 2.4.1 Classification of Hydrocarbons 32 2.4.2 Methane 34 2.4.3 Volatile Organic Compounds 36 2.4.4 Biogenic Hydrocarbons 36 2.4.5 Carbon Monoxide 39 2.4.6 Carbon Dioxide 40

18

29

vii

viii

CONTENTS

2.5

Halogen-Containing Compounds 40 2.5.1 Methyl Chloride (CH3Cl) 42 2.5.2 Methyl Bromide (CH3Br) 42 2.6 Atmospheric Ozone 44 2.7 Particulate Matter (Aerosols) 47 2.7.1 Stratospheric Aerosol 48 2.7.2 Chemical Components of Tropospheric Aerosol 48 2.7.3 Cloud Condensation Nuclei (CCN) 49 2.7.4 Sizes of Atmospheric Particles 49 2.7.5 Carbonaceous Particles 51 2.7.6 Mineral Dust 53 2.7.7 Biomass Burning 53 2.7.8 Summary of Atmospheric Particulate Matter 54 2.8 Mercury 55 2.9 Emission Inventories 55 Appendix 2.1 US Air Pollution Legislation 56 Appendix 2.2 Hazardous Air Pollutants (Air Toxics) 57 Problems 59 References 61

PART II Chapter 3

|

Atmospheric Chemistry | Chemical Kinetics

69

3.1 3.2

Order of Reaction 69 Theories of Chemical Kinetics 71 3.2.1 Collision Theory 71 3.2.2 Transition State Theory 74 3.2.3 Potential Energy Surface for a Bimolecular Reaction 75 3.3 The Pseudo-Steady-State Approximation 76 3.4 Reactions of Excited Species 77 3.5 Termolecular Reactions 78 3.6 Chemical Families 81 3.7 Gas–Surface Reactions 83 Problems 84 References 87 Chapter 4

| Atmospheric Radiation and Photochemistry 4.1 Radiation 88 4.2 Radiative Flux in the Atmosphere 91 4.3 Beer − Lambert Law and Optical Depth 93 4.4 Actinic Flux 95 4.5 Atmospheric Photochemistry 97 4.6 Absorption of Radiation by Atmospheric Gases 100 4.7 Absorption by O2 and O3 105 4.8 Photolysis Rate as a Function of Altitude 109 4.9 Photodissociation of O3 to Produce O and O(1D) 112 4.10 Photodissociation of NO2 114 Problems 117 References 117

88

ix

CONTENTS

Chapter 5

| Chemistry of the Stratosphere

119

5.1 5.2

Chapman Mechanism 122 Nitrogen Oxide Cycles 129 5.2.1 Stratospheric Source of NOx from N2O 129 5.2.2 NOx Cycles 131 5.3 HOx Cycles 134 5.4 Halogen Cycles 139 5.4.1 Chlorine Cycles 140 5.4.2 Bromine Cycles 143 5.5 Reservoir Species and Coupling of the Cycles 144 5.6 Ozone Hole 146 5.6.1 Polar Stratospheric Clouds (PSCs) 149 5.6.2 PSCs and the Ozone Hole 150 5.6.3 Arctic Ozone Hole 153 5.7 Heterogeneous (Nonpolar) Stratospheric Chemistry 155 5.7.1 The Stratospheric Aerosol Layer 155 5.7.2 Heterogeneous Hydrolysis of N2O5 155 5.7.3 Effect of Volcanoes on Stratospheric Ozone 160 5.8 Summary of Stratospheric Ozone Depletion 162 5.9 Transport and Mixing in the Stratosphere 165 5.10 Ozone Depletion Potential 167 Problems 168 References 173 Chapter 6

| Chemistry of the Troposphere 6.1 6.2 6.3

6.4 6.5 6.6

6.7

6.8

6.9 6.10

175

Production of Hydroxyl Radicals in the Troposphere 176 Basic Photochemical Cycle of NO2, NO, and O3 179 Atmospheric Chemistry of Carbon Monoxide 181 6.3.1 Low-NOx Limit 183 6.3.2 High-NOx Limit 184 6.3.3 Ozone Production Efficiency 184 6.3.4 Theoretical Maximum Yield of Ozone from CO Oxidation 188 Atmospheric Chemistry of Methane 188 The NOx and NOy Families 192 6.5.1 Daytime Behavior 192 6.5.2 Nighttime Behavior 193 Ozone Budget of the Troposphere and Role of NOx 195 6.6.1 Ozone Budget of the Troposphere 195 6.6.2 Role of NOx 195 6.6.3 Global Hydroxyl Radical Budget 197 Tropospheric Reservoir Molecules 203 6.7.1 H2O2, CH3OOH, and Hydroperoxides 203 6.7.2 Nitrous Acid (HONO) 204 6.7.3 Peroxyacyl Nitrates (PANs) 204 Relative Roles of VOC and NOx in Ozone Formation 208 6.8.1 Importance of the VOC/NOx Ratio 208 6.8.2 Ozone Isopleth Plot 209 6.8.3 Weekend Ozone Effect 211 Simplified Organic/NOx Chemistry 212 Chemistry of Nonmethane Organic Compounds in the Troposphere 214

x

CONTENTS

6.10.1 Alkanes 215 6.10.2 Alkenes 222 6.10.3 Aromatics 228 6.10.4 Aldehydes 230 6.10.5 Ketones 230 6.10.6 Ethers 231 6.10.7 Alcohols 231 6.10.8 Tropospheric Lifetimes of Organic Compounds 232 6.11 Atmospheric Chemistry of Biogenic Hydrocarbons 233 6.11.1 Atmospheric Chemistry of Isoprene 233 6.11.2 Monoterpenes (α-Pinene) 241 6.12 Atmospheric Chemistry of Reduced Nitrogen Compounds 244 6.12.1 Amines 245 6.12.2 Nitriles 246 6.12.3 Nitrites 246 6.13 Atmospheric Chemistry (Gas Phase) of Sulfur Compounds 246 6.13.1 Sulfur Oxides 246 6.13.2 Reduced Sulfur Compounds (Dimethyl Sulfide) 247 6.14 Tropospheric Chemistry of Halogen Compounds 249 6.14.1 Chemical Cycles of Halogen Species 249 6.14.2 Tropospheric Chemistry of CFC Replacements: Hydrofluorocarbons (HFCs) and Hydrochlorofluorocarbons (HCFCs) 251 6.15 Atmospheric Chemistry of Mercury 253 Appendix 6 Organic Functional Groups 254 Problems 256 References 259 Chapter 7

| Chemistry of the Atmospheric Aqueous Phase Liquid Water in the Atmosphere 265 Absorption Equilibria and Henry’s Law 268 Aqueous-Phase Chemical Equilibria 271 7.3.1 Water 271 7.3.2 Carbon Dioxide–Water Equilibrium 272 7.3.3 Sulfur Dioxide–Water Equilibrium 274 7.3.4 Ammonia–Water Equilibrium 278 7.3.5 Nitric Acid–Water Equilibrium 280 7.3.6 Equilibria of Other Important Atmospheric Gases 281 7.4 Aqueous-Phase Reaction Rates 284 7.5 S(IV)–S(VI) Transformation and Sulfur Chemistry 286 7.5.1 Oxidation of S(IV) by Dissolved O3 286 7.5.2 Oxidation of S(IV) by Hydrogen Peroxide 289 7.5.3 Oxidation of S(IV) by Organic Peroxides 290 7.5.4 Uncatalyzed Oxidation of S(IV) by O2 291 7.5.5 Oxidation of S(IV) by O2 Catalyzed by Iron and Manganese 291 7.5.6 Comparison of Aqueous-Phase S(IV) Oxidation Paths 293 7.6 Dynamic Behavior of Solutions with Aqueous-Phase Chemical Reactions 7.6.1 Closed System 296 7.6.2 Calculation of Concentration Changes in a Droplet with Aqueous-Phase Reactions 298 Appendix 7.1 Thermodynamic and Kinetic Data 301

265

7.1 7.2 7.3

295

xi

CONTENTS

Appendix 7.2 Additional Aqueous-Phase Sulfur Chemistry 305 7A.1 S(IV) Oxidation by the OH Radical 305 7A.2 Oxidation of S(IV) by Oxides of Nitrogen 308 7A.3 Reaction of Dissolved SO2 with HCHO 309 Appendix 7.3 Aqueous-Phase Nitrite and Nitrate Chemistry 311 7A.4 NOx Oxidation 311 7A.5 Nitrogen Radicals 311 Appendix 7.4 Aqueous-Phase Organic Chemistry 312 Appendix 7.5 Oxygen and Hydrogen Chemistry 313 Problems 314 References 317

PART III Chapter 8

| Aerosols | Properties of the Atmospheric Aerosol

325

8.1

The Size Distribution Function 325 8.1.1 The Number Distribution nN(Dp) 328 8.1.2 The Surface Area, Volume, and Mass Distributions 330 8.1.3 Distributions Based on ln Dp and log Dp 331 8.1.4 Relating Size Distributions Based on Different Independent Variables 333 8.1.5 Properties of Size Distributions 334 8.1.6 Definition of the Lognormal Distribution 335 8.1.7 Plotting the Lognormal Distribution 338 8.1.8 Properties of the Lognormal Distribution 339 8.2 Ambient Aerosol Size Distributions 342 8.2.1 Urban Aerosols 343 8.2.2 Marine Aerosols 344 8.2.3 Rural Continental Aerosols 347 8.2.4 Remote Continental Aerosols 348 8.2.5 Free Tropospheric Aerosols 348 8.2.6 Polar Aerosols 349 8.2.7 Desert Aerosols 349 8.3 Aerosol Chemical Composition 352 8.4 Spatiotemporal Variation 354 Problems 357 References 359 Chapter 9

| Dynamics of Single Aerosol Particles 9.1

9.2

9.3 9.4 9.5

Continuum and Noncontinuum Dynamics: the Mean Free Path 362 9.1.1 Mean Free Path of a Pure Gas 363 9.1.2 Mean Free Path of a Gas in a Binary Mixture 365 The Drag on a Single Particle: Stokes’ Law 368 9.2.1 Corrections to Stokes’ Law: the Drag Coefficient 371 9.2.2 Stokes’ Law and Noncontinuum Effects: Slip Correction Factor 371 Gravitational Settling of an Aerosol Particle 372 Motion of an Aerosol Particle in an External Force Field 376 Brownian Motion of Aerosol Particles 376 9.5.1 Particle Diffusion 379

362

xii

CONTENTS

9.5.2 Aerosol Mobility and Drift Velocity 381 9.5.3 Mean Free Path of an Aerosol Particle 384 9.6 Aerosol and Fluid Motion 385 9.6.1 Motion of a Particle in an Idealized Flow (90° Corner) 9.6.2 Stop Distance and Stokes Number 387 9.7 Equivalent Particle Diameters 388 9.7.1 Volume Equivalent Diameter 388 9.7.2 Stokes Diameter 390 9.7.3 Classical Aerodynamic Diameter 391 9.7.4 Electrical Mobility Equivalent Diameter 393 Problems 393 References 394

386

Chapter 10 | Thermodynamics of Aerosols

396

10.1

Thermodynamic Principles 396 10.1.1 Internal Energy and Chemical Potential 396 10.1.2 The Gibbs Free Energy G 398 10.1.3 Conditions for Chemical Equilibrium 400 10.1.4 Chemical Potentials of Ideal Gases and Ideal-Gas Mixtures 402 10.1.5 Chemical Potential of Solutions 404 10.1.6 The Equilibrium Constant 408 10.2 Aerosol Liquid Water Content 409 10.2.1 Chemical Potential of Water in Atmospheric Particles 411 10.2.2 Temperature Dependence of the DRH 412 10.2.3 Deliquescence of Multicomponent Aerosols 415 10.2.4 Crystallization of Single- and Multicomponent Salts 419 10.3 Equilibrium Vapor Pressure Over a Curved Surface: the Kelvin Effect 419 10.4 Thermodynamics of Atmospheric Aerosol Systems 423 10.4.1 The H2SO4–H2O System 423 10.4.2 The Sulfuric Acid–Ammonia–Water System 427 10.4.3 The Ammonia–Nitric Acid–Water System 430 10.4.4 The Ammonia–Nitric Acid–Sulfuric Acid–Water System 434 10.4.5 Other Inorganic Aerosol Species 439 10.4.6 Organic Aerosol 440 10.5 Aerosol Thermodynamic Models 440 Problems 442 References 443 Chapter 11 | Nucleation 11.1

11.2

Classical Theory of Homogeneous Nucleation: Kinetic Approach 449 11.1.1 The Forward Rate Constant βi 452 11.1.2 The Reverse Rate Constant γi 453 11.1.3 Derivation of the Nucleation Rate 453 Classical Homogeneous Nucleation Theory: Constrained Equilibrium Approach 457 11.2.1 Free Energy of i-mer Formation 457 11.2.2 Constrained Equilibrium Cluster Distribution 459 11.2.3 The Evaporation Coefficient γi 461 11.2.4 Nucleation Rate 461

448

xiii

CONTENTS

11.3 11.4

Recapitulation of Classical Theory 464 Experimental Measurement of Nucleation Rates 465 11.4.1 Upward Thermal Diffusion Cloud Chamber 466 11.4.2 Fast Expansion Chamber 466 11.4.3 Turbulent Mixing Chambers 467 11.5 Modifications of the Classical Theory and More Rigorous Approaches 467 11.6 Binary Homogeneous Nucleation 468 11.7 Binary Nucleation in the H2SO4–H2O System 473 11.8 Nucleation on an Insoluble Foreign Surface 475 11.9 Ion-Induced Nucleation 478 11.10 Atmospheric New-Particle Formation 480 11.10.1 Molecular Constituency of New Particles 481 11.10.2 New-Particle Growth Rates 482 11.10.3 CLOUD Studies of Atmospheric Nucleation 482 11.10.4 Atmospheric Nucleation by Organic Species 487 Appendix 11 The Law of Mass Action 487 Problems 489 References 490 Chapter 12 | Mass Transfer Aspects of Atmospheric Chemistry 12.1

Mass and Heat Transfer to Atmospheric Particles 493 12.1.1 The Continuum Regime 493 12.1.2 The Kinetic Regime 497 12.1.3 The Transition Regime 497 12.1.4 The Accommodation Coefficient 500 12.2 Mass Transport Limitations in Aqueous-Phase Chemistry 503 12.2.1 Characteristic Time for Gas-Phase Diffusion to a Particle 505 12.2.2 Characteristic Time to Achieve Equilibrium at the Gas–Liquid Interface 506 12.2.3 Characteristic Time of Aqueous Dissociation Reactions 508 12.2.4 Characteristic Time of Aqueous-Phase Diffusion in a Droplet 510 12.2.5 Characteristic Time for Aqueous-Phase Chemical Reactions 511 12.3 Mass Transport and Aqueous-Phase Chemistry 511 12.3.1 Gas-Phase Diffusion and Aqueous-Phase Reactions 512 12.3.2 Aqueous-Phase Diffusion and Reaction 514 12.3.3 Interfacial Mass Transport and Aqueous-Phase Reactions 515 12.3.4 Application to the S(IV)–Ozone Reaction 517 12.3.5 Application to the S(IV)–Hydrogen Peroxide Reaction 519 12.3.6 Calculation of Aqueous-Phase Reaction Rates 520 12.3.7 An Aqueous-Phase Chemistry/Mass Transport Model 525 12.4 Mass Transfer to Falling Drops 526 12.5 Characteristic Time for Atmospheric Aerosol Equilibrium 527 12.5.1 Solid Aerosol Particles 528 12.5.2 Aqueous Aerosol Particles 529 Appendix 12 Solution of the Transient Gas-Phase Diffusion Problem: Equations (12.4)–(12.7) 532 Problems 533 References 535

493

xiv

CONTENTS

Chapter 13 | Dynamics of Aerosol Populations 13.1

Mathematical Representations of Aerosol Size Distributions 13.1.1 Discrete Distribution 537 13.1.2 Continuous Distribution 538 13.2 Condensation 538 13.2.1 The Condensation Equation 538 13.2.2 Solution of the Condensation Equation 540 13.3 Coagulation 544 13.3.1 Brownian Coagulation 544 13.3.2 The Coagulation Equation 551 13.3.3 Solution of the Coagulation Equation 553 13.4 The Discrete General Dynamic Equation 557 13.5 The Continuous General Dynamic Equation 558 Appendix 13.1 Additional Mechanisms of Coagulation 560 13A.1 Coagulation in Laminar Shear Flow 560 13A.2 Coagulation in Turbulent Flow 560 13A.3 Coagulation from Gravitational Settling 561 13A.4 Brownian Coagulation and External Force Fields 562 Appendix 13.2 Solution of (13.73) 567 Problems 568 References 571

537 537

Chapter 14 | Atmospheric Organic Aerosols 14.1

Chemistry of Secondary Organic Aerosol Formation 574 14.1.1 Oxidation State of Organic Compounds 576 14.1.2 Generation of Highly Oxygenated Species by Autoxidation 579 14.2 Volatility of Organic Compounds 582 14.3 Idealized Description of Secondary Organic Aerosol Formation 583 14.3.1 Noninteracting Secondary Organic Aerosol Compounds 583 14.3.2 Formation of Binary Ideal Solution with Preexisting Aerosol 586 14.3.3 Formation of Binary Ideal Solution with Other Organic Vapor 588 14.4 Gas–Particle Partitioning 590 14.4.1 Gas–Particle Equilibrium 590 14.4.2 Effect of Aerosol Water on Gas-Particle Partitioning 594 14.5 Models of SOA Formation and Evolution 596 14.5.1 The Volatility Basis Set 597 14.5.2 Two-Dimensional SOA Models 603 14.6 Primary Organic Aerosol 605 14.7 The Physical State of Organic Aerosols 608 14.8 SOA Particle-Phase Chemistry 610 14.8.1 Particle-Phase Accretion Reactions 612 14.8.2 Heterogeneous Gas-Aerosol Reactions 612 14.9 Aqueous-Phase Secondary Organic Aerosol Formation 615 14.9.1 Gas- versus Aqueous-Phase Routes to SOA 616 14.9.2 Sources of OH Radicals in the Aqueous Phase 618 14.9.3 Glyoxal as a Source of aqSOA 619 14.10 Estimates of the Global Budget of Atmospheric Organic Aerosol 622 14.10.1 Estimate Based on Total VOC Emissions 622 14.10.2 Sulfate Lifetime and Ratio of Organic to Sulfate 622

573

xv

CONTENTS

14.10.3 Atmospheric Burden and Lifetime of SOA 14.10.4 Satellite Measurements 623 Problems 623 References 626

623

Chapter 15 | Interaction of Aerosols with Radiation Scattering and Absorption of Light by Small Particles 633 15.1.1 Rayleigh Scattering Regime 638 15.1.2 Geometric Scattering Regime 640 15.1.3 Scattering Phase Function 640 15.1.4 Extinction by an Ensemble of Particles 640 15.2 Visibility 644 15.3 Scattering, Absorption, and Extinction Coefficients From Mie Theory 647 15.4 Calculated Visibility Reduction Based on Atmospheric Data Appendix 15 Calculation of Scattering and Extinction Coefficients by Mie Theory 654 Problems 654 References 656

633

15.1

651

PART IV | Physical and Dynamic Meteorology, Cloud Physics, and Atmospheric Diffusion Chapter 16 | Physical and Dynamic Meteorology Temperature in the Lower Atmosphere 661 Atmospheric Stability 665 The Moist Atmosphere 670 16.3.1 The Gas Constant for Moist Air 671 16.3.2 Level of Cloud Formation: The Lifting Condensation Level 671 16.3.3 Dew-point and Wet-Bulb Temperatures 673 16.3.4 The Moist Adiabatic Lapse Rate 675 16.3.5 Stability of Moist Air 679 16.3.6 Convective Available Potential Energy (CAPE) 680 16.3.7 Thermodynamic Diagrams 681 16.4 Basic Conservation Equations for the Atmospheric Surface Layer 683 16.4.1 Turbulence 687 16.4.2 Equations for the Mean Quantities 688 16.4.3 Mixing-Length Models for Turbulent Transport 690 16.5 Variation of Wind with Height in the Atmosphere 692 16.5.1 Mean Velocity in the Adiabatic Surface Layer over a Smooth Surface 16.5.2 Mean Velocity in the Adiabatic Surface Layer over a Rough Surface 16.5.3 Mean Velocity Profiles in the Nonadiabatic Surface Layer 695 16.5.4 The Pasquill Stability Classes—Estimation of L 698 16.5.5 Empirical Equation for the Mean Windspeed 700 Appendix 16.1 Properties of Water and Water Solutions 701 16A.1 Specific Heat of Water and Ice 701 16A.2 Latent Heats of Vaporization and Melting for Water 701 16A.3 Water Surface Tension 701

661

16.1 16.2 16.3

693 694

xvi

CONTENTS

Appendix 16.2 Derivation of the Basic Equations of Surface-Layer Atmospheric Fluid Mechanics 702 Problems 705 References 706 Chapter 17 | Cloud Physics

708

17.1

Equilibrium of Water Droplets in the Atmosphere 708 17.1.1 Equilibrium of a Pure Water Droplet 708 17.1.2 Equilibrium of a Flat Water Solution 710 17.1.3 Atmospheric Equilibrium of an Aqueous Solution Drop 712 17.1.4 Atmospheric Equilibrium of an Aqueous Solution Drop Containing an Insoluble Substance 717 17.2 Cloud and Fog Formation 719 17.2.1 Isobaric Cooling 720 17.2.2 Adiabatic Cooling 720 17.2.3 A Simplified Mathematical Description of Cloud Formation 721 17.3 Growth Rate of Individual Cloud Droplets 723 17.4 Growth of a Droplet Population 726 17.5 Cloud Condensation Nuclei 730 17.5.1 Ambient CCN 733 17.5.2 The Hygroscopic Parameter Kappa 733 17.6 Cloud Processing of Aerosols 736 17.6.1 Nucleation Scavenging of Aerosols by Clouds 736 17.6.2 Chemical Composition of Cloud Droplets 737 17.6.3 Nonraining Cloud Effects on Aerosol Concentrations 739 17.6.4 Interstitial Aerosol Scavenging by Cloud Droplets 742 17.7 Other Forms of Water in the Atmosphere 743 17.7.1 Ice Clouds 743 17.7.2 Rain 747 Appendix 17 Extended Köhler Theory 751 17A.1 Modified Form of Köhler Theory for a Soluble Trace Gas 751 17A.2 Modified Form of Köhler Theory for a Slightly Soluble Substance 754 17A.3 Modified Form of Köhler Theory for a Surface-Active Solute 755 17A.4 Examples 756 Problems 759 References 760 Chapter 18 | Atmospheric Diffusion 18.1 18.2 18.3 18.4

18.5 18.6

Eulerian Approach 763 Lagrangian Approach 766 Comparison of Eulerian and Lagrangian Approaches 767 Equations Governing the Mean Concentration of Species in Turbulence 18.4.1 Eulerian Approaches 767 18.4.2 Lagrangian Approaches 769 Solution of the Atmospheric Diffusion Equation for an Instantaneous Source 771 Mean Concentration from Continuous Sources 772 18.6.1 Lagrangian Approach 772 18.6.2 Eulerian Approach 776 18.6.3 Summary of Continuous Point Source Solutions 777

763

767

xvii

CONTENTS

18.7

Statistical Theory of Turbulent Diffusion 778 18.7.1 Qualitative Features of Atmospheric Diffusion 778 18.7.2 Motion of a Single Particle Relative to a Fixed Axis 780 18.8 Summary of Atmospheric Diffusion Theories 783 18.9 Analytical Solutions for Atmospheric Diffusion: the Gaussian Plume Equation and Others 784 18.9.1 Gaussian Concentration Distributions 784 18.9.2 Derivation of the Gaussian Plume Equation as a Solution of the Atmospheric Diffusion Equation 786 18.9.3 Summary of Gaussian Point Source Diffusion Formulas 791 18.10 Dispersion Parameters in Gaussian Models 791 18.10.1 Correlations for σy and σz Based on Similarity Theory 791 18.10.2 Correlations for σy and σz Based on Pasquill Stability Classes 795 18.11 Plume Rise 796 18.12 Functional Forms of Mean Windspeed and Eddy Diffusivities 798 18.12.1 Mean Windspeed 800 18.12.2 Vertical Eddy Diffusion Coefficient Kzz 800 18.12.3 Horizontal Eddy Diffusion Coefficients Kxx and Kyy 803 18.13 Solutions of the Steady-State Atmospheric Diffusion Equation 803 18.13.1 Diffusion from a Point Source 804 18.13.2 Diffusion from a Line Source 805 Appendix 18.1 Further Solutions of Atmospheric Diffusion Problems 807 18A.1 Solution of (18.29)–(18.31) 807 18A.2 Solution of (18.50) and (18.51) 809 18A.3 Solution of (18.59)–(18.61) 810 Appendix 18.2 Analytical Properties of the Gaussian Plume Equation 811 Problems 815 References 823

PART V | Dry and Wet Deposition Chapter 19 | Dry Deposition 19.1 19.2 19.3 19.4

Deposition Velocity 829 Resistance Model for Dry Deposition 830 Aerodynamic Resistance 834 Quasilaminar Resistance 835 19.4.1 Gases 836 19.4.2 Particles 836 19.5 Surface Resistance 839 19.5.1 Surface Resistance for Dry Deposition of Gases to Water 841 19.5.2 Surface Resistance for Dry Deposition of Gases to Vegetation 845 19.6 Measurement of Dry Deposition 849 19.6.1 Direct Methods 849 19.6.2 Indirect Methods 850 19.6.3 Comparison of Methods 851 19.7 Some Comments on Modeling and Measurement of Dry Deposition 851 Problems 852 References 854

829

xviii

CONTENTS

Chapter 20 | Wet Deposition

856

20.1 20.2

General Representation of Atmospheric Wet Removal Processes 856 Below-Cloud Scavenging of Gases 860 20.2.1 Below-Cloud Scavenging of an Irreversibly Soluble Gas 861 20.2.2 Below-Cloud Scavenging of a Reversibly Soluble Gas 864 20.3 Precipitation Scavenging of Particles 868 20.3.1 Raindrop–Aerosol Collision Efficiency 870 20.3.2 Scavenging Rates 871 20.4 In-Cloud Scavenging 873 20.5 Acid Deposition 874 20.5.1 Acid Rain Overview 874 20.5.2 Surface Water Acidification 876 20.5.3 Cloudwater Deposition 877 20.5.4 Fogs and Wet Deposition 877 20.6 Acid Deposition Process Synthesis 878 20.6.1 Chemical Species Involved in Acid Deposition 878 20.6.2 Dry versus Wet Deposition 878 20.6.3 Chemical Pathways for Sulfate and Nitrate Production 878 20.6.4 Source–Receptor Relationships 879 20.6.5 Linearity 880 Problems 881 References 886

PART VI | The Global Atmosphere, Biogeochemical Cycles, and Climate Chapter 21 | General Circulation of the Atmosphere

891

21.1 21.2 21.3 21.4

Hadley Cell 893 Ferrell Cell and Polar Cell 893 Coriolis Force 895 Geostrophic Windspeed 897 21.4.1 Buys Ballot’s Law 899 21.4.2 Ekman Spiral 900 21.5 The Thermal Wind Relation 902 21.6 Stratospheric Dynamics 905 21.7 The Hydrologic Cycle 905 Problems 906 References 907 Chapter 22 | Global Cycles: Sulfur and Carbon 22.1 22.2

The Atmospheric Sulfur Cycle 908 The Global Carbon Cycle 912 22.2.1 Carbon Dioxide 912 22.2.2 Compartmental Model of the Global Carbon Cycle 914 22.2.3 Atmospheric Lifetime of CO2 921 22.3 Solution for a Steady-State Four-Compartment Model of the Atmosphere 923 Problems 927 References 929

908

xix

CONTENTS

Chapter 23 | Global Climate

931

23.1 23.2

Earth’s Energy Balance 931 Radiative Forcing 933 23.2.1 Climate Sensitivity 934 23.2.2 Climate Feedbacks 935 23.2.3 Timescales of Climate Change 935 23.3 The Greenhouse Effect 936 23.4 Climate-Forcing Agents 942 23.4.1 Solar Irradiance 942 23.4.2 Greenhouse Gases 945 23.4.3 Radiative Efficiencies of Greenhouse Gases 946 23.4.4 Aerosols 946 23.4.5 Summary of IPCC (2013) Estimated Forcing 947 23.4.6 The Preindustrial Atmosphere 948 23.5 Cosmic Rays and Climate 949 23.6 Climate Sensitivity 950 23.7 Simplified Dynamic Description of Climate Forcing and Response 951 23.7.1 Response to a Perturbation of Earth’s Radiative Equilibrium 951 23.7.2 Physical Interpretation of Feedback Factors 954 23.8 Climate Feedbacks 955 23.8.1 Water Vapor Feedback 955 23.8.2 Lapse Rate Feedback 956 23.8.3 Cloud Feedback 956 23.8.4 Arctic Sea Ice Feedback 958 23.8.5 Summary of Feedbacks 958 23.9 Relative Radiative Forcing Indices 960 23.10 Atmospheric Chemistry and Climate Change 961 23.10.1 Indirect Chemical Impacts 962 23.10.2 Atmospheric Lifetimes and Adjustment Times 963 23.11 Conclusion 964 Problems 965 References 967 Chapter 24 | Aerosols and Climate 24.1 24.2 24.3 24.4 24.5 24.6 24.7 24.8

Scattering–Absorbing Model of an Aerosol Layer 972 Cooling Versus Heating of an Aerosol Layer 975 Scattering Model of an Aerosol Layer for a Nonabsorbing Aerosol 977 Upscatter Fraction 979 Optical Depth and Column Forcing 981 Internal and External Mixtures 985 Top-of-the-Atmosphere Versus Surface Forcing 987 Indirect Effects of Aerosols on Climate 990 24.8.1 Stratocumulus Clouds 991 24.8.2 Simplified Model for Cloud Albedo 993 24.8.3 Albedo Susceptibility: Simplified Model 995

970

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CONTENTS

24.8.4 Albedo Susceptibility: Additional Considerations 997 24.8.5 A General Equation for Cloud Albedo Susceptibility 999 24.8.6 Estimating Indirect Aerosol Forcing on Climate 1003 Problems 1004 References 1004

PART VII

| Chemical Transport Models and Statistical Models

Chapter 25 | Atmospheric Chemical Transport Models

1011

25.1

Introduction 1011 25.1.1 Model Types 1012 25.1.2 Types of Atmospheric Chemical Transport Models 1013 25.2 Box Models 1014 25.2.1 The Eulerian Box Model 1015 25.2.2 A Lagrangian Box Model 1017 25.3 Three-Dimensional Atmospheric Chemical Transport Models 1020 25.3.1 Coordinate System—Uneven Terrain 1020 25.3.2 Initial Conditions 1022 25.3.3 Boundary Conditions 1023 25.4 One-Dimensional Lagrangian Models 1024 25.5 Other Forms of Chemical Transport Models 1026 25.5.1 Atmospheric Diffusion Equation Expressed in Terms of Mixing Ratio 1026 25.5.2 Pressure-Based Coordinate System 1029 25.5.3 Spherical Coordinates 1031 25.6 Numerical Solution of Chemical Transport Models 1031 25.6.1 Coupling Problem—Operator Splitting 1032 25.6.2 Chemical Kinetics 1037 25.6.3 Diffusion 1041 25.6.4 Advection 1042 25.7 Model Evaluation 1046 25.8 Response of Organic and Inorganic Aerosols to Changes in Emission Problems 1048 References 1050 Chapter 26 | Statistical Models 26.1 26.2

26.3

26.4

Receptor Modeling Methods 1051 Chemical Mass Balance (CMB) 1054 26.2.1 CMB Evaluation 1058 26.2.2 CMB Resolution 1059 26.2.3 CMB Codes 1059 Factor Analysis 1059 26.3.1 Principal-Component Analysis (PCA) 1061 26.3.2 Positive Matrix Factorization (PMF) 1064 Methods Incorporating Wind Information 1067 26.4.1 Potential Source Contribution Function (PSCF) 26.4.2 Empirical Orthogonal Function (EOF) 1070

1047

1051

1068

xxi

CONTENTS

26.5

Probability Distributions for Air Pollutant Concentrations 1072 26A.1 The Lognormal Distribution 1073 26A.2 The Weibull Distribution 1074 26.6 Estimation of Parameters in the Distributions 1074 26A.1 Method of Quantiles 1075 26A.2 Method of Moments 1076 26.7 Order Statistics of Air Quality Data 1078 26A.1 Basic Notions and Terminology of Order Statistics 1078 26A.2 Extreme Values 1079 26.8 Exceedances of Critical Levels 1080 26.9 Alternative Forms of Air Quality Standards 1080 26.10 Relating Current and Future Air Pollutant Statistical Distributions Problems 1085 References 1087

Appendixes Appendix A: | Units and Physical Constants 1091 A.1 SI Base Units 1091 A.2 SI Derived Units 1092 A.3 Fundamental Physical Constants 1094 A.4 Properties of the Atmosphere and Water 1094 A.5 Units for Representing Chemical Reactions 1096 A.6 Concentrations in the Aqueous Phase 1096 A.7 Symbols Denoting Concentration 1097 References 1097 Appendix B: | Rate Constants of Atmospheric Chemical Reactions References 1106 Appendix C: | Abbreviations Index

1112

1107

1098

1083

Preface to the First Edition The study of atmospheric chemistry as a scientific discipline goes back to the eighteenth century, when the principal issue was identifying the major chemical components of the atmosphere, nitrogen, oxygen, water, carbon dioxide, and the noble gases. In the late nineteenth and early twentieth centuries attention turned to the so-called trace gases, species present at less than 1 part per million parts of air by volume (1 μmol per mole). We now know that the atmosphere contains a myriad of trace species, some at levels as low as 1 part per trillion parts of air. The role of trace species is disproportionate to their atmospheric abundance; they are responsible for phenomena ranging from urban photochemical smog, to acid deposition, to stratospheric ozone depletion, to potential climate change. Moreover, the composition of the atmosphere is changing; analysis of air trapped in ice cores reveals a record of striking increases in the longlived so-called greenhouse gases, carbon dioxide (CO2), methane (CH4), and nitrous oxide (N2O). Within the last century, concentrations of tropospheric ozone (O3), sulfate SO24 , and carbonaceous aerosols in the Northern Hemisphere have increased significantly. There is evidence that all these changes are altering the basic chemistry of the atmosphere. Atmospheric chemistry occurs within a fabric of profoundly complicated atmospheric dynamics. The results of this coupling of dynamics and chemistry are often unexpected. Witness the unique combination of dynamical forces that lead to a wintertime polar vortex over Antarctica, with the concomitant formation of polar stratospheric clouds that serve as sites for heterogeneous chemical reactions involving chlorine compounds resulting from anthropogenic chlorofluorocarbons—all leading to the near total depletion of stratospheric ozone over the South Pole each spring; witness the nonlinear, and counterintuitive, dependence of the amount of ozone generated by reactions involving hydro­ carbons and oxides of nitrogen (NOx) at the urban and regional scales—although both hydrocarbons and NOx are ozone precursors, situations exist where continuous emission of more and more NOx actually leads to less ozone. The chemical constituents of the atmosphere do not go through their lifecycles independently; the cycles of the various species are linked together in a complex way. Thus a perturbation of one component can lead to significant, and nonlinear, changes to other components and to feedbacks that can amplify or damp the original perturbation. In many respects, at once both the most important and the most paradoxical trace gas in the atmosphere is ozone (O3). High in the stratosphere, ozone screens living organisms from biologically harmful solar ultraviolet radiation; ozone at the surface, in the troposphere, can produce adverse effects on human health and plants when present at levels elevated above natural. At the urban and regional scales, significant policy issues concern how to decrease ozone levels by controlling the ozone precursors—volatile organic compounds and oxides of nitrogen. At the global scale, understanding both the natural ozone chemistry of the troposphere and the causes of continually increasing back­ ground tropospheric ozone levels is a major goal. Aerosols are particles suspended in the atmosphere. They arise directly from emissions of particles and from the conversion of volatile organic compounds to particles in the atmosphere. At elevated levels they inhibit visibility and are a human health hazard. There is a growing body of epidemiological data suggesting that increasing levels of aerosols may cause a significant increase in human mortality. For many years it was assumed that atmospheric aerosols did not interact in any appreciable way with the cycles of trace gases. We now know that particles in the air affect climate and interact chemically in heretofore unrecognized ways with atmospheric gases. Volcanic aerosols in the stratosphere, for example, participate in the catalytic destruction of ozone by chlorine compounds, not directly, but

xxiii

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PREFACE TO THE FIRST EDITION

through the intermediary of NOx chemistry. Aerosols reflect solar radiation back to space and, in so doing, cool the planet Earth. Aerosols are also the nuclei around which clouds droplets form—no aerosols, no clouds. Clouds are one of the most important elements or our climate system, so the effect of increasing global aerosol levels on Earth’s cloudiness is a key problem in climate. Historically the study of urban air pollution and its effects occurred more or less separately from that of the chemistry of Earth’s atmosphere as a whole. Similarly, in its early stages, climate research focused exclusively on CO2, without reference to effects on the underlying chemistry of the atmosphere and their feedbacks on climate itself. It is now recognized, in quantitative scientific terms, that Earth’s atmosphere is a continuum of spatial scales in which the urban atmosphere, the remote troposphere, the marine boundary layer, and the stratosphere are merely points from the smallest turbulent eddies and the fastest timescales of free-radical chemistry to global circulations and the decadal timescales of the longest-lived trace gases. The object of this book is to provide a rigorous, comprehensive treatment of the chemistry of the atmosphere, including the formation, growth, dynamics, and properties of aerosols; the meteorology of air pollution; the transport, diffusion, and removal of species in the atmosphere; the formation and chemistry of clouds; the interaction of atmospheric chemistry and climate; the radiative and climatic effects of gases and particles; and the formulation of mathematical chemical/transport models of the atmosphere. Each of these elements is covered in detail in the present volume. In each area the central results are developed from first principles. In this way, the reader will gain a significant understanding of the science underlying the description of atmospheric processes and will be able to extend theories and results beyond those for which we have space here. The book assumes that the reader has completed introductory courses in thermodynamics, transport phenomena (fluid mechanics and heat and mass transfer), and engineering mathematics (differential equations). Thus the treatment is aimed at the senior or first-year graduate level in typical engineering curricula as well as in meteorology and atmospheric science programs. The book is intended to serve as a textbook for a course in atmospheric science that might vary in length from one quarter or semester to a full academic year. Aside from its use as a course textbook, the book will serve as a comprehensive reference book for professionals as well as for those from traditional engineering and science disciplines. Two types of appendixes are given: those of a general nature appear at the end of the book and are designated by letters; those of a nature specific to a certain chapter appear with that chapter and are numbered according to the associated chapter. Numerous problems are provided to enable readers to evaluate their understanding of the material. In many cases the problems have been chosen to extend the results given in the chapter to new situations. The problems are coded with a “degree of difficulty” for the benefit of the student and the instructor. The subscript designation “A” (e.g., 1.1A in the “Problems” section of Chapter 1) indicates a problem that involves a straightforward application of material in the text. Those problems denoted “B” require some extension of the ideas in the text. Problems designated “C” encourage the reader to apply concepts from the book to current problems in atmospheric science and go somewhat beyond the level of “B” problems. Finally, those problems denoted “D” are of a degree of difficulty corresponding to “C” but generally require development of a computer program for their solution. JOHN H. SEINFELD SPYROS N. PANDIS

Preface to the Third Edition The Third Edition contains a number of significant changes since the Second Edition. The treatment of tropospheric chemistry (Chapter 6) has been expanded and updated. A major section on the atmospheric chemistry of biogenic hydrocarbons has been added, including a detailed treatment of isoprene chemistry. Sections on amine chemistry and the atmospheric chemistry of mercury have been added. An expanded and updated treatment of atmospheric new-particle formation (nucleation) has been added to Chapter 11. Chapter 14, “Atmospheric Organic Aerosols,” has been completely rewritten, reflecting the major advances in our understanding of how organic aerosols form and evolve that have emerged since the publication of the Second Edition. This includes new understanding of the nature of primary organic aerosols, of particle-phase chemistry, and of secondary organic aerosol formation in the aqueous phase. This Third Edition contains a major expansion of physical and dynamic meteorology (Chapter 16), including a rigorous, self-contained treatment of atmospheric temperature profiles, atmospheric stability, and moist atmospheric thermodynamics. Treatment of the global carbon cycle (Chapter 22) has been updated and expanded. Chapter 23, “Global Climate,” is an entirely new chapter with a self-contained comprehensive treatment of radiative forcing of climate by greenhouse gases, solar output changes, and a development of Earth’s climate sensitivity and climate feedbacks, including water vapor, clouds, and atmospheric lapse rate. A significantly expanded and updated treatment of aerosol– cloud relationships in climate has been added to Chapter 24, “Aerosols and Climate.” Using strato­ cumulus clouds as the basis, the theory of the effect of aerosol perturbations on cloud properties is developed and illustrated. More examples, offset by vertical bars, have been added to the chapters to illustrate the concepts and show numerical calculations. Chapter 26 includes a new, self-contained treatment of positive matrix factorization, a widely used statistical method for analysis of aerosol data. New problems have been added in many of the chapters. A revised Problem Solution Manual is available for instructors. In order to help the reader navigate the major areas covered in the book, the chapters in this edition are grouped according to major themes: Part I: The Atmosphere and its Constituents Chapters 1 and 2 Part II: Atmospheric Chemistry Chapters 3–7 Part III: Aerosols Chapters 8–15 Part IV: Physical and Dynamic Meteorology, Cloud Physics, and Atmospheric Diffusion Chapters 16–18 Part V: Dry and Wet Deposition Chapters 19 and 20 Part VI: The Global Atmosphere, Biogeochemical Cycles, and Climate Chapters 21–24 Part VII: Chemical Transport Models and Statistical Models Chapters 25 and 26

xxv

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PREFACE TO THE THIRD EDITION

As in the First and Second Editions, many colleagues have provided important inputs. We want to especially acknowledge Kelvin Bates, Yi-Chun Chen, Neil Donahue, Christos Fountoukis, Daniel Jacob, Jesse Kroll, Mark Lawrence, Renee McVay, Sally Ng, Tran Nguyen, Allen Robinson, Rebecca Schwantes, Manabu Shiraiwa, Rainer Volkamer, Paul Wennberg, Xuan Zhang, and Andreas Zuend. Finally, we are indebted to Martha Hepworth and Yvette Grant for skillful preparation of the manuscript for the Third Edition. JOHN H. SEINFELD SPYROS N. PANDIS

P A R T I

The Atmosphere and Its Constituents

CHAPTER

1

The Atmosphere

1.1 HISTORY AND EVOLUTION OF EARTH’S ATMOSPHERE It is generally believed that the solar system condensed out of an interstellar cloud of gas and dust, referred to as the primordial solar nebula, about 4.6 billion years ago. The atmospheres of Earth and the other terrestrial planets, Venus and Mars, are thought to have formed as a result of the release of trapped volatile compounds from the planet itself. The early atmosphere of Earth is believed to have been a mixture of carbon dioxide (CO2), nitrogen (N2), and water vapor (H2O), with trace amounts of hydrogen (H2), a mixture similar to that emitted by present-day volcanoes. The composition of the present atmosphere bears little resemblance to the composition of the early atmosphere. Most of the water vapor that outgassed from Earth’s interior condensed out of the atmosphere to form the oceans. The predominance of the CO2 that outgassed formed sedimentary carbonate rocks after dissolution in the ocean. It is estimated that for each molecule of CO2 presently in the atmosphere, there are about 105 CO2 molecules incorporated as carbonates in sedimentary rocks. Since N2 is chemically inert, non-water-soluble, and noncondensable, most of the outgassed N2 accumulated in the atmosphere over geologic time to become the atmosphere’s most abundant constituent. The early atmosphere of Earth was a mildly reducing chemical mixture, whereas the present atmosphere is strongly oxidizing. Geochemical evidence points to the fact that atmospheric oxygen underwent a dramatic increase in concentration about 2300 million years ago (Kasting 2001). While the timing of the initial O2 rise is now well established, what triggered the increase is still in question. There is agreement that O2 was initially produced by cyanobacteria, the only prokaryotic organisms (bacteria and archea) capable of oxygenic photosynthesis. These bacteria emerged 2700 million years ago. The gap of 400 million years between the emergence of cyanobacteria and the rise of atmospheric O2 is still an issue of debate. The atmosphere from 3000 to 2300 million years ago was rich in reduced gases such as H2 and CH4. Hydrogen can escape to space from such an atmosphere. Since the majority of Earth’s hydrogen was in the form of water, H2 escape would lead to a net accumulation of O2. One possibility is that the O2 left behind by the escaping H2 was largely consumed by oxidation of continental crust. This oxidation might have sequestered enough O2 to suppress atmospheric levels before 2300 million years ago, the point at which the flux of reduced gases fell below the net photosynthetic production rate of oxygen. The present level of O2 is maintained by a balance between production from photosynthesis and removal through respiration and decay of organic carbon (Walker 1977).

Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, Third Edition. John H. Seinfeld and Spyros N. Pandis.  2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

3

4

THE ATMOSPHERE

Earth’s atmosphere is composed primarily of the gases N2 (78%), O2 (21%), and Ar (1%), whose abundances are controlled over geologic timescales by the biosphere, uptake and release from crustal material, and degassing of the interior. Water vapor is the next most abundant constituent; it is found mainly in the lower atmosphere and its concentration is highly variable, reaching concentrations as high as 3%. Evaporation and precipitation control its abundance. The remaining gaseous constituents, the trace gases, represent less than 1% of the atmosphere. These trace gases play a crucial role in Earth’s radiative balance and in the chemical properties of the atmosphere. Aristotle was the first to propose in his book Meteorologica in 347 BC that the atmosphere was actually a mixture of gases and that water vapor should be present to balance the water precipitation to Earth’s surface. The study of atmospheric chemistry can be traced back to the eighteenth century when chemists such as Joseph Priestley, Antoine-Laurent Lavoisier, and Henry Cavendish attempted to determine the chemical components of the atmosphere. Largely through their efforts, as well as those of a number of nineteenth-century chemists and physicists, the identity and major components of the atmosphere, N2, O2, water vapor, CO2, and the rare gases, were established. In the late nineteenth–early twentieth century focus shifted from the major atmospheric constituents to trace constituents, that is, those having mole fractions below 10 6, 1 part per million (ppm) by volume. We now know that the atmosphere contains a myriad of trace species. Spectacular innovations in instrumentation over the last several decades have enabled identification of atmospheric trace species down to levels of about 10 12 mole fraction, 1 part per trillion (ppt) by volume. The extraordinary pace of the recent increases in atmospheric trace gases can be seen when current levels are compared with those of the distant past. Such comparisons can be made for CO2 and CH4, whose histories can be reconstructed from their concentrations in bubbles of air trapped in ice in such perpetually cold places as Antarctica and Greenland. With gases that are long-lived in the atmosphere and therefore distributed rather uniformly over the globe, such as CO2 and CH4, polar ice core samples reveal global average concentrations of previous eras. Analyses of bubbles in ice cores show that CO2 and CH4 concentrations remained essentially unchanged from the end of the last ice age some 10,000 years ago until roughly 300 years ago, at mole fractions close to 260 and 0.7 ppm by volume, respectively. Activities of humans account for most of the rapid changes in the trace gases over the past 200 years–combustion of fossil fuels (coal and oil) for energy and transportation, industrial and agricultural activities, biomass burning (the burning of vegetation), and deforestation. These changes have led to the definition of a new era in Earth’s history, the Anthropocene (Crutzen and Steffen 2003). Records of atmospheric CO2, CH4, and N2O show a clear increase since the end of the eighteenth century, coinciding more or less with the invention of the steam engine in 1784. The global release of SO2, from coal and oil burning, is at least twice that of all natural emissions. More nitrogen is now fixed synthetically and applied as fertilizers in agriculture than fixed naturally in all terrestrial ecosystems. The Haber–Bosch industrial process to produce ammonia from N2, in many respects, made the human explosion possible. The emergence of the Antarctic ozone hole in the 1980s provided unequivocal evidence of the ability of trace species to perturb the atmosphere. The essentially complete disappearance of ozone in the Antarctic stratosphere during the austral spring is now recovering, owing to a global ban on production of stratospheric ozone-depleting substances. Whereas stratospheric ozone levels eroded in response to human emissions, those at ground level have, over the past century, been increasing. Paradoxically, ozone in the stratosphere protects living organisms from harmful solar ultraviolet radiation, whereas increased ozone in the lower atmosphere has the potential to induce adverse effects on human health and plants. Levels of airborne particles in industrialized regions of the Northern Hemisphere have increased markedly since the Industrial Revolution. Atmospheric particles (aerosols) arise both from direct emissions and from gas-to-particle conversion of vapor precursors. Aerosols can affect climate and have been implicated in human morbidity and mortality in urban areas. Atmospheric chemistry comprises the study of the mechanisms by which molecules introduced into the atmosphere react and, in turn, how these alterations affect atmospheric composition and properties (Ravishankara 2003). The driving force for chemical changes in the atmosphere is sunlight. Sunlight

LAYERS OF THE ATMOSPHERE

directly interacts with many molecules and is also the source of most of the atmospheric free radicals. Despite their very small abundances, usually less than one part in a billion parts of air, free radicals act to transform most species in the atmosphere. The study of atmospheric chemical processes begins with determining basic chemical steps in the laboratory, then quantifying atmospheric emissions and removal processes, and incorporating all the relevant processes in computational models of transport and transformation, and finally comparing the predictions with atmospheric observations to assess the extent to which our basic understanding agrees with the actual atmosphere. Atmospheric chemistry occurs against the fabric of the physics of air motions and of temperature and phase changes. In this book we attempt to cover all aspects of atmospheric chemistry and physics that bear on air pollution and climate change.

1.2 CLIMATE Viewed from space, Earth is a multicolored marble: clouds and snow-covered regions of white, blue oceans, and brown continents. The white areas make Earth a bright planet; about 30% of the sun’s radiation is reflected immediately back to space. The surface emits infrared radiation back to space. The atmosphere absorbs much of the energy radiated by the surface and reemits its own energy, but at lower temperatures. In addition to gases in the atmosphere, clouds play a major climatic role. Some clouds cool the planet by reflecting solar radiation back to space; others warm the earth by trapping energy near the surface. On balance, clouds exert a significant cooling effect on Earth. The temperature of the earth adjusts so that the net flow of solar energy reaching Earth is balanced by the net flow of infrared energy leaving the planet. Whereas the radiation budget must balance for the entire Earth, it does not balance at each particular point on the globe. Very little solar energy reaches the white, ice-covered polar regions, especially during the winter months. The earth absorbs most solar radiation near its equator. Over time, though, energy absorbed near the equator spreads to the colder regions of the globe, carried by winds in the atmosphere and by currents in the oceans. This global heat engine, in its attempt to equalize temperatures, generates the earth’s climate. It pumps energy into storm fronts and powers hurricanes. In the colder seasons, low-pressure and high-pressure cells push each other back and forth every few days. Energy is also transported over the globe by masses of wet and dry air. Through evaporation, air over the warm oceans absorbs water vapor and then travels to colder regions and continental interiors where water vapor condenses as rain or snow, a process that releases heat into the atmosphere. The condition of the atmosphere at a particular location and time is its weather; this includes winds, clouds, precipitation, temperature, and relative humidity. In contrast to weather, the climate of a region is the condition of the atmosphere over many years, as described by long-term averages of the same properties that determine weather.

1.3 LAYERS OF THE ATMOSPHERE In the most general terms, the atmosphere is divided into lower and upper regions. The lower atmosphere is generally considered to extend to the top of the stratosphere, an altitude of about 50 kilometers (km). Study of the lower atmosphere is known as meteorology; study of the upper atmosphere is called aeronomy. The earth’s atmosphere is characterized by variations of temperature and pressure with height. In fact, the variation of the average temperature profile with altitude is the basis for distinguishing the layers of the atmosphere. The regions of the atmosphere are (Figure 1.1) as follows: Troposphere. The lowest layer of the atmosphere, extending from Earth’s surface up to the tropo­ pause, which is at 10–15 km altitude depending on latitude and time of year; characterized by decreasing temperature with height and rapid vertical mixing.

5

6

THE ATMOSPHERE

FIGURE 1.1 Layers of the atmosphere.

Stratosphere. Extends from the tropopause to the stratopause (from ∼45 to 55 km altitude); temper­ ature increases with altitude, leading to a layer in which vertical mixing is slow. Mesosphere. Extends from the stratopause to the mesopause (from ∼80 to 90 km altitude). Its temperature decreases with altitude to the mesopause, which is the coldest point in the atmosphere. It is characterized by rapid vertical mixing. Thermosphere. The region above the mesopause characterized by high temperatures as a result of absorption of short-wavelength radiation by N2 and O2 and rapid vertical mixing. The ionosphere is a region of the upper mesosphere and lower thermosphere where ions are produced by photoionization. Exosphere. The outermost region of the atmosphere (>500 km altitude) where gas molecules with sufficient energy can escape from Earth’s gravitational attraction. Over the equator the average height of the tropopause is about 16 km; over the poles, about 8 km. By convention of the World Meteorological Organization (WMO), the tropopause is defined as the lowest level at which the rate of decrease in temperature with height is sustained at 2 K km 1 (Holton et al. 1995). The tropopause is at a maximum height over the tropics, sloping downward moving toward the poles. The troposphere—a term coined by British meteorologist, Sir Napier Shaw, from the Greek word tropi, meaning turn—is a region of ceaseless turbulence and mixing. The caldron of all weather, the troposphere contains almost all of the atmosphere’s water vapor. Although the troposphere accounts for only a small fraction of the atmosphere’s total height, it contains about 80% of its total mass. In the troposphere, the temperature decreases almost linearly with height. In dry air the rate of decrease with increasing altitude is 9.7 K km 1. The reason for this decline in temperature is the increasing distance

7

PRESSURE IN THE ATMOSPHERE

from the sun-warmed earth. At the tropopause, the temperature has fallen to an average of ∼217 K ( 56 °C). The troposphere can be divided into the planetary boundary layer, extending from Earth’s surface up to ∼1 km, and the free troposphere, extending from ∼1 km to the tropopause. The stratosphere, extending from approximately 11 km to ∼50 km, was discovered at the turn of the twentieth century by the French meteorologist Léon Philippe Teisserenc de Bort. Sending up temperature-measuring devices in balloons, he found that, contrary to the popular belief of the day, the temperature in the atmosphere did not steadily decrease to absolute zero with increasing altitude, but stopped falling and remained constant after 11 km or so. He named the region the stratosphere from the Latin word stratum, meaning layer. Although an isothermal region does exist from approximately 11–20 km at midlatitudes, temperature progressively increases from 20 to 50 km, reaching 271 K at the stratopause, a temperature not much lower than the average of 288 K at Earth’s surface. The vertical thermal structure of the stratosphere is a result of absorption of solar ultraviolet radiation by O3.

1.4 PRESSURE IN THE ATMOSPHERE 1.4.1 Units of Pressure The unit of pressure in the International System of Units (SI) is newtons per meter squared (N m 2), which is called the pascal (Pa). In terms of pascals, the atmospheric pressure at the surface of Earth, the socalled standard atmosphere, is 1.01325 × 105 Pa. Another commonly used unit of pressure in atmospheric science is the millibar (mbar), which is equivalent to the hectopascal (hPa) (see Tables A.5 and A.8). The standard atmosphere is 1013.25 mbar. Because instruments for measuring pressure, such as the manometer, often contain mercury, commonly used units for pressure are based on the height of the mercury column (in millimeters) that the gas pressure can support. The unit mm Hg is often called the torr in honor of the scientist Evangelista Torricelli. A related unit for pressure is the standard atmosphere (abbreviated atm). We summarize the various pressure units as follows: 1 Pa 1 atm 1 bar 1 mbar 1 torr

ˆ ˆ ˆ ˆ ˆ

1 N m 2 ˆ 1 kg m 1 s 1:01325  105 Pa 105 Pa 1 hPa ˆ 100 Pa 1 mm Hg ˆ 134 Pa

2

Standard atmosphere : 1:01325  105 Pa ˆ 1013:25 hPa ˆ 1013:25 mbar ˆ 760 torr The variation of pressure and temperature with altitude in the standard atmosphere is given in Table A.8. Because the millibar (mbar) is the unit most commonly used in the meteorological literature, we will use it when discussing pressure at various altitudes in the atmosphere. Mean surface pressure at sea level is 1013 mbar; global mean surface pressure, calculated over both land and ocean, is estimated as 985.5 mbar. The lower value reflects the effect of surface topography; over the highest mountains, which reach an altitude of over 8000 m, the pressure may be as low as 300 mbar. The 850 mbar level, which as we see from Table A.8, is at about 1.5 km altitude, is often used to represent atmospheric quantities, such as temperature, as the first standard meteorological level above much of the topography.

1.4.2 Variation of Pressure with Height in the Atmosphere Let us derive the equation governing the pressure in the static atmosphere. Imagine a volume element of the atmosphere of horizontal area dA between two heights, z and z + dz. The pressures exerted on the top and bottom faces are p(z + dz) and p(z), respectively. The gravitational force on the mass of air in the

8

THE ATMOSPHERE

volume = ρg dA dz, with p(z) > p(z + dz) due to the additional weight of air in the volume. The balance of forces on the volume gives …p…z† p…z ‡ dz†† dA ˆ ρg dA dz Dividing by dz and letting dz → 0 produce dp…z† ˆ ρ…z†g dz

(1.1)

where ρ(z) is the mass density of air at height z and g is the acceleration due to gravity. From the ideal-gas law, we obtain Mair p…z† ρ…z† ˆ (1.2) RT…z† where Mair is the average molecular weight of air (28.97 g mol 1). Thus dp…z† ˆ dz

Mair gp…z† RT…z†

(1.3)

which we can rewrite as d ln p…z† ˆ dz

1 H…z†

(1.4)

where H(z) = RT(z)/Mairg is a characteristic length scale for decrease of pressure with height. The temperature in the atmosphere varies by a factor of 0.001 μm); a particle may also consist of two or more such unit structures held together by interparticle adhesive forces such that it behaves as a single unit in suspension or on deposit A term derived from smoke and fog, applied to extensive contamination by aerosols; now sometimes used loosely for any contamination of the air Small gasborne particles resulting from incomplete combustion, consisting predominantly of carbon and other combustible materials, and present in sufficient quantity to be observable independently of the presence of other particles; Dp  0.01 μm Agglomerations of particles of carbon impregnated with “tar,” formed in the incomplete combustion of carbonaceous material

47

48

ATMOSPHERIC TRACE CONSTITUENTS

as a suspension of fine solid or liquid particles in a gas, common usage refers to the aerosol as the particulate component only (Table 2.19). Emitted directly as particles (primary aerosol) or formed in the atmosphere by gas-to-particle conversion processes (secondary aerosol), atmospheric aerosols are generally considered to be the particles that range in size from a few nanometers (nm) to tens of micrometers (μm) in diameter. Once airborne, particles can change their size and composition by condensation of vapor species or by evaporation, by coagulating with other particles, by chemical reaction, or by activation in the presence of water supersaturation to become fog and cloud droplets. Particles smaller than 1 μm diameter generally have atmospheric concentrations in the range from around ten to several thousand per cm3; those exceeding 1 μm diameter are usually found at concen­ trations of 1, m > 2a, and soot will form. This argument indicates that one expects soot formation when the C/O ratio in the fuel air mixture exceeds the critical value of unity. Actually both CO and CO2 are formed in the combustion even at low C/O ratios. As oxygen is tied up in the stable CO2 molecules, less of it is available for CO formation and soot starts forming at C/O ratios lower than 1. Soot forms in a flame as the result of a chain of events starting with the oxidation and/or pyrolysis of the fuel into small molecules. Acetylene, C2H2, and polycyclic aromatic hydrocarbons (PAHs) are considered the main molecular intermediates for soot formation and growth. The composition of soot that has been aged in the high-temperature region of a flame is typically C8H, but soot usually contains more hydrogen earlier in the flame. The growth of soot particles involves first the formation of soot nuclei followed by rapid growth by surface reactions. Soot particles are agglomerates of small roughly spherical elementary carbonaceous particles. While the size and morphology of the clusters vary widely, the small spherical elementary particles are remarkably consistent from one to another. These elementary particles vary in size from 20 to 30 nm and cluster with each other, forming straight or branched chains, which agglomerate and form visible soot particles with sizes up to a few micrometers.

PARTICULATE MATTER (AEROSOLS)

Global emissions of BC for the year 2000 were estimated as ∼7500 Gg BC yr 1: ∼4770 Gg BC yr–1 is estimated from energy-related burning, with ∼2760 Gg BC yr–1 from open biomass burning (Bond et al. 2013). BC-rich emissions include diesel engines, industry, residential solid fuel, and open burning.

2.7.6 Mineral Dust Mineral dust aerosol arises from wind acting on soil particles. The arid and semiarid regions of the world, which cover about one-third of the global land area, are where the major dust sources are located. The largest global source is the Sahara–Sahel region of northern Africa; central Asia is the second largest dust source. Published global dust emission estimates since 2001 range from 1000 to 2150 Tg yr 1, and atmospheric burden estimates range from 8 to 36 Tg (Zender et al. 2004). The large uncertainty range of dust emission estimates is mainly a result of the complexity of the processes that raise dust into the atmosphere. The emission of dust is controlled by both the windspeed and the nature of the surface itself. In addition, the size range of the dust particles is a crucial factor in emissions estimates. It is generally thought that a threshold windspeed as a function of particle size is required to mobilize dust particles into the atmosphere. Although clearly an oversimplification, a single value of the threshold windspeed of 6.5 m s 1 at 10 m height has frequently been used in modeling dust emissions in general circulation models (Sokolik 2002). The main species found in soil dust are quartz, clays, calcite, gypsum, and iron oxides, and the properties (especially optical) depend on the relative abundance of the various minerals. Gravitational settling and wet deposition are the major removal processes for mineral dust from the atmosphere. The average lifetime of dust particles in the atmosphere is about 2 weeks, during which dust can be transported thousands of kilometers. Saharan dust plumes frequently reach the Caribbean and Europe, and plumes of dust from Asia are detected on the west coast of North America. Mineral dust is emitted from both natural and anthropogenic activities. Natural emissions arise by wind acting on undisturbed source regions. Anthropogenic emissions result from human activity, including (1) resuspension of dust particles by traffic, (2) land-use changes that modify soil surface conditions, and (3) climate modifications that, in turn, alter dust emissions. Such modifications include changes in windspeeds, clouds and precipitation, and the amounts of airborne soluble material, such as sulfate, that may become attached to mineral dust particles and render them more susceptible to wet removal.

2.7.7 Biomass Burning The intentional burning of land is a major source of combustion products to the atmosphere. Most of this burning occurs in the tropics. Emissions from burning vegetation are typical of those from any uncontrolled combustion process and include CO2, CO, NOx, CH4 and nonmethane hydrocarbons, and elemental and organic particulate matter. The quantity and type of emissions from a biomass fire depend not only on the type of vegetation but also on its moisture content, ambient temperature, humidity, and local windspeed. Estimates of global emissions of trace gases from biomass burning appear in many of the tables in this chapter. Large-scale savanna burning in the dry season in Africa leads to regional-scale ozone levels that approach those characteristic of urban industrialized regions (60–100 ppb) (Fishman et al. 1990, 1992). Fire has been a feature of Earth over history. The occurrence of fire on paleoclimate timescales can be inferred from charcoal preserved in sedimentary sequences in lakes, bogs, and soils. The extent of total biomass burned in a particular region is registered in such sedimentary sequences through the total charcoal abundance. In high-resolution sedimentary records, individual fires are identified as peaks in charcoal accumulation. The earliest known charcoal record of vegetation fires is from the late Silurian (∼420 million years ago). Fire has been implicated in the rapid spread of tropical grasslands, which occurred simultaneously in Africa, Asia, and the Americas between 7 and 8 million years ago (Keeley and Rundel 2005). Analysis of geological charcoal records shows that global fire regimes varied continuously since the Last Glacial Maximum (∼21,000 years ago) in response to long-term changes in global climate, although it is difficult to identify the specific causes of the variation because climate affects fire regimes as well as the amount of vegetation.

53

54

ATMOSPHERIC TRACE CONSTITUENTS

Biomass burning is the major source of black carbon, organic carbon, and ozone precursors in the Southern Hemisphere and is an important contributor in the Northern Hemisphere as well. It is a seasonal source, with significant variability in emissions both temporally and geographically (Andreae and Merlet 2001; Streets et al. 2003). Aerosols emitted by biomass burning have been shown to significantly alter the radiation balance and cloud properties over large regions of the Southern Hemisphere (Artaxo et al. 2002; Andreae et al. 2004; Reid et al. 2005a, 2005b). Different approaches have been used to obtain emission factors for biomass burning, including direct measurements over fires in field experiments (Yamasoe et al. 2000), aircraft measurements (Yokelson et al. 2007), and laboratory measurements (Christian et al. 2003). Andreae and Merlet (2001) compiled emission factors for over 100 trace-gas species and aerosol compo­ nents for tropical forests, extratropical forests, savannah and grassland burning emissions. Several detailed global inventories for biomass-burning emissions have been compiled. These studies include EDGAR (Olivier et al. 1996), Bond et al. (2007), Junker and Liousse (2008), and Lamarque et al. (2010). A few inventories provide biomass-burning emissions for past decades. For the principal categories of biomass burning, the main emission datasets are (1) the RETRO inventory (Schultz et al. 2008), which provides emissions from wildfires for each year during the 1960–2000 period, on a monthly basis; (2) the GICC inventory (Mieville et al. 2010), which gives emissions from open biomass burning for the twentieth century on a decadal basis, based on Mouillot and Field (2005); and (3) the GFEDv2 inventory (van der Werf et al. 2006), which covers emissions for the 1997–2006 period.

2.7.8 Summary of Atmospheric Particulate Matter Table 2.21 presents a range of emission estimates of particles generated from natural and anthropogenic sources, on a global basis. Emissions of particulate matter attributable to the activities of humans arise

TABLE 2.21 Global Emission Estimates for Major Aerosol Classes Source Natural Primary Mineral dust 0.1–1.0 μm 1.0–2.5 μm 2.5–5.0 μm 5.0–10.0 μm 0.1–10.0 μm Seasalt Volcanic dust Biological debris Secondary Sulfates from DMS Sulfates from volcanic SO2 Organic aerosol from biogenic VOC Anthropogenic Primary Industrial dust (except black carbon) Black carbon Organic aerosol Secondary Sulfates from SO2 Nitrates from NOx a

Tg C. Tg S. c Tg NO3 . b

Estimated flux (Tg yr 1)

Reference

Zender et al. (2003) 48 260 609 573 1490 10,100 30 50 12.4 20 11.2

100 12a 81a 48.6b 21.3c

Gong et al. (2002) Kiehl and Rodhe (1995) Kiehl and Rodhe (1995) Liao et al. (2003) Kiehl and Rodhe (1995) Chung and Seinfeld (2002)

Kiehl and Rodhe (1995) Liousse et al. (1996) Liousse et al. (1996) Liao et al. (2003) Liao et al. (2004)

EMISSION INVENTORIES

primarily from: fuel combustion, industrial processes, nonindustrial fugitive sources (roadway dust from paved and unpaved roads, wind erosion of cropland, construction, etc.), biomass burning, and transportation sources (motor vehicles, etc.).

2.8 MERCURY Mercury is a neurotoxic pollutant that is released by a variety of sources. Mercury exists in the atmosphere as Hg0 (elemental mercury) and HgII (oxidized mercury). Mercury is emitted primarily in gaseous form as elemental mercury (Hg0). Hg0 is oxidized in the atmosphere to HgII, with a lifetime of ∼1 year (Lindberg et al. 2007). HgII is highly water-soluble and is removed from the atmosphere by wet and dry deposition. HgII also partitions between the gas phase and aerosol (Amos et al. 2012). Measurement of the molecular identity of HgII oxidation products is difficult; as a result, all gas-phase HgII products are generally represented as reactive gaseous mercury (RGM). Holmes et al. (2010) have obtained a global budget for mercury using the global chemical transport model GEOS-Chem. Anthropogenic Hg0 emissions are estimated as 2050 Mg Hg yr–1, and total emis­ sions are 8300 Mg Hg yr–1. Land emissions are estimated as 1200 Mg yr 1 from soils plus 260 Mg yr 1 from rapid photoreduction of HgII deposited to vegetation and 260 Mg yr 1 from snow. Terrestrial ecosystems are estimated to emit 2200 Mg yr 1, including primary geogenic sources. Atmospheric Hg is removed as HgII (5100 Mg yr 1), with wet deposition accounting for 3100 Mg yr 1 and dry deposition for 800 Mg yr 1. Sea salt particles are estimated to take up an additional 1200 Mg yr 1, which accounts for 35% of HgII deposition to the ocean. Global dry deposition of Hg0 is estimated as 3200 Mg yr 1, but emissions offset this so that oceans and soils are net sources of atmospheric Hg0. Atmospheric concentrations of mercury are reported for total gaseous mercury (TGM = Hg0 + RGM) in units of ppq (1 ppq = 10–15 mol mol = 8.97 pg m–3 at 273 K, 1013 hPa). Largest mixing ratios occur in the Northern Hemisphere, which approach 300 ppq. Global average concentration at 39 land sites over 2000–2008 was 209 ppq (Holmes et al. 2010). Trends in Hg concentration levels over the last decade or so are negligible. Mercury can cycle between Hg0 and HgII a number of times before being removed from the atmosphere. Removal of Hg0 via wet and dry deposition is slow because Hg0 is relatively insoluble in water and is estimated to have a low dry deposition velocity (∼0.01 cm s–1 over land; see Chapter 19). In the planetary boundary layer with an approximate height of 1000 m, the estimated lifetime of Hg0 by dry deposition is 2.7 months. HgII is removed from the atmosphere more rapidly than Hg0 because HgII species [e.g., HgCl2, Hg(OH)2, and HgO] are water-soluble and absorb readily on most surfaces. The atmospheric chemistry of mercury will be considered in Chapter 6.

2.9 EMISSION INVENTORIES An estimate of emissions of a species from a source is based on a technique that uses emission factors, which are based on source-specific emission measurements as a function of activity level (e.g., amount of annual production at an industrial facility) with regard to each source. For example, suppose one wants to sample emissions of SO2 or NOx from a power plant. The plant’s boiler design and its fuel consumption rate are known. The sulfur and nitrogen content of fuel burned can be used to calculate an emissions factor of kilograms (kg) of SO2 or NOx emitted per metric ton (Mg) of fuel consumed. Most governments compile emission factors for sources. For example, the US Environmental Protection Agency (EPA) reports factors since 1972 in AP-42 Compilation of Air Pollutant Emission Factors, for which supplements are issued regularly. The formulation of emission factors for mobile sources is based on rather complex emission estimation models used in conjunction with data from laboratory testing of representative groups of motor vehicles. Vehicle testing is performed with a chassis dynamometer, which determines the exhaust emission of a vehicle as a function of a specified ambient

55

56

ATMOSPHERIC TRACE CONSTITUENTS

temperature and humidity, speed, and load cycle. The specified testing cycle in the United States is called the Federal Test Procedure (FTP). On the basis of results from the vehicle emissions data, a computer model has been developed to simulate for specified speeds, temperatures, and trip profiles, for example, the emission factors to be applied for the national fleet average for all vehicles or any specified distribution of vehicle age and type. These data are then incorporated with activity data on vehicle miles traveled as a function of spatial and temporal allocation factors to estimate emissions.

APPENDIX 2.1

US AIR POLLUTION LEGISLATION

The legislative basis for air pollution abatement in the United States is the 1963 Clean Air Act and its amendments. The original act was signed into law in 1963, and major amendments were made in 1970, 1977, and 1990. The act establishes the federal–state relationship that requires the US Environmental Protection Agency (EPA) to develop National Ambient Air-Quality Standards (NAAQS) and empowers the states to implement and enforce regulations to attain them. The EPA has established NAAQS for each of six criteria pollutants: sulfur dioxide, particulate matter, nitrogen dioxide, carbon monoxide, ozone, and lead. At certain concentrations and length of exposure these pollutants are anticipated to endanger public health or welfare. The NAAQS are threshold concentrations based on a detailed review of the scientific information related to effects. Table 2A.1 presents the US national primary and secondary ambient air quality standards for ozone, carbon monoxide, nitrogen dioxide, sulfur dioxide, suspended particulate matter, and lead. The 1990 amendments of the Clean Air Act establish an interstate ozone transport region extending from the Washington, DC metropolitan area to Maine. In this densely populated region, ozone violations in one area are caused, at least in part, by emissions in upwind areas. A transport commission is authorized to coordinate control measures within the interstate transport region and to recommend to the EPA when additional control measures should be applied in all or part of the region in order to bring any area in the region into attainment. Hence areas within the transport region that are in attainment of

TABLE 2A.1 United States National Ambient Air Quality Standards Pollutant

Primary (P), secondary (S)

Carbon monoxide

P

Lead Nitrogen dioxide Ozone Particulate Matter PM2.5

PM10 Sulfur dioxide

Averaging time

Level

P and S P P and S P and S

8h 1h 3 mo 1h Annual 8h

9 ppma 35 ppma 0.15 μg m–3 100 ppbb 53 ppbc 75 ppbd

P S P and S P and S

Annual Annual 24 h 24 h

12 μg m–3 e 15 μg m–3 e 35 μg m–3 b 150 μg m–3 f

P

1h

75 ppbg

a

Not to be exceeded more than once per year. 98th percentile, averaged over 3 years. c Annual mean. d Annual fourth-highest daily maximum 8-h concentration, averaged over 3 years. e Annual mean, averaged over 3 years. f Not to be exceeded more than once per year on average over 3 years. g 99th percentile of 1-h daily maximum concentrations, averaged over 3 years. b

Source: US Environmental Protection Agency.

57

HAZARDOUS AIR POLLUTANTS (AIR TOXICS)

the ozone NAAQS might become subject to the emission controls required for nonattainment areas in that region.

APPENDIX 2.2

HAZARDOUS AIR POLLUTANTS (AIR TOXICS)

Hazardous air pollutants or toxic air contaminants (air toxics) refer to any substances that may cause or contribute to an increase in mortality or in serious illness, or that may pose a present or potential hazard to human health. The US Clean Air Act Amendments of 1990 established a list of 189 hazardous air pollutants. The initial list may be revised by the EPA to either add or remove substances. The EPA was directed to establish and promulgate emissions standards that effect the maximum degree of reduction in the listed substance. In establishing these emissions standards, the EPA may also consider health threshold levels. The California Air Resources Board (ARB) (1989) developed a list of substances of concern in California, called “Status of Toxic Air Contaminant Identification.” This list and the organization of substances within it are subject to periodic revision, as needed. Compounds from that list are summarized in Table 2A.2.

TABLE 2A.2 Substances Either Confirmed or under Study as Hazardous to Human Health by State of California Air Resources Board (1989) Concentrations Substance Benzene Ethylene dibromide Ethylene dichloride Hexavalent chromium Dioxins Asbestos

Cadmium Carbon tetrachloride Ethylene oxide

Vinyl chloride Inorganic arsenic

Methylene chloride

Qualitative health assessmenta

Manner of usage/major sources

Atmospheric residence time

Ambient averageb

Human carcinogen Probable carcinogen Probable carcinogen Human carcinogen Probable carcinogen Human carcinogen

Gasoline

12 days

Gasoline, pesticides

50 days

4.6 ppb (SoCAB)c 7.4 ppt (SoCAB)

—

Gasoline, solvents, pesticides Chrome plating, corrosion inhibitor Combustion product

42 days

19–110 ppt

—

—

0.5 ng m 3 (SoCAB) 1.0 pg m 3

—

Probable carcinogen Probable carcinogen Probable carcinogen Human carcinogen Human carcinogen Probable carcinogen

Milling, mining

Secondary smelters, fuel combustion CCl4 production, grain fumigant Sterilization agent, manufacture of surfactants Landfill byproduct Fuel combustion, pesticides Solvent

1 yr in soil

Hotspot —

—

8–80 fibers m

3

50–500 fibers m 3

Unknown; removed by deposition 7 days; removed by deposition 42 yr

0.13 ppb

0.63 ppb

200 days

50 ppt (SoCAB)

17 ppb

2 days

—

0.08–0.34 ppb

Unknown; removed by deposition 0.4 yr

2.4 ng m 3 (SoCAB)

200 ng m

1.1–2.4 ppb

10.7 ppb

1–2.5 ng m

3

40 ng m

3

3

(continued )

58

ATMOSPHERIC TRACE CONSTITUENTS

TABLE 2A.2 (Continued ) Concentrations Substance Perchloroethylene Trichloroethylene Nickel

Chloroform Formaldehyde 1,3-Butadiene Acetaldehyde Acrylonitrile Beryllium

Dialkylnitrosamines p-Dichlorobenzene Di-(2-ethylhexyl) phthalate 1,4-Dioxane Dimethyl sulfate Ethyl acrylate Hexachlorobenzene Lead

Mercury

Qualitative health assessmenta

Manner of usage/major sources

Atmospheric residence time

Ambient averageb

Hotspot

Probable carcinogen Probable carcinogen Probable carcinogen

Solvent, chemical intermediate Solvent, chemical

0.4 yr

0.71 ppb

22 ppb

5–8 days

0.22 ppb

—

Alloy, plating ceramics, dyes intermediate

7.3 ng m

Probable carcinogen Probable carcinogen Probable carcinogen Probable carcinogen Probable carcinogen Probable carcinogen

Solvent, chemical intermediate Chemical

Unknown, removed by deposition 0.55 yr

0.006–0.13 ppb

10 ppb

3.8–8.6 h

2–39 ppb

—

Chemical feedstock, resin production Motor vehicles

< 1 day

—

0.016 ppb

9h

—

35 ppb

Feedstock, resins, rubber

5.6 days

—

—

Metal alloys, fuel combustion

0.11–0.22 ng m

Probable carcinogen Probable carcinogen Probable carcinogen Probable carcinogen Probable carcinogen Possible carcinogen Probable carcinogen Blood system toxic, neurotoxicity Neurotoxic

Chemical feedstock

10 days to be removed by deposition 9.6 h

—

0.3 ppb

39 days

—

105–1700 μg m

1.3–13 h (urban)

2 μg m

Solvent stabilizer, feedstock Chemical reagent

3.9 days

—

—

—

—

—

Chemical intermediate

12 h

—

—

Solvent, pesticide

4 yr in soils

—

—

Auto exhaust, fuel additive

7–30 days; removed by deposition

270–820 ng m

Electronics, paper/pulp manufacture Chemical intermediate

0.3–2 yr; removed by deposition 6.4 h

0.37–0.49 ppb

1.2 ppb

—

—

Detergents, corrosion inhibitor Fuel combustion

242 nm, the atmosphere is transparent with respect to O2. Ozone is the dominant absorber in the range 240–320 nm. Table 4.3 gives estimated surface-level spectral actinic flux at 40°N latitude on January 1 and July 1.

TABLE 4.3 Estimated Ground-Level Actinic Flux I(λ) at 40°N Latitude I(λ) × 10

14

(photons cm

2

s 1)

Δλ (nm)

Noon January 1

Noon July 1

290–295 295–300 300–305 305–310 310–315 315–320 320–325 325–330 330–335 335–340 340–345 345–350 350–355 355–360 360–365

0.0 0.0 0.021 0.196 0.777 1.45 2.16 3.44 3.90 4.04 4.51 4.62 5.36 5.04 5.70

0.0 0.031 0.335 1.25 2.87 4.02 5.08 7.34 7.79 7.72 8.33 8.33 9.45 8.71 9.65

Source: Finlayson-Pitts and Pitts (1986).

109

PHOTOLYSIS RATE AS A FUNCTION OF ALTITUDE

A Somewhat More Accurate Estimate of Absorption of Solar Radiation by O2 and O3 We calculate the reduction in solar radiation at Earth’s surface as a result of O2 and O3 absorption in the wavelength range from 200 to 320 nm at a solar zenith angle of 45°. The approximate column abundances are 4 × 1024 molecules cm 8 × 1018 molecules cm

O2 O3

2 2

(300 DU)

Absorption cross sections for O2 and O3, as well as the solar flux at the top of the atmosphere, in 20 nm increments are λ (nm)

σO2 (cm2)

σO3 (cm2)

9 × 10 24 4.7 × 10 24 1.2 × 10 24 0 0 0 0

200 220 240 260 280 300 320

F…z; λ† τi …z; λ†

3.2 × 10 1.8 × 10 8.2 × 10 1.1 × 10 3.9 × 10 3.2 × 10 2.2 × 10 ˆ ˆ

FTOA …λ†exp mσi …λ†

1

∫0

FTOA (photons cm

2

s 1)

1.5 × 1013 9.6 × 1013 1.0 × 1014 2.6 × 1014 4.2 × 1014 1.6 × 1015 2.4 × 1015

19 18 18 17 18 19 19

…τO2 …z; λ† ‡ τO3 …z; 놆

ni …z†dz

i ˆ O2 ; O3

For solar zenith angle = 45°, m = sec 45° = 1.414. The total transmittance of the atmosphere is defined as the ratio F…z; λ†=FTOA ˆ exp τO2 ‡ τO3 ˆ exp τO2 exp τO3 , which is the product of the individual transmittances of O2 and O3. We obtain λ (nm)

200 220 240 260 280 300 320

O2 transmittance

7.8 × 10 2.8 × 10 1.1 × 10 1.0 1.0 1.0 1.0

23 12 3

O3 transmittance

2.6 × 10 1.2 × 10 2.3 × 10 3.0 × 10 4.7 × 10 2.6 × 10 8.1 × 10

2 9 41 55 20 2 2

Total transmittance

2.0 × 10 3.4 × 10 2.6 × 10 3.0 × 10 4.7 × 10 2.6 × 10 8.1 × 10

24 21 44 55 20 2 2

F(0,λ)(photons cm

2

s 1)

3.0 × 10 11 3.3 × 10 7 2.6 × 10 30 7.8 × 10 41 2.0 × 10 5 4.2 × 1013 2.0 × 1014

4.8 PHOTOLYSIS RATE AS A FUNCTION OF ALTITUDE Photolysis reactions are central to atmospheric chemistry, since the source of energy that drives the entire system of atmospheric reactions is the sun. The general expression for the first-order rate coefficient jA for photodissociation of a species A is given by (4.32). Because the rate of a photolysis reaction depends on the spectral actinic flux I and because the spectral actinic flux varies with altitude in the atmosphere, the rate of photolysis reactions depends in general on altitude. The altitude dependence of jA enters indirectly through the temperature dependence of the absorption cross section and directly through the altitude dependence of the spectral actinic flux, which can be written as I(z,λ). We have already seen that the altitude dependence of the solar irradiance in the wavelength range below ∼1 μm is essentially a

110

ATMOSPHERIC RADIATION AND PHOTOCHEMISTRY

result of absorption of solar radiation by O2 and O3. From the absorption cross sections for O2 and O3, it is possible to calculate I(z,λ) at any altitude z, which can then be used in (4.32) to calculate the photolysis rate of any species A as a function of altitude. The photolysis rate coefficient of a species A can be expressed as a function of altitude from (4.32) as jA …z† ˆ

λ2

∫ λ1

σA …T; λ†ϕA …T; λ†I…z; λ†dλ

(4.40)

Assume that the attenuation of radiation is due solely to absorption by species A. From the Beer– Lambert Law, I(z,λ) can be expressed as I…z; λ† ˆ I TOA …λ†exp… τA …z; 놆

(4.41)

If we neglect the z dependence of temperature, then τA …z; λ† ˆ mσA …T; λ†

1

∫z

nA …z´ †dz´

(4.42)

To proceed further, consider the case in which the absorbing species A has a uniform mixing ratio, ξA (e.g., O2); then, its concentration can be expressed in terms of the molecular air density as a function of altitude, nair (z): nA …z† ˆ ξA nair …z†

(4.43)

The total number concentration of air as a function of altitude falls off approximately with the scale height H of atmospheric pressure: nair …z† ˆ nair …0†exp

z H

(4.44)

Using (4.44) in (4.42), we get τA …z; λ† ˆ mσA …T; λ†ξA nair …0†H e

z=H

(4.45)

Combining (4.40), (4.41), and (4.45), the photolysis rate coefficient is given by jA …z† ˆ

λ2

∫ λ1

σA …T; λ†ϕA …T; λ†I TOA …λ†expf mσA …T; λ†ξA nair …0† e

z=H

gdλ

(4.46)

The photolysis rate at any altitude z is the product of jA (z) and the number concentration of A, nA (z): dnA …z† ˆ dt

jA …z†nA …z†

ˆ

jA …z†ξA nair …0† e

ˆ

ξA nair …0† e

ˆ

ξA nair …0†

z=H λ2

z=H

λ2

∫ λ1

∫ λ1

σA …T; λ†ϕA …T; λ†I TOA …λ†expf mσA …T; λ†ξA nair …0† e

σA …T; λ†ϕA …T; λ†I TOA …λ†exp

z H

mσA …T; λ†ξA nair …0† e

z=H

gdλ

z=H

dλ

(4.47)

As one descends toward Earth’s surface, the photolysis rate decreases because the overall actinic flux is decreasing due to increasing absorption in the layers above. As one ascends in the atmosphere, the

111

PHOTOLYSIS RATE AS A FUNCTION OF ALTITUDE

concentration of the gas nA(z) simply decreases as the atmosphere’s density thins out. These two competing effects combine to produce a maximum in the photolysis rate at a certain altitude. The altitude at which this rate is a maximum is the value of z at which d dnA …z† dz dt

ˆ0

(4.48)

We find that the photolysis rate is a maximum at the altitude z at which mσA ξA nair …0†H exp… z=H† ˆ 1

(4.49)

This is just the altitude z where τ(z,λ) = 1. The function in the integral of (4.47) is called a Chapman function in honor of Sydney Chapman, pioneer of stratospheric ozone chemistry. We will return to Chapman in the next chapter. The elegant analytical form of the Chapman function results when the mixing ratio of the absorbing gas A is constant throughout the atmosphere, so that nA(z) scales with altitude according to the scale height H of pressure. Molecular O2 is the prime example of such an absorbing gas. For a gas with a nonuniform mixing ratio with altitude, one can, of course, numerically evaluate (4.42) for the optical depth at any z on the basis of its actual profile nA(z). Figure 4.12 shows the photodissociation rate coefficient for O2, namely, jO2 , as a function of altitude above 30 km. From 30 to 60 km, the Herzberg continuum provides the dominant contribution to jO2 . At about 60 km, the contribution from the Schumann–Runge bands equals that from the Herzberg continuum; above 60 km, the Schumann–Runge bands predominate until about 80 km, where the Schumann–Runge continuum takes over. In the mid-to upper stratosphere, at solar zenith angle = 0°, an approximate value of jO2 is jO2 ≅ 10 9 s 1 .

FIGURE 4.12 Photodissociation rate coefficient for O2, jO2 , and O2 photolysis rate as a function of altitude at solar zenith angle θ0 = 0°. (Courtesy Ross J. Salawitch)

112

ATMOSPHERIC RADIATION AND PHOTOCHEMISTRY

4.9 PHOTODISSOCIATION OF O3 TO PRODUCE O AND O(1D) Ozone photodissociates to produce either ground-state atomic oxygen O3 ‡ hν ! O2 ‡ O or the first electronically excited state of atomic oxygen, O…1 D†: O3 ‡ hν ! O2 ‡ O…1 D† Ozone always dissociates when it absorbs a visible or UV photon; therefore, the quantum yield for O3 dissociation is unity. Photodissociation of O3 in the visible region, the Chappuis band, is the major source of ground state O atoms in the stratosphere. (Since these O atoms just combine with O2 to reform O3 with the release of heat, this absorption of visible radiation by O3 merely converts light into heat.) Figure 4.13 shows the photodissociation rate coefficients for O3 ‡ hν ! O…1 D† ‡ O2 [panels (a) and (b)] and for O3 + hν → O + O2 [panels (c) and (d)]. Panels (a) and (c) show jO3 !O…1 D† and jO3 !O , respectively, as a function of altitude at solar zenith angles θ = 0° and 85°. Panels (b) and (d) show the two photolysis rate coefficients at 20 km as a function of solar zenith angle. The photodissociation of

FIGURE 4.13 Photodissociation rate coefficient for O3 as a function of altitude and solar zenith angle. Panels (a) and (b) are jO3 !O…1 D† . Panels (c) and (d) are jO3 !O . (Courtesy Ross J. Salawitch)

113

PHOTODISSOCIATION OF O3 TO PRODUCE O AND O(1D)

O3 at altitudes below ∼30 km is governed mainly by absorption in the Chappuis bands, and this absorption is practically independent of altitude. Above ∼30 km, absorption in the Hartley bands dominates, the rate of which increases strongly with increasing altitude. In the troposphere, the photodissociation rate coefficients jO3 !O…1 D† and jO3 !O are virtually independent of altitude; at θ = 0°: jO3 !O…1 D† ≅ 6  10

5

s

1

jO3 !O ≅ 5:5  10

4

s

1

Because of the importance of O…1 D† to the chemistry of the lower stratosphere and entire troposphere, a great deal of effort has gone into precisely determining its production from O3 photodissociation. Since solar radiation of wavelengths below ∼290 nm does not reach the lower stratosphere and troposphere, the wavelength region just above 290 nm is critical for production of O…1 D†. Table 4.4 gives the quantum yield for production of O…1 D† as a function of wavelength in this region. On the basis of heat of reaction data, the threshold energy for photodissociation of O3 to produce O…1 D† is estimated to be 390 kJ mol 1, which is equivalent to a wavelength of 305 nm. At wavelengths below 305 nm, the quantum yield for O…1 D† production is indeed close to unity. It is expected that the threshold is shifted to somewhat longer wavelengths (∼310 nm) because a fraction of the O3 molecules will have extra vibrational energy that can help overcome the barrier. Michelson et al. (1994) showed that the effect of vibrationally excited O3 leads to nonzero quantum yields that vary with temperature up to about 320 nm. Recent data, as reflected in Table 4.4, show that O…1 D† is produced well beyond the 310-nm threshold, attributable to so-called spinforbidden channels that may account for as much as 10% of the overall rate. The critical role of wavelength in photolysis can be seen by comparing Figure 4.9 and Table 4.4. From λ = 305 to 320 nm, the absorption cross section drops by a factor of ∼10, and the quantum yield for O…1 D† formation drops from 0.9 to about 0.1. Since the σɸ product changes rapidly with λ, the actual rate of TABLE 4.4 O(1D) Quantum Yield in Photolysis of O3 Wavelength (nm)

298 K

253 K

203 K

306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328

0.884 0.862 0.793 0.671 0.523 0.394 0.310 0.265 0.246 0.239 0.233 0.222 0.206 0.187 0.166 0.146 0.128 0.113 0.101 0.092 0.086 0.082 0.080

0.875 0.844 0.760 0.616 0.443 0.298 0.208 0.162 0.143 0.136 0.133 0.129 0.123 0.116 0.109 0.101 0.095 0.089 0.085 0.082 0.080 0.079 0.078

0.872 0.836 0.744 0.585 0.396 0.241 0.152 0.112 0.095 0.090 0.088 0.087 0.086 0.085 0.083 0.082 0.080 0.079 0.078 0.078 0.077 0.077 0.077

Source: Matsumi et al. (2002).

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ATMOSPHERIC RADIATION AND PHOTOCHEMISTRY

production of O…1 D† is critically dependent on how I(λ) varies with λ. At the surface of Earth, the spectral actinic flux increases by about an order of magnitude between λ = 300 nm and λ = 320 nm (see Table 4.3).

4.10 PHOTODISSOCIATION OF NO2 Since solar radiation of wavelength shorter than ∼290 nm does not reach the troposphere, the absorbing species of interest from the point of view of tropospheric chemistry are those that absorb in the portion of the spectrum above 290 nm. Nitrogen dioxide is an extremely important molecule in the troposphere; it absorbs over the entire visible and ultraviolet range of the solar spectrum in the lower atmosphere (Figure 4.14). Between 300 and 370 nm over 90% of the NO2 molecules absorbing will dissociate into NO and O (Figure 4.15). Above 370 nm this percentage drops off rapidly and above ∼420 nm dissociation

FIGURE 4.14 Absorption cross section for NO2 at 298 K.

FIGURE 4.15 Primary quantum yield for O formation from NO2 photolysis, as correlated by Demerjian et al. (1980).

115

PHOTODISSOCIATION OF NO2

TABLE 4.5 Absorption Cross Section and Quantum Yield for NO2 Photolysis at 273 and 298 K Absorption cross section From: λ (nm) 273.97 277.78 281.69 285.71 289.85 294.12 298.51 303.03 307.69 312.5 317.5 322.5 327.5 332.5 337.5 342.5 347.5 352.5 357.5 362.5 367.5 372.5 377.5 382.5 387.5 392.5 397.5 402.5 407.5 412.5 417.5

To: λ (nm)

λ (nm)

At 273 K 1020σ (cm2)

At 298 K 1020σ (cm2)

ϕ

277.78 281.69 285.71 289.85 294.12 298.51 303.03 307.69 312.5 317.5 322.5 327.5 332.5 337.5 342.5 347.5 352.5 357.5 362.5 367.5 372.5 377.5 382.5 387.5 392.5 397.5 402.5 407.5 412.5 417.5 422.5

275.875 279.735 283.7 287.78 291.985 296.315 300.77 305.36 310.095 315 320 325 330 335 340 345 350 355 360 365 370 375 380 385 390 395 400 405 410 415 420

5.03 5.88 7 8.15 9.72 11.54 13.44 15.89 18.67 21.53 24.77 28.07 31.33 34.25 37.98 40.65 43.13 47.17 48.33 51.66 53.15 55.08 56.44 57.57 59.27 58.45 60.21 57.81 59.99 56.51 58.12

5.05 5.9 6.99 8.14 9.71 11.5 13.4 15.8 18.5 21.4 24.6 27.8 31 33.9 37.6 40.2 42.8 46.7 48.1 51.3 52.9 54.9 56.2 57.3 59 58.3 60.1 57.7 59.7 56.5 57.8

1 1 1 0.999 0.9986 0.998 0.997 0.996 0.995 0.994 0.993 0.992 0.991 0.99 0.989 0.988 0.987 0.986 0.984 0.983 0.981 0.979 0.9743 0.969 0.96 0.92 0.695 0.3575 0.138 0.0603 0.0188

does not occur. As noted earlier, the bond energy between O and NO, about 300 kJ mol 1, corresponds to the energy contained in wavelengths near 400 nm. At longer wavelengths, there is insufficient energy to promote bond cleavage. The point at which dissociation fails to occur is not sharp because the individual molecules of NO2 do not possess a precise amount of ground-state energy prior to absorption. The gradual transition area (370–420 nm) corresponds to a variation in ground-state energy of about 40 kJ mol 1. This transition curve can be shifted slightly to longer wavelengths by increasing the temperature and therefore increasing the ground-state energy of the system. Table 4.5 gives tabulated values of the NO2 absorption cross section and quantum yield at 273 and 298 K, and Table 4.6 illustrates the calculation of the photolysis rate jNO2 for noon, July 1, at 40°N. Figure 4.16 shows the NO2 photodissociation rate coefficient jNO2 as a function of altitude. The majority of NO2 photolysis occurs for λ > 300 nm. Since in this region of the spectrum there is virtually no absorption by O2 and O3, the actinic flux is essentially independent of altitude (see Figure 4.9), and as a result, jNO2 is nearly independent of altitude. For the same reason, jNO2 does not drop off with increasing solar zenith angle as rapidly as that for molecules that absorb in the region of λ > 300 nm.

116

ATMOSPHERIC RADIATION AND PHOTOCHEMISTRY

TABLE 4.6 Calculation of the Photolysis Rate of NO2 at Ground Level, July 1, Noon, 40°N, 298 K From: λ (nm) 295 300 305 310 315 320 325 330 335 340 345 350 355 360 365 370 375 380 385 390 395 400 405 410 415 420

To: λ (nm) 300 305 310 315 320 325 330 335 340 345 350 355 360 365 370 375 380 385 390 395 400 405 410 415 420 422.5

I a (photons cm 3.14 × 1012 3.35 × 1013 1.24 × 1014 2.87 × 1014 4.02 × 1014 5.08 × 1014 7.34 × 1014 7.79 × 1014 7.72 × 1014 8.33 × 1014 8.32 × 1014 9.45 × 1014 8.71 × 1014 9.65 × 1014 1.19 × 1015 1.07 × 1015 1.20 × 1015 9.91 × 1014 1.09 × 1015 1.13 × 1015 1.36 × 1015 1.64 × 1015 1.84 × 1015 1.94 × 1015 1.97 × 1015 9.69 × 1014

2

s 1)

1019 σNO2 (cm2)

ϕ

1.22 1.43 1.74 2.02 2.33 2.65 2.97 3.28 3.61 3.91 4.18 4.51 4.75 5.00 5.23 5.41 5.57 5.69 5.84 5.86 5.93 5.86 5.89 5.78 5.73 5.78

0.9976 0.9966 0.9954 0.9944 0.9934 0.9924 0.9914 0.9904 0.9894 0.9884 0.9874 0.9864 0.9848 0.9834 0.9818 0.9798 0.9762 0.9711 0.9636 0.9360 0.7850 0.4925 0.2258 0.0914 0.0354 0.0188

jNO2 (s 1) 3.83 × 10 4.78 × 10 2.16 × 10 5.77 × 10 9.31 × 10 1.34 × 10 2.16 × 10 2.53 × 10 2.76 × 10 3.22 × 10 3.43 × 10 4.21 × 10 4.07 × 10 4.75 × 10 6.09 × 10 5.68 × 10 6.51 × 10 5.48 × 10 6.13 × 10 6.18 × 10 6.34 × 10 4.74 × 10 2.45 × 10 1.03 × 10 4.00 × 10 1.05 × 10

Total jNO2 ˆ 8:14  10 3 s jNO2 ˆ 0:488 min 1

1

Actinic flux from Finlayson-Pitts and Pitts (1986).

a

FIGURE 4.16 Photodissociation rate coefficient for NO2, jNO2 , as a function of altitude. (Courtesy Ross J. Salawitch)

7 6 5 5 5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5

117

REFERENCES

PROBLEMS 4.1A The enthalpy changes for the following photodissociation reactions are O2 + hν → O + O…1 D† O3 ‡ hν ! O…1 D† ‡ O2 …1 Δg † O3 + hν → O + O2

164.54 kcal mol 93.33 kcal mol

1

25.5 kcal mol

1

1

Estimate the maximum wavelengths at which these reactions can occur. 4.2A Convert the values of the surface UV radiation flux obtained for O2 and O3 absorption of solar radiation in Section 4.7 to W m 2. 4.3A The combination of O2 and O3 absorption explains the solar spectrum at different altitudes in Figure 4.9. At the altitude where the optical depth τ = 1, the solar irradiance is reduced to 1/e of its value at the top of the atmosphere. Considering overhead sun (θ = 0°), estimate the altitude at which τO2 ˆ 1 for each of the three O2 absorption bands given in Section 4.7. 4.4B At λ = 300 nm, the total transmittance of the atmosphere at a solar zenith angle of 45° is ∼0.026, which is a result of absorption of solar radiation by O3. At λ = 300 nm, the absorption cross section of O3 is σO3 ≅ 3:2  10 19 cm2 . What column abundance of O3, expressed in Dobson units, produces this transmittance? 4.5B Consider the absorption of 240 nm sunlight by O2. The O2 cross section at this wavelength is 1 × 10 24 cm2 molecule 1. Assuming that FTOA photons of this wavelength enter the top of the atmosphere, derive the equation for the attenuation of this light as a result of absorption of these photons as a function of altitude, neglecting scattering, for a solar zenith angle of 0°. You may assume nair(0) = 2.6 × 1019 molecules cm 3 and the atmosphere scale height H = 7.4 km. As the sun moves lower in the sky, how does the absorption change? 4.6B Show that (4.49) defines the altitude of the maximum photolysis rate of a species. 4.7B Estimate the altitude at which the photolysis rate of O2 is maximum corresponding to the three spectral regions: Schumann–Runge continuum, Schumann–Runge bands, and Herzberg contin­ uum. Approximate the O2 absorption cross section in each of these three regions by its maximum value, 10 17, 10 20, and 10 23 cm2, respectively. Assume a solar zenith angle of 0°. 4.8B Plot the photodissociation rate of O2 as a function of altitude from z = 20 to z = 60 km in the Herzberg continuum. Approximate the O2 absorption cross section as 10 23 cm2 and take λ = 200 nm. Consider solar zenith angles of 0° and 60°. 4.9A What is the longest wavelength of light, absorption of which by NO2 leads to dissociation at least 50% of the time? At 20 km, what are the lifetimes of NO2 by photolysis at solar zenith angles of 0° and 85°?

REFERENCES Brasseur, G., and Solomon, S. (1984), Aeronomy of the Middle Atmosphere. Reidel, Dordrecht, The Netherlands. Burrows, J. P., et al. (1999), Atmospheric remote sensing reference data from GOME—2. Temperature-dependent absorption cross sections of O3 in the 231–794 nm range J. Quant. Spectrosc. Radiative Transf. 61, 509–517. Demerjian, K. L., et al. (1980), Theoretical estimates of actinic (spherically integrated) flux and photolytic rate constants of atmospheric species in the lower troposphere, Adv. Environ. Sci. Technol. 10, 369–459. DeMore, W. G., et al. (1994), Chemical Kinetics and Photochemical Data for Use in Stratospheric Modeling, JPL Publication 94-26, Jet Propulsion Laboratory, Pasadena, CA.

118

ATMOSPHERIC RADIATION AND PHOTOCHEMISTRY Finlayson-Pitts, B. J., and Pitts, J. N., Jr. (1986), Atmospheric Chemistry: Fundamentals and Experimental Techniques. Wiley, New York. Fröhlich, C., and London, J. (eds.) (1986), Revised Instruction Manual on Radiation Instruments and Measurements, World Climate Research Program (WCRP) Publication Series 7, World Meteorological Organization/TD No. 149, Geneva. Iqbal, M. (1983), An Introduction to Solar Radiation, Academic Press, Toronto. Kasten, F., and Young, A. T. (1989), Revised optical air mass tables and approximation formula, Appl. Opt. 28, 4735–4738. Madronich, S. (1987) Photodissociation in the atmosphere 1. Actinic flux and the effects of ground reflections and clouds, J. Geophys. Res. 92, 9740–9752. Matsumi, Y., et al. (2002), Quantum yields for production of O…1 D† in the ultraviolet photolysis of ozone: Recommendation based on evaluation of laboratory data, J. Geophys. Res. 107 (D3), 4024 (doi: 10.1029/ 2001JD000510). Michelsen, H. A., et al. (1994), Production of O…1 D† from photolysis of O3 Geophys. Res. Lett. 21, 2227–2230.

CHAPTER

5

Chemistry of the Stratosphere

Ozone is the most important trace constituent of the stratosphere. Discovered in the nineteenth century, its importance as an atmospheric gas became apparent in the early part of the twentieth century when the first quantitative measurements of the ozone column were carried out in Europe. The British scientist Dobson developed a spectrophotometer for measuring the ozone column, and the instrument is still in widespread use. In recognition of his contribution, the standard measure of the ozone column is the Dobson unit (DU) (recall the definition in Chapter 2). Sydney Chapman, also a British scientist, proposed in 1930 that ozone is continually produced in the stratosphere by a cycle initiated by photolysis of O2 in the upper stratosphere (Chapman 1930). This photochemical mechanism for ozone production bears Chapman’s name. Eventually, it was realized that the Chapman mechanism predicted too much stratospheric ozone. It was not until 1970 when the true breakthrough in understanding stratospheric chemistry emerged when Paul Crutzen elucidated the role of nitrogen oxides in stratospheric ozone chemistry, closely followed by investigation by Harold Johnston of the possible depletion of strato­ spheric ozone as a result of the catalytic effect of NOx emitted from a proposed fleet of supersonic aircraft. Shortly thereafter, the effect on stratospheric ozone of chlorine released from industrial chlorofluorocarbons was predicted by Mario Molina and F. Sherwood Rowland. For their pioneering studies of atmospheric ozone chemistry, Crutzen, Molina, and Rowland were awarded the 1995 Nobel Prize in Chemistry. About 90% of the atmosphere’s ozone is found in the stratosphere, residing in what is commonly referred to as the ozone layer. At the peak of the ozone layer the O3 mixing ratio is about 12 ppm. The total amount of O3 in Earth’s atmosphere is not great; if all the O3 molecules in the troposphere and stratosphere were brought down to Earth’s surface and distributed uniformly as a layer over the globe, the resulting layer of pure gaseous O3 would be less than 5 mm thick. Stratospheric ozone is produced naturally as a result of the photolytic decomposition of O2. The two oxygen atoms that result each react with another O2 molecule to produce two molecules of ozone. The overall process, therefore, converts three O2 molecules to two O3 molecules. The O3 molecules produced themselves react with other stratospheric molecules, both natural and anthropogenic; the balance achieved between O3 produced and destroyed leads to a steady-state abundance of O3. Rowland and Molina discovered that industrial gases containing chlorine and bromine emitted at Earth’s surface, and impervious to chemical destruction or removal by precipitation in the troposphere, eventually reach the stratosphere, where the chlorine and bromine are liberated by photolysis of the molecules. The Cl and Br atoms then attack O3 molecules, initiating catalytic cycles that remove ozone. These industrial gases include the chlorofluorocarbons (CFCs) and hydrochlorofluorocarbons (HCFCs),

Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, Third Edition. John H. Seinfeld and Spyros N. Pandis.  2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

119

120

CHEMISTRY OF THE STRATOSPHERE

FIGURE 5.1 Primary sources of chlorine and bromine in the stratosphere in 1999 (WMO 2002).

previously used in almost all refrigeration and air-conditioning systems, and the halons, which are used as fire retardants (see Chapter 2). Figure 5.1 shows the distribution of chlorine- and bromine-containing gases in the stratosphere as of 1999. Methyl chloride (CH3Cl) is virtually the only natural source of chlorine in the stratosphere, which, in 1999, accounted for 16% of the chlorine. Natural sources, on the other hand, account for nearly half of the stratospheric bromine. Even though the amount of chlorine is about 170 times that of bromine, on an atom-for-atom basis, Br is about 50 times more effective than Cl in destroying ozone. The fluorine atoms in the molecules end up in chemical forms that do not participate in O3 destruction. Because many of these halogen-containing gases have no significant removal processes in the troposphere, their atmospheric lifetimes are determined by their eventual chemical breakdown in the stratosphere, for example: CFCl3 CF2Cl2 CCl4 CH3Cl CBrClF2 CF3Br CH3Br

(CFC-11) (CFC-12) (Carbon tetrachloride) (Methyl chloride) (Halon-1211) (Halon-1301) (Methyl bromide)

52 years 102 years 35 years 1.5 years 11 years 65 years 0.8 years

121

CHEMISTRY OF THE STRATOSPHERE

Although anthropogenic CFCs are emitted mostly in the Northern Hemisphere, they become well mixed throughout the entire troposphere over the course of a couple of years. They enter the stratosphere primarily across the tropical tropopause with air pumped into the stratosphere by rising air in the tropics. Air motions in the stratosphere then transport them upward and toward the poles in both hemispheres. Those gases with long lifetimes are present in comparable amounts throughout the stratosphere in both hemispheres. Once the stratospheric circulation lifts the air above about 24 km, CFCs are photolyzed by UV radiation at wavelengths of 190–230 nm. Massive annual ozone decreases during the Antarctic spring (Sept.–Nov.) were reported in 1985 by a research team at the British Antarctic Survey led by Joseph Farman. The team reported drops of about 30% in October ozone levels over Antarctica. The stratosphere is generally cloudless because of its small amount of water vapor, but at the extremely cold temperatures of the winter Antarctic stratosphere, ice clouds called polar stratospheric clouds (PSCs) form, and these cloud particles serve as sites for ozone depletion chemistry. The spatial extent of the Antarctic ozone hole, at its peak, was almost twice the area of the Antarctic continent. The Montreal Protocol on Substances that Deplete the Ozone Layer was signed in 1987 and has since been ratified by over 180 nations. The protocol establishes legally binding controls on the national production and consumption of ozone-depleting gases. Amendments and Adjustments to the protocol added new controlled substances, accelerated existing control measures, and scheduled phaseouts of certain gases. The Montreal Protocol provides for the transitional use of HCFCs as substitutes for the major ozone-depleting gases, such as CFC-11 and CFC-12. As a result of the Montreal Protocol, the total amount of ozone-depleting gases in the atmosphere is decreasing. Because CFCs are longlived, however, their concentrations decrease very slowly, despite near-zero emissions resulting from the Montreal Protocol. This chapter will be devoted to carefully working through the chemistry that governs the stratospheric O3 layer. Some Comments on Notation and Quantities of General Use In this and succeeding chapters the molecular number concentration of a species is denoted with square brackets ([O2], [O3], etc.) in units of molecules cm 3. The rate coefficient of a reaction will carry, as a subscript, either the number of the reaction, for example O ‡ O3

4

! O2 ‡ O2

k4

or, if the reaction is not explicitly numbered, the reaction itself, for instance O ‡ O3 ! O2 ‡ O2

kO‡O3

We will have frequent occasion to require stratospheric temperature and number concentration as a function of altitude. Approximate global values based on the US Standard Atmosphere are given in Table 5.1. The number concentration of air is denoted as [M(z)]. A number of atmospheric reactions involve the participation of a molecule of N2 or O2 as a third body (recall Chapter 3), and the third body in a chemical reaction is commonly designated as M TABLE 5.1 Stratospheric Temperature, Pressure, and Atmospheric Number Density z (km) 20 25 30 35 40 45

T (K)

p (hPa)

p/p0

[M] (molecules cm 3)

217 222 227 237 251 265

55 25 12 5.6 2.8 1.4

0.054 0.025 0.012 0.0055 0.0028 0.0014

1.4 × 1018 6.4 × 1017 3.1 × 1017 1.4 × 1017 7.1 × 1016 3.6 × 1016

[M0] = 2.55 × 1019 molecules cm

a

3

(288 K).

122

CHEMISTRY OF THE STRATOSPHERE

The global mean surface temperature is usually taken as 288 K, at which [M] = 2.55 × 1019 mol­ ecules cm 3. (If one chooses 298 K, which is the standard temperature at which chemical reaction rate coefficients are usually reported, then [M] = 2.46 × 1019 molecules cm 3.) In example calculations of chemical reaction rates, in which a value of [M] is needed at Earth’s surface, we will often simply use 2.5 × 1019 molecules cm 3 as the approximate value at 298 K.

5.1 CHAPMAN MECHANISM Ozone formation occurs in the stratosphere above ∼30 km altitude, where solar UV radiation of wavelengths less than 242 nm slowly dissociates molecular oxygen: 1

O2 ‡ hν

!O ‡ O

The oxygen atoms react with O2 in the presence of a third molecule M (N2 or O2) to produce O3: 2

O ‡ O2 ‡ M

! O3 ‡ M

Reaction 2 is, for all practical purposes, the only reaction that produces ozone in the atmosphere. The O3 molecule formed in that reaction itself strongly absorbs radiation in the wavelength range of 240–320 nm (recall Chapter 4) to decompose back to O2 and O O3 ‡ hν

3 3´

! O ‡ O2

Chappuis bands

! O…1 D† ‡ O2

…400

Hartley bands

600 nm†

…< 320 nm†

where reaction 3´ leads to excited states of both O and O2 [O2 (1 Δ)]. As we recall, O…1 D† is quenched to ground-state atomic oxygen by collision with N2 or O2: O…1 D† ‡ M ! O ‡ M The rate coefficient for this reaction is (Table B.2) kO…1 D†‡M ˆ 3:2  10 ˆ 1:8  10

11 11

exp…70=T† M ˆ O2 exp…110=T† M ˆ N2

The mean lifetime of O…1 D† against reaction with M is τO…1 D† ˆ

1 kO…1 D†‡M ‰MŠ

Choosing 30 km (T = 227 K), and noting that M consists of 0.21 O2 + 0.79 N2, [M] = 3.1 × 1017 molecules cm 3 (Table 5.1), we obtain τO…1 D† ≅ 10

7

s

Consequently, O…1 D† is effectively converted instantaeously to ground-state O, and the photodissociation of O3 by both reactions 3 and 3´ (above) can be considered to produce entirely groundstate O atoms. Finally, O and O3 react to reform two O2 molecules: O ‡ O3

4

! O2 ‡ O2

123

CHAPMAN MECHANISM

The mechanism (reactions 1–4 above) for production of ozone in the stratosphere was proposed by Chapman (1930) and bears his name. The rate equations for [O] and [O3] in the Chapman mechanism are d‰OŠ ˆ 2jO2 ‰O2 Š dt

k2 ‰OŠ‰O2 Š‰MŠ ‡ jO3 ‰O3 Š

d‰O3 Š ˆ k2 ‰OŠ‰O2 Š‰MŠ dt

jO3 ‰O3 Š

k4 ‰OŠ‰O3 Š

(5.1)

k4 ‰OŠ‰O3 Š

(5.2)

Once an oxygen atom is generated in reaction 1 (above), reactions 2 and 3 proceed rapidly. The characteristic time for reaction of the O atom in reaction 2 is 1 k2 ‰O2 Š‰MŠ

τ2 ˆ

(5.3)

where k2 is assumed to be expressed as a termolecular rate coefficient (cm6 molecule coefficient is (Table B.2) k2 ˆ 6:0  10

34

…T=300†

2:4

cm6 molecule

2

s

2

s 1). The rate

1

Let us evaluate τ2 at z = 30 and 40 km. Since [O2] = 0.21 [M] at any altitude, we obtain τ2 ˆ

1 0:21 k2 ‰MŠ2

(5.4)

We find z (km)

T (K)

[M] (molecules cm 3)

30 40

227 251

3.1 × 1017 7.1 × 1016

k2 (cm6 molecule

2

τ2 (s)

s 1)

1.15 × 10 33 9.1 × 10 34

0.04 1.04

The overall photolysis rate of ozone, jO3 , is the sum of the rates of the reactions 3 and 3´ above, and the overall lifetime of O3 against photolysis is τ3 ˆ

1 jO3 !O ‡ jO3 !O…1 D†

(5.5)

Evaluating τ3 at 30 and 40 km at a solar zenith angle of 0° (see Figure 4.13) yields z (km)

jO3 !O (s 1)

30 40

∼7 × 10 ∼9 × 10

jO3 !O…1 D† (s 1)

∼5 × 10 ∼1 × 10

4 4

4 3

τ3 (s)

∼800 ∼500

Thus, the photolytic lifetime of an O3 molecule is on the order of 10 min at these altitudes. However, this lifetime is not the overall lifetime of O3. Once O3 photodissociates, the O atom formed can rapidly reform O3 in reaction 2; thus, O3 photolysis alone does not lead to a loss of O3. Only reaction 4 truly removes O3 from the system. The lifetime of O3 against reaction 4 is τ4 ˆ

1 k4 ‰OŠ

(5.6)

124

CHEMISTRY OF THE STRATOSPHERE

FIGURE 5.2 Odd-oxygen family.

where k4 = 8.0 × 10 12 exp( 2060/T) cm3 molecule 1 s 1 (Table B.1). To estimate τ4, we need [O]. Let us assume, for the moment, that once an O atom is produced in reaction 1, reactions 2 and 3 cycle many times before reaction 4 has a chance to take place. If, indeed, there is rapid interconversion between O and O3, it is useful to view the sum of O and O3 as a chemical family (see Section 3.6). The sum of O and O3 is given the designation odd oxygen and is denoted Ox. The odd-oxygen family can be depicted as shown in Figure 5.2. The rapid interconversion between O and O3 leads to a pseudo-steady-state relationship between the O and O3 concentrations: jO3 jO3 ‰OŠ ˆ ˆ ‰O3 Š k2 ‰O2 Š‰MŠ 0:21 k2 ‰MŠ2

(5.7)

From the values of k2, [M], and jO3 , we can estimate the [O]/[O3] ratio. For example, at 30 and 40 km: z (km)

[O]/[O3]

3.0 × 10 9.4 × 10

30 40

5 4

As altitude increases, [M] decreases, so the [O]/[O3] ratio becomes larger at higher altitudes. Regardless, O3 is the dominant form of odd oxygen below ∼50 km. The lifetime of odd oxygen Ox can be found from the ratio of its abundance to its rate of disappearance: τOx ˆ

‰OŠ ‡ ‰O3 Š k4 ‰OŠ‰O3 Š

But since [O3]  [O], it follows that τOx ≅

‰O3 Š 1 ˆ k4 ‰OŠ‰O3 Š k4 ‰OŠ 0:21 k2 ‰MŠ2 ˆ k4 jO3 ‰O3 Š

(5.8)

which is just the lifetime of O3 against reaction 4 (above). Now, let us obtain an estimate of the lifetime of O3 against reaction 4, τ4. Shortly we will see that O3 concentrations at 30 and 40 km are about 3 × 1012 and 0.5 × 1012 molecules cm 3, respectively. Given these concentrations and the [O]/[O3] estimates above, we find that

125

CHAPMAN MECHANISM

z (km)

T (K)

30 40

227 251

k4 (cm3 molecule

9.2 × 10 2.2 × 10

1

τOx …ˆ τ4 † (s)

s 1)

1.2 × 107 (∼140 days) 106 (∼12 days)

16 15

Whereas the photolytic lifetime of O3 at these altitudes is about 10 min, the overall lifetime of O3 is on the order of weeks to months, validating the assumption that cycling within the Ox family is rapid relative to loss of Ox. τOx varies from about 140 days at 30 km to about 12 days at 40 km. (Even though both jO3 and [O3] vary with altitude, it is the [M]2 dependence of τOx that dominates its behavior as a function of altitude.) At lower altitudes, O3 lifetime is sufficiently long for it to be transported intact. At high altitudes, ozone tends to be produced locally rather than imported. Defining the chemical family [Ox] = [O] + [O3], and by adding (5.1) and (5.2), we obtain the rate equation governing odd oxygen: d‰Ox Š ˆ 2jO2 ‰O2 Š dt

2 k4 ‰OŠ‰O3 Š

(5.9)

Since O3 constitutes the vast majority of total Ox, a change in the abundance of Ox is usually just referred to as a change in O3. Two molecules of odd oxygen are formed on photolysis of O2, and two molecules of odd oxygen are consumed in reaction 4. Using (5.7) in (5.9), the reaction rate equation for odd oxygen becomes d‰Ox Š ˆ 2jO2 ‰O2 Š dt

2 k 4 ‰O 3 Š

jO3 ‰O3 Š k2 ‰O2 Š‰MŠ

(5.10)

Since [Ox] = [O] + [O3], and [O]  [O3], d[Ox]/dt  d[O3]/dt, so (5.10) becomes d‰O3 Š ˆ 2jO2 ‰O2 Š dt

2 k4 jO3 ‰O3 Š2 k2 ‰O2 Š‰MŠ

(5.11)

After a sufficiently long time, the steady-state O3 concentration resulting from reactions 1–4 is k2 ‰MŠjO2 ‰O3 Šss ˆ ‰O2 Š k4 jO3

1=2

(5.12)

which can be rearranged as

‰O3 Šss ˆ 0:21

k2 jO2 k4 jO3

1=2

‰MŠ3=2

(5.13)

An equivalent way to express the steady-state O3 concentration is

‰O3 Šss ˆ

0:21 k2 k4 jO3

1=2

…jO2 ‰O2 Š†1=2 ‰MŠ

The steady-state O3 concentration is sustained by the continual photolysis of O2.

(5.14)

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CHEMISTRY OF THE STRATOSPHERE

The steady-state O3 concentration should be a maximum at the altitude where the product of the number density of air and the square root of the O2 photolysis rate is largest. Figure 4.12 shows jO2 as a function of altitude above 30 km. The photolysis rate of O2, jO2 ‰O2 Š (molecules cm 3 s 1), peaks at about 29 km. This maximum results because even though jO2 continues to increase with z, the number density of O2, [O2] = 0.21 [M], decreases with z. The product …jO2 ‰O2 Š†1=2 ‰MŠ peaks at about 25 km. This is a key result—the peak in the stratospheric ozone layer occurs at ∼ 25 km. In summary, at high altitudes the concentration of O3 decreases primarily as a result of a drop in the concentration of O2, the photolysis of which initiates the formation of O3. At low altitudes, the O3 concentration decreases because of a decrease in the flux of photons at the UV wavelengths at which O2 photodissociates. An issue that we have not yet addressed is how long it takes to establish the steady-state O3 concentration at any particular altitude. To determine this we need to return to the rate equation for [O3], namely, (5.11). If we let y = [O3], this differential equation is of the form dy ˆb dt

ay2

(5.15)

If y(0) = 0, the general solution of (5.15) is b a

y…t† ˆ

1=2

1

exp‰ 2…ab†1=2 tŠ

(5.16)

1 ‡ exp‰ 2…ab†1=2 tŠ

The characteristic approach time of this solution to its steady state is τˆ

1 2…ab†1=2

Applying this to (5.11), we find the characteristic time needed to establish the O3 steady state is given by τss o3

1 k2 ‰MŠ ˆ 4 k4 jO2 jO3

1=2

(5.17)

Although jO2 and jO3 decrease as altitude decreases, the exponential increase of [M] with decreasing altitude exerts the dominate influence on τss o3 . Thus, we expect this time scale to increase at lower altitudes. Let us estimate τss o3 as a function of altitude, at a solar zenith angle of 0°: z (km)

T (K)

20 25 30 35 40 45

217 222 227 237 251 265

k4 (cm3 molecule

6 × 10 7.5 × 10 9.2 × 10 1.3 × 10 2.2 × 10 3.4 × 10

16 16 16 15 15 15

1

s 1)

jO2 (s 1)

1 × 10 2 × 10 6 × 10 2 × 10 5 × 10 8 × 10

11 11 11 10 10 10

τss o3 (h)

jO3 (s 1)

0.7 × 10 0.7 × 10 1.2 × 10 1.6 × 10 1.9 × 10 .6 × 10

3 3 3 3 3 3

∼1400 ∼600 ∼160 ∼40 ∼12 ∼3

Again, we caution that τss o3 is not the overall lifetime of an O3 molecule; rather, this quantity is the characteristic time required for the Chapman mechanism to achieve a steady-state balance after some perturbation. When τss o3 is short relative to other stratospheric processes, it can be assumed that the local O3 concentration obeys the steady state, (5.12)–(5.14). When τss o3 is relatively long, the Chapman mechanism does not actually attain a steady state. We see that above ∼30 km, [O3] can be assumed

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CHAPMAN MECHANISM

TABLE 5.2 Chemical Families in Stratospheric Chemistry Symbol Ox NOx NOy HOx Cly ClOx CCly Bry

Name Odd oxygen Nitrogen oxides Oxidized nitrogen Hydrogen radicals Inorganic chlorine Reactive chlorine Organic chlorine Inorganic bromine

Components O + O3 NO + NO2 NO + NO2 + HNO3 + 2N2O5 + ClONO2 + NO3 + HOONO2 + BrONO2 OH + HO2 Sum of all chlorine-containing species that lack a carbon atom (Cl + 2Cl2 + ClO + OClO + 2Cl2O2 + HOCl + ClONO2 + HCl + BrCl) Cl + ClO CF2Cl2 + CFCl3 + CCl4 + CH3CCl3 + CFCl2CF2Cl (CFC 113) + CF2HCl (CFC Sum of all bromine-containing species that lack a carbon atom (Br + BrO + HOBr + BrONO2)

22)

to be in a local steady state. At these altitudes the O3 concentration over a year is governed by the seasonal cycle of production and loss terms. Finally, the Ox chemical family is the first of a number of chemical families that are important in stratospheric chemistry. Table 5.2 summarizes chemical families important in stratospheric chemistry. As we proceed through this chapter, each of these families will emerge. Figure 5.3 shows the ozone distribution in the stratosphere. Note that the regions of highest ozone concentration do not coincide with the location of the highest rate of formation of O3. Rates of O3 production are highest at the equator and at about 40 km altitude, whereas ozone concentrations peak at northern latitudes. Even at the equator, maximum ozone concentrations occur at about 25 km as opposed to 40 km, where the production rate is greatest. At the poles, maximum O3 production occurs at altitudes even higher than those over the equator, whereas the maximum O3 concentrations are at lower

FIGURE 5.3 Zonally averaged ozone concentration (in units of 1012 molecules cm 3) as a function of altitude, March 22 (Johnston 1975). Ozone concentration at the equator peaks at an altitude of about 25 km. Over the poles the location of the maximum is between 15 and 20 km. At altitudes above the ozone bulge, O3 formation is oxygen-limited; below the bulge, O3 formation is photon-limited.

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CHEMISTRY OF THE STRATOSPHERE

FIGURE 5.4 Zonally averaged total ozone column density (in Dobson units) as a function of latitude and time of year (Dütsch 1974). This distribution is representative of that prior to anthropogenic perturbation of stratospheric ozone.

altitudes. There is even a north–south asymmetry. Figure 5.4 shows the historical total column ozone as a function of latitude and time of year, measured in Dobson units, prior to anthropogenic ozone depletion. Highest ozone column abundances are found at high latitudes in winter, and the lowest values are in the tropics. The steady-state odd-oxygen model of (5.10)–(5.12) predicts that the O3 concentration should be proportional to the square root of the O2 photolysis rate. We see that, in fact, ozone concentration and O2 photolysis rate do not peak together. The explanation for the lack of alignment of these two lies in the role of horizontal and vertical transport in redistributing stratospheric airmasses. The ozone bulge in the northern polar regions is a result, for example, of northward and downward air movement in the NH (Northern Hemisphere) stratosphere that transports ozone from high-altitude equatorial regions where ozone production is the largest. That stratospheric O3 concentrations are maximum in areas far removed from those where O3 is being produced suggests that the lifetime of O3 in the stratosphere is longer than the time needed for the transport to occur. The stratospheric transport timescale from the equator to the poles is of order 3–4 months. Until about 1964, the Chapman mechanism was thought to be the principal set of reactions governing ozone formation and destruction in the stratosphere. First, improved measurement of the rate constant of reaction 4 (above) indicated that the reaction is considerably slower than previously thought, leading to larger abundances of O3 as predicted by (5.10)–(5.12). Then, measurements indicated that the actual amount of ozone in the stratosphere is a factor of ∼2 less than what is predicted by the

129

NITROGEN OXIDE CYCLES

FIGURE 5.5 Comparison of stratospheric ozone concentrations as a function of altitude as predicted by the Chapman mechanism and as observed over Panama (9°N) on November 13, 1970.

Chapman mechanism with the more accurate rate constant of reaction 4 (Figure 5.5). It was concluded that significant additional ozone destruction pathways must be present beyond reaction 4. For an atmospheric species to contribute to ozone destruction, it would either have to be in great excess or, if a trace species, regenerated in a catalytic cycle. Following the initial studies of the effect of NOx emissions from supersonic aircraft and chlorofluorocarbons, an intricate and highly interwoven chemistry of catalytic cycles involving nitrogen oxides, hydrogen radicals, chlorine, and bromine emerged in one stunning discovery after another. The result is a tapestry of different catalytic cycles dominating at different altitudes in the stratosphere. The remainder of this chapter develops each of these systems, culminating in a sythesis of the stratospheric ozone depletion cycles not accounted for in the Chapman chemistry.

5.2 NITROGEN OXIDE CYCLES For a stratospheric species to effectively destroy ozone, it either has to be in great excess or regenerated in a catalytic cycle. As early as 1950, Bates and Nicolet (1950) introduced the idea of a catalytic loss process involving hydrogen radicals, but the true breakthrough in understanding stratospheric ozone chemistry did not occur until the early 1970s when Crutzen (1970) and Johnston (1971) revealed the role of nitrogen oxides in stratospheric chemistry. Reactive nitrogen is produced in the stratosphere from N2O. N2O is produced by microbial processes in soils and the ocean and does not undergo reactions in the troposphere (Chapter 2).

5.2.1 Stratospheric Source of NOx from N2O The principal natural source of NOx (NO + NO2) in the stratosphere is N2O. Approximately 90% of N2O in the stratosphere is destroyed by photolysis: N2 O ‡ hν

1

! N2 ‡ O…1 D†

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CHEMISTRY OF THE STRATOSPHERE

The remainder reacts with O…1 D†: N2 O ‡ O…1 D†

2a

! NO ‡ NO

2b

! N2 ‡ O 2

k2a ˆ 6:7  10 k2b ˆ 4:9  10

11

11

cm3 molecule

cm3 molecule

1

1

s

s

1

1

Reaction 2a is the main source of NOx in the stratosphere. About 58% of the N2 O ‡ O…1 D† reaction proceeds via channel 2a, the remaining 42% by channel 2b. The main influx of tropospheric species into the stratosphere occurs in the tropics, and N2O enters via this route. On crossing the tropopause, N2O advects slowly upward. During its ascent, N2O is diluted by N2O-poor air that mixes in from outside the tropical stratosphere and, at the same time, disappears by reactions 1 and 2. The higher a molecule of N2O rises in the stratosphere, the more energetic the photons that are encountered, and the more rapid its photodissociation by reaction 1. Even though the photodissociation of N2O produces O…1 D†, the main source of O…1 D† in the stratosphere is photodissociation of O3 (reaction 3´ in Section 5.2), and the concentration of O…1 D† at any altitude is determined by its source from O3 photolysis and its sink from quenching by O…1 D† ‡ M. The concentration of O…1 D† increases with altitude, largely because [M] is decreasing. Because the O3 profile eventually decreases with altitude, the concentration of O…1 D† reaches a maximum at a certain altitude. Also, since the N2O concentration is continually decreasing with altitude, its rate of destruction, jN2 O ‰N2 OŠ, reaches a maximum at a certain altitude even though the light intensity is stronger at higher altitudes. Figure 5.6a shows vertical profiles of the N2O mixing ratio in the tropics. The circles denote balloonborne measurements at 9°N and 5°S; the squares represent aircraft measurements between 1.6°S and 9.9°N. The dashed curve is the average of satellite measurements at 5°N, equinox, between 1979 and 1981. Figure 5.6b shows the diurnally averaged 1980 loss rate for N2O (molecules cm 3 s 1) as a function of altitude and latitude for equinox calculated with a photochemical model (Minschwaner et al. 1993). The loss rate includes N2O photolysis and reaction with O…1 D†. As seen from Figure 5.6, the global loss of

FIGURE 5.6 (a) Vertical profiles of N2O over the tropics at equinox circa 1980. Circles denote balloonborne measurements at 9°N and 5°S; squares represent aircraft measurements between 1.6°S and 9.9°N. Dashed curve refers to the average of satellite measurements at 5°N, equinox, between 1979 and 1981. This compilation of data was presented by Minschwaner et al. (1993), where the original sources of data can be found. The dotted curve indicates the vertical profile used by Minschwaner et al. to estimate the lifetime of N2O. (b) Calculated diurnally averaged loss rate for N2O (in units of 102 molecules cm 3 s 1) as a function of altitude and latitude, at equinox. The loss rate includes both photolysis and reaction with O…1 D† (Minschwaner et al. 1993).

131

NITROGEN OXIDE CYCLES

N2O occurs mainly at latitudes between the equator and 30°. Also, N2O loss rates are largest in the 25–35 km altitude range. The fractional yield of NO from N2O at any altitude is the ratio of the number of molecules of NO formed to the total molecules of N2O reacted: NO yield ˆ

2k2a ‰O…1 D†Š

(5.18)

jN2 O ‡ …k2a ‡ k2b †‰O…1 D†Š

The stratospheric O…1 D† concentration at any altitude is controlled by the source from O3 photolysis and the sink by quenching to ground-state atomic oxygen 3

O3 ‡ hν

! O…1 D† ‡ O2 4

O…1 D† ‡ M

!O ‡ M

so that the O…1 D† steady-state concentration is given by O…1 D† ˆ For example, at 30 km and θo = 45°, jN2 O ≅ 5  10

8

jO3 !O…1 D† ‰O3 Š

(5.19)

k4 ‰MŠ

s 1 , and jO3 !O…1 D† ≅ 15  10

5

s 1 . With k4 = 3.2 × 10

11

cm3

molecule 1 s 1, the instantaneous steady-state concentration of O…1 D† from (5.19) at 30 km and θo = 45° is ∼45 molecules cm 3. The NO yield under these conditions from (5.18) is 0.11 molecules of NO formed per molecule of N2O removed. The NO yield at any altitude would require using 24-h averages of the two j values. The fractional yield of NO from N2O loss varies with altitude and latitude; yet, the ratio of NOy (NO + NO2 + all products of NOx oxidation) to N2O is nearly linear with altitudes (until NOy reaches its peak), suggesting a nearly constant yield with altitude and latitude (Figure 5.7). Transport smoothes out the variations of the fractional yield with altitudes and latitude. In the uppermost stratosphere, the reaction N + NO → N2 + O converts NO back to N2; this is the explanation for the lack of adherence of the data to the straight lines in Figure 5.7.

5.2.2 NOx Cycles Consider the following cycle involving NOx that converts odd oxygen (O3 + O) into even oxygen (O2): NOx cycle 1 :

NO ‡ O3

NO2 ‡ O Net : O3 ‡ O

1

! NO2 ‡ O2

k1 ˆ 3:0  10

2

! NO ‡ O2 !O2 ‡ O2

k2 ˆ 5:6  10

12

exp… 1500=T†

12

exp…180=T†

The characteristic time of reaction 1 with respect to NO removal is τ1 = (k1[O3]) 1. At 25 km (T = 222 K), [O3]  4 × 1012 molecules cm 3 and k1 = 3.6 × 10 15 cm3 molecule 1 s 1. Thus, at this altitude, τ1  70 s. The characteristic time of reaction 2 relative to reaction of NO2 is τ2 = (k2[O]) 1. At 25 km, k2 = 12.6 × 10 12 cm3 molecule 1 s 1. The O atom concentration is given by (5.7). Using an estimate of [O]  108 molecules cm 3, we find τ2  800 s. Therefore, the rate of the overall cycle of reactions 1 and 2 is controlled by reaction 2. In competition with reaction 2 is NO2 photolysis: NO2 ‡ hν

3

! NO ‡ O

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CHEMISTRY OF THE STRATOSPHERE

which is followed by O ‡ O2 ‡ M

4

! O3 ‡ M

If reaction 1 is followed by reaction 3, then no net O3 destruction takes place because the O atom formed in reaction 3 rapidly combines with O2 to re-form O3 in reaction 4. If, on the other hand, reaction 2 follows reaction 1, then two molecules of odd oxygen are converted to two O2 molecules. These reactions involving the NOx family can be depicted as in Figure 5.8. The inner cycle interconverting NO and NO2 is sufficiently rapid that a steady state can be assumed: 0 ˆ k1 ‰NOŠ‰O3 Š

jNO2 ‰NO2 Š

k2 ‰NO2 Š‰OŠ

FIGURE 5.7 Stratospheric NOy mixing ratio versus N2O mixing ratio observed by balloonborne in situ measurements at 44°N during October and November 1994 (WMO 1998). Solid line is linear fit to the data. Points labeled “AER” are results of a stratospheric twodimensional (2D) model. [Adapted from Kondo et al. (1996).]

(5.20)

133

NITROGEN OXIDE CYCLES

FIGURE 5.8 The NOx chemical family in relation to stratospheric NOx cycle 1.

The reaction rate equation for odd oxygen is d‰Ox Š ˆ k1 ‰NOŠ‰O3 Š dt

k2 ‰NO2 Š‰OŠ ‡ jNO2 ‰NO2 Š

(5.21)

Adding (5.20) and (5.21), the rate of destruction of odd oxygen by the cycle is d‰Ox Š ˆ 2 k2 ‰NO2 Š‰OŠ dt

(5.22)

Let us compare (5.22) to the rate of destruction of odd oxygen by reaction 4 in the Chapman mechanism; so we need to compare the rates of the two reactions: NO2 ‡ O ! NO ‡ O2

kNO2 ‡O ˆ 5:6  10

12

O ‡ O 3 ! O2 ‡ O2

kO‡O3 ˆ 8:0  10

exp… 2060=T†

12

exp…180=T†

The ratio of the two rates is kNO2 ‡O ‰NO2 Š kO‡O3 ‰O3 Š Let us evaluate this ratio at 35 km (237 K). From Figure 5.2, [O3]  2 × 1012 molecules cm 3 at this altitude. We will see later that a representative NO2 concentration is ∼109 molecules cm 3. The rate coefficient ratio at this altitude is kNO2 ‡O =kO‡O3 ≅ 9000. Thus, the ratio of the two rates is kNO2 ‡O ‰NO2 Š ‰NO2 Š ≅ 9000 ≅ 4:5 ‰O3 Š kO‡O3 ‰O3 Š Even though O3 is present at roughly 1000-fold greater concentration than NO2, the large value of the NO2 + O rate coefficient compensates for this difference to make the NO2 + O reaction almost 5 times more efficient in odd oxygen destruction than O + O3. The cycle shown above is most effective in the upper stratosphere, where O atom concentrations are highest. An NOx cycle that does not require oxygen atoms and therefore is more important in the lower stratosphere is as follows: NOx cycle 2 :

NO ‡ O3

1

! NO2 ‡ O2

NO2 ‡ O3

2

! NO3 ‡ O2

3

! NO ‡ O2 !O2 ‡ O2 ‡ O2

NO3 ‡ hν Net : O3 ‡ O3

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CHEMISTRY OF THE STRATOSPHERE

FIGURE 5.9 NO, NO2, HNO3, N2O5, and ClONO2 mixing ratios as a function of altitude (WMO 1998). All species except N2O5 measured at sunset; N2O5 measured at sunrise. Lines are the result of a calculation assuming photochemical steady state over a 24-h period. [Adapted from Sen et al. (1998).]

The nitrate radical NO3 formed in reaction 2, during daytime, rapidly photolyzes (at a rate of about 0.3 s 1). There are two channels for photolysis: NO3 ‡ hν

a

! NO2 ‡ O

b

! NO ‡ O2

The photolysis rate by channel a is about 8 times that for channel b. Channel b leads to an overall loss of odd oxygen and is the reaction involved in cycle 2. At night, NO3 formed by reaction 2 does not photodissociate and participates in an important series of reactions that involve stratospheric aerosol particles; we will return to this in Section 5.7. Figure 5.9 shows mixing ratios of NO, NO2, HNO3, N2O5, and ClONO2 (this compound to be discussed subsequently) measured at 35°N. At 25 km, the predominant NOy species is HNO3, whereas from 30 to 35 km, NO2 is the major compound. Above 35 km, NO is predominant. We will consider the source of HNO3 shortly.

5.3 HOx CYCLES The HOx family (OH + HO2) plays a key role in stratospheric chemistry. In fact, the first ozonedestroying catalytic cycle identified historically involved hydrogen-containing radicals (Bates and Nicolet 1950). Production of OH in the stratosphere is initiated by photolysis of O3 to produce O…1 D†, followed by O(1 D) + H2O → 2 OH O( D) + CH4 → OH + CH3 1

kO…1 D†‡H2 O ˆ 2:2  10

10

cm3 molecule

1

s

1

kO…1 D†‡CH4 ˆ 1:5  10

10

cm3 molecule

1

s

1

Whereas the troposphere contains abundant water vapor, little H2O makes it to the stratosphere; the low temperatures at the tropopause lead to an effective freezing out of water before it can be transported up (a “cold trap” at the tropopause). Mixing ratios of H2O in the stratosphere do not exceed

135

HOx CYCLES

approximately 5–6 ppm. In fact, about half of this water vapor in the stratosphere actually results from the oxidation of methane that has leaked into the stratosphere from the troposphere. Between 20 and 50 km the total rate of OH production from the two reactions above is ∼2 × 104 molecules cm 3 s 1. About 90% is the result of the O…1 D† ‡ H2 O reaction; the remaining 10% results from O…1 D† ‡ CH4 . Figure 5.10 shows the OH mixing ratio as a function of altitude at three different latitudes and the OH concentration as a function of solar zenith angle. The strong solar zenith angle dependence reflects the photolytic source of OH. Figure 5.11 shows OH, HO2, and total HOx versus altitude.

FIGURE 5.10 (a) OH mixing ratio versus altitude (pressure) and (b) OH concentration as a function of solar zenith angle (WMO 1998). Solid curve is prediction by Wennberg et al. (1994).

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CHEMISTRY OF THE STRATOSPHERE

FIGURE 5.11 OH, HO2, and total HOx versus altitude, as compared with different model predictions (Jucks et al. 1998). Concentration profiles of OH and HO2 measured during midmorning by satellite on April 30, 1997 near Fairbanks, Alaska (65°N). Model predictions: model A—JPL 1997 kinetics; model B—rate of O + HO2 decreased by 50%; model C—rate of O + OH decreased by 20% and rate of OH + HO2 increased by 30%; model D—rates of O + HO2 and OH + HO2 decreased by 25%.

OH and HO2 rapidly interconvert so as to establish the HOx chemical family (Figure 5.12). A reaction that is important in affecting the interconversion between OH and HO2 in the HOx cycle is HO2 ‡ NO ! NO2 ‡ OH

k ˆ 3:5  10

12

exp…250=T†

The NO2 formed in this reaction photolyzes NO2 ‡ hν ! NO ‡ O followed by O ‡ O2 ‡ M ! O3 ‡ M As a result, if a molecule of HO2 reacts with NO before it has a chance to react with either O or O3, the result is a do-nothing (null) cycle with respect to removal of O3. Each time a molecule of HO2 follows the

137

HOx CYCLES

FIGURE 5.12 The stratospheric HOx family. The upper panel shows only the reactions affecting the inner dynamics of the HOx system. The lower panel includes additional reactions that affect OH and HO2 levels; these reactions will be discussed subsequently.

bottom path and reacts with O or O3, two molecules of odd oxygen are removed: HO2 ‡ O ! OH ‡ O2 HO2 ‡ O3 ! OH ‡ 2O2 Regeneration of HO2 occurs by OH ‡ O3 ! HO2 ‡ O2 The resulting catalytic ozone depletion cycles are HOx cycle 1 :

OH ‡ O3 ! HO2 ‡ O2 HO2 ‡ O ! OH ‡ O2

Net : HOx cycle 2 :

O3 ‡ O ! O2 ‡ O2 OH ‡ O3 ! HO2 ‡ O2 HO2 ‡ O3 ! OH ‡ O2 ‡ O2

Net :

O3 ‡ O3 ! O2 ‡ O2 ‡ O2

The steady-state relation in the HOx family is kOH‡O3 ‰OHŠ‰O3 Š ˆ kHO2 ‡NO ‰HO2 Š‰NOŠ ‡ kHO2 ‡O ‰HO2 Š‰OŠ ‡ kHO2 ‡O3 ‰HO2 Š‰O3 Š

(5.23)

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CHEMISTRY OF THE STRATOSPHERE

The balance on odd oxygen, O + O3, is d‰Ox Š d‰O3 Š d‰OŠ ˆ ‡ dt dt dt ˆ kOH‡O3 ‰OHŠ‰O3 Š

kHO2 ‡O3 ‰HO2 Š‰O3 Š ‡ jNO2 ‰NO2 Š

kHO2 ‡O ‰HO2 Š‰OŠ

(5.24)

Using (5.23) in (5.24), we obtain d‰Ox Š ˆ dt

2kHO2 ‡O3 ‰HO2 Š‰O3 Š ‡jNO2 ‰NO2 Š

2 kHO2 ‡O ‰HO2 Š‰OŠ

kHO2 ‡NO ‰HO2 Š‰NOŠ

(5.25)

However, formation of NO2 by HO2 + NO is followed immediately by photolysis of NO2: jNO2 ‰NO2 Š ˆ kHO2 ‡NO ‰HO2 Š‰NOŠ

(5.26)

Thus (5.25) becomes d‰Ox Š ˆ 2 kHO2 ‡O3 ‰HO2 Š‰O3 Š dt

2 kHO2 ‡O ‰HO2 Š‰OŠ

(5.27)

and the rates of O3 depletion by HOx cycles 1 and 2 are governed by the rates of the HO2 + O and HO2 + O3 reactions. The characteristic time for cycling within the HOx family is controlled by the rate of which HO2 is converted back to OH. That interconversion is controlled by the HO2 + NO reaction at all altitudes. The characteristic time is τHO2 ˆ …kHO2 ‡NO ‰NOŠ† 1 . At 30 km, for example, τHO2 ≅ 100 s. This time is short relative to competing processes, establishing the validity of treating HOx as a chemical family. The two HOx cycles are important at different altitudes. First, we can examine the relative values of the two key rate coefficients, kHO2 ‡O and kHO2 ‡O3 . Neither is strongly temperature-dependent; roughly, we have kHO2 ‡O ≅ 10

10

cm3 molecule

1

s

1

kHO2 ‡O3 ≅ 10

15

cm3 molecule

1

s

1

From Figure 5.11 we note that [HO2] does not vary strongly with altitude, with a value  107 molecule cm 3. The major difference arises from the relative concentrations of O and O3. In the lower stratosphere (∼20 km), [O]/[O3] ∼ 10 7, so even with a rate coefficient five orders of magnitude larger than that of HO2 + O3, the HO2 + O reaction is negligible compared to that of HO2 + O3. Therefore, HOx cycle 2 dominates in the lower stratosphere. Conversely, at ∼50 km, [O]/[O3]  10 2, so HOx cycle 1 is predominant. Let us consider behavior of the HOx family in the lower stratosphere, where HO2 reacts preferen­ tially with O3 rather than O. The partitioning between HO2 and OH is given by kOH‡O3 ‰O3 Š ‰HO2 Š ˆ ‰OHŠ kHO2 ‡NO ‰NOŠ

(5.28)

The relative concentrations depend on altitude through the temperature dependence of the reaction rate coefficients and through the concentrations of O3 and NO. At 30 km (T = 227 K), the two reaction rate coefficients, kOH‡O3 ˆ 1:7  10 12 exp… 940=T† and kHO2 ‡NO ˆ 3:5  10 12 exp…250=T†, have the values (cm3 molecule 1 s 1), kOH‡O3 ˆ 2:7  10 14 and kHO2 ‡NO ˆ 10:5  10 12 . Using [O3]  2 × 1012 molecules cm 3 and an NO mixing ratio of 3 ppb at 30 km ([NO]  9.3 × 108 molecules cm 3), we find ‰HO2 Š ≅ 4:4 ‰OHŠ

139

HALOGEN CYCLES

This estimate is roughly consistent with the ratio at 30 km derived from the vertical profiles of OH and HO2 measured by satellite, as shown in Figure 5.11. At higher altitudes, the concentration of OH becomes roughly comparable to that of HO2. The OH concentration is basically independent of the NO concentration. This can be explained as follows. At constant [O3], the concentration of HO2 decreases with increasing NOx. The increase of OH that results from the HO2 + NO reaction is offset by a decrease of the rates of the HO2 + O3, HO2 + ClO, and HOCl + hν reactions, which themselves generate OH. (We will discuss the latter two reactions shortly.) The net result is that OH becomes more or less independent of the NO concentration. The [HO2]/[OH] ratio does, however, depend on the NOx level as in (5.28). Any process that might serve to decrease the level of NOx would have the effect of shifting this ratio in favor of HO2. And since HO2 is the key species in the rate-determining step of both HOx cycles, the overall effect of a decrease in NOx would be an increase in the effectiveness of the HOx cycles. As we did for NOx, let us compare the effectiveness of the HOx cycles with the Chapman mechanism. The appropriate comparison is with HOx cycle 1, since both involve atomic oxygen. The effectiveness of HOx cycle 1 relative to reaction 4 in the Chapman mechanism is given by the ratio kHO2 ‡O ‰HO2 Š‰OŠ kHO2 ‡O ‰HO2 Š ˆ kO‡O3 ‰O3 Š kO‡O3 ‰OŠ‰O3 Š At 35 km (237 K), the rate coefficient ratio is 3:0  10 11 exp…200=T† kHO2 ‡O ˆ kO‡O3 8:0  10 12 exp… 2060=T† ≅ 5:4  104 From Figure 5.3, we estimate [O3] at 35 km as [O3]  2 × 1012 molecules cm 3. At this altitude, [HO2]  1.5 × 107 molecules cm 3. Thus kHO2 ‡O ‰HO2 Š ≅ …5:4  104 †…0:75  10 5 † ≅ 0:4 kO‡O3 ‰O3 Š HOx cycle 1 is therefore about half as effective in destroying O3 at 35 km as the Chapman cycle.

5.4 HALOGEN CYCLES In 1974 Molina and Rowland (Molina and Rowland 1974; Rowland and Molina 1975) realized that chlorofluorocarbons (CFCs), manufactured and used by humans in a variety of technological applications from refrigerants to aerosol spray propellants, have no tropospheric sink and persist in the atmosphere until they diffuse high into the stratosphere where the powerful UV light photolyzes them. The photolysis reactions release a chlorine (Cl) atom, for example, for CFCl3 (CFC-11) and CF2Cl2 (CFC-12): CFCl3 ‡ hν ! CFCl2 ‡ Cl CF2 Cl2 ‡ hν ! CF2 Cl ‡ Cl To photodissociate, CFCs need not rise above most of the atmospheric O2 and O3; they are photodissociated at wavelengths in the 185–210 nm spectral window between O2 absorption of shorter wavelengths and O3 absorption of longer wavelengths. The maximum loss rate of CFCl3 occurs at about 23 km, whereas that for CF2Cl2 takes place in the 25–35 km range. As with N2O, the bulk of the removal of CFCl3 and CF2Cl2 is confined to the tropics, reflecting larger values for photolysis rates in this region.

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CHEMISTRY OF THE STRATOSPHERE

The only continuous natural source of chlorine in the stratosphere is methyl chloride, CH3Cl (see Chapter 2). The tropospheric lifetime for CH3Cl is sufficiently long, 1.5 years (Table 2.17), so some amount of CH3Cl is transported through the tropopause. In the stratosphere, as in the troposphere, CH3Cl is removed primarily by reaction with the OH radical. At higher altitudes in the stratosphere a portion of CH3Cl is photolyzed. Regardless of the path of reaction, the chlorine atom in CH3Cl is released as active chlorine.

5.4.1 Chlorine Cycles The chlorine atom is highly reactive toward O3 and establishes a rapid cycle of O3 destruction involving the chlorine monoxide (ClO) radical: Cl ‡ O3

1

! ClO ‡ O2

k1 ˆ 2:3  10

11

ClO ‡ O Net : O3 ‡ O

2

! Cl ‡ O2 ! O2 ‡ O2

k2 ˆ 3:0  10

11

ClOx cycle 1 :

exp… 200=T† exp…70=T†

Let us estimate the lifetime of the Cl atom against reaction 1 and the lifetime of ClO against reaction 2. To do so we need the concentrations of O3 and O; these are related by (5.7). At 40 km, for example, where T = 251 K, the O3 concentration can be taken as 0.5 × 1012 molecules cm 3, and the ratio [O]/[O3] was shown to be about 9.4 × 10 4. The two lifetimes at 40 km are τCl τClO

1 ˆ 0:2 s k1 ‰O3 Š 1 ˆ 53 s ˆ k2 ‰OŠ ˆ

Subsequent reactions of the CFCl2 and CF2Cl radicals lead to rapid release of the remaining chlorine atoms. Together, Cl and ClO establish the ClOx chemical family (Figure 5.13). The rapid inner cycle is characterized by ClO formation by reaction 1 and loss by reaction with both O and NO. If ClO reacts with O, the catalytic O3 depletion cycle above occurs. If ClO reacts with NO, the following cycle takes place: Cl ‡ O3

1

! ClO ‡ O2

ClO ‡ NO

3

! Cl ‡ NO2

NO2 ‡ hν Net :

O3 ‡ hν

k3 ˆ 6:4  10

12

exp…290=T†

! NO ‡ O ! O ‡ O2

FIGURE 5.13 The stratospheric ClOx family.

141

HALOGEN CYCLES

Because the O atom rapidly reforms O3, this is a null cycle with respect to O3 removal. The rate of net O3 removal by the ClOx cycle of reactions 1 and 2 is d‰Ox Š d‰O3 Š d‰OŠ ˆ ‡ ˆ 2 k2 ‰ClOŠ‰OŠ dt dt dt

(5.29)

Within the ClOx chemical family we can compare the relative importance of reactions 2 and 3: kClO‡O ‰OŠ kClO‡NO ‰NOŠ At 40 km, T = 251 K, [O]/[O3]  9.4 × 10 4, [O3]  0.5 × 1012 molecules cm 3, and [NO]  1 × 109 molecules cm 3. The rate coefficient values are kClO+O  4 × 10 11 cm3 molecule 1 s 1 and kClO+NO  2 × 10 11 cm3 molecule 1 s 1. Thus kClO‡O ‰OŠ ≅ 0:9 kClO‡NO ‰NOŠ The steady-state ratio [Cl]/[ClO] within the ClOx family is given by ‰ClŠ kClO‡O ‰OŠ ‡ kClO‡NO ‰NOŠ ˆ ‰ClOŠ kCl‡O3 ‰O3 Š

(5.30)

At 40 km ‰ClŠ ≅ 0:008 ‰ClOŠ so most of the ClOx is in the form of ClO; this is a result of the rapidity of the Cl + O3 reaction. Since the ClOx cycle of reactions 1 and 2 and the Chapman mechanism (Section 5.2) both involve the O atom, let us compare, as we have done with the NOx and HOx cycles, the rate of reaction 2 to that of reaction 4 in the Chapman mechanism: kClO‡O ‰ClOŠ kO‡O3 ‰O3 Š At 35 km, the rate coefficient ratio, kClO‡O =kO‡O3 ˆ 3  104 . We need the altitude dependence of [ClO]. Figure 5.14 shows the stratospheric chlorine inventory as of November 1994 (35–49°N). At 35 km, the ClO mixing ratio was about 0.5 ppb, which translates to [ClO]  0.7 × 108 molecules cm 3. Thus, at this altitude kClO‡O ‰ClOŠ ≅ 1:1 kO‡O3 ‰O3 Š Ozone removal by catalytic chlorine chemistry and the Chapman cycle are, therefore, roughly equal at about 40 km. The ClOx cycle of reactions 1 and 2 achieves its maximum effectiveness at about 40 km, roughly the altitude at which ClO achieves its maximum concentration. At higher altitudes, the Cl + CH4 reaction controls Cl/ClO partitioning, and Cl becomes sequestered as HCl. The overall set of reactions in the chlorine cycles involves a few more key reactions. Figure 5.15 shows an expanded view of stratospheric chlorine chemistry. At lower altitudes, the reservoir species, chlorine nitrate, ClONO2, ties up ClO. As altitude increases, [M] decreases, and the rate of the termolecular reaction, ClO + NO2 + M → ClONO2 + M,

142

CHEMISTRY OF THE STRATOSPHERE

FIGURE 5.14 Stratospheric chlorine inventory (35–49°N), November 3–12, 1994 (WMO 1998). [Adapted from Zander et al. (1996).]

decreases. The competing formation of the reservoir species HCl and ClONO2 produce the maximum in [ClO] at about 40 km. An increase of OH produces opposite effects on the NOx and ClOx cycles. As OH increases, HCl is converted back to active Cl, enhancing the effectiveness of the ClOx cycle in O3 destruction. On the other hand, as OH increases, more NO2 is converted to the HNO3 reservoir, decreasing the catalytic

FIGURE 5.15 Cly chemical family.

143

HALOGEN CYCLES

effectiveness of the NOx cycle. (Although HNO3 itself reacts with OH, HNO3 + OH → H2O + NO3, releasing NOx, this reaction is less effective in releasing NOx than is OH + NO2 + M → HNO3 + M in sequestering it.) From Figure 5.15 we can define Cly ˆ Cl ‡ ClO ‡ HOCl ‡ ClONO2 ‡ HCl as the chemical family of reactive chlorine. Note from Figure 5.14 that all the stratospheric chlorine is accounted for, as indicated by the constant value of Cltot at all altitudes. As CCly, the chlorine contained in CFCs, CH3Cl, CCl4, and so on decreases with increasing altitude as a result of photodissociation of the CFCs, the species in Cly appear, such that the sum of all chlorine is conserved. Together, HCl and ClONO2 store as much as 99% of the active chlorine. As a result, only a small change in the abundance of either HCl or ClONO2 can have a profound effect on the catalytic efficiency of the ClOx cycle At lower altitudes where O atom levels are significantly lower, the following cycle that involves coupling with HOx is important: HOx =ClOx cycle 1 :

Cl ‡ O3 ! ClO ‡ O2 OH ‡ O3 ! HO2 ‡ O2 ClO ‡ HO2 ! HOCl ‡ O2 HOCl ‡ hν ! OH ‡ Cl

Net :

O3 ‡ O3 ! O2 ‡ O2 ‡ O2

In addition, the following HOx/BrOx and BrOx/ClOx cycles are important in the lower stratosphere (∼20 km): HOx =BrOx cycle 1 :

Br ‡ O3 ! BrO ‡ O2 OH ‡ O3 ! HO2 ‡ O2 BrO ‡ HO2 ! HOBr ‡ O2 HOBr ‡ hν ! OH ‡ Br

Net : BrOx =ClOx cycle 1 :

O3 ‡ O3 ! O2 ‡ O2 ‡ O2 Br ‡ O3 ! BrO ‡ O2 Cl ‡ O3 ! ClO ‡ O2 BrO ‡ ClO ! BrCl ‡ O2 BrCl ‡ hν ! Br ‡ Cl

Net : BrOx =ClOx cycle 2 :

O3 ‡ O3 ! O2 ‡ O2 ‡ O2 Br ‡ O3 ! BrO ‡ O2 Cl ‡ O3 ! ClO ‡ O2 BrO ‡ ClO ! ClOO ‡ Br ClOO ‡ M ! Cl ‡ O2

Net :

O3 ‡ O3 ! O2 ‡ O2 ‡ O2

5.4.2 Bromine Cycles Historically, the total chlorine compound mixing ratio in the stratosphere peaked at approximately 3500 ppt, while that for bromine gases was only ∼20 ppt (Figure 5.1). Remarkably, with 150 times less

144

CHEMISTRY OF THE STRATOSPHERE

FIGURE 5.16 Bry chemical family.

abundance than chlorine, bromine is approximately as important as chlorine in overall ozone destruc­ tion. Methyl bromide (CH3Br) constitutes about half the source of bromine in the stratosphere (see Section 2.5). The H-atom-containing bromine compounds, CH3Br, CH2Br2, and CHBr, release their Br almost immediately on entry into the stratosphere; the halons, CBrFCl2 and CBrF3, release their Br more slowly and therefore at higher altitude. The Bry system is shown in Figure 5.16. In order to understand why bromine compounds are so much more effective in ozone destruction, we need to consider the role of reservoir species in stratospheric ozone destruction cycles.

5.5 RESERVOIR SPECIES AND COUPLING OF THE CYCLES Cycles such as HOx cycles 1 and 2, NOx cycle 1, and ClOx cycle 1 would appear to go on destroying O3 forever. Actually, the cycle can be interrupted when the reactive species, OH, NO2, Cl, and ClO, become tied up in relatively stable species so that they are not available to act as catalysts in the cycles. Knowing the competing reactions that can occur, it is possible to estimate the average number of times each catalytic O3 destruction cycle will proceed before one of the reactants participates in a competing reaction and terminates the cycle. For example, at current stratospheric concentrations, ClOx cycle 1, once initiated, loops, on average, 105 times before the Cl atom or the ClO molecule reacts with some other species to terminate the cycle. This means that, on average, one chlorine atom can destroy 100,000 molecules of ozone before it is otherwise removed. The frequency of such cycle termination reactions is thus critical to the overall efficiency of a cycle. The removal of a reactive species can be permanent if the product actually leaves the stratosphere by eventually migrating down to the troposphere, where it is removed from the atmosphere altogether. Examples of cycle-interrupting reactions that can lead to ultimate removal from the atmosphere are OH ‡ NO2 ‡ M ! HNO3 ‡ M Cl ‡ CH4 ! HCl ‡ CH3 Both nitric acid (HNO3) and hydrogen chloride (HCl) are relatively stable in the stratosphere and some fraction of each migrates back to and is removed from the troposphere by precipitation. A reactive species can also be temporarily removed from a catalytic cycle and be stored in a reservoir species, which itself is relatively unreactive but is not actually removed from the atmosphere. One of the most important reservoir species in the stratosphere is chlorine nitrate (ClONO2), formed by ClO ‡ NO2 ‡ M ! ClONO2 ‡ M Chlorine nitrate can photolyze ClONO2 ‡ hν ! ClO ‡ NO2 ! Cl ‡ NO3

145

RESERVOIR SPECIES AND COUPLING OF THE CYCLES

thereby releasing Cl or ClO back into the active ClOx reservoir. Chlorine nitrate is an especially important reservoir species because it stores two catalytic agents, NO2 and ClO. Partitioning of chlorine between reactive (e.g., Cl and ClO) and reservoir forms (e.g., HCl and ClONO2) depends on temperature, altitude, and latitude history of an air parcel. For the midlatitude, lower stratosphere, HCl and ClONO2 are the dominant reservoir species for chlorine, constituting over 90% of the total inorganic chlorine (Figure 5.14). From the Cl y family in Figure 5.15, Cl can be bled off the internal cycle by reacting with methane to produce HCl, and the chlorine atom in HCl can be returned to active Cl if HCl reacts with OH. The abundances of CH4 and OH will control the amount of Cl sequestered as HCl. ClO can be temporarily removed from active participation in the ClOx catalytic cycle by reacting with NO2 to form ClONO2 or with HO2 to form HOCl. Both ClONO2 and HOCl can photolyze and release active chlorine back into the cycle. The importance of ClONO2 and HOCl as reservoir species will depend on the abundances of NO2 and HO2 relative to atomic oxygen. Lifetime of the Reservoir Species ClONO2 and HCl Chlorine nitrate (ClONO2) photodissociates back to ClO; its lifetime against photolysis is τClONO2 ˆ jClONO2 1 . In the midstratosphere (15–30 km), jClONO2 ≅ 5  10 5 s 1 , so τClONO2 ∼5 h. The lifetime of HCl against reaction with OH is τHCl = (kOH+HCl[OH]) 1, where kOH+HCl = 2.6 × 10 12 exp ( 350/T). At 35 km (237 K), kOH+HCl  6 × 10 13 cm3 molecule 1 s 1, and [OH]  107 molecules cm 3, so τHCl  2 days. At lower altitudes, where OH levels are lower, the HCl lifetime increases to several weeks. From Figure 5.15 we note the amounts of ClONO2 and HCl relative to those of Cl and ClO. NO2 and CH4 are responsible for shifting most of the active chlorine into these reservoir species. Typically ‰HClŠ ∼0:7 ‰Cly Š

‰ClONO2 Š ∼0:3 ‰Cly Š

‰ClOŠ ∼0:02 ‰Cly Š

0:05

Any process that even modestly shifts the balance away from the reservoir species to ClO can have a large impact on ozone depletion. In the midlatitude lower stratosphere the concentration of ClO is actually controlled by the amount of ClONO2 present: ‰ClOŠ ≅

jClONO2 ‰ClONO2 Š kClO‡NO2 ‰NO2 Š‰MŠ

Note that the O atom concentration in the lower stratosphere is too low for its reaction with ClO to compete effectively with the ClO + NO2 reaction.

The HOx, NOx, and ClOx cycles are all coupled to one another, and their interrelationships strongly govern stratospheric ozone chemistry. The NOx and ClOx cycles are coupled by ClONO2. For example, increased emissions of N2O would lead to increased stratospheric concentrations of NO and hence increased ozone depletion by the NOx catalytic cycle. Likewise, increasing CFC levels will lead to increased ozone depletion by the ClOx cycle. However, increased NOx will lead to an increased level of the ClONO2 reservoir and a mitigation of the chlorine cycle. Thus the net effect on ozone of simultaneous increases in both N2O and CFCs is less than the sum of their separate effects because of increased formation of ClONO2. Increases in CH4 or stratospheric H2O would also act to mitigate the efficiency of NOx and ClOx cycles through increased formation of HCl and OH (and hence HNO3).

146

CHEMISTRY OF THE STRATOSPHERE

Whereas HCl is formed by Cl + CH4, the corresponding reaction of Br atoms with CH4 is endothermic and extremely slow.1 As a result, the only possible routes for formation of HBr involve reaction of Br with species far less abundant than CH4, such as HO2. Even if HBr is formed, the OH + HBr rate coefficient is about 12 times larger than that of OH + HCl, so that the stratospheric lifetime of HBr against OH reaction is a matter of days as compared to about 1 month for HCl. In addition, BrONO2 is considerably less stable than ClONO2 because of rapid photolysis; the photolysis lifetimes of ClONO2 and BrONO2 in the lower stratosphere are 6 h and 10 min, respectively. In summary, the extraordinary effectiveness of the Br catalytic system is a consequence of two factors: (1) Br compounds release their Br rapidly in the stratosphere; and (2) a large fraction of the total Bry exists in the form of Br + BrO. As a result, Br ozone depletion potentials (see Section 5.10) are about 50 times greater than those of the Cl system.

5.6 OZONE HOLE In 1985 a team led by British scientist Joseph Farman shocked the scientific community with reports of massive annual decreases of stratospheric ozone over Antarctica in the polar spring (Sept.–Oct.), an observation that the prevailing understanding of stratospheric chlorine chemistry was incapable of explaining (Farman et al. 1985). This phenomenon has been termed the ozone hole by the popular press. The British Antarctic Survey had, for many years, been measuring total column ozone levels from ground level at its base at Halley Bay. Data seemed to indicate that column ozone levels had been decreasing since about 1977. When the Farman et al. paper appeared, there was a concern that the instruments aboard the Nimbus 7 satellite, the total ozone mapping spectrometer (TOMS) and the solar backscattered ultraviolet (SBUV) instrument, had apparently not detected the ozone depletion seen in the ground-based data. It turned out that, on inspection of the satellite data, the low ozone concentrations were indeed observed, but were being systematically rejected in the database as being outside the reasonable range of data. Once this was discovered, the satellite data confirmed the ground-based measurements of the British Antarctic Survey. Figure 5.17a shows October mean total column ozone observed over Halley, Antarctica, from 1956 to 1994. The open triangles are the data first reported by Farman et al. (1985); the solid triangles are the subsequent data (Jones and Shanklin 1995). Figure 5.17b shows the lowest daily value of total column ozone observed over Halley, Antarctica, in October for the years between 1956 and 1994 (the Oct. minimum). The daily value is the mean of all the readings taken during the day. The monthly mean column ozone over Halley Bay, Antarctica (76° S), decreased from a high of 350 Dobson units (DU) in the mid-1970s to values approaching 100 DU. Ozone vertical profiles from balloon measurements showed that the O3 depletion was occurring at altitudes between 10 and 20 km (Hofmann et al. 1987). The discovery of the ozone hole was surprising not only because of its magnitude but also its location. It was expected that ozone depletion resulting from the ClOx catalytic cycle would manifest itself predomi­ nantly at middle and lower latitudes and at altitudes between 35 and 45 km. Figure 5.18 shows historical changes in October ozone profiles over Antarctica. Although the Antarctic has some of Earth’s highest ozone concentrations during much of the year, most of its ozone is actually formed in the tropics and delivered, along with the molecular reservoirs of chlorine, to the Antarctic by large-scale air movement. Arctic ozone is similar. The Antarctic stratosphere is deficient in atomic oxygen because of the absence of the intense UV radiation. As air cools during the Antarctic winter, it descends and develops a westerly circulation. This polar vortex develops a core of very cold air. In the winter and early spring the vortex is extremely stable, effectively sealing off air in the vortex from that outside. The polar vortex serves to keep high levels of the imported ozone trapped over Antarctica for the several-month period each year. As the sun returns in September at the end of the long polar night, the temperature rises and the vortex weakens, eventually breaking down in November. Normally the amount of ozone in the polar vortex begins to decrease somewhat as the Antarctic emerges At 260 K, the Br + CH4 rate coefficient  10 molecule 1 s 1. 1

25

cm3 molecule

1

s 1, as compared with that for Cl + CH4  10

14

cm3

147

OZONE HOLE

FIGURE 5.17 Column ozone over Halley, Antarctica (Jones and Shanklin 1995). (a) October mean total ozone (in Dobson units) observed over Halley from 1956 to 1994. The open triangles are the data of Farman et al. (1985); solid triangles, subse­ quent data. (b) The lowest daily value of total ozone observed over Halley in October for the years between 1956 and 1994 (the October minimum). The daily value is the mean of all the readings taken during the day (between 5 and 30). Lowest daily values of total ozone (in DU) for 1996–2001 were as follows: 1996

111

1998

90

2000

94

1997

104

1999

92

2001

99

(Reprinted with permission from Nature 376, Jones, A. E. and Shanklin, J. D. Copyright 1995 Macmillan Magazines Limited.)

from the months-long austral night in late August and early September. It levels off in October and increases in November. The discovery of the Antarctic ozone hole represented a significant change in historic patterns; the springtime ozone levels decreased to unprecedented levels, with each succeeding year being, more or less, worse than the year before. It was initially suggested that the Antarctic ozone hole could be explained on the basis of solar cycles or purely atmospheric dynamics. Neither explanation was consistent with observed features of the ozone hole. Chemical explanations based on the gas-phase catalytic cycles described above were advanced. As noted, little ozone is produced in the polar stratosphere as the low Sun elevation (large solar zenith angle) results in essentially no photodissociation of O2. Thus catalytic cycles that require oxygen atoms were not able to explain the massive ozone depletion. Moreover, CFCs and halons would be most effective in ozone depletion in the Antarctic stratosphere at an altitude of about 40 km, whereas the ozone hole is sharply defined between 12 and 24 km altitude. Also, existing levels of CFCs and halons could lead at most to O3 depletion at 40 km of 5–10%, far below that observed. Molina and Molina (1987) proposed that a mechanism involving the ClO dimer, Cl2O2, might be involved: ClO ‡ ClO ‡ M

1

! Cl2 O2 ‡ M

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CHEMISTRY OF THE STRATOSPHERE

FIGURE 5.18 Observations of the change in October total ozone profiles over Antarctica (WMO 1994). Historical data at South Pole and Syowa show changes in October mean profiles measured in the 1960s and 1970s as compared to more recent observations. Changes in seasonal vertical profiles are shown at the other stations. Isopleths mapped onto Antartica represent TOMS ozone column measurements on Oct. 5, 1987.

Cl2 O2 ‡ hν Net :

2‰Cl ‡ O3 2 O3 ‡ hν

2

! Cl ‡ Cl ‡ O2

3

! ClO ‡ O2 Š !3 O2

Bimolecular reactions of ClO and ClO are slow and can be neglected. The termolecular reaction 1 is facilitated at higher pressures, that is, larger M, and low temperature. Cl2O2 has been shown to have the symmetric structure ClOOCl (McGrath et al. 1990). Photolysis of ClOOCl has two possible channels: ClOOCl ‡ hν

4a

! Cl ‡ ClOO

4b

! ClO ‡ ClO

…λ∼350 nm†

149

OZONE HOLE

Only reaction 4a can lead to an ozone-destroying cycle. Indeed, reaction 4a is the main photolysis path (Molina et al., 1990). The ClOO product rapidly decomposes to yield a Cl atom and O2, ClOO ‡ M

5

! Cl ‡ O2 ‡ M

Thus reaction 2 is seen to be a composite of reactions 4a and 5, leading to the release of both chlorine atoms from Cl2O2. Two similar cycles involving both ClOx and BrOx cycles also can take place: ClO ‡ BrO ! BrCl ‡ O2 BrCl ‡ hν ! Br ‡ Cl Cl ‡ O3 ! ClO ‡ O2 Br ‡ O3 ! BrO ‡ O2 Net : 2 O3 ! 3 O2 ClO ‡ BrO ! ClOO ‡ Br ClOO ‡ M ! Cl ‡ O2 ‡ M Cl ‡ O3 ! ClO ‡ O2 Br ‡ O3 ! BrO ‡ O2 Net : 2O3 ! 3 O2 Note that atomic oxygen is not required in either cycle. If sufficient concentrations of ClO and BrO could be generated, then the three cycles shown above could lead to substantial O3 depletion. However, gas-phase chemistry alone does not produce the necessary ClO and BrO concentrations needed to sustain these cycles.

5.6.1 Polar Stratospheric Clouds (PSCs) The stratosphere is very dry and generally cloudless. The long polar night, however, produces temperatures as low as 183 K ( 90 °C) at heights of 15–20 km. At these temperatures even the small amount of water vapor condenses to form polar stratospheric clouds (PSCs), seen as wispy pink or green clouds in the twilight sky over polar regions. The conceptual breakthrough in explaining the Antarctic ozone hole occurred when it was realized that PSCs provide the surfaces on which halogen-containing reservoir species are converted to active catalytic species. Then, intense interest ensued: What are PSCs made of—pure ice or ice mixed with other species? Nitrate was detected in PSCs, and both laboratory and theoretical studies showed that nitric acid trihydrate, HNO3
3H2O, denoted NAT, is the thermodynamically stable form of HNO3 in ice at polar stratospheric temperature (Peter 1996). It was also discovered that some PSCs are liquid particles composed of supercooled ternary solutions of H2SO4, HNO3, and H2O. An important implication of the fact that PSCs contain nitrate is that, if the particles are sufficiently large, they can fall out of the stratosphere and thereby permanently remove nitrogen from the stratosphere. The removal of nitrogen from the stratosphere is termed denitrification. If nitrate-containing PSCs sediment out of the stratosphere, then that could lead to an appreciably lower supply of nitrate for possible return to NOx (and conversion of ClO to ClONO2). Stratospheric NAT particles were first detected in situ in the 1999–2000 SAGE III “ozone loss and validation experiment,” carried out in the stratosphere over northern Sweden (Voight et al. 2000; Fahey et al. 2001). The particles identified were large enough (1–20 μm in diameter) to be able to fall a substantial distance before evaporating. The fall of nitrate-containing PSC particles was therefore established as the mechanism by which the stratosphere is denitrified.

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CHEMISTRY OF THE STRATOSPHERE

5.6.2 PSCs and the Ozone Hole Gas-phase chemistry associated with the ClOx and NOx cycles is not capable of explaining the polar ozone hole phenomenon. The ozone hole is sharply defined between about 12 and 24 km altitude. Polar stratospheric clouds occur in the altitude range 10–25 km. Ordinarily, liberation of active chlorine from the reservoir species HCl and ClONO2 is rather slow, but the PSCs promote the conversion of the major chlorine reservoirs, HCl and ClONO2, to photolytically active chlorine. The first step in the process, absorption of gaseous HCl by PSCs, occurs very efficiently. This step is followed by the heterogeneous reaction of gaseous ClONO2 with the particle, 1

HCl…s† ‡ ClONO2

! Cl2 ‡ HNO3 …s†

where (s) denotes a species on the surface of the ice, with the liberation of molecular chlorine as a gas and the retention of nitric acid in the particles. This is the most important chlorine-activating reaction in the polar stratosphere. (The gas-phase reaction between HCl and ClONO2 is extremely slow.) The solubility of HCl in normal stratospheric aerosol, 50–80 wt% H2SO4 solutions, is low. When stratospheric temperatures drop below 200 K, the stratospheric particles absorb water and allow HCl to be absorbed, setting the stage for reaction 1. Gaseous Cl2 released from the PSCs in reaction 1 rapidly photolyzes, producing free Cl atoms, while the other product, HNO3, remains in the ice, leading to the overall removal of nitrogen oxides from the gas phase. This trapping of HNO3 further facilitates catalytic O3 destruction by removing NOx from the system, which might otherwise react with ClO to form ClONO2. The net result of reaction 1 is HCl…s† ‡ ClONO2

1

! Cl2 ‡ HNO3 …s†

Cl2 ‡ hν

2

! 2 Cl

2‰Cl ‡ O3

3

! ClO ‡ O2 Š

4

Net :

ClO ‡ NO2 ‡ M ! ClONO2 ‡ M HCl…s† ‡ NO2 ‡ 2 O3 ! ClO ‡ HNO3 …s† ‡ 2 O2

The reaction between ClONO2 and H2O(s), which is very slow in the gas phase, also can occur: ClONO2 ‡ H2 O…s†

5

! HOCl ‡ HNO3 …s†

The gaseous HOCl rapidly photolyzes to yield a free chlorine atom. HOCl itself can undergo a subsequent heterogeneous reaction (Abbatt and Molina 1992) HOCl ‡ HCl…s†

6

! Cl2 ‡ H2 O

The mechanism of ozone destruction in the polar stratosphere is thus as follows. Two ingredients are necessary: cold temperatures and sunlight. The absence of either one of these precludes establishing the destruction mechanism. Cold temperatures are needed to form polar stratospheric clouds to provide the surfaces on which the heterogeneous reactions take place. The reservoir species ClONO2 and N2O5 react heterogeneously with PSCs on which HCl has been absorbed to produce gaseous Cl2, HOCl, and ClNO2. Sunlight is then required to photolyze the gaseous Cl2, HOCl, and ClNO2 that are produced as a result of the heterogeneous reactions. At sunrise, the Cl2, ClNO2, and HOCl are photolyzed, releasing free Cl atoms that then react with O3 by reaction 3. Figure 5.19 depicts the catalytic cycles and the role of PSCs. At first, the ClO just accumulates (recall that the O atoms normally needed to complete the cycle by ClO + O are essentially absent in the polar stratosphere). However, once the ClO concentrations are sufficiently large, the three catalytic cycles presented in the beginning of this section occur. The ClO–ClO

151

OZONE HOLE

FIGURE 5.19 Catalytic cycles in O3 depletion involving polar stratospheric clouds (PSCs).

cycle accounts for ∼60% of the Antarctic ozone loss, and the ClO–BrO cycles account for ∼40% of the removal. Furthermore, since much of the NOx is tied up as HNO3 in PSCs, the normally moderating effect of NOx, through formation of ClONO2, is absent. In fact, massive ozone depletion requires that the abundance of gaseous HNO3 be very low. The major process removing HNO3 from the gas phase at temperatures below about 195 K is the formation of NAT PSCs. Of course, HNO3 is also removed by HCl(s) + ClONO2, but that is only the NOx associated with ClONO2, and removal of HNO3 by this reaction alone would not be sufficient to accomplish the large-scale denitrification that is required; that requires the formation of PSCs and the removal of the nitrogen associated with them. PSCs exhibit a bimodal size distribution in the Antarctic stratosphere, with most of the nitrate concentrated in particles with radii  1 μm. The bimodal size distribution sets the stage for efficient denitrification, with nitrate particles either falling on their own or serving as nuclei for condensation of ice (Salawitch et al. 1989). Figure 5.20 is a schematic of the time evolution of the polar stratospheric chlorine chemistry. In the Antarctic winter vortex, vertical transport within the vortex as well as horizontal transport across the boundaries of the vortex is slower than the characteristic time for the ozone-depleting reactions. The local rate of loss of ozone can be approximated as that of the rate-determining step: d‰O3 Š ≅ dt

2kClO‡ClO‡M ‰ClOŠ‰CIOŠ‰MŠ

ClO levels are typically elevated by a factor of 100 over their normal levels in the 12–24 km altitude range. With ClO mixing ratios in the range of 1–1.3 ppb, the above rate predicts complete O3 removal in about 60 days. Eventually, as the polar air mass warms through breakup of the polar vortex and by absorption of sunlight, the PSCs evaporate, releasing HNO3. The nitric acid vapor photolyzes and reacts with OH to restore gas-phase NOx: HNO3 ‡ hν

! OH ‡ NO2

HNO3 ‡ OH ! H2 O ‡ NO3

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CHEMISTRY OF THE STRATOSPHERE

FIGURE 5.20 Schematic of photochemical and dynamical features of polar ozone depletion [WMO (1994), as adapted from Webster et al. (1993)]. Upper panel shows the conversion of chlorine from inactive reservoir forms, ClONO2 and HCl, to active forms, Cl and ClO, in the winter in the lower stratosphere, followed by reestablishment of the inactive forms in spring. Corresponding stages of the polar vortex are indicated in the lower panel, where the temperature scale represents changes in the minimum tempera­ tures in the lower polar stratosphere.

Gaseous NO2 reacts with ClO to again tie up active chlorine as ClONO2. The ClO/HCl ratio is indicative of the course of the ozone destruction process. Most of the atomic chlorine in the stratosphere reacts either with O3, or with CH4. The ratio of rate constants, kCl‡O3 =kCl‡CH4 , is about 900 at 200 K. The Cl + CH4 reaction is the principal source of stratospheric HCl, and this reaction governs the recovery rate of HCl following its loss from the PSC-catalyzed reaction HCl(s) + ClONO2. Once PSC chlorine conversion has ceased, HCl recovers to its original amount with a characteristic time of Δ‰HClŠ=kCl‡CH4 ‰ClŠ‰CH4 Š. (The HCl recovery time is the ratio of the change in HCl, in molecules cm 3, to the rate of its production, in molecules cm 3 s 1.) This recovery time is estimated to be about ninety 12-h days, assuming a mean temperature of 200 K, a mean [Cl] of 0.015 ppt, and a mean [CH4] of 0.8 ppm during the recovery process (Webster et al. 1993). Because an important contribution to stratospheric warming is solar absorption by O3, and because O3 levels have been depleted, the usual warmup is delayed, prolonging the duration of the ozone hole. After discovery of the Antarctic ozone hole a number of field campaigns were mounted to measure concentrations of important species in the ozone depletion cycle. The key active chlorine species in the polar ozone-destroying catalytic cycle is ClO. Simultaneous measurement of ClO and O3, as shown in Figure 5.21, provided conclusive evidence linking ClO generation to ozone loss. At an altitude near 20 km, ClO mixing ratios reached 1 ppb, several orders of magnitude higher than those in the midlatitude stratosphere, indicating almost total conversion of chlorine to active species within the polar vortex (Anderson et al. 1991). Significant reductions in the column abundances of HCl, ClONO2, and NO2 are equally important evidence as elevated ClO in verifying the mechanism of catalytic ozone destruction.

153

OZONE HOLE

FIGURE 5.21 Simultaneous measurements of O3 and ClO made aboard the NASA ER-2 aircraft on a flight from Punta Arenas, Chile (53–72°S), on September 16, 1987 (Anderson et al. 1989). As the plane entered the polar vortex, concentrations of ClO increased to about 500 times normal levels while O3 plummeted.

The equivalent effective stratospheric chlorine (EESC) quantifies the effects of chlorine and bromine on Antarctic ozone depletion. The EESC is calculated by summing up the Cl and Br atoms in ground-level measurements of halogens, factoring in the time it takes for those compounds to reach the Antarctic lower stratosphere, and estimating the rate at which the CFCs and halons are destroyed. The bromine contribution is scaled by a factor of 60 to account for the more efficient destruction of ozone by Br.

5.6.3 Arctic Ozone Hole The ozone hole phenomenon was first discovered in the Antarctic stratosphere. What about the Arctic stratosphere? Recall that two ingredients are necessary to produce the ozone hole: very cold tempera­ tures and sunlight. The Arctic winter stratosphere is generally warmer than the Antarctic by about 10 K. Figure 5.22 shows the distribution of minimum temperatures in the Antarctic and Arctic stratospheres. We see the distribution of generally lower minimum temperatures in the Antarctic versus the Arctic and the overall lower frequency of PSC formation in the Arctic. Thus polar stratospheric clouds are less abundant in the Arctic and, where they do form, they tend to disappear several weeks before solar radiation penetrates the Arctic stratosphere. Also, the Arctic polar vortex is less stable than that over the Antarctic, because Antarctica is a land mass, colder than the water mass over the Arctic. As a result, wintertime transport of ozone toward the north pole from the tropics is stronger than in the Southern Hemisphere. All these factors combine to maintain relatively higher levels of ozone in the Arctic region. Ample evidence for perturbed ozone chemistry does, however, exist over the Arctic, including observed ClO levels up to 1.4 ppb (Figure 5.23) (Salawitch et al. 1993; Webster et al. 1993). ClO concentrations are routinely as high in February in the Arctic as in September in the Antarctic, but the PSCs disappear sooner in the Arctic. As a result, denitrification is the most important difference between the Antarctic and Arctic; the Arctic experiences only a modest denitrification, whereas denitrification in the Antarctic is massive. The reason for the difference is the sufficient persistence of NAT PSCs in the Antarctic to allow time to settle out of the stratosphere.

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FIGURE 5.22 Minimum air temperatures in the Arctic and Antarctic lower stratosphere. (Courtesy of Paul Newman, NASA Goddard Space Flight Center.)

FIGURE 5.23 Vertical O3 profiles determined by bal­ loonborne sensors over Spitzbergen, Norway (79°N), on March 18, 1992 and March 20, 1995 (Reprinted with permission from Chemical & Engineering News, June 12, 1995, p. 21. Copyright 1995 American Chemical Society.)

HETEROGENEOUS (NONPOLAR) STRATOSPHERIC CHEMISTRY

5.7 HETEROGENEOUS (NONPOLAR) STRATOSPHERIC CHEMISTRY Evidence now exists for significant loss of O3 in all seasons in both hemispheres over the decades of the 1980s and 1990s. It is difficult to account for the global average (69°S–69°N) total ozone decrease of 3.5% over the 11 year period, 1979–1989, based on gas-phase chemistry alone; the reduction expected on that basis is only about 1% over that period. The process of unraveling the chemistry responsible for the Antarctic ozone hole led to the realization of the pivotal importance of heterogeneous reactions in the polar stratosphere. When midlatitude ozone losses could not be explained on the basis of gas-phase chemistry alone, it was recognized that midlatitude ozone destruction is also tied to surface reactions. In this case the reactions occur on the sulfuric acid aerosols that are ubiquitous in the stratosphere (Brasseur et al. 1990; Rodriguez et al. 1991).

5.7.1 The Stratospheric Aerosol Layer The stratosphere contains a natural aerosol layer at altitudes of 12–30 km. This aerosol is composed of small sulfuric acid droplets with size on the order of 0.2 μm diameter and at number concentrations of 1–10 cm 3. In the midlatitude lower stratosphere (about 16 km) the temperature is about 220 K, and the particles in equilibrium with 5 ppm H2O have compositions of 70–75 wt% H2SO4. As temperature decreases, these particles absorb water to maintain equilibrium; at 195 K, they are about 40 wt% H2SO4. The role of carbonyl sulfide (OCS) as a source of the natural stratospheric aerosol layer was first pointed out by Crutzen (1976). Because OCS is relatively chemically inert in the troposphere, much of it is transported to the stratosphere where it is eventually photodissociated and attacked by O atoms and OH radicals. The gaseous sulfur product of the chemical breakdown of OCS is SO2, which is subsequently converted to H2SO4 aerosol. As noted in Section 2.2.2, an approximate, global average tropospheric OCS mixing ratio is 500 ppt. In the stratosphere the OCS mixing ratio decreases rapidly with altitude, from near 500 ppt at the tropopause to less than 10 ppt at 30 km. Chin and Davis (1995) evaluated existing data on stratospheric OCS concentrations and used the measured vertical profile of OCS to calculate an average OCS stratospheric mixing ratio of 380 ppt.

5.7.2 Heterogeneous Hydrolysis of N2O5 In NOx cycle 2, which is important in the lower stratosphere, the nitrate radical NO3 is formed from the NO2 + O3 reaction NO2 ‡ O3 ! NO3 ‡ O2 During daytime, NO3 rapidly photolyzes, but at night, NO3 reacts with NO2 to produce dinitrogen pentoxide, N2O5: NO3 ‡ NO2 ‡ M ! N2 O5 ‡ M N2O5 can decompose back to NO3 and NO2 either photolytically or thermally. N2O5 itself is photolyzed in the 200–400-nm region; since this wavelength region overlaps that of the strongest O3 absorption, the photolysis lifetime of N2O5 depends on the overhead column of ozone. The photolytic lifetime of N2O5 is typically on the order of hours at 40 km and many days near 30 km. The key reaction that N2O5 undergoes is with a water molecule to form two molecules of nitric acid. Whereas the gas-phase reaction between N2O5 and a H2O vapor molecule is too slow to be of any importance, the reaction proceeds effectively on the surface of aerosol particles that contain water: N2 O5 ‡ H2 O…s† ! 2 HNO3 …s† This reaction has been demonstrated in the laboratory to proceed rapidly on the surface of concentrated H2SO4 droplets (Mozurkewich and Calvert 1988; Van Doren et al. 1991).

155

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CHEMISTRY OF THE STRATOSPHERE

FIGURE 5.24 Variation of stratospheric aerosol surface area with altitude as inferred from satellite measurements. Maxi­ mum in surface area concentration occurs at about 18–20 km. (Reprinted from McElroy et al. (1992) with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 1 GB, UK.)

When an N2O5 molecule strikes the surface of an aqueous particle, not every collision leads to reaction. In Section 3.7, a reaction efficiency or uptake coefficient γ was introduced to account for the probability of reaction. Values of γ for this reaction ranging from 0.06 to 0.1 have been reported. The firstorder rate efficient for this reaction can be expressed as in (3.38) kN2 O5 ‡H2 O…s† ˆ

γ 8 kT 4 πmN2 O5

1=2

Ap

(5.31)

1=2

is the molecular mean speed of N2O5, mN2 O5 is the N2O5 molecular mass, and Ap where 8 kT=πmN2 O5 is the aerosol surface area per unit volume (cm2 cm 3). Thus, the reaction occurs at a rate governed by that at which N2O5 molecules strike the aerosol surface area times the amount of surface area times the reaction efficiency. The HNO3 formed in this reaction is assumed to be released to the gas phase immediately since the midlatitude stratosphere is undersaturated with respect to HNO3. Figure 5.24 shows a distribution of stratospheric aerosol surface area as a function of altitude from 18 to 30 km inferred from satellite measurements. The surface area units used in Figure 5.24 are cm2 cm 3, and typical values of the stratospheric surface area at, say, 18 km altitude are about 0.8 × 10 8 cm2 cm 3. This is equivalent to 0.8 μm2 cm 3. As a useful rule of thumb, stratospheric aerosol surface area in the lower stratosphere ranges between 0.5 and 1.0 μm2 cm 3. Figure 5.25 depicts the complete NO, NO2, NO3, N2O5, HNO3 system (top). During daytime, photolysis of NO3 is so rapid that [NO3]  0, and HNO3 is formed only by the gas-phase OH + NO2 + M → HNO3 + M reaction. At night, the system shifts to that shown in the bottom panel, in which HNO3 forms entirely by the heterogenous path involving stratospheric sulfate aerosol. HNO3 has a relatively long lifetime (∼ 10 days) so that the HNO3 formed during daylight remains at night, to be augmented by that formed heterogeneously at night. At night, the NO3 concentration achieves a steady state given by ‰NO3 Š ˆ

kNO2 ‡O3 ‰O3 Š kNO3 ‡NO2 ‡M ‰MŠ

(5.32)

157

HETEROGENEOUS (NONPOLAR) STRATOSPHERIC CHEMISTRY

FIGURE 5.25 Stratospheric NOy system. Lower panel shows only the paths that operate at night.

and N2O5 also attains a steady state by ‰N2 O5 Š ˆ

kNO3 ‡NO2 ‡M ‰NO2 Š‰NO3 Š‰MŠ kN2 O5 ‡H2 O…s†

(5.33)

kNO2 ‡O3 ‰NO2 Š‰O3 Š ˆ kN2 O5 ‡H2 O…s†

Because of its relatively long lifetime, HNO3 achieves a steady-state concentration over both day and night so that total production and loss are in balance: kOH‡NO2 ‡M ‰OHŠ‰NO2 Š‰MŠ ‡ 2 kN2 O5 ‡H2 O…s† ‰N2 O5 Š ˆ kOH‡HNO3 ‰OHŠ‰HNO3 Š ‡ jHNO3 ‰HNO3 Š

(5.34)

During daytime when [N2O5]  0 kOH‡HNO3 ‰OHŠ ‡ jHNO3 ‰NO2 Š ˆ kOH‡NO2 ‡M ‰OHŠ‰MŠ ‰HNO3 Š

(5.35)

As altitude increases, [M] decreases, and the [NO2]/[HNO3] ratio increases. At night, jHNO3 ˆ 0 and [OH]  0 so HNO3 builds up. However, over a full daily cycle HNO3 achieves a steady state given by ‰NO2 Š ˆ ‰HNO3 Š

kOH‡HNO3 ‰OHŠ ‡ jHNO3 kOH‡NO2 ‡M ‰OHŠ‰MŠ ‡ 2 kN2 O5 ‡H2 O…s†

‰N2 O5 Š ‰NO2 Š

(5.36)

kOH‡HNO3 ‰OHŠ ‡ jHNO3 ˆ kOH‡NO2 ‡M ‰OHŠ‰MŠ ‡ 2 kNO2 ‡O3 ‰O3 Š The “bottleneck” reaction in the heterogeneous formation of HNO3 is that of NO2 and O3. As altitude increases, temperature increases and the rate of this reaction increases; the faster this reaction, the more

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CHEMISTRY OF THE STRATOSPHERE

aerosol surface area is needed to effectively compete for the increased amount of N2O5. The conversion of NO2 to HNO3 at midlatitude saturates for aerosol loadings close to typical background levels of 0.5 μm2 cm 3 near 20 km; at 30 km, saturation occurs at aerosol loadings of about 3 μm2 cm 3. However, from Figure 5.24 we note that aerosol surface area concentration decreases with increasing altitude, so the amount of NOx present, as opposed to NOy, actually increases with altitude because the hydrolysis of N2O5 becomes aerosol surface area–limited. Returning to Figure 5.9, we can now explain the observed altitude variation of HNO3 and NO2. The stratospheric aerosol layer resides between approximately 20 and 24 km, and it is in this region that heterogeneous formation of HNO3 is most effective. At altitudes above the stratospheric aerosol layer, the daytime formation of HNO3 by OH + NO2 + M runs out of [M] as altitude increases; for this reason the HNO3 concentration drops off with altitude, and the [NO2]/[HNO3] ratio increases in accord with (5.36). The conversion of a reservoir species N2O5 into a relatively stable species HNO3 serves to remove NO2 from the active catalytic NOx system, reducing the effectiveness of O3 destruction by the NOx cycle. The reduction of NO2, on the other hand, decreases the formation of ClONO2, allowing more ClO to accumulate than in the absence of the N2O5 + H2O(s) reaction. More ClO means that the effectiveness of the ClOx cycles is increased. Although photolysis of HNO3, with the H atom derived from aerosol H2O, provides an additional source of HOx to the system, this is not predicted to have a major effect on the HOx cycles. Effectively, the hydrolysis of N2O5 on stratospheric aerosol moves NO2 from ClONO2 with a 1-day lifetime to HNO3 with a 10-day lifetime, leading to substantially lower NOx concentrations. As a result, lower stratospheric O3 becomes more sensitive to halogen and HOx chemistry and less sensitive to the addition of NOx. Effect of N2O5 Hydrolysis on the Chemistry of the Lower Stratosphere NO2 ‡ O3

1

! NO3 ‡ O2

NO3 ‡ NO2 ‡ M

2

! N2 O 5 ‡ M

NO3 ‡ hν

3a

! NO2 ‡ O

3b

! NO ‡ O2

NO3 forms from reaction 1 and is consumed by reactions 2 and 3. During daytime, photolysis of NO3 is very efficient; jNO3 from both channels 3a and 3b is about 0.3 s 1, giving a mean daytime lifetime of NO3 of 3 s. At night, reaction 2 is the only removal pathway for NO3. The reaction rate coefficient of reaction 2 from Table B.2 has k0 = 2.2 × 10 30(T/300) 3.9 and k1 = 1.5 × 10 12(T/300) 0.7 cm3 mole­ cule 1 s 1. At 20 km, T = 220 K, [M] = 2 × 1018 molecules cm 3, and the Troe formula gives k2 = 1.2 × 10 12 cm3 molecule 1 s 1. Assuming that NO2 is present at 1 ppb, the mean lifetime of NO3 against reaction 2 is 400 s. In summary, the lifetime of a molecule of NO3 at 20 km increases from 3 s in daytime to about 7 min at night. Each NO3 molecule formed at night is converted to N2O5 in about 7 min, so over the course of a 12-h night, every time reaction 1 occurs, a molecule of N2O5 is formed, with consumption of two NO2 molecules. NOx is converted to N2O5 at the following rate: d‰NOx Š ˆ 2 k1 ‰O3 Š‰NO2 Š dt At T = 220 K, k1 = 1.7 × 10

18

cm3 molecule

1

k1 ˆ 1:2  10

13

exp… 2450=T†

s 1. The lifetime of NOx against reaction 1 is

τNOx ˆ

1 2 k1 ‰O3 Š

With [O3] = 5 × 1012 molecules cm 3, τNOx ˆ 5:9  104 s. The fractional decay of NOx is given by exp t=τNOx . Over a single 12-h night (4.3 × 104 s), 52% of the NOx is converted to N2O5.

159

HETEROGENEOUS (NONPOLAR) STRATOSPHERIC CHEMISTRY

N2O5 is hydrolyzed on stratospheric aerosol to produce nitric acid: N2 O5 ‡ H2 O…s†

4

! 2HNO3

The conversion can be represented as a pseudo-first-order reaction with rate coefficient (5.31) k4 ˆ

1=2

γ 8 kT 4 πmN2 O5

Ap

which depends on the stratospheric aerosol surface area Ap and the reactive uptake coefficient γ. Previously, we noted that a representative value for γ for reaction 4 is 0.06. Values of k4 for Ap = 1 × 10 8 cm2 cm 3 and 10 × 10 8 cm2 cm 3 at T = 220 K are k4 = 3.1 × 10 6 s 1 = 3.1 × 10 5 s 1

Ap = 1 × 10 8 cm2 cm 3 Ap = 10 × 10 8 cm2 cm 3

at at

The lifetime of N2O5 against reaction 4 is just τN2 O5 ˆ 1=k4 . What fraction of NOx is converted to HNO3 over the nighttime? N2O5 is gradually produced throughout the night at a rate determined by reaction 1. As N2O5 is produced, N2O5 itself is gradually converted to HNO3 by reaction 4. The system is the classic reactions in series: A → B → C. In this case, A = NO2, B = N2O5, and C = HNO3. Here, two molecules of C are produced in the B → C reaction. To determine the amount of C formed over the night starting from a given concentration of A at sundown, one must integrate the rate equations: d‰AŠ ‰AŠ ˆ dt τNOx d‰BŠ 1 ‰AŠ ˆ dt 2 τNOx d‰CŠ ‰BŠ ˆ dt τN2 O5

‰BŠ τN2 O5

The factor of 0.5 in the rate equation for [B] reflects the fact that each molecule of B formed requires the consumption of two molecules of A. The fractional conversion of an initial concentration of A, [A]0 into C is 2[C]/[A]0, where the factor of 2 accounts for the fact that two molecules of HNO3 are formed in reaction 4. The solutions for [A], [B], and [C] are ‰A…t†Š ˆ ‰AŠ0 exp ‰B…t†Š ˆ

‰C…t†Š ˆ

t τNOx

‰AŠ0 =2τNOx exp 1 1 τN2 O5 τNOx

t τNOx

exp

1 ‰AŠ0 1‡ τN2 O5 exp 2 τNOx τN2 O5

t τN2 O5 t τN2 O5

τNOx exp

t τNOx

The fraction of NOx converted to HNO3 over a time period t is thus f NOx !HNO3 ˆ 1 ‡

1 τNOx

τN2 O5

τN2 O5 exp

t τN2 O5

τNOx exp

t τNOx

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CHEMISTRY OF THE STRATOSPHERE

We are interested in t = 12 h = 4.3 × 104 s. From before, τNOx ˆ 5:9  104 s. For Ap = 1 × 10 τN2 O5 ˆ 3:2  105 s; for Ap = 10 × 10 8 cm2 cm 3, τN2 O5 ˆ 3:2  104 s. We find f NOx !HNO3 ˆ 0:04

at

Ap ˆ 1  10

ˆ 0:26

at

Ap ˆ 10  10

8

cm2 cm 8

8

cm2 cm 3,

3

cm2 cm

3

The conversion rate of NOx to HNO3 through N2O5 depends on the two lifetimes, τNOx ˆ …2k1 …T†‰O3 Š†

1

and τN2 O5 ˆ

γ 4

8kT=π mN2 O5

1=2

Ap

1

. The smaller these two characteristic times,

the faster the conversion rate. The greatest sensitivity of the conversion is to T (k1 is a strong function of temperature) and Ap; the larger T and Ap are, the greater the conversion to nitric acid. The largest values of Ap occur when the stratospheric aerosol is perturbed by a volcano or by formation of PSCs in the polar stratosphere.

5.7.3 Effect of Volcanoes on Stratospheric Ozone Volcanoes inject gaseous SO2 and HCl directly into the stratosphere. Because of its large water solubility, HCl is rapidly removed by liquid water. The SO2 remains and is converted to H2SO4 aerosol by reaction with the OH radical. (We will discuss this reaction in Chapter 6 and the nucleation process that leads to fresh particles in Chapter 11.) Thus volcanic eruptions lead to an increase in the amount of the stratospheric aerosol layer (Figure 5.26). As we have just seen, “normal” stratospheric aerosol surface area concentrations lie in the range of 0.5–1.0 μm2 cm 3. Eruption of Mount Pinatubo in the Philippines in June 1991, the largest volcanic eruption of the twentieth century, led to average midlatitude stratospheric aerosol surface areas of 20 μm2 cm 3. In the core of the plume, 5 months after the eruption, the aerosol surface area concentration was 35 cm2 cm 3 (Grainger et al. 1995). The time required for stratospheric aerosol levels to decay back to “normal” levels following a volcanic eruption is about 2 years. With volcanoes erupting somewhere on Earth every few years or so, the stratosphere is seldom in a state totally uninfluenced by volcanic emissions. Volcanic injection of large quantities of sulfate aerosol into the stratosphere offers the opportunity to examine the sensitivity of ozone depletion and species concentrations to a major perturbation in aerosol surface area (Hofmann and Solomon 1989; Johnston et al. 1992; Prather 1992; Mills et al. 1993). The increase in stratospheric aerosol surface area resulting from a major volcanic eruption can lead to profound effects on ClOx-induced ozone depletion chemistry. Because the heterogeneous reaction of

FIGURE 5.26 Time series of the abundance of stratospheric aerosols, as inferred from integrated backscatter measurements (WMO 2002). The datastream beginning in 1976 was acquired by ground-based lidar at Garmisch, Germany (47.5°N). That beginning in 1985 is from the SAGE II satellite over the latitude band 40–50°N. Vertical arrows show major volcanic eruptions. The dashed line indicates the 1979 level. (Data from Garmisch provided courtesy of H. Jäger.)

HETEROGENEOUS (NONPOLAR) STRATOSPHERIC CHEMISTRY

FIGURE 5.27 Measured ratios NOx/NOy and ClO/Cly (solid circles) shown as a function of stratospheric aerosol surface area (Fahey et al. 1993). The open circles represent simulations of the conditions of the measurements account­ ing for gas-phase chemistry only; the crosses are the corresponding simula­ tions including the heterogeneous hydrolysis of N2O5. The vertical dashed line represents stratospheric background aerosol surface area (0.5 μm2 cm 3). The curved lines represent the modeled dependence of the two ratios on aerosol surface area for the average conditions in the September (solid) and March (dashed) datasets. (Reprinted with permission from Nature 363, Fahey, D. W., et al. Copyright 1993 Macmillan Magazines Limited.)

N2O5 and water on the surface of stratospheric aerosols effectively removes NO2 from the active reaction system, less NO2 is available to react with ClO to form the reservoir species ClONO2. As a result, more ClO is present in active ClOx cycles. Therefore an increase in stratospheric aerosol surface area, as from a volcanic eruption, can serve to make the chlorine present more effective at ozone depletion, even if no increases in chlorine are occurring. Figure 5.27 shows in situ stratospheric data (solid circles) on NOx/NOy and ClO/Cly ratios plotted as a function of stratospheric aerosol surface area concentration (Fahey et al. 1993). Figure 5.27 shows comparisons of measurements made in September 1991 in a region of the atmosphere that had not been strongly impacted by the Mount Pinatubo aerosol with those made in March 1992 at a similar latitude and solar zenith angle where the effect of the eruption is clearly present. In the March data the aerosol surface area concentration was 20 μm2 cm 3, a factor of 40 over the “natural” level of 0.5 μm2 cm 3 (vertical dashed line). The solid and dashed curves are model-calculated relationships for these two ratios for the September (solid) and March (dashed) data sets. The open circles on the NOx/NOy plot at a surface area of 20 μm2 cm 3 indicate calculations assuming gas-phase chemistry only; the crosses are calculations including heterogeneous hydrolysis of N2O5. The measured ratio of NOx/NOy decreased as aerosol surface area increased and the ClO/Cly ratio increased. Both of these effects are expected as a

161

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CHEMISTRY OF THE STRATOSPHERE

result of the heterogeneous hydrolysis of N2O5 on aerosol surfaces. Increasing aerosol surface area from about 1 to 20 μm2 cm 3 led to an increase in the ClO/Cly ratio from about 0.025 to about 0.05. Thus this increase in aerosol surface area from a large volcano renders the chlorine already present in the stratosphere twice as effective in ozone depletion as in the absence of the volcano. Even at a more modest scale, an increase of aerosol surface area from 1 to 5 μm2 cm 3 is estimated to increase ClO/Cly from 0.02 to 0.03 and thereby render the chlorine 50% more effective in ozone depletion. Thus one volcanic eruption, at least during the 2 years or so following the eruption when stratospheric aerosol levels are elevated, can produce the same ozone-depleting effect as a decade of increases in CFC emissions [e.g., see Tie et al. (1994)]. Conversely, in the absence of chlorine in the stratosphere, stratospheric ozone would increase after a volcanic eruption as a result of the removal of active NOx by the heterogeneous N2O5 + H2O reaction.

5.8 SUMMARY OF STRATOSPHERIC OZONE DEPLETION Figure 5.28 shows the total ozone loss rate as a function of altitude. Chemical destruction of O3 in the lower stratosphere ( 25 km) is slow. In this region of the stratosphere, O3 has a lifetime of months. Rates exceeding 106 molecules cm 3 s 1 are achieved only  28 km. Figure 5.29 gives the fractional contribu­ tions of the Ox, NOx, HOx, and halogen cycles to the total ozone loss rate. In many respects, Figures 5.28 and 5.29 represent the culmination of this chapter—the synthesis of how the various ozone depletion cycles interact at different altitudes in the stratosphere. Figure 5.30 summarizes global total ozone change relative to the 1964–1980 average. In order to explain the fractional contributions in Figure 5.29, we can consider both the rate coefficients of the key reactions (Figure 5.31) and the profiles of O, NO2, HO2, O3, BrO, and ClO (Figure 5.32). The rates of the rate-determining reactions in each of the catalytic cycles can then be estimated at different altitudes. For example, at 15 km (T ∼ 215 K), the rates of key reactions (molecules cm 3 s 1) can be estimated as follows: RNO2 ‡O ˆ kNO2 ‡O ‰NO2 Š‰OŠ ≅ …1  10

11

RHO2 ‡O3 ˆ kHO2 ‡O3 ‰HO2 Š‰O3 Š ≅ …1  10

†  …4  108 †  …1  103 † ≅ 5:0

15

†  …2  106 †  …5  1011 † ≅ 1  103

RClO‡BrO!BrCl ˆ kClO‡BrO!BrCl ‰ClOŠ‰BrOŠ ≅ …2  10 RClO‡BrO!ClOO ˆ kClO‡BrO!ClOO ‰ClOŠ‰BrOŠ ≅ …8  10 RBrO‡HO2 ˆ kBrO‡HO2 ‰BrOŠ‰HO2 Š ≅ …4  10

11

12

†  …1  103 †  …2  106 † ≅ 3  10

12

†  …1  103 †  …2  106 † ≅ 2  10

†  …2  106 †  …2  106 † ≅ 2  102

FIGURE 5.28 Contribution to total O3 loss rate by different catalytic cycles. (After Osterman et al. (1997) updated to current kinetic parameters.)

3 2

SUMMARY OF STRATOSPHERIC OZONE DEPLETION

FIGURE 5.29 Vertical distribution of the relative contributions by different catalytic cycles to the O3 loss rate (WMO 1998). (After Osterman et al. (1997), updated to current kinetic parameters.)

FIGURE 5.30 Change in O3 column relative to 1980 value for the atmo­ sphere between 60°S and 60°N. Measurements based on a variety of satellite and ground-based instruments (WMO 2002).

FIGURE 5.31 Rate coefficients of rate-determining reactions in O3 loss catalytic cycles.

163

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CHEMISTRY OF THE STRATOSPHERE

FIGURE 5.32 Stratospheric concentration profiles of key species involved in O3 loss catalytic cycles (35°N, Sept.).

The total estimated O3 loss rate is ∼ 1 × 103 molecules cm 3 s 1, approximately 90% of which is due to HO2 + O3 (HOx cycle 2) and 10% to BrO + HO2 (HOx/BrOx cycle 1). At 30 km (T ∼ 255 K), a similar analysis leads to RO3 ‡O ˆ kO3 ‡O ‰O3 Š‰OŠ ≅ …8  10

16

†  …3  1012 †  …3  107 † ≅ 8  104

RHO2 ‡O ˆ kHO2 ‡O ‰HO2 Š‰OŠ ≅ …7  10

11

†  …1  107 †  …3  107 † ≅ 2  104

RNO2 ‡O ˆ kNO2 ‡O ‰NO2 Š‰OŠ ≅ …1  10

11

†  …2  109 †  …3  107 † ≅ 7  105

RClO‡O ˆ kClO‡O ‰ClOŠ‰OŠ ≅ …4  10

11

†  …5  107 †  …3  107 † ≅ 6  104

RBrO‡O ˆ kBrO‡O ‰BrOŠ‰OŠ ≅ …5  10

11

†  …2  106 †  …3  107 † ≅ 3  103

The total estimated O3 loss rate is ∼ 9 × 105 molecules cm 3 s 1, about 80% of which is attributable to NO2 + O (NOx cycle 1) and 10% each to the Chapman cycle and ClO + O (ClOx cycle 1). Below ∼20 km, HOx cycle 2 is dominant in O3 removal. Between 20 and 40 km, NOx cycles dominate O3 loss; NOx cycle 2 in the lower portion of this range, and NOx cycle 1 in the upper portion. Above ∼40 km, HOx cycles are again dominant (HOx cycle 1). Because of the rapid release of Br in the lower stratosphere, BrOx cycles are approximately comparable in importance to ClOx cycles at 20 km. The coupled BrOx/ClOx cycle is itself responsible for ∼25% of the overall halogen-controlled loss in the lower stratosphere. The halogen cycles achieve two maxima: (1) one in the lowermost stratosphere due to HOx/ClOx cycle 1, HOx/BrOx cycle 1, and BrOx/ClOx cycles 1 and 2; and (2) one in the upper stratosphere due to ClOx cycle 1. NOx is the dominant O3-reducing catalyst between 25 and 35 km. Because of coupling among the HOx, ClOx, and NOx cycles, and the role that NOx plays in that coupling, the rate of O3 consumption is very sensitive to the concentrations of NO and NO2. Partitioning between OH and HO2 by HO2 + NO → NO2 + OH is controlled by the level of NO. This reaction short-circuits HOx cycle 4 by reducing the concentration of HO2. Figure 5.33 shows schematically the overall O3 loss rate for the midlatitude lower stratosphere as a function of the NOx level. At very low NOx concentrations, the O3 loss rate increases as NOx decreases. This is a result of the absence of NOx to tie up reactive hydrogen and chlorine in reservoir species. As NOx increases, a critical value is reached where O3 removal through the reaction, O + NO2, is equal to the sum of O3 loss by the HOx and ClOx cycles. At this point, increases in NOx result in comparable reductions in the concentrations of HO2 and ClO. As the chlorine content of the stratosphere has

TRANSPORT AND MIXING IN THE STRATOSPHERE

FIGURE 5.33 Qualitative behavior of the ozone removal rate in the lower stratosphere as a function of NOx level (Wennberg et al. 1994). SPADE stands for Stratospheric Photochemistry, Aerosols, and Dynamics Expedition. This is a cumulative plot; the contributions of the various pro­ cesses add to produce the upper curve.

increased over time, this crossover point has been shifted to higher NOx concentrations. As [NOx] increases further, the NOx cycle eventually becomes dominant, and O3 loss increases linearly with [NOx]. The dependence of the O3 loss rate on NOx at higher altitudes in the stratosphere is different from that in Figure 5.33. At higher altitudes, the increasing amount of atomic oxygen means that the catalytic cycles that rely on O atoms become increasingly important. As a result, substantially higher O3 loss rates occur in the upper stratosphere than in the lower stratosphere. The fractional contribution to O3 loss by HOx becomes dominant above 45 km. As altitude increases, OH and HO2 concentrations fall off slowly, while those of ClOx and NOx fall off rapidly. The ClO/HCl ratio decreases because of a shift in the Cl/ ClO ratio favoring Cl and production of HCl from Cl + CH4. The decrease in NOx at high altitudes results because the N + NO reaction (recall Figure 5.8) begins to become important, even though the NOx/NOy ratio shifts in favor of NOx.

5.9 TRANSPORT AND MIXING IN THE STRATOSPHERE A latitude–height schematic cross section of the upper troposphere and lower stratosphere is shown in Figure 5.34. Tropospheric air enters the stratosphere principally in the tropics, as a result of deep cumulus convection, and then moves poleward in the stratosphere. By overall mass conservation, stratospheric air must return to the troposphere. This return occurs in midlatitudes; the overall circulation is called the Brewer–Dobson circulation. The midlatitude stratosphere is disturbed by breaking planetary waves, forming what is referred to as a surf zone. The rising air in the tropical stratosphere has given rise to the concept of a tropical pipe, where tropical air is assumed to advect upward with no exchange with midlatitude stratospheric air. Observed vertical profiles of chemical species cannot, however, be reconciled with a strict interpretation of the tropical pipe; some mixing with midlatitude air is required, especially below ∼22 km. By considering dilution, vertical ascent, and chemical decay, these so-called leaky pipe calculations estimate that by 22 km, close to 50% of the air in the tropical upwelling region is of midlatitude origin. Volk et al. (1996) estimated that about 45% of air of extratropical origin accumulates in a tropical air parcel during its 8-month ascent from the tropopause to 21 km. This lateral mixing is depicted by the wavy arrows in Figure 5.34. On analysis of relative vertical profiles of CO2 and N2O, Boering et al. (1996) estimated the following vertical ascent rates in the stratosphere:

165

166

CHEMISTRY OF THE STRATOSPHERE

FIGURE 5.34 Stratospheric transport and mixing (UK National Environment Research Council). Temperature isotherms are labeled as potential temperature. Potential temperature θ at any altitude z is the temperature of an air parcel brought from that altitude adiabatically to the surface (see Chapter 16). It can be calculated from the relation θ = T(p/p0) 2/7, where T and p are the temperature and pressure at altitude z and p0 is the surface pressure, usually taken to be 1013 hPa. (Figure available at http://utls.nerc.ac.uk/background/sp/ utls_sp_fig1a.pdf.)

NH summer: NH winter:

0.21 mm s 0.31 mm s

1 1

(17 m day 1) (26 m day 1)

The tropical tropopause, essentially defined by the 380 K potential temperature surface (see Chapter 21), is located at about 16–17 km. Tropical convection occurs up to an altitude of about 11–12 km. Between these two levels lies a transition region between the tropopause and the stratosphere, called the tropical tropopause layer (TTL). Together with the lowermost stratosphere, the region is referred to as the upper troposphere/lower stratosphere (UT/LS). Convection does penetrate into the TTL and deliver air from the boundary layer, and within the TTL lateral mixing occurs with subtropical air of stratospheric character. Within the TTL, ozone profiles exhibit a continuous transition from low values characteristic of the upper troposphere to the higher values characteristic of the stratosphere, with no special signature at the tropopause itself. The flux of ozone from the stratosphere to the troposphere is taken as that across the 380 K potential temperature surface at mid and high latitudes. This ozone is of stratospheric origin, because the flux of air from the upper tropical troposphere into the lowermost stratosphere contains very little ozone. The ozone flux from the stratosphere to the troposphere is estimated by general circulation models. As tropospheric air enters the tropical stratosphere, the cold temperatures encountered freeze out the water vapor and dehydrate the air, on average, to an H2O mixing ratio of about 4 ppm. Once in the stratosphere, subsequent oxidation of CH4 and H2 can lead to H2O vapor mixing ratios as high as ∼8 ppm (4 ppm H2O + 2 × 1.75 ppm CH4 + 0.5 ppm H2). Water vapor crossing the extratropical

167

OZONE DEPLETION POTENTIAL

tropopause experiences higher temperatures than air that enters via the tropical tropopause, leading to water vapor mixing ratios in the lowermost stratosphere of tens of ppm (Dessler et al. 1995).

5.10 OZONE DEPLETION POTENTIAL The stratospheric ozone-depleting potential of a compound emitted at Earth’s surface depends on how much of it is destroyed in the troposphere before it reaches the stratosphere, the altitude at which it is broken down in the stratosphere, and chemistry subsequent to its dissociation. Halocarbons containing hydrogen in place of halogens or containing double bonds are susceptible to attack by OH in the troposphere. (We will consider the mechanisms of such reactions in Chapter 6.) The more effective the tropospheric removal processes, the less of the compound that will survive to reach the stratosphere. Once halocarbons reach the stratosphere, their relative importance in ozone depletion depends on the altitude at which they are photolyzed and the distribution of halogen atoms, Cl, Br, and F, contained within the molecule. Different halocarbons are photolyzed at different wavelengths. Those photolyzed at shorter wave­ lengths must reach higher altitudes in the stratosphere to be photodissociated. Generally, substitution of fluorine atoms for chlorine atoms shifts absorption to shorter wavelengths. For example, maximum photolysis of CFCl3 occurs at about 25 km, that for CF2Cl2 at 32 km, and that for CClF2CF3 at 40 km. Molecule for molecule, more heavily chlorinated halocarbons are more effective in destroying ozone than are those containing more fluorine; they are photolyzed at lower altitudes where the amount of ozone is greater, and the chlorine atom is much more efficient at ozone destruction than fluorine. Atom for atom, however, bromine is, as noted earlier, an even more efficient catalyst for ozone destruction than chlorine. Also, photolysis of bromine-containing compounds occurs lower in the stratosphere, where impact is the greatest. The atmospheric photolytic lifetime is shortest for molecules containing relatively more bromine than chlorine or fluorine, followed next by those with relatively more chlorine. The effect of a compound on stratospheric ozone depletion is usually assessed by simulating transport and diffusion, chemistry, and photochemistry in a one-dimensional vertical column of air from Earth’s surface to the top of the stratosphere at a given latitude. Two-dimensional models have also been used. One-dimensional models calculate the global averaged vertical variation of the chemical degra­ dation processes. Two-dimensional models allow examination of the calculated effects as a function of latitude and time of year. Ozone changes calculated with two-dimensional models are then averaged with respect to both latitude and season to obtain a single ozone depletion value for the compound of interest. If a constant rate of emission is assumed, then eventually a steady state will be reached such that the concentration of the compound is unchanging in time at all levels of the atmosphere. The ozone depletion potential (ODP) of a compound is usually defined as the total steady-state ozone destruction, vertically integrated over the stratosphere, that results per unit mass of species i emitted per year relative to that for a unit mass emission of CFC-11: ODPi ˆ

ΔO3i ΔO3CFC

11

Thus ODPi is a relative measure. Table 5.3 presents ozone depletion potentials calculated with two-dimensional models, as presented by WMO (2002). Tropospheric OH reaction is the primary sink for the hydrogen-containing species. As seen in Table 5.3, the ODPs for the hydrogen-containing species are considerably smaller than those of the CFCs, the difference reflecting the more effective chemical removal in the troposphere. Although Table 5.3 gives the ODPs of several brominated halocarbons relative to CFC-11, the ODPs for brominecontaining compounds should really be compared relative to each other because of the strong coupling of bromine catalytic cycles to chlorine levels in the lower stratosphere. The halon that is generally selected as the reference compound for ODPs of bromine-containing halocarbons is H-1301 (CF3Br), the halon with the longest atmospheric lifetime and largest ODP.

168

CHEMISTRY OF THE STRATOSPHERE

TABLE 5.3 Steady-State Ozone Depletion Potentials (ODPs) Derived from Two-Dimensional Modelsa Trace gas

ODP

CFC-11 (CFCl3) CFC-12 (CF2Cl2) CFC-113 (CFCl2CF2Cl) CFC-114 (CF2ClCF2Cl) CFC-115 (CF2ClCF3) CCl4 CH3CCl3 HCFC-22 (CF2HCl) HCFC-123 (CF3CHCl2) HCFC-124 (CF3CHFCl) HCFC-141b (CH3CFCl2) HCFC-142b (CH3CF2Cl) HCFC-225ca (CF3CF2CHCl2) HCFC-225cb (CF2ClCF2CHFCl) CH3Br H-1301 (CF3Br) H-1211 (CF2ClBr) H-1202 (CF2Br2) H-2402 (CF2BrCF2Br) CH3Cl

1.0 0.82 0.90 0.85 0.40 1.20 0.12 0.034 0.012 0.026 0.086 0.043 0.017 0.017 0.37 12 5.1 1.3 H CO2

(7.26)

and therefore the total amount of CO2 dissolved always exceeds that predicted by Henry’s law for CO2 alone. Note that while the Henry’s law constant H CO2 depends only on temperature, the effective Henry’s law constant H ∗CO2 depends on both temperature and the solution pH. For pH < 5, the dissolved carbon dioxide does not dissociate appreciably and its effective Henry’s law constant is, for all practical purposes, equal to its Henry’s law constant. For a gas-phase CO2 mixing ratio equal to 330 ppm, the equilibrium aqueous-phase concentration is 11.2 μM (Figure 7.4). As the pH increases to values higher than 5, CO2
H2O starts dissociating and the dissolved total carbon dioxide increases exponentially. However, even at pH 8, H ∗CO2 is only 1.5 M atm 1, and practically all the available carbon dioxide is still in the gas phase. The aqueous-phase concentration of total carbon dioxide increases to hundreds of μM for alkaline water. Let us assume momentarily that the atmosphere contains only CO2 and water. What is the pH of cloud and rainwater in this system? For example, consider an ambient mixing ratio of 350 ppm and a temperature of 298 K. The concentrations of the ions in solution will satisfy the electroneutrality equation ‰H‡ Š ˆ ‰OH Š ‡ HCO3 ‡ 2 CO23

(7.27)

which can be used to calculate [H+]. Using (7.12), (7.21), and (7.22) for [OH ], HCO3 , and CO23 , respectively, we can place (7.27) in the form of an equation with a single unknown, the hydrogen

274

CHEMISTRY OF THE ATMOSPHERIC AQUEOUS PHASE

FIGURE 7.4 Effective Henry’s law constant for CO2 as a function of the solution pH. Also shown is the corresponding equilibrium total dissolved CO2 concentration COT2 for a CO2 mixing ratio of 330 ppm.

ion concentration H‡ ˆ

H CO2 Kc1 pCO2 2 H CO2 Kc1 Kc2 pCO2 Kw ‡ ‡ ‡ ‰H Š ‰H‡ Š ‰H‡ Š2

(7.28)

or after rearranging: ‰H‡ Š3

Kw ‡ H CO2 Kc1 pCO2 ‰H‡ Š

2 H CO2 Kc1 Kc2 pCO2 ˆ 0

(7.29)

Even if some CO2 dissolves in the water, this amount is too small to change the gas-phase concentration of CO2. Given the temperature, which determines the values of Kw, H CO2 , Kc1, and Kc2, [H+] can be computed from (7.29), from which all other ion concentrations can be obtained. At an ambient CO2 mixing ratio of 350 ppm at 298 K the solution pH is 5.6. This value is often cited as the pH of “pure” rainwater.

7.3.3 Sulfur Dioxide–Water Equilibrium The behavior of SO2 in aqueous solution is qualitatively similar to that of carbon dioxide. Absorption of SO2 in water results in SO2 …g† ‡ H2 O ƒ SO2 ? H2 O

(7.30)

SO2 ? H2 O ƒ H‡ ‡ HSO3

(7.31)

‡

HSO3 ƒ H ‡

SO32

(7.32)

with H SO2 ˆ

‰SO2 ? H2 OŠ pSO2

(7.33)

Ks1 ˆ

‰H‡ Š HSO3 ‰SO2 ? H2 OŠ

(7.34)

Ks2 ˆ

‰H‡ Š SO23 HSO3

(7.35)

275

AQUEOUS-PHASE CHEMICAL EQUILIBRIA

FIGURE 7.5 Concentrations of S(IV) species and total S(IV) as a function of solution pH for an SO2 mixing ratio of 1 ppb at 298 K.

where Ks1 = 1.3 × 10 given by

2

M and Ks2 = 6.6 × 10

8

M at 298 K. The concentrations of the dissolved species are

‰SO2 ? H2 OŠ ˆ HSO2 pSO2

HSO3 ˆ

SO23

ˆ

(7.36)

Ks1 ‰SO2 ? H2 OŠ H SO2 Ks1 pSO2 ˆ ‰H‡ Š ‰H‡ Š

(7.37)

H SO2 Ks1 Ks2 pSO2 Ks2 HSO3 ˆ ‰H‡ Š ‰H‡ Š2

(7.38)

The corresponding ionic concentrations in equilibrium with 1 ppb of SO2 are shown as functions of the pH in Figure 7.5. The total dissolved sulfur in solution in oxidation state 4, referred to as S(IV) (see Section 2.2), is equal to ‰S…IV†Š ˆ ‰SO2 ? H2 OŠ ‡ HSO3 ‡ SO23

(7.39)

The total dissolved sulfur, S(IV), can be related to the partial pressure of SO2 over the solution by ‰S…IV†Š ˆ H SO2 pSO2 1 ‡

Ks1 Ks1 Ks2 ‡ ‰H‡ Š ‰H‡ Š2

(7.40)

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CHEMISTRY OF THE ATMOSPHERIC AQUEOUS PHASE

FIGURE 7.6 Effective Henry’s law constant for SO2 as a function of solution pH at 298 K.

or, if we define the effective Henry’s law coefficient for SO2, H∗S…IV† , as H ∗S…IV† ˆ H SO2 1 ‡

Ks1 Ks1 Ks2 ‡ ‰H‡ Š ‰H‡ Š2

(7.41)

the total dissolved sulfur dioxide is given by ‰S…IV†Š ˆ H∗S…IV† pSO2

(7.42)

The effective Henry’s law constant for SO2 increases by almost seven orders of magnitude as the pH increases from 1 to 8 (Figure 7.6). The effect of the acid–base equilibria is to “pull” more material into solution than predicted on the basis of Henry’s law alone. The Henry’s law coefficient for SO2 alone, H SO2 , is 1.23 M atm 1 at 298 K, while for the same temperature, the effective Henry’s law coefficient for S(IV), H ∗S…IV† , is 16.4 M atm 1 for pH = 3, 152 M atm 1 for pH = 4, and 1524 M atm 1 for pH = 5. Equilibrium S(IV) concentrations for SO2 gas-phase mixing ratios of 0.2 to 200 ppb, and over a pH range of 1–6, vary approximately from 0.001 to 1000 μM (Figure 7.7). S(IV) Concentrations in Different Environments Calculate [S(IV)] as a function of pH for SO2 mixing ratios of 0.2 and 200 ppb over the range from pH = 0–6 and for temperatures of 0°C and 25°C. The concentration of S(IV) is given by (7.40). As we have seen, at 298 K, H SO2 ˆ 1:23 M atm 1 , Ks1 = 1.3 × 10 2 M, and Ks2 = 6.6 × 10 8 M. We can calculate the values of these constants for T = 273 K using (7.5). The required parameters are given in Table 7A.1 in Appendix 7 in the end of this chapter. We find that at 273 K, H SO2 ˆ 3:2 M atm 1 , Ks1 = 2.55 × 10 2 M, and Ks2 = 10 7 M. Figure 7.7 shows [S(IV)] as a function of pH for these two SO2 concentrations at 0°C and 25°C. The concentration of [S(IV)] increases dramatically as pH increases. The concentration of SO2
H2O does not depend on the pH, and the abovementioned increase (for constant T and ξSO2 ) is due exclusively to the increased concentrations of HSO3 and SO23 . Temperature has a significant effect on the [S(IV)] concentration. For example for pH = 5 and 0.2 ppb SO2, the equilibrium concentration increases from 0.16 to 0.82 μM, that is by a factor of ∼5 as the temperature decreases from 298 to 273 K.

277

AQUEOUS-PHASE CHEMICAL EQUILIBRIA

FIGURE 7.7 Equilibrium dissolved S(IV) as a function of pH, gasphase mixing ratio of SO2, and temperature.

S(IV) Composition and pH Let us compute the mole fractions of the three S(IV) species as a function of solution pH. The mole fractions are found by combining (7.36)–(7.38) with (7.40):1 xSO2 ? H2 O ˆ

‰SO2 ? H2 OŠ ˆ ‰S…IV†Š

xHSO3 ˆ xSO2 ˆ 3

HSO3 ˆ ‰S…IV†Š SO23 ˆ ‰S…IV†Š

1‡

‰H‡ Š Ks2 ‡ Ks1 ‰H‡ Š

1

‰H‡ Š ‰H‡ Š2 ‡ Ks2 Ks1 Ks2

1

1‡

1‡

Ks1 Ks1 Ks2 ‡ ‰H‡ Š ‰H‡ Š2

1

(7.43)

(7.44)

(7.45)

Figure 7.8 shows these three mole fractions as a function of pH at 298 K. At pH values lower than 2, S(IV) is mainly in the form of SO2
H2O. At higher pH values the HSO3 fraction increases, and in the pH range from 3 to 6 practically all S(IV) occurs as HSO3 . At pH values higher than 7, SO23 dominates. Since these different S(IV) species can be expected to have different chemical reactivities, if a chemical reaction occurs in solution involving either HSO3 or SO23 , we expect that the rate of the reaction will depend on pH since the concentration of these species depends on pH. In aquatic chemistry a common notation for these mole fractions is α0, α1, and α2, respectively (Stumm and Morgan 1996). 1

278

CHEMISTRY OF THE ATMOSPHERIC AQUEOUS PHASE

FIGURE 7.8 Concentrations of S(IV) species expressed as S(IV) mole frac­ tions. These fractions are independent of the gas-phase SO2 concentration.

7.3.4 Ammonia–Water Equilibrium Ammonia is the principal basic gas in the atmosphere. Absorption of NH3 in H2O leads to NH3 ‡ H2 O ƒ NH3 ? H2 O

(7.46)

NH3 ? H2 O ƒ NH‡4 ‡ OH

(7.47)

with ‰NH3 ? H2 OŠ pNH3

(7.48)

NH‡4 ‰OH Š ‰NH3 ? H2 OŠ

(7.49)

H NH3 ˆ Ka1 ˆ

where at 298 K, H NH3 ˆ 62 M atm 1 and Ka1 = 1.7 × 10 5 M. NH4OH is an alternative notation often used instead of NH3
H2O. The concentration of NH‡4 is given by NH‡4 ˆ

Ka1 ‰NH3 ? H2 OŠ HNH3 Ka1 ˆ pNH3 H‡ Kw ‰OH Š

(7.50)

The total dissolved ammonia NHT3 is simply NHT3 ˆ ‰NH3 ? H2 OŠ ‡ NH‡4 ˆ H NH3 pNH3 1 ‡

Ka1 ‰H‡ Š Kw

(7.51)

and the ammonium fraction is given by NH‡4 NHT4

ˆ

Ka1 ‰H‡ Š Kw ‡ Ka1 ‰H‡ Š

(7.52)

279

AQUEOUS-PHASE CHEMICAL EQUILIBRIA

FIGURE 7.9 Ammonium concentration as a function of pH for a gasphase ammonia mixing ratio of 1 ppb at 298 K.

Note that for pH values lower than 8, Ka1[H+]  Kw and NHT3  NH‡4 So under atmospheric conditions practically all dissolved ammonia in clouds is in the form of ammonium ion. The aqueous-phase concentrations of NH‡4 in equilibrium with 1 ppb of NH3(g) are shown in Figure 7.9. The partitioning of ammonia between the gas and aqueous phases inside a cloud can be calculated using (7.9) and the effective Henry’s law coefficient for ammonia, H∗NH3 ˆ H NH3 Ka1 ‰H‡ Š=Kw . If the cloud pH is less than 5 practically all the available ammonia will be dissolved in cloudwater (Figure 7.10).

FIGURE 7.10 Equilibrium fraction of total ammonia in the aqueous phase as a function of pH and cloud liquid water content at 298 K.

280

CHEMISTRY OF THE ATMOSPHERIC AQUEOUS PHASE

7.3.5 Nitric Acid–Water Equilibrium Nitric acid is one of the most water-soluble atmospheric gases with a Henry’s law constant (at 298 K) of H HNO3 ˆ 2:1  105 M atm 1 . After dissolution HNO3 …g† ƒ HNO3 …aq†

(7.53)

the nitric acid dissociates readily to nitrate HNO3 …aq† ƒ NO3 ‡ H‡

(7.54)

increasing further its solubility. The dissociation constant for nitrate is Kn1 = 15.4 M at 298 K, where by definition Kn1 ˆ

NO3 ‰H‡ Š ‰HNO3 …aq†Š

(7.55)

The total dissolved nitric acid HNOT3 will then be HNOT3 ˆ ‰HNO3 …aq†Š ‡ NO3

(7.56)

‰HNO3 …aq†Š ˆ H HNO3 pHNO3

(7.57)

and using Henry’s law, we obtain

Substituting this into the dissociation equilibrium equation, we have NO3 ˆ

H HNO3 Kn1 pHNO3 ‰H‡ Š

(7.58)

The last three equations can be combined to provide an expression for the total dissolved nitric acid in equilibrium with a given nitric acid vapor concentration, HNOT3 ˆ H ∗HNO3 pHNO3 ˆ H HNO3 1 ‡

Kn1 p ‰H‡ Š HNO3

(7.59)

where H ∗HNO3 ˆ H HNO3 …1 ‡ …Kn1 =‰H‡ Š†† is the effective Henry’s law coefficient for nitric acid. Note that because the dissociation constant Kn1 has such a high value, it follows that …Kn1 =‰H‡ Š†  1

(7.60)

for any cloud pH of atmospheric interest. Therefore NO3  ‰HNO3 …aq†Š for all atmospheric clouds and one can safely assume that dissolved nitric acid exists in clouds exclusively as nitrate. In other words, nitric acid is a strong acid and upon dissolution in water it dissociates completely to nitrate ions. The effective Henry’s law constant for nitrate is then H ∗HNO3  H HNO3 where H ∗HNO3 is in M atm

1

and [H+] in M.

3:2  106 Kn1 ˆ ‰H‡ Š ‰H‡ Š

(7.61)

281

AQUEOUS-PHASE CHEMICAL EQUILIBRIA

Partitioning of Nitric Acid inside a Cloud Calculate the fraction of nitric acid that will exist in the aqueous phase inside a cloud as a function of the cloud liquid water content (L = 0.001–1 g m 3) and pH. What does one expect in a typical cloud with liquid water content in the 0.1–1 g m 3 and pH in the 2–7 ranges? Using (7.9) for the aqueous fraction and substituting H A ˆ H ∗HNO3 ˆ 3:2  106 =‰H‡ Š (the Henry’s law coefficient in that equation is the effective Henry’s law coefficient for all the dissociating species), we can calculate the aqueous fraction of nitric acid as a function of pH for different cloud liquid water contents. The results are shown in Figure 7.11. For L = 1 g m 3 the aqueous fraction is practically 1 for all pH values of interest. For all clouds of atmospheric interest (L > 0.1 g m 3) and for all pH values (pH > 2), nitric acid at equilibrium is completely dissolved in cloudwater and its gas-phase concentration inside a cloud is expected to be practically zero. For example, for pH = 3, H∗HNO3 ˆ 3:2  109 M atm 1 , and for a nitrate concentration of 100 μM, the corresponding equilibrium gas-phase mixing ratio is only 0.03 ppt (3 × 10 14 atm).

FIGURE 7.11 Equilibrium fraction of total nitric acid in the aqueous phase as a function of pH and cloud liquid water content at 298 K assuming ideal solution.

7.3.6 Equilibria of Other Important Atmospheric Gases 7.3.6.1 Hydrogen Peroxide Hydrogen peroxide is soluble in water with a Henry’s law constant H H2 O2 ˆ 1  105 M atm It dissociates to produce HO2 and H+ H2 O2 …aq† ƒ HO2 ‡ H‡ but is a rather weak electrolyte with a dissociation constant of Kh1 = 2.2 × 10 dissociation equilibrium HO2 K h1 ˆ ‡ ‰H2 O2 …aq†Š ‰H Š

1

at 298 K. (7.62)

12

M at 298 K. Using the

(7.63)

282

CHEMISTRY OF THE ATMOSPHERIC AQUEOUS PHASE

FIGURE 7.12 Equilibrium fraction of total hydrogen peroxide in the aqueous phase as a function of cloud liquid water content at 298 K.

and this concentration ratio remains less than 10 4 for pH values less than 7.5. Therefore, for most atmospheric applications, the dissociation of dissolved hydrogen peroxide can be neglected. The equilibrium partitioning of H2O2 between the gas and aqueous phases can be calculated from (7.9) using the Henry’s law coefficient H H2 O2 and is shown in Figure 7.12. H2O2 exists in appreciable amounts in both the gas and aqueous phases inside a typical cloud. For example, for cloud liquid water content of 0.2 g m 3, roughly 30% of the H2O2 will be dissolved in the cloudwater, whereas the remaining 70% will remain in the cloud interstitial air. 7.3.6.2 Ozone Ozone is slightly soluble in water with a Henry’s law constant of 0.011 M atm 1 at 298 K. For a cloud gasphase mixing ratio of 100 ppb O3, the equilibrium ozone aqueous-phase concentration is 1.1 nM and practically all ozone remains in the gas phase. 7.3.6.3 Oxides of Nitrogen Nitric oxide (NO) and nitrogen dioxide (NO2) are also characterized by small solubility in water (Henry’s law coefficients 0.0019 and 0.012 M atm 1 at 298 K). A negligible fraction of these species is dissolved in cloudwater, and their aqueous-phase concentrations are estimated to be on the order of 1 nM or even smaller. 7.3.6.4 Formaldehyde Formaldehyde, upon dissolution in water, undergoes hydrolysis to yield the gem diol (for definition, see Section 14.8), methylene glycol HCHO…aq† ‡ H2 O ƒ H2 C…OH†2

(7.64)

with a hydration constant KHCHO defined by KHCHO ˆ

‰H2 C…OH†2 Š ‰HCHO…aq†Š

(7.65)

283

AQUEOUS-PHASE CHEMICAL EQUILIBRIA

FIGURE 7.13 Equilibrium fraction of total formaldehyde in the aqueous phase as a function of cloud liquid water content at 298 K.

that has a value of 2530 at 298 K (Le Henaff 1968). This rather large value of KHCHO suggests that the hydration is essentially complete (>99.9%) and that practically all dissolved formaldehyde will exist in its gem diol form. Formaldehyde has a Henry’s law constant HHCHO = 2.5 M atm 1 at 298 K (Betterton and Hoffmann 1988), but its water solubility is enhanced by several orders of magnitude as a result of the diol formation. Combining the Henry’s law equilibrium and the hydrolysis relationships, we can calculate the effective Henry’s law coefficient for the total dissolved formaldehyde H ∗HCHO as H ∗HCHO ˆ H HCHO …1 ‡ KHCHO †  H HCHO KHCHO

(7.66)

with a value of 6.3 × 103 M atm 1. The effective Henry’s law coefficient for formaldehyde is often quoted instead of the intrinsic constant. Therefore, for thermodynamic equilibrium assuming that all dissolved formaldehyde exists as H2C(OH)2: ‰H2 C…OH†2 Š ˆ H ∗HCHO pHCHO

(7.67)

Most of the available formaldehyde remains in the gas phase inside a typical cloud (Figure 7.13). 7.3.6.5 Formic and Other Atmospheric Acids The most abundant carboxylic acids in the atmosphere are formic and acetic acid, although more than 100 aliphatic, olefinic, and aromatic acids have been detected in the atmosphere (Graedel et al. 1986). These acids are weak electrolytes, and their partial dissociation enhances their solubility. Formic acid has a Henry’s law coefficient of 880 M atm 1 at 298 K and a dissociation constant Kf = 1.8 × 10 4 M. From the dissociation equilibrium HCOOH…aq† ƒ HCOO ‡ H‡

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CHEMISTRY OF THE ATMOSPHERIC AQUEOUS PHASE

one can calculate the dissociated fraction of formic acid as Kf ‰HCOO Š ˆ ‡ ‰HCOOH…aq†Š ‰H Š

(7.68)

At pH = 3.74, 50% of the dissolved formic acid is dissociated. Above pH 4, most of the dissolved formic acid exists in its ionic form, while for pH values below 3 most of it is undissociated. These results illustrate that significant formic acid concentrations may be found in cloudwater or rainwater that has not been acidified. For pH values below 4 most of the formic acid remains in the cloud interstitial air and only a small fraction (less than 10%) dissolves in the cloud water. For pH values >7 practically all the available formic acid is transferred into the aqueous phase and only traces remain in the interstitial air. In the intermediate pH regime from 4 to 7 the formic acid equilibrium partitioning varies considerably depending on the cloud liquid water content and pH. Acetic acid has a lower Henry’s law coefficient (400 M atm 1 at 298 K) than formic acid and dissociates less (dissociation constant 1.7 × 10 5 M at 298 K). 7.3.6.6 OH and HO2 Radicals The hydroperoxyl radical, HO2, is a weak acid and on dissolution in the aqueous phase dissociates, HO2 …aq† ƒ O2 ‡ H‡ with an equilibrium constant KHO2 ˆ 3:5  10 dissolved concentration as

5

(7.69)

M at 298 K (Perrin 1982). For HO2 one defines the total

HOT2 ˆ ‰HO2 …aq†Š ‡ O2

(7.70)

and then its effective Henry’s law constant is given by H ∗HO2 ˆ H HO2 1 ‡

KHO2 ‰H‡ Š

(7.71)

where H ∗HO2 ˆ 5:7  103 M atm 1 is the Henry’s law constant for HO2 at 298 K. The dissociation of HO2 enhances its dissolution in the aqueous phases for pH values above 4.5 and H ∗HO2 ˆ 2  105 M atm 1 for pH = 6. The Henry’s law constant of OH at 298 K is 30 M atm 1, so at equilibrium and for pH values lower than 6 most OH and HO2 will be present in the gas phase inside a cloud (see Figure 7.3). For gas-phase mixing ratios of OH and HO2 of 0.3 and 40 ppt, respectively, the corresponding equilibrium concentra­ tions in the aqueous phase are, for pH = 4, [OH] = 7.5 × 10 12 M and [HO2] = 0.08 μM. The Henry’s law equilibrium concentration of HO2 increases to 2.9 μM at pH = 6.

7.4 AQUEOUS-PHASE REACTION RATES Rates of reaction of aqueous-phase species are generally expressed in terms of moles per liter (M) of solution per second. It is often useful to express aqueous-phase reaction rates on the basis of the gasphase properties, especially when comparing gas-phase and aqueous-phase reaction rates. In this way both rates are expressed on the same basis. To place our discussion on a concrete basis, let us say we have a reaction of S(IV) with a dissolved species A S…IV† ‡ A…aq† ! products

(7.72)

285

AQUEOUS-PHASE REACTION RATES

the rate of which is given by Ra ˆ k‰A…aq†Š‰S…IV†Š

…M s 1 †

(7.73)

where Ra is in M s 1, the aqueous-phase concentrations [S(IV)] and [A(aq)] are in M, and the reaction constant k is in M 1 s 1. The reaction rate can be expressed in moles of SO2 per liter of air per second by multiplying Ra by the liquid water mixing ratio wL = 10 6 L: R´a ˆ kwL ‰A…aq†Š‰S…IV†Š ˆ 10 6 k L‰A…aq†Š‰S…IV†Š

…mol…L of air†

1

s 1†

(7.74)

The moles per liter of air can be then converted to equivalent SO2 partial pressure for 1 atm total pressure by applying the ideal-gas law to obtain R´´a ˆ 3:6  106 L RT Ra

…ppb h 1 †

(7.75)

where L is in g m 3, R = 0.082 atm L K 1 mol 1, and T is in K. For example, an aqueous-phase reaction rate of 1 μM s 1 in a cloud with liquid water content of 0.1 g m 3 at 288 K is equivalent to a gas-phase oxidation rate of 8.5 ppb h 1. A nomogram relating aqueous-phase reaction rates in μM s 1 to equivalent gas-phase reaction rates in ppb h 1 as a function of the cloud liquid water content L at 288 K is given in Figure 7.14. Reaction rates are sometimes also expressed as a fractional conversion rate in % h 1. The rate R´´a given above can be converted to a fractional conversion rate by dividing by the mixing ratio, ξSO2 , of SO2 in ppb and multiplying by 100 Ra ˆ 3:6  108

Ra LRT ξSO2

…% h 1 †

(7.76)

where ξSO2 is in ppb. Let us assume that both S(IV) and A are in thermodynamic equilibrium with Henry’s law constants H∗S…IV† and HA, respectively. Combining (7.73), (7.76), and (7.3), one obtains Ra ˆ 3:6  10

10

k H A H ∗S…IV† ξA LRT

…% h 1 †

FIGURE 7.14 Nomogram relating aqueous reaction rates, in μM s 1, to equivalent gas-phase reaction rates in ppb h 1, at T = 288 K, p = 1 atm, and given liquid water content.

(7.77)

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CHEMISTRY OF THE ATMOSPHERIC AQUEOUS PHASE

where k is in M s 1, HA and H∗S…IV† are in M atm 1, ξA is in ppb, L is in g m 3, R = 0.082 atm L K 1 mol 1, and T is in K. Note that the SO2 oxidation rate given by (7.77) is independent of the SO2 concentration and depends only on the mixing ratio of A, the cloud liquid water content, and the temperature. Equation (7.77) should be used only if A exists in both gas and aqueous phases and Henry’s law equilibrium is satisfied by both S(IV) and A. If the two species do not satisfy Henry’s law, (7.76) can still be used. The characteristic time for SO2 oxidation τSO2 (s) can be calculated from (7.76) as τSO2 ˆ

ξ 100 ˆ 3 SO2 Ra 10 Ra LRT

(7.78)

7.5 S(IV)–S(VI) TRANSFORMATION AND SULFUR CHEMISTRY The aqueous-phase conversion of dissolved SO2 to sulfate, also referred to as S(VI) since sulfur is in oxidation state 6, is considered the most important chemical transformation in cloudwater. As we have seen, dissolution of SO2 in water results in the formation of three chemical species: hydrated SO2 (SO2
H2O), the bisulfite ion HSO3 , and the sulfite ion SO23 . At the pH range of atmospheric interest (pH = 2–7) most of the S(IV) is in the form of HSO3 , whereas at low pH (pH < 2), all of the S(IV) occurs as SO2
H2O. At higher pH values (pH > 7), SO23 is the preferred S(IV) state (Figure 7.8). Since the individual dissociations are fast, occurring on timescales of milliseconds or less (see Chapter 12), during a reaction consuming one of the three species, SO2
H2O, HSO3 , or SO23 , the corresponding aqueous­ phase equilibria are reestablished instantaneously. As we saw earlier, the dissociation of dissolved SO2 enhances its aqueous solubility so that the total amount of dissolved S(IV) always exceeds that predicted by Henry’s law for SO2 alone and is quite pH dependent. Several pathways for S(IV) transformation to S(VI) have been identified involving reactions of S(IV) with O3, H2O2, O2 (catalyzed by Mn(II) and Fe(III) ), OH, SO5 , HSO5 , SO4 , PAN, CH3OOH, CH3C(O)OOH, HO2, NO3, NO2, N(III), HCHO, and Cl2 (Pandis and Seinfeld 1989a). There is a large literature on the reaction kinetics of aqueous sulfur chemistry. We consider here only a few of the most important reactions.

7.5.1 Oxidation of S(IV) by Dissolved O3 Although ozone reacts very slowly with SO2 in the gas phase, the aqueous-phase reaction S…IV† ‡ O3 ! S…VI† ‡ O2

(7.79)

is rapid. This reaction has been studied by many investigators (Erickson et al. 1977; Larson et al. 1978; Penkett et al. 1979; Harrison et al. 1982; Maahs 1983; Martin 1984; Hoigné et al. 1985; Lagrange et al. 1994; Botha et al. 1994). A detailed evaluation of existing experimental kinetic and mechanistic data suggested the following expression for the rate of the reaction of S(IV) with dissolved ozone for a dilute solution (subscript 0) (Hoffmann and Calvert 1985): R0 ˆ

d‰S…IV†Š ˆ …k0 ‰SO2 ? H2 OŠ ‡ k1 HSO3 ‡ k2 SO23 †‰O3 Š dt

(7.80)

with k0 = 2.4 ± 1.1 × 104 M 1 s 1, k1 = 3.7 ± 0.7 × 105 M 1 s 1 and, k2 = 1.5 ± 0.6 × 109 M 1 s 1. The activa­ tion energies recommended by Hoffmann and Calvert (1985) are based on the work of Erickson et al. (1977) and are 46.0 kJ mol 1 for k1 and 43.9 kJ mol 1 for k2. The complex pH dependency of the effective rate constant of the S(IV)–O3 reaction is shown in Figure 7.15. Over the pH range of 0 2 the slope is close to 0.5 (corresponding to an [H+] 0.5 dependence), while over the pH range of 5 7 the slope of the plot is close to 1 (corresponding to an [H+] 1 dependence).

287

S(IV)–S(VI) TRANSFORMATION AND SULFUR CHEMISTRY

FIGURE 7.15 Second-order reaction rate constant for the S(IV)–O3 reaction defined according to d[S(VI)]/dt = k[O3(aq)] [S(IV)], as a function of solution pH at 298 K. The curve shown is the three-component expression of (7.80) and the symbols are the corresponding measure­ ments by a number of investigators.

In the transition regime between pH 2 and 4 the slope is 0.34. Some studies focusing on pH values lower than 4 have produced rate laws that yield erroneous rates when extrapolated to higher pH values. Hoffmann and Calvert (1985) proposed that the S(IV)–O3 reaction proceeds by nucleophilic attack on ozone by SO2
H2O, HSO3 , and SO32 . Considerations of nucleophilic reactivity indicate that SO23 should react more rapidly with ozone than HSO3 , and HSO3 should react more rapidly in turn than SO2
H2O. This order is reflected in the relative numerical values of k0, k1, and k2. An increase in the aqueous-phase pH results in an increase of HSO3 and SO23 equilibrium concentrations, and therefore in an increase of the overall reaction rate. For an ozone gas-phase mixing ratio of 30 ppb, the reaction rate varies from less than 0.001 μM h 1 (ppb SO2) 1 at pH 2 (or less than 0.01% SO2(g) h 1 (g water/m3 air) 1) to 3000 μM h 1 (ppb SO2) 1 at pH 6 (7000% SO2(g) h 1 (g water/m3 air) 1) (Figure 7.16). A typical gas-phase SO2 oxidation rate by OH is on the order of 1% h 1 and therefore S(IV) heterogeneous oxidation by ozone is significant for pH values greater than 4. The strong increase of the reaction rate with pH often renders this reaction self-limiting; production of sulfate by this reaction lowers the pH and slows down further reaction. The ubiquity of atmospheric ozone guarantees that this reaction will play an important role both as a sink of gas-phase SO2 and as a source of cloudwater acidification as long as the pH of the atmospheric aqueous phase exceeds about 4. The S(IV)–O3 reaction rate is not affected by the presence of metallic ion traces: Cu2+, Mn2+, Fe2+, 3+ Fe , and Cr2+ (Maahs 1983; Martin 1984; Lagrange et al. 1994). Ionic Strength The ionic strength of a solution is a measure of the concentration of dissolved ions in that solution. The total electrolyte concentration in solution affects properties such as the dissociation or the solubility of different salts. The ionic strength of a solution I is defined as Iˆ

1 2

n

mi z2i

(7.81)

iˆ1

where n is the number of different ions in solution, mi and zi are the molality (solute concentration in mol kg 1 of solvent) and number of ion charges of ion i, respectively. Because in fairly concentrated nonideal solutions volumes are not strictly additive, it is customary to define the ionic strength in

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CHEMISTRY OF THE ATMOSPHERIC AQUEOUS PHASE

FIGURE 7.16 Rate of aqueous-phase oxidation of S(IV) by ozone (30 ppb) and hydrogen peroxide (1 ppb), as a function of solution pH at 298 K. Gas aqueous equilibria are assumed for all reagents. R=ξSO2 represents the aqueous phase reaction rate per ppb of gasphase SO2. R/L represents rate of reaction referred to gas-phase SO2 pressure per (g m 3) of cloud liquid water content.

terms of molality rather than molarity. Ionic strength values for relatively dilute atmospheric droplets (expressed in terms of molarity) are (Herrmann 2003): Remote cloud Polluted cloud Polluted fog

(0.75 – 7.5) × 10 (0.5 – 1) × 10 4 (0.07 – 4) × 10 2

4

mol L

1

While chemical reactions in clouds and fog can be treated as reactions in dilute electrolyte solutions, this is not the case for aerosol particles in which high ionic strengths can be reached. Lagrange et al. (1994) suggested that the rate of the S(IV)–O3 reaction increases linearly with the ionic concentration of the solution. Experiments at 18°C supported the following ionic strength correction R ˆ …1 ‡ FI†R0

(7.82)

where F is a parameter characteristic of the ions of the supporting electrolyte, I is the ionic strength of the solution, defined in this case in units of mol L 1, and R and R0 are the reaction rates at ionic strengths I and zero, respectively. The measured values of F are F ˆ 1:59  0:3 F ˆ 3:71  0:7

for NaCl for Na2 SO4

(7.83)

These results indicate that the ozone reaction can be 2.6 times faster in a seasalt particle with an ionic strength of 1 M than in a solution with zero ionic strength.

289

S(IV)–S(VI) TRANSFORMATION AND SULFUR CHEMISTRY

7.5.2 Oxidation of S(IV) by Hydrogen Peroxide Hydrogen peroxide, H2O2, is one of the most effective oxidants of S(IV) in clouds and fogs (Pandis and Seinfeld 1989a). H 2O2 is very soluble in water and under typical ambient conditions its aqueousphase concentration is approximately six orders of magnitude higher than that of dissolved ozone. The S(IV)–H2O2 reaction has been studied in detail by several investigators (Hoffmann and Edwards 1975; Penkett et al. 1979; Cocks et al. 1982; Martin and Damschen 1981; Kunen et al. 1983; McArdle and Hoffmann 1983) and the published rates all agree within experimental error over a wide range of pH (0 8) (Figure 7.17). The reproducibility of the measurements suggests a lack of susceptibility of this reaction to the influence of trace constituents. The rate expression is (Hoffmann and Calvert 1985) d‰S…IV†Š k‰H‡ Š‰H2 O2 Š HSO3 ˆ dt 1 ‡ K‰H‡ Š

(7.84)

with k = 7.5 ± 1.16 × 107 M 2 s 1 and K = 13 M 1 at 298 K. Noting that H2O2 is a very weak electrolyte, that ‰H‡ Š HSO3 ˆ H SO2 Ks1 pSO2 , and that, for pH > 2, 1 + K[H+]  1, one concludes that the rate of this reaction is practically pH independent over the pH range of atmospheric interest. For a H2O2(g) mixing ratio of 1 ppb, the rate is roughly 300 μM h 1 (ppb SO2) 1 (700% SO2(g) h 1 (g water/m3 air) 1). The near pH independence is due to the fact that the pH dependences of the S(IV) solubility and of the reaction rate constant cancel each other. The reaction is very fast and indeed both field measurements (Daum et al. 1984) and theoretical studies (Pandis and Seinfeld 1989b) suggest that, as a result, H2O2(g) and SO2(g) rarely coexist in clouds and fogs. The species with the lower concentration before cloud or fog formation is the limiting reactant and is rapidly depleted inside the cloud or fog layer. The reaction proceeds via a nucleophilic displacement by hydrogen peroxide on bisulfite as the principal reactive S(IV) species (McArdle and Hoffmann 1983) HSO3 ‡ H2 O2 ƒ SO2 OOH ‡ H2 O

FIGURE 7.17 Second-order rate constant for oxidation of S(IV) by hydro­ gen peroxide defined according to d[S(VI)]/dt = k[H2O2(aq)] [S(IV)] as a function of solution pH at 298 K. The solid curve corresponds to the rate expression (7.84). Dashed lines are arbitrarily placed to encompass most of the experimental data shown.

(7.85)

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CHEMISTRY OF THE ATMOSPHERIC AQUEOUS PHASE

and then the peroxymonosulfurous acid, SO2OOH , reacts with a proton to produce sulfuric acid: SO2 OOH ‡ H‡ ! H2 SO4

(7.86)

The latter reaction becomes faster as the medium becomes more acidic.

7.5.3 Oxidation of S(IV) by Organic Peroxides Organic peroxides have also been proposed as potential aqueous-phase S(IV) oxidants (Graedel and Goldberg 1983; Lind and Lazrus 1983; Hoffmann and Calvert 1985; Lind et al. 1987). Methylhydroxyperoxide, CH3OOH, reacts with HSO3 according to HSO3 ‡ CH3 OOH ‡ H‡ ! SO24 ‡ 2H‡ ‡ CH3 OH

(7.87)

with a reaction rate given in the atmospheric relevant pH range 2.9–5.8 by R7:87 ˆ k7:87 ‰H‡ Š‰CH3 OOHŠ‰HSO3 Š

(7.88)

where k7.87 = 1.7 ± 0.3 × 107 M 2 s 1 at 18°C and the activation energy is 31.6 kJ mol 1 (Hoffmann and Calvert 1985; Lind et al. 1987). Because of the inverse dependence of HSO3 on [H+] the overall rate is pH independent (Figure 7.18). The corresponding reaction involving peroxyacetic acid is HSO3 ‡ CH3 C…O†OOH ! SO24 ‡ H‡ ‡ CH3 COOH

(7.89)

with a rate law for pH 2.9–5.8 given by R7:89 ˆ …ka ‡ kb ‰H‡ Š†‰HSO3 Š‰CH3 C…O†OOHŠ

(7.90)

where ka = 601 M 1 s 1 and kb = 3.64 ± 0.4 × 107 M 2 s 1 at 18°C (Hoffmann and Calvert 1985; Lind et al. 1987). The overall rate of the reaction increases with increasing pH (Figure 7.18).

FIGURE 7.18 Reaction rates for the oxidation of S(IV) by CH3OOH and CH3C(O)OOH as a function of pH for mixing ratios [SO2(g)] = 1 ppb, [CH3OOH(g)] = 1 ppb, and [CH3C(O)OOH(g)] = 0.01 ppb.

291

S(IV)–S(VI) TRANSFORMATION AND SULFUR CHEMISTRY

The Henry’s law constants of methylhydroperoxide and peroxyacetic acid are more than two orders of magnitude lower than that of hydrogen peroxide. Typical atmospheric mixing ratios are on the order of 1 ppb for CH3OOH and 0.01 ppb for CH3C(O)OOH. Applying Henry’s law, the corresponding equilibrium aqueous-phase concentrations are ∼ 0.2 μM and ∼ 5 nM, respectively. These rather low concentrations result in relatively low S(IV) oxidation rates on the order of 1 μM h 1 (for a SO2 mixing ratio of 1 ppb), or equivalently in rates of less than 1% SO2 h 1 for a typical cloud liquid water content of 0.2 g m 3 (Figure 7.18). As a result, these reactions are of minor importance for S(IV) oxidation under typical atmospheric conditions and represent only small sinks for the gas-phase methyl hydroperoxide (0.2% CH3OOH h 1) and peroxyacetic acid (0.7% CH3C(O)OOH h 1) (Pandis and Seinfeld 1989a).

7.5.4 Uncatalyzed Oxidation of S(IV) by O2 The importance of the reaction of S(IV) with dissolved oxygen in the absence of any metal catalysts (iron, manganese) has been an unsettled issue. Solutions of sodium sulfite in the laboratory oxidize slowly in the presence of oxygen (Fuller and Crist 1941; Martin 1984). However, observations of Tsunogai (1971) and Huss et al. (1978) showed that the rate of the uncatalyzed reaction is negligible. The observed rates can be explained by the existence of very small amounts of catalyst such as iron (concentrations lower than 0.01 μM) that are extremely difficult to exclude. It is interesting to note that for actual cloud droplets traces of catalyst are usually present (Table 7.5).

7.5.5 Oxidation of S(IV) by O2 Catalyzed by Iron and Manganese 7.5.5.1 Iron Catalysis S(IV) oxidation by O2 is known to be catalyzed by Fe(III) and Mn(II): Mn2‡ ;Fe3‡ 1 ! S…VI† S…IV† ‡ O2 2

(7.91)

This reaction has been the subject of considerable interest (Hoffmann and Boyce 1983; Hoffmann and Jacob 1984; Martin 1984; Hoffmann and Calvert 1985; Clarke and Radojevic 1987), but significantly different measured reaction rates, rate laws, and pH dependences have been reported (Hoffmann and Jacob 1984). Martin and Hill (1987a,b) showed that this reaction is inhibited by increasing ionic strength, the sulfate ion, and various organics and is even self-inhibited. They explained most of the literature discrepancies by differences in these factors in various laboratory studies. In the presence of oxygen, iron in the ferric state, Fe(III), catalyzes the oxidation of S(IV) in aqueous solutions. Iron in cloudwater exists in both the Fe(II) and Fe(III) states, and there are a series of oxidation– reduction reactions cycling iron between these two forms (Stumm and Morgan 1996). Fe(II) appears not to directly catalyze the reaction and is first oxidized to Fe(III) before S(IV) oxidation can begin (Huss et al.

TABLE 7.5 Manganese and Iron Concentrations in Aqueous Particles and Drops Medium Aerosol (haze) Clouds Rain Fog Source: Martin (1984).

Manganese (μM)

Iron (μM)

0.1–100 0.01–10 0.01–1 0.1–10

100–1000 0.1–100 0.01–10 1–100

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CHEMISTRY OF THE ATMOSPHERIC AQUEOUS PHASE

1982a,b). The equilibria involving Fe(III) in aqueous solution are Fe3‡ ‡ H2 O ƒ FeOH2‡ ‡ H‡ FeOH2‡ ‡ H2 O ƒ Fe…OH†‡2 ‡ H‡ Fe…OH†‡2 ‡ H2 O ƒ Fe…OH†3 …s† ‡ H‡ 2 FeOH2‡ ƒ Fe2 …OH†4‡ 2 3+ with Fe3+, FeOH2‡ Fe…OH†‡2 , and Fe2 …OH†4‡ 2 soluble and Fe(OH)3 insoluble. The concentration of Fe can be calculated from the equilibrium with solid Fe(OH)3 (Stumm and Morgan 1996)

Fe…OH†3 …s† ‡ 3 H‡ ƒ Fe3‡ ‡ 3 H2 O as ‰Fe3‡ Š  103 ‰H‡ Š3

in M at 298 K

and for a pH of 4.5, [Fe3+] = 3 × 10 11 M. For pH values from 0 to 3.6 the iron-catalyzed S(IV) oxidation rate is first-order in iron, is first-order in S(IV), and is inversely proportional to [H+] (Martin and Hill 1987a): d‰S…IV†Š ‰Fe…III†Š‰S…IV†Š ˆ k7:91 dt ‰H‡ Š

R7:91 ˆ

(7.92)

This reaction is inhibited by ionic strength and sulfate. Accounting for these effects, the reaction rate constant is given by k7:91 ˆ

k∗7:91

10

p p 2 I=…1‡ I†

(7.93)

1 ‡ 150‰S…VI†Š2=3

where [S(VI)] is in M. A rate constant k∗7:91 ˆ 6 s 1 was recommended by Martin and Hill (1987a). Sulfite appears to be almost equally inhibiting as sulfate. This does not pose a problem for regular atmospheric conditions ([S(IV)] < 0.001 M), but the preceding rate expressions should not be applied to laboratory studies where the S(IV) concentrations exceed 0.001 M. This reaction, according to the rate expressions presented above is very slow under typical atmospheric conditions in this pH regime (0–3.6). The rate expression for the same reaction changes completely above pH 3.6. This suggests that the mechanism of the reaction differs in the two pH regimes and is probably a free radical chain at high pH and a nonradical mechanism at low pH (Martin et al. 1991). The low solubility of Fe(III) above pH 3.6 poses special experimental problems. At high pH the reaction rate depends on the actual amount of iron dissolved in solution, rather than on the total amount of iron present in the droplet. In this range the reaction is second-order in dissolved iron (zero-order above the solution iron saturation point) and firstorder in S(IV). Martin et al. (1991) proposed the following phenomenological expressions (in M s 1) d‰S…IV†Š ˆ 1  109 ‰S…IV†Š‰Fe…III†Š2 dt d‰S…IV†Š pH 5:0 to 6:0 : ˆ 1  10 3 ‰S…IV†Š dt

pH 4:0 :

pH 7:0 :

d‰S…IV†Š ˆ 1  10 4 ‰S…IV†Š dt

(7.94) (7.95) (7.96)

for the following conditions: ‰S…IV†Š  10 μM;

‰Fe…III†Š > 0:1 μM;

‰S…VI†Š < 100 μM;

and

I < 0:01 M; T ˆ 298 K:

(7.97)

293

S(IV)–S(VI) TRANSFORMATION AND SULFUR CHEMISTRY

Note that iron does not appear in the pH 5 7 range because it is assumed that a trace of iron will be present under normal atmospheric conditions. This reaction is important in this high pH regime (Pandis and Seinfeld 1989b; Pandis et al. 1992). Martin et al. (1991) also found that noncomplexing organic molecules (e.g., acetate, trichlor­ oacetate, ethyl alcohol, isopropyl alcohol, formate, allyl alcohol) are highly inhibiting at pH values of 5 and above and are not inhibiting at pH values of 3 and below. They calculated that for remote clouds formate would be the main inhibiting organic, but by less than 10%. On the contrary, near urban areas formate could reduce the rate of the catalyzed oxidation by a factor of 10–20 in the highpH regime. 7.5.5.2 Manganese Catalysis The manganese-catalyzed S(IV) oxidation rate was initially thought to be inversely proportional to the [H+] concentration. Martin and Hill (1987b) suggested that ionic strength, not hydrogen ion, accounts for the pH dependence of the rate. The manganese-catalyzed reaction obeys zero-order kinetics in S(IV) in the concentration regime above 100 μM S(IV), d‰S…IV†Š ˆ k0 ‰MnŠ2 dt k0 ˆ k∗0 10 with k∗0 ˆ 680 M order in S(IV),

1

s

1

p p 4:07 I=…1‡ I†

(7.98) (7.99)

(Martin and Hill 1987b). For S(IV) concentrations below 1 μM the reaction is first d‰S…IV†Š ˆ k0 ‰MnŠ‰S…IV†Š dt

(7.100)

k0 ˆ k∗0 10

(7.101)

p p 4:07 I=…1‡ I†

with k∗0 ˆ 1000 M 1 s 1 (Martin and Hill 1987b). It is still not clear which rate law is appropriate for use in atmospheric calculations, although Martin and Hill (1987b) suggested the provisional use of the firstorder, low S(IV) rate. 7.5.5.3 Iron/Manganese Synergism When both Fe3+ and Mn2+ are present in atmospheric droplets, the overall rate of the S(IV) reaction is enhanced over the sum of the two individual rates. Martin (1984) reported that the rates measured were 3 to 10 times higher than expected from the sum of the independent rates. Martin and Good (1991) obtained at pH 3.0 and for [S(IV)] < 10 μM the following rate law: d‰S…IV†Š ˆ 750‰Mn…II†Š‰S…IV†Š ‡ 2600‰Fe…III†Š‰S…IV†Š dt ‡ 1:0  1010 ‰Mn…II†Š‰Fe…III†Š‰S…IV†Š

(7.102)

and a similar expression for pH 5.0 in agreement with the work of Ibusuki and Takeuchi (1987).

7.5.6 Comparison of Aqueous-Phase S(IV) Oxidation Paths We will now compare the different routes for SO2 oxidation in aqueous solution as a function of pH and temperature. In doing so, we will set the pH at a given value and calculate the instantaneous rate of S(IV) oxidation at that pH. The rate expressions used and the parameters in the rate expressions are given in Table 7.6. Figure 7.19 shows the oxidation rates in μM h 1 for the different paths at 298 K for the

294

CHEMISTRY OF THE ATMOSPHERIC AQUEOUS PHASE

TABLE 7.6 Rate Expressions for Sulfate Formation in Aqueous Solution Used in Computing Figure 7.19 Oxidant O3

H2O2

Fe(III) Mn(II) NO2

Rate expression, d[S(IV)]/dt

Reference

k0 ‰SO2 ? H2 OŠ ‡ k1 HSO3 ‡ k2 SO23 ‰O3 …aq†Š k0 = 2.4 × 104 M 1 s 1 k1 = 3.7 × 105 M 1 s 1 k2 = 1.5 × 109 M 1 s 1 k4 ‰H‡ Š HSO3 ‰H2 O2 …aq†Š=…1 ‡ K‰H‡ Š† k4 = 7.45 × 107 M 1 s 1 K = 13 M 1 k5 ‰Fe…III†Š SO23 a k5 = 1.2 × 106 M 1 s 1 for pH  5 k6[Mn(II)][S(IV)] k6 = 1000 M 1 s 1 (for low S(IV) ) k7[NO2(aq)][S(IV)] k7 = 2 × 106 M 1 s 1

Hoffmann and Calvert (1985)

Hoffmann and Calvert (1985)

Hoffmann and Calvert (1985) Martin and Hill (1987b) Lee and Schwartz (1983)

a

This is an alternative reaction rate expression for the low-pH region. Compare with (7.92) and (7.95).

FIGURE 7.19 Comparison of aqueous-phase oxidation paths. The rate of conversion of S(IV) to S(VI) as a function of pH. Conditions assumed are [SO2(g)] = 5 ppb; [NO2(g)] = 1 ppb; [H2O2(g)] = 1 ppb; [O3(g)] = 50 ppb; [Fe(III)] = 0.3 μM; [Mn(II)] = 0.03 μM.

DYNAMIC BEHAVIOR OF SOLUTIONS WITH AQUEOUS-PHASE CHEMICAL REACTIONS

following conditions: ‰SO2 …g†Š ˆ 5 ppb ‰NO2 …g†Š ˆ 1 ppb ‰Fe…III†…aq†Š ˆ 0:3 μM

‰H2 O2 …g†Š ˆ 1 ppb ‰O3 …g†Š ˆ 50 ppb ‰Mn…II†…aq†Š ˆ 0:03 μM

We see that under these conditions oxidation by dissolved H2O2 is the predominant pathway for sulfate formation at pH values less than roughly 4–5. At pH  5 oxidation by O3 starts dominating and at pH 6 it is 10 times faster than that by H2O2. Also, oxidation of S(IV) by O2 catalyzed by Fe and Mn may be important at high pH, but uncertainties in the rate expressions at high pH preclude a definite conclusion. Oxidation of S(IV) by NO2 is unimportant at all pH for the concentration levels above. The oxidation rate of S(IV) by OH cannot be calculated using the simple approach outlined above. Since the overall rate depends on the propagation and termination rates of the radical chain, it depends, in addition to the S(IV) and OH concentrations, on those of HO2, HCOOH, HCHO, and so on, and its determination requires a dynamic chemical model. The inhibition of most oxidation mechanisms at low pH results mainly from the lower overall solubility of SO2 with increasing acidity. H2O2 is the only identified oxidant for which the rate is virtually independent of pH. The effect of temperature on oxidation rates is a result of two competing factors. First, at lower temperatures, higher concentrations of gases are dissolved in equilibrium (see Figure 7.7 for SO2), which lead to higher reaction rates. On the other hand, rate constants in the rate expressions generally decrease as temperature decreases. The two effects therefore act in opposite directions. Except for Fe- and Mn­ catalyzed oxidation, the increased solubility effect dominates and the rate increases with decreasing temperature. In the transition-metal-catalyzed reaction, the consequence of the large activation energy is that as temperature decreases the overall rate of sulfate formation for a given SO2 concentration decreases. As we have noted, it is often useful to express aqueous-phase oxidation rates in terms of a fractional rate of conversion of SO2. Assuming cloud conditions with 1 g m 3 of liquid water content, we find that the rate of oxidation by H2O2 can exceed 500% h 1 (Figure 7.16).

7.6 DYNAMIC BEHAVIOR OF SOLUTIONS WITH AQUEOUS-PHASE CHEMICAL REACTIONS To compare the rates of various aqueous-phase chemical reactions we have been calculating instanta­ neous rates of conversion as a function of solution pH. In the atmosphere a droplet is formed, usually by water condensation on a particle, and subsequently exposed to an environment containing reactive gases. Gases then dissolve in the droplet, establishing an initial pH and composition. Aqueous-phase reactions ensue and the pH and composition of the droplet change accordingly. In this section we calculate that time evolution. We neglect any changes in droplet size that might result; for dilute solutions such as those considered here, this is a good assumption. We focus on sulfur chemistry because of its atmospheric importance. In our calculations up to this point we have simply assumed values for the gas-phase partial pressures. When the process occurs over time, one must consider also what is occurring in the gas phase. Two assumptions can be made concerning the gas phase: 1. Open System. Gas-phase partial pressures are maintained at constant values, presumably by continous infusion of new air. This is a convenient assumption for order of magnitude calcula­ tions, but it is often not realistic. Up to now we have implicitly treated the cloud environment as an open system. Note that mass balances (e.g., for sulfur) are not satisfied in such a system, because there is continuous addition of material.

295

296

CHEMISTRY OF THE ATMOSPHERIC AQUEOUS PHASE

2. Closed System. Gas-phase partial pressures decrease with time as material is depleted from the gas phase. One needs to describe mathematically both the gas- and aqueous-phase concentra­ tions of the species in this case. Calculations are more involved, but the resulting scenarios are more representative of actual atmospheric behavior. The masses of the various elements (sulfur, nitrogen, etc.) are conserved in a closed system.

7.6.1 Closed System The basic assumption in the closed system is that the total quantity of each species is fixed. Consider, as an example, the cloud formation (liquid water mixing ratio wL) in an air parcel that has initially a H2O2 partial pressure p0H2 O2 and assume that no reactions take place. If we treat the system as open, then at equilibrium the aqueous-phase concentration of H2O2 is given by ‰H2 O2 …aq†Šopen ˆ H H2 O2 p0H2 O2

(7.103)

Note that for an open system, the aqueous-phase concentration of H2O2 is independent of the cloud liquid water content. For a closed system, the total concentration of H2O2 per liter (physical volume) of air, [H2O2]total, is equal to the initial amount of H2O2 or ‰H2 O2 Štotal ˆ

p0H2 O2 RT

(7.104)

After the cloud is formed, this total H2O2 is distributed between gas and aqueous phases and satisfies the mass balance ‰H2 O2 Štotal ˆ

pH2 O2 ‡ ‰H2 O2 …aq†Šclosed wL RT

(7.105)

where pH2 O2 is the vapor pressure of H2O2 after the dissolution of H2O2 in cloudwater. If, in addition, Henry’s law is assumed to hold, then ‰H2 O2 …aq†Šclosed ˆ H H2 O2 pH2 O2

(7.106)

Combining the last three equations, we obtain

‰H2 O2 …aq†Šclosed ˆ

H H2 O2 p0H2 O2 1 ‡ H H2 O2 wL RT

(7.107)

The aqueous-phase concentration of H2O2, assuming a closed system, decreases as the cloud liquid water content increases, reflecting the increase in the amount of liquid water content available to accommodate H2O2. Figure 7.20 shows [H2O2(aq)] as a function of the liquid water content L = 10 6 wL, for initial H2O2 mixing ratios of 0.5, 1, and 2 ppb. For soluble species, such as H2O2, the open and closed system assumptions lead to significantly different concentration estimates, with the open system resulting in the higher concentration. The fraction of the total quantity of H2O2 that resides in the liquid phase is given by (7.9) and is shown in Figure 7.12. A species like H2O2 with a large Henry’s law coefficient (1 × 105 M atm 1) will have a significant fraction of the fixed total quantity in the aqueous phase. As wL increases, more liquid is

297

DYNAMIC BEHAVIOR OF SOLUTIONS WITH AQUEOUS-PHASE CHEMICAL REACTIONS

FIGURE 7.20 Aqueous-phase H2O2 as a function of liquid water content for 0.5, 1, and 2 ppb H2O2. Dashed lines correspond to an open system and solid lines to a closed one.

available to accommodate the gas and the aqueous fraction increases, while its aqueous-phase concentration, as we saw above, decreases. For species with low solubility, such as ozone, 1 + HwLRT  1, and the two approaches result in essentially identical concentration estimates (recall that wL  10 6 for most atmospheric clouds). Aqueous-Phase Concentration of Nitrate inside a Cloud (Closed System) A cloud with liquid water content wL forms in an air parcel with HNO3 partial pressure equal to p0HNO3 . Assuming that the air parcel behaves as a closed system, calculate the aqueous-phase concentration of NO3 . Using the ideal-gas law, we obtain ‰HNO3 Štotal ˆ

p0HNO3 RT

(7.108)

The available HNO3 will be distributed between gas and aqueous phases ‰HNO3 Štotal ˆ

pHNO3 ‡ ‰HNO3 …aq†Š ‡ NO3 RT

wL

(7.109)

where Henry’s law gives ‰HNO3 …aq†Š ˆ H HNO3 pHNO3

(7.110)

and dissociation equilibrium is Kn1 ˆ

NO3 ‰H‡ Š ‰HNO3 …aq†Š

(7.111)

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CHEMISTRY OF THE ATMOSPHERIC AQUEOUS PHASE

Solving these equations simultaneously yields H HNO3 p0 1 ‡ wL H ∗HNO3 RT HNO3

(7.112)

Kn1 H HNO3 p0 ‡ ‰H Š 1 ‡ wL H ∗HNO3 RT HNO3

(7.113)

HNO3 …aq† ˆ NO3 ˆ

The last equation can be simplified by using our insights from Section 7.3.5, that is: wL H ∗HNO3 RT  1 for typical clouds and that H HNO3  H HNO3 Kn1 =‰H‡ Š. Using these two simplifications, we find that NO3 ˆ p0HNO3 =…wL RT††. This is the expected result for a highly water soluble strong electrolyte. All the nitric acid will be dissolved in the cloud water and all of it will be present in its dissociated form, NO3 .

7.6.2 Calculation of Concentration Changes in a Droplet with Aqueous-Phase Reactions Let us consider a droplet that at t = 0 is immersed in air containing SO2, NH3, H2O2, O3, and HNO3. Equilibrium is immediately established between the gas and aqueous phases. As the aqueous-phase oxidation of S(IV) to S(VI) proceeds, the concentrations of all the ions adjust so as to satisfy electro­ neutrality at all times ‰H‡ Š ‡ NH‡4 ˆ ‰OH Š ‡ HSO3 ‡ 2 SO23 ‡ 2‰SO24 Š ‡ HSO4 ‡ NO3

(7.114)

where the weak dissociation of H2O2 has been neglected. The concentrations of each ion except sulfate and bisulfate can be expressed in terms of [H+] using the equilibrium constant expressions: NH‡4 ˆ

H NH3 Ka1 pNH3 H‡ Kw

(7.115)

‰OH Š ˆ

Kw ‰H‡ Š

(7.116)

HSO3 ˆ

H SO2 Ks1 p ‰H‡ Š SO2

(7.117)

H SO2 Ks1 Ks2 pSO2 ‰H‡ Š2

(7.118)

H HNO3 Kn1 pHNO3 ‰H‡ Š

(7.119)

‡ HSO4 ‡ ‰H2 SO4 …aq†Š

(7.120)

HSO4 ˆ

K7:123 ‰H‡ Š‰S…VI†Š ‰H Š ‡ K7:123 ‰H‡ Š ‡ K7:123 K7:124

(7.121)

ˆ

K7:123 K7:124 ‰S…VI†Š ‰H Š ‡ K7:123 ‰H‡ Š ‡ K7:123 K7:124

(7.122)

ˆ

SO23

NO3 ˆ If we define ‰S…VI†Š ˆ SO24 then

SO24

‡ 2

‡ 2

299

DYNAMIC BEHAVIOR OF SOLUTIONS WITH AQUEOUS-PHASE CHEMICAL REACTIONS

where K7.123 and K7.124 are the equilibrium constants for the reactions H2 SO4 …aq† ƒ H‡ ‡ HSO4

(7.123)

HSO4 ƒ H‡ ‡ SO42

(7.124)

respectively. For all practical purposes K7.123 may be considered to be infinite since virtually no undissociated sulfuric acid will exist in solution, and K7.124 = 1.02 × 10 2 M at 298 K. Therefore the equations can be simplified to HSO4 ˆ

‰H‡ Š‰S…VI†Š ‰H‡ Š ‡ K7:124

(7.125)

ˆ

K7:124 ‰S…VI†Š ‰H‡ Š ‡ K7:124

(7.126)

SO24

The concentration changes are computed as follows assuming that all compounds in the system remain always in thermodynamic equilibrium. At t = 0, assuming that there is an initial sulfate concentration [S(VI)] = [S(VI)]0, the electroneutrality equation is solved to determine [H+] and, thereby, the concentrations of all dissolved species. If an open system is assumed, then the gas-phase partial pressures will remain constant with time. For a closed system, the partial pressures change with time as outlined above. The concentration changes are computed over small time increments of length Δt. The sulfate present at any time t is equal to that of time t Δt plus that formed in the interval Δt: ‰S…VI†Št ˆ ‰S…VI†Št

Δt

‡

d‰S…VI†Š dt

t Δt

Δt

(7.127)

The sulfate formation rate d[S(VI)]/dt is simply the sum of the reaction rates that produce sulfate in this system, namely, that of the H2O2–S(IV) and O3–S(IV) reactions. The value of S(VI) at time t is then used to calculate HSO4 and SO24 at time t. These concentrations are then substituted into the electro­ neutrality equation to obtain the new [H+] and the concentrations of other dissolved species. This process is then just repeated over the total time of interest. Let us compare the evolution of open and closed systems for an identical set of starting conditions. We choose the following mixing ratios at t = 0: ‰S…IV†Štotal ˆ 5 ppb; ‰HNO3 Štotal ˆ 1 ppb; ‰NH3 Štotal ˆ 5 ppb ‰O3 Štotal

ˆ 5 ppb; ‰H2 O2 Štotal ˆ 1 ppb;

‰S…VI†Š0

ˆ0

wL ˆ 10

6

For the closed system, there is no replenishment, whereas in the open system the partial pressures of all species in the gas phase are maintained constant at their initial values and aqueous-phase concentrations are determined by equilibrium. Solving the equilibrium problem at t = 0 we find that the initial pH = 6.17, and that the initial gasphase mixing ratios are ξSO2 ˆ 3:03 ppb;ξNH3 ˆ 1:87 ppb; ξHNO3 ˆ 8:54  10 ξO3 ˆ 5 ppb;

9

ppb

ξH2 O2 ˆ 0:465 ppb

We will use the initial conditions presented above for both the open and closed systems. In the open system these partial pressures remain constant throughout the simulation, whereas in the closed system they change. A comment concerning the choice of O3 concentration is in order. We selected the uncharacteristically low value of 5 ppb so as to be able to show the interplay possible between both the H2O2 and O3 oxidation rates and in the closed system, the role of depletion as the reaction proceeds. Figures 7.21 and 7.22 show the sulfate concentration and pH, respectively, in the open and closed systems over a 60-min period.

300

CHEMISTRY OF THE ATMOSPHERIC AQUEOUS PHASE

FIGURE 7.21 Aqueous sulfate concentration as a function of time for both open and closed systems. The conditions for the simulation are [S(IV)]total = 5 ppb; [NH3]total = 5 ppb; [HNO3]total = 1 ppb; [O3]total = 5 ppb; [H2O2]total = 1 ppb; wL = 10 6; pH0 = 6.17.

FIGURE 7.22 pH as a function of time for both open and closed systems. Same conditions as in Figure 7.21.

301

THERMODYNAMIC AND KINETIC DATA

More sulfate is produced in the open system than in the closed system, but the pH decrease is less in the open system. This behavior is a result of several factors: 1. Sulfate production due to H2O2 and O3 is less in the closed system because of depletion of H2O2 and O3. 2. Continued replenishment of NH3 provides more neutralization in the open system. 3. The low pH in the closed system drives S(IV) from solution. 4. The lower pH in the closed system depresses the rate of sulfate formation by O3. 5. S(IV) is continually depleted in the closed system. 6. Total HNO3 decreases in the open system due to the decrease in pH. In the open system the importance of H2O2 in sulfate formation grows significantly, from 20% initially to over 80%, whereas in the closed system the increasing importance of H2O2 is enhanced at lower pH but suppressed by the depletion of H2O2. In the closed system, after about 30 min, the H2O2 has been depleted, the pH has decreased to 4.8 slowing down the ozone reaction, and sulfate formation ceases. The fractions of SO2, O3, and H2O2 reacted in the closed system at 60 min are 0.43, 0.23, and 0.997, respectively. The ozone reaction produces 44.4% and 53.9% of the sulfate in the open and closed systems, respectively.

APPENDIX 7.1

THERMODYNAMIC AND KINETIC DATA

Tables 7A.1–7A.7 present selected thermodynamic and kinetic data on aqueous-phase atmospheric chemis­ try. For a comprehensive compilation of aqueous-phase chemistry we refer the reader to the CAPRAM (Chemical Aqueous Phase RAdical Mechanism) website: projects.tropos.de/capram/capram_intro.html. TABLE 7A.1 Equilibrium Reactions Equilibrium reaction

K298(M or M atm 1)a

ΔH/R (K)

1.3 × 10 2 6.6 × 10 8 1000 1.02 × 10 2 2.2 × 10 12 15.4 5.1 × 10 4 4.3 × 10 7 4.68 × 10 11 1.7 × 10 5 1.0 × 10 14

1960 1500

SO2 ? H2 O ƒ HSO3 ‡ H‡ HSO3 ƒ SO23 ‡ H‡ H2 SO4 …aq† ƒ HSO4 ‡ H‡ HSO4 ƒ SO24 ‡ H‡ H2 O2 …aq† ƒ HO2 ‡ H‡ HNO3 …aq† ƒ NO3 ‡ H‡ HNO2 …aq† ƒ NO2 ‡ H‡ CO2 ? H2 O ƒ HCO3 ‡ H‡ HCO3 ƒ CO23 ‡ H‡ NH4 OH ƒ NH‡4 ‡ OH H2O ƒ H+ + OH

3

H O

2 H2 C…OH†2 …aq† HCHO…aq† „ HCOOH(aq) ƒ HCOO + H+ HCl(aq) ƒ H+ + Cl Cl2 ƒ Cl ‡ Cl NO3(g) ƒ NO3(aq) HO2 …aq† ƒ H‡ ‡ O2 HOCH2 SO3 ƒ OCH2 SO3 ‡ H‡

a

2720 3730 8700b 1260 1000 1760 450 6710

2.53 × 10 1.8 × 10 4 1.74 × 106 5.26 × 10 6 2.1 × 105 3.50 × 10 5 2.00 × 10 12

4020 20 6900 8700

The temperature dependence is represented by

K ˆ K298 exp where K is the equilibrium constant at temperature T (in K). b Value for equilibrium: HNO3 …g† ƒ NO3 ‡ H‡ .

ΔH 1 R T

1 298

Reference Smith and Martell (1976) Smith and Martell (1976) Perrin (1982) Smith and Martell (1976) Smith and Martell (1976) Schwartz (1984) Schwartz and White (1981) Smith and Martell (1976) Smith and Martell (1976) Smith and Martell (1976) Smith and Martell (1976) Le Hanaf (1968) Martell and Smith (1977) Marsh and McElroy (1985) Jayson et al. (1973) Jacob (1986) Perrin (1982) Sorensen and Andersen (1970)

302

CHEMISTRY OF THE ATMOSPHERIC AQUEOUS PHASE

TABLE 7A.2 Oxygen − Hydrogen Chemistry k298a or j (s–1)

Reaction 1.

hν

H2 O2

! 2 OH

7.2 × 10

6

! H2 O2 ‡ O2

O3

3. 4. 5. 6.

OH + HO2 → H2O + O2 OH ‡ O2 ! OH ‡ O2 OH + H2O2 → H2O + HO2 HO2 + HO2 → H2O2 + O2

7.

HO2 ‡ O2

Graedel and Weschler (1981)

! H2 O2 ‡ O2 ‡ OH

2H2 O

! H2 O2 ‡ O2 ‡ 2OH

9. 10. 11. 12.

HO2 + H2O2 → OH + O2 + H2O O2 ‡ H2 O2 ! OH ‡ O2 ‡ OH OH + O3 → HO2 + O2 HO2 + O3 → OH + 2O2

13.

O2 ‡ O3

H2 O

! OH ‡ 2O2 ‡ OH

H2 O

14.

OH ‡ O3

15. 16.

HO2 ‡ O3 ! OH ‡ O2 ‡ O2 H2O2 + O3 → H2O + 2O2

1.0 × 10 1.1 × 1010 2.7 × 107 8.6 × 105

2120 1700 2365

Elliott and Buxton (1992) Christensen et al. (1989) Christensen et al. (1982) Bielski (1978)

1.0 × 108

1500

Bielski (1978)

10

H2 O

O2 ‡ O2

Reference Zellner et al. (1990)

hν;H2 O

2.

8.

E/R (K)

! H2 O2 ‡ O2 ‡ OH

0.1 μm, it cannot provide a reasonable approximation of the typical volume distributions (Figure 8.9). Atmospheric aerosol volume distributions always have multiple modes for Dp > 0.1 μm, and the power-law volume distribution that we calculated above is also a monotonic function (continuously decreasing for α > 3 and increasing for α < 3). For the distribution of Figure 8.9, the power-law distribution grossly overpredicts the volume of the submicrometer particles and seriously underpredicts the volume of the coarse aerosol. Therefore use of a fitted power-law distribution for the calculation of aerosol properties that depend on powers of the diameter (e.g., optical properties, condensation rates) should be avoided.

342

PROPERTIES OF THE ATMOSPHERIC AEROSOL

8.2 AMBIENT AEROSOL SIZE DISTRIBUTIONS As a result of particle emission, in situ formation, and the variety of atmospheric processes, the atmospheric aerosol distribution is characterized by a number of modes. The volume or mass distribution is dominated in most areas by two modes (Figure 8.10, lower panel): the accumulation (from ∼0.1 to ∼2 μm) and the coarse mode (from ∼2 to ∼50 μm). Accumulation-mode particles are the result of primary emissions; condensation of secondary sulfates, nitrates, and organics from the gas phase; and coagulation of smaller particles. Processing of accumulation- and coarse-mode aerosols by activation to form clouds (Chapter 17) can also modify the concentration and composition of these modes. Aqueous-phase chemical reactions take place in cloud and fog droplets, and in aerosol particles at relative humidities approaching 100%. These reactions can lead to production of nonvolatile sulfate (Chapter 7) and after evaporation of water, a larger aerosol particle is left in the atmosphere. This transformation can lead to the formation of the condensation mode and the droplet mode (Figure 8.10, lower panel) (John et al. 1990; Meng and Seinfeld 1994). Particles in the coarse mode generally result from mechanical processes such as wind or erosion (dust, seasalt, pollens, etc.). Terms used to describe the aerosol mass concentration include total suspended particulate matter (TSP) and PMx (particulate matter with diameter smaller than x μm). TSP refers to the mass concentration of atmospheric particles smaller than 40–50 μm, while PM2.5 and PM10 are routinely monitored. A different picture of the ambient aerosol distribution is obtained if one focuses on the number of particles instead of their mass (Figure 8.10, upper panel). Particles with diameters exceeding 0.1 μm, which contribute practically all the aerosol mass, are negligible in number compared to the particles smaller than 0.1 μm. Two modes usually dominate the aerosol number distribution in urban and rural areas: the nucleation mode (particles smaller than ∼20 nm) and the Aitken mode (particles with diameters between ∼20 and ∼100 nm). Nucleation mode particles are usually fresh aerosols created in situ from the gas phase by nucleation. The nucleation mode may or may not be present depending on the atmospheric conditions. Most of the Aitken nuclei start their atmospheric lives as primary particles,

FIGURE 8.10 Typical number and volume distributions of atmospheric particles with the different modes.

343

AMBIENT AEROSOL SIZE DISTRIBUTIONS

and secondary material condenses on them as they are transported through the atmosphere. The nucleation-mode particles have negligible mass (e.g., 100,000 particles cm 3 with a diameter equal to 10 nm have a mass concentration of 10 nm. Particles with diameters larger than 2.5 μm are identified as coarse particles, while those with diameters less than 2.5 μm are called fine particles. The fine particles include most of the total number of particles and a large fraction of the mass. The fine particles with diameters smaller than 0.1 μm are often called ultrafine particles. Coarse particles are generated by mechanical processes and consist of soil dust, sea salt, fly ash, tire wear particles, and so on. Aitken and accumulation mode particles contain primary particles from combustion sources and secondary aerosol material (sulfate, nitrate, ammonium, secondary organics) formed by chemical reactions resulting in gas-to-particle conversion (see Chapters 10 and 14). The main mechanisms of transfer of particles from the Aitken to accumulation mode are coagulation (Chapter 13) and growth by condensation of vapors formed by chemical reactions (Chapters 12 and 14) onto existing particles. Coagulation among accumulation mode particles is a slow process and does not effectively transfer particles to the coarse-particle mode. Atmospheric aerosol size distributions are often described as the sum of n lognormal distributions n

n N …log Dp † ˆ iˆ1

Ni …2π†1=2 log σi

exp

log Dp

log Dpi

2

(8.54)

2 log2 σi

where Ni is the number concentration, Dpi is the median diameter, and σi is the standard deviation of the ith lognormal mode. In this case 3n parameters are necessary for the description of the full aerosol distribution. Some parameters of fine-mode model aerosol distributions are presented in Table 8.3. In the remainder of this section we will present generic aerosol distributions characteristic of different regions of the globe. We do this mainly to emphasize the ranges of sources and atmospheric processes that affect airborne particles. In reality, owing to long-range transport of aerosols and a lifetime of roughly a week, particles emitted into or formed in one region of the atmosphere tend to be carried to other regions, so classifications like rural and remote aerosols are somewhat idealized.

8.2.1 Urban Aerosols Urban aerosols are mixtures of primary particulate emissions from industries, transportation, power generation, and natural sources and secondary material formed by gas-to-particle conversion mecha­ nisms. The number distribution is dominated by particles smaller than 0.1 μm, while most of the surface

TABLE 8.3 Parameters for Model Aerosol Distributions Expressed as the Sum of Three Lognormal Modes Mode I Type Urban Marine Rural Remote continental Free troposphere Polar Desert

Mode II

Mode III

N (cm 3)

Dpg (μm)

log σ

N (cm 3)

Dpg (μm)

log σ

N (cm 3)

Dpg (μm)

log σ

7100 133 6650 3200 129 21.7 726

0.0117 0.008 0.015 0.02 0.007 0.138 0.002

0.232 0.657 0.225 0.161 0.645 0.245 0.247

6320 66.6 147 2900 59.7 0.186 114

0.0373 0.266 0.054 0.116 0.250 0.75 0.038

0.250 0.210 0.557 0.217 0.253 0.300 0.770

960 3.1 1990 0.3 63.5 3 × 10 4 0.178

0.151 0.58 0.084 1.8 0.52 8.6 21.6

0.204 0.396 0.266 0.380 0.425 0.291 0.438

Source: Urban distribution (Helsinki, Finland 1997 2003) are from Hussein et al. (2004); the other distributions from Jaenicke (1993).

344

PROPERTIES OF THE ATMOSPHERIC AEROSOL

FIGURE 8.11 Typical urban aerosol number, surface, and volume distributions.

area is in the 0.1–0.5 μm size range. On the contrary, the aerosol mass distribution usually has two distinct modes, one in the submicrometer regime (referred to as the accumulation mode) and the other in the coarse-particle regime (Figure 8.11). The aerosol size distribution is quite variable in an urban area. High concentrations of ultrafine particles ( 300 nm) SSA particles. The ambient marine aerosol number distribution is more complex than the fresh SSA distribution and is usually characterized by multiple modes (Figure 8.15). The size distribution is affected by cloud processing causing the separation of the two modes and the characteristic minimum (see Chapter 17). Also there is often significant entrainment of particles from the free troposphere that can be responsible for most of the smaller particles and can represent the dominant source of particle number in the remote marine boundary layer (Raes 1995).

FIGURE 8.14 Normalized seaspray aerosol size distribution measured over two sampling regions in an ocean water wave channel. (Source: Prather, K.A., et al. (2013), Proc. Natl. Acad. Sci. USA, 110, 7550 7555, Figure S3 (B). Reprinted by permission.)

AMBIENT AEROSOL SIZE DISTRIBUTIONS

FIGURE 8.15 Measured marine aerosol number distributions and a model distribution used to represent average conditions.

8.2.3 Rural Continental Aerosols Aerosols in rural areas are mainly of natural origin but with a moderate influence of anthropogenic sources. The number distribution is characterized by two modes at diameters of approximately 0.02 and 0.08 μm, respectively (Jaenicke 1993), while the mass distribution is dominated by the coarse mode centered at around 7 μm (Figure 8.16). The mass distribution of continental aerosol not influenced by

FIGURE 8.16 Typical rural continental aerosol number, surface, and volume distributions.

347

348

PROPERTIES OF THE ATMOSPHERIC AEROSOL

local sources has a small accumulation mode and no nuclei mode. The PM10 concentration of rural aerosols is around 20 μg m 3.

8.2.4 Remote Continental Aerosols Primary particles (e.g., dust, pollens, plant waxes) and secondary oxidation products are the main components of remote continental aerosol (Deepak and Gali 1991). Aerosol number concentrations average around 1000–10,000 cm 3, and PM10 concentrations are around 10 μg m 3 (Bashurova et al. 1992; Koutsenogii et al. 1993; Koutsenogii and Jaenicke 1994). For the continental United States PM10 concentrations in remote areas vary from 5 to 25 μg m 3 and PM2.5 from 3 to 17 μg m 3 (USEPA 1996). Particles smaller than 2.5 μm in diameter represent 40–80% of the PM10 mass and consist mainly of sulfate, ammonium, and organics. The aerosol number distribution may be characterized by three modes at diameters 0.02, 0.1, and 2 μm (Jaenicke 1993) (Figure 8.17). Much interest has focused on aerosols in the Amazon (Martin et al. 2010).

8.2.5 Free Tropospheric Aerosols Background free tropospheric aerosol is found in the mid/upper troposphere above cloud level. The modes in the number distribution correspond to mean diameters of 0.01 and 0.25 (Jaenicke 1993) (Figure 8.18). The middle troposphere spectra typically indicate more particles in the accumulation mode relative to lower tropospheric spectra, suggesting precipitation scavenging and deposition of smaller and larger particles (Leaitch and Isaac 1991). The low temperature and low aerosol surface area render the upper troposphere suitable for new particle formation, and a nucleation mode is often present in the number distribution.

FIGURE 8.17 Typical remote continental aerosol number, surface, and volume distributions.

AMBIENT AEROSOL SIZE DISTRIBUTIONS

FIGURE 8.18 Typical free tropospheric aerosol number, surface, and volume distributions.

8.2.6 Polar Aerosols The northern polar region is characterized by a prevailing “Arctic haze”, which was suggested to originate from anthropogenic emissions in northern midlatitudes (Rahn et al. 1977; Barrie 1986). Aerosol abundances increase in winter and early spring (February to April) each year. The aerosols comprise nonseasalt (nss) sulfate and seasalt with ammonium, nitrate, dust, and organic matter. A comprehensive 2008 measurement campaign in the Arctic, the Polar Study using Aircraft, Remote Sensing, Surface Measurements and Models, Climate, Chemistry, Aerosols and Transport (POLARCAT) (Law et al. 2014), documented the strong influence of Eurasian fire emissions, especially from springtime agricul­ tural fires. The main aerosol source near the Arctic surface was determined to be northern Eurasia, while organics and sulfate from longer range transport dominate the free tropospheric aerosol. Aerosol size distribution data in the Arctic have been presented by a number of investigators (Rahn 1981; Shaw 1985; Heintzenberg 1989; Ottar 1989). The number distribution appears practically monodisperse (Ito and Iwai 1981) with a mean diameter of approximately 0.15 μm; two additional modes at 0.75 and 8 μm (Shaw 1986; Jaenicke et al. 1992) dominate the mass distribution (Figure 8.19). During the spring period the aerosol number concentration increases to >200 cm 3. The nucleation mode mean diameter is at ∼0.05 μm and the accumulation mode, at ∼0.2 μm (Covert and Heintzenberg 1993) (Figure 8.20). Similar measurements have been reported by Heintzenberg (1980), Radke et al. (1984), and Shaw (1984). Aerosol PM10 concentrations in the polar regions are 0 and at all times the concentration of particles right at the surface is zero as a result of their removal at the surface. 1. What is the particle concentration as a function of height and time, N(z, t)? 2. What is the removal rate of particles at the surface? The concentration distribution of aerosol particles in a stagnant fluid in which the particles are subject to Brownian motion and in which there is a velocity vt in the z direction is described by @N @t

vt

@N @2N ˆD 2 @z @z

(9.80)

subject to the conditions N…z; 0† ˆ N 0 N…0; t† ˆ 0 N…z; t† ˆ N 0

(9.81) z!1

where the z coordinate is taken as vertically upward. The solution of (9.80) and (9.81) for the vertical profile of the number distribution N(z, t) is N…z; t† ˆ

N0 z ‡ vt t 1 ‡ erf p 2 2 Dt

vt z z vt t erfc p D 2 Dt

exp

(9.82)

We can calculate the deposition rate of particles on the z = 0 surface from the expression for the flux of particles at z = 0, JˆD

@N @z

‡ vt N…0; t†

(9.83)

zˆ0

Recall that N(0, t) = 0 in (9.83). Combining (9.82) and (9.83) we obtain J ˆ N0

vt 1 2

erf

vt pt 2 Dt

‡

D πt

1=2

exp

v2t t 4D

(9.84)

According to (9.84), there is an infinite removal flux at t = 0, because of our artificial specification of an infinite concentration gradient at z = t = 0. We can identify a characteristic time τds for the system τds ˆ

4D v2t

(9.85)

and observe the following limiting behavior for the particle flux at short and long times: J…t† ˆ N 0 J…t† ˆ N 0 vt

D πt

1=2

‡

vt 2

t  τds t  τds

(9.86)

BROWNIAN MOTION OF AEROSOL PARTICLES

FIGURE 9.9 Evolution of the concentration profile (times 1, 10, 100, and 400 s) of an aerosol population settling and diffusing in stationary air over a flat perfectly absorbing surface. The particles are assumed to be monodisperse with Dp = 0.2 μm and have an initial concentration of 1000 cm 3.

FIGURE 9.10 Removal rate of particles as a function of time for the conditions of Figure 9.9.

Thus, at very short times, the deposition flux is that resulting from diffusion plus one-half that due to settling, whereas for long times the deposition flux becomes solely the settling flux. For particles of radii 0.1 μm and 1 μm in air (at 1 atm, 298 K), τds is about 80 s and 0.008 s, respectively, assuming a density of 1 g cm 3. For times longer than that, Brownian motion does not have any effect on the particle motion. The aerosol number concentration and removal flux are shown in Figures 9.9 and 9.10. The system reaches a steady state after roughly 100 s and at this state 0.23 particles are deposited per second on each cm2 of the surface (Figure 9.10). Note that the concentration profile changes only over a shallow layer of approximately 1 mm above the surface (Figure 9.9). The depth of this layer is proportional to D/vt. Note that the example above is not representative of the ambient atmosphere, where there is turbulence and possibly also sources of particles at the ground.

383

384

DYNAMICS OF SINGLE AEROSOL PARTICLES

FIGURE 9.11 A two-dimensional projection of the path of (a) an air molecule and (b) the center of a 1-μm particle. Also shown is the apparent mean free path of the particle.

9.5.3 Mean Free Path of an Aerosol Particle The concept of mean free path is an obvious one for gas molecules. In the Brownian motion of an aerosol particle there is not an obvious length that can be identified as a mean free path. This is depicted in Figure 9.11 showing plane projections of the paths followed by an air molecule and an aerosol particle of radius roughly equal to 1 μm. The trajectories of the gas molecules consist of straight segments, each of which represents the path of the molecule between collisions. At each collision the direction and speed of the molecule are changed abruptly. With aerosol particles the mass of the particle is so much greater than that of the gas molecules with which it collides that the velocity of the particle changes negligibly in a single collision. Appreciable changes in speed and direction occur only after a large number of collisions with molecules, resulting in an almost smooth particle trajectory. The particle motion can be characterized by a mean thermal speed cp : cp ˆ

8kT πmp

1=2

(9.87)

To obtain the mean free path λp, we recall that in Section 9.1, using kinetic theory, we connected the mean free path of a gas to measured macroscopic transport properties of the gas such as its binary diffusivity. A similar procedure can be used to obtain a particle mean free path λp from the Brownian diffusion coefficient and an appropriate kinetic theory expression for the diffusion flux. Following an argument identical to that in Section 9.1, diffusion of aerosol particles can be viewed as a mean free path phenomenon so that 1 D ˆ c p λp 2

(9.88)

and the mean free path λp combining (9.73), (9.87), and (9.88) is then

λp ˆ

Cc 6μ

ρkTDp 3

(9.89)

Certain quantities associated with the Brownian motion and the dynamics of single aerosol particles are shown as a function of particle size in Table 9.5. All tabulated quantities in Table 9.5 depend strongly on

385

AEROSOL AND FLUID MOTION

TABLE 9.5 Characteristic Quantities in Aerosol Brownian Motion D, (μm)

D (cm2 s 1)

0.002 0.004 0.01 0.02 0.04 0.1 0.2 0.4 1.0 2.0 4.0 10.0 20.0

1.28 × 10 3.23 × 10 3 5.24 × 10 4 1.30 × 10 4 3.59 × 10 5 6.82 × 10 6 2.21 × 10 6 8.32 × 10 7 2.74 × 10 7 1.27 × 10 7 6.1 × 10 8 2.38 × 10 8 1.38 × 10 8

τ (s)

cp (cm s 1)

2

4965 1760 444 157 55.5 14.0 4.96 1.76 0.444 0.157 5.55 × 10 1.40 × 10 4.96 × 10

1.33 × 10 2.67 × 10 6.76 × 10 1.40 × 10 2.98 × 10 9.20 × 10 2.28 × 10 6.87 × 10 3.60 × 10 1.31 × 10 5.03 × 10 3.14 × 10 1.23 × 10

2 2 3

λp (μm) 9 9 9 8 8 8 7 7 6 5 5 4 3

6.59 × 10 2 4.68 × 10 2 3.00 × 10 2 2.20 × 10 2 1.64 × 10 2 1.24 × 10 2 1.13 × 10 2 1.21 × 10 2 1.53 × 10 2 2.06 × 10 2 2.8 × 10 2 4.32 × 10 2 6.08 × 10 2

particle size with the exception of the apparent mean free path λp, which is of the same order of magnitude right down to molecular sizes, with atmospheric values λp  10–60 nm.

9.6 AEROSOL AND FLUID MOTION In our discussion so far, we have assumed that the aerosol particles are suspended in a stagnant fluid. In most atmospheric applications, the air is in motion and one needs to describe simultaneously the air and particle motion. Equation (9.36) will be the starting point of our analysis. Actually (9.36) is a simplified form of the full equation of motion, which is (Hinze 1959) mp

dv 3πμDp ˆ …u dt Cc ‡

3D2p 2

v† ‡ V p ρ

…πρμ†

1=2

t

∫0

du du ‡ Vp ρ dt dt

…du=dt´ † …t

…dv=dt´ † t´ †1=2

dv dt (9.90) ´

dt ‡

Fei i

where Vp is the particle volume. The second term on the RHS is due to the pressure gradient in the fluid surrounding the particle, caused by acceleration of the gas by the particle. The third term is the force required to accelerate the apparent mass of the particle relative to the fluid. Finally, the fourth term, the Basset history integral, accounts for the force arising as a result of the deviation of fluid velocity from steady state. In most situations of interest for aerosol particle in air, the second, third, and fourth terms on the RHS of (9.90) are neglected. Assuming that gravity is the only external force exerted on the particle, we again obtain (9.36). Neglecting the gravitational force and particle inertia leads to the zero-order approximation that v  u; that is, the particle follows the streamlines of the airflow. This approximation is often sufficient for most atmospheric applications, such as turbulent dispersion. However, it is often necessary to quantify the deviation of the particle trajectories from the fluid streamlines (Figure 9.12). A detailed treatment of particle flow around objects, in channels of various geometries, and so on, is beyond the scope of this book. Treatments are provided by Fuchs (1964), Hinds (1999), and Flagan and Seinfeld (1988). We will focus our analysis on a few simple examples demonstrating the important concepts.

386

DYNAMICS OF SINGLE AEROSOL PARTICLES

FIGURE 9.12 Schematic diagram of particles and fluid motion around a cylinder. Streamlines are shown as solid lines, while the dashed lines depict aerosol paths.

9.6.1 Motion of a Particle in an Idealized Flow (90° Corner) Let us consider an idealized flow, shown in Figure 9.13, in which an airflow makes an abrupt 90° turn in a corner maintaining the same velocity (Crawford 1976). We would like to determine the trajectory of an aerosol particle originally on the streamline y = 0, which turns at the origin x = 0. The trajectory of the particle is governed by (9.37). Neglecting gravity, the x and y components of the equation of motion are after the turning point dvx ‡ vx ˆ 0 dt dvy ‡ vy ˆ U τ dt

(9.91)

d2 x dx ‡ ˆ0 dt2 dt d2 y dy τ 2‡ ˆU dt dt

(9.92)

τ

and as vx = dx/dt and vy = dy/dt, we get τ

FIGURE 9.13 Motion of an air particle in a flow making a 90° turn with no change in velocity.

387

AEROSOL AND FLUID MOTION

FIGURE 9.14 Idealized flow toward a plate.

subject to x…0† ˆ 0;

dx dt

y…0† ˆ 0;

ˆ U; tˆ0

dy dt

ˆ0

(9.93)

tˆ0

Solving (9.92) subject to (9.93) gives the particle coordinates as a function of time: x…t† ˆ Uτ‰1 y…t† ˆ Uτ‰1

exp… t=τ†Š

(9.94)

exp… t=τ†Š ‡ Ut

(9.95)

We see that for t  τ, the particle trajectory is described by x(t) = Uτ and y(t) = Ut. Thus the particle eventually ends up at a distance Uτ to the right of its original fluid streamline. Larger particles with high relaxation times will move, because of their inertia, significantly to the right, while small particles, with τ → 0, will follow closely their original streamline. For example, for a 2 μm diameter particle moving with a speed U = 20 m s 1 and having density ρp = 2 g cm 3, we find that the displacement is 0.48 mm, while for a 20 μm diameter particle Uτ = 4.83 cm. The flow depicted in Figure 9.13 is the most idealized one representing the stagnation flow of a fluid toward a flat plane (see also Figure 9.14). If we imagine that in Figure 9.14 there is a flat plate at x = 0 and that y = 0 is the line of symmetry, then all the particles that initially are a distance smaller than x0 = Uτ from the line of symmetry will collide with the flat plate. On the other hand, particles outside this area will be able to turn and, avoiding the plane, continue flowing parallel to it. A more detailed treatment of the flow in such situations is presented by Flagan and Seinfeld (1988) considering more realistic fluid streamlines.

9.6.2 Stop Distance and Stokes Number Let us consider a particle moving with a speed U in a stagnant fluid with no forces acting on it. The particle will slow down because of the drag force exerted on it by the fluid and eventually stop after moving a distance sp. We can calculate this stop distance of the particle employing (9.39), neglecting gravity, and noting that v = ds/dt. The motion of the particle is described by τ

ds2 ds ‡ ˆ 0; dt2 dt

s…0† ˆ 0;

ds dt

ˆU

(9.96)

tˆ0

with solution s…t† ˆ τU 1

exp

t τ

(9.97)

388

DYNAMICS OF SINGLE AEROSOL PARTICLES

Note that as t  τ, s(t) → sp, where sp ˆ τU

(9.98)

is the particle stop distance. For a 1 μm diameter particle, with an initial speed of 10 m s 1, for example, the stop distance is 36 μm. Let us write the equation of motion (9.37) without the gravitational term, in dimensionless form. To do so we introduce a characteristic fluid velocity u0 and a characteristic length L both associated with the flow of interest. We define dimensionless time t∗, distance x∗, and velocity u∗x by t∗ ˆ

tu0 ; L

x∗ ˆ

x ; L

ux∗ ˆ

ux u0

(9.99)

Placing (9.37) in dimensionless form gives τu0 d2 x∗ dx∗ ‡ ˆ u∗x L dt∗2 dt∗

(9.100)

Note that all the variables of the system have been combined in the dimensionless group τu0/L, which is called the Stokes number (St):

St ˆ

2 τu0 Dp ρp Cc u0 ˆ L 18μL

(9.101)

Note that the Stokes number is the ratio of the particle stop distance sp to the characteristic length of the flow L. As particle mass decreases, the Stokes number also decreases. A small Stokes number implies that the particle is able to adopt the fluid velocity very quickly. Since the dimensionless equation of motion depends only on the Stokes number, equality between two geometrically similar flows indicates similarity of the particle trajectories.

9.7 EQUIVALENT PARTICLE DIAMETERS Up to this point we have considered spherical particles of a known diameter Dp and density ρp. Atmospheric particles are sometimes nonspherical and we seldom have information about their density. Also a number of techniques used for atmospheric aerosol size measurement actually measure the particle’s terminal velocity or its electrical mobility. In these cases we need to define an equivalent diameter for the nonspherical particles or even for the spherical particles of unknown density or charge. These equivalent diameters are defined as the diameter of a sphere, which, for a given instrument, would yield the same size measurement as the particle under consideration. A series of diameters have been defined and are used for such particles.

9.7.1 Volume Equivalent Diameter The volume equivalent diameter Dve is the diameter of a sphere having the same volume as the given nonspherical particle. If the volume Vp of the nonspherical particle is known, then

389

EQUIVALENT PARTICLE DIAMETERS

Dve ˆ

6 Vp π

1=3

(9.102)

For a spherical particle the volume equivalent diameter is equal to its physical diameter, Dve = Dp. To account for the shape effects during the flow of nonspherical particles, Fuchs (1964) defined the shape factor χ as the ratio of the actual drag force on the particle FD to the drag force Fve D on a sphere with diameter equal to the volume equivalent diameter of the particle: χˆ

FD Fve D

(9.103)

The dynamic shape factor is almost always greater than 1.0 for irregular particles and flows at small Reynolds numbers and is equal to 1.0 for spheres. For a nonspherical particle of a given shape χ is not a constant but changes with pressure, particle size, and as a result of particle orientation in electric or aerodynamic flow fields. The dynamic shape factor for flow in the continuum regime is equal to 1.08 for a cube, 1.12 for a 2-sphere cluster, 1.15 for a compact 3-sphere cluster, and 1.17 for a compact 4-sphere cluster (Hinds 1999). These values are averaged over all orientations of the particle, which is the usual situation for atmospheric aerosol flows (Re < 0.1) because of the Brownian motion of the particles. Liquid atmospheric particles are spherical for all practical purposes. Dry NaCl crystals have cubic shape (Figure 9.15), while dry (NH4)2SO4 is approximately but not exactly spherical (Figure 9.16). Dynamic

FIGURE 9.15 A micrograph of a single NaCl particle with electrical mobility equivalent diameter of 550 nm. The dry NaCl is almost cubic with rounded edges. Also shown is a polystyrene latex (PSL) particle with diameter of 491 nm (Zelenyuk et al. 2006).

FIGURE 9.16 Ammonium sulfate particles with electrical mobility diameters of 200 and 322 nm (Zelenyuk et al. 2006).

390

DYNAMICS OF SINGLE AEROSOL PARTICLES

shape factors ranging from 1.03 to 1.07 have been measured in the laboratory (Zelenyuk et al. 2006) with the higher values observed for larger particles with diameters of ∼500 nm. Nonspherical particles are subjected to a larger drag force compared to their volume equivalent spheres because χ > 1 and therefore settle more slowly. The terminal settling velocity of a nonspherical particle is then [following the same approach as in the derivation of (9.42)] vt ˆ

2 1 Dve ρp gCc …Dve † 18 χμ

(9.104)

Volume Equivalent Diameter An approximately cubic NaCl particle with density 2.2 g cm 3 has a terminal settling velocity of 1 mm s 1 in air at ambient conditions. Calculate its volume equivalent diameter and its physical size using the continuum regime shape factor. The terminal settling velocity is given by (9.104), so after some rearrangement; we obtain D2ve Cc …Dve † ˆ …18 vt μχ†=…ρp g† and substituting vt = 10 NaCl we find that

3

m s 1, μ = 1.8 × 10

5

(9.105)

s 1, χ = 1.08 for a cube, ρp = 2200 kg m

kg m

1

D2ve Cc …Dve † ˆ 1:62  10

11

3

for

m2 ˆ 0:162 μm2

This equation needs to be solved numerically using (9.34) for calculation of the slip correction, Cc(Dve), to obtain Dve = 0.328 μm. One can also estimate the value iteratively assuming as a first guess that Cc = 1 and then Dve = 0.402 μm. This suggests that for this particle Dve  2λ and the exponential term in (9.34) will be much smaller than the 1.257 term. With this simplification we are left with the following quadratic equation for Dve D2ve ‡ 0:163 Dve

0:162 ˆ 0

with Dve = 0.329 μm as the positive solution. Our simplification of the slip correction expression resulted in an error of only 1 nm. To calculate the physical size L of the particle, we only need to equate the volume of the cube to the volume of the sphere: L3 ˆ …π=6†D3ve and L ˆ 0:264 μm This calculation requires knowledge about the shape of the particle and its density.

9.7.2 Stokes Diameter The diameter of a sphere having the same terminal settling velocity and density as the particle is defined as its Stokes diameter DSt. For irregularly shaped particles, DSt is the diameter of a sphere that would have the same terminal velocity. The Stokes diameter for Re < 0.1 can then be calculated using (9.42) as

DSt ˆ

18vt μ ρp gCc …DSt †

1=2

(9.106)

where vt is the terminal velocity of the particle, μ is viscosity of air, and ρp is the density of the particle that needs to be known. Evaluation of (9.106) because of the dependence of Cc on DSt requires in general the

391

EQUIVALENT PARTICLE DIAMETERS

solution of a nonlinear algebraic equation with one unknown. For spherical particles by its definition DSt = Dp and the Stokes diameter is equal to the physical diameter. Stokes Diameter What is the relationship connecting the volume equivalent diameter and the Stokes diameter of a nonspherical particle with dynamic shape factor χ for Re < 0.1? Let us calculate the Stokes diameter of the NaCl particle of the previous example. The two approaches (dynamic shape factor combined with the volume equivalent diameter and the Stokes diameter) are different ways to describe the drag force and terminal settling velocity of a nonspherical particle. The terminal velocity of a nonspherical particle with a volume equivalent diameter Dve is given by (9.104): 2 1 Dve ρp gCc …Dve † vt ˆ 8 χμ By definition this settling velocity can be also written as a function of its Stokes diameter as 1 DSt ρp gCc …DSt † 18 μ 2

vt ˆ

Combining these two and simplifying we find that

DSt ˆ Dve

Cc …Dve † χCc …DSt †

1=2

(9.107)

The Stokes diameter of the NaCl particle can be calculated from (9.106): D2St Cc …DSt † ˆ 0:150 μm2 and solving numerically DSt = 0.313 μm. This value corresponds to the volume equivalent diameter that we would calculate based on the terminal velocity of the particle if we did not know that the particle was nonspherical. For most atmospheric aerosol measurements the density of the particles is not known and the Stokes diameter cannot be calculated from the measured particle terminal velocity. This makes the use of the Stokes diameter more difficult and requires the introduction of other equivalent diameters that do not require knowledge of the particle density.

9.7.3 Classical Aerodynamic Diameter The diameter of a unit density sphere, ρ°p ˆ 1 g cm 3 , having the same terminal velocity as the particle is defined as its classical aerodynamic diameter, Dca. The classical aerodynamic diameter is then given by

Dca ˆ

18vt μ ρp° gCc …Dca †

1=2

(9.108)

Dividing (9.108) by (9.106), one can then find the relationship between the classical aerodynamic and the Stokes diameter as Dca ˆ DSt …ρp =ρp°†1=2 ‰Cc …DSt †=Cc …Dca †Š1=2

(9.109)

392

DYNAMICS OF SINGLE AEROSOL PARTICLES

For spherical particles we replace DSt with Dp in this equation to find that

Dca ˆ Dp

ρp ρp°

1=2

Cc …Dp † Cc …Dca †

1=2

(9.110)

For a spherical particle of nonunit density the classical aerodynamic diameter is different from its physical diameter and it depends on its density. Spherical particles with density exceeding 1 gcm 3 have a larger aerodynamic than physical diameter. The opposite is true for particles with density less than 1 g cm 3. Aerosol instruments like the cascade impactor and aerodynamic particle sizer measure the classical aerodynamic diameter of atmospheric particles, which is in general different from the physical diameter of the particles even if they are spherical. Aerodynamic Diameter Calculate the aerodynamic diameter of spherical particles of diameters equal to 0.01, 0.1, and 1 μm. Assume that their density is 1.5 g cm 3, which is a typical average density for multicomponent atmospheric particles. Using (9.110), we obtain D2ca Cc …Dca † ˆ 1:5 D2p Cc …Dp † for Dp = 0.01 μm, Cc = 22.2, and D2ca Cc …Dca † ˆ 3:33  10 3 μm2 with solution Dca = 0.015 μm. Repeating the same calculation we find that for Dp = 0.1 μm, Cc = 2.85, and D2ca Cc …Dca † ˆ 0:043 μm2 , resulting in Dca = 0.135 μm. Finally, for Dp = 1 μm, we find that Dca = 1.242 μm. For typical atmospheric particles, the aerodynamic and physical diameters are quite different (more than 20% in this case) with the discrepancy increasing for smaller particles. Vacuum Aerodynamic Diameter Calculate the ratio of the aerodynamic to the physical diameter of a spherical particle of density ρp in the continuum and the free molecular regimes. Using equation (9.110) yields Dca =Dp ˆ …ρp =ρ°p†1=2 ‰Cc …Dp †=Cc …Dca †Š1=2

(9.111)

For conditions in the continuum regime the slip correction factor is practically unity and Dca =Dp ˆ …ρp =ρp°†1=2

(9.112)

If the particle is in the free molecular regime, then 2λ  Dp and the second term dominates the RHS of (9.34), which simplifies to Cc …Dp † ˆ 2:514 λ=Dp Combining this and simplifying, we find that in the free molecular regime: Dca =Dp ˆ ρp =ρp°

(9.113)

The aerodynamic diameter of a spherical particle with diameter Dp and nonunit density will therefore depend on the mean free path of the air molecules around it and thus also on pressure [see (9.6)]. For low pressures resulting in high Knudsen numbers, the particle will be in the free molecular regime

393

PROBLEMS

and the aerodynamic diameter will be proportional to the density of the particle and given by (9.113). This diameter is often called the vacuum aerodynamic diameter or the free molecular regime aerodynamic diameter of the particle. A number of aerosol instruments that operate at low pressures such as the aerosol mass spectrometer measure the vacuum aerodynamic diameter of particles. In the other extreme, for high pressures resulting in low Knudsen numbers the particle will be in the continuum regime and its aerodynamic diameter will be proportional to the square root of its density (9.112). This is known as the continuum regime aerodynamic diameter. The aerodynamic diameter of the particle changes smoothly from its vacuum to its continuum value as the Knudsen number decreases.

9.7.4 Electrical Mobility Equivalent Diameter Electrical mobility analyzers, like the differential mobility analyzer, classify particles according to their electrical mobility Be given by (9.50). The electrical mobility equivalent diameter Dem is defined as the diameter of a sphere having the same electrical mobility as the given particle. Particles with the same Dem have the same migration velocity in an electric field. Particles with equal Stokes diameters that carry the same electrical charge will have the same electrical mobility. For spherical particles, assuming that the particle and its mobility equivalent sphere have the same charge, then Dem = Dp = Dve. For nonspherical particles one can show that Dem ˆ Dve χ

Cc …Dem † Cc …Dve †

(9.114)

Therefore, Dem is greater than Dve for nonspherical particles because χ > 1 and Cc is a monotonically increasing function of diameter. The electrical mobility equivalent diameter does not depend on the density of the particle. Instruments such as the differential mobility analyzer (DMA) (Liu et al. 1979) size particles according to their electrical mobility equivalent diameter.

PROBLEMS 9.1A (a) Knowing a particle’s density ρp and its settling velocity vt, show how to determine its diameter. Consider both the non-Stokes and the Stokes law regions. (b) Determine the size of water droplet that has vt = 1 cm s 1 at T = 20°C, 1 atm. 9.2A (a) A unit density sphere of diameter 100 μm moves through air with a velocity of 25 cm s 1. Compute the drag force offered by the air. (b) A unit density sphere of diameter 1 μm moves through air with a velocity of 25 cm s 1. Compute the drag force offered by the air. 9.3A Calculate the terminal settling velocities of silica particles (ρp = 2.65 g cm 3) of 0.05, 0.1, and 0.5, and 1.0 μm diameters. 9.4A Calculate the terminal settling velocities of 0.001, 0.1, 1, 10, and 100 μm diameter water droplets in air at a pressure of 0.1 atm. 9.5A Develop a table of terminal settling velocities of water drops in still air at T = 20°C, 1 atm. Consider drop diameters ranging from 1.0 to 1000 μm. (Note that for drop diameters exceeding about 1 mm the drops can no longer be considered spherical as they fall. In this case one must resort to empirical correlations. We do not consider that complication here.) 9.6A What is the stop distance of a spherical particle of 1 μm diameter and density 1.5 g cm still air at 298 K with a velocity of 1 m s 1?

3

moving in

9.7B A 0.2 μm diameter particle of density 1 g cm 3 is being carried by an airstream at 1 atm and 298 K in the y-direction with a velocity of 100 cm s 1. The particle enters a charging device and acquires a

394

DYNAMICS OF SINGLE AEROSOL PARTICLES

charge of two electrons (the charge of a single electron is 1.6 × 10 19 C) and moves into an electric field of constant potential gradient Ex = 1000 V cm 1 perpendicular to the direction of flow. a. Determine the characteristic relaxation time of the particle. b. Determine the particle trajectory, assuming that it starts at the origin at time zero. c. Repeat the calculation for a 50 nm diameter particle. 9.8C At t = 0 a uniform concentration N0 of monodisperse particles exists between two horizontal plates separated by a distance h. Assuming that both plates are perfect absorbers of particles and the particles settle with a settling velocity vt, determine the number concentration of particles as a function of time and position. The Brownian diffusivity of the particles is D. 9.9B The dynamic shape factor of a chain that consists of 4 spheres is 1.32. The diameter of each sphere is 0.1 μm. Calculate the terminal settling velocity of the particle in air at 298 K and 1 atm. What is the error if the shape factor is neglected? Assume the density of the spheres is 2 g cm 3. 9.10B Derive (9.110) using the appropriate force balance for the motion of a charged particle in an electric field. 9.11B Spherical particles with different diameters can have the same electrical mobility if they have a different number of elementary charges. Calculate the diameters of particles that have an electrical mobility equal to that of a singly charged particle with Dp = 100 nm, assuming that they have 2, 3, or 4 charges. Assume T = 298 K and 1 atm. 9.12A The density of a spherical particle can be estimated if its size is measured with different instruments relying on different principles. The differential mobility analyzer (a component of the scanning mobility particle sizer) measures the electrical mobility equivalent diameter Dem of the particle (assume for simplicity that the particle is singly charged during the measurement). The aerosol particle sizer measures the classical aerodynamic particle diameter Dca : Calculate the particle density, ρp , if Dca ˆ 600 nm and Dem ˆ 500 nm: 9.13A For a non-spherical particle with shape factor χ and density ρp derive relationships linking its: a. Electrical mobility equivalent diameter Dem with its classical aerodynamic diameter Dca : b. Electrical mobility equivalent diameter with its vacuum aerodyamic diameter.

REFERENCES Allen, M. D., and Raabe, O. G. (1982), Reevaluation of Millikan’s oil drop data for the motion of small particles in air, J. Aerosol Sci. 13, 537–547. Bird, R. B., et al. (2002), Transport Phenomena, Second edition, Wiley, New York. Chandrasekhar, S. (1943), Stochastic problems in physics and astronomy, Rev. Modern Phys. 15, 1–89. Chapman, S., and Cowling, T. G. (1970), The Mathematical Theory of Non-uniform Gases, Cambridge Univ. Press, Cambridge, UK. Crawford, M. (1976), Air Pollution Control Theory, McGraw-Hill, New York. Davis, E. J. (1983), Transport phenomena with single aerosol particles, Aerosol Sci. Technol. 2, 121–144. Flagan, R. C., and Seinfeld, J. H. (1988), Fundamentals of Air Pollution Engineering, Prentice-Hall, Englewood Cliffs, NJ (reprinted by Dover Publications, 2012). Fuchs, N. A. (1964), The Mechanics of Aerosols, Pergamon, New York. Fuchs, N. A., and Sutugin, A. G. (1971), High dispersed aerosols, in Topics in Current Aerosol Research, G. M. Hidy and J. R. Brock (eds.), Pergamon, New York, pp. 1–60. Hinds, W. C. (1999), Aerosol Technology, Properties, Behavior, and Measurement of Airborne Particles Wiley, New York. Hinze, J. O. (1959), Turbulence, McGraw-Hill, New York. Hirschfelder, J. O., et al. (1954), Molecular Theory of Gases and Liquids, Wiley, New York.

REFERENCES Liu, B. Y. H., et al. (1979), Electrical aerosol analyzer: History, principle, and data reduction, in Aerosol Measurement, D. A. Lundgren (ed.), Univ. Presses of Florida, Gainesville. Loyalka, S. K., et al. (1989), Isothermal condensation on a spherical particle, Phys. Fluids A 1, 358–362. Tang, M. J., et al. (2014), Compilation and evaluation of gas phase diffusion coefficients of reactive trace gases in the atmosphere: Volume 1. Inorganic compounds, Atmos. Chem. Phys. 14, 9233–9247. Tang, M. J., et al. (2015), Compilation and evaluation of gas-phase diffusion coefficients of reactive trace gases in the atmosphere: Volume 2. Diffusivities of organic compounds, pressure normalized free paths, and average Knudsen numbers for gas uptake calculations, Atmos. Chem. Phys. 15, 5585–5598. Twomey, S. (1977), Atmospheric Aerosols, Elsevier, New York. Wax, N. (1954), Selected Papers on Noise and Stochastic Processes, Dover, New York. Zelenyuk, A., et al. (2006), From agglomerates of spheres to irregularly shaped particles: Determination of dynamic shape factors from measurements of mobility and vacuum aerodynamic diameters, Aerosol Sci. Technol. 40, 197–217.

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CHAPTER

10

Thermodynamics of Aerosols

A number of chemical compounds (water, ammonia, nitric acid, organics, etc.) can exist in both the gas and aerosol phases in the atmosphere. Understanding the partitioning of these species between the vapor and particulate phases requires an analysis of the thermodynamic properties of aerosols. Since the most important “solvent” for constituents of atmospheric particles and drops is water, we will pay particular attention to the thermodynamic properties of aqueous solutions.

10.1 THERMODYNAMIC PRINCIPLES An atmospheric air parcel can be viewed thermodynamically as a homogeneous system that may exchange energy, work, and mass with its surroundings. Let us assume that an air parcel contains k chemical species and has a temperature T, pressure p, and volume V. There are ni moles of species i in the parcel. This chapter begins with a review of fundamental chemical thermodynamic principles focusing on the chemical potential of species in the gas, aqueous, and solid phases. Further discussion of funda­ mentals of chemical thermodynamics can be found in Denbigh (1981). Chemical potentials form the basis for the development of a rigorous mathematical framework for the derivation of the equilibrium conditions between different phases. This framework is then applied to the partitioning of inorganic aerosol components (sulfate, nitrate, chloride, ammonium, and water) between the gas and particulate phases. The behavior of organic aerosol components will be discussed in Chapter 14.

10.1.1 Internal Energy and Chemical Potential In addition to the macroscopic kinetic and potential energy that the air parcel may have, it has internal energy U, arising from the kinetic and potential energy of the atoms and molecules in the system. Let us assume that the state of the air parcel changes infinitesimally (e.g., it rises slightly) but there is no mass exchange between the air parcel and its surroundings, namely, that the air parcel is a closed system. Then, according to the first law of thermodynamics, the infinitesimal change of internal energy dU is given by dU ˆ dQ ‡ dW

(10.1)

Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, Third Edition. John H. Seinfeld and Spyros N. Pandis.  2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

396

397

THERMODYNAMIC PRINCIPLES

where dQ is the infinitesimal amount of heat that is absorbed by the system and dW is the infinitesimal amount of work that is done on the system. Equation (10.1) can also be viewed as a definition of the internal energy of the system U. The infinitesimal work done on the system by its surroundings is equal to dW ˆ p dV

(10.2)

where p is the pressure of the system and dV its infinitesimal volume change. Note that, if the parcel expands, dV is positive and the work done on the system is negative (or alternatively the work done by the system is positive). Because of this expansion, if there is no heat exchange, dU < 0 and the internal energy of the system will decrease. On the contrary, if the parcel volume decreases, dV < 0, dW > 0, and if dQ = 0, the internal energy of the system increases. A thermodynamically reversible process is defined as one in which the system changes infinites­ imally slowly from one equilibrium state to the next. According to the second law of thermodynamics, the heat added to a system during a reversible process dQrev is given by dQrev ˆ T dS

(10.3)

where S is the entropy of the system. The entropy is another property of the system (like the temperature, volume, and pressure) measuring the degree of disorder of the elements of the system; the more disorder, the greater the entropy. Combining (10.1), (10.2), and (10.3)): dU ˆ T dS

p dV

(10.4)

for a closed system. This equation contains the whole of knowledge obtained from the basic thermo­ dynamic laws for a closed system undergoing a reversible change. In our discussion so far we have assumed that the system is closed. According to (10.4), if the number of moles of all system species n1, n2, . . ., nk remains constant, the change in internal energy U of the system depends only on changes in S and V. However, for variable composition we have U ˆ U…S; V; n1 ; n2 ; . . .; nk † and thus the total differential of U is dU ˆ

@U @S

@U @V

dS ‡ V;ni

k

dV ‡ S;ni

iˆ1

@U @ni

dni

(10.5)

S;V;nj

In this expression, the subscript ni in the first two partial derivatives implies that the amounts of all species are constant during the variation in question. On the other hand, the last partial derivative assumes that all but the ith substance are constant. Note that for a closed system dni = 0 and from (10.5) dU ˆ

@U @S

dS ‡ V;ni

@U @V

dV

(10.6)

S;ni

Comparing (10.4) and (10.6) that are both valid for a closed system, we obtain Tˆ

@U @S

pˆ

and V;ni

@U @V

(10.7) S;ni

Finally, (10.5) can be rewritten as k

dU ˆ T dS

p dV ‡ iˆ1

@U @ni

dni

(10.8)

S;V;nj

Let us define the chemical potential of species i, specifically, μi, as μi ˆ

@U @ni

(10.9) S;V;nj

398

THERMODYNAMICS OF AEROSOLS

so that (10.8) can be written as k

dU ˆ T dS

p dV ‡

μi dni

(10.10)

iˆ1

The chemical potential μi has an important function in the system’s thermodynamic behavior analogous to pressure or temperature. A temperature difference between two bodies determines the tendency of heat to pass from one body to another, while a pressure difference determines the tendency for bodily movement. We will show that a difference in chemical potential can be viewed as the cause for chemical reaction or for mass transfer from one phase to another. The chemical potential μi greatly facilitates the discussion of open systems, or of closed systems that undergo chemical composition changes.

10.1.2 The Gibbs Free Energy G Calculation of changes dU of the internal energy of a system U requires the estimation of changes of its entropy S, volume V, and number of moles ni. For chemical applications, including atmospheric chemistry, it is inconvenient to work with entropy and volume as independent variables. Temperature and pressure are much more useful. The study of atmospheric processes can therefore be facilitated by introducing other thermodynamic variables in addition to the internal energy U. One of the most useful is the Gibbs free energy G, defined as G ˆ U ‡ pV

TS

(10.11)

Differentiating (10.11) dG ˆ dU ‡ p dV ‡ V dp

T dS

S dT

(10.12)

and combining (10.12) with (10.10), one obtains k

μi dni

dG ˆ S dT ‡ V dp ‡

(10.13)

iˆ1

Note that one can propose using (10.13) as an alternative definition of the chemical potential: μi ˆ

@G @ni

(10.14) T;p;nj

Both definitions (10.9) and (10.14) are equivalent. Equation (10.13) is the basis for chemical thermodynamics. For a system at constant temperature (dT = 0) and pressure (dp = 0) k

μi dni

dG ˆ

(10.15)

iˆ1

For a system under constant temperature, pressure, and with constant chemical composition (dni = 0), dG = 0, or the system has a constant Gibbs free energy. Equation (10.13) provides the means of calculating infinitesimal changes in the Gibbs free energy of the system. Let us assume that the system

399

THERMODYNAMIC PRINCIPLES

under discussion is enlarged m times in size, its temperature, pressure, and the relative proportions of each component remaining unchanged. Under such conditions the chemical potentials, which do not depend on the overall size of the system, remain unchanged. Let the original value of the Gibbs free energy of the system be G and the number of moles of species i, ni. After the system is enlarged m times, these quantities are now mG and mni. The change in Gibbs free energy of the system is ΔG ˆ mG

G ˆ …m

1†G

(10.16)

ni ˆ …m

1†ni

(10.17)

and the changes in the number of moles are Δni ˆ mni and as T, P, and μi are constant using (10.15) k

ΔG ˆ

μi Δni iˆ1

or k

μi n i

Gˆ

(10.18)

iˆ1

Equation (10.18) applies in general and provides additional significance to the concept of chemical potential. The Gibbs free energy of a system containing k chemical compounds can be calculated by G ˆ μ1 n1 ‡ μ2 n2 ‡


‡ μk nk that is, by summation of the products of the chemical potentials and the number of moles of each species. Note that for a pure substance μi ˆ

G ni

(10.19)

and thus the chemical potential is the value of the Gibbs free energy per mole of the substance. One should note that both (10.13) and (10.18) are applicable in general. It may appear surprising that T and p do not enter explicitly in (10.18). To explore this point a little further, differentiating (10.18) k

dG ˆ

k

ni dμi ‡ iˆ1

μi dni

(10.20)

iˆ1

and combining with (10.13), we obtain k

SdT ‡ V dp ˆ

ni dμi

(10.21)

iˆ1

This relation, known as the Gibbs–Duhem equation, shows that when the temperature and pressure of a system change there is a corresponding change of the chemical potentials of the various compounds.

400

THERMODYNAMICS OF AEROSOLS

10.1.3 Conditions for Chemical Equilibrium The second law of thermodynamics states that the entropy of a system in an adiabatic (dQ = 0) enclosure increases for an irreversible process and remains constant in a reversible one. This law can be expressed as dS  0

(10.22)

Therefore a system will try to increase its entropy, and when the entropy reaches its maximum value, the system will be at equilibrium. One can show that for a system at constant temperature and pressure the criterion corresponding to (10.22) is dG  0

(10.23)

or that a system will tend to decrease its Gibbs free energy. For a proof the reader is referred to Denbigh (1981). Consider the reaction A ⇌ B, and let us assume that initially there are nA moles of A and nB moles of B. The Gibbs free energy of the system is, using (10.18) G ˆ nA μA ‡ nB μB If the system is closed nT ˆ nA ‡ nB ˆ constant and this equation can be rewritten as G ˆ nT …xA μA ‡ …1

xA †μB †

where xA ˆ

nA nA ˆ nA ‡ nB nT

the mole fraction of A in the system. Let us assume that the Gibbs free energy of the system is that shown in Figure 10.1. If at a given moment the system is at point K, (10.23) suggests that dG  0 and G will tend to decrease, so xA will increase, B will be converted to A, and the system will move to the right. If the system at a given moment is at point M, once more dG  0, so the system will move to the left (A will be converted to B). At point L, the Gibbs free energy is at a minimum. The system cannot spontaneously move to the left or right because then the Gibbs free energy would increase, violating (10.23). If the system is forced to move, then it will return to this equilibrium state. Therefore, for a constant T and p, the point L and the corresponding composition is the equilibrium state of the system and (xA)L the corresponding mole fraction of A. At this point dG = 0. Let us consider a general chemical reaction aA ‡ bB ƒ cC ‡ dD which can be rewritten mathematically as aA ‡ bB

cC

dD ˆ 0

The Gibbs free energy of the system is given by G ˆ nA μA ‡ nB μB ‡ nC μC ‡ nD μD

(10.24)

401

THERMODYNAMIC PRINCIPLES

FIGURE 10.1 Sketch of the Gibbs free energy for a closed system where the reaction A ⇌ B takes place versus the mole fraction of A.

If dnA moles of A react, then, according to the stoichiometry of the reaction, they will also consume (b/a) dnA moles of B and produce (c/a) dnA moles of C and (d/a) dnA moles of D. The corresponding change of the Gibbs free energy of the system at constant T and p is, according to (10.15) dG ˆ μA dnA ‡ μB dnB ‡ μC dnC ‡ μD dnD b ˆ μA dnA ‡ μB dnA a b c ˆ μA ‡ μB μ a a C

c d μC dnA μ dnA a a D d μ dnA a D

(10.25)

At equilibrium dG = 0 and therefore the condition for equilibrium is

or

b μA ‡ μB a

c μ a C

d μ ˆ0 a D

aμA ‡ bμB

cμC

dμD ˆ 0

(10.26)

Let us try to generalize our conclusions so far. The most general reaction can be written as k

νi A i ˆ 0

(10.27)

iˆ1

where k is the number of species Ai participating in the reaction and νi is the corresponding stoichiometric coefficients (positive for reactants, negative for products). We can easily extend our arguments for the single reaction (10.24) to show that the general condition for equilibrium is k

νi μi ˆ 0 iˆ1

(10.28)

402

THERMODYNAMICS OF AEROSOLS

This is the most general condition of equilibrium of a single reaction and is applicable regardless of whether the reactants and products are solids, liquids, or gases. If there are multiple reactions taking place in a system with k species k

νi1 Ai ˆ 0 iˆ1 k

νi2 Ai ˆ 0

(10.29)

iˆ1

.. . k

νin Ai ˆ 0 iˆ1

the equilibrium condition applies to each one of these reactions and therefore at equilibrium k

νij μi ˆ 0;

j ˆ 1; . . .; n

(10.30)

iˆ1

where νij is the stoichiometric coefficient of species i in reaction j (there are n reactions and k species). Reactions in the H2SO4–NH3–HNO3 System Let us assume that the following reactions take place: 2 NH3 …g† ‡ H2 SO4 …g† ƒ …NH4 †2 SO4 …s† NH3 …g† ‡ HNO3 …g† ƒ NH4 NO3 …s† Calculate the chemical potential of sulfuric acid as a function of the chemical potentials of nitric acid and the two solids. At equilibrium the chemical potentials of gas-phase NH3, and H2SO4 and solids (NH4)2SO4 and NH4NO3 satisfy 2 μNH3 ‡ μH2 SO4 μ…NH4 †2 SO4 ˆ 0 (10.31) μNH3 ‡ μHNO3 μNH4 NO3 ˆ 0 From the second equation μNH3 ˆ μNH4 NO3

μHNO3 . Substituting this into the first, we find that

μH2 SO4 ˆ 2 μHNO3

2 μNH4 NO3 ‡ μ…NH4 †2 SO4

(10.32)

Determination of the equilibrium composition of this multiphase system therefore requires determi­ nation of the chemical potentials of all species as a function of the corresponding concentrations, temperature, and pressure.

10.1.4 Chemical Potentials of Ideal Gases and Ideal-Gas Mixtures In this section we will discuss the chemical potentials of species in the gas, aqueous, and aerosol phases. In thermodynamics it is convenient to set up model systems to which the behavior of ideal systems approximates under limiting conditions. The important models for atmospheric chemistry are the ideal gas and the ideal solution. We will define these ideal systems using the chemical potentials and then discuss other definitions.

403

THERMODYNAMIC PRINCIPLES

10.1.4.1 The Single Ideal Gas We define the ideal gas as a gas whose chemical potential μ(T, p) at temperature T and pressure p is given by μ…T; p† ˆ μ°…T; 1 atm† ‡ RT ln p

(10.33)

where μ° is the standard chemical potential defined at a pressure of 1 atm and therefore is a function of temperature only. R is the ideal-gas constant. Pressure p actually stands for the ratio (p/1 atm) and is dimensionless. This definition suggests that the chemical potential of an ideal gas at constant tempera­ ture increases logarithmically with its pressure. Differentiating (10.33) with respect to pressure at constant temperature, we obtain @μ @p

ˆ T

@μ° @p

‡ RT T

d ln p dp

(10.34)

But μ° is not a function of pressure and therefore its derivative with respect to pressure is zero, so for an ideal gas @μ RT ˆ (10.35) @p T p Using (10.13), one can calculate the derivative of G with respect to pressure @G @p

ˆV

(10.36)

T

but for a system consisting of n moles of a single gas [see (10.18)] G ˆ nμ

(10.37)

and therefore from (10.36) and (10.37) @μ @p

ˆ T

V n

(10.38)

Combining (10.35) and (10.38), the ideal gas satisfies pV ˆ n RT

(10.39)

the traditional ideal-gas law. Deviations from ideal-gas behavior are customarily expressed in terms of the compressibility factor C = pV/(nRT). Both dry air and water vapor have compressibility factors C, in the range 0.998 < C < 1 for the pressure and temperature ranges of atmospheric interest (Harrison 1965). Hence both dry air and water vapor can be treated as ideal gases with an error of less than 0.2% for all the conditions of atmospheric interest. 10.1.4.2 The Ideal-Gas Mixture A gaseous mixture is defined as ideal if the chemical potential of its ith component satisfies μi ˆ μi°…T† ‡ RT ln p ‡ RT ln yi

(10.40)

404

THERMODYNAMICS OF AEROSOLS

where μi°…T† is the standard chemical potential of species i, p is the total pressure of the mixture, and yi is the gas mole fraction of compound i. For yi = 1 (pure component), (10.40) is simplified to (10.33) and therefore μi° is precisely the same for both equations. This standard chemical potential is the Gibbs free energy per mole (recall for pure compounds G = μn) of the gas in the pure state and pressure of 1 atm. By defining the partial pressure of compound i as p i ˆ yi p

(10.41)

μi ˆ μi° …T† ‡ RT ln pi

(10.42)

a more compact form of (10.40) is

One can also prove that (10.42) is equivalent to the more traditional ideal-gas mixture definition: pi V ˆ ni RT The atmosphere can be treated as an ideal-gas mixture with negligible error.1

10.1.5 Chemical Potential of Solutions Atmospheric aerosols at high relative humidities are aqueous solutions. For the moment, let us consider solutions of inorganic species such as ammonium, nitrate, sulfate, chloride, and sodium. 10.1.5.1 Ideal Solutions A solution is defined as ideal if the chemical potential of every component is a linear function of the logarithm of its aqueous mole fraction xi, according to the relation μi ˆ μ∗i …T; p† ‡ RT ln xi

(10.43)

A multicomponent solution is ideal only if (10.43) is satisfied by every component. A solution, in general, approaches ideality as it becomes increasingly dilute in all except one component (the solvent). The standard chemical potential μ∗i is the chemical potential of pure species i (xi = 1) at the same temperature and pressure as the solution under discussion. Note that in general μ∗i is a function of both T and p but does not depend on the chemical composition of the solution. Let us consider the relationship of the preceding definition with Henry’s and Raoult’s laws, which are often used to define ideal solutions. Assuming that an ideal solution of i is in equilibrium with an ideal gas mixture, we have I…g† ƒ I…aq† and at equilibrium

μi …g† ˆ μi …aq†

according to (10.28). Using (10.42) and (10.43) μi°…T† ‡ RT ln pi ˆ μ∗i …T; p† ‡ RT ln xi

1

For a discussion of non-ideal-gas mixtures, the reader is referred to Denbigh (1981). Discussion of these mixtures is not necessary here, as the behavior of all gases in the atmosphere can be considered ideal for all practical purposes.

405

THERMODYNAMIC PRINCIPLES

or pi ˆ exp

μi∗ μi° xi ˆ Ki …T; p†xi RT

(10.44)

The standard chemical potentials μ∗i and μi° are functions only of temperature and pressure, and therefore the constant Ki is independent of the solution’s composition. If xi = 1 in (10.44), then Ki(T, p) is equal to the vapor pressure of the pure component i, pi°, and the equation can be rewritten as pi ˆ pi° xi

(10.45)

Equation (10.45) states that the vapor pressure of a gas over a solution is equal to the product of the pure component vapor pressure and its mole fraction in the solution. The lower the mole fraction in the solution, the more the vapor pressure of the gas over the solution drops. Thus (10.45) is the same as Raoult’s law. Most solutions of practical interest satisfy (10.43) only in certain chemical composition ranges and not in others. Let us focus on a binary solution of A and B. If the solution is ideal for every composition, then the partial pressures of A and B will vary linearly with the mole fraction of B (Figure 10.2). When xA = 0, the mixture consists of pure B and the equilibrium partial pressure of B over the solution is pB° and of A zero. The opposite is true at the other end, where xA = 1 and pA ˆ pA°. The partial pressures of A and B in an actual mixture generally behave like either of the two panels of Figure 10.3. This type of behavior arises when the adhesive forces between molecules of A and B differ from the cohesive forces between A and A or B and B. If the adhesive forces between A and B are greater than the cohesive forces between A and A or B and B, then the vapor pressures of A and B in the solution are less than those predicted by Raoult’s law (the straight dashed lines in Figure 10.3). This situation, depicted in the top panel of Figure 10.3, is referred to as a negative deviation from Raoult’s law. The cohesive forces are lessened not only by dilution but also by the forces of attraction of A and B. The result

FIGURE 10.2 Equilibrium partial pressures of the components of an ideal binary mixture as a function of the mole fraction of A, xA.

406

THERMODYNAMICS OF AEROSOLS

FIGURE 10.3 Equilibrium partial pressures of the components of a nonideal mixture of A and B. Dashed lines correspond to ideal behavior.

is a reduced tendency of A and B to evaporate from the solution. On the other hand, if the cohesive forces between like molecules are greater than the adhesive forces between A and B, both A and B escape the solution more readily, and their vapor pressures over the solution will be greater than predicted by Raoult’s law (not shown). In this case, there is a positive deviation from Raoult’s law. For most mixtures the relationships between pA, pB, and xA are nonlinear with the exceptions of the limits of xA → 0 and xA → 1. When xA → 1, we have a dilute solution of B in A. In this regime pA ≅ pA° xA

(10.46)

and Raoult’s law applies to A. In the same regime pB ˆ H ´B xB

(10.47)

where H ´B is a constant calculated from the slope of the pB line as xA → 1. This relationship corresponds to Henry’s law, and H ´B is the Henry’s law constant2 (based on mole fraction) for B in A (equal to the slope of the line BN). At the other end (xA → 0) we have pB ≅ pB° xB pA ˆ H ´A xA and B obeys Raoult’s law while A obeys Henry’s law. Summarizing, if a solution is ideal over the whole composition range (sometimes called “perfect” solution), (10.44) is satisfied for every xi. In this case Ki is equal to the vapor pressure of pure i, which is also equal in this case to the Henry’s law constant of i. Nonideal solutions approach ideality when the concentrations of all components but one approach zero. In that case the solutes satisfy Henry’s law pi ˆ H´i xi , where the solvent satisfies Raoult’s law pj ˆ pj°xj . For a dilute aqueous solution (10.47) is equivalent to ‰B…aq†Š ˆ …0:018 H´B † 1 pB . The Henry’s law constant HB defined in Chapter 7 is then related to H´B by HB ˆ …0:018 H ´B † 1 .

2

407

THERMODYNAMIC PRINCIPLES

10.1.5.2 Nonideal Solutions Atmospheric aerosols are usually concentrated aqueous solutions that deviate significantly from ideality. This deviation from ideality is usually described by introducing the activity coefficient γi, and the chemical potential is given by μi ˆ μ∗i …T; p† ‡ RT ln …γi xi †

(10.48)

The activity coefficient γi is in general a function of pressure and temperature together with the mole fractions of all substances in solution. For an ideal solution γi = 1. The standard chemical potential μ∗i is defined as the chemical potential at the hypothetical state for which γi → 1 and xi → 1 (Denbigh 1981). Following the same procedure as in the derivation of (10.45) one can show that in this case pi ˆ pi° γi xi

(10.49)

For negative derivations from Raoult’s law γi < 1 while for positive ones γi > 1. The product of the mole fraction xi of a solution component and its activity coefficient γi is defined as the activity, αi, of the component αi ˆ γi xi

(10.50)

and the chemical potential of a species i is then given by μi ˆ μ∗i …T; p† ‡ RT ln αi

(10.51)

For convenience, the amount of a species in solution is often expressed as a molality rather than as a mole fraction. The molality of a solute is its amount in moles per kilogram of solvent. For an aqueous solution containing ni moles of solute and nw moles of water (molecular weight 0.018 kg mol 1) the molality, mi, of the solute is mi ˆ

ni 0:018 nw

(10.52)

Another measure of solution concentration is the molarity, expressed in moles of solute per liter of solvent (denoted by M). For water solutions at ambient conditions, because 1 liter weighs 1 kilogram, the molality and molarity of a solution are practically equal. Traditionally, the activity of the solvent is almost always defined on the mole fraction scale ((10.40) and (10.50)), but the activity coefficient of the solute is often expressed on the molality scale: μi ˆ μ⋄i ‡ RT ln …γi mi †

(10.53)

In this case μ⋄i is the value of the chemical potential as mi → 1 and γi → 1. The activity coefficients γi are determined experimentally by a series of methods including vapor pressure, freezing-point depression, osmotic pressure, and solubility measurements (Denbigh 1981). 10.1.5.3 Pure Solid Compounds The chemical potential of a pure solid compound i can be derived from (10.43) by setting xi = 1, so that μi …solid† ˆ μ∗i …T; p†

(10.54)

408

THERMODYNAMICS OF AEROSOLS

The chemical potential of the solid is therefore equal to its standard potential and is a function only of temperature and pressure. 10.1.5.4 Solutions of Electrolytes Most of the inorganic aerosol components dissociate on dissolution; for example, NH4NO3 dissociates forming NH‡4 and NO3 . The concentration of each ion in the aqueous solution is traditionally expressed on the molality scale and the chemical potential of each ion in a NH4NO3 solution is μNH‡4 ˆ μ⋄NH‡ ‡ RT ln γNH‡4 mNH‡4 4

μNO3 ˆ μ⋄NO3 ‡ RT ln γNO3 mNO3 where mNH‡4 and mNO3 are the ion molalities and γNH4‡ and γ NO3 the corresponding activity coefficients. For the dissociation reaction NH4 NO3 ƒ NH‡4 ‡ NO3 at equilibrium, the chemical potential of NH4NO3 satisfies μNH4 NO3 ˆ μNH4‡ ‡ μNO3 or μNH4 NO3 ˆ μ⋄NH‡ ‡ μ⋄NO3 ‡ RT ln γ NH4‡ γ NO3 mNH‡4 mNO3 4

(10.55)

The binary activity coefficient for NH4NO3 can be defined as γ 2NH4 NO3 ˆ γ NH‡4 γ NO3

(10.56)

and (10.55) can be rewritten as μ∗NH4 NO3 ˆ μ⋄NH‡ ‡ μ⋄NO3 ‡ RT ln γ2NH4 NO3 mNH‡4 mNO3 4

If the electrolyte dissociates completely and the initial molality of NH4NO3 is mNH4 NO3 , then mNO3 ˆ mNH‡4 ˆ mNH4 NO3 and μ∗NH4 NO3 ˆ μ⋄NH‡ ‡ μ⋄NO3 ‡ RT ln γ 2NH4 NO3 m2NH4 NO3 4

(10.57)

10.1.6 The Equilibrium Constant The equilibrium expression (10.28) can be used to obtain a useful expression for aerosol equilibrium calculations. Let us consider the general reaction k

νi μ i ˆ 0 iˆ1

409

AEROSOL LIQUID WATER CONTENT

Then substituting (10.51) into (10.28) k

νi μ∗i ‡ RT ln αi ˆ 0

iˆ1

or k

∏ανi i ˆ K

(10.58)

iˆ1

1 RT

K ˆ exp

k

νi μ∗i

(10.59)

iˆ1

If, for example, the species participating in the reaction are all gases, then the activity can be replaced with the partial pressures and k

∏pνi i ˆ K

(10.60)

iˆ1

Equilibrium Constant for Ammonium Sulfate Formation Let us determine the equilibrium constant for the following reaction: H2 SO4 …g† ‡ 2 NH3 …g† ƒ …NH4 †2 SO4 …s† For this reaction νH2 SO4 ˆ 1, νNH3 ˆ 2, and ν…NH4 †2 SO4 ˆ 1. The condition for equilibrium is 2 μNH3 ‡ μH2 SO4

μ…NH4 †2 SO4 ˆ 0

or following (10.59) and noting that α = 1 for solids exp

μ∗…NH4 †2 SO4

2 μNH ° 3

μH ° 2 SO4

RT

ˆ K…T† ˆ p2NH3 pH2 SO4

and therefore at equilibrium the product of the square of ammonia partial pressure and of sulfuric acid partial pressure should equal a constant. The value of the constant is a strong function of temperature.

10.2 AEROSOL LIQUID WATER CONTENT Water is an important component of atmospheric aerosols. Most of the water associated with atmo­ spheric particles is chemically unbound (Pilinis et al. 1989). At very low relative humidities, atmospheric aerosol particles containing inorganic salts are solid. As the ambient relative humidity increases, the particles remain solid until the relative humidity reaches a threshold value characteristic of the particle composition (Figure 10.4). At this RH, the solid particle spontaneously absorbs water, producing a

410

THERMODYNAMICS OF AEROSOLS

FIGURE 10.4 Diameter change of (NH4)2SO4, NH4HSO4, and H2SO4 particles as a function of relative humidity. Dp0 is the diameter of the particle at 0% RH.

saturated aqueous solution. The relative humidity at which this phase transition occurs is known as the deliquescence relative humidity (DRH). Further increase of the ambient RH leads to additional water condensation onto the salt solution to maintain thermodynamic equilibrium (Figure 10.4). On the other hand, as the RH over the wet particle is decreased, evaporation of water occurs. However, the solution generally does not crystallize at the DRH, but remains supersaturated until a much lower RH at which crystallization occurs (Junge 1952; Richardson and Spann 1984; Cohen et al. 1987). This hysteresis phenomenon with different deliquescence and crystallization points is illustrated in Figure 10.4 for (NH4)2SO4. The relative humidities of deliquescence for some inorganic salts, which are common constituents of ambient aerosols, are given in Table 10.1. One should also note that some aerosol species do not exhibit deliquescent behavior. Species like H2SO4 are highly hygroscopic, and therefore the water content associated with them changes smoothly as the RH increases or decreases (Figure 10.4). For each relative humidity a single salt can exist in either of two states: as a solid or as an aqueous solution. For relative humidities lower than the deliquescence relative humidity, the Gibbs free energy of

TABLE 10.1 Deliquescence Relative Humidities of Electrolyte Solutions at 298 K Salt

DRH (%)

KCl Na2SO4 NH4Cl (NH4)2SO4 NaCl NaNO3 (NH4)3H(SO4)2 NH4NO3 NaHSO4 NH4HSO4

84.2 ± 0.3 84.2 ± 0.4 80.0 79.9 ± 0.5 75.3 ± 0.1 74.3 ± 0.4 69.0 61.8 52.0 40.0

Sources: Tang (1980) and Tang and Munkelwitz (1993).

411

AEROSOL LIQUID WATER CONTENT

FIGURE 10.5 Gibbs free energy of a solid salt and its aqueous solution as a function of RH. At the DRH these energies become equal.

the solid salt is lower than the energy of the corresponding solution and the salt remains in the solid state (Figure 10.5). As the relative humidity increases, the Gibbs free energy of the corresponding solution state decreases, and at the DRH it becomes equal to the energy of the solid. When the RH increases further, the solution represents the lower energy state and the particle spontaneously absorbs water to form a saturated salt solution. This deliquescence transition is accompanied by a significant increase in the mass of the particle (Figure 10.4). For even higher RHs, the solution state is the preferable one. When the RH decreases reaching the DRH the energies of the two states become once more equal. However, as the RH decreases further, for the particle to attain the lower energy state (solid), all the water in the particle needs to evaporate. This is physically difficult, as salt nuclei need to be formed and salt crystals to grow around them. In the atmosphere, where these salts are suspended in air, this transition does not occur at this point and the particle remains liquid. As the RH continues to decrease, the water in the particle continues to evaporate and the particle becomes a supersaturated solution. The solution eventually reaches a critical supersaturation, and nucleation (crystallization) takes place, forming at last a solid particle at RH significantly lower than the DRH (Figure 10.4).

10.2.1 Chemical Potential of Water in Atmospheric Particles Water vapor exists in the atmosphere in concentrations on the order of grams per m3 of air while its concentration in the aerosol phase is less than 1 mg m 3 of air. As a result, transport of water to and from the aerosol phase does not affect the ambient vapor pressure of water in the atmosphere. This is in contrast to the cloud phase, where a significant amount of water exists in the form of cloud droplets (see Chapter 17). Thus the ambient RH can be treated as a known constant in aerosol thermodynamic calculations. Considering the equilibrium H2 O…g† ƒ H2 O…aq† and using the criterion for thermodynamic equilibrium and the corresponding chemical potentials μH2 O …g† ˆ μH2 O …aq†

412

THERMODYNAMICS OF AEROSOLS

or

μH ° 2 O ‡ RT ln pw ˆ μ∗H2 O ‡ RT ln αw

(10.61)

where pw is the water vapor pressure (in atm) and αw is the water activity in solution. For pure water in equilibrium with its vapor, αw = 1 and pw ˆ pw° (the saturation vapor pressure of water at this tempera­ ture); therefore μ∗H2 O

μH ° 2 O ˆ RT ln p°w

(10.62)

Using (10.62) in (10.61) yields αw ˆ

pw RH ˆ pw° 100

(10.63)

because the ratio pw =pw° is by definition equal to the relative humidity expressed in the 0–1 scale. Thus the water activity in an atmospheric aerosol solution is equal to the RH (in the 0.0–1.0 scale). This result simplifies significantly equilibrium calculations for atmospheric aerosol, because for each RH, the water activity for any liquid aerosol solution is fixed. Water equilibrium between the gas and aerosol phases at the point of deliquescence requires that the deliquescence relative humidity of a salt will then satisfy DRH ˆ αws 100

(10.64)

where αws is the water activity of the saturated solution of the salt at that temperature. The water activity values can be calculated from thermodynamic arguments using aqueous salt solubility data (Cohen et al. 1987; Pilinis and Seinfeld 1987; Pilinis et al. 1989).

10.2.2 Temperature Dependence of the DRH The DRH for a single salt varies with temperature. The vapor–liquid equilibrium of a salt S can be expressed by the following reactions H2 O…g† ƒ H2 O…aq†

(a)

H2 O…aq† ‡ n S…s† ƒ n S…aq†

(b)

where n is the solubility of S in water in moles of solute per mole of water (Table 10.2). The energy that is released in reaction (a) is the heat of condensation of water vapor, which is equal to the negative value of its heat of vaporization, ΔHv. The heat that is absorbed in reaction (b) is the enthalpy of solution of the salt ΔHs. This enthalpy can be readily calculated from the heats of formation tabulated in standard thermodynamic tables. Values of ΔHs are shown in Table 10.3 (Wagman et al. 1966). The overall enthalpy change ΔH for the two reactions is ΔH ˆ n ΔH s

ΔH v

(10.65)

The change of the vapor pressure of water over a solution with temperature is given by the Clausius– Clapeyron equation (Denbigh 1981) d ln pw ˆ dT

ΔH RT 2

(10.66)

413

AEROSOL LIQUID WATER CONTENT

TABLE 10.2 Solubility of Common Aerosol Salts in Water as a Function of Temperature (n = A + BT + CT2, n = mol of solute per mol of water) Salt (NH4)2SO4 Na2SO4 NaNO3 NH4NO3 KCl NaCl

n (at 298 K) 0.104 0.065 0.194 0.475 0.086 0.111

A

B

C

0.1149 0.3754 0.1868 4.298 0.2368 0.1805

4.489 × 10 1.763 × 10 1.677 × 10 3.623 × 10 1.453 × 10 5.310 × 10

4 3 3 2 3 4

1.385 × 10 2.424 × 10 5.714 × 10 7.853 × 10 1.238 × 10 9.965 × 10

6 6 6 5 6 7

TABLE 10.3 Enthalpy of Solution for Common Aerosol Salts at 298 K ΔHs (kJ mol 1)

Salt (NH4)2SO4 Na2SO4 NaNO3 NH4NO3 KCl NaCl

6.56 2.43 20.37 25.69 15.21 3.77

Source: Calculated from the corresponding heats of formation in Wagman et al. (1982).

which for this case becomes d ln pw ΔH v ˆ dT RT 2

n

ΔH s RT 2

(10.67)

Applying the Clausius–Clapeyron equation to pure water, we also obtain d ln pw° ΔH v ˆ dT RT 2

(10.68)

where p°w is the saturation vapor pressure of water at temperature T. Combining (10.67) and (10.68) d ln pw =pw° ΔH s ˆ n dT RT 2

(10.69)

and substituting (10.63) into (10.69) and applying it to the DRH, we obtain d ln …DRH=100† ΔH s ˆ n dT RT 2

(10.70)

The solubility n can be written as a polynomial in T (see Table 10.2), and the equation can be integrated from T0 = 298 K to T to give ln

DRH…T† ΔH s 1 A ˆ R T DRH…T 0 †

1 T0

B ln

T T0

C…T

T0 †

(10.71)

414

THERMODYNAMICS OF AEROSOLS

or DRH…T† ˆ DRH…298†exp

ΔH s 1 A R T

1 298

B ln

T 298

C…T

(10.72)

298†

The only assumption used in (10.72) is that the heat of solution is almost constant from 298 K to T. Wexler and Seinfeld (1991) proposed a similar expression assuming constant solubility, namely, B = C = 0 in (10.72), while expression (10.72) was derived by Tang and Munkelwitz (1993). Dependence of the (NH4)2SO4, NH4NO3, and NaNO3 DRH on Temperature Calculate the DRH of (NH4)2SO4, NH4NO3, and NaNO3 at 0 °C, 15 °C, and 30 °C. The DRH at 25 °C for the three salts is given in Table 10.1. Using (10.72) and the corresponding parameter values from Tables 10.3 and 10.2, we find that Deliquescence relative humidity Salt

0 °C

15 °C

30 °C

(NH4)2SO4 NH4NO3 NaNO3

81.8 76.6 80.9

80.6 68.1 76.9

79.5 58.5 73.0

These results indicate that the DRH for (NH4)2SO4 is practically constant within the temperature range of atmospheric interest while for the other two salts it varies significantly. The predictions of (10.72) for ammonium sulfate are compared with measurements from a series of investigators in Figure 10.6.

FIGURE 10.6 Deliquescence RH as a function of temperature for (NH4)2SO4. (Reprinted from Atmos. Environ. 27A, Tang, I. N., and Munkelwitz, H. R., Composition and temperature dependence of the deliquescence properties of hygroscopic aerosols, 467–473. Copyright 1993, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 1GB, UK.)

415

AEROSOL LIQUID WATER CONTENT

10.2.3 Deliquescence of Multicomponent Aerosols Multicomponent aerosol particles exhibit behavior similar to that of single-component salts. As the ambient RH increases, the salt mixture is solid, until the ambient RH reaches the deliquescence point of the mixture, at which the aerosol absorbs atmospheric moisture and produces a saturated solution. A typical set of data of multicomponent particle deliquescence, growth, evaporation, and then crystalli­ zation is shown in Figure 10.7 for a KCl–NaCl particle. Note that deliquescence for the mixed-salt particle occurs at 72.7% RH, which is lower than the DRH of either NaCl (75.3%) or KCl (84.2%). Following Wexler and Seinfeld (1991), let us consider two electrolytes in a solution exposed to the atmosphere. The change of the DRH of a single-solute aqueous solution when another electrolyte is added can be calculated using the Gibbs–Duhem equation, (10.21). For constant T and p and for a solution containing two electrolytes (1 and 2) and water (w): n1 dμ1 ‡ n2 dμ2 ‡ nw dμw ˆ 0

(10.73)

where n1, n2, and nw are the numbers of moles of electrolytes 1 and 2, and water, respectively, while μ1, μ2, and μw are the corresponding chemical potentials. Let us assume that initially electrolyte 1 is in equilibrium with solid salt 1 and the solution does not yet contain electrolyte 2. As electrolyte 2 is added to the solution, the chemical potential of electrolyte 1 does not change, because it remains in equilibrium

FIGURE 10.7 Hygroscopic growth and evaporation of a mixed-salt particle composed initially of 66% mass KCl and 34% mass NaCl. (Reprinted from Atmos. Environ., 27A, Tang I. N., and Munkelwitz, H. R., Composition and temperature dependence of the deliquescence properties of hygroscopic aerosols, 467–473. Copyright 1993, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 1GB, UK.)

416

THERMODYNAMICS OF AEROSOLS

with its solid phase. Thus dμ1 = 0 in (10.73). The chemical potentials of electrolyte 2 and water can be expressed using (10.51) to get n2 d ln α2 ‡ nw d ln αw ˆ 0 (10.74) Accounting for the fact that n2/nw = Mwm2/1000, where m2 is the molality of electrolyte 2 and Mw is the molecular weight of water, we obtain 1000 d ln αw ˆ 0 (10.75) m2 d ln α2 ‡ Mw Integration of the last equation from m´2 ˆ 0 to m2´ ˆ m2 gives ln

αw …m2 † ˆ αw …0†

Mw m2 m´2 dα2 m´2 dm2´ 1000 ∫ 0 α2 m´2 dm´2

(10.76)

Wexler and Seinfeld (1991) have argued that dα2/dm2  0 and therefore the integral is positive, and then αw …m2 †  αw …m2 ˆ 0†

(10.77)

Hence the activity of water decreases as electrolyte 2 is added to the system, until the solution becomes saturated in that electrolyte, too. The aerosol is exposed to the atmosphere and therefore its DRH also decreases. The preceding analysis can be extended to aerosols containing more than two salts. Thus one can prove that water activity reaches a minimum at the deliquescence point of the aerosol. Another consequence of this analysis is that the DRH of a mixed salt is always lower than the DRH of the individual salts in the particle. Wexler and Seinfeld (1991) solved (10.76) for the case of the system containing NH4NO3 and NH4Cl. Their calculations at 303 K are depicted in Figure 10.8.

FIGURE 10.8 Water activity at saturation for an aqueous solution of NH4NO3 and NH4Cl at 303 K. (Reprinted from Atmos. Environ., 25A, Wexler, A. S., and Seinfeld, J. H., Second-generation inorganic aerosol model, 2731–2748. Copyright 1991, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 1GB, UK.)

AEROSOL LIQUID WATER CONTENT

When there is only NH4Cl in the particle …xNH4 NO3 ˆ 0† the DRH of the particle is 77.4%. If the RH is below 77.4%, the particle is solid; if it is above 77.4% it is an aqueous solution of NH‡4 and Cl . (Note that the calculations in Figure 10.8 are at 303 K.) There are seven different RH composition regimes in Figure 10.8: (a) RH > DRH(NH4Cl) > DRH(NH4NO3). When the RH exceeds the DRH of both compounds, the aerosol is an aqueous solution of NH‡4 , NO3 , and Cl . (b1) DRH(NH4Cl) > RH > DRH(NH4NO3)—left of [1]. In this regime the aerosol consists of solid NH4Cl in equilibrium with an aqueous solution of NH‡4 , NO3 , and Cl . (b2) DRH(NH4Cl) > RH > DRH(NH4NO3)—right of [1]. If the aerosol contains enough NH4NO3 so that its composition is to the right of line [1], the aerosol is an aqueous solution of NH‡4 , NO3 , and Cl . In this regime addition of NH4NO3 results in complete dissolution of NH4Cl even if the RH is higher than its DRH. (c1) DRH(NH4Cl) > DRH(NH4NO3) > RH > DRH∗—left of [1]. The aerosol consists of solid NH4Cl in equilibrium with a solution of NH‡4 , NO3 , and Cl . (c2) Right of [1], left of [2]. In this regime there is no solid phase and the aerosol consists exclusively of an aqueous solution of NH‡4 , NO3 , and Cl . (c3) Right of [2]. Here the aerosol consists of solid NH4NO3 in equilibrium with an aqueous solution of NH‡4 , NO3 , and Cl . (d) RH < DRH∗. If the relative humidity is below DRH∗ = 51%, the aerosol consists of solid NH4NO3 and solid NH4Cl. In conclusion, for this mixture of salts, if both are present, the aerosol will contain some liquid water for RH > 51%. However, if we are below the solid line in Figure 10.8, the aerosol will also contain a solid phase in equilibrium with the solution. Only above the solid line is the particle completely liquid. It is instructive to follow the changes in a particle of a given composition as RH increases. For example, consider a particle consisting of 40% NH4NO3 and 60% NH4Cl as RH increases from 40 to 90%, assuming that there is no evaporation or condensation of these salts. At the beginning (RH = 40%) the particle is solid and it remains solid until RH reaches 51.4% (= DRH∗). At this point, the particle deliquesces and consists of two phases—a solid phase consisting of solid NH4Cl and an aqueous solution with the composition corresponding to the eutonic point …xNH4 NO3 ˆ 0:811; xNH4 Cl ˆ 0:189†. As the RH increases further, more NH4Cl dissolves in the solution and the composition of the aqueous solution follows line [1]. For example, at RH = 60% the aerosol consists of solid NH4Cl and a solution of composition xNH4 NO3 ˆ 0:73, xNH4 Cl ˆ 0:27. At 70% RH, most of the NH4Cl has dissolved and the composition of the solution is close to the net particle composition xNH4 NO3 ˆ 0:42, xNH4 Cl ˆ 0:58. Finally, when the RH reaches 71%, all the NH4Cl dissolves and the particle consists of only one phase (the aqueous one) with composition equal to the particle composition. Note that there is only one step change in the mass of the particle corresponding to the mutual DRH∗ (e.g., see Figure 10.7). Further increases in the RH cause continuous changes of the aerosol mass and not step changes. On the contrary, step changes are observed during particle evaporation and will be discussed subsequently. This discussion of phase transitions has been extended to mixtures of more than three salts by Potukuchi and Wexler (1995a,b). The phase diagrams become now three dimensional (Figure 10.9). Note that the solid phases that appear are (NH4)2SO4, NH4HSO4, letovicite [(NH4)3H(SO4)2], NH4NO3, and the double salt (NH4)2SO4
2NH4NO3. The labels on the contours show relative humidities at deliques­ cence. The bold lines in this figure are the phase boundaries separating the various solid phases. On these lines both phases coexist. For example, let us assume that there is enough ammonia present so X = 1 in Figure 10.9. If sulfate dominates then Y = 1 and the system is in the upper right-hand corner of the diagram with a DRH of 80% corresponding to (NH4)2SO4. If there is enough nitrate available so that Y = 0.3 then the DRH is 68% corresponding to (NH4)2SO4
2NH4NO3. The mutual deliquescence points of a series of pairs are given in Table 10.4.

417

418

THERMODYNAMICS OF AEROSOLS

FIGURE 10.9 Deliquescence relative humidity contours (solid lines) for aqueous solutions of H‡ NH‡4 HSO4 SO24 NO3 shown together with the lines (dashed) describing aqueous-phase composition with relative humidity. Labels on the contours represent the deliquescence relative humidity values. Total hydrogen is total moles of protons and bisulfate ions. Total sulfate is number of moles of sulfate and bisulfate ions. X = ammonium (ammonium + total hydrogen); Y = total sulfate (total sulfate + nitrate). (Reprinted from Atmos. Environ. 29, Potukuchi, S., and Wexler, A. S., Identifying solid– aqueous phase transitions in atmospheric aerosols. II Acidic solutions, 3357–3364. Copy­ right 1995, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 1GB, UK.)

TABLE 10.4 Deliquescence RH (DRH∗) at Mutual Solubility Point at 303 K Compound 1 NH4NO3 NH4NO3 NH4NO3 NaNO3 NH4NO3 NaNO3 NaCl NH4Cl

Compound 2

DRH∗

DRH1

DRH2

NaCl NaNO3 NH4Cl NH4Cl (NH4)2SO4 NaCl NH4Cl (NH4)2SO4

42.2 46.3 51.4 51.9 52.3 67.6 68.8 71.3

59.4 59.4 59.4 72.4 59.4 72.4 75.2 77.2

75.2 72.4 77.2 77.2 79.2 75.2 77.2 79.2

Source: Wexler and Seinfeld (1991).

EQUILIBRIUM VAPOR PRESSURE OVER A CURVED SURFACE: THE KELVIN EFFECT

10.2.4 Crystallization of Single- and Multicomponent Salts The behavior of inorganic salts when RH is decreased is different from that discussed in Sections 10.2.2 and 10.2.3. For example, for (NH4)2SO4, as RH decreases below 80% (the DRH of (NH4)2SO4), the particle water evaporates, but not completely. The particle remains liquid until a RH of approximately 35%, where crystallization finally occurs (Figure 10.4). The RH at which the particle becomes dry is often called the efflorescence RH (ERH). This hysteresis phenomenon is characteristic of most salts. For such salts, knowledge of the RH alone is insufficient for determining the state of the aerosol in the RH between the efflorescence and deliquescence RH. One needs to know the RH history of the particle. The ERH of a salt cannot be calculated from thermodynamic principles such as the DRH; rather, it must be measured in the laboratory. These measurements are challenging because of the potential effects of impurities, observation time, and size of the particle. Measured ERH values at ambient temperatures are shown in Table 10.5 (Martin 2000). Pure NH4HSO4, NH4NO3, and NaNO3 aqueous particles do not readily crystallize even at RH values close to zero. TABLE 10.5 Efflorescence Relative Humidity at 298 K Salt (NH4)2SO4 NH4HSO4 (NH4)H(SO4)2 NH4NO3 Na2SO4 NaCl NaNO3 NH4Cl KCl

ERH (%) 35 ± 2 Not observed 35 Not observed 56 ± 1 43 ± 3 Not observed 45 59

Source: Martin (2000).

Particles consisting of multiple salts exhibit a more complicated behavior (Figure 10.7). For example, the evaporation of a KCl–NaCl particle is characterized by two-step changes: the first at 65% with the formation of KCl crystals and the second at 62% with the complete drying of the particles and the crystallization of the remaining NaCl. The ERH of a particle consisting of multiple salts depends on the composition of the particle. For example, as H2SO4 is added to an (NH4)2SO4 particle, its ERH decreases from approximately 35% to around 20% for a molar ratio of NH‡4 : SO42 ˆ 1:5 to approximately zero for NH‡4 : SO42 ˆ 1 (ammonium bisulfate) (Martin et al. 2003). For a particle consisting of approximately 1:1 (NH4)2SO4:NH4NO3, the ERH is around 20%, while for a 1:2 molar ratio it decreases to around 10%. As a result, particles that are either very acidic (containing ammonium bisulfate) or contain significant amounts of ammonium nitrate tend to remain liquid during their atmospheric lifetime even when the ambient RH is quite low (Shaw and Rood 1990). Inclusions of insoluble dust minerals (CaCO3, Fe2O3, etc.) can increase the efflorescence RH of salts (Martin 2000). For example, for (NH4)2SO4, the ERH can increase from 35% to 49% when CaCO3 inclusions are present. These mineral inclusions provide well-ordered atomic arrays and thus assist in the formation of crystals at higher RH and lower solution supersaturations. Soot, on the other hand, does not appear to be an effective nucleus for crystallization of salts because it does not contain a regular array of atoms that can induce order at least locally in the aqueous medium (Martin 2000).

10.3 EQUILIBRIUM VAPOR PRESSURE OVER A CURVED SURFACE: THE KELVIN EFFECT In our discussion of aerosol thermodynamics so far, we have assumed that the aqueous aerosol solution has a flat surface. However, a key aspect that distinguishes the thermodynamics of atmospheric particles

419

420

THERMODYNAMICS OF AEROSOLS

and drops is their curved interface. In this section we investigate the effect of curvature on the vapor pressure of a species A over the particle surface. Our approach will be to relate this vapor pressure to that over a flat surface. To do so we begin by considering the change of Gibbs free energy accompanying the formation of a single drop of pure A of radius Rp containing n molecules of the substance: ΔG ˆ Gdroplet

Gpure vapor

Let us assume that the total number of molecules of vapor initially is NT; after the drop forms, the number of vapor molecules remaining is N1 = NT n. Then, if gv and gl are the Gibbs free energies of a molecule in the vapor and liquid phases, respectively ΔG ˆ N 1 gv ‡ ngl ‡ 4π Rp2 σ

N T gv

where 4πR2p σ is the free energy associated with forming an interface with radius of curvature Rp and surface tension σ. Surface tension is the amount of energy required to increase the area of a surface by 1 unit. This equation can be rewritten as gv † ‡ 4πR2p σ

ΔG ˆ n…gl

(10.78)

Note that the number of molecules in the drop n and the drop radius Rp are related by nˆ

4πR3p

(10.79)

3vl

where vl is the volume occupied by a molecule in the liquid phase. Thus, combining (10.78) and (10.79), we obtain ΔG ˆ

4πR3p 3vl

…gl

gv † ‡ 4πR2p σ

(10.80)

We now need to evaluate gl gv, the difference in the Gibbs free energy per molecule of the liquid and vapor states. Using (10.13) at constant temperature and because dni = 0, it follows that dg = v dp or gl gv = (vl vv)dp. Since vv  vl for all conditions of interest to us, we can neglect vl relative to vv in this equation, giving gl gv  vv dp. The vapor phase is assumed to be ideal so vv = kT/p. Thus, integrating, we obtain gl

gv ˆ kT

pA

dp ∫ p° p A

(10.81)

where pA° is the vapor pressure of pure A over a flat surface and pA is the actual equilibrium partial pressure over the liquid. Then p (10.82) gl gv ˆ kT ln A pA° We can define the ratio pA =pA° as the saturation ratio S. Substituting (10.82) into (10.80), we obtain the following expression for the Gibbs free energy change: ΔG ˆ

4 3 kT ln S ‡ 4πR2p σ πR 3 p vl

(10.83)

Figure 10.10 shows a sketch of the behavior of ΔG as a function of Rp. We see that if S < 1, both terms in (10.83) are positive and ΔG increases monotonically with Rp. On the other hand, if S > 1, ΔG contains

421

EQUILIBRIUM VAPOR PRESSURE OVER A CURVED SURFACE: THE KELVIN EFFECT

FIGURE 10.10 Gibbs free energy change for formation of a droplet of radius Rp from a vapor with saturation ratio S.

both positive and negative contributions. At small values of Rp the surface tension term dominates and the behavior of ΔG is similar to the S < 1 case. As Rp increases, the first term on the RHS of (10.83) becomes more important so that ΔG reaches a maximum value ΔG∗ at Rp ˆ R∗p and then decreases. The radius corresponding to the maximum can be calculated by setting @ΔG/@Rp = 0 in (10.83) to obtain R∗p ˆ

2σvl kT ln S

(10.84)

Since ΔG is a maximum, as opposed to a minimum, at Rp ˆ R∗p , the equilibrium at this point is metastable. Equation (10.84) relates the equilibrium radius of a droplet of a pure substance to the physical properties of the substance, σ and vl, and to the saturation ratio S of its environment. Equation (10.84) can be rearranged recalling the definition of the saturation ratio as pA ˆ pA° exp

2σvl kT Rp

(10.85)

Expressed in this form, the equation is referred to as the Kelvin equation. The Kelvin equation can also be expressed in terms of molar units as pA ˆ pA° exp

2σM RT ρl Rp

(10.86)

where M is the molecular weight of the substance, and ρl is the liquid-phase density. The Kelvin equation tells us that the vapor pressure over a curved interface always exceeds that of the same substance over a flat surface. A rough physical interpretation of this so-called Kelvin effect is depicted in Figure 10.11. The vapor pressure of a liquid is determined by the energy necessary to separate a molecule from the attractive forces exerted by its neighbors and bring it to the gas phase. When a curved interface exists, as in a small droplet, there are fewer molecules immediately adjacent to a molecule on the surface than when the surface is flat. Consequently, it is easier for the molecules on the surface of a small drop to escape into the vapor phase and the vapor pressure over a curved interface is always greater than that over a plane surface.

422

THERMODYNAMICS OF AEROSOLS

FIGURE 10.11 Effect of radius of curvature of a drop on its vapor pressure.

Table 10.6 contains surface tensions for water and a series of organic molecules. At 298 K the organic surface tensions are about one-third that of water and their molecular volumes (M/ρl) range from three to six times that of water. Figure 10.12 gives the magnitude of the Kelvin effect pA =pA° for a pure water droplet and a typical organic compound (DOP) as a function of droplet diameter. We see that for water the vapor pressure is

TABLE 10.6 Surface Tensions and Densities of Selected Compounds at 298 Ka Compound

Molecular Weight

Water Benzene Acetone Ethanol Styrene Carbon tetrachloride σ has units in the SI system of N m

a

18 78.11 58.08 46.07 104.2 153.82 1

Density, g cm

3

Surface Tension, dyn cm

1.0 0.879 0.787 0.789 0.906 1.594

(= kg s 2). In the cgs system σ has units dyn cm 1. 1 dyn cm

72 28.21 23.04 22.14 31.49 26.34 1

= 10

FIGURE 10.12 Saturation ratio versus droplet size for pure water and an organic liquid (dioctyl phthalate DOP) at 20°C.

3

kg s 2.

1

423

THERMODYNAMICS OF ATMOSPHERIC AEROSOL SYSTEMS

increased by 2.1% for a 0.1 μm and 23% for a 0.01 μm drop over that for a flat interface. Roughly, we may consider a 50 nm diameter as the point at which the Kelvin effect begins to become important for aqueous particles. For higher-molecular-weight organics, the Kelvin effect is significant for even larger particles and should be included in calculations for particles with diameters smaller than about 200 nm.

10.4 THERMODYNAMICS OF ATMOSPHERIC AEROSOL SYSTEMS 10.4.1 The H2SO4–H2O System The importance of sulfuric acid in aerosol formation in a humid environment has been the subject of a number of studies (e.g., Kiang et al. 1973; Mirabel and Katz 1974; Jaecker-Voirol and Mirabel 1989; Kulmala and Laaksonen 1990; Laaksonen et al. 1995). It is instructive to study first this simplified binary system before starting the discussion of more complicated atmospheric aerosol systems. Sulfuric acid is very hygroscopic, absorbing significant amounts of water even at extremely low relative humidities. In Figure 10.13, sulfuric acid aqueous solution properties are shown as a function of the mass fraction of H2SO4 in solution. Included in Figure 10.13 are the equilibrium relative humidity over a flat solution surface RH, the solution density ρ, the boiling point B.P., and the surface tension σ. In addition, the solution normality (in N or equivalents per liter), the mass concentration xH2 SO4 ρ, the particle growth factor (the ratio of the actual diameter of the droplet Dp to that if the water associated with that were completely evaporated Dp0) Dp/Dp0, and the Kelvin parameter Dp0 ln (pH2 SO4 =pH° 2 SO4) are shown in the same figure. Calculation of the Properties of a H2SO4 Droplet (Liu and Levi 1980) Consider a 1 μm H2SO4-H2O droplet in equilibrium at 50% RH. Using Figure 10.13, calculate the following: 1. 2. 3. 4. 5. 6. 7. 8. 9.

The The The The The The The The The

H2SO4 concentration in the solution density of the solution boiling point of the solution droplet surface tension solution normality mass concentration of H2SO4 in the solution size of the droplet if all the water were removed Kelvin effect parameter size of this droplet at 90% RH.

Using the curve labeled RH and the corresponding scale at the right of the graph, we find that the solution concentration is 42.5% H2SO4 or that xH2 SO4 ˆ 0:425 g H2SO4 per g of solution. The rest of the solution properties can be calculated using this solution concentration, the appropriate curve, and the appropriate scale. At this concentration the solution has a density, ρ, of 1.32 g cm 3, a boiling point of 115 °C, a surface tension, σ, of 76 dyn cm 1, a normality of 11 N, and a mass concentration, xH2 SO4 ρ, of 0.55 g H2SO4 cm 3 of solution. The particle size growth factor, Dp/Dp0, is 1.48. Thus the H2SO4–H2O solution droplet would become a droplet of pure H2SO4 of 1/1.48 = 0.68 μm if all the water were removed. The Kelvin effect parameter Dp0 ln (pH2 SO4 =pH° 2 SO4 ) is 11.3 × 10 4 μm, indicating that ln

pH2 SO4 pH° 2 SO4

ˆ

11:3  10 0:68

4

ˆ 16:62  10

4

THERMODYNAMICS OF ATMOSPHERIC AEROSOL SYSTEMS

Thus pH2 SO4 =pH° 2 SO4 ˆ 1:0017, and in this case the Kelvin effect is negligible; the increase in water vapor pressure due to the droplet curvature is only 0.17% above that of a flat surface. The particle size growth factor at 90% relative humidity is 2.12. Thus the 1 μm diameter drop at 50% RH will grow to become a drop of (2.12/1.48) (1) = 1.43 μm at 90% RH. The saturation vapor pressure of pure sulfuric acid pH° 2 SO4 has been difficult to determine because it is so small. Roedel (1979), Ayers et al. (1980), and Kulmala and Laaksonen (1990) have argued that pH° 2 SO4 ˆ 1:3  1:0  10 8 atm (13 ± 10 ppb) at 296 K. The temperature dependence of the saturation vapor pressure as a function of temperature is shown in Figure 10.14. The variation of the vapor pressure of H2SO4 in a mixture with water over a flat surface as a function of composition at ambient temperatures is shown in Figure 10.15. Information in Figures 10.13, 10.14, and 10.15 can be combined to obtain most of the required information about sulfuric acid properties in the atmosphere. For example, let us calculate the expected H2SO4 concentrations, focusing only on the H2SO4–H2O system. If the relative humidity exceeds 50%, the H2SO4 concentration in solution will be less than 40% by mass, and the H2SO4 mole fraction is less than 0.1 (Figure 10.13). For a temperature equal to 20 °C, this corresponds to equilibrium vapor pressures less than 10 12 mm Hg. Under all conditions the concentration of H2SO4 in the gas phase is much less than the aerosol sulfate concentration. The role of the Kelvin effect on the composition of atmospheric H2SO4–H2O droplets is illustrated in Figure 10.16. If the effect were negligible, the equilibrium aerosol composition would not be a function of its size. This is the case for particles larger than 0.1 μm in diameter. For smaller particles the H2SO4 mole fraction in the droplet is highly dependent on particle size. Also notice that for a fixed droplet size, the water concentration increases as the relative humidity increases. The sulfuric acid–water system reactions are shown in Table 10.7. Note that the reaction H2SO4(g) ⇌ H2SO4(aq) is not included as H2SO4(aq) can be assumed to completely dissociate to HSO4 in the atmosphere for all practical purposes.

FIGURE 10.14 Vapor pressure of sulfuric acid as a function of temperature using the Ayers et al. (1980) and Kulmala and Laaksonen (1990) estimates. (Reprinted with permission from Kulmala, M. and Laaksonen, A. Binary nucleation of water sulfuric acid system. Comparison of classical theories with different H2SO4 saturation vapor pressure, J. Chem. Phys. 93, 696–701. Copyright 1990 American Institute of Physics.)

425

426

THERMODYNAMICS OF AEROSOLS

FIGURE 10.15 Equilibrium vapor pressure of H2SO4 in a mixture with water over a flat surface as a function of composition and temperature.

FIGURE 10.16 Equilibrium concentration of H2SO4 in a spherical droplet of H2SO4 and H2O as a function of relative humidity and droplet diameter.

427

THERMODYNAMICS OF ATMOSPHERIC AEROSOL SYSTEMS

TABLE 10.7 Inorganic Chemical Reactions Occurring in Atmospheric Aerosols Equilibrium constant value K (298)a

Reaction NaCl(s) + HNO3(g) ⇌ NaNO3(s) + HCl(g) HSO4 ƒ H‡ ‡ SO24 NH3 …g† ‡ HNO3 …g† ƒ NH‡4 ‡ NO3 HCl(g) ⇌ H+ + Cl NH3 …g† ‡ HCl…g† ƒ NH‡4 ‡ Cl Na2 SO4 …s† ƒ 2Na‡ ‡ SO24 …NH4 †2 SO4 …s† ƒ 2NH‡4 ‡ SO24 HNO3 …g† ƒ H‡ ‡ NO3 NH4Cl(s) ⇌ NH3(g) + HCl(g) NH4 NO3 …s† ⇌ NH3 …g† ‡ HNO3 …g† NaCl(s) ⇌ Na+ + Cl NaHSO4 …s† ƒ Na‡ ‡ HSO4 NaNO3 …s† ƒ Na‡ ‡ HNO3 where

a

3.96 1.01 × 10 2 (mol kg 1) 4.0 × 1017 (mol2 kg 2 atm 2) 1.97 × 106 (mol2 kg 2 atm 1) 2.12 × 1017 (mol2 kg 2 atm 2) 0.48 (mol3 kg 3) 1.82 (mol3 kg 3) 2.51 × 106 (mol2 kg 2 atm 1) 1.09 × 10 16 (atm2) 4.20 × 10 17 (atm2) 37.74 (mol2 kg 2) 2.41 × 104 (mol2 kg 2) 11.97 (mol2 kg 2)

K…T† ˆ K…298†exp a

298 T

1 ‡ b 1 ‡ ln

298 T

5.50 8.85 64.7 30.20 65.08 0.98 2.65 29.17 71.00 74.74 1.57 0.79 8.22

b 2.18 25.14 11.51 19.91 14.51 39.57 38.57 16.83 2.40 6.03 16.89 14.75 16.0

298 T

a

Equilibrium constant values in this table are based on products divided by reactants.

Source: Kim et al. (1993).

The reaction

H2 SO4 …g† ƒ H‡ ‡ HSO4

can usually be neglected in atmospheric aerosol calculations as the vapor pressure of H2SO4(g) is practically zero over atmospheric particles. The whole system is then described by the bisulfate dissocia­ tion reaction, HSO4 ƒ H‡ ‡ SO24 with an equilibrium constant at 298 K (Table 10.7) given by 1:01  10

2

…mol kg † ˆ 1

γ2H‡ ;SO2 mH‡ mSO2 4

4

γHSO4 mHSO4

The molar ratio of HSO4 to SO24 is obtained by γH‡ ;SO2 mHSO4 4 ˆ 99 mH‡ mSO2 γHSO4 2

4

As the ratio is proportional to the hydrogen ion concentration, it is expected that HSO4 will be present in acidic particles.

10.4.2 The Sulfuric Acid–Ammonia–Water System Consider a simplified system containing only H2SO4, NH3, and water. The problem we are interested in is the following: given the temperature (T), RH, and total available ammonia and sulfuric acid, determine the aerosol composition.

428

THERMODYNAMICS OF AEROSOLS

FIGURE 10.17 Aerosol composition for a system containing 10 μg m 3 H2SO4 at 30% RH and T = 298 K as a function of the total (gas plus aerosol) concentration. The composition was calculated using the approach of Pilinis and Seinfeld (1987).

A series of compounds may exist in the aerosol phase including solids like letovicite [(NH4)3H(SO4)2], (NH4)2SO4, and NH4HSO4, but also aqueous solutions of NH‡4 , SO24 , HSO4 , and NH3(aq). The possible reactions in the system are shown in Table 10.7. For environments with low NH3 availability, sulfuric acid exists in the aerosol phase in the form of H2SO4. As the NH3 available increases, H2SO4 is converted to HSO4 and its salts, and finally, if there is an abundance of NH3, to SO24 and its salts. Figure 10.17 illustrates this behavior for an environment with low relative humidity (30%) and a fixed amount of total H2SO4 available equal to 10 μg m 3. For total (gas and aerosol) ammonia concentrations less than 0.8 μg m 3 the aerosol phase consist mainly of H2SO4(aq), while some NH4HSO4(s) is also present. A significant amount of water accompanies the H2SO4(aq) even at this low relative humidity. This regime is characterized by an ammonia/sulfuric acid molar ratio of approximately less than 0.5. When the ammonia/sulfuric acid molar ratio is between 0.5 and 1.25, NH4HSO4(s) is the dominant aerosol component for this system. When the ratio reaches unity (ammonia concentration 1.8 μg m 3), the salt (NH4)3H(SO4)2(s) (letovicite) is formed in the aerosol phase, gradually replacing NH4HSO4(s). For a molar ratio of 1.25, (NH4)3H(SO4)2(s) is the dominant aerosol component, and for a value equal to 1.5, the aerosol phase consists almost exclusively of letovicite. For even higher ammonia concentrations (NH4)2SO4 starts forming. At an ammonia concentration equal to 3.6 μg m 3 (molar ratio 2), the aerosol consists of only (NH4)2SO4(s). If the total ammonia concentration is lower than 3.6 μg m 3, practically all exists in the aerosol phase in the form of sulfate salts. Further increases in the available ammonia do not change the aerosol composition, but rather the excess ammonia remains in the gas phase as NH3(g).

THERMODYNAMICS OF ATMOSPHERIC AEROSOL SYSTEMS

The total aerosol mass during the variation of total ammonia in the system is also shown in Figure 10.17. One would expect that an increase of the availability of NH3, an aerosol precursor, would result in a monotonic increase of the total aerosol mass. This is not the case for at least the ammonia-poor conditions (ammonia/sulfuric acid molar ratio less than 1). The increase of NH3 in this range results in a reduction of the H2SO4(aq) and the accompanying water. The overall aerosol mass decreases mainly because of the loss of water, reaching a minimum for an ammonia concentration of 1.8 μg m 3. Further increases of ammonia result in increases of the overall aerosol mass. This nonlinear response of the aerosol mass to changes in the concentration of an aerosol precursor is encountered often in atmospheric aerosol thermodynamics. The behavior of the same system at a higher relative humidity, 75%, is illustrated in Figure 10.18. Recall that the deliquescence relative humidities for the NH4HSO4 and (NH4)3H(SO4)2 salts at this temperature are 40% and 69%, respectively (Table 10.1). As a result, the aerosol is a liquid solution of H2SO4, HSO4 , SO24 , and NH‡4 for the ammonia concentration range where the formation of these salts is favored. Once more, for ammonia/sulfuric acid molar ratios less than 0.5, H2SO4 dominates; for ratios between 0.5 and 1.5, HSO4 is the main aerosol component; while for higher ammonia concentrations, sulfate is formed. When there is enough ammonia to completely neutralize the available sulfate, forming (NH4)2SO4, the liquid aerosol loses its water, becoming solid (Figure 10.18). The deliquescence relative humidity of (NH4)2SO4 is 80% at this temperature, and assuming that we are on the deliquescence branch of the hysteresis curve, the aerosol will exist as a solid. These changes in the aerosol chemical composition and the accompanying phase transformation change significantly the hygroscopic

FIGURE 10.18 Aerosol composition for a system containing 10 μg m 3 H2SO4 at 75% RH and T = 298 K as a function of the total (gas plus aerosol) concentration. The composition was calculated using the approach of Pilinis and Seinfeld (1987).

429

430

THERMODYNAMICS OF AEROSOLS

character of the aerosol, resulting in a monotonic decrease of its water mass. The net result is a significant reduction of the overall aerosol mass with increasing ammonia for this system with an abrupt change accompanying the phase transition from liquid to solid. Summarizing, in very acidic atmospheres (NH3/H2SO4 molar ratio less than 0.5) the aerosol particles exist primarily as H2SO4 solutions. For acidic atmospheres (NH3/H2SO4 molar ratio in the 0.5–1.5 range) the particles exist mainly as bisulfate. If there is sufficient ammonia to neutralize the available sulfuric acid, (NH4)2SO4 (or an NH‡4 and SO24 solution) is the preferred composition of the aerosol phase. For acidic atmospheres all the available ammonia is taken up by the aerosol phase and only for NH3/H2SO4 molar ratios above 2 can ammonia also exist in the gas phase.

10.4.3 The Ammonia–Nitric Acid–Water System Ammonia and nitric acid can react in the atmosphere to form ammonium nitrate, NH4NO3: NH3 …g† ‡ HNO3 …g† ƒ NH4 NO3 …s†

(10.87)

Ammonium nitrate is formed in areas characterized by high ammonia and nitric acid concentrations and, as we will see in the next section, low sulfate concentrations. Depending on the ambient RH, ammonium nitrate may exist as a solid or as an aqueous solution of NH‡4 and NO3 . Equilibrium concentrations of gaseous NH3 and HNO3, and the resulting concentration of solid or aqueous NH4NO3, can be calculated from fundamental thermodynamic principles using the method presented by Stelson and Seinfeld (1982a). The procedure is composed of several steps, requiring as input the ambient temperature and relative humidity. First, the equilibrium state of NH4NO3 is defined. If the ambient relative humidity is less than the deliquescence relative humidity (DRH), given by ln …DRH† ˆ

723:7 ‡ 1:6954 T

(10.88)

then the equilibrium state of NH4NO3 is a solid. For example, at 298 K the corresponding DRH is 61.8%, while at 288 K it increases to 67%. At low RH, when NH4NO3 is a solid, the equilibrium condition for (10.87) is μNH3 ‡ μHNO3 ˆ μNH4 NO3

(10.89)

and using the definitions of the chemical potentials of ideal gases and solids: exp

μ∗NH4 NO3

μNH ° 3 RT

μHNO ° 3

ˆ Kp …T† ˆ pNH3 pHNO3

(10.90)

The dissociation constant, Kp(T), is therefore equal to the product of the partial pressures of NH3 and HNO3. Kp can be estimated by integrating the van’t Hoff equation (Denbigh 1981). The resulting equation for Kp in units of ppb2 (assuming 1 atm of total pressure) is (Mozurkewich 1993) ln Kp ˆ 118:87

24084 T

6:025 ln …T=298†

(10.91)

This dissociation constant is shown as a function of temperature in Figure 10.19. Note that the constant is quite sensitive to temperature changes, varying over more than two orders of magnitude for typical ambient conditions. Lower temperatures correspond to lower values of the dissociation constant, and therefore lower equilibrium values of the NH3 and HNO3 gas-phase concentrations. Thus lower temperatures shift the equilibrium of the system toward the aerosol phase, increasing the aerosol mass of NH4NO3 (Figure 10.20). Equation (10.91) is accurate within 10–15%. If higher accuracy is required the more complex expression of Table 10.7 should be used.

THERMODYNAMICS OF ATMOSPHERIC AEROSOL SYSTEMS

FIGURE 10.19 NH4NO3 equilibrium dissociation constant as a function of tempera­ ture at RH = 50%.

FIGURE 10.20 NH4NO4(s) concentration as a function of temperature for a system containing 7 μg m 3 NH3 and 26.5 μg m 3 HNO3 at 30% RH. The difference between the total available mass (dashed line) and the NH4NO3 mass remains in the gas phase as NH3(g) and HNO3(g).

Solid Ammonium Nitrate Formation Calculate the NH4NO3(s) mass for a system containing 17 μg m 3 of NH3 and 63 μg m 3 of HNO3 at 308 K, 30% RH, and 1 atm. At 308 K, according to (10.88), the DRH of NH4NO3 is 57.1%, and therefore at 30% RH the NH4NO3, if it exists in stable equilibrium, will be solid. Let us first determine whether the atmosphere is saturated with NH3 and HNO3 by calculating the product of the corresponding mixing ratios. For 308 K and 1 atm, we obtain Mixing ratio of species i in ppb ˆ

25:6  …concentration of i in μg m 3 † Mi

and ξNH3 ˆ ξHNO3 ˆ 25:6 ppb. Thus ξNH3 ξHNO3 ˆ 655 ppb2 . Using (10.91), we find that Kp(308 K) = 486 ppb2 (in mixing ratio units) and

431

432

THERMODYNAMICS OF AEROSOLS

ξNH3 ξHNO3 > Kp The system is therefore supersaturated with ammonia and nitric acid and a fraction of them will be transferred to the aerosol phase to establish equilibrium, given mathematically by (10.90). Let us assume that x ppb of NH3 is transferred to the aerosol phase. Because reaction (10.87) requires 1 mole of HNO3 for each mole of NH3 reacting, x ppb of HNO3 will also be transferred to the aerosol phase. The gas phase will contain (25.6 x) ppb of HNO3 and (25.6 x) ppb of NH3. The system will be in equilibrium and therefore the new gas-phase mixing ratios will satisfy (10.90), or x†2 ˆ Kp ˆ 486

…25:6

resulting in x = 3.6 ppb. Therefore 3.6 ppb of NH3 and 3.6 ppb of HNO3 will form solid ammonium nitrate. These correspond to 11.3 μg m 3 of NH4NO3(s). 10.4.3.1 Ammonium Nitrate Solutions At relative humidities above that of deliquescence, NH4NO3 will be found in the aqueous state. The corresponding dissociation reaction is then NH3 …g† ‡ HNO3 …g† ƒ NH‡4 ‡ NO3

(10.92)

Stelson and Seinfeld (1981) have shown that solution concentrations of 8–26 M can be expected in wetted atmospheric aerosol. At such concentrations the solutions are strongly nonideal, and appro­ priate thermodynamic activity coefficients are necessary for thermodynamic calculations. Tang (1980), Stelson and Seinfeld (1982a–c), and Stelson et al. (1984) have developed activity coefficient expressions for aqueous systems of nitrate, sulfate, ammonium, nitric acid, and sulfuric acid at concentrations exceeding 1 M. Following the approach illustrated in Sections 10.1.5 and 10.1.6, one can show that the condition for equilibrium of reaction (10.92) is μHNO3 ‡ μNH3 ˆ μNH‡4 ‡ μNO3

(10.93)

or after using the definitions of the corresponding chemical potentials exp

μ⋄NO3

° 3 ‡ μNH ° 3 μHNO

RT

μ⋄NH‡ 4

ˆ

γNH‡4 γNO3 mNH‡4 mNO3 pHNO3 pNH3

(10.94)

The right-hand side is the equilibrium constant, and defining γ2NH4 NO3 ˆ γNH‡4 γNO3

(10.95)

the equilibrium relationship becomes KAN ˆ

γ2NH4 NO3 mNH‡4 mNO3

(10.96)

pHNO3 pNH3

The equilibrium constant can be calculated by KAN ˆ 4  1017 exp 64:7 where KAN is in mol2 kg

2

atm 2.

298 T

1 ‡ 11:51 1 ‡ ln

298 T

298 T

(10.97)

433

THERMODYNAMICS OF ATMOSPHERIC AEROSOL SYSTEMS

Solution of (10.96) for a given temperature requires calculation of the corresponding molalities. These concentrations depend not only on the aerosol nitrate and ammonium but also on the amount of water in the aerosol phase. Therefore calculation of the aerosol solution composition requires estimation of the aerosol water content. As we have seen in Section 10.2.1, the water activity will be equal to the relative humidity (expressed in the 0–1 scale). While this is very useful information, it is not sufficient for the water calculation. One needs to relate the tendency of the aerosol components to absorb moisture with their availability and the availability of water given by the relative humidity. In atmospheric aerosol models (Hanel and Zankl 1979; Cohen et al. 1987; Pilinis and Seinfeld 1987; Wexler and Seinfeld 1991) the water content of aerosols is usually predicted using the ZSR relationship (Zdanovskii 1948; Stokes and Robinson 1966) Wˆ i

Ci mi;0 …aw †

(10.98)

where W is the mass of aerosol water in kg of water per m3 of air, Ci is the aqueous-phase concentration of electrolyte i in moles per m3 of air, and mi,0(aw) is the molality (mol kg 1) of a single-component aqueous solution of electrolyte i that has a water activity aw = RH/100. Equations (10.96) and (10.98) form a nonlinear system that has as a solution the equilibrium composition of the system. The calculation requiring the activity coefficient functions for (10.96) is quite involved and is usually carried out by appropriate thermodynamic codes. Figure 10.21 depicts the results of such a computation, namely, the product of the mixing ratios of ammonia and nitric acid over a solution as a function of relative humidity. Assuming that there is no (NH4)2SO4 present, this corresponds to the line Y = 1.00. The product of the mixing ratios decreases rapidly with relative humidity, indicating the shifting of the equilibrium toward the aerosol phase. The presence of water allows NH4NO3 to dissolve in the liquid aerosol particles and increases its aerosol concentration. Defining the equilibrium dissociation constant as ξHNO3 ξNH3 shown in Figure 10.21, the equilibrium concentrations of NH3(g), HNO3(g), NH‡4 , and NO3 (all in mol m 3 of air) can be calculated for a specific relative humidity. Note that by using (10.96) Kp is related to KAN by Kp ˆ

γ2NH4 NO3 mNH‡4 mNO3 KAN

(10.99)

Aqueous Ammonium Nitrate Formation Calculate the ammonium nitrate concentration for a system containing 5 ppb of total (gas and aerosol) NH3 and 6 ppb of total HNO3 at 25 °C at 80% RH. Using Figure 10.21, we find that the equilibrium product is approximately 15 ppb2. The actual product of the total concentrations is 5 × 6 = 30 ppb2, and therefore the system is supersaturated with ammonia and nitric acid and some ammonium nitrate will form. If x ppb of ammonium nitrate is formed, then there will be 5 x ppb of NH3 and 6 x ppb of HNO3 left. At equilibrium, the product of these remaining concentrations will be the equilibrium product and therefore …5

x† …6

x† ˆ 15 or x ˆ 1:6 ppb

There is one more mathematical solution of the problem (x = 9.4 ppb), but obviously it is not acceptable because it exceeds the total available concentrations of NH3 and HNO3 in the system. Therefore the amount of aqueous ammonium nitrate will be 1.6 ppb or approximately 5.2 μg m 3. Figure 10.22 shows the calculated ammonium and nitrate for a closed system as a function of its RH. The ammonium nitrate concentration increases by a factor of 4.3 in this example as the RH increases from 65% to 95%. Further increases of the RH will lead to the complete dissolution of practically all the available nitric acid to the particulate phase (see also Chapter 7).

434

THERMODYNAMICS OF AEROSOLS

FIGURE 10.21 NH4NO3 equilibrium dissociation constant for an ammonium sulfate–nitrate solution as a function of relative humidity and ammonium nitrate ionic strength fraction defined by

Yˆ

‰NH4 NO3 Š ‰NH4 NO3 Š ‡ 3‰…NH4 †2 SO4 Š

at 298 K (Stelson and Seinfeld 1982c). Pure NH4NO3 corresponds to Y = 1.

It is important to note the variation of the product of pNH3 pHNO3 and the corresponding aerosol composition over several orders of magnitude under typical atmospheric conditions. This variation may introduce significant difficulties in the measurement of aerosol NH4NO3 as small changes in RH and T will shift the equilibrium and cause evaporation/condensation of the aerosol sample.

10.4.4 The Ammonia–Nitric Acid–Sulfuric Acid–Water System The system of interest consists of the following possible components: Gas phase: NH3, HNO3, H2SO4, H2O. Solid phase: NH4HSO4, (NH4)2SO4, NH4NO3, (NH4)2SO4
2NH4NO3, (NH4)2SO4
3NH4NO3, (NH4)3H(SO4)2. Aqueous phase: NH‡4 , H+, HSO4 , SO24 , NO3 , H2O. Two observations are useful in determining a priori the composition of the aerosol that exists in such a system: 1. Sulfuric acid in the presence of water vapor possesses an extremely low vapor pressure. 2. (NH4)2SO4 solid or aqueous is the preferred form of sulfate.

THERMODYNAMICS OF ATMOSPHERIC AEROSOL SYSTEMS

FIGURE 10.22 Calculated aerosol ammonium and nitrate concentra­ tions as a function of relative humidity for an atmosphere with [TA] = 5 μg m 3, [TN] = 10 μg m 3, and T = 295 K.

The second observation means that, if possible, each mole of sulfate will remove 2 moles of ammonia from the gas phase, and the first observation implies that the amount of sulfuric acid in the gas phase will be negligible. On the basis of these observations, we can define two regimes of interest: ammonia-rich and ammonia-poor. If [TA], [TS], and [TN] are the total (gas + aqueous + solid) molar concentrations of ammonia, sulfate, and nitrate, respectively, then the two cases are 1. Ammonia-poor, [TA] < 2[TS]. 2. Ammonia-rich, [TA] > 2[TS]. Case 1: Ammonia-Poor. In this case there is insufficient NH3 to neutralize the available sulfate. Thus the aerosol phase will be acidic. The vapor pressure of NH3 will be low, and the sulfate will tend to drive the nitrate to the gas phase. Since the NH3 partial pressure will be low, the NH3–HNO3 partial pressure product will also be low so ammonium nitrate levels will be low or zero. Sulfate may exist as bisulfate. Case 2: Ammonia-Rich. In this case there is excess ammonia, so that the aerosol phase will be neutralized to a large extent. The ammonia that does not react with sulfate will be available to react with nitrate to produce NH4NO3. The behavior in these two different regimes is depicted in Figure 10.23. The transition occurs at approximately 3.5 μg m 3. At very low ammonia concentrations, sulfuric acid and bisulfate constitute the aerosol composition. As ammonia increases, ammonium nitrate becomes a significant aerosol constituent. The aerosol liquid water content responds nonlinearly to these changes, reaching a minimum close to the transition between the two regimes. The accurate calculation of the ammonium nitrate concentrations for this rather complicated system is usually performed using computational thermodynamic models (see Section 10.5). However, one can often estimate the ammonium nitrate concentration in an air parcel with a rather simple approach taking advantage of appropriate diagrams that summarize the results of the computational models. For ammonia-poor systems one can assume as a first-order approximation that the ammonium nitrate concentration is very small. For ammonia-rich systems when the particles are solid, the approach is the same as that of Section 10.4.3 with one important difference. In these calculations one uses the free ammonia instead of the total ammonia. The free ammonia in the system is defined as the total ammonia

435

436

THERMODYNAMICS OF AEROSOLS

FIGURE 10.23 Calculated aerosol composition as a function of total ammonia for an atmosphere with [TS] = 10 μg m 3, [TN] = 10 μg m 3, T = 298 K, and RH = 70%.

minus the ammonia required to neutralize the available sulfate. So in molar units, we have ‰Free ammoniaŠ ˆ ‰TAŠ

2‰TSŠ

NH4NO3 Calculation for Solid Particles in the NH3–HNO3–H2SO4 System Let us repeat the first example of Section 10.4.3 (for a system with 17 μg m 3 of NH3, 63 μg m 3 HNO3, T = 308 K, RH = 30%) assuming now that there are also 10 μg m 3 of sulfuric acid. We first need to estimate the NH3 necessary for the full neutralization of the available sulfate, which is equal to 10 × 2 × 17/98 = 3.5 μg m 3 NH3. Therefore, from the available ammonia, only 17 3.5 = 13.5 μg m 3 will be free for the potential formation of NH4NO3. Converting the concentra­ tions to ppb, the nitric acid mixing ratio will be 25.6 ppb, while the free ammonia mixing ratio will now be 20.3 ppb. The mixing ratio product will then be 520 ppb2, exceeding the equilibrium mixing ratio product Kp(308 K) = 486 ppb2. The system will therefore be supersaturated with nitric acid and free ammonia, and x ppb of ammonium nitrate will be formed. The ammonium nitrate concentration can be found from the equilibrium condition for the gas phase: …25:6

x† …20:3

x† ˆ Kp ˆ 486

resulting in x = 0.75 ppb or 2.3 μg m 3 of NH4NO3. For zero sulfate, we found in the previous example that 11.3 μg m 3 of NH4NO3 will be formed. Therefore, for this system, the removal of 10 μg m 3 of

437

THERMODYNAMICS OF ATMOSPHERIC AEROSOL SYSTEMS

sulfuric acid results in the removal of 10 × (96 + 36)/98 = 13.5 μg m 3 of (NH4)2SO4 and the increase of the NH4NO3 by 11.3 2.3 = 9 μg m 3. Roughly 67% of the concentration of the reduced ammonium sulfate will be replaced by ammonium nitrate. For ammonia-rich aqueous systems the ammonium nitrate calculation is complicated by the dependence of Kp not only on T and RH but also on the sulfate concentration. This dependence is usually expressed by the parameter Y (Figure 10.21). The parameter Y is the ionic strength fraction of ammonium nitrate and is calculated as Yˆ

‰NH4 NO3 Š ‰NH4 NO3 Š ‡ 3‰…NH4 †2 SO4 Š

(10.100)

As the (NH4)2SO4 in the particles increases compared to the ammonium nitrate, the parameter Y decreases and the equilibrium product of ammonia and nitric acid decreases. The additional ammonium and sulfate ions make the aqueous solution a more favorable environment for ammonium nitrate, shifting its partitioning toward the particulate phase. Therefore, addition of ammonium sulfate to aqueous aerosol particles will tend to increase the concentration of ammonium nitrate in the particulate phase, and, vice versa, reductions of ammonium sulfate can lead to decreases of ammonium nitrate for aqueous aerosol. The parameter Y depends on the unknown ammonium nitrate concentration, making an iterative solution necessary. For this approach, one assumes a value of the ammonium nitrate concentration, calculates the corresponding Y, and finds from Figure 10.21 the corresponding Kp. Then one can follow the same approach as that for solid ammonium nitrate and calculate the actual ammonium nitrate concentration. If the assumed value and the calculated one are the same, then this value is the solution of the problem. If not, these values are used to improve the initial guess and the process is repeated. Calculation of Aqueous Ammonium Nitrate Concentration Estimate the ammonium nitrate concentration in aqueous atmospheric particles in an area charac­ terized by the following conditions: Total (gas and aerosol) ammonia: 10 μg m 3

Total (gas and aerosol) nitric acid: 10 μg m 3

Sulfuric acid: 10 μg m 3

T = 298 K and RH = 75%

We first need to estimate the NH3 necessary for the full neutralization of the available sulfuric acid, which is equal to 10 × 2 × 17/98 = 3.5 μg m 3 NH3. Therefore from the available ammonia only 10 3.5 = 6.5 μg m 3 NH3 will be available for the potential formation of ammonium nitrate. Converting all the concentrations to ppb: Free NH3 ˆ 8:314  298  6:5=…100  17† ˆ 9:5 ppb Total HNO3 ˆ 8:314  298  10=…100  53† ˆ 4:7 ppb Sulfuric acid ˆ 8:314  298  10=…100  98† ˆ 2:5 ppb Assuming that all sulfuric acid will be in the form of sulfate in the particulate phase, there will also be 2.5 ppb of sulfate. The product of the free ammonia and total nitric acid concentrations is ξNH3 ξHNO3 ˆ 9:5  4:7 ˆ 44:6 ppb2 . This value exceeds the corresponding equilibrium concentration products at 298 K and 75% RH for all values of Y (Figure 10.21), and therefore some ammonium nitrate will be formed. As a first guess, we assume that 2 ppb of NH4NO3 will be formed from 2 ppb of NH3 and 2 ppb of HNO3 leaving in the gas phase 7.5 ppb NH3 and 2.7 ppb HNO3. The corresponding gas-phase concentration product ξNH3 ξHNO3 will then be 20.3 ppb2. We need to estimate now the equilibrium

438

THERMODYNAMICS OF AEROSOLS

concentration production for the aqueous aerosol and compare it to this value. The Y for this composition is Y = 2/(2 + 3 × 2.5) = 0.2, and using Figure 10.21, we find that Kp is approximately 8 ppb2. Obviously, the use of approximate values from a graph introduces some uncertainty in these calculations. For this trial, we were left with a gas-phase concentration product of 20.3 ppb2 and an aqueous aerosol phase with an equilibrium product of 8 ppb2. These two are not equal, and therefore we need to increase the assumed ammonium nitrate concentration. Let us now improve our first guess by assuming that 3 ppb of NH4NO3 will be formed from 3 ppb NH3 and 3 ppb HNO3. There will be 6.5 ppb NH3 and 1.7 ppb HNO3 remaining in the gas phase, resulting in a concentration product of 11 ppb2. For the particulate phase the corresponding Y = 3/ (3 + 3 × 2.5) = 0.3, and using Figure 10.21, Kp is approximately 10 ppb2. The assumed composition brings the system quite close to equilibrium, and therefore approximately 3 ppb of ammonium nitrate is dissolved in the aqueous particles. This corresponds to roughly 2.2 μg m 3 of NH‡4 and 6 μg m 3 of NO3 . The total concentration of inorganic particles will be 9.8 + 3.5 + 2.2 + 6 = 21.5 μg m 3. For more accurate results, these calculations are routinely performed today by suitable thermodynamic codes (see Section 10.4.6). The formation of ammonium nitrate is often limited by the availability of one of the reactants. Figure 10.24 shows the ammonium concentration as a function of the total available ammonia and the total available nitric acid for a polluted area. The upper left part of the figure (area A) is characterized by relatively high total nitric acid concentrations and relatively low ammonia. Large urban areas are often in this regime. The isopleths are almost parallel to the y axis in this area, so decreases in nitric acid availability do not significantly affect the NH4NO3 concentration in the area. On the other hand, decreases in ammonia result in significant decreases in ammonium nitrate. NH4NO3 formation is limited by ammonia in area A. The opposite behavior is observed in area B. Here, NH4NO3 is not sensitive to changes in ammonia concentrations but responds readily to changes in nitric acid availability. Rural areas with significant ammonia emissions are often in this regime, where NH4NO3 formation is nitric acid–limited. The boundary between the nitric acid–limited and ammonia-limited regimes depends on temperature, relative humidity, and sulfate concentrations and also the concentrations of other major inorganic aerosol components. Summarizing, reductions of sulfate concentrations affect the inorganic particulate matter concen­ trations in two different ways. Part of the sulfate may be replaced by nitric acid, leading to an increase in the ammonium nitrate content of the aerosol. The sulfate decrease frees up ammonia to react with nitric acid and transfers it to the aerosol phase. If the particles are aqueous, there is also a second effect.

FIGURE 10.24 Isopleths of predicted particulate nitrate concen­ tration (μg m 3) as a function of total (gaseous + particulate) ammonia and nitric acid at 293 K and 80% RH for a system containing 25 μg m 3 of sulfate.

THERMODYNAMICS OF ATMOSPHERIC AEROSOL SYSTEMS

FIGURE 10.25 Calculated aerosol composition as a function of total sulfate for an atmosphere with [TA] = 10 μg m 3, [TN] = 20 μg m 3, T = 298 K, and RH = 70%.

The reduction of the sulfate ions in solution increases the equilibrium vapor pressure product of ammonia and nitric acid of the particles (see Figure 10.21), shifting the ammonium nitrate partitioning toward the gas phase. So for aqueous particles, the substitution of sulfate by nitrate is in some cases relatively small. These interactions are illustrated in Figure 10.25. Sulfate reductions are accompanied by an increase in the aerosol nitrate and a reduction in the aerosol ammonium, water, and total mass. However, the reduction in mass is nonlinear. For example, a reduction in total sulfate of 20 μg m 3 (from 30 to 10 μg m 3) results in a net reduction in the dry aerosol mass (excluding water) of only 12.9 μg m 3, because of the increase in the aerosol nitrate by 10 μg m 3. Ammonium decreases by only 2.9 μg m 3.

10.4.5 Other Inorganic Aerosol Species Aerosol Na+ and Cl are present in substantial concentrations in regions close to seawater. Sodium and chloride interact with several aerosol components (Table 10.7). Various solids can be formed during these reactions, including ammonium chloride, sodium nitrate, sodium sulfate, and sodium bisulfate, while HCl(g) may be released to the gas phase. Addition of NaCl to an urban aerosol can have a series of interesting effects, including the reaction of NaCl with HNO3: NaCl…s† ‡ HNO3 …g† ƒ NaNO3 …s† ‡ HCl…g†

439

440

THERMODYNAMICS OF AEROSOLS

As a result of this reaction, more nitrate is transferred to the aerosol phase and is associated with the large seasalt particles. At the same time, hydrochloric acid is liberated and the aerosol particles appear to be “chloride”-deficient. This deficiency may also be a result of the reactions 2NaCl…s† ‡ H2 SO4 …g† ƒ Na2 SO4 …s† ‡ 2 HCl…g† NaCl…s† ‡ H2 SO4 …g† ƒ NaHSO4 …s† ‡ HCl…g†

10.4.6 Organic Aerosol Atmospheric aerosols comprise inorganic material, such as mineral dust, seasalt, sulfates, nitrates, ammonia, and a vast variety of organic compounds. With the advent of modern analytical chemistry as applied to atmospheric aerosols, a view into the composition of ambient particles emerged (e.g., Rogge et al. 1993; Middlebrook et al. 1998; Lee et al. 2002; Kanakidou et al. 2005; Decesari et al. 2006; Murphy et al. 2006; Zhang et al. 2007; Hallquist et al. 2009; Jimenez et al. 2009). Globally, roughly 50% of the submicrometer-size aerosol mass is organic material (Murphy et al. 2006; Zhang et al. 2007; Hallquist et al. 2009; Jimenez et al. 2009). A substantial fraction of the organic aerosol is formed by condensation of low-volatility products of hydrocarbon oxidation reactions in the gas phase, so-called secondary organic aerosol (see Chapter 14). Individual particle measurements reveal that organic and inorganic species typically coexist in the same particle (but not necessarily as a single mixed phase). The extent to which organic and inorganic species partition between the gas and particle phases depends on temperature and relative humidity (RH), pure component vapor pressures, the physical state of the condensed phase, and the thermodynamic interactions among aerosol components. The prevalence of the few main inorganic aerosol components, sulfate, nitrate, ammonium, chloride, and water, is well characterized by a wealth of ambient measurements (e.g., Zhang et al. 2007). In contrast, the organic aerosol fraction generally consists of a multitude of compounds (e.g., Rogge et al. 1993; Saxena and Hildemann 1996; Decesari et al. 2000, 2006; Goldstein and Galbally 2007; Russell et al. 2009). Thus, a single organic compound is generally present only as a small fraction of the entire aerosol mixture. Nevertheless, most organic aerosol compounds consist of different combina­ tions of a small number of functional groups and, as a result, share similar chemical and physical properties. This commonality allows the use of so-called group contribution methods that represent the properties of organic molecules as combinations of those of a relatively small set of functional groups. Bowman and Melton 2004; Chang and Pankow 2006; Zuend et al. 2010; Zuend and Seinfeld 2012). In addition to these physicochemical effects, liquid and solid phases and surfaces present in aerosol particles provide reaction media for heterogeneous and multiphase chemistry (Ravishankara 1997; Herrmann et al. 2005).

10.5 AEROSOL THERMODYNAMIC MODELS Thermodynamic models constitute a powerful tool for the computation of phase behavior and gas– particle equilibrium of organic–inorganic mixtures. Such models can effectively address the overall gas– particle distribution of chemical species and the extent to which phase separation occurs in the particle. Thermodynamic models are based on minimization of the Gibbs free energy with respect to the composition and phase state of the particle. This is a constrained nonlinear optimization problem. For discussion of numerical approaches to solving these problems we refer the reader, for example, to Zuend and Seinfeld (2012, 2013). The first thermodynamic models for inorganic atmospheric aerosols were developed in the mid­ 1980s: EQUIL (Bassett and Seinfeld 1983), MARS (Saxena et al. 1986), and SEQUILIB (Pilinis and Seinfeld 1987). These original models evolved with the use of more accurate activity coefficient models and improved numerical algorithms (Kim et al. 1993; Nenes et al. 1998). Two models that are widely used are AIM (aerosol inorganics model), developed by Clegg, Wexler, and Brimblecombe (Wexler and Clegg, 2002; Clegg and Seinfeld 2004, 2006; Clegg et al. 2008) (www.aim.env.uea.ac.uk), and ISORROPIA,

441

AEROSOL THERMODYNAMIC MODELS

developed by Nenes, Pilinis, and Pandis (Nenes et al. 1998, 1999; Fountoukis and Nenes 2007) (www .isorropia.eas.gatech.edu). AIM is considered an accurate benchmark model while ISORROPIA has been optimized for use in chemical transport models. AIM calculates gas/liquid/solid partitioning in aerosol systems containing inorganic and selected organic components and water, and solute and solvent activities in aqueous solutions and liquid mixtures. The principal models included within the AIM family are listed here: (I) (II) (III) (IV)

H‡ H‡ H‡ H‡

SO24 NH‡4 NH‡4 NH‡4

NO3 Cl Br H2 O SO24 NO3 H2 O Na‡ SO24 NO3 Cl Na‡ SO24 NO3 Cl

H2 O H2 O

( 1. (We will see later that when two vapor species A and B are nucleating together, their individual saturation ratios need not exceed 1.0 for nucleation to occur.) When the saturation ratio is raised above unity, there is an excess of monomer molecules over that at S = 1. These excess monomers bombard clusters and produce a greater number of clusters of larger size than exist at S = 1. If the value of S is sufficiently large, then sufficiently large clusters can be formed so that some clusters exceed a critical size, enabling them to grow rapidly to form a new phase. The socalled critical cluster will be seen to be of a size such that its rate of growth is equal to its rate of decay. Clusters that fluctuate to a size larger than the critical size will likely continue to grow to macroscopic size, whereas those smaller than the critical size most likely will shrink. The nucleation rate is the net number of clusters per unit time that grow past the critical size.

11.1 CLASSICAL THEORY OF HOMOGENEOUS NUCLEATION: KINETIC APPROACH The theory of nucleation is based on a set of rate equations for the change of the concentrations of clusters of different sizes as the result of the gain and loss of molecules (monomers). It is usually assumed that the temperatures of all clusters are equal to that of the background gas. This is not strictly correct; phase change involves the evolution of heat. What this assumption implies is that clusters undergo sufficient collisions with molecules of the background gas so that they become thermally equilibrated on a timescale that is short compared to that associated with the gain and loss of monomers. Wyslouzil and Seinfeld (1992) extended the dynamic cluster balance to include an energy balance, but we will not consider this aspect here. Generally the assumption of an isothermal process is a reasonable one for atmospheric nucleation. In this chapter it will prove to be advantageous to use psA as the saturation vapor pressure rather than p°A as used up to this point.

1

450

NUCLEATION

FIGURE 11.1 Cluster growth and evaporation processes.

To develop the rate equation for clusters it is assumed that clusters grow and shrink via the acquisition or loss of single molecules. Cluster–cluster collision events are so rare that they can be ignored, as can those in which a cluster fissions into two or more smaller clusters. Moreover, from the principle of microscopic reversibility, that is, that at equilibrium every forward process has to be matched by its corresponding reverse process, it follows that if clusters grow only by the addition of single molecules, evaporation also occurs one molecule at a time. Let Ni(t) be the number concentration of clusters containing i molecules (monomers) at time t. By reference to Figure 11.1, Ni(t) is governed by the following rate equation d Ni ˆ βi dt

1

N i 1 …t†

γi N i …t†

βi N i …t† ‡ γi‡1 N i‡1 …t†

(11.2)

where βi is the forward rate constant for the collision of monomers with a cluster of size i (a so-called i-mer) and γi is the reverse rate constant for the evaporation of monomers from an i-mer. Equation (11.2) provides the basis for studying transient nucleation. For example, if the monomer concentration is abruptly increased at t = 0, (11.2) governs the time-dependent development of the cluster distribution. Physically, in such a case there is a transient period over which the cluster concentrations adjust to the perturbation in monomer concentration, followed eventually by the establishment of a pseudo-steady-state cluster distribution. Since the characteristic time needed to establish the steady-state cluster distribution is generally short compared to the timescale over which typical monomer concentrations might be changing in the atmosphere, we can assume that the distribution of clusters is always at a steady state corresponding to the instantaneous monomer concentration. There are instances, generally in liquid-to-solid phase transitions, where transient nucleation can be quite important (Shi et al. 1990), although we do not pursue this aspect here. We define Ji+1/2 as the net rate (cm 3 s 1) at which clusters of size i become clusters of size i + 1. This net rate is given by J i‡1=2 ˆ βi N i

γi‡1 N i‡1

(11.3)

If, for a given monomer concentration (or saturation ratio S), the cluster concentrations can be assumed to be in a steady state, then the left-hand side (LHS) equals zero in (11.2). At steady state from (11.2) we see that all the fluxes must be equal to a single, constant flux J: J i‡1=2 ˆ J;

all i

(11.4)

451

CLASSICAL THEORY OF HOMOGENEOUS NUCLEATION: KINETIC APPROACH

Let us define a quantity fi by βi f i ˆ γi‡1 f i‡1

(11.5)

and then, after setting Ji+1/2 = J, divide (11.3) by (11.5) J Ni ˆ βi f i f i

N i‡1 f i‡1

(11.6)

where we have used f i‡1 ˆ

βi f γi‡1 i

(11.7)

Then, setting f1 = 1 and summing (11.6) from i = 1 to some maximum value imax gives imax

J iˆ1

N imax f imax

1 ˆ N1 βi f i

(11.8)

Let us examine the behavior of fi with i: fi ˆ

β β 1 β2


i 1 γi γ2 γ3 i 1

ˆ ∏

(11.9)

βj

jˆ1 γj‡1

The ratio βi 1/γi is the ratio of the rate at which a cluster of size i 1 gains a monomer to the rate at which a cluster of size i loses a monomer. Under nucleation conditions (S > 1) this ratio must be larger than 1; that is, the forward rate must exceed the reverse rate. For large i, fi must increase with i by a factor larger than 1 for each increment of i. Also, based on the relative cluster concentrations, N imax < N 1 , since even under nucleation conditions monomers are still by far the most populous entities. Thus, by choosing imax sufficiently large, the second term on the right-hand side (RHS) of (11.8) will be negligible compared to the first. Then J ˆ N1

imax iˆ1

1 βi f i

1

(11.10)

Since fi is an increasing function of i, the summation in (11.10) can be extended to infinity, with convergence guaranteed by the rapid falloff of 1/fi. Thus

J ˆ N1

1 iˆ1

1 βi f i

1

(11.11)

The net flux J is the nucleation rate, and (11.11) is a direct expression for the nucleation rate as a function of the forward and reverse rate constants for the clusters.

452

NUCLEATION

11.1.1 The Forward Rate Constant βi From the kinetic theory of gases the number of collisions occurring between molecular entities A and B in a gas per unit time and unit volume of the gas is given by (3.6), with slightly different notation 8π kT mAB

ZAB ˆ

1=2

σ2AB N A N B

(11.12)

where mAB, the reduced mass, equals mAmB/(mA + mB), and σAB, the collision diameter, equals rA + rB, the sum of the molecular radii. To determine the collision rate between a monomer and an i-mer, let A represent the monomer and B the i-mer: 1=2

1 1 ‡ m1 mi

Z1i ˆ …8π kT†1=2

…r1 ‡ ri †2 N 1 N i

(11.13)

The i-mer mass is equal to i times the monomer mass (mi = im1). Assuming the monomer and i-mer densities are equal, vi = iv1. Thus Z1i ˆ

8π kT m1

1=2

3v1 4π

2=3

1‡

1=2

1 i

…1 ‡ i1=3 †2 N 1 N i

(11.14)

Since the surface area of a monomer 3v1 4π

2=3

1 a1 1‡ 4π i

1=2

a1 ˆ 4π

(11.15)

then Z1i ˆ

8π kT m1

Z1i ˆ

kT 2π m1

1=2

…1 ‡ i1=3 †2 N 1 N i

(11.16)

…1 ‡ i1=3 †2 N 1 N i

(11.17)

or 1=2

1=2

1 i

a1 1 ‡

The number of collisions occurring between monomer and a single i-mer per unit time (the forward rate constant) is βi ˆ

kT 2π m1

1=2

1‡

1 i

1=2

…1 ‡ i1=3 †2 a1 N 1

(11.18)

which can be written in terms of the monomer partial pressure, p1 = N1kT, as

βi ˆ

p1 …2π m1 kT†1=2

1‡

1 i

1=2

…1 ‡ i1=3 †2 a1

(11.19)

The expression for βi is sometimes multiplied by an accommodation coefficient, that is, the fraction of monomer molecules impinging on a cluster that stick. The accommodation coefficient is generally an

453

CLASSICAL THEORY OF HOMOGENEOUS NUCLEATION: KINETIC APPROACH

unknown function of cluster size. Its effect on the nucleation rate is relatively small since, as will be shown later, the nucleation rate depends on the ratio of forward to reverse rates and in this ratio the accommodation coefficient cancels. Henceforth, we will assume the accommodation coefficient is unity. s in (11.1). Replacing p1 by p1s in (11.19) gives The saturation ratio, S ˆ p1 =ps1 , where p1s is the same as pA s the forward rate constant at saturation, β1 . Then, the forward rate constant under nucleation conditions (S > 1) can be written as βi ˆ Sβsi

(11.20)

11.1.2 The Reverse Rate Constant γi The reverse rate constant γi governs the rate at which molecules leave a cluster. This quantity is more difficult to evaluate theoretically than the forward rate constant βi. What we do know is that, as long as the number concentration of monomer molecules is much smaller than that of the carrier gas, the rate at which a cluster loses monomers should depend only on the cluster size and temperature and be independent of monomer partial pressure. Thus the evaporation rate constant under nucleation conditions (S > 1) should be identical to that in the saturated (S = 1) vapor: γi ˆ γsi

(11.21)

Two different approaches are commonly employed to obtain γsi . One is based on the knowledge of the distribution of clusters in the saturated vapor, and the second is based on the Kelvin equation. These two approaches lead to similar results. In what immediately follows we will employ the cluster distribution at saturation to infer γsi ; the approach based on the Kelvin equation is given in Section 11.2. It is important at this point to define three different cluster distributions that arise in nucleation theory. First is the saturated (S = 1) equilibrium cluster distribution, N si , which we will always denote with a superscript s. Second is the steady-state cluster distribution at S > 1 and a constant net growth rate of J, Ni. At saturation (S = 1) all Ji+1/2 = 0, whereas at steady-state nucleation conditions all Ji+1/2 = J. There is a third distribution that we will not explicitly introduce until the next section. It is the hypothetical, equilibrium distribution of clusters corresponding to S > 1. Thus it corresponds to all Ji+1/2 = 0, but S > 1. Because of the constraint of zero flux, this third distribution is called the constrained equilibrium distribution, N ei . We will distinguish this distribution by a superscript e.

11.1.3 Derivation of the Nucleation Rate From (11.9), fi can be written as i 1

fi ˆ ∏

βj

jˆ1 γj‡1

(11.22)

Now, using (11.20) and (11.21), this becomes f i ˆ Si

1

i 1

∏

βsj

s jˆ1 γj‡1

(11.23)

We now have a direct expression for the nucleation rate, (11.11), with fi given by (11.23). The challenge in using this equation lies, as noted, in properly evaluating the evaporation coefficients γsi . To approach this problem consider the formation of a dimer from two monomers at saturation (S = 1) as a reversible chemical reaction: A1 ‡ A1 ƒ A2

454

NUCLEATION

This is not a true chemical reaction but rather a physical agglomeration of two monomers. The forward rate constant is βs1 , and the reverse rate constant is γs2 . The ratio βs1 =γs2 is just the equilibrium constant for the reaction. Similarly, the formation of a trimer can be written as a reversible reaction between a monomer and a dimer A1 ‡ A2 ƒ A3 with an equilibrium constant βs2 =γs3 . The product βs1 γs2

βs2 γs3

is just the equilibrium constant for the overall reaction 3A1 ƒ A3 Thus the product of rate constant ratios in (11.23) is just the equilibrium constant for the formation of an i-mer from i individual molecules under saturated equilibrium conditions: iA1 ƒ Ai If ΔGi is the Gibbs free-energy change for the “reaction” above, then (11.23) can be written as f i ˆ Si

1

exp… ΔGi =kT†

(11.24)

We have thus expressed the quantity fi in terms of the Gibbs free-energy change for formation of a cluster of size i at saturation (S = 1). Now we need to determine ΔGi. It is at this point that the major assumption of classical nucleation theory is invoked. It is supposed that the i molecules are converted to a cluster by the following pathway. First, the molecules are transferred from the gas phase (at partial pressure ps1 ) to the liquid phase at the same pressure. The free energy change for this step is zero because the two phases are at saturation equilibrium at the same pressure. Next, a droplet of i molecules is “carved out” of the liquid and separated from it. This step involves the creation of an interface between the gas and liquid and does entail a free energy change. It is assumed that the free-energy change associated with this step is given by2 ΔGi ˆ σai

(11.25)

where σ is the surface tension of the solute A, and, as before, ai is the surface area of a cluster of size i. If the cluster is taken to be spherical and to have the same volume per molecule v1 as the bulk liquid, then 2=3

ai ˆ …36π†1=3 v1 i2=3

(11.26)

The surface tension σ in (11.25) is taken as that of the bulk liquid monomer; thus it is assumed that clusters of a small number of molecules exhibit the same surface tension as the bulk liquid. This is the major assumption underlying classical nucleation theory and it has been given the name of the capillarity approximation. If we define the dimensionless surface tension 2=3

θ ˆ …36π†1=3 v1 σ=kT 2

(11.27)

The free-energy change associated with the formation of an i-mer is also referred to as the reversible work and is given the symbol Wi.

455

CLASSICAL THEORY OF HOMOGENEOUS NUCLEATION: KINETIC APPROACH

then the Gibbs free energy change at saturation is just ΔGi ˆ θkTi2=3

(11.28)

Combining (11.11), (11.20), (11.24), and (11.28) produces the following direct expression for the nucleation rate: 1

J ˆ N1

1

1

iˆ1

(11.29)

βsi Si exp… θi2=3 †

If we examine the terms in the denominator of the summation, some mathematical approximations are possible. The first term, βsi , increases as i2/3 for large i [see (11.19)]. The second term, Si, grows exponentially with i since S > 1. The third term, exp( θi2/3), decreases exponentially as i increases, as i2/3. The product, Si exp( θi2/3), initially decreases rapidly as i increases, reaches a minimum, and then begins to increase as Si begins to dominate. The terms in the summation are largest near the minimum in the denominator. The minimum point of the denominator, i∗, can be found from solving d s i β S exp… θi2=3 † ˆ 0 di i

(11.30)

In practice, because βsi varies slowly with i relative to the other two terms, with little loss of accuracy it is removed from the summation and replaced with its value at the minimum, βsi∗ : J ˆ N1

1 βsi∗

1 iˆ1

1

1

(11.31)

Si exp… θi2=3 †

If we let gi ˆ θi2=3

i ln S

(11.32)

then (11.31) becomes J ˆ N1

1 βsi∗

1 iˆ1

1 exp… gi †

1

(11.33)

and (11.30) can be replaced by d exp… gi † ˆ 0 di

(11.34)

Setting the derivative equal to zero yields i∗ ˆ

2θ 3 ln S

3

32π v21 σ3 ˆ 3 …kT†3 …ln S†3 This is the critical cluster size for nucleation.

(11.35)

456

NUCLEATION

If i∗ is sufficiently large, then the summation in (11.33) can be replaced with an integral 1

J ˆ N 1 βsi∗

1

exp…g…i††di

∫0

(11.36)

where the lower limit can be changed to zero. The function g(i) can be expanded in a Taylor series around its minimum: g…i† ≅ g…i∗ † ‡

dg di

i∗ † ‡

…i ∗

i

1 d2 g …i 2 di2 ∗

i∗ †2 ‡ ∙ ∙ ∙

(11.37)

i

By definition of i∗, the second term on the RHS is zero, so g…i† ≅ g…i∗ † ‡

1 d2 g …i 2 di2 ∗

i∗ †2

(11.38)

i

Then 1 d2 g …i 2 di2 ∗

exp…g…i†† ≅ exp…g…i∗ ††exp

i ∗ †2

(11.39)

i

From (11.32) we obtain d2 g di2

2 ∗ θi 9

ˆ ∗

i

4=3

(11.40)

Then (11.36), with the aid of (11.39) and (11.40), becomes J ˆ N 1 βsi∗ exp… g…i∗ ††

1

∫0

exp

b …i 2

i∗ †2 di

1

(11.41)

where b ˆ 29 θ i∗ 4=3 . The integral in (11.41) has the form of the error function (see (8.38)). To convert the integral into the standard form of the error function, we define a new integration variable, y = i i∗. In so doing, the lower limit of integration becomes i∗. We again invoke the approximation that i∗ is large and replace the lower limit by 1. We obtain J ˆ N 1 βsi∗ exp… g…i∗ ††

b 2π

1=2

(11.42)

or J ˆ N 1 βsi∗

1 θ 3 π

1=2

i∗

2=3

exp

4 θ3 27 …ln S†2

(11.43)

At saturation (S = 1), the monomer partial pressure is ps1 . At nucleation conditions (S > 1), p1 ˆ Sps1 . Thus 1 βsi∗ ˆ βi∗ S

(11.44)

457

CLASSICAL HOMOGENEOUS NUCLEATION THEORY: CONSTRAINED EQUILIBRIUM APPROACH

so Jˆ

N1 1 θ β∗ S i 3 π

1=2

i∗

2=3

4 θ3 27 …ln S†2

exp

(11.45)

is the classical homogeneous nucleation theory expression for the nucleation rate. The nucleation rate (11.45) can be written in terms of the original variables by using the definitions of βi, θ, and i∗. In doing so, it is customary to approximate βi∗ as follows: βi∗ ˆ ≅

p1

1‡

…2πm1 kT†1=2 p …2πm1 kT†

1 i∗

1=2

…1 ‡ i∗1=3 †2 a1 (11.46)

i∗2=3 a1 1=2

With (11.46) and the definitions of θ and i∗, (11.45) becomes

Jˆ

2σ πm1

1=2

v1 N 21 exp S

16π v21 σ3 3 …kT†3 …ln S†2

(11.47)

Equation (11.47) is the classical homogeneous nucleation theory expression for the nucleation rate. It is the expression that we will use to predict nucleation rates of different pure substances as a function of saturation ratio. The next section presents an alternate derivation of a classical homogeneous nucleation rate expression. The development in the next section provides a fuller appreciation of classical theory and its underlying assumptions, but the section may be skipped by those interested primarily in application of the theory. We note that several useful tables of physical properties appear in Section 11.2.

11.2 CLASSICAL HOMOGENEOUS NUCLEATION THEORY: CONSTRAINED EQUILIBRIUM APPROACH 11.2.1 Free Energy of i-mer Formation In the previous chapter we derived the Kelvin, or Gibbs–Thomson, equation [see (10.82) and (10.85)] gv

gl ˆ μ v

μl ˆ kT ln

pA 2σv1 ˆ psA R

(11.48)

where μv μl is the difference in chemical potential between a molecule in the vapor and liquid phases. Thus the i-mer’s vapor pressure exceeds that of the same substance at the same temperature over a flat surface. Letting S ˆ pA =psA (since the i-mer must exist in a vapor of partial pressure pA) yields μv

μl ˆ kT ln S

(11.49)

A partial pressure less than psA …S < 1† indicates μv < μl. In this case the vapor is stable and will not condense. If the partial pressure exceeds psA …S > 1†, μv > μl, the i-mer will tend to grow at the vapor’s expense since the system gravitates toward its lowest chemical potential. A transfer of i molecules from the vapor phase forms an i-mer of radius r. The accompanying free energy change is ΔGi ˆ …μl

μv †i ‡ 4πσr2

(11.50)

458

NUCLEATION

Thus, from (11.49), and (11.50), we obtain ΔGi ˆ 4πσr2

4π kT ln S 3 r 3 v1

(11.51)

The free-energy change of i-mer formation contains two terms. The first is the free-energy increase as a result of the formation of the i-mer surface area. The second term is the free-energy decrease from the change in chemical potential on going from the vapor to the condensed phase. For small r, the increase in free energy resulting from formation of surface area (∼r2) mathematically dominates the free-energy decrease from formation of the condensed phase (∼r3). ΔGi is sketched in Figure 10.10. Free energy increases for a subsaturated vapor (S < 1) only as the i-mer radius is increased. On the other hand, the free energy of a supersaturated vapor (S > 1) initially increases until the bulk free energy decrease overtakes the surface free-energy increase. ΔGi achieves a maximum at a certain critical i-mer radius r∗ (and corresponding i∗): 2σv1 (11.52) r∗ ˆ kT ln S The corresponding value of the number of molecules at the critical size is given by (11.35). The effective vapor pressure on an i-mer with critical radius r∗ is the prevailing partial pressure pA. This critical size i-mer is therefore in equilibrium with the vapor. It is, however, only a metastable equilibrium; the i∗-mer sits precariously on the top of the ΔG curve. If another vapor molecule attaches to it, it will plunge down the curve’s right-hand side and grow. If a molecule evaporates from the i∗-mer, it will slide back down the left-hand side and evaporate. The free energy barrier height is obtained by substituting (11.52) into (11.51): 4π ∗2 σr 3 16π v21 σ3 ˆ 3 …kT ln S†2

ΔG∗ ˆ

(11.53)

Increasing the saturation ratio S decreases both the free-energy barrier and the critical i-mer radius. From (11.35) we already knew that it decreases the value of i∗. Equation (11.52) relates the equilibrium radius of a droplet of pure substance to the physical properties of the substance, σ and v1, and to the saturation ratio S of its environment. Equation (11.52) can be rearranged so that the equilibrium saturation ratio is given as a function of the radius of the drop: ln S ˆ

2σv1 kTr∗

(11.54)

Expressed in this form, the equation is referred to as the Kelvin equation. Table 11.1 gives i∗ and r∗ for water at T = 273 K and 298 K as a function of S. As expected, we see that as S increases both i∗ and r∗ decrease. Table 11.2 contains surface tension and density data for five organic TABLE 11.1 Critical Number and Radius for Water Droplets T = 273 Ka

T = 298 Kb

S

r∗ , Å

i∗

r∗ , Å

i∗

1 2 3 4 5

1 17.3 10.9 8.7 7.5

1 726 182 87 58

1 15.1 9.5 7.6 6.5

1 482 121 60 39

σ = 75.6 dyn cm 1; v1 = 2.99 × 10 23 cm3 molecule 1. σ = 72 dyn cm 1; v1 = 2.99 × 10 23 cm3 molecule 1.

a

b

459

CLASSICAL HOMOGENEOUS NUCLEATION THEORY: CONSTRAINED EQUILIBRIUM APPROACH

TABLE 11.2 Surface Tensions and Densities of Five Organic Species at 298 K Species Acetone (C3H6O) Benzene (C6H6) Carbon tetrachloride (CCl4) Ethanol (C2H6O) Styrene (C8H8)

M

ρℓ (g cm 3)

v1 × 1023 (cm3 molecule 1)

σ (dyn cm 1)a

58.08 78.11 153.82 46.07 104.2

0.79 0.88 1.60 0.79 0.91

12.25 14.75 16.02 9.694 19.10

23.04 28.21 26.34 22.14 31.49

a

The traditional unit for surface tension is dyn cm 1. This unit is equivalent to g s 2.

Source: Handbook of Chemistry and Physics, 56th ed.

molecules, and values of i∗ and r∗ for these five substances at T = 298 K are given in Table 11.3. The critical radius r∗ depends on the product σv1. For the five organic liquids σ < σwater but v1 > v1water. The organic surface tensions are about one-third that of water, and their molecular volumes range from 3 to 6 times that of water. The product σv1 for ethanol is approximately the same as that of water, and consequently we see that the r∗ values for the two species are virtually identical. Since i∗ involves an additional factor of v1, even though the r∗ values coincide for ethanol and water, the critical numbers i∗ differ appreciably because of the large size of the ethanol molecule.

11.2.2 Constrained Equilibrium Cluster Distribution Equation (11.51) can be written in terms of the dimensionless surface tension θ as ΔGi ˆ kTθi2=3

ikT ln S

(11.55)

Recall that the constrained equilibrium cluster distribution is the hypothetical equilibrium cluster distribution at a saturation ratio S > 1. The constrained equilibrium cluster distribution obeys the Boltzmann distribution N ei ˆ N 1 exp… ΔGi =kT†

(11.56)

TABLE 11.3 Critical Number and Radius for Five Organic Species at 298 K S Species Acetone Benzene Carbon tetrachloride Ethanol Styrene

i∗ r∗ (Å) i∗ r∗ (Å) i∗ r∗ (Å) i∗ r∗ (Å) i∗ r∗ (Å)

2

3

4

5

265 19.8 706 29.2 678 29.6 147 15.1 1646 42.2

67 12.5 177 18.4 170 18.7 37 9.5 413 26.6

33 9.9 88 14.6 85 14.8 18 7.5 206 21.1

21 8.5 56 12.6 54 12.7 12 6.5 132 18.2

460

NUCLEATION

where N1 is the total number of solute molecules in the system. Using (11.55) in (11.56) yields N ei ˆ N 1 exp… θi2=3 ‡ i ln S†

(11.57)

For small i and very large i, (11.57) departs from reality. When i = 1, N ei should be identical to N1, but (11.57) does not produce this identity. The failure to approach the proper limit as i → 1 is a consequence of the capillarity approximation and the specific form, θi2/3, chosen for the surface free energy of an i-mer. It is not a fundamental problem with the theory. There have been several attempts to modify the formula for N ei so that N ei equals N1 when i = 1 (Girshick and Chiu 1990; Wilemski 1995). Some of these take the form of replacing i2/3 in (11.57) by (i2/3 1) or (i 1)2/3. Such modifications can be regarded as making the surface tension σ an explicit function of i. We will not deal further with these corrections here. The mismatch at i = 1 turns out not to be a problem because we are generally interested in N ei for values of i quite a bit larger than 1. At the other extreme, N ei cannot exceed N1, but (11.57) predicts that at sufficiently large i, N ei can be greater than N1. N ei and the actual cluster distribution Ni are sketched as a function of i in Figure 11.2. Equation (11.57) is no longer valid when the accumulated value of the product N ei begins to approach N1 in magnitude. Another consistency issue arises with (11.57). The law of mass action (Appendix 11) requires that the equilibrium constant for formation of an i-mer, iA1 ƒ Ai Ki …T† ˆ

N ei

(11.58)

…N 1 †i

depend on i and T but not on N1 or the total pressure if the gas behaves ideally. The distribution N ei ˆ N s1 exp… θi2=3 ‡ i ln S†

(11.59)

where N s1 is the monomer number density at saturation, satisfies mass action, in that Ki …T† ˆ …N s1 †1 i exp…θ…i

i2=3 ††

FIGURE 11.2 Constrained equilibrium and steady-state cluster distributions, N ei and Ni, respectively.

(11.60)

461

CLASSICAL HOMOGENEOUS NUCLEATION THEORY: CONSTRAINED EQUILIBRIUM APPROACH

is a function only of i and T. This correction (11.59), originally due to Courtney (1961), is equivalent to introducing the factor (1/S) in (11.44). Since i∗ is at the maximum of ΔGi, the N ei distribution should be a minimum at i∗. Physically, we do not expect that for i > i∗, the cluster concentrations should increase with increasing i. Thus N ei given by (11.57) is taken to be valid only up to i∗. Even though N ei as given by (11.57) is not accurate for i → 1 and i → 1, it can be considered as accurate in the region close to and below i∗. Fortunately, it is precisely this region in which the nucleation flux is desired.

11.2.3 The Evaporation Coefficient γi To determine the monomer escape frequency from an i-mer, assume that the i-mer is in equilibrium with the surrounding vapor. Then the i-mer vapor pressure equals the system vapor pressure. By (11.48), we obtain p1 ˆ ps1 exp

2σv1 kTr

(11.61)

At equilibrium the escape frequency equals the collision frequency. Thus using (11.19) yields γi ˆ βi ˆ ˆ

p1 …2πm1 kT†1=2 ps1 …2πm1 kT†

1=2

1‡ 1‡

1 i

1=2

1 i

1=2

…1 ‡ i1=3 †2 ai

(11.62)

…1 ‡ i1=3 †2 ai exp

2σv1 kTr

γi is thus a function of ps1 , m1, σ, and v1 and is independent of the actual species partial pressure. Thus (11.62) holds at any value of the saturation ratio. By combining (11.62) with the definition of i∗, γi can be written as γi ˆ

ps1 …2πm1 kT†

1=2

1‡

1 i

1=2

∗

…1 ‡ i1=3 †2 a1 S…i

=i†1=3

(11.63)

Taking γi from (11.63), based on equilibrium, and using (11.19) with p1 ˆ Sps1 , we can form the nonequilibrium ratio of γi to βi as 1=3 ∗ γi ˆ Sf…i =i† βi

1g

(11.64)

We see from (11.64) that subcritical i-mers (i < i∗) tend to evaporate since their evaporation frequency exceeds their collision frequency. Critical size i-mers (i = i∗) are in equilibrium with the surrounding vapor (γi = βi). Supercritical clusters (i > i∗) grow since monomers tend to condense on them faster than monomers evaporate. Figure 11.3 shows γi and βi as a function of i for various values of S. The evaporation and growth curves intersect at the i∗ values for the particular value of S. As expected, i∗ decreases as S increases.

11.2.4 Nucleation Rate At constrained equilibrium βi N ei ˆ γi‡1 N ei‡1

(11.65)

462

NUCLEATION

FIGURE 11.3 βi and γi as a function of i for various values of the saturation ratio S. The two quantities are equal at the critical i values.

The ratio of equilibrium cluster concentrations between neighboring clusters is N ei‡1 βi e ˆ γi‡1 Ni

(11.66)

Using (11.19) and (11.63), we obtain 1=2

e 1 ‡ 1i …1 ‡ i1=3 †2 N i‡1 ˆ S 1=2 N ei 1 ‡ i‡11 …1 ‡ …i ‡ 1†1=3 †2

f‰i∗ =…i‡1†Š1=3 1g

(11.67)

When (11.67) is successively multiplied from i = i to i = 1, it yields i N ie 25=2 ˆ ∏S e 1=2 N1 1 ‡ 1i …1 ‡ i1=3 †2 jˆ2

f…i∗ =j†1=3 1g

(11.68)

This equation is essentially an alternate way of writing (11.57) in terms of i∗. It differs slightly from (11.57) in that we have retained the (1 + 1/i)1/2 and (1 + i1/3)2 terms instead of approximating them simply by i2/3. Figure 11.4 shows N ei =N 1 as a function of i for S = 6 and i∗ = 40. The constrained equilibrium cluster distribution, N ei , is based on a supposed equilibrium existing for S > 1. In actuality, when S > 1, a nonzero cluster current J exists, which is the nucleation rate. Let the actual steady-state cluster distribution be denoted by Ni. When imax is sufficiently larger than i∗, the actual cluster distribution approaches zero, lim N i ˆ 0

i!imax

(11.69)

463

CLASSICAL HOMOGENEOUS NUCLEATION THEORY: CONSTRAINED EQUILIBRIUM APPROACH

FIGURE 11.4 N ei =N 1 as a function of i for S = 6 and i∗ = 40. The increase for i > i∗ is physically unrealistic.

The traditional condition used at i = 1 is that the constrained equilibrium distribution and the actual distribution start from the same value: Ni lim e ˆ 1 (11.70) i!1 N i From (11.3) and (11.65), we have J ˆ βi N i ˆ βi N ei

γi‡1 N i‡1 Ni N ei

Summing successive applications of (11.71) from i = imax imax 1 jˆ1

J N1 ˆ βj N je N e1

N i‡1 N ei‡1

(11.71)

1 to i = 1 yields N imax N eimax

(11.72)

Using (11.69) and (11.70), we obtain Jˆ

imax 1 jˆ1

1

1 βj N ej

(11.73)

464

NUCLEATION

Thus the unknown steady-state cluster distribution Ni, disappears, allowing the nucleation rate to be expressed in terms of the constrained equilibrium distribution N ei . We see that (11.73) is equivalent to (11.11), in that (11.65), which defines the constrained equilibrium distribution N ei , is identical to (11.5) that defines fi. We have demonstrated the essential equivalence of the two approaches to deriving the nucleation rate. We can obtain a corresponding expression to (11.47) for the nucleation rate from (11.73) and the expressions for βi, (11.19), and N ei , (11.68). When expressions for βi and N ei are substituted into (11.73), the following equation is obtained for the nucleation rate: N 2 kT J ˆ a1 1 S 2πm1

1=2

25=2 1 ‡

imax 1 j

1 ∗

∏ Sf…i

=k†

1=3

1g

(11.74)

jˆ2 kˆ2

From the point of view of computationally implementing (11.74), J varies little with the value of imax as long as imax  i∗.

11.3 RECAPITULATION OF CLASSICAL THEORY The classical theory of homogeneous nucleation dates back to pioneering work by Volmer and Weber (1926), Farkas (1927), Becker and Döring (1935), Frenkel (1955), and Zeldovich (1942). The classical theory is based on a blend of statistical and thermodynamic arguments and can be approached from a kinetic viewpoint (Section 11.1) or that of constrained equilibrium cluster distributions (Section 11.2). In either case, the defining assumption of the classical theory is reliance on the capillarity approximation wherein bulk thermodynamic properties are used for clusters of all sizes. Let us summarize what we have obtained in our development of homogeneous nucleation theory. In order for homogeneous nucleation to occur, it is necessary that clusters surmount the free-energy barrier at i = i∗. The larger the saturation ratio S, the smaller the critical cluster size i∗. At saturation conditions (S = 1) clusters containing more than a very few molecules are exceedingly rare and nucleation does not occur. At supersaturated conditions (S > 1) the saturated equilibrium cluster distribution N si is perturbed to produce a steady-state cluster distribution Ni. The nucleation flux J results from a chain of bombardments of vapor molecules on clusters that produces a net current through the cluster distribution. In deriving the nucleation rate expression from the thermodynamic approach, it is useful to define a hypothetical equilibrium cluster distribution, called the constrained equilibrium distribution. To maintain the nucleation rate at a constant value J, it is necessary that the saturation ratio of vapor be maintained at a constant value.3 Without outside reinforcement of the vapor concentration, the saturation ratio will eventually fall as a result of depletion of vapor molecules to form stable nuclei. The case in which the vapor concentration is augmented by a source of fresh vapor is referred to as nucleation with a continuously reinforced vapor. Let us evaluate the homogeneous nucleation rate for water as a function of the saturation ratio S. To do so, we use (11.47). Table 11.4 gives the nucleation rate at 293 K for saturation ratios ranging from 2 to 10. By comparing Tables 11.1 and 11.4, the effect of temperature on i∗ can be seen also. We see that the nucleation rate of water varies over 70 orders of magnitude from S = 2 to S = 10. As S increases, the critical cluster size i∗ at 293 K decreases from 523 at S = 2 to 14 at S = 10. From Tables 11.1 and 11.4 we see that at S = 5, i∗ varies from 58 to 42 to 39 as T increases from 273 K to 293 K to 298 K. The question arises as to how closely the predictions from (11.47) and (11.74) agree. Derivations of these two expressions are based on similar assumptions, but these expressions do not agree exactly 3

It is sometimes supposed that to maintain a constant S, imax monomers reenter the system at the same rate that imax-mers leave. The imax-mers are said to be broken up by Maxwell demons.

465

EXPERIMENTAL MEASUREMENT OF NUCLEATION RATES

TABLE 11.4 Homogeneous Nucleation Rate and Critical Cluster Size for Water at T = 293 Ka S

i∗

2 3 4 5 6 7 8 9 10

523 131 65 42 30 24 19 16 14

J(cm

3

s 1)

5.02 × 10 54 1.76 × 10 6 1.05 × 106 1.57 × 1011 1.24 × 1014 8.99 × 1015 1.79 × 1017 1.65 × 1018 9.17 × 1018

σ = 72.75 dyn cm 1 v1 = 2.99 × 10 23 cm3 molecule 1 m1 = 2.99 × 10 23 g molecule 1 psH2 O ˆ 2:3365  104 g cm 1 s 2 a

because of approximations that are made in each derivation. It is useful to compare nucleation rates of water predicted by (11.74) with those in Table 11.4 determined from (11.47): S

(11.47)

(11.74)

3 5 7 9

1.76 × 10 6 1.57 × 1011 8.99 × 1015 1.65 × 1018

4.52 × 10 3 4.03 × 1014 2.06 × 1019 3.26 × 1021

We see that (11.74) predicts a nucleation rate about 3 orders of magnitude larger than that predicted by (11.47). In most areas of science a discrepancy of prediction of 3 orders of magnitude between two equations representing the same phenomenon would be a cause for serious concern. As we noted, nucleation rates of water vary over 70 orders of magnitude for the range of saturation ratios considered; in addition, the rate expressions are extremely sensitive to small changes in parameters. Thus, a difference of 3 orders of magnitude, while certainly not trivial, is well within the sensitivity of the rate expressions. As we noted at the end of Section 11.1, we will use (11.47) henceforth as the basic classical theory expression for the homogeneous nucleation rate of a single substance.

11.4 EXPERIMENTAL MEASUREMENT OF NUCLEATION RATES The traditional method of studying gas–liquid nucleation involves the use of a cloud chamber. In such a chamber the saturation ratio S is changed until, at a given temperature, droplet formation is observable. Because once clusters reach the critical size for nucleation, subsequent droplet growth is rapid, the rate of formation of macroscopically observable droplets is assumed to be that of formation of critical nuclei. In such a device it is difficult to measure the actual rate of nucleation because the nucleation rate changes so rapidly with S. J is very small for S values below a critical saturation ratio Sc and very large for S > Sc. Thus what is actually measured is the value of Sc, defined rather arbitrarily by the point where the rate of appearance of droplets is 1 cm 3 s 1.

466

NUCLEATION

11.4.1 Upward Thermal Diffusion Cloud Chamber An improvement on the cloud chamber is the upward thermal diffusion cloud chamber (Katz 1970; Heist 1986; Hung et al. 1989). In this device a liquid pool at the bottom of a container is heated from below to vaporize it partially. The upper surface of the container is held at a lower temperature, establishing a temperature gradient in the vapor between bottom and top. The total pressure of gas in the container (background carrier gas plus nucleating vapor) is approximately uniform. However, the partial pressure of the nucleating vapor pA decreases linearly with height in the chamber because of the diffusion gradient between the pool of liquid at the bottom and the top of the chamber. The saturation vapor pressure of the vapor, psA , depends exponentially on temperature, and therefore as the temperature decreases with height, the saturation vapor pressure decreases rapidly. The result is that the saturation ratio, S ˆ pA =psA , has a rather sharp peak at a height about three-fourths of the way up in the chamber, and nucleation occurs only in the narrow region where S is at its maximum (Figure 11.5). The temperatures at the bottom and top of the chamber are then adjusted so that the nucleation rate at the point of maximum S is about 1 cm 3 s 1. As droplets form and grow they fall under the influence of gravity; continued evaporation of the liquid pool serves to replenish the vapor molecules of the nucleating substance so that the experiment can be run in steady state. Nucleation rates that can be measured in the upward thermal diffusion cloud chamber vary roughly from 10 4 to 103 cm 3 s 1.

11.4.2 Fast Expansion Chamber Another effective technique to study vapor-to-liquid nucleation is an expansion cloud chamber (Schmitt 1981, 1992; Wagner and Strey 1981; Strey and Wagner 1982; Strey et al. 1994). Initially an enclosed volume saturated with the vapor under consideration is allowed to come to thermal equilibrium. The volume is then abruptly expanded, and the carrier gas and vapor decrease in temperature (Figure 11.6). While the expansion slightly decreases the pressure (and density) of the vapor, the partial pressure is still much greater than the saturation vapor pressure at the lower temperature. The vapor is thus super­ saturated, and quite large supersaturations can be achieved with the expansion technique. After expansion, the chamber is immediately recompressed to an intermediate pressure (and saturation ratio). Typical durations of the expansion pulse are the order of milliseconds. In order for a condition of steady-state nucleation to be achieved, the time needed to establish the steady-state cluster distribution corresponding to the value of S at the peak must be significantly shorter than the time the system is held at its highest supersaturation. Since J is such a strong function of S, the immediate recompression quenches the nucleation that occurs during the brief expansion pulse. The temperature at the peak supersaturation is calculated from the properties of the carrier gas and the vapor based on the initial temperature, the initial pressure, and the pressure after expansion. The expansion is adiabatic, because of the short duration of the expansion. After recompression the system remains supersaturated and

FIGURE 11.5 Upward thermal diffusion cloud chamber. Typical cloud chamber profiles of total gasphase density, temperature, partial pressure p (n-nonane in this example), equilibrium vapor pressure ps, saturation ratio S, and nucleation rate J, as a function of dimensionless chamber height at T = 308.4 K, S = 6.3, and total p = 108.5 kPa. (Reprinted with permission from Katz, J. L., Fisk, J. A., and Chakarov, V. M., Condensation of a Supersaturated vapor IX. Nucleation of ions, J. Chem. Phys. 101. Copyright 1994 American Institute of Physics.)

MODIFICATIONS OF THE CLASSICAL THEORY AND MORE RIGOROUS APPROACHES

FIGURE 11.6 Fast expansion chamber. Initially undersaturated (S < 1) at p0, T0; for a time interval of typically 1 ms supercritically supersaturated (S > Scrit), where Scrit is the saturation ratio corresponding to a threshold nucleation rate, at pmin, Tmin; finally still supersaturated, but subcritical (1 < S < Scrit) to permit droplet growth only. Sketches at right indicate the distribution of clusters with size at the three stages of operation.

particles nucleated during the supercritical nucleation pulse grow by condensation. The growing drops are illuminated by a laser and their number concentration is measured by an appropriate optical technique. From the measured droplet concentration the nucleation rate during the supercritical nucleation pulse can be determined. The duration of the nucleation pulse (about 10 3 s) is short compared to characteristic times for growth of the nucleated droplets; thus nucleation is decoupled from droplet growth. The fast expansion chamber is capable of allowing measurement of nucleation rates up to 1010 cm 3 s 1.

11.4.3 Turbulent Mixing Chambers Another method of nucleation measurement that differs from both diffusion and expansion chambers involves the rapid turbulent mixing of two gas streams (Wyslouzil et al. 1991a,b). This method is particularly suited to studies of binary nucleation. Two carrier gas-vapor streams are led to a device where rapid turbulent mixing takes place. The two-component vapor mixture is supersaturated and begins to nucleate immediately. The stream passes to a tube where nucleation may continue and the nucleated particles grow. Residence time in the flow tube is the order of seconds. When the nucleated particle concentration is sufficiently low, droplet growth does not deplete the vapor appreciably, and constant nucleation conditions can be assumed.

11.5 MODIFICATIONS OF THE CLASSICAL THEORY AND MORE RIGOROUS APPROACHES The principal limitation to the classical theory is seen as the capillarity approximation, the attribution of bulk properties to the critical cluster (see Figure 11.7). A number of modifications to the classical theory retain the basic capillarity approximation but introduce corrections to the model [e.g., see Hale (1986), Dillmann and Meier (1989, 1991), and Delale and Meier (1993)].

467

468

NUCLEATION

FIGURE 11.7 Three approaches to nucleation theory and the nature of the cluster that results.

Computer simulation offers a way to avoid many of the limitations inherent in classical and related theories. In principle, nucleation can be simulated by starting with a supersaturated gas-like configuration and solving the equations of motion of each molecule. Eventually a liquid drop would emerge from the simulation. Because typical simulation volumes are the order of 10 20 cm3 and simulation times are the order of 10 10 s, for any particle formation to result, nucleation rates must be of order 1030 cm 3 s 1, far higher than anything amenable to laboratory study. Research has focused on simulating an isolated cluster of a given size to determine its free energy. The variation of that free energy with number of monomers can then be used in nucleation theory to relate forward and backward rate constants or to estimate the height of the free energy barrier to nucleation (Reiss et al. 1990; Weakleim and Reiss 1993). In classical nucleation theory the density of a cluster is assumed to be equal to the bulk liquid density, and the free energy of the surface (surface tension) is taken as that of a bulk liquid with a flat interface. A cluster is then completely defined by its number of molecules or its radius. In general, there is no reason why the density at the center of a cluster should equal the bulk liquid density, nor is there any reason why the density profile should match that at a planar interface (Figure 11.8). Whereas the condition for the critical nucleus in classical theory is that the derivative of the free energy with respect to radius r be equal to zero, when ρ is allowed to vary with position r, the free energy is now a function of that density. The free energy has a minimum at the uniform vapor density and a second, lower minimum at the uniform liquid density. Between these two minima lies a saddle point at which the functional derivative of the free energy with respect to the density profile is zero. The density functional theory has been applied to nucleation by Oxtoby and colleagues (Oxtoby 1992a,b).

11.6 BINARY HOMOGENEOUS NUCLEATION In homogeneous–homomolecular nucleation, nucleation does not occur unless the vapor phase is supersaturated with respect to the species. When two or more vapor species are present, neither of which is supersaturated, nucleation can still take place as long as the participating vapor species are super­ saturated with respect to a liquid solution droplet. Thus heteromolecular nucleation can occur when a

469

BINARY HOMOGENEOUS NUCLEATION

FIGURE 11.8 Clusters in binary nucleation. Proceeding vertically up the left side of the figure are clusters of pure water; proceeding horizontally along the bottom of the figure are clusters of pure acid molecules. The darker circles denote water molecules; the dashed lines indicate the boundary of the cluster.

mixture of vapors is subsaturated with respect to the pure substances as long as there is supersaturation with respect to a solution of these substances. The theory of homogeneous–heteromolecular nucleation parallels that of homogeneous–homomolecular nucleation extended to include two or more nucleating vapor species. We consider here only two such species, focusing on binary homogeneous nucleation. Figure 11.8 depicts clusters in binary nucleation of acid and water. In classical homogeneous-homomolecular nucleation theory the rate of nucleation can be written in the form J ˆ C exp… ΔG∗ =kT†

(11.75)

where ΔG∗ is the free energy required to form a critical nucleus. This same form will apply in binary homogeneous nucleation theory. The first theoretical treatment of binary nucleation dates back to Flood (1934), but it was not until 1950 that Reiss (1950) published a complete treatment of binary nucleation. The free energy of formation of a nucleus containing nA molecules of species A and nB molecules of species B is given by ΔG ˆ nA …μAℓ

μAg † ‡ nB …μBℓ

μBg † ‡ 4 πr2 σ

(11.76)

where r is the radius of the droplet and μAℓ , μBℓ and μAg, μBg are the chemical potentials of components A and B in the mixed droplet (ℓ) and in the gas phase (g), respectively. From before we can express the

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NUCLEATION

difference in chemical potential between the liquid and vapor phases as pj μjℓ μjg ˆ kT ln ; j ˆ A; B pj

(11.77)

sol

where pj is the partial pressure of component j in the gas phase and pj is the vapor pressure of component sol j over a flat solution of the same composition as the droplet. The following quantities can be defined: aAg ˆ pA =psA

ˆ activity of A in gas phase

aBg ˆ pB =psB

ˆ activity of B in gas phase

aAℓ ˆ pAsol =psA ˆ activity of A in the liquid phase aBℓ ˆ pBsol =psB ˆ activity of B in the liquid phase Here psA and pBs are the saturation vapor pressures of A and B over a flat surface of pure A and B, respectively. Thus ΔG can be written as aAg aBg nB kT ln ‡ 4πr2 σ (11.78) ΔG…nA ; nB ; T† ˆ nA kT ln aBℓ aAℓ where aAg and aBg correspond to the saturation ratios of A and B, respectively, in the homomolecular system. Again, pAsol and pBsol are the partial vapor pressures of A and B over a flat surface of the solution. The radius r is given by 4 3 πr ρ ˆ nA mA ‡ nB mB 3

(11.79)

Whereas in the case of a single vapor species, ΔG is a function of the number of molecules, i, here ΔG depends on both nA and nB. Nucleation will occur only if μjℓ μjg is negative, that is, if both vapors A and B are supersaturated with respect to their vapor pressures over the solution (not with respect to their pure component vapor pressure as in the case of homomolecular nucleation). In such a case, the first two terms on the RHS of (11.76) are negative. The last term on the RHS is always positive. For very small values of nA and nB, ΔG is dominated by the surface energy term, and ΔG increases as nA and nB (and size) increase. Eventually a point is reached where ΔG achieves a maximum. Instead of considering ΔG as a function only of i, it is now necessary to view ΔG as a surface in the nA nB plane. Reiss (1950) showed that the three-dimensional surface ΔG(nA, nB) has a saddle point that represents the minimum height of the free-energy barrier. The saddle point is one at which the curvature of the surface is negative in one direction (that of the path) and positive in the direction at right angles to it. The saddle point represents a “mountain pass” that clusters have to surmount in order to grow and become stable (Figure 11.9). The height of the free-energy barrier at the saddle point is denoted by ΔG∗. This point may be found by solving the two equations: @ΔG @nA

ˆ nB

@ΔG @nB

ˆ0

(11.80)

nA

Differentiating (11.78) and expressing nA and nB in terms of partial molar volumes vA and vB , we obtain (Mirabel and Katz 1974) RT ln

RT ln

pA 2σv A ˆ pAsol r

3xB v dσ r dxB

pB 2σv B 3…1 xB †v dσ ˆ ‡ pBsol r r dxB

(11.81)

(11.82)

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BINARY HOMOGENEOUS NUCLEATION

FIGURE 11.9 Free energy of cluster formation, ΔG(nA, nB), for binary nucleation: (a) schematic diagram of saddle point in the ΔG surface; (b) ΔG surface for H2SO4–H2O system at 298 K.

where xB = nB/(nA + nB) and the molar volume of solution v ˆ …1 xB †vA ‡ xB vB . (Note that this definition of v is an approximation since in mixing two species in general 1 cm3 of A mixed with 1 cm3 of B does not produce precisely 2 cm3 of solution.) It has been shown that (11.81) and (11.82) need to be revised somewhat to locate the saddle point [e.g., see Mirabel and Reiss (1987)]. In short, the revised theory distinguishes between the composition of the surface layer and that of the interior of the cluster. The “revised” theory leads to the following equations to locate the saddle point ΔμA ‡

2σv A ˆ0 r∗

(11.83)

ΔμB ‡

2σv B ˆ0 r∗

(11.84)

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NUCLEATION

or r∗ ˆ

2σv ‡ x∗B ΔμB

x∗B † ΔμA

(11.85)

vB ΔμA ˆ vA ΔμB

(11.86)

…1

and

Both the classical and the revised classical theory lead to 4 ΔG∗ ˆ πσr ∗2 3

(11.87)

where ΔG∗ ˆ

v 2 16 3 πσ 3 …kT ln S∗ †2

(11.88)

…1 x B∗ † x∗B SB

(11.89)

and S∗ ˆ SA

Under conditions for which dσ/dxB is small, which is the case for many of the binary systems of interest, the results given by (11.81), (11.82) and (11.83), (11.84) are only slightly different. The expression for the pre-exponential factor C in (11.75) was first derived by Reiss (1950) Cˆ

β A βB …N A ‡ N B † a∗ Z βA sin ϕ ‡ βB cos2 ϕ 2

(11.90)

where NA and NB are the gas-phase number concentrations of A and B, respectively, and βj ˆ

pj …2πmj kT†1=2

;

j ˆ A; B

(11.91)

where a∗ is the surface area of the critical sized cluster, ϕ is the angle between the direction of growth at the saddle point and the nA axis, and Z is the so-called Zeldovich nonequilibrium factor. In Reiss’ (1950) theory ϕ is given by the direction of steepest descent on the free-energy surface. Stauffer (1976) subsequently showed that the direction of the nucleation path is not simply given by that of the steepest descent of the free-energy surface alone, but rather by a combination of both the impinging rate of the condensing species and the free-energy surface. In particular, he showed that if the arrival rates of the two species at the cluster are different, the direction of the nucleation path is shifted somewhat from the steepest-descent direction. For practical purposes, ϕ can be determined from (Määttänen et al. 2007) tan ϕ ˆ

n∗B n∗A

(11.92)

The Zeldovich factor generally has a value ranging from 0.1 to 1 and thus has only a slight effect on the rate of nucleation. Mirabel and Clavelin (1978) have derived the limiting behavior of binary homogeneous nucleation theory when the concentration of one of the vapor species becomes very small. For a low concentration of

473

BINARY NUCLEATION IN THE H2SO4–H2O SYSTEM

one species, say, B, the preexponential factor C simplifies from (11.90) but one has to distinguish two cases: 1. If the new phase (critical cluster) is dilute with respect to component B (e.g., the critical cluster contains 1 or 2 molecules of B), then ϕ = 0, and the flow of critical clusters through the saddle point is parallel to the nA axis. In this case, C reduces to C ˆ βA N A a ∗

(11.93)

which is similar to the pre-exponential factor for one-component nucleation of A. 2. If the critical cluster is not dilute with respect to component B, but B is still present at a very small concentration in the vapor phase, then in general βA sin2 ϕ  βB cos2 ϕ and C reduces to Cˆ

βB N A a ∗ sin2 ϕ

(11.94)

The rate of nucleation is then controlled by species B. After one molecule of B strikes a cluster, many A molecules impinge such that an equilibrium with respect to A is achieved until another B molecule arrives and the process is repeated all over again.

11.7 BINARY NUCLEATION IN THE H2SO4–H2O SYSTEM The first system studied in an effort to explain atmospheric nucleation was that of sulfuric acid and water. Doyle (1961), the first to publish predicted nucleation rates for the H2SO4–H2O system, showed that, even in air of relative humidity less than 100%, extremely small amounts of H2SO4 vapor are able to induce nucleation. Subsequently, Mirabel and Katz (1974), Heist and Reiss (1974) Jaecker-Voirol and Mirabel (1989), Kulmala and Laaksonen (1990), Laaksonen et al. (1995), and Noppel et al. (2002), among others, studied the the H2SO4–H2O binary nucleation system. For the H2SO4–H2O system, let A refer to H2O and B to H2SO4. In air containing water vapor, H2SO4 molecules exist in various states of hydration: H2SO4
H2O, H2SO4
2H2O, . . . , H2SO4
hH2O. Nucleation of H2SO4 and H2O is then not just the nucleation of individual molecules of H2SO4 and H2O, but involves cluster formation between individual H2O molecules and already-hydrated molecules of H2SO4 (Jaecker-Voirol et al. 1987; Jaecker-Voirol and Mirabel 1988). The Gibbs free energy of cluster formation, including hydrate formation, is ΔGhyd ˆ ΔGunhyd

kT ln Ch

(11.95)

where the correction factor Ch due to hydration is Ch ˆ

1 ‡ K1 pAsol ‡ ∙ ∙ ∙ ‡ K1 K2 ∙ ∙ ∙ Kh phAsol

nB

1 ‡ K1 pA ‡ ∙ ∙ ∙ ‡ K1 K2 ∙ ∙ ∙ Kh phA

(11.96)

where pAsol is the partial vapor pressure of water (species A) over a flat surface of the solution, pA is the partial pressure of water vapor in the gas, nB is the number of H2SO4 molecules (species B) in the cluster, h is the number of water molecules per hydrate, and Ki are the equilibrium constants for hydrate formation. The equilibrium constants are those for the reactions H2 O ‡ H2 SO4
…h

1†H2 Oƒ H2 SO4
h H2 O

and can be evaluated using the following expression Kh ˆ exp… ΔGh °=kT†

(11.97)

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NUCLEATION

and ΔGh ° ˆ kT ln pAsol ‡

2σmA 1 XB @ρ ‡ r ρ ρ2 @XB

(11.98)

where ρ is the density of the solution and XB is the mass fraction of B in the cluster, XB ˆ

nB m B nA mA ‡ nB mB

(11.99)

The @ρ/@XB term in (11.98) can often be neglected as small, so (11.98) becomes ΔGh ° ˆ kT ln pAsol ‡

2σvA r

(11.100)

where vA is the partial molecular volume of water. Hydrate formation can have a relatively large effect on the nucleation rate by changing the shape of the free-energy surface and therefore the location and value of ΔG at the saddle point. The first 10 hydration constants for H2SO4 and H2O at 298 K are given in Table 11.5. Constants at 50°C, 25°C, 0°C, 50°C, 75°C, and 100°C, in addition, are given by JaeckerVoirol and Mirabel (1989). Sulfuric acid–water nucleation falls into the category where component B (H2SO4) is present at a very small concentration relative to that of A (H2O), yet the critical cluster is not dilute with respect to H2SO4. Thus the preexponential factor (11.94) applies. The rate of binary nucleation in the H2SO4–H2O system, accounting for the effect of hydrates and using (11.94), is given by Jˆ

…8πkT†1=2 Ch N A Z hmax 2 …δ μ sin2 ϕ hˆ0

1=2

N h † exp … ΔG∗ =kT†

(11.101)

where δ is the sum of the radii of the critical nucleus and the hydrate (or a free acid molecule), μ is the reduced mass of the critical nucleus and the hydrate (or free acid molecule), and Nh is the number concentration of hydrates. As before, ϕ is determined from (11.92). Note that in (11.101), ΔG∗ refers to ΔGunhyd. To calculate the nucleation rate of H2SO4–H2O, first ΔG(nA, nB, T) has to be evaluated to locate the saddle point. One can construct a table of ΔG as a function of nA and nB and the saddle point can be found by inspection of the values. To calculate ΔG, one needs values for activities, densities, and surface tensions as functions of temperature. The value of the saturation vapor pressure of H2SO4 has a major effect on the nucleation rate. Kulmala and Laaksonen (1990) have evaluated available expressions for the H2SO4 saturation vapor pressure and derived a recommended expression (Table 11.6). Table 11.7 illustrates a particular calculation of ΔG as a function of nA and nB (neglecting hydration). Predicting the nucleation rate of H2SO4–H2O on the basis of the theory presented above is somewhat involved; appropriate thermodynamic data need to be assembled, including the equilibrium constants for H2SO4 hydration. Figure 11.10a–d show predicted nucleation rates (cm 3 s 1) as a function of N H2 SO4 , the total number concentration of H2SO4 molecules in the gas phase at relative humidities of 20%, 50%, TABLE 11.5 Equilibrium Constants Kh at 298 K for the Successive Addition of a Water Molecule to a Hydrated Sulfuric Acid Molecule H2O + H2SO4
(h h Kh

1 1430

2 54.72

3 14.52

Source: Jaecker-Voirol and Mirabel (1989).

4 8.12

1)H2O ƒ H2SO4
h H2O 5 5.98

6 5.04

7 4.62

8 4.41

9 4.28

10 4.27

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NUCLEATION ON AN INSOLUBLE FOREIGN SURFACE

TABLE 11.6 Saturation Vapor Pressure of H2SO4 ln psH2 SO4 ˆ ln psH2 SO4 ;0 ‡

ΔH v …T0 † R

ΔHv(T0)/R = 10156 K Tc = 905 K T0 = 360 K

1 1 0:38 T0 ‡ ‡ 1 ‡ ln T T0 Tc T0 T

T0 T

ln psH2 SO4 ;0 ˆ …10156=T 0 † ‡ 16:259 …atm†

Source: Kulmala and Laaksonen (1990).

TABLE 11.7 Values of ΔG (×1012 erg) as a Function of nA and nB(A = H2O; B = H2SO4). Pressures used are pA = 2 × 10 − 4 and pB = 1.15 × 10 − 11 mm Hg nA\nB

75

76

77

78

79

80

81

82

128 129 130 131 132

5.64 5.51 a 5.63 5.73 5.91

5.82 5.73 5.61 5.54 5.65

5.94 5.88 5.80 5.70 5.56

6.0 5.97 5.93 5.86 5.77

6.02 6.01 6.00 5.96 5.91

5.99 6.02 6.02 6.01 5.99

5.94 5.98 6.01 6.02 6.02

5.86 5.91 5.96 6.00 6.02

133 134 135 136 137 138 139

5.89 5.95 6.00 6.04 6.06 6.08 6.07

5.75 5.83 5.91 5.97 6.01 6.05 6.07

5.57 5.68 5.77 5.85 5.92 5.98 6.02

5.66 5.51 5.60 5.70 5.79 5.87 5.94

5.84 5.74 5.62 5.51 5.63 5.72 5.81

5.95 5.89 5.81 5.71 5.57 5.54 5.64

6.01 5.98 5.93 5.87 5.78 5.67 5.52

6.08 6.02 6.00 5.97 5.91 5.84 5.74

a

Single underline represents the path of minimum free energy; double underlined value is the saddle-point free energy.

Source: Hamill et al. (1977).

80%, and 100% at temperatures ranging from 50°C to 25°C. Figure 11.11 gives the composition of the critical nucleus, for four different temperatures, in terms of the mole fraction of H2SO4 in the droplet, at a rate of nucleation of J = 1 cm 3 s 1. These curves give a good indication of the composition for other rates since the composition of the critical cluster is quite insensitive to the nucleation rate. Defining J = 1 cm 3 s 1 as the critical rate above which nucleation is significant, the critical gas-phase sulfuric acid concentration that produces such a rate can be evaluated from the following empirical fit to nucleation rate calculations (Jaecker-Voirol and Mirabel 1989; Wexler et al. 1994) Ccrit ˆ 0:16 exp…0:1 T

3:5 RH

27:7†

(11.102)

where T is in K, RH is the relative humidity on a scale of 0 to 1, and Ccrit is in μg m 3. Thus, when the gas-phase concentration of H2SO4 exceeds Ccrit, one can assume that H2SO4–H2O nucleation commences.

11.8 NUCLEATION ON AN INSOLUBLE FOREIGN SURFACE Let us consider nucleation of a liquid droplet on an insoluble foreign surface (Figure 11.12). The heterogeneous nucleation rate is defined as the number of critical nuclei appearing on a unit surface area

476

NUCLEATION

FIGURE 11.10 H2SO4–H2O binary nucleation rate (cm 3 s 1) as a function of N H2 SO4 , calculated from classical binary nucleation theory (Kulmala and Laaksonen 1990): (a) relative humidity = 20%; (b) RH = 50%; (c) RH = 80%; (d) RH = 100%.

per unit time. The classical expression for the free energy of critical cluster formation on a flat solid (insoluble) surface is (11.103) ΔG∗ ˆ ΔG∗hom f …m† where ΔG∗hom is the free energy for homogeneous critical nucleus formation and the contact parameter m is the cosine of the contact angle θ between the nucleus and the substrate (Figure 11.12) σSV σSL m ˆ cos θ ˆ (11.104) σLV

FIGURE 11.11 Composition of the critical H2SO4–H2O nucleus, calculated for J = 1 cm 3 s 1, as a function of RH. The different curves correspond to the temperatures shown. (Reprinted from Atmos. Environ. 23, Jaecker-Voirol, A. and Mirabel, P., Heteromolecular nucleation in the sulfuric acid-water system, p. 2053, 1989, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 1 GB, UK.)

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NUCLEATION ON AN INSOLUBLE FOREIGN SURFACE

FIGURE 11.12 Cluster formation on an insoluble flat surface; θ is the contact angle between the cluster and the flat substrate.

where the σ terms are the interfacial tensions between the different phases, V denoting the gas phase, L the nucleus, and S the substrate. The function f(m), given by 1 f …m† ˆ …2 ‡ m†…1 4

m†2

(11.105)

assumes values between 0 and 1. When the contact angle θ is zero, m = 1, and f(m) = 0, and nucleation begins as soon as the vapor is supersaturated; that is, the free-energy barrier to critical cluster formation is zero. If θ = 180°, f(m) = 1, and heterogeneous nucleation is not favored over homogeneous nucleation. The discussion above applies to heterogeneous nucleation on flat surfaces. For atmospheric particles the curvature of the surface complicates the situation. Fletcher (1958) showed that the free energy of formation ΔG∗ of an embryo of critical radius r∗ on a spherical nucleus of radius R is given by ΔG∗ ˆ ΔGhom f …m; x† ˆ

16π v21 σ 3LV f …m; x† 3 …kTlnS†2

(11.106)

where σLV is the surface tension between the embryo phase and the surrounding phase, v1 represents the volume per molecule, S is the supersaturation, and f(m,x) accounts for the geometry of the embryo and is now given by f …m; x† ˆ

1 1 mx 1‡ 2 g

3

‡ x3 2

3

x

m g

‡

x

m g

3

‡ 3mx2

x

m g

1

(11.107)

with 2mx†1=2

g ˆ …1 ‡ x2

(11.108)

where m is given by (11.104) and R r∗

(11.109)

2v1 σLV kT ln S

(11.110)

xˆ The critical radius for the embryo is given by r∗ ˆ

which corresponds to (11.52). One can show that for x = 0 (11.106) corresponds to homogeneous nucleation and for x approaching infinity it corresponds to (11.103). The contact angle with the

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NUCLEATION

nucleating liquid is an important parameter for heterogeneous nucleation. Small contact angles make the corresponding insoluble particles good sites for nucleation while larger contact angles inhibit nucleation. The direct relationship between the contact angle and the critical supersaturation for small contact angles and supersaturations was estimated by McDonald (1964) as cos θ ˆ 1

x

1 x

…0:662 ‡ 0:022 ln R†1=2 ln Sc

(11.111)

where Sc is the critical supersaturation corresponding to a nucleation rate of 1 embryo per particle per second. For formation of water droplets on insoluble particles in clouds in ambient supersaturations (usually less than 3%), equation (11.111) suggests that the contact angle should be less than 12°.

11.9 ION-INDUCED NUCLEATION Phase transition from supersaturated vapor to a more stable liquid phase necessarily involves the formation of clusters and their growth within the vapor phase. In the case of homogeneous nucleation, a cluster is formed among the molecules of condensing substance themselves, while in the case of ioninduced nucleation, the cluster will be formed preferentially around the ion, for the electrostatic interaction between the ion and the condensing molecules always lowers the free energy of formation of the cluster. The presence of ions has been shown experimentally to enhance the rate of nucleation of liquid drops in a supersaturated vapor. Katz et al. (1994) showed, for example, that the nucleation rate of n-nonane, measured in an upward thermal diffusion cloud chamber, at an ion density of 16 × 106 ions cm 3, increased by a factor of 2500 over that in the absence of ions. These investigators also confirmed experimentally that the nucleation rate is directly proportional to the ion density. While both positive and negative ions increase the nucleation rate, many substances exhibit a preference for ions of one sign over the other. Understanding the effect of ions on nucleation would be very difficult if the ions’ specific chemical characteristics have a significant effect on their nucleating efficiency. Since frequently the number of molecules of the nucleating species in the critical-size cluster is on the order of tens of molecules or more, it is reasonable to assume that an ion buried near its center has little effect on the cluster’s surface properties other than those due to its charge. Thus the ion creates a central force field that makes it more difficult for a molecule to evaporate than it would be from an uncharged but otherwise identical cluster. Ion self-repulsion and the typical low density of ions ensure that almost all such clusters contain only one ion. Molecules arrive at and evaporate from the ion–vapor molecule clusters. The critical cluster size for nucleation is that for which the evaporation rate of single vapor molecules is equal to their arrival rate. The effect of the ion imbedded in the cluster on the arrival rate of vapor molecules is considerably weaker than that on the evaporation rate. This difference leads to a significant decrease in the critical cluster size in the presence of ions relative to that for homogeneous nucleation because the reduced evaporation rate means that the two rates become equal at a smaller monomer arrival rate. The smaller monomer arrival rate translates into a lower saturation ratio of vapor molecules needed to achieve the same rate of nucleation as that in the absence of ions. From a kinetic perspective, a cluster is formed and changes its size by the condensation and evaporation of molecules. At a given temperature, the evaporation rate of molecules (per unit time and unit area) is independent of S, but determined primarily by i, while the condensation rate (per unit time and unit area) is approximately proportional to S and independent of i. Now, the intermolecular interaction energy per molecule in the cluster increases with i. Thus, in the case of homogeneous nucleation, the evaporation rate decreases monotonically with increasing i, approaching the bulk liquid value equal to the condensation rate at saturation (S = 1). When S > 1, there is only one cluster size i∗ at which the rate of evaporation and that of condensation balance to yield a critical nucleus at unstable equilibrium with the vapor phase. On the other hand, when an ion is introduced into the cluster, it attracts molecules, thereby decreasing the evaporation rate. Since the electrostatic field decays as r 2, the

ION-INDUCED NUCLEATION

evaporation rate decrease should be most significant for small i and become negligible as i approaches infinity. When the ion–molecule interaction energy is sufficiently large, the condensation rate balances the evaporation rate even if S  1. This implies that the evaporation rate first increases with i, achieves a maximum at a certain value of i, and then decreases while approaching the bulk value. Thus, for S > 1, there are two values of i at which the evaporation rate and the condensation rate balance, the smaller of which is a stable subcritical cluster and the larger is the unstable critical nucleus (Figure 11.13). For S  S´max , the condensation rate is always larger than the evaporation rate for any cluster size i; hence the cluster formation and its growth are spontaneous. The cluster free energy, as a function of number of molecules i within the cluster, is shown for the typical case of ion-induced nucleation in Figure 11.13. The free-energy curves are in fact consistent with the kinetic point of view given above. Thus, for an appropriate value of the supersaturation, the freeenergy curve shows a local minimum and a maximum for the case of ion-induced nucleation, corresponding, respectively, to the stable subcritical cluster and the unstable critical cluster. The local minimum disappears for the case of homogeneous nucleation.

FIGURE 11.13 Ion-induced nucleation: (a) free energy of cluster formation in homogeneous nucleation; (b) free energy of cluster formation in ion-induced nucleation. Smax and S´max are the maximum values of the saturation ratio for which no barrier to nucleation exists in homogeneous and ion-indued nucleation, respectively.

479

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The earliest attempt to calculate the Gibbs free-energy change of cluster formation in ion-induced nucleation is due to Thomson (1906), based on the theory of capillarity, where a nucleus is represented as a bulk liquid enclosed by an interface of zero thickness with an ion placed at the center. The free- energy change from the Thomson theory depends on q2, where q is the ion charge, and therefore is incapable of expressing a dependence on the sign of q. Physically, the dependence of the ion-induced nucleation rate of a substance on the sign of the ion charge must arise from some asymmetry in the molecular interactions. Such asymmetry should, in principle, manifest itself in a sign dependence of the relevant thermodynamic quantities such as the surface tension. To account for the ion charge preference in ion-induced nucleation requires a statistical mechanical theory, which assumes an intermolecular potential as the fundamental information required to evaluate the relevant thermodynamic properties. Kusaka et al. (1995a) presented a density functional theory for ion-induced nucleation of dipolar molecules. Asymmetry is introduced into a molecule by placing a dipole moment at some fixed distance from its center. As a result of the asymmetric nature of the molecules and their interactions with the ion, the free-energy change acquires a dependence on the ion charge. The predicted freeenergy change shows a sign preference, resulting in a difference in the nucleation rate by a factor of 10–102, for realistic values of model parameters. The sign effect is found to decrease systematically as the supersaturation is increased. The asymmetry of a molecule is shown to be directly responsible for the sign preference in ion-induced nucleation. Kusaka et al. (1995b) also present a density functional theory for ion-induced nucleation of polarizable multipolar molecules. For a fixed orientation of a molecule, the ion–molecule interaction through the molecular polarizability is independent of the sign of the ion charge, while that through the permanent multipole moments is not. As a result of this asymmetry, the reversible work acquires a dependence on the sign of the ion charge.

11.10 ATMOSPHERIC NEW-PARTICLE FORMATION The first evidence of particle formation in the atmosphere was provided by John Aitken at the end of the nineteenth century (Aitken 1897). Aitken built the first apparatus to measure the number of dust and fog particles in the air. Little progress was made in understanding the source of new-particle formation or how widespread it might be for almost a century. In the 1990s, the development of instruments capable of measuring the size distribution of particles as small as 3 nm led to the discovery that nucleation of new particles is common in many areas around the world (Kulmala et al. 2004). New-particle formation occurs frequently in the atmosphere. It is estimated that nucleation generates about one-half of global cloud condensation nuclei (CCN) (Merikanto et al. 2009). All nucleated particles and most combustion particles are initially too small to serve as CCN (∼50 nm diameter and larger). Model calculations and in situ measurements show that most of the mass on atmospheric CCN results from condensation of vapor molecules onto particles originally much smaller than CCN sizes. Even so, only a fraction of originally formed nanometer-size particles grow large enough to act as CCN. Most disappear by scavenging; nanoparticles are highly diffusive and prone to coagulating with other particles. Unless they grow quickly, they are unlikely to survive to become CCN. The distinctive feature of atmospheric new-particle formation events is the detection of a “nucleation burst,” the rapid appearance of a high concentration of particles with diameters of ∼3 nm. A nucleation event that occurred in Pittsburgh, PA on August 11, 2001 is shown in Figure 11.14. Stanier et al. (2004), who reported the event shown in Figure 11.14, observed similar occurrences on 30% of the days in a full year in Pittsburgh. The characteristic shape on a logarithmic scale of particle size versus time plot for a nucleation burst, such as that in Figure 11.14, is described as a “banana plot.” The relatively smooth growth of the particles, even as different air parcels pass by the measurement site, suggests that these events take place over relatively large spatial scales.

ATMOSPHERIC NEW-PARTICLE FORMATION

FIGURE 11.14 Evolution of particle size distribution on August 11, 2001 in Pittsburgh, PA, a day with new-particle formation and growth. Particle number concentration (z axis) is plotted against time of day (x axis) and particle diameter (y axis). [Measurements by Stanier et al. (2004).]

11.10.1 Molecular Constituency of New Particles The molecular constituency of new particles in the atmosphere remained elusive ever since the first evidence of ambient nucleation. Sulfuric acid is a prime candidate as a nucleation precursor because it is readily produced in the gas phase via the oxidation of sulfur dioxide by the hydroxyl radical. Sulfuric acid has a vapor pressure vastly lower than that of its precursor SO2. This single-step drastic reduction in vapor pressure is essential to achieve a supersaturation that allows it to nucleate rather than simply condense onto preexisting particles. As a result, new-particle formation in the atmosphere was long thought to be the result of binary nucleation of H2SO4–H2O clusters. Water vapor is a plausible partner for sulfuric acid because sulfuric acid is hygroscopic, and water vapor is orders of magnitude more plentiful in the atmosphere than other candidate partners. Ion-induced nucleation does occur in the atmosphere, but its overall role is estimated to account for only about 10% of overall particle formation (Hirsikko et al. 2011). Marine aerosol nucleation has been confirmed from biogenic iodine emissions (O’Dowd et al. 2002). Daytime concentrations of H2SO4 in the atmospheric boundary layer are typically around 105–107 molecules cm 3 (0.004–0.4 ppt). Despite its tendency to nucleate, at this level of H2SO4, the rate of H2SO4 – H2O binary homogeneous nucleation alone is far below that needed to account for observed rates of atmospheric new particle formation (Kirkby et al. 2011). The inescapable conclusion is that other species must be involved. Consequently, a major effort ensued to identify other species that can explain atmospheric nucleation and to determine whether sulfuric acid itself is even a necessary ingredient for atmospheric new particle formation. Because of the ubiquity of the base NH3 in the atmosphere, the focus of attention turned to ternary H2SO4–NH3 – H2O nucleation. Indeed, trace amounts of NH3 in experiments designed to study binary H2SO4 – H2O nucleation were found to exert a strong effect on the nucleation rate. In an assessment of six competing nucleation theories applied to 10 days in Pittsburgh, Jung et al. (2008) showed that ammoniadriven ternary H2SO4–NH3–H2O nucleation could discriminate successfully between nucleation and nonnucleation days. But, ternary H2SO4 – H2O – NH3 nucleation has been estimated to account in many cases for not more than about one-tenth of the particle formation rates observed in the continental boundary layer (Kirkby et al. 2011). Amines are closely related chemically to ammonia; these compounds, which are known to form strong chemical bonds with sulfuric acid, are largely derived from anthropogenic activities, but are emitted as well by the oceans, the soil, and from biomass burning (see Section 2.3.5). Indeed, amines strongly enhance formation of stable H2SO4 – H2O clusters (Kurten et al. 2008; Loukonen et al. 2010; Almeida et al. 2013). While atmospheric amine levels are well below those of ammonia (typical mixing

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ratios of individual amine species vary from 1 10 ppt), it was hypothesized that amines, as stronger bases than ammonia, may form new particles with sulfuric acid more efficiently than ammonia. Oxidized, low-volatility organic molecules are ubiquitous in the atmosphere (see Chapters 6 and 14). Another potential nucleation mechanism involves clustering of these organic molecules with sulfuric acid via hydrogen bonding, stabilizing the clusters, or even nucleation of these low-volatility oxidized organic species in the absence of sulfuric acid and ammonia/amines. And since electrostatic attraction between ions enhances nucleation rates over what is predicted for neutral species alone, any of these neutral molecule nucleation mechanisms could be enhanced by the presence of ions. Summarizing, the possible atmospheric nucleation mechanisms that can potentially contribute to observed new particle formation are as follows: 1. 2. 3. 4. 5.

Ternary nucleation of H2SO4–NH3–H2O Ternary nucleation of H2SO4–amine–H2O Ternary nucleation of H2SO4–oxidized organic–H2O Nucleation of oxidized organics alone Each of the above processes occurring in the presence of ions

11.10.2 New-Particle Growth Rates It has been estimated that up to half of all cloud droplets globally form on cloud condensation nuclei (CCN) that originated as nucleated particles rather than being directly emitted into the atmosphere as particulate matter (Merikanto et al. 2009). In order for atmospheric nucleation to have a climatic effect, it is necessary for the nascent clusters to grow into the 50–100 nm diameter size range, where they can act as CCN. This growth occurs through the uptake of vapor molecules. Growth from 1 nm to 50 nm in diameter corresponds to more than a 100,000-fold increase in particle mass, so the composition of a particle that reaches this size is determined virtually completely by the condensing vapors that caused the growth. Moreover, the growth needs to be fast enough for the growing clusters to effectively compete with coagulation scavenging by larger particles (Riipinen et al. 2011, 2012; Donahue et al. 2013). The lifetime of nanoparticles against coagulation is quite short (see Chapter 13). The typical size of the smallest stable clusters in the atmosphere is ∼1 nm diameter. Traditionally, the smallest size of a nascent particle that could be detected was ∼3 nm diameter, and measurements of the rate of new particle formation were based on the rate of appearance of such-size particles. The recent advent of powerful new mass spectrometry allows detection of particles down to the smallest nuclei. Typical growth rates for stable clusters between 1 and 3 nm are of order 1–2 nm h 1. Significant loss by coagulation with larger particles occurs before the particles reach the 3-nm threshold. Under typical atmospheric conditions the lifetime of a 1-nm particle owing to coagulation scavenging by larger particles is of order 10 min; that for a 3-nm particle is less than 1 h; the coagulation lifetime of a particle in the 10–100 nm size range is usually many hours to days. Above a diameter of ∼5 nm, owing to the decreased efficiency of coagulation scavenging, condensation of organic molecules onto growing particles becomes overwhelmingly important, due to the predominance of organics (Wang et al. 2010; Riipinen et al. 2011, 2012; Zhang et al. 2012). The competitive processes involved in the growth of nucleated particles to CCN size are depicted in Figure 11.15.

11.10.3 CLOUD Studies of Atmospheric Nucleation Molecular ions formed in the atmosphere by galactic cosmic rays, high-energy subatomic particles that bombard Earth’s atmosphere from space, can, in principle, play a role in facilitating new-particle formation through ion-induced nucleation. As described in Section 11.9, ions attract polar molecules, such as H2SO4, and help stabilize small molecular clusters. Since molecular ions in the troposphere result mainly from galactic cosmic rays, the role of ions in atmospheric nucleation is of particular interest as a possible mechanism that links fluctuations in solar output to formation of CCN, and hence, to cloud formation itself (Kirkby 2007). Figure 11.16 shows the altitude dependence of ion pair production in the

ATMOSPHERIC NEW-PARTICLE FORMATION

FIGURE 11.15 Schematic diagram showing the influence of ammonia, organics, and ions on aerosol nucleation (Pierce 2011). (a) Sulfuric acid vapor is considered a ubiquitous ingredient in atmospheric nucleation, but H2SO4 – H2O nucleation alone is too slow to explain observed rates. (b) Ammonia, amines, and gas-phase ions stabilize molecular clusters of H2SO4, preventing evaporation. (c) Stabilized particles can proceed to nucleation. (d) Condensation of vapor molecules onto the stable clusters causes particles to grow to sizes of order 100 nm diameter. (e) This growth occurs simultaneously with coagulation with preexisting particles. (Source: Pierce, J., Nature Geosci. 4, 665–666 (2011). Reprinted by permission.)

troposphere from galactic cosmic rays, as well as a typical altitude-dependent sulfuric acid concentra­ tion. Even though ions can enhance nucleation rates via clusters that include HSO4 , ion-induced nucleation rates of H2SO4 H2O measured in the laboratory are still well below those observed in the atmosphere near the surface. The solar–climatic linkage via ion-induced nucleation, which will be discussed further in Chapter 23, spurred the initial development in 2009 of the world’s most sophisticated atmospheric nucleation experimental facility: the CLOUD system at CERN (the European Organization for Nuclear Research) (Kirkby et al. 2011). The CLOUD experimental facility was constructed to measure nucleation rates of H2SO4 and associated compounds under highly controlled, ultraclean conditions. The original objective was to constrain the role of cosmic-ray-induced ions in atmospheric nucleation. CLOUD was con­ structed with both precise control over electric fields and the potential to expose the chamber to a controlled flux of pions, which mimic cosmic rays. In a typical measurement sequence in CLOUD, nucleation rates are measured under ion-free (neutral) conditions and at ion pair concentrations ∼400 cm 3 for boundary-layer galactic cosmic ray conditions, and ∼3000 cm 3 representative of the top of the troposphere. The initial results from CLOUD (Kirkby et al. 2011) showed that even though ions can indeed strongly enhance new-particle formation rates of sulfuric acid via molecular clusters involving HSO4– the ion-induced nucleation rates are still well below those observed in the atmosphere near the surface of

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FIGURE 11.16 Predictions of a numerical model of ion-induced aerosol formation in the troposphere. Left panel: Temperature (crosses), relative humidity (pluses), preexisting aerosol surface area concentration (squares). RIght panel: H2SO4 concentration profile (triangles), solar maximum (open circles) and minimum (solid circles) ion pair production rate Q between solar maximum and minimum. (Source: Kazil, J., and Lovejoy, E. R., J. Geophys. Res. 109, D19206 (doi:10.1029/2004JD004852) (2004). Reprinted by permission.)

Earth. Despite the ultraclean CLOUD experiments, minute traces of ammonia were found to be present in the chamber, as evidenced from high-resolution mass spectra of growing clusters. The ammonia traces in the system were below levels found in the atmosphere, but the presence of ammonia molecules in the growing clusters confirmed the importance of this base molecule in atmospheric nucleation. As noted above, only recently has it been possible to observe directly the nascent molecular clusters that constitute stable nuclei. With the development of the atmospheric-pressure interface time-of-flight (APiTOF) mass spectrometer capable of accessing the range of particle sizes corresponding to critical nuclei, it has become possible to measure the concentration and composition of molecular clusters and particles in the 1–2 nm diameter range (Kulmala et al. 2013). The high-resolution mass spectrometer allows an unambiguous determination of the cluster composition. The growth of clusters in the H2SO4–NH3 system initially proceeds by simple addition of H2SO4 molecules to a bisulfate anion but then switches to neutral growth in which each H2SO4 addition is quickly matched by a NH3 molecule. Results from CLOUD confirmed that binary H2SO4–H2O nucleation does not take place at appreciable rates in the atmospheric boundary layer except under very cold conditions (Kirkby et al. 2011). The presence of ∼100 ppt NH3 enhances the binary nucleation rate by about a factor of 100, while additional amounts of NH3 result in only small further increases in nucleation rate. Additional experiments in the CLOUD facility showed that ternary nucleation involving H2SO4 and dimethylamine (DMA), (CH3)2NH, can lead to new-particle formation rates consistent with surface-level measurements (1–10 particles cm 3 s 1 for H2SO4 concentrations between 106 and 107 molecules cm 3) (Almeida et al. 2013). Growth of the clusters occurs by nearly stoichiometric addition of one sulfuric acid molecule and one DMA molecule. Dimethylamine is just the first in the series of amine molecules. Electronic energies computed for H2SO4 + amine dimer-forming reactions increase according to the sequence: NH3 < CH3NH2 < CH3CH2NH2 < (CH3)2NH < (CH3)3N < (CH3CH2)2NH < (CH3CH2)3N (Kurten et al. 2008). Nucleation rates in the presence of only 5 ppt of dimethylamine are more than 1000 times faster than those with NH3 (Almeida et al. 2013). Amine mixing ratios of a few ppt occur in the continental boundary layer (Ge et al. 2011). Additional DMA, up to 140 ppt, results in less than a factor of three further rate increase, indicating that amine levels of ∼5 ppt are sufficient to reach the limit for amine ternary nucleation under atmospheric conditions …‰H2 SO4 Š ≲ 3  107 molecules cm 3 †. The modest increase in nucleation rate observed above 5 ppt DMA indicates that DMA-promoted nucleation at ambient H2SO4 concentrations

ATMOSPHERIC NEW-PARTICLE FORMATION

is limited by the availability of H2SO4, not DMA. The presence of ions corresponding to surface-level galactic cosmic ray fluxes enhances the nucleation rates by a maximum factor of 2–10 (Almeida et al. 2013). An important consequence of the CLOUD experiments was the discovery of the base stabilization mechanism for NH3 ternary nucleation, involving the formation of strongly bound acid–base pairs that reduce cluster evaporation. Sulfuric acid–amine nucleation proceeds by the same base stabilization mechanism observed for NH3, in which each additional acid molecule in the cluster is stabilized by one (or occasionally, two) base molecule(s). Moreover, it has been found that the rate of new-particle formation in the sulfuric acid–amine system does not require passage of the cluster over a free-energy barrier. Cluster formation and growth appear to be a chemical reaction acid–base-driven process. Figure 11.17 summarizes a number of measurements of boundary-layer nucleation rates as a function of the concentration of H2SO4 vapor (Chen et al. 2012). Observed new-particle formation rates of ∼1.5 nm diameter particles cover the range of 0.1–1000 cm 3 s 1 for sulfuric acid vapor concentrations of 106 to 108 molecules cm 3. A log–log plot of observed J1.5 versus [H2SO4] exhibits a slope between 1 and 2. The considerable scatter in the measured nucleation rates J at a given H2SO4 concentration likely reflects dependence on other nucleation precursor gases, temperature, relative humidity, as well as uncertainties introduced in computing J from ambient aerosol measurements. For all the studies shown in Figure 11.17, nucleation rates range from 1 × 10 2 to 5 × 10 6 times the H2SO4–H2SO4 molecular collision rate, 0.5 k11 [H2SO4]2, where k11 is the hard-sphere collision rate constant for two H2SO4 molecules (see Chapter 3). Note that classical binary H2SO4–H2O homogeneous nucleation theory underpredicts atmospheric observations by over 10 orders of magnitude for typical ambient daytime H2SO4 concentrations (106 to 3 –108 molecules cm 3). Given the essential role played by base species, like amines, in sulfuric acid nucleation, an acid–base chemical model is now considered to be a better representation of the sulfuric acid–ammonia/amine nucleation process than a classical nucleation barrier formulation.

FIGURE 11.17 Dependence of nucleation rate J observed in the atmospheric boundary layer on the concentration of H2SO4, [H2SO4] (molecules cm 3) (Chen et al. 2012). The solid diagonal lines provide bounds for the observations and show the nucleation rates from 10 2 to 5 × 10 6 times the hard-sphere collision rate of two H2SO4 vapor molecules. The figure includes data from Mexico City (428 data points), Atlanta, Georgia 2009 (31 data points), Atlanta, Georgia 2002 (115 data points), Hyytiälä, Finland (602 data points), Idaho Hill, Colorado (86 data points), Mauna Loa, Hawaii (107 data points), Macquarie Island (11 data points), and Boulder, Colorado (180 data points). (Source: Chen, M., et al., Proc. Natl. Acad. Sci. USA 109, 18713–18718 (2012). Reprinted by permission.)

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Acid–Base Chemical Model for Atmospheric Nucleation (Chen et al. 2012) Let A1, A2, A3, and so on denote clusters that contain, respectively, one, two, three, and higher H2SO4 molecules. Each of these clusters may contain molecules of H2O and base species, as well. The acid– base model for nucleation can be represented chemically by k11

A1 ‡ A1 „A2MV E2MV

A2MV ‡ B

k´21

! A2LV

k21

A2LV ‡ A1 „A3 E3

k31

A3 ‡ A1

! A4

For clusters containing two H2SO4 molecules, A2MV and A2LV denote “more volatile” and “less volatile” dimers. A2MV is converted to A2LV by reacting with the base vapor B (e.g., NH3 and amines), leading to a more stable cluster. E2MV denotes the evaporation rate of A2MV back to two H2SO4 monomers. E3 is the rate of evaporation of the A3 trimer back to the dimer. In this model, E2MV and E3 are regarded as empirical parameters that can be adjusted to fit measurements of dimer and trimer concentrations. k11 represents the hard-sphere monomer–monomer collision rate, k21 the monomer–dimer collision rate, and so on. It is assumed that the tetramer A4 is stable, that is, its rate of evaporation, E4 = 0. Chen et al. (2012) determined values of E2MV = 400 s 1 and E3 = 0.4 s 1 based on fitting dimer and trimer data from chamber and field experiments. The clusters are also, in general, subject to removal through scavenging by preexisting particles. Since the scavenging rates for the nascent clusters tend to be smaller than the growth and evaporation rates, we have not included scavenging by preexisting particles in the present formulation. [These are included in the treatment by Chen et al. (2012).] The nucleation rate is taken to be equal to the tetramer (A4) formation rate. The kinetic equation for A2MV is d‰A2MV Š 1 ˆ k11 ‰A1 Š2 dt 2

E2MV ‰A2MV Š

k´21 ‰A2MV Š‰BŠ

(11.112)

where k´21 is the bimolecular rate constant for A2MV + B. The rapidity of these reactions allows one to make the pseudo-steady-state approximation. The pseudo-steady-state concentration of A2MV is ‰A2MV Š ˆ

1 1
k11 ‰A1 Š2 E2MV ‡ k´21 ‰BŠ 2

(11.113)

The rate of formation of A2LV is d‰A2LV Š ˆ k´21 ‰A2MV Š‰BŠ dt

k21 ‰A1 Š ‰A2LV Š ‡ E3 ‰A3 Š

(11.114)

Under typical conditions, the A3 evaporation term, E3[A3], in (11.114) will be small as compared to the other terms in the equation, so the steady-state concentration of A2LV is approximated by ‰A2LV Š ˆ

k´21 ‰BŠ
‰A2MV Š k21 ‰A1 Š

(11.115)

The base stabilization of A2MV is very effective, so it may be assumed that ‰A2 Š ≅ ‰A2LV Š, that is most of the A2 “family” resides in A2LV.

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THE LAW OF MASS ACTION

The trimer balance is d‰A3 Š ˆ k21 ‰A1 Š‰A2LV Š dt

k31 ‰A1 Š‰A3 Š

E3 ‰A3 Š

(11.116)

and the steady state concentration of A3 is ‰A3 Š ˆ

k21 ‰A1 Š‰A2LV Š k31 ‰A1 Š ‡ E3

(11.117)

Finally, the nucleation rate is taken to be the rate of formation of A4 J 4 ˆ k31 ‰A1 Š‰A3 Š

(11.118)

1 J 4 ˆ P
k11 ‰A1 Š2 2

(11.119)

which gives

where Pˆ

k31 ‰A1 Š k ´ ‰B Š
´ 21 k31 ‰A1 Š ‡ E3 k21 ‰BŠ ‡ E2MV

(11.120)

Note that P < 1. For sufficiently high monomer concentration (e.g. [A1] > 109 cm 3), the first term on the right-hand side of (11.120) becomes independent of [A1] because the denominator is dominated by the k31 [A1] term. Likewise for sufficiently high base concentration, the second term in (11.120) becomes independent of [B]. In these cases, the overall rate of nucleation is controlled by the rate of collision of two H2SO4 molecules. This gives rise to the two solid lines proportional to [H2SO4]2 encompassing the data in Figure 11.17.

11.10.4 Atmospheric Nucleation by Organic Species A wide variety of oxidized, low-volatility organics are generated in the atmospheric oxidation of volatile organic compounds, especially those emitted by vegetation (see Chapters 6 and 14), and condensation of such organics plays a key role in the growth of particles larger than ∼3 nm (Figure 11.15). In addition, organics of sufficiently low volatility form stable molecular clusters with H2SO4 (Zhang et al. 2009; Metzger et al. 2010; Schobesberger et al. 2013; Riccobono et al. 2014). Unlike inorganic bases, which condense onto growing particles at essentially a 1:1 ratio with H2SO4, these highly oxidized organic molecules can condense on mixed sulfate–amine–organic clusters at rates that are not constrained by the 1:1 stoichiometry associated with sulfuric acid–base clusters. In fact, owing to the relatively high ambient concentration of the great variety of oxidized organic molecules, it is highly likely that low-volatility organic vapors are involved in all stages of atmospheric nucleation and growth.

APPENDIX 11

THE LAW OF MASS ACTION

The rate of an irreversible chemical reaction can generally be written in the form r = kF(ci), where k is the rate constant of the reaction and F(ci) is a function that depends on the composition of the system as expressed by concentrations ci. The rate constant k does not depend on the composition of the system— hence the term rate constant. Frequently, the function F(ci) can be written as F…ci † ˆ ∏ cαi i

(11A.1)

i

where the product is taken over all components of the system. The exponent αi is called the order of the reaction with respect to species i. The algebraic sum of the exponents is the total order of the reaction. If the αi are equal to the stoichiometric coefficients |νi| for reactants and equal to zero for all other

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components, then F…ci † ˆ ∏ cijνi j

for reactants only

(11A.2)

i

This form is termed the law of mass action. When a reaction is reversible, its rate can generally be expressed as the difference between the rates in the forward and reverse directions. The net rate can be written as r ˆ kf ∏ cαi i

αr

f

kr ∏ ci i

i

(11A.3)

i

At equilibrium, r = 0 and αr

kf ci ˆ ∏ if kr i cαi i

(11A.4)

Now the ratio kf/kr depends only on temperature. The equilibrium constant Kc, defined by Kc ˆ ∏ cνi i

(11A.5)

i

also depends only on temperature. Consequently αr

∏ i

ci i

(11A.6)

f

α

ci i

must be a certain function of ∏ cνi i i

that is αr

∏

ci i f

i cαi i

ˆ f ∏cνi i i

This relation must hold regardless of the value of the concentrations. This is so if αr

∏ i

with αri

ci i α

f

ci i

ˆ

∏cνi

n

(11A.7)

i

f

αi ˆ νi n for all i. In particular, then kf ˆ Knc kr

(11A.8)

kf ˆ Kc kr

(11A.9)

Ordinarily n = 1, so

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PROBLEMS

PROBLEMS 11.1A Calculate the homogeneous nucleation rate of ethanol at 298 K for saturation ratios from 2 to 7 using (11.47) and (11.74). Comment on any differences. 11.2B For the homogeneous nucleation of water at 20°C at a saturation ratio S = 3.5, calculate the sensitivity of the nucleation rate to small changes in saturation ratio and surface tension; that is, find x and y in ΔJ ΔS Δσ ˆx ‡y σ J S 11.3B The nucleation rate can be expressed in the form J ˆ C exp… ΔG∗ =kT† Assuming that the preexponential term C is not a strong function of S, show that d ln J ≅ i∗ d ln S This result is called the nucleation theorem. 11.4A a. Calculate the H2SO4 gas-phase concentration (in molecules cm 3) that produces a binary homogeneous nucleation rate of 1 cm 3 s 1 at RHs of 50% and 100% at 273 K and 298 K, using (11.102). b. Maximum tropospheric H2SO4 concentrations are ∼ 5 × 107 molecules cm 3. What is the likelihood of binary homogeneous nucleation of H2SO4-H2O occurring at a rate exceeding 1 cm 3 s 1 over this range of conditions? 11.5B A simple conceptual model of H2SO4 nucleation in the presence of an amine molecule is 1f

A1 ‡ A1 „ A2 1b

A2 ‡ B

2

! A2
B

where A1 denotes a molecule of sulfuric acid, and B denotes a molecule of amine. Let us assume that the rate of nucleation J is equivalent to the rate of reaction 2. a. Assuming that the sulfuric acid dimer A2 is in a pseudosteady state, show that the rate of formation of the A2
B entity is k2 ‰BŠ 1 Jˆ
k1f ‰A1 Š2 k1b ‡ k2 ‰BŠ 2 b. Assuming k1b  104 s 1 k2 = 3.4 × 10 10 cm3 molecule 1 s 1 (hard-sphere collision rate constant) [B] = 2.34 × 109 to 2.34 × 1010 molecules cm 3 (10 100 ppt dimethyl amine) calculate the range of values of the prefactor Pˆ

k2 ‰BŠ k1b ‡ k2 ‰BŠ

c. From the straight lines bounding the atmospheric nucleation rates in Figure 11.17, compare the range of the prefactor values from Figure 11.17 with those estimated in part b.

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REFERENCES Aitken, J. (1897), On some nuclei of cloudy condensation, Trans. Royal Soc. XXXIX. Almeida, J., et al. (2013), Molecular understanding of amine-sulphuric acid particle nucleation in the atmosphere, Nature 502, 359–363. Becker, R., and Döring, W. (1935), Kinetische behandlung der keimbildung in übersättigten dämpfen, Ann. Phys. (Leipzig) 24, 719–752. Chen, M., et al. (2012), Acid-base chemical reaction model for nucleation rates in the polluted atmospheric boundary layer, Proc. Natl. Acad. Sci. USA 109, 18713–18718. Courtney, W. G. (1961), Remarks on homogeneous nucleation, J. Chem. Phys. 35, 2249–2250. Delale, C. F., and Meier, G. E. A. (1993), A semiphenomenological droplet model of homogeneous nucleation from the vapor phase, J. Chem. Phys. 98, 9850–9858. Dillmann, A., and Meier, G. E. A. (1989) Homogeneous nucleation of supersaturated vapors, Chem. Phys. Lett. 160, 71–74. Dillmann, A., and Meier, G. E. A. (1991) A refined droplet approach to the problem of homogeneous nucleation from the vapor phase, J. Chem. Phys. 94, 3872–3884. Donahue, N. M., et al. (2013), How do organic vapors contribute to new-particle formation? Faraday Discuss. 165, 91–104. Doyle, G. J. (1961), Self-nucleation in the sulfuric acid–water system, J. Chem. Phys. 35, 795–799. Farkas, L. (1927), Keimbildungsgeschwindigkeit in übersättigen dämpfen, Z. Phys. Chem. 125, 236–242. Fletcher, N. H. (1958), Size effect in heterogeneous nucleation, J. Chem. Phys. 29, 572–576. Flood, H. (1934), Tröpfenbildung in übersättigten äthylalkohol-wasserdampfgemischen, Z. Phys. Chem. A170 286–294. Frenkel, J. (1955), Kinetic Theory of Liquids. Dover, New York (first published in 1946). Ge, X., et al. (2011), Atmospheric amines – Part II. Thermodynamic properties and gas/particle partitioning, Atmos. Environ. 45, 561–577. Girshick, S. L., and Chiu, C.-P. (1990), Kinetic nucleation theory: A new expression for the rate of homogeneous nucleation from an ideal supersaturated vapor, J. Chem. Phys. 93, 1273–1277. Hale, B. N. (1986), Application of a scaled homogeneous nucleation rate formalism to experimental data at T < TC, Phys. Rev. A 33, 4156–4163. Hamill, P., et al. (1977), The nucleation of H2SO4–H2O solution aerosol particles in the stratosphere, J. Atmos. Sci. 34, 150–162. Heist, R. H. (1986), Nucleation and growth in the diffusion cloud chamber, in Handbook of Heat and Mass Transfer, Gulf Publishing, Houston, TX, pp. 487–521. Heist, R. H., and Reiss, H. (1974), Hydrates in supersaturated sulfuric acid–water vapor, J. Chem. Phys. 61, 573–581. Hirsikko, A., et al. (2011), Atmospheric ions and nucleation: A review of observations, Atmos. Chem. Phys. 11, 767–798. Hung, C. H., et al. (1989), Condensation of a supersaturated vapor. 8. The homogeneous nucleation of n-nonane, J. Chem. Phys. 90, 1856–1865. Jaecker-Voirol, A., and Mirabel, P. (1988), Nucleation rate in a binary mixture of sulfuric acid and water vapor, J. Phys. Chem. 92, 3518–3521. Jaecker-Voirol, A., and Mirabel, P. (1989), Heteromolecular nucleation in the sulfuric acid–water system, Atmos. Environ. 23, 2053–2057. Jaecker-Voirol, A., et al. (1987), Hydrates in supersaturated binary sulfuric acid–water vapor: A reexamination, J. Chem. Phys. 87, 4849–4852. Jung, J. G., et al. (2008), Evaluation of nucleation theories in a sulfur-rich environment, Aerosol Sci. Technol. 42, 495– 504. Katz, J. L. (1970), Condensation of a supersaturated vapor. I. The homogeneous nucleation of the n-alkanes, J. Chem. Phys. 52, 4733–4748. Katz, J. L., et al. (1994), Condensation of a supersaturated vapor IX. Nucleation on ions, J. Chem. Phys. 101, 2309–2318. Kazil, J., and Lovejoy, E. R. (2004), Tropospheric ionization and aerosol production: A model study, J. Geophys. Res. 109, D19206 (doi: 10.1029/2004JD004852).

REFERENCES Kirkby, J. (2007), Cosmic rays and climate, Surv. Geophys. 28, 333–375. Kirkby, J., et al. (2011), Role of sulphuric acid, ammonia and galactic cosmic rays in atmospheric aerosol nucleation, Nature 476, 429–433. Kulmala, M., and Laaksonen, A. (1990), Binary nucleation of water–sulfuric acid system: Comparison of classical theories with different H2SO4 saturation vapor pressures, J. Chem. Phys. 93, 696–701. Kulmala, M., et al. (2004), Formation and growth rates of ultrafine particles: A review of observations, J. Aerosol Sci. 35, 143–176. Kulmala, M., et al. (2013), Direct observations of atmospheric aerosol nucleation, Science 339, 943–946. Kurten, T., et al. (2008), Amines are likely to enhance neutral and ion-induced sulfuric acid-water nucleation in the atmosphere more efficiently than ammonia, Atmos. Chem. Phys. 8, 4095–4103. Kusaka, I., et al. (1995a), Ion-induced nucleation: A density functional approach, J. Chem. Phys. 102, 913–924. Kusaka, I., et al. (1995b), Ion-induced nucleation. II. Polarizable multipolar molecules, J. Chem. Phys. 103, 8993–9009. Laaksonen, A., et al. (1995), Nucleation: Measurements, theory, and atmospheric applications, Annu. Rev. Phys. Chem. 46, 489–524. Loukonen, V., et al. (2010), Enhancing effect of dimethylamine in sulfuric acid nucleation in the presence of water–a computational study, Atmos. Chem. Phys. 10, 4961–4974. Määttänen, A., et al. (2007), Two-component heterogeneous nucleation kinetics and an application to Mars, J. Chem. Phys. 127 (13), 134710. McDonald, J. E. (1964), Cloud nucleation on insoluble particles, J. Atmos. Sci. 21, 109. Merikanto, J., et al. (2009), Impact of nucleation on global CCN, Atmos. Chem. Phys. 9, 8601–8616. Metzger, A., et al. (2010), Evidence for the role of organics in aerosol particle formation under atmospheric conditions, Proc. Natl. Acad. Sci. USA 107, 6646–6651. Mirabel, P., and Clavelin, J. L. (1978), On the limiting behavior of binary homogeneous nucleation theory, J. Aerosol Sci. 9, 219–225. Mirabel, P., and Katz, J. L. (1974), Binary homogeneous nucleation as a mechanism for the formation of aerosols, J. Chem. Phys. 60, 1138–1144. Mirabel, P., and Reiss, H. (1987), Resolution of the “Renninger–Wilemski problem” concerning the identification of heteromolecular nuclei, Langmuir 3, 228–234. Noppel, M., et al. (2002), An improved model for hydrate formation in sulfuric acid–water nucleation, J. Chem. Phys. 116, 4221–4227. O’Dowd, C. D., et al. (2002), Marine aerosol formation from biogenic iodine emissions, Nature 417, 632–636. Oxtoby, D. W. (1992a), Homogeneous nucleation: Theory and experiment, J. Phys. Condensed Matter 4, 7627–7650. Oxtoby, D. W. (1992b), Nucleation, in Fundamentals of Inhomogeneous Fluids, D. Henderson (ed.), Marcel Dekker, New York, pp. 407–442. Pierce, J. (2011), Atmospheric chemistry: Particulars of particle formation, Nature Geosci. 4, 665–666. Reiss, H. (1950), The kinetics of phase transition in binary systems, J. Chem. Phys. 18, 840–848. Reiss, H., et al. (1990), Molecular theory of vapor phase nucleation: The physically consistent cluster, J. Chem. Phys. 92, 1266–1274. Riccobono, F., et al. (2014), Oxidation products of biogenic emissions contribute to nucleation of atmospheric particles, Science 344, 717–721. Riipinen, I., et al. (2011), Organic condensation: A vital link connecting aerosol formation to cloud condensation nuclei (CCN) concentrations, Atmos. Chem. Phys. 11, 3865–3878. Riipinen, I., et al. (2012), The contribution of organics to atmospheric nanoparticle growth, Nature Geosci. 5, 453–458. Schmitt, J. L. (1981), Precision expansion cloud chamber for homogeneous nucleation studies, Rev. Sci. Instrum. 52, 1749–1754. Schmitt, J. L. (1992), Homogeneous nucleation of liquid from the vapor phase in an expansion cloud chamber, Metall. Trans. A 23A, 1957–1961. Schobesberger, S., et al. (2013), Molecular understanding of atmospheric particle formation from sulfuric acid and large oxidized organic molecules, Proc. Natl. Acad. Sci. USA 110, 17223–17228. Shi, G., et al. (1990), Transient kinetics of nucleation, Phys. Rev. A 41, 2101–2108. Stanier, C. O., et al. (2004), Nucleation events during the Pittsburgh Air Quality Study: Description and relation to key meteorological, gas phase, and aerosol parameters, Aerosol Sci. Technol. 38, 1–12.

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NUCLEATION Stauffer, D. (1976), Kinetic theory of two-component (“heteromolecular”) nucleation and condensation, J. Aerosol Sci. 7, 319–333. Strey, R., and Wagner, P. E. (1982), Homogeneous nucleation of 1-pentanol in a two-piston expansion chamber for different carrier gases, J. Phys. Chem. 86, 1013–1015. Strey, R., et al. (1994), The problem of measuring homogeneous nucleation rates and molecular contents of nuclei: Progress in the form of nucleation pulse measurements, J. Phys. Chem. 98, 7748–7758. Thomson, J. J. (1906), Conduction of Electricity through Gases, Cambridge Univ. Press, Cambridge, UK. Volmer, M., and Weber, A. (1926), Keimbildung in übersättigten gebilden, Z. Phys. Chem. 119, 277–301. Wagner, P. E., and Strey, R. (1981), Homogeneous nucleation rates of water vapor measured in a two-piston expansion chamber, J. Phys. Chem. 85, 2694–2698. Wang, L., et al. (2010), Atmospheric nanoparticles formed from heterogeneous reactions of organics, Nature Geosci. 3, 238–242. Weakleim, C. L., and Reiss, H. (1993), Toward a molecular theory of vapor-phase nucleation. III. Thermodynamic properties of argon clusters from Monte Carlo simulations and a modified liquid drop theory, J. Chem. Phys. 99, 5374–5383. Wexler, A. S., et al. (1994), Modelling urban and regional aerosols: I. Model development, Atmos. Environ. 28, 531–546. Wilemski, G. (1995), The Kelvin equation and self-consistent nucleation theory, J. Chem. Phys. 103, 1119–1126. Wyslouzil, B. E., and Seinfeld, J. H. (1992), Nonisothermal homogeneous nucleation, J. Chem. Phys. 97, 2661–2670. Wyslouzil, B. E., et al. (1991a) , Binary nucleation in acid–water systems. I. Methanesulfonic acid–water, J. Chem. Phys. 94, 6827–6841. Wyslouzil, B. E., et al. (1991b), Binary nucleation in acid–water systems. II. Sulfuric acid–water and a comparison with methanesulfonic acid–water, J. Chem. Phys. 94, 6842–6850. Zeldovich, Y. B. (1942), Theory of new-phase formation: Cavitation, J. Exp. Theor. Phys. (USSR) 12, 525–538. Zhang, R., et al. (2009), Formation of nanoparticles of blue haze enhanced by anthropogenic pollution, Proc. Natl. Acad. Sci. 106, 17650–17654. Zhang, R., et al. (2012), Nucleation and growth of nanoparticles in the atmosphere, Chem. Rev. 112, 1957–2011.

CHAPTER

12

Mass Transfer Aspects of Atmospheric Chemistry

12.1 MASS AND HEAT TRANSFER TO ATMOSPHERIC PARTICLES Mass and energy transport to or from atmospheric particles accompanies their growth or evaporation. We would like to develop mathematical expressions describing the mass transfer rates between condensed and gas phases. The desired expressions for the vapor concentrations and temperature profiles around a growing or evaporating particle can be obtained by solving the appropriate mass and energy conservation equations. Let us consider a particle of pure species A in air that also contains vapor molecules of A. Particle growth or evaporation depends on the direction of the net flux of vapor molecules relative to the particle. As we saw in Chapter 8, the mass transfer process will depend on the particle size relative to the mean free path of A in the surrounding environment. We will therefore start our discussion from the simpler case of a relatively large particle (mass transfer in the continuum regime) and then move to the other extreme (mass transfer in the kinetic regime).

12.1.1 The Continuum Regime The unsteady-state diffusion of species A to the surface of a stationary particle of radius Rp is described by @c ˆ @t

1 @ 2~ …r J A;r † r2 @r

(12.1)

where c(r, t) is the concentration of A, and ~J A;r …r; t† is the molar flux of A (moles area 1 time 1) at any radial position r. This equation is simply an expression of the mass balance in an infinitesimal spherical shell around the particle. The molar flux of species A through stagnant air is given by Fick’s law (Bird et al. 2002) ~J A;r ˆ xA …J~A;r ‡ ~J air;r †

Dg

@c @r

(12.2)

where xA is the mole fraction of A, ~J air;r the radial flux of air at position r, and Dg the diffusivity of A in air. Since air is not transferred to or from the particle, ~J air;r ˆ 0 at all r. Assuming dilute conditions, an Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, Third Edition. John H. Seinfeld and Spyros N. Pandis.  2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

493

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MASS TRANSFER ASPECTS OF ATMOSPHERIC CHEMISTRY

assumption applicable under almost all atmospheric conditions, xA  0 and (12.2) can be rewritten as ~J A;r ˆ Dg @c @r

(12.3)

Combining (12.1) and (12.3), we obtain @ 2 c 2 @c @c ‡ ˆ Dg @r2 r @r @t

(12.4)

which is valid for transfer of A to a particle under dilute conditions. If c1 is the concentration of A far from the particle, cs is its vapor-phase concentration at the particle surface, and the particle is initially in an atmosphere of uniform A with a concentration equal to c1, the corresponding initial and boundary conditions for (12.4) are c…r; 0† ˆ c1 ;

r > Rp

(12.5)

c…1; t† ˆ c1

(12.6)

c…Rp ; t† ˆ cs

(12.7)

The solution of (12.4) subject to (12.5) to (12.7) is (Appendix 12) as follows: c…r; t† ˆ c1

Rp …c1 r

2Rp cs † ‡ p …c1 r π

cs †

p …r Rp †=2 Dg t

∫0

e

ξ2

dξ

(12.8)

The time dependence of the concentration at any radial position r is given by the third term on the right-hand side (RHS) of (12.8). Note that for large values of t, the upper limit of integration approaches zero and the concentration profile approaches its steady state given by c…r† ˆ c1

Rp …c1 r

cs †

(12.9)

In Section 12.2.1 we will show that the characteristic time for relaxation to the steady-state value is on the order of 10 3 s or smaller for all particles of atmospheric interest. Rearranging (12.9), at steady state, we obtain c…r† c1 Rp ˆ cs c1 r

(12.10)

The total flow of A (moles time 1) toward the particle is denoted by Jc, the subscript c referring to the continuum regime, and is given by J c ˆ 4πR2p …J~A †rˆRp

(12.11)

or using (12.9) and (12.3): J c ˆ 4πRp Dg …c1

cs †

(12.12)

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MASS AND HEAT TRANSFER TO ATMOSPHERIC PARTICLES

If c1 > cs, the flow of molecules of A is toward the particle and if c1 < cs vice versa. The result given above was first obtained by Maxwell (1877) and (12.12) is often called the Maxwellian flux. Note that as c is the molar concentration of A, the units of Jc are moles per time. On the contrary, the units of ~J A are moles per area per time. A mass balance on the growing or evaporating particle is ρp d 4 3 πR MA dt 3 p

ˆ Jc

(12.13)

where ρp is the particle density and MA the molecular weight of A. Combining (12.12) with (12.13) gives dRp Dg MA ˆ …c1 dt ρp R p

cs †

(12.14)

When c1 and cs are constant, (12.14) can be integrated to give R2p ˆ R2p0 ‡

2 D g MA …c1 ρp

cs †t

(12.15)

The use of the time-independent steady-state profile given by (12.9) to calculate the change of the particle size with time in (12.15) may seem inconsistent. Use of the steady-state diffusional flux to calculate the particle growth rate implies that the vapor concentration profile near the particle achieves steady state before appreciable growth occurs. Since growth does proceed hundreds of times more slowly than diffusion, the profile near the particle in fact remains at its steady-state value at all times. Growth of atmospheric particles for a constant gradient of MA(c1 cs) = 1 μg m 3 between the bulk and surface concentrations of A is depicted in Figure 12.1. During the condensation/evaporation of a particle, latent heat is released/absorbed at the particle surface. As mass transfer continues, the particle surface temperature changes until the rate of heat transfer balances the rate of heat generation/consumption. The steady-state external temperature and vapor concentration profiles are related by a steady-state energy balance that detemines the surface temperature at all times during the particle growth or evaporation.

FIGURE 12.1 Growth of aerosol particles of different initial radii as a function of time for a constant concentration gradient of 1 μg m 3 between the aerosol and gas phases (Dg = 0.1 cm2 s 1, ρp = 1 g cm 3).

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MASS TRANSFER ASPECTS OF ATMOSPHERIC CHEMISTRY

The steady-state temperature distribution around a particle is governed by

ur

dT 1 d 2 dT ˆα 2 r dr dr r dr

(12.16)

where α = k/ρcp is the thermal diffusivity of air and ur is the mass average velocity at radial position r. The convective velocity ur is the net result of the fluid motion due to the concentration gradients (Pesthy et al. 1981). Equation (12.16) should be solved subject to T…Rp † ˆ T s T…1† ˆ T 1 For a dilute system the first term in (12.16) can be neglected and (12.16) is simplified to the pure conduction equation d2 T 2 dT ‡ ˆ0 dr2 r dr

(12.17)

with solution Rp …T s T1 † r The criterion for neglecting the convective term in (12.16) is T ˆ T1 ‡

Dg M A 1 xAs ln α Mair 1 xA1

(12.18)

 1

(12.19)

where Mair is the molecular weight of air and xAs and xA1 are the mole fractions of A at the particle surface and far away from it. This convective flow is often referred to as Stefan flow and (12.19) provides a quantitative criterion for determining when it can be neglected. In most applications involving mass and heat transfer to atmospheric particles it can be neglected (Davis 1983). Up to this point we have been avoiding the complications of the coupled mass and energy balances by treating cs and Ts as known. The surface temperature is in general unknown and cs depends on it. To determine Ts we need to write an energy balance on the particle ~J A;rˆR ΔH v 4πR2 ˆ k dT p p dr

rˆRp

4πR2p ‡ kp

dTp dr

rˆRp

4πR2p

(12.20)

where k and kp are the thermal conductivities of air and the particle, respectively; T and Tp are the air and particle temperatures; and ΔHv is the molar heat released. The left-hand side (LHS) is the latent heat contribution to the energy balance, while the RHS includes the rates of heat conduction outward into the gas and inward from the particle surface. Chang and Davis (1974) solved the coupled mass and energy balances numerically. Their numerical solution shows that the last term in (12.20) can generally be neglected, indicating that the energy ΔHv is transferred entirely to the gas phase. Combining (12.18) and (12.9) with (12.20), we obtain k…T s

T 1 † ˆ ΔH v Dg …c1

cs †

(12.21)

where cs is in general a function of Ts. For convenience this equation can be written as Ts

T1 T1

ˆ

ΔH v Dg …c1 kT1

cs †

(12.22)

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MASS AND HEAT TRANSFER TO ATMOSPHERIC PARTICLES

If c1  cs, then the temperature difference between the particle and the ambient gas is ΔHv Dgc1/k. For slowly evaporating species and modest heat of vaporization this temperature change is sufficiently small so that isothermal conditions are approached and (12.9) can be used.

12.1.2 The Kinetic Regime For molecules in three-dimensional random motion the number of molecules ZN striking a unit area per unit time is (Moore 1962) 1 (12.23) ZN ˆ NcA 4 where cA is the mean speed of the molecules: cA ˆ

8 kT π mA

1=2

(12.24)

Under these conditions the molar flow Jk (moles time 1) to a particle of radius Rp is J k ˆ π R2p cA α …c1

cs †

(12.25)

where α is the molecular accommodation coefficient [not to be confused with the thermal diffusivity in (12.16)]. The ratio of this kinetic regime flow to the continuum regime flow Jc is J k αcA ˆ Rp J c 4 Dg

(12.26)

The accommodation coefficient will be assumed equal to unity in the next section, and the implications of this assumption will be discussed in Section 12.1.4.

12.1.3 The Transition Regime The steady-state flow of vapor molecules to or from a sphere, when the particle is sufficiently large compared to the mean free path of the diffusing vapor molecules, is given by Maxwell’s equation (12.12). Since this equation is based on the solution of the continuum transport equation, it is no longer valid when the mean free path of the diffusing vapor molecules becomes comparable to the particle diameter. At the other extreme, the expression based on the kinetic theory of gases (12.25) is also not valid in this intermediate regime where λ  Dp. When Kn  1, the phenomena are said to lie in the transition regime. The concentration distributions of the diffusing species and background gas in the transition regime are governed rigorously by the Boltzmann equation. Unfortunately, there does not exist a general solution to the Boltzmann equation valid over the full range of Knudsen numbers for arbitrary masses of the diffusing species and the background gas. Consequently, most investigations of transport phe­ nomena in the transition regime follow an approach based on so-called flux matching. Flux matching assumes that the noncontinuum behavior is limited to a region Rp  r  Δ + Rp beyond the particle surface and that continuum theory applies for r  Δ + Rp. The distance Δ is then of the order of the mean free path λ and within this inner region the basic kinetic theory of gases is assumed to apply. 12.1.3.1 Fuchs Theory The matching of continuum and free molecule fluxes dates back to Fuchs (1964), who suggested that by matching the two fluxes at r = Δ + Rp, one may obtain a boundary condition on the continuum diffusion

498

MASS TRANSFER ASPECTS OF ATMOSPHERIC CHEMISTRY

equation. This condition is, assuming unity accommodation coefficient 4π R2p

1 cA 4

c…Rp ‡ Δ†

cs ˆ D

dc dr

rˆRp ‡Δ

4π…Rp ‡ Δ†2

(12.27)

Then, the steady-state continuum transport equation for a dilute system is d2 c 2 dc ‡ ˆ0 dr2 r dr

(12.28)

Using as boundary conditions (12.27) and c(1) = c1 one obtains the solution c…r† ˆ c1

Rp …c1 r

cs †βF

(12.29)

where the correction factor βF is given by βF ˆ

‰1 ‡ …Δ=Rp †Š cA Rp cA Rp ‡ 4 D‰1 ‡ …Δ=Rp †Š

(12.30)

Relating the binary diffusivity and the mean free path using DAB =λAB cA ˆ 1=3 and letting Kn = λAB/Rp, one obtains J 1 ‡ KnΔ=λAB ˆ 0:75 0:75 ‡ Kn ‡ …Δ=λAB †Kn2 Jc

(12.31)

Note that the definition of the mean free path by DAB =λAB cA ˆ 1=3 implies, using (12.26), that, for α=1 3 Jk ˆ J c 4 Kn

(12.32)

and the Fuchs relation (12.31) also implies, using (12.32), that J 1 ‡ KnΔ=λAB ˆ J k 1 ‡ KnΔ=λAB ‡ 0:75 Kn

1

(12.33)

The value of Δ used in the expressions above was not specified in the original theory and must be adjusted empirically or estimated by independent theory. Several choices for Δ have been proposed; the simplest, due to Fuchs, is Δ = 0. Other suggestions include Δ = λAB and Δ ˆ 2DAB =cA (Davis 1983). 12.1.3.2 Fuchs–Sutugin Approach Fuchs and Sutugin (1971) fitted Sahni’s (1966) solution to the Boltzmann equation for z  1, where z = MA/Mair is the molecular weight ratio of the diffusing species and air, to produce the following transition regime interpolation formula: J 1 ‡ Kn ˆ J c 1 ‡ 1:71 Kn ‡ 1:33 Kn2

(12.34)

499

MASS AND HEAT TRANSFER TO ATMOSPHERIC PARTICLES

Equation (12.34) is based on results for z  1 and therefore is directly applicable to light molecules in a heavier background gas. The mean free path included in the definition of the Knudsen number in (12.34) is given by λAB ˆ

3 DAB cA

For Kn → 0 both (12.31) and (12.34) reduce to the correct limit J/Jc = 1. For the kinetic limit Kn → 1 both (12.31) and (12.34) give Jk/Jc = 3/(4 Kn). 12.1.3.3 Dahneke Approach Dahneke (1983) used the flux matching approach of Fuchs but, assuming that Δ = λAB and defining DAB =…λAB cA † ˆ 1=2, obtained J 1 ‡ Kn ˆ J c 1 ‡ 2 Kn…1 ‡ Kn†

(12.35)

where Kn = λAB/Rp. The mean free path included in the definition of the Knudsen number in (12.35) is given by λAB ˆ

2 DAB cA

Note here that for Kn → 0, J/Jc → 1 as expected. On the other hand, for Kn → 1, J/Jc → 1/(2 Kn). This limit is in agreement with (12.26) because DAB =…λAB cA † ˆ 1=2 and therefore the expressions are consistent. 12.1.3.4 Loyalka Approach Loyalka (1983) constructed improved interpolation formulas for mass transfer in the transition regime by solving the BGK model [Bhatnagar, Gross, and Krook (Bhatnagar et al. 1954)] of the Boltzmann equation to obtain p πKn…1 ‡ 1:333 Kn† J p ˆ J k 1 ‡ 1:333 Kn ‡ …1:333 πKn ‡ ζc †Kn

(12.36)

The mean free path used by Loyalka was defined by 4 DAB λAB ˆ p π cA

(12.37)

and the mass transfer jump coefficient has a value ζc = 1.0161. Williams and Loyalka (1991) pointed out that (12.36) does not exhibit the correct shape near the free-molecule limit. 12.1.3.5 Sitarski–Nowakowski Approach None of the approaches given above describes the dependence of the transition regime mass flux on the molecular mass ratio z of the condensing/evaporating species and the surrounding gas. Sitarski and

500

MASS TRANSFER ASPECTS OF ATMOSPHERIC CHEMISTRY

Nowakowski (1979) applied the 13-moment method of Grad (Hirschfelder et al. 1954) to solve the Boltzmann equation to obtain J Kn…1 ‡ a Kn† ˆ J k b ‡ c Kn ‡ d Kn2 aˆ

3 β…1 ‡ z†2 ; 4…3 ‡ 5z†

bˆ

β…1 ‡ 2z† 1 ‡ ; cˆ π…3 ‡ 5z† 2β

(12.38)

4…9 ‡ 10z† 15π…1 ‡ z†2

(12.39)

9…1 ‡ z†2 dˆ 8…3 ‡ 5z†

where β = 1 for unity accommodation coefficient and z = MA/Mair is the molecular weight ratio. This result is obviously incorrect near the free-molecule regime, because in the limit Kn → 1, (12.38) yields J → 0.666 Jk. Therefore we expect (12.38) to be in error for relatively high values of Kn. Table 12.1 summarizes the transition regime expressions that we have presented in this section. Predictions of mass transfer rates of the preceding four theories are shown as a function of the particle diameter in Figure 12.2. All approaches give comparable results for particle diameters larger than 0.2 μm, even if they employ different definitions of the Knudsen number and different functional dependencies of the mass transfer rate on the Knudsen number. This agreement indicates that as long as one uses a mean free path consistent with the mass transfer theory, the final result will differ little from theory to theory. The theory of Sitarski and Nowakowski (1979), although it is the only one that includes an explicit dependence of the mass transfer rate on z, gives erroneous results for particles smaller than 0.2 μm in this case (Figure 12.2). The dependence of the rate itself on z is rather weak and for z = 5 15, the Fuchs, Dahneke, and Loyalka formulas are in agreement with the Sitarski–Nowakowski results. Li and Davis (1995) compared the results of the above mentioned theories with measurements of the evaporation rates of dibutyl phthalate (DBP) in air (Figure 12.3). All theories are in agreement with the data with the exception of the theory of Sitarski and Nowakowski (1979), which exhibits deviations for Kn > 0.2.

12.1.4 The Accommodation Coefficient Up to this point we have assumed that once a vapor molecule encounters the surface of a particle, its probability of sticking is unity. This assumption can be relaxed by introducing an accommodation coefficient α, where 0  α  1. The flux of a gas A to a spherical particle in the kinetic regime is then given by (12.25). TABLE 12.1 Transition Regime Formulas for Diffusion of Species A in a Background Gas B to an Aerosol J/Jc

Mean free path definition

0:75 α …1 ‡ KnΔ=λAB † 0:75α ‡ Kn ‡ …Δ=λAB †Kn2 Kn2

0:75 α …1 ‡ Kn† ‡ Kn ‡ 0:283 Knα ‡ 0:75 α

1 ‡ Kn 1 ‡ 2 Kn…1 ‡ Kn†=α 1 ‡ 1:333 Kn p 1 ‡ 1:333 Kn ‡ …1:333 πKn ‡ 1†Kn b…1 ‡ aKn† b ‡ cKn ‡ d Kn2

Reference

3DAB cA

Fuchs (1964)

3DAB cA

Fuchs and Sutugin (1971)

2DAB cA

Dahneke (1983)

4 D p AB π cA

Loyalka (1983)

DAB cA [see (12.39) for a, b, c, and d] 4b

Sitarski and Nowakowski (1979)

MASS AND HEAT TRANSFER TO ATMOSPHERIC PARTICLES

FIGURE 12.2 Mass transfer rate predictions for the transition regime by the approaches of (a) Fuchs and Sutugin (1971), (b) Dahneke (1983), (c) Loyalka (1983), and (d) Sitarski and Nowakowski (1979) (z = 15) as a function of particle diameter. Accommodation coefficient α = 1.

FIGURE 12.3 Comparison of experimental dibutyl phthalate evaporation data with the theories of Loyalka et al. (1983), Sitarski and Nowakowski (1979) (for z = 15), and the equation of Fuchs and Sutugin (1971). (Reprinted from Aerosol Science and Technology, 25, Li and Davis, 11–21. Copyright 1995, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 1 GB, UK.)

501

502

MASS TRANSFER ASPECTS OF ATMOSPHERIC CHEMISTRY

The transition regime formulas can then be extended to account for imperfect accommodation by multiplying the LHS of (12.27) by α. The Fuchs expression in (12.31) becomes 1 ‡ KnΔ=λAB J ˆ 0:75 α 0:75 α ‡ Kn ‡ …Δ=λAB †Kn2 Jc

(12.40)

3α Jk ˆ J c 4 Kn

(12.41)

and

The expression (12.35) by Dahneke (1983) becomes J 1 ‡ Kn ˆ J c 1 ‡ 2 Kn…1 ‡ Kn†=α

(12.42)

whereas the Fuchs–Sutugin (1971) approach gives J 0:75 α…1 ‡ Kn† ˆ J c Kn2 ‡ Kn ‡ 0:283 Knα ‡ 0:75 α

(12.43)

The formula of Loyalka is applicable only for α = 1, but the theory of Sitarski and Nowakowski (1979) can be used for any accommodation coefficient setting α (12.44) ∠2 α Figure 12.4 shows mass transfer rates as a function of particle diameter for the three approaches for accommodation coefficient values of 1, 0.1, and 0.01.

FIGURE 12.4 Mass transfer rates as a function of particle diameter for accommodation coefficient values 1.0, 0.1, and 0.01 for the approaches of Sitarski and Nowakowski (1979), Fuchs and Sutugin (1971), and Dahneke (1983).

503

MASS TRANSPORT LIMITATIONS IN AQUEOUS-PHASE CHEMISTRY

12.2 MASS TRANSPORT LIMITATIONS IN AQUEOUS-PHASE CHEMISTRY Dissolution of atmospheric species into cloud droplets followed by aqueous-phase reactions involves the following series of steps: 1. 2. 3. 4. 5.

Diffusion of the reactants from the gas phase to the air–water interface Transfer of the species across the interface Possible hydrolysis/ionization of the species in the aqueous phase Aqueous-phase diffusion of the ionic and nonionic species inside the cloud drop Chemical reaction inside the droplet

These steps must occur during the production of sulfate or for other aqueous-phase chemical reactions within cloud drops. In Chapter 7 we studied the aqueous-phase kinetics of a number of reactions, corresponding to step 5 above. Chapter 7 provided the tools for the calculation of the sulfate production rate at a given point inside a cloud drop, provided that we know the reactant concentra­ tions at this point. For some of the Chapter 7 calculations we assumed that the cloud droplets are saturated with reactants, or equivalently that the concentration of the species inside the droplet is uniform and satisfies at any moment Henry’s law equilibrium with the bulk gas phase. In mathemati­ cal terms we assumed that the aqueous-phase concentration of a species A at location r, at time t, satisfies ‰A…r; t†Š ˆ H ∗A p1 …t†

(12.45)

where H ∗A is the effective Henry’s law coefficient for A in M atm 1 and p1 is the A partial pressure in the bulk gas phase. Use of (12.45) simplified significantly our order of magnitude estimates in Chapter 7, but its validity relies on the following assumptions: 1. The time to establish a gas-phase steady-state concentration profile of A around the cloud droplet is short. 2. Gas-phase diffusion is sufficiently rapid that the concentration of A around the droplet is approximately constant. Mathematically, this implies that p1(t) = p(Rp, t), where p(Rp, t) is the partial pressure of A at the droplet surface. 3. The transfer of A across the interface is sufficiently rapid, so that local Henry’s law equilibrium is always satisfied. Mathematically, this is equivalent to ‰A…Rp ; t†Š ˆ H ∗A p…Rp ; t†. 4. The time to establish a steady-state concentration profile inside the droplet is very short. 5. The time to establish the ionization/hydrolysis equilibria is very short. 6. Aqueous-phase diffusion is rapid enough, so that the concentration of A inside the droplet is approximately uniform. If all six of these assumptions are satisfied, then (12.45) is valid and the calculations are simplified considerably. Our goal now will be to quantify the rates of the five necessary process steps (gas-phase diffusion, interfacial transport, ionization, aqueous-phase diffusion, reaction) calculating appropriate timescales. Then, we will compare these rates to those of aqueous-phase chemical reactions. Finally, we will integrate our conclusions, developing overall reaction rate expressions that factor in, when necessary, the effects of the mass transport limitations. Figure 12.5 depicts schematically the gas- and aqueous-phase concentrations of A in and around a droplet. The aqueous-phase concentrations have been scaled by H ∗A RT, to remove the difference in the units of the two concentrations. This scaling implies that the two concentration profiles should meet at the interface if the system satisfies Henry’s law at that point. In the ideal case, described by (12.45),

504

MASS TRANSFER ASPECTS OF ATMOSPHERIC CHEMISTRY

FIGURE 12.5 Schematic of gas- and aqueous-phase steady-state concentration profiles for the case where there are gas-phase, interfacial, and aqueous-phase mass transport limitations. Also shown is the ideal case where there are no mass transport limitations.

the concentration profile after the scaling should be constant for any r. However, in the general case the gas-phase mass transfer resistance results in a drop of the concentration from cA(1) to cA(Rp) at the air– droplet interface. The interface resistance to mass transfer may also cause deviations from Henry’s law equilibrium indicated in Figure 12.5 by a discontinuity. Finally, aqueous-phase transport limitations may result in a profile of the concentration of A in the aqueous phase from [A(Rp)] at the droplet surface to [A(0)] at the center. All these mass transfer limitations, even if the system can reach a pseudo-steady state, result in reductions of the concentration of A inside the droplet, and retard the aqueous-phase chemical reactions. The importance of these mass transfer processes occurring simultaneously can be assessed by understanding their corresponding timescales. We start by analyzing the various processes independently.

505

MASS TRANSPORT LIMITATIONS IN AQUEOUS-PHASE CHEMISTRY

12.2.1 Characteristic Time for Gas-Phase Diffusion to a Particle Let us assume that we introduce a particle of radius Rp consisting of a species A in an atmosphere with a uniform gas-phase concentration of A equal to c1. Initially, the concentration profile of A around the particle will be flat and eventually after time τdg it will achieve its steady state. This timescale, τdg, corresponds to the time required by gas-phase diffusion to establish a steady-state profile around a particle. It is not the same as the timescale of equilibration of the particle with the surrounding atmosphere. We assume that c1 remains constant and that the concentration of A at the particle surface (equilibrium) concentration is cs and also remains constant. This problem was studied in Section 12.1.1, and the change of the concentration profile with time is given by (12.8). The characteristic timescale of the problem can be derived by nondimensionalizing the differential equation describing the problem, namely, (12.4). The characteristic lengthscale of the problem is the particle radius Rp, the characteristic concentration c1, and the characteristic timescale, our unknown τdg. We define dimensionless variables by dividing the problem variables by their characteristic values: r c c1 t xˆ ϕˆ (12.46) τˆ cs c1 Rp τdg Equation (12.4) can be rewritten as R2p

@ϕ @ 2 ϕ 2 @ϕ ˆ ‡ @τ @x2 x @x

τdg Dg ϕ…x; 0† ˆ 0

ϕ…1; τ† ˆ 0

(12.47)

ϕ…1; τ† ˆ 1

Note that all the scaling in the problem is included in the dimensionless term R2p =τdg Dg and, as the rest of the terms in the differential equation are of order one, this term should also be of order unity. Therefore R2p (12.48) τdg ≅ Dg This timescale can also be obtained from the complete solution of the problem in (12.8). Note that the time-dependent term approaches zero as the upper limit of the integration approaches zero or when p 2 Dg t  r Rp or equivalently when t  (r Rp)2/4Dg. For a point close to the particle surface, …r

Rp †2  R2p and

τdg ˆ

R2p

(12.49)

4 Dg

Note that for a point where r  Rp, the time-dependent term is practically zero because of the existence of the term (Rp/r) in front of the integral. One should not be bothered by the different numerical factor in the two timescales in (12.48) and (12.49). These timescales are, by definition, order-of-magnitude estimates and as such either value is sufficient for estimation purposes. The characteristic timescale of (12.49) can be evaluated for a typical molecular gas-phase diffusivity of Dg = 0.1 cm2 s 1 as a function of the particle radius Rp: Rp(μm) τdg(s)

0.1 2.5 × 10

10

10 2.5 × 10

6

100 2.5 × 10

4

We conclude that the characteristic time for attaining a steady-state concentration profile around particles of atmospheric size is smaller than 1 ms. Since we are interested in changes that occur in

506

MASS TRANSFER ASPECTS OF ATMOSPHERIC CHEMISTRY

atmospheric particles and droplets over timescales of seconds to minutes, we can safely neglect this millisecond transition and assume that the gas-phase concentration is always at steady state and the concentration profile is given in the continuum regime by (12.9) and the flux by (12.12).

12.2.2 Characteristic Time to Achieve Equilibrium at the Gas–Liquid Interface We next want to obtain an expression for the characteristic time associated with establishing phase equilibrium at a gas–liquid interface. We should note that earlier we assumed a gas-phase concentration just above the interface, which presumably is in equilibrium with the liquid-phase composition. It is customary to assume implicitly Henry’s law equilibrium at the gas–liquid interface. This, in effect, is tantamount to assuming that the characteristic time to achieve interfacial phase equilibrium is short compared with the timescale for changes in the concentration of dissolved A in the drop. We will now see if such an assumption is justified. Suppose that at t = 0, a droplet of pure water is immersed in air containing species A at concentration c1 . As A is absorbed into the liquid, the concentration in the liquid will eventually reach that in equilibrium with the bulk gas phase C∗. If the partial pressure correspond­ ing to c1 is p1 , then that equilibrium can be expressed by Henry’s law, C∗ ˆ H ∗A p1 . Our interest here is in determining the characteristic time for equilibrium at the gas-liquid interface to be established. To describe the process leading to equilibrium, we first note that to a gas molecule, the particle is not distinguishable from a flat surface, so within the liquid phase, diffusion of A can be described by @C @2C ˆ Daq 2 @t @x

(12.50)

where x is the distance into the liquid from the interface and Daq is the aqueous-phase diffusion coefficient of A. At the gas–liquid interface molecules of A are arriving at the liquid from the gas, those molecules not absorbed are returning from the liquid surface to the gas, and other molecules are diffusing into the liquid phase. Our object is to solve (12.50) subject to appropriate initial and boundary conditions. At t = 0 there is no A in the liquid phase. Then A starts to dissolve in the liquid and diffuse away from the interface into the bulk liquid. Far from the interface in the liquid the concentration of dissolved A will be zero, although right at the interface equilibrium, described by Henry’s law, will be established after a certain characteristic time. By solving (12.50) we want to determine an expression for that characteristic time. For molecules in three-dimensional random motion the number of molecules striking a unit area in unit time is …1=4†Nc, where N is the molecular number density and c is the mean speed, 8kT πm

cˆ

1=2

ˆ

8RT πM

1=2

(12.51)

where m = the mass of a molecule and M = molecular weight. Since the partial pressure of the vapor is related to N by p = NkT, the number of molecules striking unit area per unit time is, in molar units, 1 8RT 4 πM

1=2

p p ˆ RT …2πMRT †1=2

(12.52)

At the liquid surface molecules are arriving from the gas, molecules are leaving the surface back to the gas, and molecules are diffusing into the liquid phase. Let us call these fluxes R–g, R+g, and R+l, respectively. Since the surface is presumed to have no thickness, these three fluxes must just balance each other. The flux of molecules to the interface from the gas is, as we have just seen, p/(2πMRT)1/2. If the fraction of the incoming molecules that is incorporated into the liquid is α, the accommodation coefficient, then the net flux from the gas to the interface is R

g

ˆ

αp …2πMRT †1=2

(12.53)

507

MASS TRANSPORT LIMITATIONS IN AQUEOUS-PHASE CHEMISTRY

The rate of evaporation, R+g, of A from the liquid depends on the surface concentration of A. Suppose the surface concentration is Cs(t) = C(0,t). The evaporation process does not “know” whether equilibrium has been reached; it simply expels molecules at a rate dependent on Cs(t). If the gas phase were at equilibrium with Cs(t), then the flux into the liquid phase would be zero, and R

g

αps

ˆ R‡g ˆ

(12.54)

…2πMRT†1=2

where ps is the partial pressure of A in equilibrium with Cs. Thus, the evaporative flux at any time is R‡g ˆ

αps …t†

The flux of A into the liquid is given by R+l = R–g R‡l ˆ

(12.55)

…2πMRT†1=2 R+g or

p1

ps … t † α

(12.56)

…2πMRT†1=2

At equilibrium, Henry’s law holds, so C∗ ˆ H ∗A p1 and Cs …t† ˆ H ∗A ps …t†, where C∗ is the surface concentration in equilibrium with p1 . The flux R+l is then equated to the diffusive flux to give Daq

@C @x

ˆ xˆ0

α ∗ HA …2πMRT†1=2

…C∗

C†

Thus, the transient mass transfer leading to phase equilibrium is described by (12.50) subject to C…x; 0† ˆ 0 C…x; t† ˆ 0 Daq

(12.57)

x!1

@C α ˆ …C @x H ∗A …2πMRT†1=2

(12.58) C∗ † x ˆ 0

(12.59)

The solution of (12.50) subject to (12.57)–(12.59) is C…x; t† ˆ C∗ erfc

x p 2 Daq t

C∗ exp

κx t exp erfc Daq τp

t τp

1=2

x ‡ p 2 Daq t

(12.60)

where τp ˆ Daq =κ2 , κ ˆ α H ∗A …2πMRT† 1=2 . As t ! 1; C…0; t† ˆ C∗ . The characteristic time scale of the approach to equilibrium is τp , given by 1

τp ˆ

2

∗ 2πMRTDaq H A α2

(12.61)

Using c ˆ …8RT=πM†1=2 , we can express τp as τp ˆ Daq

4RTH ∗A αc

2

(12.62)

∗ We note that as the Henry’s law coefficient H ∗A increases, τp increases. Physically, a larger H A implies a more soluble gas, so a longer time is required to establish phase equilibrium since more of the gas must cross the interface before equilibrium is achieved. The accommodation coefficient α is interpreted as the fraction of impinging gas molecules that enter the liquid phase. In the absence of other information, α is

508

MASS TRANSFER ASPECTS OF ATMOSPHERIC CHEMISTRY

usually taken to be unity. We see, however, that as α decreases from 1.0, the characteristic time to achieve interfacial phase equilibrium increases as α 2, since fewer of the incoming molecules enter the liquid phase. Let us evaluate the value of τp for O3 and H2O2. In estimating τp we take Daq = 10 5 cm2 s 1, a value characteristic of many dissolved gases in water. For T = 298 K, (12.62) becomes τp ˆ 1:51  10 12 2 MH 2 A =α s. Evaluating this expression for α = 1.0, we get

O3 H2O2

M

H∗A (Table 7.2)

48 34

1.1 × 10 2 1 × 105

τp , s

8.8 × 10 0.52

15

We see that τp varies strongly depending on the Henry’s law coefficient. In the case of these two compounds dissolving in water, with α = 1.0 and Daq having the value for a typical diffusion coefficient of a solute in water, the time scale is so short that the assumption of instantaneous interfacial equilibrium is valid. As we will see in Chapter 14, organic atmospheric aerosols have been found in many instances to be highly viscous and characterized by diffusivities that are substantially smaller than that for liquid-phase particles. Figure 12.6 shows particle-phase profiles, C(x,t)/C∗, based on (12.60) at t = τp for two different values of Daq, one characterizing a liquid-phase particle and one characterizing a particle with high viscosity, and for α = 1.0 and 0.01. Keep in mind that even at the same value of α, τp depends on the value of Daq so that the actual time scales for the two cases are not comparable. At t = τp the interfacial concentration in each case has reached ∼57% of its equilibrium value.

12.2.3 Characteristic Time of Aqueous Dissociation Reactions The next process in the chain is ionization. For SO2, for example, we know that subsequent to absorption we have SO2
H2 O ƒ H‡ ‡ HSO3 We are interested in determining the characteristic time to establish this equilibrium. To do this, we wish to derive an expression for the characteristic time to reach equilibrium of the general reversible reaction AƒB ‡ C with kf the reaction constant for the first-order forward reaction and kr for the second-order reverse reaction. At equilibrium kf ‰AŠe ˆ kr ‰BŠe ‰CŠe Let us define the extent of reaction η by [A] = [A]0 η, [B] = [B]0 + η, and [C] = [C]0 + η, and the equilibrium extent ηe by [A]e = [A]0 ηe, [B]e = [B]0 + ηe, and [C]e = [C]0 + ηe. Then we can define Δη = η ηe to get ‰AŠ ˆ ‰AŠe

Δη

‰BŠ ˆ ‰BŠe ‡ Δη

(12.63)

‰CŠ ˆ ‰CŠe ‡ Δη The rate of disappearance of A is given by d‰AŠ ˆ kf ‰AŠ ‡ kr ‰BŠ‰CŠ dt

(12.64)

509

MASS TRANSPORT LIMITATIONS IN AQUEOUS-PHASE CHEMISTRY

FIGURE 12.6 Normalized concentration profiles in the aqueous phase at t = τp for two values of the aqueous-phase diffusion coefficient: (a) accom­ modation coefficient = 1; (b) accommodation coefficient = 0.01.

Using (12.63) together with the equilibrium relation, we obtain from (12.64) dΔη ˆ a Δη dt

b Δη2

(12.65)

where a = kf + kr[B]e + kr[C]e and b = kr. Integrating this equation from Δη = Δη0 at t = 0, we obtain Δη a ‡ b Δη0 ˆe Δη0 a ‡ b Δη

at

(12.66)

Combining this result together with (12.63) gives ‰AŠ ‰AŠe 1 ‡ …b=a†…‰AŠe ‰AŠ0 † ˆe ‰AŠ0 ‰AŠe 1 ‡ …b=a†…‰AŠe ‰AŠ†

at

(12.67)

510

MASS TRANSFER ASPECTS OF ATMOSPHERIC CHEMISTRY

where b 1 ˆ a …kf =kr † ‡ ‰BŠe ‡ ‰CŠe

(12.68)

where kf/kr is the equilibrium constant K. Let us apply the foregoing analysis to the system of SO2
H2 O ƒ H‡ ‡ HSO3 We note from (12.67) and (12.68) that for all atmospheric conditions of interest the terms [SO2
H2O]e [SO2
H2O]0 and [SO2
H2O]e [SO2
H2O] have the same order of magnitude as [SO2
H2O] and also that ‰SO2
H2 OŠ  1 (12.69) Ks1 ‡ ‰H‡ Še ‡ ‰HSO3 Še For example, this term for pH = 4 and pSO2 ˆ 10 9 atm (1 ppb) is approximately equal to 0.01. Therefore, setting the expression in brackets in (12.67) equal to unity, this expression for the approach to equilibrium simplifies to ‰SO2
H2 OŠ ‰SO2
H2 OŠe ˆe ‰SO2
H2 OŠ0 ‰SO2
H2 OŠe

at

(12.70)

and the characteristic time that we are seeking is just a 1. Thus the characteristic time to achieve equilibrium is τi ˆ

1 kf ‡ kr ‰H Še ‡ kr ‰HSO3 Še ‡

For pH = 4, kf = 3.4 × 106 s 1, and kr = 2 × 108 M

1

(12.71)

s 1, we find that

τi ≅ 2  10

7

s

Repeating the calculation at other pH values, we find that this timescale depends only weakly on pH, remaining below 1 μs. Schwartz and Freiberg (1981) estimated the timescale for the second dissociation of S(IV). Using an approach similar to the one outlined above, they found that this timescale increases with increasing pH and remains below 10 3 s for pH values below 8. Thus, in the case of the sulfur equilibria, it can be assumed that ionization equilibrium is achieved virtually instantaneously upon absorption of SO2.

12.2.4 Characteristic Time of Aqueous-Phase Diffusion in a Droplet Consider unsteady-state diffusion of a dissolved species in a droplet, initially free of solute, when the surface concentration is raised to Cs at t = 0. The problem can then be stated mathematically as @ 2 C 2 @C @C ˆ Daq ‡ @t @r2 r @r C…r; 0† ˆ 0 @C ˆ0 @r rˆ0 C…Rp; t† ˆ Cs

(12.72)

(12.73)

511

MASS TRANSPORT AND AQUEOUS-PHASE CHEMISTRY

Note that the problem statement is very similar to that solved for interfacial transport with a different boundary condition at the droplet surface. A standard separation of variables solution of (12.72) and (12.73) gives Rp C…r; t† ˆ1‡ r Cs

1

… 1†n

nˆ1

2 nπr sin exp nπ Rp

n2 π2 Daq t R2p

(12.74)

The characteristic time for aqueous-phase diffusion τda is obtained from the first term in the exponential in the solution:

τda ˆ

R2p π2 Daq

(12.75)

Note that the timescale is proportional to the square of the droplet radius. A typical value of Daq is 10 5 cm2 s 1, so we can evaluate τda as function of Rp. For a typical cloud droplet of 10 μm radius, τda = 0.01 s. For a rather large cloud droplet of 100 μm radius the timescale is 1 s. For raindrops of size 1 mm the timescale becomes appreciable as it is equal to 100 s.

12.2.5 Characteristic Time for Aqueous-Phase Chemical Reactions To develop an expression for the characteristic time for aqueous-phase chemical reaction, let us consider as an example the oxidation of S(IV). The characteristic time can be calculated by dividing the S(IV) aqueous-phase concentration by its oxidation rate: τra ˆ

‰S…IV†Š d‰S…IV†Š=dt

(12.76)

It is also possible to define a characteristic time for chemical reaction relative to the gas-phase SO2 concentration: τrg ˆ

‰SO2 …g†Š d‰S…IV†Š=dt

(12.77)

If the aqueous-phase concentration is uniform and satisfies Henry’s law equilibrium, then ‰S…IV†Š ˆ H ∗S…IV† RT‰SO2 …g†Š, and the two timescales are related by τrg ˆ

τra H ∗S…IV† RT

(12.78)

12.3 MASS TRANSPORT AND AQUEOUS-PHASE CHEMISTRY The reactants for aqueous-phase atmospheric reactions are transferred to the interior of cloud droplets from the gas phase by a series of mass transport processes. We would like to compare the rates of mass transport in the gas phase, at the gas–water interface, and in the aqueous phase in an effort to quantify the mass transport effects on the rates of aqueous-phase reactions. If there are no mass transport limitations, the gas and aqueous phases will remain at Henry’s law equilibrium at all times. Our objective will be to identify cases where mass transport limits the aqueous-phase reaction rates and then to develop approaches to quantify these effects.

512

MASS TRANSFER ASPECTS OF ATMOSPHERIC CHEMISTRY

First, we note that the characteristic times of the aqueous-phase dissociation reactions tend to be short as compared with all timescales of interest. Thus the aqueous ionic equilibria can be assumed to hold at all points in the droplet, and these characteristic times no longer need be considered. We now compare the rate of each mass transport step to the aqueous-phase reaction rate. If the mass transport rate exceeds the aqueous-phase reaction rate, then mass transport does not limit the aqueousphase kinetics.

12.3.1 Gas-Phase Diffusion and Aqueous-Phase Reactions In the atmosphere under standard conditions the mean free path of most reacting molecules is on the order of 0.1 μm, much less than the radius of cloud droplets (1 μm). Hence the mass transport of gas molecules to (or from) cloud droplets may be treated as a continuum process. The characteristic time τdg for the establishment of a steady-state concentration profile around the droplets in a cloud is, as we saw, less than a millisecond. We can therefore assume that the concentration profile of a reactant gas in the air surrounding a droplet will be given by (12.9) and the flux to the droplet by (12.12). The flux to the droplet per unit volume of the droplet Jv can be calculated by dividing the molar flux Jc (with units mol s 1) by the droplet volume, as Jv ˆ

3Dg R2p

…c1

cs †

(12.79)

where Jv is in M s 1. For a given bulk gas-phase concentration c1, this flux to a droplet reaches a maximum value when cs = 0; that is, the surface concentration becomes zero. In this case the maximum is given by flux J max v 3Dg ˆ 2 c1 (12.80) J max v Rp Equation (12.80) provides an important insight. For a given droplet size, gas-phase diffusion can provide a reactant to the aqueous phase at a rate that cannot exceed J max v . This maximum uptake rate is shown in Figure 12.7 as a function of the drop diameter. For a droplet of diameter equal to 10 μm, assuming a gasphase diffusivity Dg = 0.1 cm2s 1, gas-phase diffusion can provide as much as 50 μM s 1 ppb 1 of reagent. This rate is quite substantial, indicating that gas-phase diffusion is a rather efficient process for the

FIGURE 12.7 Maximum molar uptake rate per ppb of gas-phase reagent allowed by gas-phase diffusion at T = 298 K and for Dg = 0.1 cm2 s 1.

513

MASS TRANSPORT AND AQUEOUS-PHASE CHEMISTRY

average cloud droplet. The larger the droplet, the smaller the maximum molar uptake rate for any reagent. Let us return to our reacting system, where a species A diffuses around a droplet of radius Rp and then reacts with an aqueous-phase reaction rate Raq(M s 1). We assume here that interfacial and aqueous-phase transport are so rapid that we can neglect them. At steady-state, the diffusion rate to the particle will be equal to the reaction rate, or 3Dg R2p

…c1

cs † ˆ Raq

(12.81)

and in terms of partial pressure of A, using the ideal-gas law, we obtain pA …1†

pA …Rp † ˆ

R2p RT 3Dg

Raq

(12.82)

where pA(1) and pA(Rp) are the partial pressures of A far from the droplet and at the droplet surface, respectively. Equation (12.82) permits us to formulate a criterion for the absence of gas-phase mass transport limitation (Schwartz 1986) pA …1† pA …Rp †  εg pA …1†

(12.83)

where εg is an arbitrarily selected small number close to zero. This criterion states that there is no gasphase mass transport limitation when the partial pressure of the reactant is approximately constant around the drop (see also Figure 12.5). Combining (12.82) and (12.83), this criterion sets an upper bound for the aqueous-phase reaction rate that can be maintained by gas-phase diffusion: Raq  εg

3Dg pA …1† R2p RT

(12.84)

As long as the aqueous-phase reaction rate is less than the value specified by (12.84), the reaction will not be limited by gas-phase diffusion. If there is no deviation from equilibrium, Henry’s law will be satisfied for A. For first-order kinetics, we obtain Raq ˆ k1 ‰AŠ ˆ k1 H ∗A pA …1† where k1(s 1) is the first-order rate constant. We obtain the bound as k1 H ∗A  εg

3Dg R2p RT

(12.85)

This criterion for gas-phase diffusion limitation is illustrated in Figure 12.8 for a series of droplet diameters and an arbitrarily chosen εg = 0.1. In Figure 12.8 the inequality (12.85) corresponds to the area below and to the left of the lines. For a given situation if the point …k1 ; H ∗A † is to the left of the corresponding line in Figure 12.8, then gas-phase mass transport limitation does not exceed 10%. Similar plots, introduced by Schwartz (1984), provide a convenient way to ascertain the extent of mass transport limitation for a given condition of interest.

514

MASS TRANSFER ASPECTS OF ATMOSPHERIC CHEMISTRY

FIGURE 12.8 Gas-phase and aqueous-phase mass transport limitation for a species with Dg = 0.1 cm2 s 1, Daq = 10 5 cm2 s 1. The lines represent onset (10%) of mass transport limitation for the indicated values of drop diameter. Diagonal sections represent mass transport limitation; vertical sections represent aqueous-phase limitation (Schwartz 1986).

12.3.2 Aqueous-Phase Diffusion and Reaction In Section 12.2.4 we showed that after time τda ˆ R2p =π2 Daq the concentration profile inside a cloud droplet becomes uniform. The characteristic time for aqueous-phase reaction was found to be equal to τra = [A]/Raq. If the characteristic aqueous-phase diffusion time is much less than the characteristic reaction time, then aqueous-phase diffusion will be able to maintain a uniform concentration profile inside the droplet. In this case, there will be no concentration gradients inside the drop and therefore no aqueous-phase mass transport limitations on the aqueous-phase kinetics. This criterion can be written as τda  ετra where ε is another arbitrarily chosen small number (i.e., ε = 0.1). Assuming first-order kinetics, Raq = k1[A], and the criterion immediately above becomes k1  ε

For example, for Daq = 10

5

π2 Daq

(12.86)

R2p

cm2 s 1, ε = 0.1, and Rp = 5 μm, the criterion corresponds to k1  39:5 s

1

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MASS TRANSPORT AND AQUEOUS-PHASE CHEMISTRY

and does not depend on the value of the Henry’s law constant for A. This criterion corresponds to vertical lines in Figure 12.8 at values of k1 calculated by (12.86). Points to the left of these lines are limited to an extent less than 10% by aqueous-phase mass transport.

12.3.3 Interfacial Mass Transport and Aqueous-Phase Reactions For typical cloud droplets, although gas-phase mass transport is in the continuum regime, mass transport across the air–water interface is, ultimately, a process involving individual molecules. Therefore the kinetic theory of gases sets an upper limit to the flux of a gas to the air–water interface. This rate is given by (12.25) and depends on the value of the accommodation coefficient. Following our approach in Section 12.3.1 the molar flux per droplet volume through the interface, Jkv, will be J J kv ˆ 4 k 3 πR p 3 or using (12.25) J kv ˆ

3cA α …c1 4Rp

cs †

(12.87)

This rate reaches a maximum value for cs = 0. Converting to partial pressure, J max kv , the maximum molar uptake rate through the interface will be J max kv ˆ

3cA αpA …1† 4Rp RT

(12.88)

It is instructive to compare this rate with the maximum gas-phase diffusion rate J max given by (12.80). As v shown in Figure 12.9, for α  0.1 the maximum interfacial mass transport rate exceeds the maximum gasphase diffusion rate. Therefore for α  0.1, gas-phase diffusion is more restrictive for atmospheric cloud droplets, while for α  10 3, interfacial mass transport is the rate-limiting step. The net rate of material transfer across the interface into solution per droplet volume Rl is given, modifying (12.54), as Rl ˆ

3α ∗ Rp …2πMA RT†1=2 HA

…H ∗A p0

‰AŠs †

(12.89)

where p0 is the partial pressure of A at the interface, and [A]s is the aqueous-phase concentration of A at the droplet surface. The difference between H ∗A p0 and [A]s is the step change at the droplet surface shown in Figure 12.5. Equation (12.89) can be used to define a criterion for the absence of interfacial mass transport limitation (Schwartz 1986) by H ∗A p0 ‰AŠs ε H ∗A p0

(12.90)

where ε is once more an arbitrary small number (i.e., ε = 0.1). The physical interpretation of this criterion is that as long as the deviation from Henry’s law at the interface is less than, say, 10%, there is no interfacial mass transport limitation. Considering a system at steady state with only interfacial mass transport and aqueous-phase reactions taking place (the rest of the mass transport steps are assumed to be fast), then Raq = Rl, or H ∗A p0

‰AŠs ˆ

H ∗A p0 …2πMA RT†1=2 Raq 3α

(12.91)

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MASS TRANSFER ASPECTS OF ATMOSPHERIC CHEMISTRY

FIGURE 12.9 Maximum molar uptake rate per ppb of gas-phase reagent as a function of cloud drop diameter, as controlled by gas-phase diffusion, or interfacial transport for various accom­ modation coefficient values at T = 298 K and for Dg = 0.1 cm2 s 1 and MA = 30 g mol 1 (Schwartz 1986).

Combining (11.90) and (11.91) the criterion for absence of interfacial mass transport limitation can be rewritten as Raq  ε

3 αp0 Rp …2πMA RT†1=2

(12.92)

For first-order kinetics and no interfacial transport limitation, Raq ˆ k1 ‰AŠ ≅ k1 H ∗A p0 and therefore, using (12.92), we obtain k1 H ∗A  ε

3α Rp …2πMA RT†1=2

(12.93)

This criterion is illustrated in Figure 12.10 for ε = 0.1. A range of bounds is indicated for droplet diameters varying from 3 to 100 μm. Comparing Figures 12.10 and 12.8, one concludes that interfacial mass transport may be more or less controlling than gas-phase diffusion, depending on the value of the accommodation coefficient α. In the criteria (12.85), (12.86), and (12.93), ε is the fractional reduction in aqueous-phase reaction rate due to mass transport limitations. Satisfaction of these inequalities for, say, ε = 0.1, indicates

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MASS TRANSPORT AND AQUEOUS-PHASE CHEMISTRY

FIGURE 12.10 Interfacial mass transport limitation. The sev­ eral bands represent the onset (10%) of mass transport limitation for the indicated values of the accommodation coefficient α. The width of each band corresponds to the drop diameter range 3–100 μm for MA = 30 g mol 1 (Schwartz 1986).

that the decrease of the aqueous-phase reaction rate by the corresponding mass transfer process is less than 10%.

12.3.4 Application to the S(IV)–Ozone Reaction We would like to use the tools developed above to determine whether aqueous-phase reaction between ozone and S(IV) is limited by mass transport in a typical cloud. The subsequent analysis was first presented by Schwartz (1988). Typical ambient conditions that need to be examined include droplet diameters in the 10–30 μm range, cloud pH values in the 2–6 range, and cloud temperatures varying from 0°C to 25°C. The reaction rate for the S(IV)–O3 reaction is given by R ˆ kO3 ‰O3 Š‰S…IV†Š

(12.94)

where kO3 is the temperature- and pH-dependent effective rate constant for the reaction (see Chapter 7). This rate constant is plotted as a function of the pH in Figure 7.15. We need now to evaluate the pseudo-first-order coefficient for ozone and S(IV), which is included in the criteria of (12.85), (12.86), and (12.93). The effective first-order rate coefficient of ozone, k1;O3 is k1;O3 ˆ kO3 ‰S…IV†Š ˆ kO3 H ∗S…IV† pSO2

(12.95)

k1;S…IV† ˆ kO3 ‰O3 Š ˆ kO3 H O3 pO3

(12.96)

and similarly that for S(IV) is

518

MASS TRANSFER ASPECTS OF ATMOSPHERIC CHEMISTRY

FIGURE 12.11 Examination of mass transport limitation in the S(IV)–O3 reaction. Mass transport limitation is absent for points below and to the left of the indicated bounds (Schwartz 1988).

assuming Henry’s law equilibrium. The partial pressures used in (12.95) and (12.96) are those after establishing phase equilibrium in the cloud. Using the graphical analysis technique of Schwartz (1984, 1986, 1987), we can now examine the mass transport limitation in the S(IV)–O3 reaction for the above range of ambient conditions assuming ξO3 ˆ 30 ppb and ξSO2 ˆ 1 ppb (Figure 12.11). Each of the criteria (12.85), (12.86), and (12.93) appears in Figure 12.11 as a straight line in the log k1 log H∗ space. The gas-phase and interfacial mass transport lines have slope 1, while the aqueous-phase transport line is vertical. Mass transport limitation is less than 10% for points (k1, H∗) to the left of the inequality lines. Lines are plotted for two temperatures and two droplet diameters. The lines given for interfacial mass transport limitations reflect the accommodation coefficient measurements of αO3 ˆ 5  10 4 (Tang and Lee 1987) and αSO2 ˆ 0:08. We turn our attention first to ozone (Figure 12.11). Since ozone solubility does not depend on pH, all points k1;O3; H O3 at a given temperature for different pH values fall on a horizontal line shown on the lower part of the graph. The increase with pH of both the S(IV) solubility and the effective reaction constant kO3 leads to an increase of k1;O3 with increasing pH. The points k1;O3; H O3 lie well below the bounds of gas-phase mass transport limitation, and hence such limitation is unimportant. Similarly, no interfacial mass transport limitation is indicated, despite the low ozone accommodation coefficient. Interfacial mass transport limitation may occur only under rather unusual circumstances, corresponding to pH values higher than 7 and/or high SO2 concentrations. In contrast, significant aqueous-phase mass transport limitation is predicted for 10 μm drops at pH  5.3 and for 30 μm at pH  4.8 (Schwartz 1988). Thus one can assume cloud droplet saturation with O3 at Henry’s law equilibrium with gaseous O3 at pH values lower than 4.8, but aqueous-phase mass transport limitations should be considered for calcula­ tions for higher pH values. For higher SO2 concentrations the points k1;O3; H O3 are shifted to the right. For example, for ξSO2 ˆ 10 ppb all points for ozone shift to the right by one log unit on the abscissa scale. For a 10 ppb SO2 mixing ratio the onset of aqueous-phase mass transport limitation occurs at pH 4.8 for 10-μm droplets and at 4.4 for 30-μm droplets.

519

MASS TRANSPORT AND AQUEOUS-PHASE CHEMISTRY

The values of …k1;S…IV† ; H ∗S…IV† † are also shown in Figure 12.11. As the pH increases, both the SO2 solubility H ∗S…IV† and the effective reaction constant kO3 increase and the corresponding points start approaching the lines reflecting mass transport limitations. However, the location of the points indicates that there is essentially no such limitation over the entire range of conditions examined; the only exception is gas-phase limitation for 30 μm drops at 0°C and pH  5.9. Once more, for higher gas-phase ozone concentrations, the effective reaction constant k1;O3 increases and the points shift to the right. For an increase of O3 to 300 ppb and a shifting of the corresponding S(IV) points by a full log unit, there would be no appreciable mass transport limitation for 30 μm drops for pH  5.4.

12.3.5 Application to the S(IV)–Hydrogen Peroxide Reaction An analogous examination of mass transport limitations can be carried out for the aqueous-phase reaction of S(IV) with H2O2 with a reaction rate given by R ˆ kH2 O2 ‰H2 O2 Š‰S…IV†Š

(12.97)

where kH2 O2 is the pH- and temperature-dependent effective rate constant for the reaction (see Chapter 7). The effective first-order rate coefficients are then defined as k1;H2 O2 ˆ kH2 O2 ‰S…IV†Š ˆ kH2 O2 H ∗S…IV† pSO2 k1;S…IV† ˆ kH2 O2 ‰H2 O2 Š ˆ kH2 O2 H H2 O2 pH2 O2

(12.98)

The points …k1;S…IV† ; H ∗S…IV† † and …k1;H2 O2 ; H H2 O2 † are plotted in Figure 12.12 following the analysis of Schwartz (1988). The bound for H2O2 interfacial limitation was drawn using a value of αH2 O2 ˆ 0:2 (Gardner et al. 1987).

FIGURE 12.12 Examination of mass transport limitation in the S(IV)–H2O2 reaction. Mass transport limitation is absent for points below and to the left of the indicated bounds (Schwartz 1988).

520

MASS TRANSFER ASPECTS OF ATMOSPHERIC CHEMISTRY

Let us first examine the points for H2O2. Both the H2O2 solubility and the effective first-order rate constant, k1;H2 O2 , depend only weakly on pH, resulting in tightly clustered points. The reason for the pH independence of k1;H2 O2 is, as we saw in Chapter 7, the nearly opposite pH dependences of kH2 O2 and H ∗S…IV† that cancel each other in (12.97). Comparison of the location of these points plotted for ξSO2 ˆ 1 ppb shows that gas-phase, aqueous-phase, and interfacial mass transport limitations are absent for H2O2 even for 30-μm drops for all cases. For higher partial pressures of SO2, the points k1;H2 O2 ; H H2 O2 should be displaced appropriately to the right. The point denoted by the solid circle in Figure 12.12 corresponds to ξSO2 ˆ 10 ppb at 0°C and is therefore one log unit to the right of the corresponding ξSO2 ˆ 1 ppb point. In this case there is appreciable gas-phase mass transport limitation for 30-μm droplets and no limitation for 10-μm droplets. For S(IV) both k1,S(IV) and H ∗S…IV† are pH-dependent (Figure 12.12). At 25°C there is no gas-phase or interfacial mass transport limitation for ξH2 O2 ˆ 1 ppb.1 Aqueous-phase limitations occur for pH < 2.8 according to Figure 12.12. For temperature 0°C the points …k1;S…IV† ; H ∗S…IV† † are displaced upward and to the right of the 25°C, indicating an increase with decreasing temperature not only of solubility but also of the first-order rate constant. In this lowtemperature case, there are significant mass transport limitations for all pH values for 30 μm drops. For 10 μm drops, even at 0°C, there are no mass transport limitations.

12.3.6 Calculation of Aqueous-Phase Reaction Rates The discussion above indicates that appreciable mass transport limitation is absent under most atmospheric conditions of interest, although instances of substantial limitations under certain conditions of pH, temperature, and reagent concentrations exist. In the latter cases we would like to estimate the aqueous-phase reaction rate taking into account reductions in these rates due to the mass transport limitations. The graphical method presented in the previous sections is a useful approach to estimate if there are mass transport limitations present. Our goal in this section is to derive appropriate mathemati­ cal expressions to quantify these rates. 12.3.6.1 No Mass Transport Limitations In this case there are no appreciable concentration gradients outside or inside the droplet, or across the interface. As a result, Henry’s law is applicable and for the first-order reaction A…aq† ! B…aq† the rate Raq is given by Raq ˆ k‰A…aq†Š ˆ k H ∗A pA

(12.99)

where k is the corresponding rate constant, H ∗A the Henry’s law coefficient of A, and pA its gas-phase partial pressure. 12.3.6.2 Aqueous-Phase Mass Transport Limitation The concentration of a species A, specifically, C(r,t), undergoing aqueous-phase diffusion and irreversible reaction inside a cloud droplet, is governed by @C @ 2 C 2 @C ‡ ˆ Daq ‡ Raq …r; t† @t @r2 r @r

(12.100)

Schwartz (1988) noted that because the above calculation was made for an open system, with a fixed ξH2 O2 ˆ 1 ppb, because of the solubility of H2O2 it actually corresponds to a total amount of approximately 9.3 ppb.

1

521

MASS TRANSPORT AND AQUEOUS-PHASE CHEMISTRY

where Daq is its aqueous-phase diffusivity and Raq(r, t) the aqueous-phase reaction rate. This equation was solved in Section 12.2.4 in the absence of the reaction term. Let us focus here on the case of a firstorder reaction (12.101) Raq …r; t† ˆ kC…r; t† and assume that the system is at steady state. Equation (12.100) then becomes k d2 C 2 dC ‡ ˆ C dr2 r dr Daq

(12.102)

with boundary conditions C…Rp † ˆ Cs dC dr

ˆ0

(12.103)

rˆ0

where Cs is the concentration at the droplet surface. The solution of (12.102) under these conditions is C…r† ˆ Cs

Rp sinh…qr=Rp † r sinh q

(12.104)

where we have introduced the dimensionless parameter q ˆ Rp

k Daq

(12.105)

Steady-state concentration profiles calculated from (12.104) are shown in Figure 12.13. Note that as q → 0, the profiles become flat, while for larger q values significant concentration gradients develop inside the drop.

FIGURE 12.13 Steady-state radial concentration profiles of concentra­ tion of reactant species C(r) relative to the concentration at the surface of the drop Cs for a drop radius Rp as a function of the dimensionless parameter q (Schwartz and Freiberg 1981).

522

MASS TRANSFER ASPECTS OF ATMOSPHERIC CHEMISTRY

Because of the linearity of the system (first-order reaction), the average reaction rate over the drop hRi, will be proportional to the average concentration of A, namely, hCi hRi ˆ khCi

(12.106)

where hCi ˆ

3 Rp 2 4πr C…r†dr 4πR3p ∫ 0

(12.107)

coth q q

(12.108)

Integration of C(r) given in (12.104) yields hCi ˆ 3Cs

1 q2

The average concentration hCi for q  1 is significantly less than the surface concentration Cs, taking the value 0.34 Cs for q = 1 and 0.1 Cs for q  20. The overall rate of aqueous-phase reaction of the droplet Raq can now be calculated as function of the droplet concentration by Raq ˆ QkCs

(12.109)

where

Qˆ3

coth q q

1 ; q2

q ˆ Rp

k Daq

(12.110)

Note that when there are no other mass transfer limitations the droplet surface concentration will be in equilibrium with the bulk gas phase and Cs ˆ H ∗A pA

(12.111)

so that Raq ˆ QkH ∗A pA

(12.112)

In conclusion, when aqueous-phase mass transport limitations are present, one needs to calculate the correction factor Q and then multiply the overall reaction rate by it. This correction factor is shown in the inset of Figure 12.12 as a function of the distance of the point of interest from the line indicating the onset of mass transport limitation, Δlog k1. For example, the O3 point corresponding to pH 6, 25°C falls 1.2 log units to the right of the bound for 10 μm drops. Using Δlog k1 = 1.2 in the inset, we find that Q = 0.4, and the rate will be 40% of the rate corresponding to a uniform ozone concentration in the drop. For further discussion of the correction factor Q and applications to other species, the reader is referred to Jacob (1986). Our analysis is based on a steady-state solution of the problem. The solution to the time-dependent problem has been presented by Schwartz and Freiberg (1981). These authors concluded that the rate of uptake exceeds the steady-state rate by less than 7% after t = 1/k and 1% after

523

MASS TRANSPORT AND AQUEOUS-PHASE CHEMISTRY

t = 2/k. This approach to steady state is quite rapid and therefore the conclusions reached above are robust. 12.3.6.3 Gas-Phase Limitation The problem of coupled gas-phase mass transport and aqueous-phase chemistry was solved in Section 12.3.1 resulting in (12.82). Solving for the aqueous-phase reaction term Raq and noting that in this case Henry’s law will be satisfied at the interface …pA …Rp † ˆ Caq =H ∗A †:

Raq ˆ

3 Dg R2p RT

pA …1†

Caq H∗A

(12.113)

This equation indicates that when the aqueous-phase reaction rate is limited by gas-phase mass transport, at steady-state the aqueous-phase reaction rate is only as fast as the mass transport rate. 12.3.6.4 Interfacial Limitation The solution of the steady-state problem for interfacial limitation only is given by (12.91). The aqueousphase reaction rate is then

Raq ˆ

3α H ∗A Rp …2πMA RT†1=2

H ∗A pA …1†

Caq

(12.114)

12.3.6.5 Gas-Phase Plus Interfacial Limitation The expressions developed above can be combined to develop a single expression that includes both gasphase and interfacial mass transport effects. Schwartz (1986) showed that in this case

Raq ˆ kmt pA …1†

Caq H ∗A

(12.115)

where

kmt

R2p

Rp ˆ ‡ 3 Dg 3α

2πMA RT

1

(12.116)

The mass transfer coefficient kmt for gas-phase plus interfacial mass transport has units s 1. The rate Raq in (12.115) is in equivalent gas-phase concentration units, but the conversion to aqueousphase units is straightforward, multiplying by H ∗A . The mass transfer coefficient kmt as a function of the accommodation coefficient α and the droplet radius is shown in Figure 12.14. For values of α > 0.1, the mass transfer rate is not sensitive to the exact value of α. However, for α < 0.01, surface accommoda­ tion starts limiting the mass transfer rate to the drop, and kmt decreases with decreasing α for all droplet sizes.

524

MASS TRANSFER ASPECTS OF ATMOSPHERIC CHEMISTRY

FIGURE 12.14 Mass transfer constant kmt accounting for gas-phase diffu­ sion and interfacial transport as a function of the droplet diameter and the accommodation coefficient (Schwartz 1986).

Uptake Coefficient γ We have seen that a fundamental parameter that determines the transfer rate of trace gases into droplets is the mass accommodation coefficient α, which is defined as αˆ

number of molecules entering the liquid phase number of molecular collisions with the surface

In most circumstances, other processes, such as gas-phase diffusion to the droplet surface and liquidphase solubility, also limit gas uptake. In a laboratory experiment α can generally not be measured directly. Rather, what is accessible is the measured overall flux ~J A of gas A to the droplet (mol A per area per time). That flux can be expressed in terms of a measured uptake coefficient, γ  1, that multiplies the kinetic collision rate per unit of droplet surface area as ~J A ˆ 1 cA …1† cA γ 4

(12.117)

where cA(1) is the concentration of A in the bulk gas. Since ~J A is measured and cA(1) and cA are known, γ can be determined from the experimental uptake data. The uptake coefficient γ inherently contains all the processes that affect the rate of gas uptake, including the mass accommodation coefficient at the surface. The experimental task is to separate these effects and determine α from γ. To analyze laboratory uptake data it is necessary to determine how γ depends on the other parameters of the system. Let us assume that potentially any or all of gas-phase diffusion, interfacial transport, and aqueous-phase diffusion may be influential. We assume steady-state conditions and that species A is consumed by a first-order aqueous phase reaction (12.101). At steady state the rate of

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MASS TRANSPORT AND AQUEOUS-PHASE CHEMISTRY

transfer of species A across the gas–liquid interface, given by (12.115), must be equal to that as a result of simultaneous aqueous-phase diffusion and reaction, (12.112): QkCaq ˆ kmt pA …1†

Caq H ∗A

(12.118)

This equation can be solved to obtain an expression for Caq, the aqueous-phase concentration of A just at the droplet surface. The correction factor Q depends on the dimensionless parameter q as given in (12.110). In the regime of interest where q  1, Q  3/q, and this is the case we will consider. Thus (12.118) yields kmt pA …1† p p (12.119) Caq ˆ kmt 3 k Daq ∗ ‡ Rp HA Then the rate of transfer of A across the gas–liquid interface, in mol s 1, …3 kCaq =q†…43 πR3p †, is equated to the defining equation for γ: p p 3 k Daq 4 1 p …1† Caq πR3p ˆ A (12.120) cA γ…4πR2p † Rp 3 4 RT Using (12.116) and (12.119) in (12.120), we can obtain the basic equation that relates γ to the other parameters of the system: cA 1 Rp c A 1 p ˆ ‡ ‡ ∗ γ 4 Dg α 4 RTH A kDaq

(12.121)

The terms on the RHS of (12.121) show the three contributions to overall resistance to uptake: gasphase diffusion, mass accommodation at the surface, and interfacial transport/aqueous-phase diffusion, respectively. Thus we see that the uptake coefficient γ is equal to the mass accommodation coefficient α only if the other resistances are negligible. In experimental evaluation, γ is measured and all other quantities in (12.121) except for α are known or can be measured separately; this provides a means to determine α. Table 12.2 presents values of measured mass accommodation and uptake coefficients for a variety of atmospheric gases. For in-depth treatment of uptake of gas molecules by liquids, we refer the reader to Davidovits et al. (1995) and Nathanson et al. (1996). A general kinetic model framework for the description of mass transport and chemical reactions at the gas–particle interface has been developed by Pöschl et al. (2007).

12.3.7 An Aqueous-Phase Chemistry/Mass Transport Model The following equations can be used to describe the evolution of the aqueous-phase concentration of a species A, Caq, and each partial pressure, incorporating mass transfer effects dp 1 ˆ kmt wL p ‡ ∗ kmt Caq wL dt HA dCaq kmt ˆ p dt RT

kmt Caq H ∗A RT

QRaq

(12.122)

(12.123)

where wL is the cloud liquid water volume fraction, p (atm) is the bulk partial pressure of A in the cloud, Caq is the corresponding aqueous-phase concentration at the surface of the drop, H ∗A is the effective Henry’s law coefficient, and kmt and Q are the coefficients given by (12.116) and (12.110), respectively.

526

MASS TRANSFER ASPECTS OF ATMOSPHERIC CHEMISTRY

TABLE 12.2 Measured Mass Accommodation (α) and Uptake (γ) Coefficients on Aqueous Surfaces Species

α

HNO3 HCl N2O5 SO2 H2O2 CH3COOH HCHO CH3CHO CH3C(O)CH3 MSA O3 HO2 NO3 NH4NO3 H2SO4b

γ

T, K

Reference

0.07 0.19

268 293 283 283 273–290 273 255 295 267 267 260 285 260 285

Van Doren et al. (1990, 1991)

0.15 0.04 0.11 0.23 0.17 0.03 0.02 >0.03 0.066 0.013 0.17 0.1 0.0005 >0.2a 0.0002 0.8 0.5 0.7 0.2

293 300 298 298

Watson et al. (1990) Van Doren et al. (1991) Worsnop et al. (1989) Worsnop et al. (1989) Jayne et al. (1991) Jayne et al. (1992) Duan et al. (1993) DeBruyn et al. (1994) Tang and Lee (1987) Mozurkewich et al. (1987) Rudich et al. (1996) Dassios and Pandis (1999) Jefferson et al. (1997)

a

On 20M (NH4)2SO4. On NaCl aerosol with high stearic acid coverage.

b

Note that the concentrations described by these equations are the surface aqueous-phase concentrations, and in cases where there are aqueous-phase mass transport limitations Raq should be expressed as a function of the surface concentrations. The equations above have been the basis of most atmospheric aqueous-phase chemistry models that include mass transport limitations [e.g., Pandis and Seinfeld (1989)]. These equations simply state that the partial pressure of a species in the cloud interstitial air changes due to mass transport to and from the cloud droplets (incorporating both gas and interfacial mass transport limitations). The aqueous-phase concentrations are changing also due to aqueous-phase reactions that may be limited by aqueous-phase diffusion included in the factor Q.

12.4 MASS TRANSFER TO FALLING DROPS For a stationary drop of radius Rp for continuum regime transport, we have seen that steady-state transport is rapidly achieved and the flux to the drop is given by (12.12). The flux per unit droplet surface, ~J A ˆ J c =…4πR2 †, is then p ~J A ˆ Dg …c1 Rp

cs †

(12.124)

When the drop is in motion, calculation of the flux of gas molecules to the droplet surface is considerably more involved. The flux is usually defined in terms of a mass transfer coefficient kc as ~J A ˆ kc …c1

cs †

(12.125)

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CHARACTERISTIC TIME FOR ATMOSPHERIC AEROSOL EQUILIBRIUM

For a stationary drop, comparing (12.124) and (12.125), kc = Dg/Rp. In an effort to estimate kc for a moving drop one defines the dimensionless Sherwood number in terms of kc as Sh ˆ

kc Dp Dg

(12.126)

For diffusion to a stationary drop Sh = 2. When the drop is falling, one usually resorts to empirical correlations for Sh as a function of the other dimensionless groups of the problem, for example (Bird et al. 2002) Sh ˆ 2 ‡ 0:6 Re1=2 Sc1=3

(12.127)

where the Reynolds number, Re = vtDp/νair, and the Schmidt number Sc = νair/Dg, where vt is the terminal velocity of the droplet and νair is the kinematic viscosity of air. We have seen that for a 1 mm drop the characteristic time for aqueous-phase diffusion τda is on the order of 100 s. Thus, if the only mechanism for mixing is molecular diffusion, for larger drops, aqueousphase diffusion may create significant concentration gradients inside the drop. However, for falling drops of radii Rp > 0.1 mm, internal circulations develop (Pruppacher and Klett 1997), the timescale for which is on the order of Rpμl/vtμair, with μl the viscosity of the drop. For a 1 mm radius drop, this timescale is on the order of 10 2 s, very short compared with that taken by the drop falling to the ground. Therefore it can be assumed that for drops larger than about 0.1 mm internal circulations will produce a well-mixed interior.

12.5 CHARACTERISTIC TIME FOR ATMOSPHERIC AEROSOL EQUILIBRIUM Aerosol particles in the atmosphere contain a variety of volatile compounds (ammonium, nitrate, chloride, volatile organic compounds) that can exist either in the particulate or in the gas phase. We estimate in this section the timescales for achieving thermodynamic equilibrium between these two phases and apply them to typical atmospheric conditions. The problem is rather different compared to the equilibration between the gas and aqueous phases in a cloud discussed in the previous section. Aerosol particles are solid or concentrated solutions (cloud droplets are dilute aqueous solutions); they are relatively small, and aqueous-phase reactions in the aerosol phase can be neglected to a first approximation because of the small liquid water content. If the air surrounding an aerosol population is supersaturated with a compound A, the compound will begin to condense on the surface of the particles in an effort to establish thermodynamic equilibrium. The gas-phase concentration of A will decrease, and its particulate-phase concentration will increase until equilibrium is achieved. The characteristic time for the two phases to equilibrate due to the depletion of A in the gas phase will be inversely proportional to the total flux of A to the aerosol phase. For solid-phase aerosol particles, the equilibrium concentration of A is constant and does not change as A is transferred to the aerosol phase. However, if the aerosol contains water and A is water-soluble, the equilibrium concentration of A will increase as condensation proceeds, and equilibration between the two phases will be accelerated. This change in equilibrium concentration is a result of changes in the chemical composition of the particle as A is transferred to the particulate phase. The cases of solid and aqueous phases will be examined separately in subsequent sections, based on the analysis of Wexler and Seinfeld (1990). Recently it has been shown that organic aerosol particles may exist in a highly viscous state, such that diffusion of condensing organics within the particle itself plays an important role in the time needed for the particle to equilibrate with the ambient concentration. We will address this situation in Chapter 14.

528

MASS TRANSFER ASPECTS OF ATMOSPHERIC CHEMISTRY

12.5.1 Solid Aerosol Particles If a species A is transferred to the solid aerosol phase, as A…g† ƒ A…s† the system will achieve thermodynamic equilibrium when the gas-phase concentration of A becomes equal to the equilibrium concentration ceq. Recall that ceq will be equal to the saturation concentration of A at this temperature, which is also equal to the concentration of A at the particle surface. The equilibrium concentration ceq depends only on temperature and not on the particle composition. If the concentration of A far from the aerosol surface is c1, then the flux of A to a single particle is given by J 1 ˆ 4πRp DA f …Kn; α†…c1

ceq †

(12.128)

where Rp is the particle radius, DA is the gas-phase diffusivity of A, and f(Kn, α) is the correction to the mass transfer flux owing to noncontinuum effects and imperfect accommodation; for example, if the Fuchs–Sutugin (1971) approach is used, then f(Kn, α) is given by (12.43). Note that, if c1 = ceq, the flux between the two phases will be zero, and the aerosol will be in equilibrium with the surrounding gas phase. If the aerosol population is monodisperse and consists of N particles per cm3, then the total flux J from the gas phase to the aerosol phase will be J ˆ NJ 1 ˆ 4πNRp DA f …Kn; α†…c1

ceq †

(12.129)

This flux will be equal to the rate of change of the concentration c1: dc1 ˆ 4πNRp DA f …Kn; α†…c1 ceq † (12.130) dt The characteristic time for gas-phase concentration change and equilibrium establishment τs can be estimated by nondimensionalizing (12.130). The characteristic concentration is ceq and the characteristic timescale is τs, so we define ϕˆ

c1 ; ceq

τˆ

t τs

and (12.130) becomes dϕ ˆ u…ϕ dτ

1†

(12.131)

with u ˆ 4πNRp DA f …Kn; α†τs

(12.132)

Once more, all the physical information about the system has been included in the dimensionless parameter u. Setting u  1, we find that τs 

1 4πNRp DA f …Kn; α†

(12.133)

We can express this timescale in terms of the aerosol mass concentration mp given by 4 mp ˆ πR3p ρp N 3

(12.134)

where ρp is the aerosol density, to get

τs ˆ

ρp R2p 3 DA mp f …Kn; α†

(12.135)

529

CHARACTERISTIC TIME FOR ATMOSPHERIC AEROSOL EQUILIBRIUM

Equation (12.135) suggests that the equilibration timescale will increase for larger aerosol particles and cleaner atmospheric conditions (lower mp). The timescale does not depend on the thermodynamic properties of A, as it is connected solely to the gas-phase diffusion of A molecules to a particle. The timescale in (12.135) varies from seconds to several hours as the particle radius increases from a few nanometers to several micrometers. We can extend the analysis from a monodisperse aerosol population to a population with a size distribution n(Rp). The rate of change for the bulk gas-phase concentration of A in that case is dc1 ˆ 4πDA …c1 dt

ceq †

1

∫0

n…Rp †Rp f …Kn; α†dRp

(12.136)

and the equilibration timescale will be given by τs ˆ

1

4πDA

∫0

1

(12.137)

n…Rp †Rp f …Kn; α†dRp

Wexler and Seinfeld (1992) estimated this timescale for measured aerosol size distributions in southern California and found values lower than 10 min for most cases. The largest values of τs was 15 min. We can approximate (12.137) by assuming that 1

∫0

n…Rp †Rp f …Kn; α†dRp ≅ NRp f

(12.138)

where Rp is the number mean radius of the distribution and f is the value of f(Kn, α) corresponding to this radius. Using this rough approximation, the timescale τs for a polydisperse aerosol population can be approximated by τs 

1 4πNRp DA f

(12.139)

For a polluted environment, N  105 cm 3, Rp  0:01 μm, and assuming α = 0.1, τs  800 s in agreement with the detailed calculations of Wexler and Seinfeld (1992). Therefore, for typical polluted conditions, this timescale is expected to be a few minutes or less. For aerosol populations characterized by low number concentrations, however, this timescale can be on the order of several hours (Wexler and Seinfeld 1990).

12.5.2 Aqueous Aerosol Particles Equation (12.130) is also applicable to aqueous aerosol particles, but with one significant difference. During the condensation of A and the reduction of c1, the concentration ceq at the particle surface changes. The timescale τs calculated in the previous section remains applicable, but there is an additional timescale τa characterizing the change of ceq. The molality of A in the aqueous phase mA is given (see Chapter 10) by mA = nA/(0.018 nw), where nA and nw are the molar aerosol concentrations per m3 of air of A and water, respectively. In general, the water concentration nw changes during transport of soluble species between the two phases. For the purposes of this calculation, let us assume that this change is small (i.e., only a small amount of A is transferred) and assume that nw is approximately constant. The rate of change of the molality of A is given in this case by 1 dnA dmA ≅ dt 0:018 nw dt

(12.140)

530

MASS TRANSFER ASPECTS OF ATMOSPHERIC CHEMISTRY

The rate of change of moles of A in the aerosol phase dnA/dt will be equal to the flux J given by (12.129), so dmA 4πNRp DA f …Kn; α†  …c1 dt 0:018 nw

ceq †

(12.141)

In this equation, c1 and ceq are the gas-phase concentrations of A far from the particle and at the particle surface expressed in mol m 3. The gas-phase equilibrium concentration ceq is related to the liquid-phase molality by an equilibrium constant KA (kg m 3), which is a function of the composition of the particle and the ambient temperature, such that ceq ˆ KA γA mA

(12.142)

where γA is the activity coefficient of A. For activity coefficients of order unity, ceq ˆ KA mA

(12.143)

dceq dmA  KA dt dt

(12.144)

and after differentiation

Combining (12.144) with (12.141), we find that dceq 4πNRp DA f …Kn; α†KA ˆ …c1 dt 0:018 nw

ceq †

(12.145)

Assuming that the system is open and c1 remains constant during the condensation of A, then we can calculate the timescale τa from (12.145). This timescale corresponds to the establishment of equilibrium between the gas and liquid aerosol phases as a result of changes in the aqueous-phase concentration of A. Following the nondimensionalization procedure of the previous section, we find τa ˆ

0:018 nw 4πNRp DA f …Kn; α†KA

(12.146)

Noting that mw = 0.018 nw is the mass concentration of aerosol water in kg m 3, (12.146) can be rewritten using (12.133) as τa ˆ

mw τs KA

(12.147)

The timescale τa increases with increasing aerosol water content. The higher the aerosol water concentration, the slower the change of the aqueous-phase concentration of A resulting from the condensation of a given mass of A, and the slower the adjustment of the equilibrium concentration ceq to the background concentration c1. The timescale τa will be larger for high RH conditions that result in high aerosol liquid water content. This timescale also depends on the thermodynamic properties of A, namely, the equilibrium constant KA. Equation (12.142) suggests that the more soluble a species is, the lower its KA and the larger the timescale calculated from (12.147). Let us consider, for example, dissolution of NH3 and HNO3 NH‡4 ‡ NO3 ƒ NH3 …g† ‡ HNO3 …g†

CHARACTERISTIC TIME FOR ATMOSPHERIC AEROSOL EQUILIBRIUM

with an equilibrium constant K of 4 × 10 15 kg2 m 6 at 298p K (Stelson and Seinfeld 1982). The equilibrium constant KA used in (12.147) can be calculated as KA ˆ K (Wexler and Seinfeld 1990), and therefore KA = 63 μg m 3 (and is a strong function of temperature). The liquid water content of urban aerosol masses at high RH is of the same magnitude as this KA and therefore for NH4NO3, τa  τp. For lower temperatures KA decreases and the timescale τa increases. Summarizing, diffusive transport between the gas and aerosol phases eventually leads to their equilibration, with two timescales governing the approach to equilibrium. One timescale, τs, character­ izes the approach due to changes in the gas-phase concentration field, and the other timescale, τa, due to changes in the aqueous-phase concentrations (if the aerosol is not solid). These equilibration timescales are not necessarily the same, because during condensation the partial pressures at the particle surface (equilibrium concentrations) may change more slowly or rapidly than the ambient concentrations. For solid particles, the surface concentrations do not change, τa is infinite, and the approach to equilibrium is governed by τs. For aqueous aerosols both timescales are applicable as the system approaches equilibrium through both pathways, with the shorter timescale governing the equilibration process. The preceding analysis suggests that for polluted airmasses, high temperatures, and small aerosol sizes, the equilibrium timescale is on the order of a few seconds. On the other hand, under conditions of low aerosol mass concentrations, low temperatures, and large particle sizes, the timescale can be on the order of several hours and the aerosol phase may not be in equilibrium with the gas phase. For these conditions, the submicrometer particles are in equilibrium with the gas phase while the coarse particles may not be in equilibrium (Capaldo et al. 2000). The existence of equilibrium between the atmospheric gas and aerosol phases is generally supported by available field measurements (Doyle et al. 1979; Stelson et al. 1979; Hildemann et al. 1984; Pilinis and Seinfeld 1988; Russell et al. 1988). For example, the comparison of field measurements with theory by Takahama et al. (2004) is shown in Figure 12.15. Our analysis so far has assumed that all particles in the aerosol population have similar chemical composition. Meng and Seinfeld (1996) numerically investigated cases in which the aerosol population consists of two groups of particles with different compositions. They concluded that while the timescale of equilibration of the fine aerosol particles is indeed on the order of minutes or less, the coarse particles may require several hours to achieve thermodynamic equilibrium with the surrounding atmosphere.

FIGURE 12.15 Time series of observed and predicted PM2.5 nitrate concentrations for selected periods in July 2001 and January 2002 in Pittsburgh, PA. Error bars extend to the fifth and 95th percentiles of the distribution function associated with each prediction. The shaded areas bound the interval between the fifth and 95th percentiles of the observations. (Reprinted from Takahama et al. (2004). Reproduced by permission of American Geophysical Union.)

531

532

MASS TRANSFER ASPECTS OF ATMOSPHERIC CHEMISTRY

12 APPENDIX 12 SOLUTION OF THE TRANSIENT GAS-PHASE DIFFUSION PROBLEM: EQUATIONS (12.4)–(12.7) To solve (12.4)–(12.7), let us define a new dependent variable u(r, t) by u…r; t† ˆ

r c1 c…r; t† c1 Rp

Therefore c…r; t† ˆ1 c1

u…r; t†

Rp r

and the problem is transformed to @u @2u ˆ Dg 2 @t @r u…r; 0† ˆ 0; r > Rp u…1; t† ˆ 0 u…Rp ; t† ˆ 1 We can solve this problem by Laplace or similarity transforms. Let us use the latter. Let u…η† ˆ u…r; t† and r Rp ηˆ p 2 Dg t Then the equation reduces to d2 u du ‡ 2η ˆ0 2 dη dη the solution of which is η

u…η† ˆ A ‡ B e ∫0

ξ2

dξ

to be evaluated subject to u…0† ˆ 1 u…1† ˆ 1 Therefore, determining A and B, we have u…η† ˆ 1

2 η p e π ∫0

ξ2

dξ

or equivalently u…r; t† ˆ 1

2 …r p π ∫0

p Rp †=2 Dg t

e

ξ2

dξ

533

PROBLEMS

By definition c…r; t† ˆ c1 1

u…r; t†

Rp r

resulting in (12.8). The molar flow of species A into the droplet at any time t is 4πR2p Dg

@c @r

ˆ rˆRp

4πRp2 c1 Dg

Rp r2

2 Rp …r p r2 π∫ 0 …r

2 Rp 1 exp ‡ p p r π 2 Dg t ˆ 4πR2p c1 Dg

p Rp †=2 Dg t

e

Rp †2 4 Dg t

ξ2

dξ

rˆRp

1 1 ‡p Rp πDg t

Then the total quantity of A that has been transferred into the particle from t = 0 to t is M…t† ˆ

t

∫0

1 1 ‡p dt´ Rp πDg t´ p 2Rp t t‡ p πDg

4πRp2 c1 Dg

ˆ 4πRp Dg c1

PROBLEMS 12.1A Consider the growth of a particle of dibutyl phthalate (DBP) in air at 298 K containing DBP at a background mole fraction of 0.10. The vapor pressure of DBP is sufficiently low that it may be assumed to be negligible. a. Compute the steady-state mole fraction and temperature profiles around a DBP particle of 1 μm diameter. Assume continuum regime conditions to hold. b. Compute the flux of DBP (mol cm 2 s 1) at the particle surface. c. Can you neglect the Stefan flow in this calculation? d. Evaluate the temperature rise resulting from the condensation of the vapor on the particle. e. Plot the particle radius as a function of time starting at 0.5 μm and assuming that the background mole fraction of DBP remains constant at 0.10. The following parameters may be used in the calculation ΔHv =R ˆ 8930 K D ˆ 0:0282 cm2 s

1

R=Mair^c p ˆ 0:274 MDBP =Mair ˆ 9:61 12.2B In Problem 12.1, for the purpose of computing the mass and energy transport rates to the particle, it was assumed that continuum conditions hold. a. Evaluate the validity of using continuum transport theory for the conditions of Problem 12.1. b. Plot the particle radius as a function of time under the conditions of Problem 12.1 for a DBP particle initially of 0.5 μm radius assuming that the accommodation coefficient is α = 0.1, 0.01, and 0.001. c. Plot the particle radius as a function of time under the conditions of Problem 12.1 for a DBP particle initially of 0.01 μm radius given that the accommodation coefficient is α = 0.1, 0.01, and 0.001.

534

MASS TRANSFER ASPECTS OF ATMOSPHERIC CHEMISTRY

12.3C Consider the transient absorption of SO2 by a droplet initially free of any dissolved SO2. Calculate the time necessary for a water droplet of radius 10 μm to fill up with SO2 in an open system (constant ξSO2 ˆ 1 ppb), for constant droplet pH 4 and 6: a. Assume that the absorption is irreversible. b. Assume that the absorption is reversible. c. Discuss your results. d. Discuss qualitatively how your results would change if the system were closed and the cloud liquid water content were 0.1 g m 3. 12.4B Consider the transient dissolution of HNO3 ξHNO3 ˆ 1 ppb by a monodisperse cloud droplet population (Rp = 10 μm, L = 0.2 g m 3) at 298 K, assuming that the cloudwater has initially zero nitrate concentration. As a simplification, assume that the cloud has constant pH equal to 4.5. a. How long will it take for the cloud to reach equilibrium, if the cloud is described as an open system? b. How long will it take if the system is described as closed? c. Discuss the implications of your results for the mathematical modeling of clouds. 12.5B Solve (12.72) and (12.73) to obtain (12.74). 12.6B A (NH4)2SO4 aerosol particle of initial diameter equal to 10 nm grows as a result of condensation of H2SO4 and NH3 at 50% RH and 298 K. a. Assuming that the H2SO4 vapor has a constant mixing ratio of 1 ppt and that the vapor pressure of ammonium sulfate can be neglected, calculate the time required for the particle to grow to 0.05, 0.1, and 1 μm in diameter. Assume that there is always an excess of ammonia available to neutralize the condensing sulfuric acid. b. What are the corresponding times if the atmospheric relative humidity is 90%? 12.7B Estimate the time required for the complete evaporation of an organic aerosol particle as a function of its initial diameter Dp0 = 0.05–10 μm and its equilibrium vapor pressure p0 = 10 11– 10 8 atm. Assume that the organic has a surface tension of 30 dyn cm 1, density 0.8 g cm 3, Dg = 0.1 cm2 s 1, α = 1, and the background gas-phase concentration is zero. Assume MA = 200 g mol 1. 12.8C Fresh seasalt particles in the marine atmosphere are alkaline and can serve as sites for the heterogeneous oxidation of SO2 by ozone. Assuming that the diameter of these particles varies from 0.5 to 20 μm, their pH varies from 5 to 8, their liquid water content is 100 μg m 3, and typical mixing ratios are 50 ppt for SO2 and 30 ppb for ozone: a. Calculate the sulfate production rate assuming that both reactants are in Henry’s law equilibrium between the gas and aerosol phases. b. Is this rate limited by gas-phase diffusion, or interfacial transport? Quantify the effects of these limitations on the sulfate production rates estimated in (a). 12.9A Plot the mass transfer coefficient kc for raindrops falling through air as a function of their diameter. Assume T = 298 K and the diffusing species to be SO2. Use the theory of terminal velocities of drops developed in Chapter 9. Consider drop sizes of Rp = 0.1 and 1 mm. 12.10B Table 12P.1 gives the forward rate constants and parameters A, B, and C in the equilibrium constant expression log K ˆ A ‡ B

1000 1000 ‡C T T

2

for a number of aqueous-phase equilibria. For each of the reactions given in the table evaluate the characteristic time to reach equilibrium at T = 298 K and T = 273 K. Discuss your results.

535

REFERENCES

TABLE 12P.1 Forward Rate Constants kf and Parameters A, B, and C in the Equilibrium Constant Expression log K = A + B(1000/T) + C(1000/T)2 for Aqueous Equilibria Reaction HSO4 ƒ SO42 ‡ H‡ SO2
H2 O ƒ HSO3 ‡ H‡ HSO3 ƒ SO23 ‡ H‡ CO2
H2 O ƒ HCO3 ‡ H‡ HCO3 ƒ CO23 ‡ H‡ HO2 ƒ O2 ‡ H‡

kf (s 1) 1.8 × 107 6.1 × 104 5.4 × 100 2.1 × 104 4.3 × 10 2 2.3 × 103

A 5.95 4.84 8.86 14.25 13.80 4.9

B 1.18 0.87 0.49 5.19 2.87 0

C 0 0 0 0.85 0.58 0

REFERENCES Bhatnagar, P. L., et al. (1954), A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev. 94, 511–525. Bird, R. B., et al. (2002), Transport Phenomena, Second edition, John Wiley & Sons, Inc., New York. Capaldo, K. P., et al. (2000), A computationally efficient hybrid approach for dynamic gas/aerosol transfer in air quality models, Atmos. Environ. 34, 3617–3627. Chang X., and Davis, E. J. (1974), Interfacial conditions and evaporation rates of a liquid droplet, J. Colloid Interface Sci. 47, 65–76. Dahneke, B. (1983), Simple kinetic theory of Brownian diffusion in vapors and aerosols, in Theory of Dispersed Multiphase Flow, R. E. Meyer (ed.), Academic Press, New York, pp. 97–133. Dassios, K. G., and Pandis, S. N. (1999), The mass accommodation coefficient of ammonium nitrate aerosol, Atmos. Environ. 33, 2993–3003. Davidovits, P., et al. (1995), Entry of gas molecules into liquids, Faraday Discuss. 100, 65–82. Davis, E. J. (1983), Transport phenomena with single aerosol particles, Aerosol Sci. Technol. 2, 121–144. DeBruyn, W. J., et al. (1994), Uptake of gas phase sulfur species methanesulfonic acid, dimethylsulfoxide, and dimethyl sulfone by aqueous surfaces, J. Geophys. Res. 99, 16927–16932. Doyle, G. J., et al. (1979), Simultaneous concentrations of ammonia and nitric acid in a polluted atmosphere and their equilibrium relationship to particulate ammonium nitrate, Environ. Sci. Technol. 13, 1416–1419. Duan, S. X., et al. (1993), Uptake of gas-phase acetone by water surfaces, J. Phys. Chem. 97, 2284–2288. Fuchs, N. A. (1964), Mechanics of Aerosols, Pergamon, New York. Fuchs, N. A., and Sutugin, A. G. (1971), High dispersed aerosols, in Topics in Current Aerosol Research (Part 2) G. M. Hidy and J. R. Brock, eds., Pergamon, New York, pp. 1–200. Gardner, J. A., et al. (1987), Measurement of the mass accommodation coefficient of SO2(g) on water droplets, J. Geophys. Res. 92, 10887–10895. Hildemann, L. M., et al. (1984), Ammonia and nitric acid concentrations in equilibrium with atmospheric aerosols: Experiment vs. theory, Atmos. Environ. 18, 1737–1750. Hirschfelder, J. O., et al. (1954), Molecular Theory of Gases and Liquids, John Wiley & Sons, Inc., New York. Jacob, D. J. (1986), Chemistry of OH in remote clouds and its role in the production of formic acid and peroxymonosulfate, J. Geophys. Res. 91, 9807–9826. Jayne, J. T., et al. (1991), Uptake of gas-phase alcohol and organic acid molecules by water surfaces, J. Phys. Chem. 95, 6329–6336. Jayne, J. T., et al. (1992), Uptake of gas-phase aldehydes by water surfaces, J. Phys. Chem. 96, 5452–5460. Jefferson, A., et al. (1997), Measurements of the H2SO4 mass accommodation coefficient onto polydisperse aerosol, J. Geophys. Res. 102, 19021–19028. Li, W., and Davis E. J. (1995), Aerosol evaporation in the transition regime, Aerosol Sci. Technol. 25, 11–21. Loyalka, S. K. (1983), Mechanics of aerosols in nuclear reactor safety: A review, Prog. Nucl. Energy 12, 1–56.

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MASS TRANSFER ASPECTS OF ATMOSPHERIC CHEMISTRY Maxwell, J. C. (1877), in Encyclopedia Britannica, Vol. 2, p. 82. Meng, Z., and Seinfeld, J. H. (1996), Timescales to achieve atmospheric gas-aerosol equilibrium for volatile species, Atmos. Environ. 30, 2889–2900. Moore, W. J. (1962), Physical Chemistry, 3rd ed., Prentice-Hall, Englewood Cliffs, NJ. Mozurkewich, M., et al. (1987), Mass accommodation coefficient for HO2 radicals on aqueous particles, J. Geophys. Res. 192, 4163–4170. Nathanson, G. M., et al. (1996), Dynamics and kinetics at the gas-liquid interface, J. Phys. Chem. 100, 13007–13020. Pandis, S. N., and Seinfeld, J. H. (1989), Sensitivity analysis of a chemical mechanism for aqueous-phase atmospheric chemistry, J. Geophys. Res. 94, 1105–1126. Pesthy, A. J., et al. (1981), The effect of a growing aerosol on the rate of homogeneous nucleation of a vapor, J. Colloid Interface Sci. 82, 465–479. Pilinis, C., and Seinfeld, J. H. (1988), Development and evaluation of an Eulerian photochemical gas-aerosol model, Atmos. Environ. 22, 1985–2001. Pöschl, U., et al. (2007), Kinetic model framework for aerosol and cloud surface chemistry and gas-particle interactions: Part 1—general equations, parameters, and terminology, Atmos. Chem. Phys. 7, 5989–6023. Pruppacher, H. R., and Klett, J. D. (1997), Microphysics of Clouds and Precipitation, 2nd ed., Kluwer, Boston. Rudich, Y., et al. (1996), Reactive uptake of NO3 on pure water and ionic solutions, J. Geophys. Res. 101, 21023–21031. Russell, A. G., et al. (1988), Mathematical modeling of the formation of nitrogen-containing air pollutants. 1. Evaluation of an Eulerian photochemical model, Environ. Sci. Technol. 22, 263–271. Sahni, D. C. (1966), The effect of a black sphere on the flux distribution of an infinite moderator, J. Nucl. Energy 20, 915–920. Schwartz, S. E. (1984), Gas aqueous reactions of sulfur and nitrogen oxides in liquid water clouds, in SO2, NO, and NO2 Oxidation Mechanisms: Atmospheric Considerations, J. G. Calvert (ed.), Butterworth, Boston, pp. 173–208. Schwartz, S. E. (1986), Mass transport considerations pertinent to aqueous-phase reactions of gases in liquid-water clouds, in Chemistry of Multiphase Atmospheric Systems, W. Jaeschke (ed.), Springer-Verlag, Berlin, pp. 415–471. Schwartz, S. E. (1987), Both sides now: The chemistry of clouds, Ann. NY Acad. Sci. 502, 83–114. Schwartz, S. E. (1988), Mass transport limitation to the rate of in-cloud oxidation of SO2: Re-examination in the light of new data, Atmos. Environ. 22, 2491–2499. Schwartz, S. E., and Freiberg, J. E. (1981), Mass transport limitation to the rate of reaction of gases and liquid droplets: Application to oxidation of SO2 and aqueous solution, Atmos. Environ. 15, 1129–1144. Sitarski, M., and Nowakowski, B. (1979), Condensation rate of trace vapor on Kundsen aerosols from solution of the Boltzmann equation, J. Colloid Interface Sci. 72, 113–122. Stelson, A. W., et al. (1979), A note on the equilibrium relationship between ammonia and nitric acid and particulate ammonium nitrate, Atmos. Environ. 13, 369–371. Stelson, A. W., and Seinfeld, J. H. (1982), Relative humidity and temperature dependence of the ammonium nitrate dissociation constant, Atmos. Environ. 16, 983–992. Takahama, S., et al. (2004), Modeling the diurnal variation of nitrate during the Pittsburgh Air Quality Study, J. Geophys. Res. 109, D16S06 (doi: 10.1029/2003JD004149). Tang, I. N., and Lee, J. H. (1987), Accommodation coefficients for ozone and SO2: Implications for SO2 oxidation in cloudwater, ACS Symp. Ser. 349, 109–117. Van Doren, J. M., et al. (1990), Temperature dependence of the uptake coefficients of HNO3, HCl, and N2O5 by water droplets, J. Phys. Chem. 94, 3265–3272. Van Doren, J. M., et al. (1991), Uptake of N2O5 and HNO3 by aqueous sulfuric acid droplets, J. Phys. Chem. 95, 1684–1689. Watson, L. R., et al. (1990), Uptake of HCl molecules by aqueous sulfuric acid droplets as a function of acid concentration, J. Geophys. Res. 95, 5631–5638. Wexler, A. S., and Seinfeld, J. H. (1990), The distribution of ammonium salts among a size and composition dispersed aerosol, Atmos. Environ. 24A, 1231–1246. Wexler, A. S., and Seinfeld, J. H. (1992), Analysis of aerosol ammonium nitrate: Departures from equilibrium during SCAQS, Atmos. Environ. 26A, 579–591. Williams, M. M. R., and Loyalka, S. K. (1991) Aerosol Science: Theory and Practice, Pergamon, New York. Worsnop, D. R., et al. (1989), Temperature dependence of mass accommodation of SO2 and H2O2 on aqueous surfaces, J. Phys. Chem. 93, 1159–1172.

CHAPTER

13

Dynamics of Aerosol Populations

Up to this point we have considered the physics of atmospheric aerosols in terms of the behavior of a single particle. In this chapter we focus our attention on a population of particles that interact with one another. We will treat changes that occur in the population when a vapor compound condenses on the particles and when the particles collide and adhere. We begin by discussing a series of mathematical approaches for the description of the particle distribution. Next, we calculate the rate of change of a particle distribution due to mass transfer of material from or to the gas phase. We continue by calculating the rate at which two spherical particles in Brownian motion collide and then, in an appendix, consider coagulation induced by velocity gradients in the fluid, by differential settling of particles, and compute the enhancement or retardation to the coagulation rate due to interparticle forces. The remainder of the chapter is devoted to solution of the equations governing the size distribution of an aerosol and examination of the physical regimes of behavior of an aerosol population.

13.1 MATHEMATICAL REPRESENTATIONS OF AEROSOL SIZE DISTRIBUTIONS Observed aerosol number distributions are usually expressed as n°(log Dp), that is, using log Dp as an independent variable (see Chapter 8). However, it is often convenient to use alternative definitions of the particle size distribution to facilitate the calculations of its rate of change. The distribution can be expressed in several forms, assuming a spatially homogeneous aerosol of uniform chemical composition.

13.1.1 Discrete Distribution An aerosol distribution can be described by the number concentrations of particles of various sizes as a function of time. Let us define Nk(t) as the number concentration (cm 3) of particles containing k monomers, where a monomer can be considered as a single molecule of the species representing the particle. Physically, the discrete distribution is appealing since it is based on the fundamental nature of the particles. However, a particle of size 1 μm contains on the order of 1010 monomers, and description of the submicrometer aerosol distribution requires a vector …N 2 ; N 3 ; . . . ; N 1010 † containing 1010 numbers. This renders the use of the discrete distribution impractical for most atmospheric aerosol applications. We will use it in the subsequent sections for instructional purposes and as an intermediate step toward development of the continuous general dynamic equation. Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, Third Edition. John H. Seinfeld and Spyros N. Pandis.  2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

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13.1.2 Continuous Distribution The continuous distribution, introduced in Chapter 8, is usually a more useful concept in practice. The volume of the particle is often chosen for convenience as the independent variable for the continuous size distribution function n(v, t) (μm 3 cm 3), where v ˆ 16 π D3p is the volume of a particle with diameter Dp. Thus n(v, t)dv is defined as the number of particles per cubic centimeter having volumes in the range from v to v + dv. Note that the total aerosol number concentration Nt(t) (cm 3) is then given by N t …t† ˆ

1

∫0

n…v; t† dv

(13.1)

The mass of a particle m or the diameter Dp can also be used as the independent variable for the mathematical description of the aerosol distribution. We saw in Chapter 8 that these functions are not equal to each other but can be easily related. For example, if the radius Rp is used as an independent variable for the distribution function nR(Rp, t), then the concentration of particles dN in the size range Rp to (Rp + dRp) is given by dN = nR(Rp, t) dRp. But the same number of particles is equal to n(v, t) dv, where v ˆ …4=3†πR3p or dv ˆ 4πRp2 dRp , so nR …Rp ; t† ˆ 4πR2p n…v; t†

(13.2)

Similar expressions can easily be developed for other functions describing size distributions using (8.25).

13.2 CONDENSATION When a vapor condenses on a particle population or when material evaporates from the aerosol to the gas phase, the particle diameters change and the size distribution of the population n(v, t) changes shape. Assuming that the particles are not in equilibrium with the gas phase—that is, the vapor pressure of the particles is not equal to the partial pressure of this compound in the gas phase—we want to calculate the rate of change ( @ n(v, t)/ @ t)growth/evap.

13.2.1 The Condensation Equation We saw in Chapter 12 that the rate of change of the mass of a particle of diameter Dp as a result of transport of species i between the gas and aerosol phases is dm 2πDp Di Mi ˆ f …Kn; α†…pi dt RT

peq;i †

(13.3)

where Di is the diffusion coefficient for species i in air, Mi is its molecular weight, and f(Kn, α) is the correction due to noncontinuum effects and imperfect surface accommodation, for example, Equation (12.43). The difference between the vapor pressure of i far from the particle pi and the equilibrium vapor pressure peq,i constitutes the driving force for the transport of species i. Let us define the condensation growth rate Iv(v) as the rate of change of the volume of a particle of volume v. Assuming that the aerosol particle contains only one species, it will have constant density ρp; therefore from (13.3) and since v ˆ 16 πD3p Iv ˆ

dv 2π2=3 …6v†1=3 Di Mi ˆ f …Kn; α†…pi dt ρp RT

peq;i †

(13.4)

Let us assume that a particle population is growing due to condensation, so the size distribution is moving to the right (Figure 13.1). We focus our attention on an infinitesimal slice Δv of the distribution

539

CONDENSATION

FIGURE 13.1 Schematic representation of differential mass balance for the derivation of growth equation for a particle population.

centered at v that extends from v (Δv/2) to v + (Δv/2). The number of particles in this slice at time t is n(v, t)Δv. As a result of condensation and subsequent growth, particles enter the slice from the left and exit from the right. The instantaneous rate of entry from the left is proportional to the number concentration of particles at the left boundary of the slice, n(v (Δv/2), t), and their volume growth rate, Iv(v (Δv/2), t). So particles enter the slice from the left with a rate equal to Iv(v (Δv/2), t)n(v (Δv/2), t). Note that the units of the product are, indeed, the number of particles per time. Applying the same arguments, particles grow out of this distribution slice from the right boundary with a rate Iv(v + (Δv/2), t) n(v + (Δv/2), t). After a period Δt the number of particles inside this fixed slice will change from n(v, t)Δv to n(v, t + Δt) Δv. This number change will equal the net flux into the slice during this period, or mathematically n…v; t ‡ Δt†Δv n…v; t†Δv Δv Δv ;t n v ; t Δt ˆ Iv v 2 2

Iv v ‡

Δv Δv ;t n v ‡ ; t Δt 2 2

(13.5)

Rearranging the terms and considering the limits Δt → 0 and Δv → 0 yield lim n…v; t ‡ Δt† Δt

Δt!0

n…v; t†

ˆ

lim

Δv!0

I v …v ‡ …Δv=2†; t†n…v ‡ …Δv=2†; t† I v …v Δv

…Δv=2†; t†n…v

…Δv=2†; t†

(13.6)

and noting that the limits are by definition equal to partial derivatives, finally yields @ n…v; t† ˆ @t

@ ‰I v …v; t†n…v; t†Š @v

(13.7)

where Iv(v, t) is given by (13.4). Equation (13.7) is called the condensation equation and describes mathematically the rate of change of a particle size distribution n(v, t) due to the condensation or evaporation flux Iv(v, t), neglecting other processes that may influence the distribution shape (sources, removal, coagulation, nucleation, etc). A series of alternative forms of the condensation equation can be written depending on the form of the size distribution used or the expression for the condensation flux. For example, if the mass of a particle is used

540

DYNAMICS OF AEROSOL POPULATIONS

as the independent variable, one can show that the condensation equation takes the form @ n…m; t† @ ‡ ‰I m …m; t†n…m; t†Š ˆ 0 @t @m

(13.8)

where Im(m, t), the rate of mass change of a particle due to condensation, based on (13.3), is 2πDi Mi I m …m; t† ˆ RT

6m ρp π

1=3

f…Kn; α†…pi

peq;i †

(13.9)

Finally, if the diameter is the independent variable of choice, the number distribution is given by nD(Dp, t) and the condensation equation is @ nD …Dp ; t† @ I D …Dp ; t†nD …Dp ; t† ˆ 0 ‡ @t @ Dp

(13.10)

where ID(Dp) is the rate of diameter change of a particle as a result of condensation or evaporation: I D …Dp ; t† ˆ

dDp 4Di Mi ˆ f …Kn; α†…pi dt RTDp ρp

peq;i †

(13.11)

Equations (13.7) (13.8) and (13.10) are equivalent descriptions of the same process using different independent variables for the description of the size distribution. For sufficiently large particles (the so-called continuum regime) and assuming unity accommoda­ tion coefficient, f(Kn, α) = 1, the continuum diffusion growth law based on particle diameter is I D …Dp ; t† ˆ

1 4Di Mi …p Dp RTρp i

peq;i †

(13.12)

Assuming that the condensation driving force is maintained constant externally (e.g., constant supersaturation of the gas phase), we can write (13.12) as I D …Dp ; t† ˆ where A = 4Di Mi(pi I D …Dp †  Dp 1 .

A Dp

(13.13)

peq,i)/RTρp is a constant. This case therefore corresponds to a growth law

13.2.2 Solution of the Condensation Equation The condensation equation assuming continuum regime growth, unity accommodation coefficient, and constant gas-phase supersaturation can be written using (13.10) and (13.13) as @ nD …Dp ; t† AnD …Dp ; t† @ ‡ Dp @t @ Dp

ˆ0

(13.14)

or, after expanding the derivative, as @ nD …Dp ; t† A @ nD …Dp ; t† A ‡ ˆ 2 nD …Dp ; t† Dp @t @ Dp Dp

(13.15)

541

CONDENSATION

This partial differential equation (PDE) is to be solved subject to the following initial and boundary conditions: nD …Dp ; 0† ˆ n0 …Dp †

(13.16)

nD …0; t† ˆ 0

(13.17)

Equation (13.16) provides simply the initial size distribution for the aerosol population, while (13.17) implies that there are no particles of zero diameter. If we define A nD …Dp ; t† (13.18) F…Dp ; t† ˆ Dp then (13.14) can be rewritten as @ F…Dp ; t† A @ F…Dp ; t† ˆ0 ‡ Dp @ Dp @t

(13.19)

Using the method of characteristics for partial differential equations, F(Dp, t) will be constant along the characteristic lines given by dDp A ˆ dt Dp

(13.20)

or equivalently along the curves given by the solution of (13.20) with Dp(t = 0) = Dp0, D2p ˆ D2p0 ‡ 2 At

(13.21)

where Dp0 is a constant corresponding to the initial diameter for a given characteristic. The characteristic curves for a typical set of parameters are given in Figure 13.2. Note that if 2 At  D2p0 , then Dp  Dp0; therefore for a timescale of 1 h, particles larger than 0.6 μm will not grow appreciably under these conditions. The smaller particles grow much more rapidly as the characteristics are moving rapidly to larger diameters. It is also interesting to note that as the diameters of the small particles change much faster than the diameters of the large ones, condensation tends to “shrink” the aerosol distribution. Note that as t → 1 the distribution tends to become more and more monodisperse. Physically, the characteristics of the partial differential equation give us the trajectories in the time– diameter coordinate system of the various particle sizes. If, for a moment, we think of the particle population not as a continuous mathematical function but as groups of particles of specific diameters, the characteristic equations describe quantitatively the evolution of the sizes of these particles with time. The characteristic curves in Figure 13.2 are not the complete solution of the PDE, but they still convey information about the evolution of the size distribution for this case. Returning to the solution of the condensation equation, the constant value of F(Dp, t) along each characteristic curve can be determined by its value of t = 0, that is F…Dp ; t† ˆ F…Dp0 ; 0† ˆ

A n0 …Dp0 † Dp0

(13.22)

and recalling the definition of F(Dp, t) in (13.18), we obtain n…Dp ; t† ˆ

Dp n0 …Dp0 † Dp0

(13.23)

542

DYNAMICS OF AEROSOL POPULATIONS

FIGURE 13.2 Growth of particles of initial diameters 0.2, 0.5, and 2 μm assuming Di = 0.1 cm2 s 1, Mi = 100 g mol 1, (pi peq,i) = 10 9 atm (1 ppb), T = 298 K, and ρp = 1 g cm 3.

But Dp, Dp0, and t are related through the characteristic equation (13.21), so Dp0 can be written as Dp0 ˆ D2p

2 At

1=2

(13.24)

and substituting into (13.23) Dp

n…Dp ; t† ˆ Dp2

2 At

1=2

n0

Dp2

2 At

1=2

Note that the solution in (13.25) carries implicitly the constraint Dp >

p

(13.25)

2 At.

Growth of a Lognormal Aerosol Size Distribution Aerosol size distribution data are sometimes represented by the lognormal distribution function [see also (8.33)] n…Dp † ˆ p

N0 2πDp ln σg

exp

ln2 …Dp =Dpg †

(13.26)

2 ln2 σg

For such an initial distribution using the solution obtained in (13.25), we get

n…Dp ; t† ˆ

Dp D2p

N p 0 exp 2π ln σg 2 At

ln2

D2p

2 At 2 ln2 σg

1=2

=Dpg (13.27)

543

CONDENSATION

For an aerosol population undergoing pure growth with no particle sources or sinks, the total particle number is preserved. If the solution in (13.27) is valid, it must satisfy the following integral relation at all times: N0 ˆ

1

∫ p2 At

n…Dp ; t†dDp

(13.28)

Note that the lower limit of this integral isp determined by the characteristic curve through the origin, since the solution is valid only for Dp > 2 At. Equation (13.28) is readily simplified by using the substitutions u ˆ D2p

2 At

1=2

(13.29)

and Dp

du ˆ D2p

2 At

1=2

dDp

(13.30)

The lower and upper limits of the integral become 0 and 1, respectively. The resulting integral is identical at all times in form with the integral of the original lognormal distribution, which we know is equal to the total number concentration. Figure 13.3 shows the growth of a lognormally distributed aerosol as a function of time. The difference in growth between the small and large particles, seen from the characteristic curves of Figure 13.2, is even more evident in Figure 13.3.

FIGURE 13.3 Evolution of a lognormal distribution (initially Dp ˆ 0:2 μm, σg = 1.5) assuming Di = 0.1 cm2 s 1, Mi = 100 g mol 1, (pi peq,i) = 10 9 atm (1 ppb), T = 298 K, and ρp = 1 g cm 3.

544

DYNAMICS OF AEROSOL POPULATIONS

13.3 COAGULATION Aerosol particles suspended in a fluid may come into contact because of their Brownian motion or as a result of their motion produced by hydrodynamic, electrical, gravitational, or other forces. Brownian coagulation is often referred to as thermal coagulation. The theory of coagulation will be presented in two steps. At first, we will develop an expression describing the rate of collisions between two monodisperse particle populations consisting of N1 particles with diameter Dp1 and N2 with diameter Dp2. In the next step a differential equation describing the rate of change of a full coagulating aerosol size distribution will be derived.

13.3.1 Brownian Coagulation Let us start by considering the simplest coagulation problem, that of a monodisperse aerosol population of radius Rp at initial concentration N0. Imagine one of the particles to be stationary with its center at the origin of the coordinate system. We wish to calculate the rate at which particles collide with this stationary particle as a result of their Brownian motion. Let us neglect for the time being the size and shape change of the particles as other particles collide with it. We will show later on that this assumption does not lead to appreciable error in the early stages of coagulation. Moreover, since spherical particles come into contact when the distance between their centers is equal to the sum of their radii, from a mathematical point of view, we can replace our stationary particle of radius Rp with an absorbing sphere of radius 2Rp and the other particles by point masses (Figure 13.4). At this point, we need to describe mathematically the distribution of aerosols around the fixed particle. We need to consider, as with mass transfer to a particle, the continuum, free molecular, and transition regimes. 13.3.1.1 Continuum Regime Assuming that the distribution of particles around our fixed particle can be described by the continuum diffusion equation, then that distribution N(r, t) satisfies @ N…r; t† ˆD @t

2

@ N…r; t† 2 @ N…r; t† ‡ r @ r2 @r

FIGURE 13.4 The collision of two particles of radii Rp is equivalent geometrically to collisions of point particles with an absorbing sphere of radius 2 Rp; also shown in the corresponding coordinate system.

(13.31)

545

COAGULATION

where r is the distance of the particles from the center of the fixed particle, N is the number concentration, and D is the Brownian diffusion coefficient for the particles. The initial and boundary conditions for (13.31) are N…r; 0† ˆ N 0 N…1; t† ˆ N 0 N…2Rp ; t† ˆ 0

(13.32)

The initial condition assumes that the particles are initially homogeneously distributed in space with a number concentration N0. The first boundary condition requires that the number concentration of particles infinitely far from the particle absorbing sphere not be influenced by it. Finally, the boundary condition at r = 2Rp expresses the assumption that the fixed particle is a perfect absorber, that is, that particles adhere at every collision. Although little is known quantitatively about the sticking probability of two colliding aerosol particles, their low kinetic energy makes bounce-off unlikely. We shall therefore assume here a unity sticking probability. The solution of (13.31) and (13.32) is N…r; t† ˆ N 0 1 ˆ N0 1

2 Rp 4Rp …r ‡ p r r π ∫0

p 2Rp †=2 Dt

e

η2

dη

2Rp r 2Rp p erfc r 2 Dt

(13.33)

and the rate (particles s 1) at which particles collide with our stationary particle, which has an effective surface area of 16 π R2p , is J col ˆ 16 π R2p D

@N @r

rˆ2Rp

2Rp ˆ 8πRp DN 0 1 ‡ p πDt

(13.34)

The collision rate is initially extremely fast (actually it starts at infinity) but for t  4R2p =πD, it approaches a steady-state value of Jcol = 8πRp DN0. Physically, at t = 0, other particles in the vicinity of the absorbing one collide with it, immediately resulting in a mathematically infinite collision rate. However, these particles are soon absorbed by the stationary particle, and the concentration profile around our particle relaxes to its steady-state profile with a steady-state collision rate. One can easily calculate, given the Brownian diffusivities in Table 9.5, that such a system reaches steady state in 10 4 s for particles of diameter 0.1 μm and in roughly 0.1 s for 1-μm particles. Therefore neglecting the transition to this steady state is a good assumption for atmospheric applications. Our analysis so far includes two major assumptions, that our particle is stationary and that all particles have the same radius. Let us relax these two assumptions by allowing our particle to undergo Brownian diffusion and also let it have a radius Rp1 and the others in the fluid have radii Rp2. Our first challenge, as we want to maintain the diffusion framework of (13.31), is to calculate the diffusion coefficient that characterizes the diffusion of particles of radius Rp2 relative to those of radius Rp1. As both particles are undergoing Brownian motion, suppose that in a time interval dt they experience displacements dr1 and dr2. Then their mean square relative displacement is jdr1

2

dr2 j

ˆ dr21 ‡ dr22

2hdr1 ? dr2 i

(13.35)

Since the motions of the two particles are assumed to be independent, hdr1
dr2i = 0. Thus jdr1

2

dr2 j

ˆ dr21 ‡ dr22

(13.36)

546

DYNAMICS OF AEROSOL POPULATIONS

Referring back to our discussion of Brownian motion, we can identify coefficient D12 by (9.72): jdr1

2

dr2 j

ˆ 6D12 dt

(13.37)

On the other hand, the two individual Brownian diffusion coefficients are defined by dr21 ˆ 6D1 dt and dr22 ˆ 6D2 dt. Consequently, we find that D12 ˆ D1 ‡ D2

(13.38)

Taking advantage of this result, we can now once more regard the particle with radius Rp1 as stationary and those with radius Rp2 as diffusing toward it. The diffusion equation (13.31) then governs N2(r, t), the concentration of particles with radius Rp2, with the diffusion coefficient D12. The system is then described mathematically by @ N 2 …r; t† ˆ D12 @t

2

@ N 2 …r; t† 2 @ N 2 …r; t† ‡ r @ r2 @r

(13.39)

with conditions N 2 …r; 0† ˆ N 02 N 2 …1; t† ˆ N 02

(13.40)

N 2 Rp1 ‡ Rp2 ; t ˆ 0 Note that the boundary condition at r = 2Rp is now applied at r = Rp1 + Rp2. The solution of (13.39) and (13.40) is N 2 …r; t† ˆ N 02 1

Rp1 ‡ Rp2 r erfc r

…Rp1 ‡ Rp2 † p 2 D12 t

(13.41)

and the rate (s 1) at which #2 (secondary) particles arrive at the surface of the absorbing sphere at r = Rp1 + Rp2 is Rp1 ‡ Rp2 J col ˆ 4π…Rp1 ‡ Rp2 †D12 N 02 1 ‡ p πD12 t

(13.42)

The steady-state rate of collision is J col ˆ 4π…Rp1 ‡ Rp2 †D12 N 02

(13.43)

The collision rate we have derived is the rate, expressed as the number of #2 particles per second, at which the #2 particles collide with a single #1 (primary) particle. When there are N 01 #1 particles, the total collision rate between #1 and #2 particles per volume of fluid is equal to the collision rate derived above multiplied by N 01 . Thus the steady-state coagulation rate (cm 3 s 1) between #1 and #2 particles is J 12 ˆ 4π…Rp1 ‡ Rp2 †D12 N 01 N 02

(13.44)

547

COAGULATION

At this point there is no need to retain the superscript 0 on the number concentrations so the coagulation rate can be expressed as J 12 ˆ K12 N 1 N 2

(13.45)

where K12 ˆ 2π…Dp1 ‡ Dp2 †…D1 ‡ D2 †

(13.46)

In the continuum regime the Brownian diffusivities are given by (9.73) Di ˆ

kT 3π μ Dpi

(13.47)

where the slip correction Cc is set equal to 1, μ is the viscosity of air, and Dpi is the particle diameter. Using the expressions for D1 and D2, (13.46) becomes

K12 ˆ

2kT …Dp1 ‡ Dp2 †2 Dp1 Dp2 3μ

(13.48)

The coagulation coefficient K12 can be expressed in terms of particle volume as 1=3

K12

1=3

2kT v1 ‡ v2 ˆ 3μ …v1 v2 †1=3

2

(13.49)

The development described above is based on the assumption that continuum diffusion theory is a valid description of the concentration distribution of particles surrounding a central absorbing sphere. 13.3.1.2 Transition and Free Molecular Regime When the mean free path λp of the diffusing aerosol particle is comparable to the radius of the absorbing particle, the boundary condition at the absorbing particle surface must be corrected to account for the nature of the diffusion process in the vicinity of the surface. The physical meaning of this can be explained as follows. As we have seen, the diffusion equations can be applied to Brownian motion only for time intervals that are large compared to the relaxation time τ of the particles or for distances that are large compared to the aerosol mean free path λp. Diffusion equations cannot describe the motion of particles inside a layer of thickness λp adjacent to an absorbing wall. If the size of the absorbing sphere is comparable to λp, this layer has a substantial effect on the kinetics of coagulation. Fuchs (1964) suggested that this effect can be corrected by the introduction of a generalized coagulation coefficient K12 ˆ 2π…Dp1 ‡ Dp2 †…D1 ‡ D2 †β

(13.50)

where Di is as given by (9.73). The Fuchs form of K12 is defined in Table 13.1. It is instructive to compare the values of the corrected coagulation coefficient K versus that neglecting the kinetic effect, K0, corresponding to β = 1 (Table 13.2) for monodisperse aerosols in air. The correction factor β varies from 0.014 for particles of 0.001 μm radius, to practically 1 for 1-μm particles.

548

DYNAMICS OF AEROSOL POPULATIONS

TABLE 13.1 Fuchs Form of the Brownian Coagulation Coefficient K12 K12 ˆ 2π…D1 ‡ D2 †…Dp1 ‡ Dp2 †

Dp1 ‡ Dp2 Dp1 ‡ Dp2 ‡ 2 g21 ‡ g22

‡ 1=2

8…D1 ‡ D2 † c21 ‡ c22

1=2

1

…Dp1 ‡ Dp2 †

where ci ˆ

8kT π mi

1=2

8Di a π ci p 2 gi ˆ …Dpi ‡ ℓi †3 3 Dpi ℓi ℓi ˆ

Di ˆ

2 Dpi ‡ ℓ2i

3=2

Dpi

kTCc 3 πμDpi

The mean free path ℓi is defined slightly differently than in (9.89).

a

Let us examine the two asymptotic limits of this coagulation rate. In the continuum limit, Kn → 0, β = 1 and the collision rate reduces to that given by (13.48). In the free-molecule limit Kn → 1 and one can show that K12 ˆ π…Rp1 ‡ Rp2 †2 c12 ‡ c22

1=2

(13.51)

This collision rate is, of course, equal to that obtained directly from kinetic theory. Values of K12 are given in Table 13.3 and Figure 13.5. In using Figure 13.5, we find the smaller of the two particles as the abscissa and then locate the line corresponding to the larger particle. 13.3.1.3 Coagulation Rates Table 13.3 indicates that the smallest value of the coagulation coefficient occurs when both particles are of the same size. The coagulation coefficient rises rapidly when the ratio of the particle diameters increases. This increase is due to the synergism between the two particles. A large particle may be

TABLE 13.2 Coagulation Coefficients of Monodisperse Aerosols in Air at 25 °Ca Dp (μm) 0.002 0.004 0.01 0.02 0.04 0.1 0.2 0.4 1.0 2.0 4.0 a

K0 (cm3 s 1)b 690 × 10 340 × 10 140 × 10 72 × 10 38 × 10 18 × 10 11 × 10 8.6 × 10 7.0 × 10 6.5 × 10 6.3 × 10

Parameter values given in Figure 13.5. Coagulation coefficient neglecting kinetic effects. c Coagulation coefficient including the kinetic correction. b

10 10 10 10 10 10 10 10 10 10 10

K (cm3 s 1)c 8.9 × 10 13 × 10 19 × 10 24 × 10 23 × 10 15 × 10 11 × 10 8.2 × 10 6.9 × 10 6.4 × 10 6.2 × 10

10 10 10 10 10 10 10 10 10 10 10

549

COAGULATION

TABLE 13.3 Coagulation Coefficients (cm3 s − 1) of Atmospheric Particlesa Dp1 (μm) Dp2 (μm) 0.002 0.01 0.1 1 10 20

0.002 8.9 × 10 5.7 × 10 3.4 × 10 7.8 × 10 8.5 × 10 17 × 10

0.01 10 9 7 6 5 5

5.7 × 10 19 × 10 2.5 × 10 3.4 × 10 3.5 × 10 7.0 × 10

0.1 9 10 8 7 6 6

3.4 × 10 2.5 × 10 15 × 10 5.0 × 10 4.5 × 10 9.0 × 10

1.0 7 8 10 9 8 8

7.8 × 10 3.4 × 10 5.0 × 10 6.9 × 10 2.1 × 10 3.9 × 10

6 7 9 10 9 9

10

20

8.5 × 10 5 3.5 × 10 6 4.5 × 10 8 2.1 × 10 9 6.1 × 10 10 6.8 × 10 10

17 × 10 5 7.0 × 10 6 9.0 × 10 8 3.9 × 10 9 6.8 × 10 10 6.0 × 10 10

a

Parameter values given in Figure 13.5.

FIGURE 13.5 Brownian coagulation coefficient K12 for coagulation in air at 25 °C of particles of diameters Dp1 and Dp2. The curves were calculated using the correlation of Fuchs in Table 13.1. To use this figure, find the smaller of the two particles as the abscissa and then locate the line corresponding to the larger particle.

550

DYNAMICS OF AEROSOL POPULATIONS

sluggish from the Brownian motion perspective but, because of its large surface area, provides an ample target for the small, fast particles. Collisions between large particles are slow because both particles move slowly. On the other hand, whereas very small particles have relatively high velocities, they tend to miss each other because their cross-sectional area for collision is small. Note that target area is proportional to D2p , whereas the Brownian diffusion coefficient decreases only as Dp. We see from Figure 13.5 that for collisions among equal-size particles a maximum coagulation coefficient is reached at about 0.02 μm. In the continuum regime, Kn  1 or Dp > 1 μm, for Dp1 = Dp2 using (13.48), the coagulation coefficient is K11 ˆ

8kT 3μ

(13.52)

and is independent of the particle diameter. On the other hand, in free molecular regime coagulation with Dp1 = Dp2, using (13.51), one can show that

K11

1=2

6kT ˆ4 ρp

1=2

Dp1

(13.53)

and the coagulation coefficient increases with increasing diameter. The maximum in the coagulation coefficient for equal-size particles reflects a balance between particle mobility and cross-sectional area for collision. In the continuum regime if Dp2  Dp1, the coagulation coefficient approaches the limiting value lim K12 ˆ

Dp2  Dp1

2kT Dp2 3μ Dp1

(13.54)

In the free molecular regime if Dp2  Dp1, the coagulation coefficient has the following asymptotic behavior:

lim K12 ˆ

Dp2  Dp1

3kT ρp

1=2

D2p2

(13.55)

3=2

Dp1

By comparing (13.54) and (13.55) we see that, with Dp1 fixed, K12 increases more rapidly with Dp2 for free molecular regime than for continuum regime coagulation. 13.3.1.4 Collision Efficiency In our discussion so far, we have assumed that every collision results in coagulation of the two colliding particles. Let us relax this assumption by denoting the coagulation efficiency (fraction of collisions that result in coagulation) by α. Then Fuchs (1964) showed that the coagulation coefficient becomes K12 ˆ 2π…D1 ‡ D2 †…Dp1 ‡ Dp2 † 

Dp1 ‡ Dp2 Dp1 ‡ Dp2 ‡ 2 g21 ‡

in place of the Table 13.1 expression.

1=2 g22

‡

8…1=α†…D1 ‡ D2 † c12

‡

1=2 c22 …Dp1

‡ Dp2 †

1

(13.56)

551

COAGULATION

One can show that in the free molecular regime the coagulation rate is proportional to the collision efficiency. In the continuum regime, coagulation is comparatively insensitive to changes in α. For example, for Rp = 1 μm, the coagulation rate decreases by only 5% as α goes from 1.0 to 0.25, while at Rp = 0.1 μm a 5% decrease of the coagulation rate needs an α = 0.60 (Fuchs 1964). Our development has assumed that all particles behave as spheres. It has been found that, when coagulating, certain particles form chain-like aggregates whose behavior can no longer be predicted on the basis of ideal, spherical particles. Lee and Shaw (1984), Gryn and Kerimov (1990), and Rogak and Flagan (1992), among others, have studied the coagulation of such particles.

13.3.2 The Coagulation Equation In the previous sections, we focused our attention on calculation of the instantaneous coagulation rate between two monodisperse aerosol populations. In this section, we will develop the overall expression describing the evolution of a polydisperse coagulating aerosol population. A good place to start is with a discrete aerosol distribution. 13.3.2.1 The Discrete Coagulation Equation A spatially homogeneous aerosol of uniform chemical composition can be fully characterized by the number densities of particles of various monomer contents as a function of time, Nk(t). The dynamic equation governing Nk(t) can be developed as follows. We have seen that the rate of collision between particles of two types (sizes) is J12 = K12N1N2, expressed in units of collisions per cm3 of fluid per second. A k-mer can be formed by collision of a (k j)-mer with a j-mer. The overall rate of formation, Jfk, of the k-mer will be the sum of the rates that produce the k-mer or k 1

J f k …t† ˆ

Kk

j;j N k j …t†N j …t†

jˆ1

To investigate the validity of this formula let us apply it to a trimer, which can be formed only by collision of a monomer with a dimer J f 3 ˆ K21 N 2 N 1 ‡ K12 N 1 N 2 Note that because K12 = K21, the overall rate is J f 3 ˆ 2K12 N 1 N 2 This expression appears to be incorrect, as we have counted the collisions twice. To correct for that we need to multiply the rate by a factor of ½ so that collisions are not counted twice. Therefore Jf k ˆ

1 2

k 1

Kk

j;j N k j …t†N j …t†

(13.57)

jˆ1

What about equal-size particles? For a dimer our corrected formation rate is 1 J f 2 ˆ K11 N 21 2 Why do we need to divide by a factor of 2 here? Let us try to persuade ourselves that this factor of ½ is always needed. Assume that we have particles consisting only of (k/2) monomers. If one-half of these

552

DYNAMICS OF AEROSOL POPULATIONS

particles are painted red and one-half painted blue, then the rate of coagulation of red and blue particles is J red;blue ˆ Kk=2;k=2

1 N k=2 2

1 N k=2 2

1 ˆ Kk=2;k=2 N 2k=2 4

To calculate the overall coagulation rate we need to add to this rate the collision rate of red particles among themselves and blue particles among themselves. Let us now paint one-half the red particles as green and one-half the blue particles as yellow. The rate of coagulation within each of the initial particle categories is 1 1 1 N k=2 N k=2 ‡ N k=2 4 4 4 …green† ‡ …blue† …yellow†

1 N k=2 4 …red†

J red;green ‡ J blue;yellow ˆ Kk=2;k=2

We can continue this painting process indefinitely (provided that we do not run out of colors), and the result for the total rate of coagulation occurring in the population of equal-size particles is Kk=2;k=2 N 2k=2

1 1 1 ‡ 3 ‡ 4 ‡ ∙∙∙ 2 2 2 2

Since 1

n

2

ˆ

nˆ2

1 2

the total coagulation rate is 1 Kk=2;k=2 N 2k=2 2 The factor of ½ in the equal-size particles is thus a result of the indistinguishability of the coagulating particles. The rate of depletion of a k-mer by agglomeration with all other particles is J dk …t† ˆ N k …t†

1

Kkj N j …t†

(13.58)

jˆ1

Question: Is this rate correct? Why do we not divide here by ½ if j = k? Answer: We do need to divide by 2, due to the indistinguishability argument. However, because each collision removes 2 k-mers, we also need to multiply by 2. These factors cancel each other out and therefore the expression above is correct. Combining the coagulation production and depletion terms, we get the discrete coagulation equation: dN k …t† 1 ˆ dt 2

1

k 1

Kj;k j N j N k jˆ1

j

Nk jˆ1

Kk;j N j ;

k2

(13.59)

553

COAGULATION

In this formulation, it is assumed that the smallest particle is one of size k = 2. In reality, there is a minimum number of monomers in a stable nucleus g∗ and generally g∗  2 (see Chapter 11). This simply means that N 2 ˆ N 3 ˆ ∙ ∙ ∙ ˆ N g∗

1

ˆ0

in (13.59), and being rigorous, we can write the following equation, replacing the summation limits, k g∗

dN k …t† 1 Kj;k j N j N k ˆ dt 2 jˆg∗

1 j

Nk

Kk;j N j ;

k2

(13.60)

jˆg∗

13.3.2.2 The Continuous Coagulation Equation Although (13.59) and (13.60) are rigorous representations of the coagulating aerosol population, they are impractical because of the enormous range of k associated with the equation set above. It is customary to replace Nk(t)(cm 3) with the continuous number distribution function n(v, t) (μm 3 cm 3), where v = kv1 is the particle volume with v1 the volume of the monomer. If we let v0 = g∗v1, then (13.60) becomes, in the limit of a continuous distribution of sizes @ n…v; t† 1 v v0 ˆ K…v 2 ∫ v0 @t

q; q†n…v

q; t†n…q; t†dq (13.61)

1

n…v; t†

∫ v0

K…q; v†n…q; t†dq

The first term in (13.61) corresponds to the production of particles by coagulation of the appropriate combinations of smaller particles, while the second term corresponds to their loss with collisions with all available particles.

13.3.3 Solution of the Coagulation Equation While the coagulation equation cannot be solved analytically in its most general form, solutions can be obtained assuming constant coagulation coefficients. While these solutions have limited applicability to ambient atmospheric conditions, they still provide useful insights and can be used as benchmarks for the development of numerical methods for the solution of the full coagulation equation. Such situations are also applicable to the early stages of coagulation of a monodisperse aerosol in the continuous regime when K(v, q) = 8 kT/3μ. 13.3.3.1 Discrete Coagulation Equation Assuming Kk,j = K in (13.59), we obtain dN k …t† 1 ˆ K dt 2

k 1

N j …t†N k j …t†

K N k …t†

jˆ1

1

N j …t†

jˆ1

Noting that the last term in this equation is the total number of particles N…t† ˆ

1 jˆ1

N j …t†

(13.62)

554

DYNAMICS OF AEROSOL POPULATIONS

(13.62) is simplified to dN k …t† 1 ˆ K dt 2

k 1

N j …t†N k j …t†

K N k …t†N…t†

(13.63)

jˆ1

To solve (13.63), we need to know N(t). By summing equations (13.63) from k = 1 to 1, we obtain dN…t† 1 ˆ K dt 2

1

k 1

N k j …t†N j …t†

K N 2 …t†

(13.64)

kˆ1 jˆ1

The double summation on the right-hand side is equal to N2(t). So (13.64) becomes dN ˆ dt

1 K N 2 …t† 2

(13.65)

N…t† ˆ

N0 1 ‡ …t=τc †

(13.66)

If N(0) = N0, the solution of (13.65) is

where τc is the characteristic time for coagulation given by τc ˆ

2 K N0

(13.67)

Characteristic Timescale for Coagulation τc ˆ

2 KN 0

At t = τc, N…τc † ˆ 1 =2 N 0 . Thus, τc is the time necessary for reduction of the initial number concentration to half its original value. The timescale shortens as the initial number concentration increases. Consider an initial population of particles of about 0.2 μm diameter, for which K = 10 × 10 10 cm3 s 1. The coagulation timescales for N0 = 104 cm 3 and 106 cm 3 are N 0 ˆ 104 cm N 0 ˆ 106 cm

3 3

τc ≅ 55 h τc ≅ 33 min

We can now solve (13.63) inductively assuming that N1(0) = N0; that is, at t = 0 all particles are of size k = 1. Let us start from k = 1 dN 1 ˆ K N1N dt to be solved subject to N1(0) = N0. Using (13.66) and solving (13.68) gives N 1 …t† ˆ

N0 ‰1 ‡ …t=τc †Š2

(13.68)

555

COAGULATION

Similarly, solving dN 2 1 ˆ K N 21 dt 2

K N2N

(13.69)

subject to N2(0) = 0 gives N 2 …t† ˆ

N 0 …t=τc † ‰1 ‡ …t=τc †Š3

(13.70)

Continuing, we find N k …t† ˆ

N 0 …t=τc †k

1

‰1 ‡ …t=τc †Šk‡1

;

k ˆ 1; 2; . . .

(13.71)

Note that when t/τc  1, it follows that N k …t† ≅ N 0

τc t

2

so the number concentration of each k-mer decreases as t 2. In the short time period during which t/τc  1 N k …t† ≅ N 0

t τc

k 1

and Nk increases as tk 1. 13.3.3.2 Continuous Coagulation Equation We now consider the solution to the continuous coagulation equation (13.61) assuming a constant coagulation coefficient K(q, v) = K, assuming that v0  0: v @ n…v; t† 1 ˆ K n…v 2 ∫0 @t

q; t† n…q; t† dq

Kn…v; t† N…t†

(13.72)

In order to solve (13.72) we also need to know N(t). We have seen that the total number concentration N(t) for any aerosol distribution assuming constant coagulation coefficient is given by (13.66). Substi­ tuting this expression for N(t) into (13.72), we find that the continuous coagulation equation becomes K N0 1 v @ n…v; t† n…v; t† ˆ K n…v ‡ 2 ∫0 1 ‡ t=τc @t

q; t†n…q; t†dq

(13.73)

Let us solve (13.73) assuming the initial distribution n…v; 0† ˆ A exp… Bv†

(13.74)

where A and B are parameters. For this particular initial distribution, an analytical solution of (13.73) can be determined. In this case the total number and volume concentrations are initially N0 ˆ

1

∫0

n…v; 0† dv ˆ

A B

(13.75)

556

DYNAMICS OF AEROSOL POPULATIONS

and V0 ˆ

1

∫0

vn…v; 0† dv ˆ

A B2

(13.76)

Combining (13.75) and (13.76) we find that the parameters A and B are related to the initial aerosol number and volume concentrations by A ˆ N 20 =V 0 and B = N0/V0; therefore the distribution given by (13.74) is equivalent to n…v; 0† ˆ

N 02 exp V0

vN 0 V0

(13.77)

To solve (13.73) with the initial condition given by (13.77), we can either use an integrating factor and an assumed form of the solution or attack it directly with the Laplace transformation. Both approaches are illustrated in Appendix 13.2. The desired solution is then n…v; t† ˆ

N 20 exp V 0 …1 ‡ t=τc †2

vN 0 V 0 …1 ‡ t=τc †

(13.78)

This size distribution is shown in Figure 13.6. The exponential shape of the distribution persists, but while the number of small particles decreases with time, the availability of larger particles slowly increases. For additional solutions of the discrete and continuous coagulation equations, the interested reader may wish to consult Drake (1972), Mulholland and Baum (1980), Tambour and Seinfeld (1980), and Pilinis and Seinfeld (1987).

FIGURE 13.6 Solution of the continuous coagulation equation for an exponential initial distribution given by (13.78). The following parameters are used: N0 = 106 cm 3, V0 = 4189 μm3 cm 3, and τc = 2000 s.

557

THE DISCRETE GENERAL DYNAMIC EQUATION

13.4 THE DISCRETE GENERAL DYNAMIC EQUATION A spatially homogeneous aerosol of uniform chemical composition can be fully characterized by the number concentrations of particles of various sizes Nk(t) as a function of time. These particle concentra­ tions may undergo changes due to coagulation, condensation, evaporation, nucleation, emission of fresh particles, and removal. The dynamic equation governing Nk(t) for k  2 can be developed as follows. Consider first the contribution of coagulation to this concentration change. We showed in Section 13.3.2 that the production rate of k-mers due to coagulation of j-mers with (k j)-mers is given by 1 2

k 1

Kj;k j N j N k

j

jˆ1

where Kj,k j is the corresponding coagulation coefficient for these collisions. The loss rate of k-mers due to their coagulation with all other particles is given by (13.58) as 1

Kk;j N j

Nk jˆ1

To calculate the rate of change of Nk(t) due to evaporation, let γ´k be the flux of monomers per unit area leaving a k-mer. Then, the overall rate of escape of monomers from a k-mer, γk, with surface area ak will be given by γk ˆ γ´k ak

(13.79)

The rate of loss of k-mers from evaporation can then be written as γk N k ;

k2

The rate of formation of k-mers by evaporation of (k + 1)-mers is then just γk‡1 N k‡1 ;

k2

Note that if a dimer dissociates, then two monomers are formed and the formation rate of monomers will be equal to 2γ2N2. We can account for this introducing the Krönecker delta, δj,k, defined by δj;k ˆ 1; j ˆ k δj;k ˆ 0; j ≠ k and generalizing the formation rate of k-mers by evaporation as …1 ‡ δ1;k †γk‡1 N k‡1 To account for the process of accretion of monomers by other particles, we define pkNk as the rate of gain of (k + 1)-mers due to collisions of k-mers with monomers, where pk (s 1) is the frequency with which a monomer collides with a k-mer. This is the process we commonly refer to as condensation. Note that then pk ˆ K1;k N 1

(13.80)

and the rate of gain of k-mers will be pk 1 N k

1

ˆ K1;k 1 N 1 N k

1

558

DYNAMICS OF AEROSOL POPULATIONS

The loss rate of k-mers due to the addition of a monomer and subsequent growth will then be pk N k ˆ K1;k N 1 N k Summarizing, the rate of change of the k-mer concentration accounting for coagulation, condensa­ tion, and evaporation will be given by 1 dN k ˆ dt 2

1

k 1

Kj;k j N j N k

j

Nk

Kk;j N j

(13.81)

jˆ1

jˆ1

‡pk 1 N k

…pk ‡ γk †N k ‡ γk‡1 N k‡1 …1 ‡ δ1;k †

1

In the preceding formulation of the discrete general dynamic equation it is implicitly assumed that the smallest particle is of size k = 2. No distinction has yet been made among the processes of coagulation, homogeneous nucleation, and heterogeneous condensation; they are all included in (13.81). In reality, there is a minimum number of monomers in a stable nucleus g∗, and generally g∗  2. In the presence of a supersaturated vapor, stable clusters of size g∗ will form continuously at a rate given by the theory of homogeneous nucleation. Let us denote the rate of formation of stable clusters containing g∗ monomers from homogeneous nucleation as J0(t). Then coagulation, heterogeneous condensation of vapor on particles of size k  g∗, and nucleation of g∗-mers become distinct processes in (13.81). By changing the smallest particle from 2 to g∗ and adding emission and removal terms, we find that (13.81) becomes k g∗

dN k 1 ˆ Kj;k j N j N k dt 2 jˆg∗

1 j

Nk jˆg∗

Kk;j N j ‡ pk 1 N k

…pk ‡ γk †N k ‡ γk‡1 N k‡1 ‡ J 0 …t†δg∗ ;k ‡ Sk

1

(13.82) Rk

k ˆ g∗ ; g∗ ‡ 1; . . . where Sk is the emission rate of k-mers by sources and Rk is their removal rate. Note that in (13.82) the nucleation term is used only in the equation for the g∗-mers, and it is zero for all other particles.

13.5 THE CONTINUOUS GENERAL DYNAMIC EQUATION Although (13.82) is a rigorous representation of the system, it is impractical to deal with discrete equations because of the enormous range of k. Thus it is customary to replace the discrete number concentration Nk(t) (cm 3) by the continuous size distribution function n(v, t) (μm 3 cm 3), where v = kv1, in which v1 is the volume associated with a monomer. Thus n(v, t)dv is defined as the number of particles per cubic centimeter having volumes in the range from v to v + dv. If we let v0 = g∗v1 be the volume of the smallest stable particle, then (13.82) becomes in the limit of a continuous distribution of sizes v v0 @ n…v; t† 1 ˆ K…v 2 ∫ v0 @t

1

q; q†n…v

q; t†n…q; t†dq 2

@ @ ‰I 1 …v†n…v; t†Š ‰I 0 …v†n…v; t†Š ‡ @ v2 @v

‡ J 0 …v†δ…v

v0 † ‡ S…v†

R…v†

n…v; t† K…q; v†n…q; t†dq ∫ v0 (13.83)

559

THE CONTINUOUS GENERAL DYNAMIC EQUATION

where I 0 …v† ˆ v1 …pk I 1 …v† ˆ

γk †

(13.84)

v21 …p ‡ γk † 2 k

(13.85)

are the condensation/evaporation-related terms. Since pk γk is the frequency with which a k-mer experiences a net gain of one monomer, I0(v) is the rate of change of the volume of a particle of size v = kv1. The sum pk + γk is the total frequency by which monomers enter and leave a k-mer. I1(v) assumes the role of a diffusion coefficient from kinetic theory. Brock (1972) and Ramabhadran et al. (1976) have shown that the term in (13.83) involving I1 can ordinarily be neglected. We shall do so and simply write I0 as I, which can be calculated using (13.4). In (13.83), a stable particle has been assumed to have a lower limit of volume of v0. From the standpoint of the solution of (13.83), it is advantageous to replace the lower limits v0 of the coagulation integrals by zero. Ordinarily this does not cause any difficulty, since the initial distribution n(v, 0) can be specified as zero for v < v0, and no particles of volume v < v0 can be produced for t > 0. Homogeneous nucleation provides a steady source of particles of size v0 according to the rate defined by J0(t). Then the full equation governing n(v, t) is as follows: @ n…v; t† 1 v ˆ K…v 2 ∫0 @t

q; q†n…v

q; t†n…q; t†dq

1

n…v; t†

∫0

‡ J 0 …v†δ…v

@ ‰I…v†n…v; t†Š @v R…v†

(13.86)

K…q; v†n…q; t†dq v0 † ‡ S…v†

This equation is called the continuous general dynamic equation for aerosols (Gelbard and Seinfeld 1979). Its initial and boundary conditions are n…v; 0† ˆ n0 …v†

(13.87)

n…0; t† ˆ 0

(13.88)

In the absence of nucleation (J0(v) = 0), sources (S(v) = 0), sinks (R(v) = 0), and growth [I(v) = 0], we have the continuous coagulation equation (13.61). If particle concentrations are sufficiently small, coagulation can be neglected. If there are no sources or sinks of particles then the general dynamic equation is simplified to the condensation equation (13.7). Table 13.4 compares the properties of coagulation and condensation with respect to the total number of particles and the total particle volume. Coagulation reduces particle number, but the total volume remains constant. Condensation, since it involves gas-to-particle conversion, increases total particle volume, but the total number of particles remains constant. If both processes are occurring

TABLE 13.4 Properties of Coagulation, Condensation, and Nucleation Process

Number Concentration

Volume Concentration

Coagulation Condensation Nucleation Coagulation and condensation

Decreases No change Increases Decreases

No change Increases Increases Increases

560

DYNAMICS OF AEROSOL POPULATIONS

simultaneously, total particle number decreases and total particle volume increases. Detailed studies of the interplay between coagulation and condensation have been presented by Ramabhadran et al. (1976) and Peterson et al. (1978), to which we refer the interested reader.

APPENDIX 13.1

ADDITIONAL MECHANISMS OF COAGULATION

13A.1 Coagulation in Laminar Shear Flow Particles in a fluid in which a velocity gradient du/dy exists have a relative motion that may bring them into contact and cause coagulation (Figure 13A.1). Smoluchowski in 1916 first studied this coagulation type assuming a uniform shear flow, no fluid dynamic interactions between the particles, and no Brownian motion. This is a simplification of the actual physics since the particles affect the shear flow and the streamlines have a curvature around the particles. Smoluchowski showed that for a velocity gradient Γ = du/dy perpendicular to the x axis, the coagulation rate is LS N1N2 ˆ J coag ˆ K12

Γ …Dp1 ‡ Dp2 †3 N 1 N 2 6

(13A.1)

The relative magnitudes of the laminar shear and Brownian coagulation coefficients for equal-size 3 particles in the continuum regime are given by the ratio KLS 12 =K 12 ˆ …Γμ=2kT†Dp . Good agreement between this shear coagulation theory and experiments was reported by Swift and Friedlander (1964). From this ratio, we find that high shear rates are required to make the shear coagulation rate comparable to the Brownian even for particles of size around 1 μm. For example, for Dp = 2 μm, a Γ  60 s 1 is necessary to make the two rates equal.

13A.2 Coagulation in Turbulent Flow Velocity gradients in turbulent flows cause relative particle motion and induce coagulation just as in laminar shear flow. The difficulty in analyzing the process rests with identifying an appropriate velocity gradient in the turbulence. In fact, a characteristic turbulent shear rate at the small lengthscales applicable to aerosols is εk/ν, where εk is the rate of dissipation of kinetic energy per unit mass and ν is the kinematic viscosity of the fluid (Tennekes and Lumley 1972). The analysis by Saffman and Turner (1956) gives the coagulation coefficient for turbulent shear as KTS 12 ˆ

πεk 120ν

1=2

…Dp1 ‡ Dp2 †3

FIGURE 13A.1 Schematic illustrating collisions of particles in shear flow.

(13A.2)

561

ADDITIONAL MECHANISMS OF COAGULATION

FIGURE 13A.2 Comparison between coagulation mechanisms for a particle of 1 μm radius as a function of the particle radius of the second interacting particle.

The relative magnitudes of the turbulent shear and Brownian coagulation coefficients for equal-size particles in the continuum regime is given by the ratio 3μ…πεk =120ν† KTS 12 ˆ K12 kT

1=2

D3p

(13A.3)

Typical measured values of (εk/ν)1/2 are on the order of 10 s 1, so turbulent shear coagulation is significantly slower than Brownian for submicrometer particles, and the two rates become approxi­ mately equal for particles of about 5 μm in diameter (Figure 13A.2). The calculations indicate that coagulation by Brownian motion dominates the collisions of submicrometer particles in the atmosphere. Turbulent shear contributes to the coagulation of large particles under conditions characterized by intense turbulence.

13A.3 Coagulation from Gravitational Settling Coagulation results in a settling particle population when heavier particles move faster, catching up with lighter particles, and collide with them. This process will be discussed again in Chapter 20, where we will be interested in the scavenging of particles by falling raindrops. Suffice it to say at this point that the coagulation coefficient is proportional to the product of the target area, the relative distance

562

DYNAMICS OF AEROSOL POPULATIONS

swept out by the larger particle per unit time, and the collision efficiency between particles of sizes Dp1 and Dp2 π 2 KGS 12 ˆ …Dp1 ‡ Dp2 † …vt1 4

vt2 †E…Dp1 ; Dp2 †;

(13A.4)

where vt1 and vt2 are the terminal settling velocities for the large and small particles, respectively. The collision efficiency E(Dp1, Dp2) is discussed in Chapters 17 and 20. Particles with equal diameters have the same terminal settling velocities and KGS 12 approaches zero as Dp1 approaches Dp2 (Figure 13A.2). Gravitational coagulation can be neglected for submicrometer particles but becomes significant for particle diameters exceeding a few micrometers.

13A.4 Brownian Coagulation and External Force Fields In our discussion of Brownian motion and the resulting coagulation, we did not include the effect of external force fields on the particle motion. Here we extend our treatment of coagulation to include interparticle forces. The force F12 between primary and secondary particles can be written as the negative of the gradient of a potential ϕ: F12 ˆ rϕ

(13A.5)

Now let us assume that ϕ = ϕ(r), where r is the distance between the particle centers. Then one can show that the steady-state coagulation flux is given by J 12 ˆ

1 Brownian N1 N2 K W 12

(13A.6)

where the correction factor W is a result of the interparticle force as represented by the potential ϕ(r) and can be calculated by W ˆ …Rp1 ‡ Rp2 †

1

1 ϕ…x† exp dx 2 ∫ Rp1 ‡Rp2 x kT

(13A.7)

Let us now apply the above theory to two specific types of interactions: van der Waals forces and Coulomb forces. 13A.4.1 Van der Waals Forces Van der Waals forces are the result of the formation of momentary dipoles in uncharged, nonpolar molecules. They are caused by fluctuations in the electron cloud, which can attract similar dipoles in other molecules. The potential of the attractive force can be expressed as ϕv ˆ 4ϕ1

ϕ2 r

6

(13A.8)

where ϕ1 and ϕ2 are constants with units of energy and length, respectively, that depend on the particular species involved. Values of these constants have been tabulated by Hirschfelder et al. (1954). The van der Waals potential between two spherical particles of radii Rp1 and Rp2 whose centers are separated by a distance r is (Hamaker 1937) ϕv …r† ˆ

4π2 ϕ1 ϕ62 6v2m r2

2Rp1 Rp2 …Rp1 ‡ Rp2 †

‡ 2

2Rp1 Rp2 r2

…Rp1

Rp2 †

‡ ln 2

r2

…Rp1 ‡ Rp2 †2

r2

…Rp1

Rp2 †2

(13A.9)

563

ADDITIONAL MECHANISMS OF COAGULATION

where vm is the molecular volume. Let us investigate the case of equal-size particles, Rp1 = Rp2. Then, if s = 2Rp/r, we have ϕv …s† ˆ

4π2 ϕ1 ϕ62 s2 s2 ‡ ‡ ln…1 6v2m 2…1 s2 † 2

s2 †

(13A.10)

Using (13A.7), the correction factor Wv for the van der Waals force between equal-size particles is Wv ˆ

1

∫0

exp

x2 x2 2π2 ϕ1 ϕ62 ‡ ‡ ln…1 2 2 3vm kT 2…1 x † 2

x2 †

dx

(13A.11)

The correction factor Wv is independent of particle size for identically sized particles and depends only on the van der Waals parameters ϕ1 and ϕ2 and the temperature T. Wv can be evaluated as a function of Q 4ϕ1 ϕ62 ˆ kT v2m kT

(13A.12)

by numerically evaluating the integral of (13.A.11). The results of this integration are shown in Figure 13A.3. Note that a 50% correction is necessary for a Q/kT value of roughly 10. The Hamaker constant is defined as Aˆ

4π2 ϕ1 ϕ62 v2m

(13A.13)

For particles of radii Rp1 and Rp2, the continuum correction factor is given by (13A.7) (Friedlander 1977; Schmidt-Ott and Burtscher 1982) W v ˆ …Rp1 ‡ Rp2 †

1

1 ϕ …r† exp v dr ∫ Rp1 ‡Rp2 r2 kT

FIGURE 13A.3 Correction factor to the Brownian coagulation coefficient for equal-size particles in the continuum regime from van der Waals forces.

(13A.14)

564

DYNAMICS OF AEROSOL POPULATIONS

Marlow (1981) has extended this development to the transition and free-molecule regimes and determined that the effect of van der Waals forces on aerosol coagulation rates can be considerably more pronounced in these size ranges than in the continuum regime. 13A.4.2 Coulomb Forces Charged particles may experience either enhanced or retarded coagulation rates depending on their charges. The potential energy of interaction between two particles containing z1 and z2 charges (including sign) whose centers are separated by a distance r is ϕc ˆ

z1 z2 e 2 4πε0 εr

(13A.15)

where e is the electronic charge, ε0 is the permittivity of vacuum (8.854 × 10 12 F m 1), and ε is the dielectric constant of the medium. For air at 1 atm, ε = 1.0005. Substituting (13.A.15) into (13.A.7) and evaluating the integral Wc ˆ

eκ

1

(13A.16)

κ

where κˆ

z1 z2 e 2 4πε0 ε…Rp1 ‡ Rp2 †kT

(13A.17)

The constant κ can be interpreted as the ratio of the electrostatic potential energy at contact to the thermal energy kT. For like charge, κ is greater than zero, so Wc > 1 and the coagulation is retarded from that of pure Brownian motion. Conversely, for κ < 0 (unlike charges), Wc < 1 and the coagulation rate is enhanced. If we consider a situation where there are equal numbers of positively and negatively charged particles, then we can estimate the coagulation correction factor as the average of the like—and unlike— correction factors. This estimate is shown in Table 13A.1. We see that an aerosol consisting of an equal number of positively and negatively charged particles will exhibit an overall enhanced rate of coagulation. The enhanced rate of unlike charged particles more than compensates for the retarded rate of like charged particles. Pruppacher and Klett (1997) show that the average number of charges on an atmospheric particle at equilibrium z, regardless of sign, is given by zˆ

1 Dp kT e π

(13A.18)

TABLE 13A.1 Coagulation Correction Factors for Coulomb Forces between Equal-Size Particles κ 0.1 0.25 0.5 0.75 1.0 5.0 10.0

W c 1 (like)

W c 1 (unlike)

1 W 1 …like† ‡ W 1 …unlike† c c 2

0.9508 0.8802 0.7708 0.6714 0.5820 0.0339 0.00045

1.0508 1.1302 1.2708 1.4214 1.5820 5.0339 10.00045

1.0008 1.0052 1.0208 1.0464 1.0820 2.5339 5.00045

565

ADDITIONAL MECHANISMS OF COAGULATION

resulting in fewer than 10 charges even for a 10-μm particle. For these particles κ is on the order of 10 11 or smaller, so the correction factor Wc is equal to unity. Consequently, it is not expected that Coulomb forces will be important in affecting the coagulation rates of atmospheric particles. Only if an aerosol is charged far in excess of its equilibrium charge described by (13A.18) will there be any effect on the coagulation coefficient. 13A.4.3 Hydrodynamic Forces Fluid mechanical interactions between particles arise because a particle in motion in a fluid induces velocity gradients in the fluid that influence the motion of other particles when they approach its vicinity. Because the fluid resists being “squeezed” out from between the approaching particles, the effect of socalled viscous forces is to retard the coagulation rate from that in their absence. In our calculations of the common diffusivity D12 as D12 = D1 + D2, we have implicitly assumed that each particle approaches the other oblivious of the other’s existence. In reality the velocity gradients around each particle influence the motion of an approaching particle. Spielman (1970) proposed that these effects can be considered by introducing a corrected diffusion coefficient D´ 12 . Alam (1987) has obtained an analytical solution for the ratio D12 =D´ 12 that closely approximates Spielman’s solution: 2:6Rp1 Rp2 Rp1 Rp2 D12 ˆ 1‡ ´ 2 D 12 …r† …Rp1 ‡ Rp2 † …Rp1 ‡ Rp2 †…r Rp1 Rp1 Rp2 ‡ …Rp1 ‡ Rp2 †…r Rp1

1=2

Rp2 †

(13A.19)

Rp2 †

Note that D´ 12 ≅ D12 for sufficiently large separation r and D´ 12 decreases as the particle separation decreases. Figure 13A.4 shows that the enhancement due to van der Waals forces is decreased when viscous forces are included in the calculations. For some values of the Hamaker constant there is an overall retardation of the coagulation rate due to the viscous forces.

FIGURE 13A.4 Enhancement to Brownian coagulation when van der Waals and viscous forces are included in the calculation.

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DYNAMICS OF AEROSOL POPULATIONS

FIGURE 13A.5 Coagulation coefficient at 300 K, ρp = 1 g cm 3, A/kT = 20 as a function of Knudsen number (Knp) for a particle radii ratios of 1, 2, and 5 in both presence and absence of interparticle forces.

Alam (1987) has proposed the following interpolation formula for hydrodynamic and van der Waals forces that is a function of a particle Knudsen number Knp and attains the proper limiting forms in the continuum and kinetic regimes. The result is

K12 ˆ 4π…Rp1 ‡ Rp2 †D12 D12 ˆ D1 ‡ D2 ;

Wc 1 1 ‡ W k Knp δ=W c

δˆ

4D12 c λ21 ‡ λ22

1=2

λ2 ‡ λ22 Knp ˆ 1 ; Rp1 ‡ Rp2

cˆ

(13A.20)

1=2

8kT 8kT ‡ πm1 πm2

1=2

W c 1 and W k 1 are the enhancement factors for van der Waals and viscous forces in the continuum and kinetic regimes, respectively. Figure 13A.5 shows the coagulation coefficient K at 300 K, 1 atm, ρp = 1 g cm 3, and A/kT = 20 as a function of Knp for particle radii ratios of 1, 2, and 5 both in the presence and in the absence of interparticle forces. Figure 13A.6 shows the enhancement over pure Brownian coagulation as a function of particle Knudsen number for A/kT = 20. In the continuum regime (Kn  1) there is a retardation of coagulation because of the viscous forces. As shown in Figure 13A.6 for sufficiently large Hamaker constant A, the effect of van der Waals forces overtakes that of viscous forces to lead to an enhancement factor greater than 1. In the kinetic regime, coagulation is always enhanced because of the absence of viscous forces.

567

SOLUTION OF (13.73)

FIGURE 13A.6 Enhancement over Brownian coagulation as a function of particle Knudsen number for A/kT = 20. In the continuum regime there is a retardation of coagulation because of the viscous forces.

APPENDIX 13.2 SOLUTION OF (13.73) We solve (13.73) subject to (13.77) using the integrating factor method. The integrating factor will be given by the exponential of the integral of the term multiplying n(v, t) on the left-hand side of (13.73): t

exp

KN 0 dt´ ∫ 0 1 ‡ t´ =τc

ˆ

1‡

t τc

2

(13A.21)

If we multiply both sides of (13.73) with the factor calculated in (13A.21) and combine the resulting terms of the left-hand side, we find that @ @t

If we let y ˆ N 0 1 …1 ‡ t=τc †

1‡

1

1 τc

2

1 t n…v; t† ˆ K 1 ‡ τc 2

2 v

∫0

n…v

q; t†n…q; t†dq

(13A.22)

and w = (1 + t/τc)2n(v, t), then (13.A.22) can be transformed to v @ w…v; y† ˆ w…v ∫0 @y

q; y†w…q; y†dq

(13A.23)

As a solution let us try w(v, y) = a exp( bv), where a and b are to be determined, and we find that w…v; y† ˆ a exp… ayv†

(13A.24)

568

DYNAMICS OF AEROSOL POPULATIONS

so returning to the original variables …1 ‡ t=τc †2 n…v; t† ˆ a exp

av N 0 …1 ‡ t=τc †

(13A.25)

av N0

(13A.26)

For t = 0 (13.A.25) is simplified to n…v; 0† ˆ a exp

and comparing it with the initial condition given by (13.77) we find that a ˆ N 20 =V 0 . Therefore the solution of our problem is (13.78) n…v; t† ˆ

N 02 exp V 0 …1 ‡ t=τc †2

vN 0 V 0 …1 ‡ t=τc †

(13A.27)

PROBLEMS 13.1A Derive the general dynamic equation describing the evolution of the aerosol size distribution based on diameter n(Dp, t). 13.2A By what factor does the average particle size of tobacco smoke increase as a result of coagulation during the 2 s that it takes for the smoke to travel from the cigarette to the smoker’s lungs? Assume that the inhaled concentration is 1010 cm 3 and that the initial aerosol diameter is 20 nm. 13.3B Estimate the approximate coagulation lifetimes of 0.01 μm and 0.1 μm particles in an urban atmosphere. Table 8.3 may be used as a source of urban size distribution parameters. 13.4B Show that for an aerosol population undergoing growth by condensation only with no sources or sinks: a. The total number of particles is constant. b. The average particle size increases with time. c. The total aerosol volume increases with time. 13.5B Show that the solution of the coagulation equation with a constant coagulation coefficient, including first-order particle removal @ n…v; t† 1 v ˆ K n…v 2 ∫0 @t

v; t†n…v; t†dv

Kn…v; t†N…t†

R…t†n…v; t†

with n0(v) = (N0/v0) exp( v/v0) is (Williams 1984) N0 n…v; t† ˆ e v0 where P(t) = R(t) + KN(t).

t v=v0

exp

∫0

´

P…t †

vN 0 K exp 2v0

t´

∫0

P…t´´ †dt´´

dt´

569

PROBLEMS

13.6B Given particle growth laws expressed in terms of particle diameter for the three modes of gas-to­ particle conversion hd D p 1 hs hv D p

dDp ˆ dt

diffusion surface reaction volume reaction

show that an aerosol having an initial size distribution n0(Dp) evolves under the three growth mechanisms according to Dp2

n0 n…Dp ; t† ˆ

2hd t

n0 …Dp hs t† n0 Dp e hv t e

Dp

1=2

D2p

2hd t

1=2

hv t

13.7B Consider the steady-state size distribution of an aerosol subject to growth by condensation, sources, and first-order removal. The steady-state size distribution in this situation is governed by d …I…v†n…v†† ˆ S…v† dv

R…v†n…v†

a. Show that n…v† ˆ

1 v ´ S…v †exp I…v† ∫ 0

b. Taking I(v) = hava, R(v) = Rmvm, and S(v) = S0δ(v n…v† ˆ

S0 exp va ha

Rm ha …m ‡ 1

R…v´´ † ´´ dv dv´ ∫ v´ I…v´´ † v

v0), show that

a†

vm‡1

a

vm‡1 0

a

valid for v  v0. 13.8B It is of interest to compare the relative magnitudes of processes that remove or add particles from one size regime to another in an aerosol size spectrum. Let us assume that a “typical” atmospheric aerosol size distribution can be divided into three size ranges, 0.01 μm < Dp < 0.1 μm, 0.1 μm < Dp < 1.0 μm, and 1 μm < Dp < 10 μm, in which the particle number concentrations are assumed to be as follows: 0.01 μm < Dp < 0.1 μm 105 cm 3

0.1 μm < Dp < 1.0 μm 102 cm 3

1.0 μm < Dp < 1.0 μm 10 1 cm 3

It is desired to estimate the relative contributions of the processes listed below to the rates of change of the aerosol number concentrations in each of the three size ranges: 1. Coagulation a. Brownian b. Turbulent shear c. Differential sedimentation 2. Heterogeneous condensation For the condition of the calculation assume air at T = 298 K, p = 1 atm. For the purposes of the coagulation calculation, assume that the particles in the three size ranges have diameters 0.05 μm, 0.5 μm, and 5 μm. For turbulent shear coagulation, assume εk = 1000 cm2 s 3. In the differential

570

DYNAMICS OF AEROSOL POPULATIONS

TABLE 13P.1 Estimated Rates of Change of Aerosol Number Densities (cm − 3 s − 1) Size range 0:01 μm < Dpi < 0:1 μma

Process

0:1 μm  Dpi  1:0 μmb

1:0 μm < Dpi < 10 μmc

Coagulation Brownian Turbulent Sedimentation Heterogeneous condensation S = 1.0 S = 2.0 S = 10.0 Assuming that Dpi ˆ 0:05 μm for all particles, and N = 105 cm 3. Assuming that Dpi ˆ 0:5 μm for all particles, and N = 102 cm 3. c Assuming that Dpi ˆ 5 μm for all particles, and N = 10 1 cm 3. a

b

sedimentation calculation, assume that the particles with which those in the three size ranges collide have Dp = 10 μm with a number concentration of 10 1 cm 3. For heterogeneous condensa­ tion, the flux of particles into and out of the size ranges can be estimated by calculating the rate of change of the diameter, dDp/dt, assuming zero vapor pressure of the condensing vapor species and assuming that the number concentration in each size range is uniform across the range. In addition, assume that the condensing species has the properties of sulfuric acid: D MA p°A ρp

ˆ ˆ ˆ ˆ

0:07 cm2 s 1 98 g mol 1 1:3  10 4 Pa 1:87 g cm 3

Examine saturation ratio values, S ˆ pA =pA °…T†, of 1.0, 2.0, and 10.0. Show all calculations and carefully note any assumptions you make. Present your results in Table 13P.1. 13.9C We want to explore the dynamics of aerosol size distributions undergoing simultaneous growth by condensation and removal at a rate dependent on the aerosol concentration, with a continuous source of new particles. The size distribution function in such a case is governed by @ n…v; t† @ ‡ …I…v; t†n…v; t†† ˆ S…v; t† @v @t

R…v; t†n…v; t†

where R(v, t) is the first-order removal constant. Let us assume that I(v, t) = hava, R(v, t) = Rmvm, S(v, t) = S0δ(v v0), and n0(v) = N0Δ(v v∗). a. Show that under these conditions n…v; t† ˆ

Rm ha …m ‡ 1

S0 exp va ha

Rm ha …m ‡ 1

‡ N 0 exp 

v1∗

δ v

a

a†

‡ …1 v1∗

a

a†ha t ‡ …1

vm‡1

a

vm‡1 0

a†

m‡1 a=1 a

a†ha t

1=1 a

vm‡1 ∗

a

a

571

REFERENCES

[Hint: The equation for n(v, t) can be solved by assuming a solution of the form n(v, t) = n1(v) + n0(v, t), where n1(v) is the steady-state solution of the equation and n0(v, t) is the transient solution corresponding to the intial condition n0(v) in the absence of the source S(v, t).] b. Express the solution determined in part (a) in terms of the following dimensionless groups θˆ

…1

a†ha t v1∗

a

yˆ

v v0

~ ˆ n

n N

χˆ

v∗ v0

λˆ

‡1 a Rm vm 0 ha …m ‡ 1 a†

ρˆ

N 0 ha va∗ v0 S0

where N is the total number of particles. c. Examine the limiting cases of ρ → 1 and ρ → 0. Show that the proper forms of the solution are obtained in these two cases. d. Plot the steady-state solution n1(y) for a ˆ 13 and 23, m ˆ 23, and λ = 0.001, 0.01, 0.1, 1.0, 10.0. Discuss the behavior of the solutions. e. Plot ñ(y, θ) as a function of y for a ˆ 13, m ˆ 23, λ = 0.01, ρ = 100, and χ = 1.1 for θ = 0.1, 1, 10, and 100. Discuss.

REFERENCES Alam, M. K. (1987), The effect of van der Waals and viscous forces on aerosol coagulation, Aerosol Sci. Technol. 6, 41–52. Brock, J. R. (1972), Condensational growth of atmospheric aerosols, J. Colloid Interface Sci. 39, 32–36. Drake, R. L. (1972), A general mathematical survey of the coagulation equation, in Topics in Current Aerosol Research (Part 2), G. M. Hidy and J. R. Brock (eds.), Pergamon Press, New York, pp. 201–376. Friedlander, S. K. (1977), Smoke, Dust, and Haze: Fundamentals of Aerosol Behavior, Wiley, New York. Fuchs, N. A. (1964), Mechanics of Aerosols, Pergamon Press, New York. Gelbard, F., and Seinfeld J. H. (1979), The general dynamic equation for aerosols—theory and application to aerosol formation and growth, J. Colloid Interface Sci. 68, 363–382. Gryn, V. I., and Kerimov, M. K. (1990), Integrodifferential equations for nonspherical aerosol coagulation, USSR Comput. Math. Math. Phys. 30, 221–224. Hamaker, H. C. (1937), The London–van der Waals attraction between spherical particles, Physica 4, 1058–1072. Hirschfelder, J. O., et al. (1954), Molecular Theory of Gases and Liquids, John Wiley & Sons, Inc., New York. Lee, P. S., and Shaw, D. T. (1984), Dynamics of fibrous type particles: Brownian coagulation and the charge effect, Aerosol Sci. Technol. 3, 9–16. Marlow, W. H. (1981), Size effects in aerosol particle interactions: The van der Waals potential and collision rates, Surf. Sci. 106, 529–537. Mulholland, G. W., and Baum, H. R. (1980), Effect of initial size distribution on aerosol coagulation, Phys. Rev. Lett. 45, 761–763. Peterson, T. W., et al. (1978), Dynamics of source-reinforced, coagulating, and condensing aerosols, J. Colloid Interface Sci. 63, 426–445. Pilinis, C., and Seinfeld, J. H. (1987), Asymptotic solution of the aerosol general dynamic equation for small coagulation, J. Colloid Interface Sci. 115, 472–479. Pruppacher, H. R., and Klett, J. O. (1997), Microphysics of Clouds and Precipitation, 2nd ed., Kluwer, Boston. Ramabhadran, T. E., et al. (1976), Dynamics of aerosol coagulation and condensation, AIChE (Am. Inst. Chem. Eng.) J. 22, 840–851. Rogak, S. N., and Flagan, R. C. (1992), Coagulation of aerosol agglomerates in the transition regime, J. Colloid Interface Sci. 151, 203–224. Saffman, P. G., and Turner, J. S. (1956), On the collision of drops in turbulent clouds, J. Fluid Mech. 1, 16–30. Schmidt-Ott, A., and Burtscher H. (1982), The effect of van der Waals forces on aerosol coagulation, J. Colloid. Interface. Sci. 89, 353–357.

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DYNAMICS OF AEROSOL POPULATIONS Spielman, L. (1970), Viscous interactions in Brownian coagulation, J. Colloid Interface Sci. 33, 562–571. Swift, D. L., and Friedlander, S. K. (1964), The coagulation of hydrosols by Brownian motion and laminar shear flow, J. Colloid Sci. 19, 621–647. Tambour, Y., and Seinfeld, J. H. (1980), Solution of the discrete coagulation equation, J. Colloid Interface Sci. 74, 260–272. Tennekes, H., and Lumley, J. O. (1972), A First Course in Turbulence, MIT Press, Cambridge, MA. Williams, M. M. R. (1984), On some exact solutions of the space- and time-dependent coagulation equation for aerosols, J. Colloid. Interface Sci. 101, 19–26.

CHAPTER

14

Atmospheric Organic Aerosols

For thousands of years the smoke produced during incomplete combustion has been the most visible indication of the existence of atmospheric aerosols. The terms soot and smoke have been traditionally used to describe those particles that contain thousands of organic compounds, elemental (or black) carbon, as well as a series of inorganic components (Friedlander 2000). The fact that these primary particles and their sources are easily visible has led both scientists and laypersons to conclude that most organic aerosol in the atmosphere is directly emitted from its sources and is therefore primary. However, even in the early 1950s, some pioneers started considering atmospheric chemistry as a source of organic aerosol. In his landmark paper on urban ozone formation, “Chemistry and physiology of Los Angeles smog,” Haagen-Smit (1952) wrote the following concerning the aerosol formation that was noted to accompany high-ozone episodes: “These effects are especially noticeable with ring compounds having a double bond in the ring, such as cyclohexene, indene, and dicyclopentadiene. In these cases the opening of the ring will yield practically nonvolatile oxidation products. Because of the introduction of several polar groups, the volatility decreases so radically that aerosol formation is inevitable.” A few years later, Went (1960) suggested that oxidation of organic vapors emitted by plants was responsible for the blue haze in the atmosphere above many forested regions. Aerosol produced by the oxidation of volatile organic compounds (VOCs) to low-volatility condensable products was eventually termed secondary organic aerosol (SOA). Organic particulate matter emitted directly as particles is known as primary organic aerosol (POA). Despite these early insights and a few pioneering publications (Doyle and Jones 1963; Goetz and Pueschel 1967; O’Brien et al. 1975a,b), systematic studies of SOA formation started only in the 1980s mainly at the California Institute of Technology and the University of North Carolina smog chambers but also in Japan (Grosjean and Friedlander 1980; Kamens et al. 1981; Leone et al. 1985; Hatakeyama et al. 1985; Stern et al. 1987). These laboratory studies demonstrated that the atmospheric oxidation of organic vapors can create products with low volatility that can then condense into the particulate phase forming SOA. Since ambient measurements generally quantify only the total organic particulate matter, at first there was no clear way to separate the primary from the secondary organic aerosol (OA). Some of the first identified ambient OA components (diacids including adipic and glutaric acid, organonitrates) were clearly of secondary origin (O’Brien et al. 1975b). The relatively high concentrations of these compounds and the high SOA yields in the experiments performed at that time suggested that SOA was the dominant OA component in polluted areas (Grosjean 1984). The first chemical transport models simulating atmospheric OA included only the secondary formation (Pilinis and Seinfeld 1988; Pandis et al. 1992).

Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, Third Edition. John H. Seinfeld and Spyros N. Pandis.  2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

573

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ATMOSPHERIC ORGANIC AEROSOLS

In the early 1990s, the organic tracer approach was developed, in which individual organic molecules are used as fingerprints for major primary organic PM sources. For example, cholesterol in PM is a tracer for meat cooking emissions. Ambient measurements of these compounds were used in combination with the chemical mass balance (CMB) technique (see Chapter 26) to quantify the contributions of the various primary organic PM sources to the total organic PM. The organic aerosol that could not be attributed directly to a source was considered as SOA (Schauer et al. 1996). The technique of elemental carbon (EC) tracer analysis (Turpin and Huntzicker 1991, 1995; Cabada et al. 2004; Polidori et al. 2006) allowed one to track ambient aerosol containing EC back to primary emissions. The use of these observation-based techniques often suggested that most of the OA in polluted regions is primary. The next major development in our understanding of atmospheric OA came from a new instrument: the Aerodyne aerosol mass spectrometer (AMS) (Jayne et al. 2000). The AMS is able to measure the OA concentration and its mass spectrum. Analysis of the spectra suggested that the OA even in heavily polluted areas is much more oxygenated than expected (Alfarra et al. 2004; Zhang et al. 2005a). Zhang et al. (2005b) developed an approach for estimating the contributions of two surrogate components that they named hydrocarbon-like organic aerosol (HOA) and oxygenated organic aerosol (OOA). Comparisons of the HOA mass spectrum with those of vehicle emissions led to a strong association of HOA and POA. Analysis of the AMS data collected at a large number of sites around the world revealed that the OOA component dominated everywhere, even in heavily polluted urban regions (Zhang et al. 2007). At this point it became clear that the pieces of the atmospheric OA puzzle did not fit together and that several of the underlying assumptions about POA and SOA were problematic. For many years it had been assumed implicitly that POA emitted directly as particles from combustion sources such as transportation and biomass burning remained in particulate form once in the atmosphere and served simply as inert nuclei on which the secondary organic components condensed. Eventually it was discovered that most of the POA is actually sufficiently volatile that it evaporates after emission (Robinson et al. 2007). Once the emitted POA is diluted and evaporates, the resulting semivolatile organic vapors can then react in the gas phase with the OH radical and other atmospheric oxidants, forming low-volatility oxidation products that recondense to the particulate phase over timescales of hours to days. This evaporation–reaction–recondensation cycle leads to significant changes in the chemical nature of the primary OA and is one of the reasons why the organic aerosol in large urban areas with considerable organic particulate emissions is dominated by oxygencontaining compounds rather than those characteristic of the original POA emissions. Organic aerosol components can also continue to react in the atmosphere in the gas, particulate, or even aqueous phase. Semivolatile OA components can undergo multiple generations of oxidation by OH in the gas phase, forming more oxygenated lower-volatility compounds, thereby increasing the concentration of SOA. After condensing into the particle phase, the low-volatility products can continue to undergo chemistry. One such route is dimerization; with double the carbon number of the condensing compound, dimers are essentially nonvolatile. Water-soluble organic compounds can dissolve in droplets or moist aerosols wherein they undergo oxidation and oligomerization reactions to products that remain in the aqueous phase as dissolved SOA. When a droplet containing dissolved SOA evaporates, a residual aerosol particle rich in highly oxidized organics remains. As we saw in Chapter 2, elucidating the lifecycles of atmospheric constituents provides a great deal of information: residence times, sources, and sinks. Because of the complex molecular constituency of organic aerosols and the wide variety of their sources and sinks, obtaining closure on the lifecycle of organic aerosol is challenging. In the final section of this chapter, we review current estimates of the lifecycle of organic aerosol.

14.1 CHEMISTRY OF SECONDARY ORGANIC AEROSOL FORMATION Virtually all organic compounds in the atmosphere, whether in the gas phase or the particle phase, are susceptible to oxidation. As we saw in Chapter 6, the principal atmospheric oxidants are the hydroxyl

CHEMISTRY OF SECONDARY ORGANIC AEROSOL FORMATION

FIGURE 14.1 Idealized depiction of SOA formation and evolu­ tion, showing multiple steps of gas-phase and particle-phase reactions. “S” denotes semivolatile compounds; “P” compounds formed in the particle phase (which can be semivolatile, or, in the case of high-molecular-weight species, nonvolatile); and “V” fully volatile compounds (CO2, CO, and light organics). Each reaction is likely accompanied by a change in volatility, the magnitude and sign of which is a strong function of chemical structure and reaction conditions. Only one product from each reaction is shown; in reality most reactions will lead to several products spanning a range of volatilities. Oxidation in the particle phase (oxp) generally involves different mechanisms than in the gas phase (oxg). (Source: Kroll, J., and Seinfeld, J. H., Atmos. Environ. 42, 3593 (2008), Figure 12.)

radical (OH), ozone (O3), and the nitrate radical (NO3). The oxidation of an organic molecule leads to the formation of functional groups, such as hydroxyl ( OH), carbonyl ( C O), nitrooxy ( ONO2), hydroperoxy ( OOH), carboxyl ( C(O)OH), and ester ( C(O)OR). This so-called functionalization pathway can lead to products that are more polar and less volatile than the parent VOC with a greater tendency to condense into the particle phase. It was originally thought that only relatively large VOCs, such as aromatics, long-chain alkanes, and biogenic terpenes, could, on atmospheric oxidation, produce compounds of sufficiently low volatility to condense into the particle phase. The discovery that atmospheric oxidation of isoprene (C5H8) leads to SOA revealed that even small organic molecules, on sufficient oxidation, are organic aerosol precursors. An idealized reaction scheme of SOA formation from a parent VOC is depicted in Figure 14.1. In SOA formation several generations of gas-phase oxidation generally occur, with each generation producing compounds of decreasing volatility. The simplified scheme in Figure 14.1 shows only a single product from each oxidation step. The time dependence of SOA formation, as observed in a laboratory chamber, can be visualized by growth curves, the mass of SOA formed versus the mass of precursor VOC reacted (Figure 14.2). Shown in Figure 14.2a are SOA growth curves from several α-pinene + O3 chamber experiments, each one carried out with a different initial α-pinene concentration. In each experiment, we see that SOA growth ceases when the α-pinene is completely consumed, and the production of SOA in each experiment follows the same general curve regardless of the initial concentration of α-pinene. This behavior is consistent with the alkene ozonolysis mechanism, in which O3 reacts only with the single double bond in α-pinene. Many of the terpenes have more than one double bond; examples include terpinolene, myrcene, limonene, α-humulene, and β-caryophyllene (Table 2.13). For such molecules, O3 reacts first with one of the double bonds, and the first-generation products, which still have an intact double bond, are then available to react with O3 again. Initial O3 reaction with the endocyclic (in the ring) double bond in terpinolene, for example, leads to oxidized products, and then subsequent reaction of O3 with the

575

576

ATMOSPHERIC ORGANIC AEROSOLS

FIGURE 14.2 Growth curves, plots of SOA growth (COA) versus hydrocarbon reacted over the course of an experiment, from the ozonolysis of (a) α-pinene and (b) terpinolene (Ng et al. 2006). Different symbols denote individual experiments with differing initial hydrocarbon concentrations. Large black circles indicate “final” SOA growth, from which SOA yields are typically determined. The vertical sections (“hooks”) in the terpinolene experiments indicate continued SOA formation after the initial terpinolene is completely reacted, likely ozonolysis of the second double bond. This is an example of multigenerational oxidation. Only one generation of products results from α-pinene ozonolysis, owing to the presence of a single double bond in α-pinene, and SOA is formed promptly. (Source: Kroll, J., and Seinfeld, J. H., Atmos. Environ. 42, 3593 (2008), Figure 9.)

remaining exocyclic (outside the ring) double bond leads to even more oxidized, lower volatility products than those from the first-generation (Figure 14.2b). In contrast to α-pinene, substantial aerosol growth of terpinolene SOA occurs even after the complete consumption of the parent molecule (the vertical section or “hook” at the end of each curve). Such behavior is indicative of the multi-generation nature of the oxidation mechanism. The secondary organic aerosol yield is defined as the ratio of the mass concentration of secondary organic aerosol produced to the mass concentration of the parent VOC reacted, each usually expressed in units of μg m 3. The SOA yield is expressed in terms of mass concentrations of aerosol products and VOC reacted rather than molar quantities because SOA mass is the quantity that can be measured experimentally. Identification and quantification of all the individual compounds present in SOA is generally not possible, so the actual number of moles present in the SOA is unknown. The SOA yields for the systems in Figure 14.2 are determined simply as the ratio of the mass concentration of aerosol at the termination of growth to the mass concentration of the parent VOC reacted.

14.1.1 Oxidation State of Organic Compounds The oxidation of a VOC to yield products of low volatility is the hallmark of SOA generation. Therefore, it is useful to have a means to express the degree to which the molecular products of SOA are oxidized, the oxidation state of the products (Figure 14.3). The oxidation state of carbon can vary widely from 4 in CH4 to +4 in CO2. An approximate expression for the average carbon oxidation state of an organic molecule containing mainly hydrogen and oxygen is as follows (Kroll et al. 2011): OSC ˆ 2…O : C†

…H : C†

CHEMISTRY OF SECONDARY ORGANIC AEROSOL FORMATION

FIGURE 14.3 Oxidation state of organic species. (Courtesy of J. Kroll)

The average carbon oxidation state of SOA can be determined from an analytical method that measures its elemental ratios, O:C and H:C. Figure 14.4 depicts the oxidation of a monoterpene (C10H16) on a carbon number versus carbon oxidation state plot. Gas-phase oxidation of the parent compound that maintains the 10-carbon structure intact and adds oxygen-containing functional groups to the molecule increases the average carbon oxidation state of the molecule. Fragmentation of the oxygen-containing reaction product generally leads to even a higher oxidation state since the O:C ratio increases owing to fewer C atoms in the surviving molecule. Formation of a dimer by addition of two of the 10-carbon oxidized products to each other does not change the molecule’s oxidation state, as long as the O:C and H:C ratios do not change, even though the carbon number doubles to 20. The number of reactions and products in a chemical mechanism that accounts for every possible reaction in oxidation of a VOC grows exponentially with the length of the carbon skeleton (Aumont et al. 2005). For example, a mechanism for the complete OH oxidation of n-octane, C8H18, to CO2 and H2O, involves >106 species, comprising successive reactions involving peroxy and alkoxy radicals as well as possible OH attack on each carbon atom of the array of oxidized products. Indeed, longchain alkanes (Cn, n  10) are an important class of atmospheric VOCs, as constituents of gasoline and products of biomass and fossil fuel combustion (Schauer et al. 2002; Gentner et al. 2012). The long-chain alkanes are well-established precursors to SOA (Lim and Ziemann 2005, 2009; Presto et al. 2009, 2010; Ziemann 2011; Craven et al. 2012; Yee et al. 2012, 2013; Loza et al. 2014). Aumont et al. (2013) describe the atmospheric oxidation of C10–C22 alkanes, both straight- chain and branched, under conditions in which the NO concentration is sufficiently high that peroxy radicals react essentially exclusively with NO. Each peroxy radical-NO reaction leads to an organic nitrate or alkoxy radical. Successive steps lead to various combinations of nitrate, ketone, and hydroxyl groups along the original carbon backbone. Each of the stable products can react, in turn, with OH. [Individual rate constants are rarely available for the avalanche of reactions in a mechanism for species as large as longchain alkanes. Structure–activity relationships (SARs) are generally used to estimate rate constants, carbon atom by carbon atom, for OH reactions and alkoxy radical chemistry; see, for example, Atkinson (2007)

FIGURE 14.4 Schematic of oxidation state and volatility of the products of oxidation of a monoterpene, C10H16.

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ATMOSPHERIC ORGANIC AEROSOLS

FIGURE 14.5 A few early products of atmospheric oxidation of n-nonane under high-NO conditions.

and Vereecken and Peeters (2009).] To illustrate the progression of oxidation state, Figure 14.5 shows a few products from the early steps of OH oxidation of n-nonane (C9H20) in which the original carbon backbone is preserved. From their atomic O:C and H:C ratios the oxidation state of each of the products can be computed. The van Krevelen Plot Dirk Willem van Krevelen (1950), professor of fuel technology at the Technical University of Delft, The Netherlands, developed a graphical plot to assess the origin and maturity of kerogen and petroleum deposits. The diagram is a cross-plot of H:C and O:C atomic ratios of carbon compounds. Because the O:C and H:C ratios of a compound are related to the average carbon oxidation state, the van Krevelen plot (Figure 14.6) was introduced into atmospheric chemistry as a means to compactly display the progressive oxidation of a precursor VOC (Heald et al. 2010a). The slope of data on the van Krevelen plot provides information on the functional group formation in the SOA. A slope of zero is consistent with addition of alcohol or peroxide functional groups to a carbon backbone, in the absence of fragmentation of the molecule. A slope of –1 indicates simultaneous increases in carbonyl and alcohol groups, either on separate carbon atoms or due to the addition of carboxylic acid groups, whereas a slope of 2 corresponds to the addition of aldehyde or ketone functional groups. A slope of 0.5 in the van Krevelen diagram is consistent with the addition of 3 OH groups and 1 C O group or the addition of 2 OH groups and 1 COOH group. If peroxides are formed, the same slope can arise from the addition of 1 OOH group, 1 OH group, and 1 C O group, or the addition of 1 OOH group and 1 COOH group. The replacement of a CH2 group with a COOH group at a C C bond breakage (without loss of oxygen during the fragmentation process) will also result in a slope of 0.5.

CHEMISTRY OF SECONDARY ORGANIC AEROSOL FORMATION

FIGURE 14.6 van Krevelen plot. Dashed lines show effect of adding different functional groups to a compound with no oxygen and an H:C ratio of 2. Addition of alcohol ( OH) or peroxide ( OOH) groups to a carbon atom (replacing an H atom) does not change the H:C ratio of the compound. Addition of a carboxylic acid ( COOH) group (replacing an H atom) leads to a line with a slope of 1. To see this, consider starting with a compound CnH2n. Add one COOH group and remove one H atom. The new compound has H:C = 2n/(n + 1) and O:C = 2/(n + 1). Then ΔH:C = 2n/(n + 1) 2, and ΔO:C = 2/(n + 1). So the slope of the line on the van Krevelen diagram is ΔH:C/ΔO:C = 1. Shown on the plot are lines of constant carbon oxidation state.

While atmospheric organic aerosol formation and evolution can be depicted as movement in the two-dimensional van Krevelen space of O:C and H:C, such a plot is difficult to interpret in terms of underlying chemistry, since the carbon number of the organic molecules is not explicitly repre­ sented. If molecules were to be defined in the three-dimensional chemical space of O:C, H:C, and carbon number nC, any molecular formula CxHyOz can be represented as a unique point (Daumit et al. 2013). Then, if the chemical formula can be related to an estimate of volatility (vapor pressure), the propensity of the molecule to partition to the particle phase can be predicted. (Isomers of such a molecule involving different functional groups can, of course, exist, which will affect their vapor pressures.)

14.1.2 Generation of Highly Oxygenated Species by Autoxidation Central to the chemistry of SOA formation are the processes by which the reaction products become highly oxidized. Virtually all atmospheric VOC oxidation mechanisms involve peroxy (RO2) radicals, the well-established atmospheric chemistry of which involves reaction with NO as well as other peroxy radicals. We noted in Chapter 6 that if peroxy radicals have a sufficiently long lifetime, they can undergo autoxidation chemistry (Crounse et al. 2012, 2013). In this process, an RO2 undergoes sequential intramolecular H-atom abstractions, each followed by O2 addition at the alkyl radical site that remains. With each abstraction–addition step, a hydroperoxide is formed, and a new peroxy radical group arises from O2 addition on the C-atom at which the hydrogen abstraction took place.

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To illustrate this chemistry, Crounse et al. (2013) performed experiments involving OH oxidation of 3-pentanone, CH3CH2C(O)CH2CH3, shown here:

A peroxy radical is formed from the OH abstraction of an H–atom on the second C–atom, followed by O2 addition. That peroxy radical then abstracts a H–atom from the fourth C–atom via the formation of a 6-membered transition state ring. This leaves a hydroperoxide group OOH and an alkyl radical that rapidly adds another O2. This process is repeated as the new peroxy group abstracts the H atom on the second C–atom. Starting with C5H10O, the result is a molecule with C5H8O4, one that has a greatly reduced volatility as compared with its parent molecule. In regions of the atmosphere where NO mixing ratios are below ∼100 ppt, peroxy radicals are calculated to survive for 10s to 100s of seconds before reacting with either NO, HO2, or RO2. In this situation, the sequential H-shift autoxidation mechanism shown above effectively competes with the other channels to lead to increasingly oxygenated radicals. (Even in the absence of hydroperoxide, aldehyde, or alcohol groups, autoxidation may be important for highly branched peroxy radicals that have secondary and tertiary H atoms.) In their study of the formation of extremely-low volatility organic compounds from the O3 and OH oxidation of α-pinene, Ehn et al. (2014) confirmed that autoxidation chemistry is largely responsible for the observed highly oxidized products. Rissanen et al. (2014) proceeded to demonstrate RO2 autoxidation in cyclohexene ozonolysis. Aerosol Mass Spectrometry Advances in the field of atmospheric chemistry have been driven in many respects by advances in analytical chemistry. As noted above, the development of the Aerodyne aerosol mass spectrometer (AMS) (Jayne et al. 2000) facilitated the real-time measurement of the composition of ambient aerosol. The AMS measures the aerosol concentration and size distribution, as well as the mass spectrum of the fragments that result when the individual organic molecules of the aerosol are fragmented by electron impact. Zhang et al. (2007) analyzed 43 Northern Hemisphere AMS datasets (see Figure 8.22) and found that at most sites AMS organic aerosol (OA) spectra can be separated into oxygenated OA (denoted OOA) and hydrocarbon-like OA (denoted HOA) components (Figure 14.7). For many of the datasets, OOA could be further subcategorized into low-volatility OOA (LV-OOA) and semi-volatile OOA (SV-OOA) (Jimenez et al. 2009). LV-OOA has been described as aged OOA, with spectra dominated by mass fragment CO2+ at m/z 44, and SV-OOA, described as “fresh” OOA with a signal at C2H3O+ at m/z 43 as well. The m/z 43 fragment is mainly C2H3O+ for the OOA component and C3H7+ for the HOA component. Mass fragment CO2+ is a marker ion for organic acids (Aiken et al. 2007; Takegawa et al. 2007; Duplissy et al. 2011). Fragment ion C2H3O+, conversely, is expected to form from nonacid oxygen-containing organic compounds. Mass fragments m/z 43 and 44 are the dominant ions in SV- and LV-OOA spectra and represent different functionalities. Ng et al. (2010) plotted measured f44 vs. f43 for all OOA spectra from different sites, where f44 and f43 are the ratios of m/z 44 (mostly CO2+) and m/z 43 to the total OA signal in the spectrum, respectively (Figure 14.8). When f44, the ratio of m/z 44 to the total signal in the component

CHEMISTRY OF SECONDARY ORGANIC AEROSOL FORMATION

FIGURE 14.7 Average mass concentrations of aerosol mass spectrometer– derived HOA and OOA (sum of several OOA types) at sites in the Northern Hemisphere. The winter data of the three urban winter/summer pairs are placed to the right of the summer data and are shown in a lighter shade. Within each category, sites are ordered from left to right as Asia, North America, and Europe. Areas of the pie charts are scaled by the average concentrations of total organics (HOA + OOA). (Source: Zhang, Q., et al., Geophys. Res. Lett. 34, L13801 (2007), Figure 2.)

FIGURE 14.8 Triangle plot. f44 versus f43 for all the oxygenated organic aerosol (OOA) components from 43 ambient aerosol mass spectrometer datasets. In the inset, “HULIS” refers to humic-like substances. The peak of the triangle corresponds to fulvic acid. Humic acid is a principal component of humic substances, which are the major organic constituents of soil, produced by biodegradation of dead organic matter. Humic acid is a complex mixture of many acids, of which fulvic acids are humic acids of lower molecular weight and higher oxygen content than other humic acids. The dotted lines are added to guide the eye and define the triangular space in which ambient OOA components fall (Ng et al. 2010, 2011).

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spectrum, is plotted against f43 (defined similarly), all OA components were found to fall within a triangular region. The base of the triangle, where f44 is at its lowest values, spans the variability in HOA and SV-OOA composition. As the aerosol ages and becomes more oxidized, the data points migrate progressively toward the vertex of the triangular region, and the organic aerosol becomes more chemically similar. LV-OOA data tend to group in the top half of the triangle plot, and SV-OOA in the lower half, with the vertex of the triangle representing essentially an ultimate state of atmospheric organic aerosol oxidation.

14.2 VOLATILITY OF ORGANIC COMPOUNDS The volatility of an organic compound, as measured by its pure compound vapor pressure, is the most important property governing the extent to which it will be transferred to the particulate phase and become OA. The vapor pressure of a molecule is determined largely by its polarity and size. Owing to variations in polarity, the identity of the added functional groups plays a major role in volatility. Table 14.1 shows approximate effects of common functional groups on the vapor pressure of organics (Pankow and Asher 2008). The addition of any oxygen-containing functional group has a greater effect on vapor pressure than a one-carbon increase in the size of the carbon skeleton. This change in vapor pressure is moderate for aldehydes and ketones, which are relatively nonpolar. Larger incremental decreases in volatility result from the addition of hydroxyl, hydroperoxyl, nitrate, and acid groups, each of which may lower compound vapor pressure by over two orders of magnitude. Vapor pressures are even lower for multifunctional compounds, such as diols, and extremely low for dicarboxylic acids. Table 14.2 presents the measured vapor pressures of several OA components at 298 K. Generally, atmospheric compounds will reside in the gas phase for vapor pressures >10 5 atm, be present in both gas and particle phases if vapor pressures are between 10 5 and 10 11 atm, and be largely in the particle phase at vapor pressures ceq;i

(14.4) (14.5)

The total concentration of a product can be calculated using the reacted precursor concentration ΔVOC (in μg m 3) and the reaction stoichiometry ct;i ˆ ai

Mi ΔVOC MVOC

(14.6)

where MVOC is the molecular weight of the parent compound. Combining (14.5) with (14.6), we find that cp;i ˆ ai

Mi ΔVOC MVOC

ceq;i

if

ΔVOC > ΔVOC∗i

(14.7)

where ΔVOC∗i is the threshold reacted precursor concentration for formation of the aerosol compound i calculated from (14.2, 14.6) and (14.7) as ΔVOC∗i ˆ

pi °MVOC ai RT

(14.8)

The lower the vapor pressure of the compound and the higher its molar yield, the smaller the amount of reacted VOC needed to saturate the gas phase and initiate the formation of secondary organic aerosol. For a compound with a vapor pressure of 10 10 atm (0.1 ppb), produced by a VOC with MVOC = 100 g mol 1, and a yield equal to 0.1; ΔVOC∗ = 4.1 μg m 3. Partitioning of Noninteracting SOA Compounds Find the mass fraction of the secondary organic aerosol species i that will exist in the particulate phase, Xp,i, as a function of reacted VOC. Plot the Xp,i versus ΔVOC curves for ai = 0.05, T = 298 K, and MVOC = 150 g mol 1 for saturation mixing ratios equal to 0.1, 1, and 5 ppb.

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IDEALIZED DESCRIPTION OF SECONDARY ORGANIC AEROSOL FORMATION

FIGURE 14.9 Mass fraction of reactive organic gas (ROG) oxidation product in the aerosol phase as a function of the reacted ROG concentration ΔROG and the product saturation mixing ratio. The product is assumed to be insoluble in the aerosol phase. Other conditions are ai = 0.05, T = 298 K, and MVOC = 150 g mol 1.

Defining Xp,i as the mass fraction of the species in the aerosol phase Xp;i ˆ

cp;i ct;i

(14.9)

we find that for this case Xp;i ˆ 0

if ΔVOC  ΔVOC∗i

Xp;i ˆ 1

pi °MVOC ˆ1 ai RT ΔVOC

ΔVOCi∗ ΔVOC

if ΔVOC > ΔVOC∗i

(14.10)

This fraction is shown as a function of ΔVOC for various saturation mixing ratios and molar yields in Figure 14.9. The organic compound will exist virtually exclusively in the aerosol phase if ΔVOC  ΔVOC∗i . As noted above, the defining quantity for atmospheric secondary organic aerosol formation is the aerosol mass yield defined as Yi ˆ

cp;i ΔVOC

(14.11)

This yield should be contrasted with the molar yield ai that refers to the total concentration of i. For this case the yield Yi can be calculated combining (14.11) and (14.7): if ΔVOC  ΔVOC∗i pi °Mi Mi Y i ˆ ai MVOC RT ΔVOC Yi ˆ 0

(14.12)

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ATMOSPHERIC ORGANIC AEROSOLS

The aerosol yield is therefore a function of the compound concentration that has reacted, starting from zero and approaching asymptotically the mass yield of the compound aiMi/MVOC. Aerosol Yield Calculation Find the aerosol mass yield Yi as a function of the concentration of i in the particulate phase cp,i. What happens when cp,i → 1? Plot the aerosol mass yield as a function of ΔVOC for the conditions of the previous example. The aerosol yield can be expressed as a function of the organic aerosol concentration by solving (14.7) for ΔVOC and substituting into (14.12) to obtain Yi ˆ

cp;i ai Mi RT …cp;i RT ‡ pi °Mi †MVOC

(14.13)

If cp,i → 1, then from (14.13), Yi → aiMi/MVOC, that is, it approaches the stoichiometric mass yield of the compound. In this case, the mass yield is aiMi/MVOC = 0.06, so all yield curves approach asymptotically this value (Figure 14.10). For low-vapor-pressure products this asymptotic limit is reached as soon as a small amount of the parent VOC reacts. For more volatile products the actual yield will be less for all conditions relevant to the atmosphere.

FIGURE 14.10 Aerosol mass yield during ROG oxidation as a function of the reacted ROG concentration ΔROG and the product saturation mixing ratio. The product is assumed to be insoluble in the aerosol phase. Other conditions are ai = 0.05, T = 298 K, MVOC = 150 g mol 1, and Mi = 180 g mol 1.

14.3.2 Formation of Binary Ideal Solution with Preexisting Aerosol Let us repeat the analysis assuming that now the organic vapor i produced during the VOC oxidation can form an ideal solution with at least some of the preexisting organic aerosol material. The preexisting organic aerosol mass concentration is m0 (in μg m 3) and has an average molecular weight M0. Using the

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IDEALIZED DESCRIPTION OF SECONDARY ORGANIC AEROSOL FORMATION

same notation as above, the mole fraction of the species in the aerosol phase xi is xi ˆ

cp;i =Mi cp;i =Mi ‡ …m0 =M0 †

(14.14)

If the solution is ideal, then the vapor pressure of i over the solution pi satisfies

p i ˆ xi pi °

(14.15)

and converting to mass concentration units (μg m 3) cg;i ˆ

xi pi °Mi RT

(14.16)

Combining (14.14), (14.16), and (14.6), we can eliminate xi and cg,i to obtain an expression for the aerosol concentration of i, cp;i ‡

M0 Mi pi °cp;i ai Mi ΔVOC ˆ MVOC RT…M0 cp;i ‡ Mi m0 †

(14.17)

This equation can be simplified assuming cp;i  m0

(14.18)

that is, compound i is only a minor component of the overall organic phase. This assumption is valid for most atmospheric conditions where no single organic compound dominates the composition of organic aerosol. Using (14.17) and (14.18), we obtain cp;i ˆ

ai RT Mi m o ΔVOC MVOC m0 RT ‡ M0 pi °

(14.19)

Note that in this case if ΔVOC > 0, then caer,i > 0, that is, secondary aerosol is formed as soon as the VOC starts reacting. In this case the threshold ΔVOC∗i is zero. The aerosol concentration of i is inversely related to its saturation vapor pressure and also depends on the preexisting organic material concentration. The mass fraction of i in the aerosol phase can be calculated combining (14.19), (14.9), and (14.6) as follows: Xp;i ˆ

m0 RT m0 RT ‡ pi °M0

(14.20)

For high-organic-aerosol loadings, or very low saturation vapor pressures, Xp,i  1, and virtually all i will be in the aerosol phase. The aerosol mass yield resulting from the VOC reaction defined by (14.6) is then Yi ˆ

ai RT Mi m 0 MVOC m0 RT ‡ M0 pi °

(14.21)

As expected, for high-organic-aerosol loadings or products of very low volatility, the yield Yi becomes equal to the stoichiometric mass yield of i in the reaction aiMi/MVOC.

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ATMOSPHERIC ORGANIC AEROSOLS

14.3.3 Formation of Binary Ideal Solution with Other Organic Vapor If the products of a given VOC cannot form a solution with the existing aerosol material, but they can dissolve in each other, then the formation of secondary aerosol can be described by a modification of the above theories. Assuming that two semivolatile products are formed VOC ! a1 P1 ‡ a2 P2 with molecular weights M1 and M2, then each compound satisfies (14.3) and (14.6), resulting in cg;1 ‡ cp;1 ˆ

a1 M 1 ΔVOC MVOC

(14.22)

cg;2 ‡ cp;2 ˆ

a2 M 2 ΔVOC MVOC

(14.23)

Assuming that the two species form an ideal organic solution, their gas-phase concentrations satisfy cg;1 ˆ x1 cg;2 ˆ …1

p1 °M1 ˆ x1 c 1 ° RT

x1 †

(14.24)

p2 °M2 ˆ x2 c2 ° RT

(14.25)

where x1 is the mole fraction of the first compound in the binary solution, and c2 ° is the saturation concentration in μg m 3 of pure i. The mole fraction of the second compound is simply x2 = 1 x1. The mole fraction x1 is given by x1 ˆ

cp;1 =M1 cp;1 =M1 ‡ cp;2 =M2

(14.26)

The five equations (14.22)–(14.26) form an algebraic system for the five unknowns cp,1, cp,2, cg,1, cg,2, and x1. Combining them we are left with one equation for x1, describing the equilibrium organic solution composition. Assuming that M1  M2  MVOC, we have x21 ‡

…a1 ‡ a2 †ΔVOC c2 ° ‡ c1 ° x1 c2 ° c1 °

a1 ΔVOC ˆ0 c2 ° c1 °

(14.27)

One can show that (14.27) always has a real positive solution. However, this solution should not only exist but should also satisfy the constraints cp;1  0

and

cp;2  0

(14.28)

or equivalently a1 ΔVOC a2 ΔVOC

x1 c 1 °  0

(14.29)

c2 ° ‡ x1 c2 °  0

(14.30)

Instead of finding the most restrictive constraint, let us add (14.29) and (14.30) to simplify the algebraic calculations. The result of this simplification may be a slight underestimation of the threshold ΔVOC∗. After summation, we obtain …a1 ‡ a2 †ΔVOC  x1 c1 ° ‡ c2 °

x1 c2 °

(14.31)

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IDEALIZED DESCRIPTION OF SECONDARY ORGANIC AEROSOL FORMATION

For small ΔVOC the inequality is not satisfied and the organic aerosol phase will not exist. However, when ΔVOC = ΔVOC∗, the equality will hold, or equivalently …a1 ‡ a2 †ΔVOC∗ ˆ x1 c1° ‡ c2°

x1 c2°

(14.32)

Substituting (14.32) into (14.27) and simplifying, we find the following relationship x∗1 c1° ˆ a1 ΔVOC∗

(14.33)

which is satisfied at the point of formation of the organic aerosol phase and that relates the threshold concentration of the hydrocarbon with the composition of the organic phase at this point. Combining (14.33) with (14.32) and eliminating the composition variable x1, we obtain a 1 a2 1 ‡ ∗ ˆ ΔVOC c1° c2°

(14.34)

or in terms of vapor pressures 1 ˆ ΔVOC∗

a1 a2 ‡ RT p1°M1 p2°M1

(14.35)

Odum et al. (1996) showed that secondary organic aerosol production by m-xylene at 40 °C can be represented by the formation of two products with molar yields a1 = 0.03 and a2 = 0.17 and saturation concentrations c1 ° ˆ 31:2 μg m 3 and c2 ° ˆ 525 μg m 3 . If the two species do not interact with each other, then (14.8) would be applicable, and assuming that MVOC  Mi, the corresponding thresholds would be ΔVOC∗1 ˆ 1040 μg m 3 and ΔVOC∗2 ˆ 3090 μg m 3 , respectively. If the two species form a solution, then from (14.35), ΔVOC∗ = 780 μg m 3. Therefore, after the reaction of ΔVOC∗, a binary solution will be formed containing both compounds with a composition given by (14.33) as x∗1 ˆ 0:75. The solution formation lowers the corresponding thresholds significantly and facilitates the creation of the secondary organic aerosol phase. Note that because we have not used the most restrictive constraint from (14.30) and (14.31), our solution could underestimate the threshold ΔVOC∗. An exact calculation would involve repeating the above algorithm first using only (14.29) and then only (14.30), calculating two different ΔVOC∗ values and then selecting the higher value (see Problem 14.5). The value calculated from (14.35) is an accurate approximation for most atmospheric applications. The aerosol mass yield Y for such a reaction is given by Yˆ

cp;1 ‡ cp;2 ΔVOC

(14.36)

or defining the total organic aerosol concentration produced during the VOC oxidation, cp, as cp ˆ cp;1 ‡ cp;2

(14.37)

and assuming that MVOC  M1  M2 and combining (14.22) to (14.26) with (14.36) and (14.37), we obtain Y ˆ cp

a1 a2 ‡ cp ‡ c1° cp ‡ c2°

(14.38)

This equation, first proposed by Odum et al. (1996) in a slightly different form, indicates that for low­ organic-mass concentrations, the aerosol yield is directly proportional to the total aerosol organic mass concentration caer. For very nonvolatile products and/or for large organic mass concentrations, the

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ATMOSPHERIC ORGANIC AEROSOLS

overall aerosol yield will be independent of the organic mass concentrations and equal to a1 + a2. Also, (14.38) suggests that the aerosol yields will be sensitive to temperature since c1 ° and c2 ° depend exponentially on temperature (via the Clausius–Clapeyron equation).

14.4 GAS–PARTICLE PARTITIONING 14.4.1 Gas–Particle Equilibrium Fundamental to the formation of secondary organic aerosol is the partitioning of organic oxidation products between the gas and particle phases. In this section we generalize the simplified treatments of the previous section (e.g., assumption of ideal solutions). Consider a species i that partitions at equilibrium between the gas and particle phases. The mass concentration of i, cg,i (g m 3) in the gas phase can be expressed in terms of its partial pressure pi by the ideal-gas law cg;i ˆ

pi M i RT

(14.39)

where Mi is the molecular weight of i. At equilibrium, the partial pressure of i is equal to its vapor pressure above the aerosol phase and using (10.49), we obtain pi ˆ xi γi pi°

(14.40)

where pi ° is the vapor pressure of pure liquid i, xi is the mole fraction, and γi is the activity coefficient of i in the aerosol phase. The aerosol phase generally contains a number of other species, so the mole fraction of i can be expressed as xi ˆ N p;i =N p;t , where Np,i is the number of moles of i in the particle phase per m3 of air, and Np,t is the total molar concentration of all particle-phase species per m3 of air. Np,t can be expressed as COA =Mavg , where COA is the total organic aerosol mass concentration (g m–3) and Mavg is the mean molecular weight of the mixture (g mol 1). Then the relationship between cg;i and cp;i , both in units (g m 3), is (Pankow 1994) cg;i ˆ

γi pi°Mavg cp;i COA RT

(14.41)

A lower value of pi ° translates into greater tendency for the compound to be found in the particle phase. Also, values of γi < 1 (compound i is “comfortable” in the particle phase) favor condensation. For SOA, values of γi typically lie in the range 0.3 < γi < 3. If Mi is approximately equal to Mavg, the quantity pi°Mavg =RT is equal to Ci°, the saturation mass concentration of pure compound i. It is a bit cumbersome for the activity coefficient γi to appear explicitly in the various partitioning expressions so it is customary to define the effective saturation mass concentration of i in the gas phase C∗i as C∗i ˆ γi Ci°

(14.42)

Both C∗i and Ci° are usually expressed in units of μg m–3. Either C∗i or Ci° is used as a measure of the volatility of i. Since values of the activity coefficient γi tend to lie between about 0.3 and 3, the difference between C∗i and Ci° is not overly large and the two numbers have the same order of magnitude. The fraction of the total amount of i present in the aerosol phase, Xp;i ˆ cp;i = cp;i ‡ cg;i is then

Xp;i ˆ

Note that at C∗i ˆ COA , Xp,i = 0.5.

1‡

C∗i COA

1

(14.43)

GAS–PARTICLE PARTITIONING

FIGURE 14.11 Fraction Xp of the total amount of a species in the aerosol phase as a function of C∗ at two fixed values of COA: 1 μg m 3 (left panel) and 10 μg m 3 (right panel). In each case, Xp = 0.5 when C∗ = COA.

Figure 14.11 shows how Xp,i depends on C∗ at fixed COA = 1 μg m 3 (left panel) and 10 μg m 3 (right panel). For COA = 1 μg m 3, if Ci∗ < 0.1, most of i is in the particle phase, and if Ci∗ > 10, most of i is in the gas phase. From the right panel, for COA = 10 μg m 3, the halfway point for partitioning is at C∗ = 10 μg m 3. The relationship between Ci∗ and COA in (14.43) is central in gas–particle partitioning equilibrium. To summarize, essential features of gas–particle equilibrium partitioning are 1. Fifty percent of the material is in each phase when C∗ = COA. 2. Within about one order of magnitude on either side of this equipartition point the response curve is roughly linear. 3. Beyond this linear region almost all of the material is in one phase or the other, mostly in the condensed phase for C∗ < 0.1COA and mostly in the vapor phase for C∗ > 10COA. 4. Changing the total organic mass concentration COA shifts the partitioning. Consequently, gas– particle partitioning close to a source of OA emissions is very different from that in the remote atmosphere. Therefore, if little organic aerosol is present, only the least volatile products of VOC oxidation will partition to the particle phase. In this case, gas-phase oxidation of the more volatile products may continue to proceed. Conversely, when preexisting aerosol levels are high, gas–particle partitioning effectively traps most of the semivolatile oxidation products in the particle phase as soon as they are formed, removing them from the possibility of further gas-phase oxidation. Idealized Kinetic Model of Oxidation over Two Generations Figure 14.12 depicts an idealized kinetic model of SOA formation involving two generations of oxidation products, in which the second-generation product S2 is an order of magnitude less volatile than the first-generation product S1 …C1°=C2° ˆ 10†. An initial amount of absorbing aerosol of 1 μg m 3 is assumed to be present, and no other reactions are occurring. Three cases are considered, in which the rate constant of the second oxidation reaction (k2) is varied relative to that of the first oxidation reaction (k1). When the second oxidation step is slow relative to the first (k2/k1 = 0.1), much of the SOA formation takes place after the parent hydrocarbon has been completely reacted. On the other hand, when the second step is relatively fast (k2/k1 = 10), SOA formation is governed largely by the rate of the initial oxidation reaction. In this simple example, regardless of the relative magnitudes of the rate g g p constants, the amount of SOA generated eventually is identical, because as S1 is oxidized to S2 , S1 is

591

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ATMOSPHERIC ORGANIC AEROSOLS

FIGURE 14.12 Simulated aerosol growth curves illustrating the effect of two generations of oxidation on SOA formation, based on the mechanism shown in the figure (Chan et al. 2007). It is assumed that the second-generation product S2 is an order of magnitude less volatile than the first-generation product S1. Kp1 and Kp2 are gasparticle partitioning equilibrium constants for S1 and S2 (Kp2/Kp1 = 10). 1 μg m 3 of absorbing aerosol is assumed to be present initially. The relative rates of the two oxidation steps (k2/k1) determine the depen­ dence of aerosol growth (ΔM) on the amount of hydrocarbon reacted (ΔHC). (Source: Kroll, J., and Seinfeld, J. H., Atmos. Environ. 42, 3593 (2008), Figure 10.)

drawn back out of the aerosol phase in order to maintain gas–particle equilibrium. (This process by which early-generation condensed semivolatile products are drawn back out of the aerosol and continue to be oxidized in the gas phase has been referred to as the “gas-phase pump.”)

Relationship between a Compound’s Latent Heat of Vaporization and Its Volatility The variation in vapor pressure of a compound i with temperature is governed by the Clausius– Clapeyron equation (10.68): d ln pi° ˆ dT

ΔH vi RT 2

(14.44)

ln pi° ˆ Ai

ΔHvi RT

(14.45)

Equation (14.44) is integrated to give

or pi° ˆ exp…Ai †exp… ΔHvi =RT †. Let us say that p1° refers to the vapor pressure of α-pinene and p2° refers to the vapor pressure of an oxidation product of α-pinene that is a factor of 1010 less volatile than α-pinene. Then, assuming that A1 ≅ A2 (if this is not the case, the estimate that follows will not hold), we can write p1 ° ∼ p2 °

exp exp

ΔH v1 RT ΔH v2 RT

ˆ exp

ΔH v1 RT

ΔH v2 RT

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GAS–PARTICLE PARTITIONING

or 1010 ∼ exp Let Δ(ΔHv) denote ΔHv2

ΔH v2 ΔH v1 RT

ΔHv1 and solve for Δ(ΔHv): Δ…ΔH v † ∼ RT ln 1010 ∼ 57 kJ mol 1

ΔHv1 for α-pinene is 45 kJ mol 1, so ΔHv2 = 102 kJ mol 1. At an atmospheric mass concentration of α-pinene at 298 K of 108 μg m 3, its oxidation product would have a corresponding mass concentra­ tion of 10 2 μg m 3. Estimating the Volatility of Organic Aerosols from Evaporation Rates Since the complete chemical composition of atmospheric organic aerosols is seldom known, it is not possible to predict the volatility properties of the mixture on the basis of those of its individual components. Experimentally, if one subjects the aerosol to heating and measures the rate at which the particles evaporate, it is possible to infer an effective value of the latent heat of vaporization ΔH for the mixture. One of the instruments in which this is carried out is called a volatility tandem differential mobility analyzer (VTDMA). The VTDMA (Figure 14.13) consists of a heated tube through which air containing particles of a known initial size flows. At the outlet of the tube, the size of the partially evaporated particles is measured, from which the effective latent heat of vaporization of the organic mixture in the particles is inferred. Consider particles of initial diameter Dpo treated as a single pseudo compound A, fed to the VTDMA. The inlet temperature of the feed is To, while the VTDMA is maintained at temperature Tu. The residence time of particles in the tube is τ, and the particle diameter exiting the tube is Dp(τ). [In the analysis of the VTDMA presented here, for simplicity, we assume that each particle has identical residence time τ in the tube. Actual systems may be operated in a laminar flow regime where this is not the case, see Orsini et al. (1999).] The vapor pressure of the particles at To is poA , whereas that at temperature Tu is pA. It is desired to infer the value of ΔH for the particles from their change in size in the VTDMA. The fundamental relationship between vapor pressure and temperature is the Clausius–Clapeyron equation, the integrated form of which is pA …T† ˆ poA …T o †exp

ΔH v 1 R To

1 T

(14.46)

The temperature T in (14.46) is that in the VTDMA, Tu. The rate of shrinkage of the particles as they flow through the VTDMA is given by equation (13.11). For simplicity, we will assume that the particles are sufficiently large that the noncontinuum correction term f(Kn) in Table 12.1 is not needed. We assume an accommodation coefficient of unity.

FIGURE 14.13 Volatility tandem differential mobility analyzer (VTDMA).

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ATMOSPHERIC ORGANIC AEROSOLS

The rate of change of particle diameter in the tube is dDp 4DA MA o ˆ p dt RTρp Dp A

pA †

(14.47)

where ρp is the density of the particle and MA is its mean molecular weight. The partial pressure of A in the background air in the tube taken as that at the entrance, poA . The temperature T in (14.47) is Tu. Substituting (14.46) into (14.47) and defining ρo ˆ

poA MA RT o

(14.48)

we obtain dDp 4DA ρo T o ˆ 1 dt D p ρp T u

exp

ΔH v 1 R To

1 Tu

(14.49)

Integrating (14.49) from t = 0 to t = τ, and solving for ΔHv/R, we obtain ΔH v To Tu ˆ ln 1 R Tu To

1 ρp T u …Dp …τ†2 8DA τ ρo To

2 Dpo †

(14.50)

To estimate ΔHv for a given system, it is necessary to specify To, Tu, ρp/ρo, Dpo and DA, and τ. Experimentally, Tu can be varied at fixed Dpo to obtain the corresponding values of Dp(τ), from which a mean value of ΔHv can be inferred. (If one retains the noncontinuum correction factor f(Kn) in (14.47), the analytical solution (14.50) will contain an integral over Dp that must be evaluated numerically.) As an example, consider the following parameter values: Dp(0) = 500 nm, Dp(τ) = 50 nm, To = 298 K, Tu = 348 K, τ = 10 s, ρp/ρo = 1010, and DA = 2 × 10 7 m2 s 1. For this set of conditions, a value of ΔHv = 113.6 kJ mol 1 is inferred. Typical organic compounds have ΔHv values between 50 and 120 kJ mol 1. For example, ΔHv = 67 kJ mol 1 for glutaric acid and 109 kJ mol 1 for pinic acid. The 20-carbon alkane, eicosane C20H42, has ΔHv = 101.8 kJ mol 1.

14.4.2 Effect of Aerosol Water on Gas-Particle Partitioning The gas–particle partitioning equilibrium constant is customarily defined as Kp;i ˆ

cp;i =COA cg;i

m3 g

1

(14.51)

assuming that i is absorbing into an organic phase of mass concentration COA. The absorbing phase need not be purely organic; typically the absorbing phase is a mixture of organics, inorganics, and water. We will retain the COA notation, but with the understanding that COA may consist of species other than organics and that Ai can absorb into the entire phase. If only a fraction of the particulate matter phase is available for partitioning, then one can introduce a fraction f of the total particulate matter phase that is available for partitioning. Then, using (14.41), Kp;i ˆ

RT γi pi Mavg

(14.52)

With this definition of the partitioning constant, Kp,i does not depend on the total aerosol organic concentration of COA. If Mi is approximately equal to Mavg, then Kp;i ˆ …γi Ci † 1 or using the effective

595

GAS–PARTICLE PARTITIONING

saturation concentration defined in (14.42), Kp;i ˆ

1 C∗i

(14.53)

In general, atmospheric organic aerosol comprises a large number of different molecules, any one individual species of which may constitute only a small fraction of the total OA mass. The one species that significantly alters the thermodynamic nature of OA is water. As RH increases, eventually water becomes a significant fraction of the volume of a particle. The particle-phase water will form a high-ionic­ strength aqueous phase that attracts a fraction of the more water-soluble organic molecules. When aerosol water is present, both the activity coefficient γi and the mean molecular weight Mavg of the absorbing phase are affected. We consider first the effect of the change in the amount of liquid water in the solution on γi, using α-pinene SOA as an example. We assume that all the SOA dissolves in water at this RH so that only one phase exists in the particles. Let the mean molecular weight of α-pinene oxidation products be 185 g mol 1 and the ratio of wet to dry particle diameters of the OA at RH = 85% be 1.08. Then at this RH, the ratio of wet to dry particle volumes is 1.26. If we take the dry particle volume to be 1.0, then at 85% RH, the volume fraction of aerosol that is water is 0.26/1.26 = 0.21 and that of organic is 0.79. For 100 cm3 of particle volume, 21 cm3 of water = 1.17 mol H2O, and 79 cm3 of organic = 0.42 mol of organic. With 1.59 total moles in the particle, xw = 1.17/1.59 = 0.74 and xorg = 0.26. Because of its relatively low molecular weight, despite having a volume fraction of only 0.21, water constitutes 74% of the total moles in the mixture. At RH = 85% and xw = 0.74, from RH = 0.85 = γw xw, we obtain γw = 1.15. Since γw > 1, a molecule of water is “less comfortable” in the mixture than in pure water. From (14.52), if either the mean molecular weight of the mixture Mavg or the activity coefficient γi decreases, the quantity of particulate organic material will increase in order to continue to balance the equation. Equation (14.52) can be used as a basis to understand the effect of increasing RH on Kp through the underlying Mavg and γi. The total derivative of Kp is given by dKp ˆ

@Kp @Kp dMavg ‡ dγ @Mavg @γ

(14.54)

By (14.52), we obtain @Kp ˆ @Mavg @Kp ˆ @γ

Kp Mavg

(14.55)

Kp γ

so that dKp ˆ Kp

dMavg dγ ‡ Mavg γ

(14.56)

or d ln Mavg d ln γ dKp ˆ Kp ‡ dRH dRH dRH

(14.57)

d ln γ dRH

(14.58)

d ln Kp ˆ dRH

d ln Mavg dRH

The first term in (14.58) (including the minus sign) is a measure of the RH-induced effect on Kp caused by changes in Mavg. The second term (including the minus sign) is a measure of the RH-induced

596

ATMOSPHERIC ORGANIC AEROSOLS

effect on Kp caused by changes in γ. For secondary PM formed from a variety of monoterpenes and cyclohexene, Seinfeld et al. (2001) showed that dKp =dRH is positive over the entire RH range for all of the compounds considered. That is, increasing RH leads to increases in Kp and therefore an increase in the overall extent of partitioning to the aerosol PM. This is a result in large part of the very low molecular weight of water as compared to the molecular weights of the oxidation products; adding water to the secondary PM phase uniformly reduces Mavg. The contribution to dKp =dRH from the γ term is more complex than the contribution from the Mavg term, since the activity coefficient variation with RH depends on the hydrophilic nature of the partitioning organic components. Three types of behavior can be identified: • Type I: uniformly positive (d ln γ=dRH uniformly < 0) across the RH range. • Type II: at first positive (d ln γ=dRH < 0) at low RH, then negative (d ln γ=dRH > 0) as an increasing amount of water is taken up by the secondary PM phase. • Type III: uniformly negative (d ln γ=dRH > 0). A hydrophilic compound is expected to exhibit type I behavior; a less hydrophilic compound is more likely to exhibit type III behavior. As a VOC is oxidized, the first oxidation products that partition to the aerosol phase are those with the lowest volatility, since the mass of absorbing aerosol in the condensed phase will not yet have built up to a substantial value. This low volatility is generally a result of the addition of oxygen-containing functional groups. Such groups tend to be polar and therefore are somewhat water-friendly. In this case, the activity coefficient for water is close to unity. As more and more organic material is added to the particle phase, the oxidation products that condense will tend to be somewhat more volatile and less polar. As a result, the aerosol solution nonideality will increase, and the activity coefficient for water will move in the direction of exceeding 1.0. Therefore, the change of the ultimate concentration of condensed organic material with change in RH will be the result of a balance between the effects of changes in RH on both the mean molecular weight of the mixture and its water activity coefficient.

14.5 MODELS OF SOA FORMATION AND EVOLUTION Formation of SOA involves a series of complex processes in both the gas and particulate phases. There have been a number of efforts to develop physicochemical models describing SOA formation (Kanakidou et al. 2005). Simplicity and computational efficiency were two main requirements of these models so that they could be used in three-dimensional chemical transport models. Because of the complexity of SOA formation, no model can simulate the entire process in the absence of experimental data. The experimental standard is the environmental chamber (sometimes referred to as a “smog chamber”). In a typical chamber experiment, run in time-dependent “batch” mode, an initial amount of VOC is reacted under conditions as close to atmospheric as possible, and the quantity of SOA produced is measured as a function of reaction time, along with as much other information as possible, including the time-dependent compositions of the gas and particle phases. Experiments of this type can also be carried out in a tubular flow reactor operated at steady state. The residence time in the reactor can be chosen so that the integrated exposure to OH (expressed in units of molecules cm–3 s) corresponds to that achieved over the duration of a batch chamber experiment. Particle-phase composition is generally obtained in real time as O:C and H:C ratios, and detailed particle-phase speciation is sometimes acquired from filter samples. The suitability of a model is judged by the extent to which it replicates the most important features of experimentally observed SOA formation, such as the mass concentration of SOA, and the distribution of volatility and oxidation state of the VOC oxidation products. Because no model can completely represent the chemistry of SOA formation, each model is characterized by certain physicochemical parameters that are determined so as to provide the best fit of the model prediction to the observed experimental data. An essential set of such parameters is the volatilities of the oxidation products.

MODELS OF SOA FORMATION AND EVOLUTION

In an early effort to describe quantitatively the formation of SOA, Pandis et al. (1992) assumed that each VOC precursor formed a single SOA surrogate species that did not interact with the other aerosol components. When the concentration of each surrogate reached saturation in the gas phase, any additional amount of it produced was transferred to the particle phase to maintain equilbrium. This case was developed in Section 14.3. It was assumed that the SOA components had relatively small saturation concentrations (roughly 0.1 μg m 3 at 298 K), based on the limited available experimental information from laboratory studies (Tao and McMurry 1989; Pandis et al. 1991). The work of Odum et al. (1996) resulted in a significant step forward in describing SOA formation. The Odum model assumed that the complex oxidation mechanism of each precursor VOC could be represented by formation of a few surrogate products, each with a constant yield ai, the moles of product formed per mole of VOC reacted. A fraction of each product would partition at equilibrium into the particle phase, forming SOA. Two products were typically employed, and so the approach is often called the “Odum 2-product model.” This model explains the observed behavior in laboratory chambers that SOA yields increase with increasing organic aerosol concentrations, owing to the fact that semivolatile organic oxidation products dissolve in the organic aerosol that has already condensed, and the greater the amount of condensed products, the greater the partitioning of each species. This model is also described in Section 14.3. The resulting fits of the experimental organic aerosol data using four parameters (two stoichiometric yields and two saturation concentrations) became widely used. One of the surprises that emerged from this fitting of chamber data was that the saturation concentrations needed to fit experimental data were unexpectedly high; usually one was of the order of 10 μg m 3 and the other around 200 μg m 3. These estimated high volatilities for the surrogate SOA components were inconsistent with the observed difficulty of evaporating these particles both in the laboratory and under ambient conditions (Pandis et al. 1991; Subramanian et al. 2004).

14.5.1 The Volatility Basis Set The traditional Odum model represented SOA formation by fitting an empirical two-product gas–particle partitioning model to chamber SOA yield data. The need to treat explicitly the volatility of primary organic aerosol and to model the continued oxidation of semivolatile compounds once in the atmosphere motivated the development of improved descriptions of OA formation. Stanier and Pandis (2004) proposed dividing the volatility range of the organic oxidation products into a set of bins to describe the full range of saturation vapor pressures in the system. Donahue et al. (2006) and Stanier et al. (2008) expanded on this idea, using fixed logarithmically spaced saturation concentration bins, called the volatility basis set (VBS). The volatility bins, separated by powers of 10, typically range from C∗ = 0.01 μg m 3 to 106 μg m 3 at 298 K. (These can shift with temperature according to the Clausius–Clapeyron equation.) Typical atmospheric COA levels are between 1 and 100 μg m 3. For this range, the extent to which a particular compound exists in the gas or particle phases depends on its effective saturation mass concentration: C∗ < 0.01 μg m 3: These compounds always reside in the particle phase. C∗ = {0.01, 0.1} μg m 3: These compounds are mostly in the particle phase in all except the most remote (and warm) parts of the atmosphere. ∗ C = {1, 10, 100} μg m 3: Significant fractions of these compounds will be found in both gas and particle phases under typical conditions. C∗ = {103, 104, 105, 106} μg m 3: These compounds are almost entirely in the gas phase; they constitute a large number of compounds and (probably) a small but important fraction of the total atmospheric burden of organic aerosol. ∗ C > 106 μg m 3: The vast majority of gas-phase emissions and atmospheric organics fall in this category. These volatility bins allow one to categorize compounds as follows: Extremely low-volatility organic compounds (ELVOCs with volatility bins 10 4 μg m 3, 10 5 μg m 3, etc.): These compounds immediately condense into the particle phase and, in addition, can be

597

598

ATMOSPHERIC ORGANIC AEROSOLS

very important for the formation and growth of new particles created in situ in the atmosphere by nucleation. Low-volatility organic compounds (LVOCs, with volatility bins 10 3, 10 2 and 10 1 μg m 3): These compounds reside in the particle phase at typical atmospheric concentration levels. Semivolatile organic compounds (SVOCs, with volatility bins 1, 10 and 100 μg m 3): The corresponding compounds exist in both the gas and particle phases under typical ambient conditions. Intermediate-volatility organic compounds (IVOCs, with volatility bins 103, 104, 105 and 106 μg m 3): These compounds exist in the gas phase in the atmosphere but, on oxidation, can be readily converted to condensable compounds producing secondary organic aerosol. Volatile organic compounds (VOCs, with C∗ > 106 μg m 3): Most of the emissions of gas-phase organics fall in this traditional category. As an example of volatility binning, Figure 14.14 shows estimated anthropogenic and biogenic nonmethane organic fluxes over the continental United States as a function of volatility. The inset shows the same data on an expanded scale. In applying the VBS to SOA formation in a particular system, one desires to determine the quantities of oxidation products of different volatilities that, when summed, best predict the total amount of OA produced. To do this, the VOC oxidation is expressed in terms of a set of products of different volatilities VOC ‡ oxidant ! a1 S1 ‡ a2 S2 ‡ ∙ ∙ ∙ ‡ a9 S9 ‡ volatile products

FIGURE 14.14 Anthropogenic and biogenic nonmethane organic fluxes in the continental United States as a function of volatility. The inset shows the same data on an expanded scale only in the range of the standard volatility basis set (VBS). The anthropogenic emissions are from the USEPA national emissions inventory (NEI), grouped by predicted volatility for VOCs and a projection of volatility for the POA emissions within the VBS (Shrivastava et al. 2006). Emissions of isoprene, monoterpenes, and 2-methyl-3-buten-2-ol were calculated using the MEGAN inventory (Guenther et al. 2006), and those of acetone are based on Jacob et al. (2002). Biogenic alkene emissions are scaled to 10% of isoprene based on work of Goldstein et al. (1996, 1998). Monoterpene speciation is based on Griffin et al. (1999a). Biogenic emission rates are calculated using assimilated meteorology (surface air tempera­ ture, direct and diffuse photosynthetically active radiation) at 2° × 2.5° horizontal resolution and 3-h temporal resolution from the NASA Goddard Earth Observing System (GEOS-4), using monthly leaf area index (LAI) values derived from the Advanced Very High Resolution Radiometer (AVHRR) Pathfinder Normalized Difference Vegetation Index dataset (Myneni et al. 1997). (Source: Donahue, N. M., et al., Atmos. Environ. 43, 94–106 (2009), Figure 2.)

MODELS OF SOA FORMATION AND EVOLUTION

599

FIGURE 14.15 Increased condensation with increasing quantities of α-pinene + ozone products, with the shaded portion of each bar showing the condensed-phase material and the remainder showing the material in the vapor phase: (a) partitioning when a relatively low concentration of α-pinene is oxidized, resulting in 1 μg m 3 of SOA mass or a fraction of SOA produced to α-pinene consumed (the SOA yield) of 0.04; (b) partitioning with 18 times as much precursor, resulting in 100 times as much SOA and a SOA yield of 0.21. Note that the bin with 50:50 partitioning shifts to 100 μg m 3. (Source: Donahue, N. M., et al., Atmos. Environ. 43, 94–106 (2009), Figure 1.)

where ai are the mass yields for products over the nine volatility bins. (The chemistry that leads to SOA is almost never a single-step reaction (recall Figure 14.1), so this representation depicts the outcome of the individual chemical steps that lead to products of different volatilities.) Imagine that we have carried out a chamber experiment in which an initial concentration of α-pinene is reacted with ozone and the amount of SOA formed is measured. On the VBS plot (Figure 14.15), the height of the bar at each volatility bin indicates the total concentration of material generated in that volatility range, and the shaded area within each bar shows the fraction of compounds in that volatility bin that is in the aerosol phase. The left panel in Figure 14.15 is drawn for an experiment in which the initial concentration of α-pinene is 26 μg m 3, leading to a total SOA concentration of 1 μg m 3, and the right panel depicts an experiment in which 468 μg m 3 of α-pinene is reacted, giving 100 μg m 3 of organic aerosol. The curve in Figure 14.16 shows the cumulative aerosol mass yield relative to the initial VOC concentration for the volatility bins for ozone oxidation of α-pinene. Available experimental data are shown, together with a SOA yield curve drawn through the data. This curve is then extrapolated over the complete range of volatility. The increase in mass of α-pinene completely reacted is assumed to be 1.4, so complete oxidation of the α-pinene would give an aerosol yield of 1.4. However, the experimental data are sufficient only to constrain the product yields up to the 103 μg m 3 VBS bin. To make up this difference, SOA production is inferred to occur in the three VBS bins (104, 105, 106 μg m 3) of {0.4, 0.35, 0.2}, shown with dark bars in Figure 14.15. The exact distribution of mass in these three VBS bins is uncertain, so the dark bars represent only a guess of those if the experiment were to have been carried out further. Thus, the overall mass yield of SOA is assumed to be 1.4, and this mass comprises products over the range of volatilities. To use the VBS as a means to represent the process of SOA formation, the OA mass concentrations must be apportioned to the volatility bins in such a way as to match as closely as possible total measured OA formation. Figures 14.15 and 14.16 show quantities of OA in each volatility bin. How are these quantities arrived at? The gas–particle partitioning for each volatility class depends on the total amount of condensed organic aerosol across all the nine bins, according to (14.43). The values of C∗ for each bin are fixed, but the amount of a reaction product with that C∗ that will partition into OA for that volatility class depends on total COA. Thus, in principle, in order to determine the partitioning in each bin, it is

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ATMOSPHERIC ORGANIC AEROSOLS

FIGURE 14.16 SOA formation from the α-pinene + ozone reaction. The data points are measurements from Griffin et al. (1999b), Presto et al. (2005), and Presto and Donahue (2006). The y axis is the aerosol yield assuming unit density of the OA. The rightmost bar is the initial mass of α-pinene, which is assumed to be unity; the light gray bars are the corresponding masses of the VBS products. The light gray bars are constrained by the measurements, while the dark gray bars are guesses constrained by the system mass balance. (Source: Donahue, N. M., et al., Atmos. Environ. 43, 94–106 (2009), Figure 3.)

necessary to solve simultaneously for the amounts of condensed material in each of the nine bins. However, because of the steep nature of the Xp,i curve, where Xp,i is essentially 1 for C∗ values less than 0.1COA and 0 for C∗ values greater than 10 COA, it is possible to approximate the gas–particle partitioning in each bin by starting at the 0.01 μg m 3 bin and moving progressively to the right, filling each bin one-by-one. Example: Partitioning in the VBS To illustrate the process, let us consider SOA formation in the α-pinene + O3 system. Let us assume that 25 μg m 3 of α-pinene is reacted and that for each 1 μg m 3 reacted, 1.47 μg m 3 of products result; this accounts for the mass increase owing to addition of oxygen. (The value is slightly different than the 1.4 used in Figures 14.13 and 14.14.) It is necessary to specify the fraction of the mass of reaction products in each volatility class to the total mass of 1.47. Say that this distribution is as follows: C∗

0:01

0:1

1:0

10

100

103

104

105

Mass fraction

0:004

0:002

0:05

0:09

0:12

0:18

0:4

0:35 0:27

106

The mass fractions add up to 1.47. Now, begin at the 0.01 μg m 3 bin. The mass of products in this bin is 0.004 × 25 = 0.1 μg m 3. The fraction of that product in the particle phase is computed using (14.43). With C∗ = 0.01 and COA = 0.1, Xp = 1/1.1 = 0.91, which we say is ∼1.0. Now move to the 0.1 bin. The mass in the 0.1 bin is 0.002 × 25 = 0.05 μg m 3. In calculating the fraction of OA in that bin, we have to consider the OA from both the 0.01 and 0.1 bins. This total is taken as 0.1 + 0.05 = 0.15, where, for the moment, we put all the mass from the 0.1 bin into OA. Then, for the 0.1 bin, C∗/COA = 0.1/0.15, and Xp = 0.6. Let’s call that 1.0. Now go to the 1.0 bin. The mass of products is 0.05 × 25 = 1.25 μg m 3. Since we know that we are getting close to the 50/50 point in the Xp curve, let us arbitrarily put half of this product, 0.625 μg m 3, into this bin. Doing so, the C∗/COA = 1.0/0.775 and Xp = 0.44, which is close enough to 0.5, so our assumption of putting half the mass into the OA phase was a good one. If we proceed to the 10 μg m 3 bin, the mass of products = 2.25 μg m 3. It would seem reasonable to assume that one-tenth of this will partition into the particle phase, so

MODELS OF SOA FORMATION AND EVOLUTION

we add 0.225 μg m 3 to the accumulated COA, giving a total of 1.0 μg m 3. This results in Xp = 1/11 for this bin, which is sufficiently close to 0.1. For the 100 μg m 3 bin, the product is 0.12 × 25 = 3.0 μg m 3. If we place 0.01 of this into the OA, then the total OA up to this bin = 1.03, and Xp ∼ 0.01. From this bin on, we assume that the products remain entirely in the gas phase. With 25 μg m 3 of α-pinene reacted and a total of 1.03 μg m 3 OA formed, the SOA yield = 1.03/25 = 0.04. What is the result if 500 μg m 3 of α-pinene is reacted? We can follow the same procedure of apportioning the oxidation products bin-by-bin. We find that the COA produced = 118 μg m 3 and the SOA yield = 118/500 = 0.236. (This calculation is the subject of Problem 14.6 at the end of this chapter.) By increasing the initial VOC concentration 20-fold, we find that the SOA yield increases from 4% to 23.6%. This example highlights the fact that with a greater amount of VOC reacted, the ultimate SOA yield increases owing to the fact that the larger mass of SOA shifts the gas–particle partitioning of more volatile compounds toward the particle phase. In fact, most of the particle-phase SOA compounds at a mass loading of 118 μg m 3 are not in the particle phase at a loading of 1 μg m 3. In the calculations above, we started by assuming the distribution of volatilities of the oxidation products. Ordinarily, these will not be known a priori. It is necessary to make an initial guess, go through the bin-by-bin calculation, and then adjust the distribution by trial-and-error to fit the observed data. The curve in Figure 14.16 is a result of that process. Note that for the case in Figure 14.16 the SOA contained in the 104, 105, and 106 bins, as indicated by the dark gray bars, had to be assumed to close the mass balance on SOA. The vapor material in the different volatility bins will likely continue to react over the course of SOA generation. Lane et al. (2008) modeled the chemical aging of the vapors by OH reaction in the VBS assuming a reaction of the form OAn …g† ‡ OH ! OOAn 1 …g† where OAn(g) is the gas-phase concentration of the components in the nth volatility bin of the VBS, and OOAn 1(g) is the gas-phase concentration of an oxidized component in the (n 1)th volatility bin. This reaction (with rate constant kOH) therefore assumes that each oxidation step reduces the volatility of the compounds by one order of magnitude. (If the oxidation is occurring by OH, for compounds with a fairly large number of CH2 groups, the OH rate constants kOH ∼ 3 × 10 11 cm3 molec 1 s 1.) Other formulations of gas-phase chemical aging are possible, for example, reduction of the saturation concentration by two orders of magnitude in each reaction. Molecular SOA Corridors The upper panel of Figure 14.17 shows molecular weight or molar mass (M) as a function of saturation mass concentration for 909 identified oxidation products from seven different SOA systems (Shiraiwa et al. 2014). The negative slopes of the lines bounding the data points represent the increase in molar mass required to decrease volatility by one order of magnitude, dM/dlogCo. Slopes for SOA products from individual VOCs depend on the molecular size of the SOA precursor and the O:C ratio of the reaction products; these range from ∼10 g mol 1 for glyoxal and methylglyoxal SOA to ∼25 g mol 1 for dodecane and cyclododecane SOA. The mean value of dM/dlogCo averaged over all seven systems is 20 ± 4 g mol 1. The lines bounding all the data correspond to the volatility of n-alkanes CnH2n+2 and sugar alcohols CnH2n+2On. These lines illustrate the regular dependence of volatility on the molar mass; the slopes of 30 g mol 1 for CnH2n+2 and 12 g mol 1 for CnH2n+2On reflect that the decrease of volatility with increasing molar mass is stronger for polar compounds. Early-generation gas-phase oxidation products of alkanes as well as dimers or oligomers with low O:C ratio fall into a molecular corridor close to the CnH2n+2 line, which one can designate as the low O:C corridor ( dM/dlogCo  ∼25 g mol 1, shaded area). Aqueous-phase reaction and autoxida­ tion products with high O:C ratio, on the other hand, tend to fall into a corridor near the CnH2n+2On line, which one can designate as the high O:C corridor ( dM/dlogCo of  ∼15 g mol 1, shaded area). The area in between is characterized by intermediate O:C ratios and accordingly designated as the

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FIGURE 14.17 Molecular corridors and kinetic regimes of SOA evolution (Shiraiwa et al. 2014). Upper panel: molar mass versus saturation mass concentration (Co) at 298 K for gas-phase (open) and particle-phase (solid) oxidation products of anthropogenic precursors (dodecane, cyclododecane, hexylcyclohexane) under low/high-NO conditions, biogenic precursors (α-pinene, limonene, isoprene) and aqueous-phase reaction products of glyoxal and methylglyoxal. The dotted lines represent linear alkanes CnH2n+2 (upper with O:C = 0) and sugar alcohols CnH2n+2On (lower with O:C = 1). Chemical structures of some representative products are shown. Lower panel: characteristic reaction pathways with probable kinetic regimes. The split of molecular corridors between high and low O:C compounds (HOC, lightly shaded area; LOC, darker shaded area) reflects the median correlation fitted lines for each of the individual SOA systems. SOA products evolve over the molecular corridor driven by three key reaction types of functionalization, oligomerization and fragmentation as illustrated in the insert (the different lengths of arrows indicate different intensities of effects on volatility).

intermediate O:C corridor ( dM/dlogCo  20 g mol 1). The small precursor VOCs, glyoxal, methyl glyoxal, and isoprene (C2-C5), evolve through the high-O:C corridor; and the terpenes, α-pinene and limonene (C10), through the intermediate-O:C corridor. The alkanes, dodecane and cyclododecane (C12), evolve through the low-O:C corridor. Characteristic reaction pathways and kinetic regimes are indicated in the lower panel of Figure 14.17. SOA precursors with high volatility and low molar mass are located in the lower right corner of the molecular corridor ensemble. Single-step functionalization usually leads to a small increase in molar mass, corresponding to one order of decrease in volatility, while dimerization and oligomerization decrease volatility by several orders of magnitude. Fragmentation can lead to a substantial decrease of molar mass and increase in volatility. Simple gas-phase oxidation products

MODELS OF SOA FORMATION AND EVOLUTION

are confined to the lower right area in the two-dimensional space. Particle-phase dimerization and oligomerization lead to the formation of compounds with low volatility and high molar mass lying in the upper left area in the space. Highly oxidized, extremely low-volatility organic compounds (ELVOCs) have been detected in field and chamber experiments (Ehn et al. 2012, 2014; Schobesberger et al. 2013; Zhang et al. 2015). Such compounds, which can be formed, for example, via autoxidation in the gas and particle phases, fall into the upper left corner of the high O:C corridor.

14.5.2 Two-Dimensional SOA Models The single most important property of SOA-forming compounds is their volatility, and the volatility basis set and earlier models sort compounds according to this physical property. The oxidation state of SOA-forming compounds is, as well, a key property of SOA-forming compounds. For example, tricosane (C23H48) and levoglucosan (C6H10O5) each has a saturation concentration of the order of 10 μg m 3. (Tricosane is a constituent of lubricating oil, while levoglucosan is a tracer for wood burning.) Although these compounds have essentially the same vapor pressure, tricosane is hydrophobic and levoglucosan is hydrophilic; in addition, their oxidation pathways can be quite different. The SOA formation can be described by a so-called two-dimensional SOA model that tracks the evolution of SOA on a two-dimensional space with coordinates representing the volatility of the aerosol, either Co or C∗, and its oxygen content, expressed as either O:C ratio, oxidation state OSC, or numbers of O and C atoms. The two-dimensional models describe SOA formation in the two-dimensional volatility–oxidation state space, starting from the VOC precursor itself. Two of the most fundamental chemical pathways in SOA evolution are functionalization and fragmentation. Each of these steps can occur, in principle, on any of the carbon atoms of the molecule. A model cannot realistically keep track of all the individual reactions, so it is designed to represent the average behavior of the products resulting from each step of oxidation. As before, the models contain parameters that ultimately must be determined for each VOC system by fitting of the model predictions to laboratory data. 14.5.2.1 Two-Dimensional VBS The two-dimensional VBS (Donahue et al. 2011, 2012; Murphy et al. 2011) space is shown in Figure 14.18. The 2D-VBS does not require specific knowledge of molecular structures. As the oxidation products undergo each generation of reaction, products are assigned to a volatility class.

FIGURE 14.18 Two-dimensional volatility basis set. Solid lines extending from lower right to upper left are average carbon number nC. The carbon number of each line is indicated by numbers in the boxes. Dashed curves descending from the top of the plot to the lower left are average oxygen number nO, which is indicated in the box on each curve. (Source: Pandis, S. N., et al., Faraday Discuss. 165, 9–24 (2013), Figure 1.)

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In order to assess the lifetimes of individual organic products, it is necessary to estimate the OH oxidation rate constants of organics in the 2D-VBS bins. There is usually some information on the C, H, and O numbers, as well as the O:C and H:C ratios. Hydroxyl radical rate constants can be estimated from structure–activity relationships. It is known that (1) OH rate constants typically increase with increasing carbon number, (2) adding oxygenated functional groups for lightly oxygenated molecules typically increases the OH reactivity, and (3) highly oxygenated molecules ultimately become less reactive simply because of a scarcity of H atoms that can be abstracted. Donahue et al. (2013) proposed the following approximate gas-phase OH rate constant as a function of carbon number, oxygen number, and O:C ratio to be used to advance generation by generation in the 2D-VBS: kOH ∼ 1:2  10

12

‰nC ‡ 9nO

10…O : C†2 Š

cm3 molecule

1

s

1

(14.59)

The first two terms in the brackets express the increase of the rate constant with both carbon and oxygen numbers, while the third term represents the retardation effect due to the scarcity of H atoms as O:C increases. For nC = 10, nO = 4, and an ambient OH concentration of 2 × 106 molecules cm 3, for example, this rate constant gives a lifetime against OH reaction of 2.6 h. As organic oxidation products evolve, two overall pathways are possible: (1) functionalization of the molecule through addition of oxygen-containing groups and (2) fragmentation of the molecule through C C bond breaking. Both functionalization and fragmentation are described by generalized “kernels.” The functionalization kernel assumes that one generation of oxidation by OH results in products with Co lowered by a certain number of decades and is accompanied by addition of a certain number of O atoms. For example, if the vapor pressure of the functionalization products is assumed to be distributed over two to four lower decades accompanied by the addition of one to three oxygen atoms, the function­ alization kernel is a 3 × 3 matrix of probabilities, that is, element (1,1) is the probability that the vapor pressure of the functionalization product is lower by two orders of magnitude and the product contains one oxygen atom. The fragmentation scheme in the 2D-VBS assumes that C C bonds cleave at a random position along the carbon backbone (on average), because fragmentation is presumed to be driven by the presence of functional groups on the backbone and these functional groups are, in turn, assumed to be randomly distributed on average. With this scheme, fragmentation disperses products over a wide Co and O:C range. More heavily oxidized molecules tend to fragment more easily. Fragmentation thus generally leads to a reduction in the ability to form organic aerosol. The probability of fragmentation f for a stable molecule on reaction with OH has been proposed to be represented as a function of the O:C ratio of the molecule (Jimenez et al. 2009) by f ˆ …O : C†ν

(14.60)

where ν is a parameter to be determined by fitting of chamber data. Fragmentation leads to products with fewer carbons and, generally, higher O:C, although the ultimate contribution of these products to the SOA mass depends on their volatilities. Species with a higher O:C ratio have a greater probability of fragmentation (i.e., ν < 1). For example, a molecule with a C10 backbone with on average 1.5 oxygen atoms added per generation, according to (14.60) with ν = 0.25, will have a fragmentation probability of 0.62, 0.74, 0.82, 0.88, and 0.93 for generations 1–5. Different implementations of the 2D-VBS can be found in Murphy et al. (2011). 14.5.2.2 Statistical Oxidation Model (SOM) The SOM (Cappa and Wilson 2012; Cappa et al. 2013) describes SOA formation as a statistical evolution in the nC nO space. Equilibrium partitioning between the gas and particle phases is assumed to hold at each timestep of the model. It is assumed that the properties of each nC nO pair can be represented by mean values that account for the actual distribution of functional groups within the group of molecules that make up an SOM species. The adjustable parameters within the SOM are (1) the number of oxygen atoms added per reaction, which is represented as an array of probabilities of adding one, two, three, or

PRIMARY ORGANIC AEROSOL

four oxygen atoms, termed Pfunc; (2) the decrease in volatility on addition of an oxygen atom, ΔLVP, which is the difference in the logarithm of the saturation concentration per oxygen added; and (3) the probability Pfrag that a given reaction leads to fragmentation into two smaller molecules. Positive values of ΔLVP correspond to a decrease in vapor pressure on oxygen addition. These parameters are adjusted to determine a best fit of the model to chamber observations. The SOM also includes the possibility of heterogeneous reactions with OH radicals; it is assumed that each heterogeneous OH reaction leads to the addition of one oxygen atom, that the fragmentation probability is equivalent to that in the gas phase, and that the reactive uptake coefficient is unity. 14.5.2.3 Carbon Number–Polarity Grid (CNPG) In the CNPG (Pankow and Barsanti 2009) model, the original Odum 2-product model is expanded to n products with m possible types of low-volatility compounds and accounting for effects of solution nonideality, variation of particle-phase mean molecular weight, water uptake, and phase separation. Products are tracked on a carbon number–polarity grid. Compounds with similar structures can be lumped together in the same grid bin even if formed by different routes. 14.5.2.4 Functional Group Oxidation Model (FGOM) The FGOM (Zhang and Seinfeld 2013) is based, in part, on explicit chemical information on the types of functional groups that result from the oxidation of a parent VOC. A comprehensive chemical mechanism such as the Master Chemical Mechanism (www.mcm.leeds.ac.uk/MCM/) or GECKO-A (Aumont et al. 2005, 2012; Camredon et al. 2007; Lee-Taylor et al. 2011; Valorso et al. 2011) is used to predict the types of functional groups that characterize the oxidation of the parent organic molecule, yet the detailed mechanism is not employed in the model itself. With the carbon number and the spectrum of functional groups specified, a derived structure–activity relationship is used to predict the volatility of the products. The model merges group contribution and element contribution methods for estimating Co to relate the carbon number to the saturation concentration of a compound i. A set of four surrogate functional groups characterizes the progressive oxidation. Particle-phase reactions are assumed to proceed in parallel with the gas-phase reactions. Fragmentation occurs according to (14.60). The FGOM contains six parameters that are to be determined by fitting of the model to chamber data on total organic aerosol concentration, and O:C and H:C ratios, the ratio of the particle-phase oxidative reaction rate constant to that in the gas phase; the fragmentation parameter ν; ka, the accretion reaction rate constant in the particle phase; and a matrix [Cx, Hy, Oz] of the carbon, hydrogen, and oxygen numbers of the nonvolatile products. 14.5.2.5 Conclusion Summarizing, two-dimensional SOA models represent SOA formation and evolution over a twodimensional space of volatility and oxidation state in terms of the competition between functionalization and fragmentation pathways, the extent of oxygen atom addition per step, and the resulting change in volatility. Each of the models discussed above includes parameters that must be determined for each VOC system by fitting of experimental data. The parameters are incorporated in a set of reaction rate equations governing the time-dependent evolution of the species in the model. The differential equations are solved numerically, with gas–particle partitioning calculated at each timestep. Values of the parameters are determined by optimal fitting of model simulations of organic mass growth and elemental compositions with laboratory chamber data (organic aerosol mass, O:C, and H:C).

14.6 PRIMARY ORGANIC AEROSOL As noted at the beginning of this chapter, most organic particulate matter emitted directly from sources, termed primary organic aerosol (POA), evaporates as dilution lowers the partial pressure of vapors over it. It is now understood that the majority of POA emissions fall in this category, specifically compounds in the

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SVOC and IVOC volatility ranges (Shrivastava et al. 2006; Lipsky and Robinson 2006; Grieshop et al. 2009). Emissions with C∗ > 1 μg m 3, corresponding roughly to a saturation mixing ratio > 0.1 ppbv, will be found, at least partially, in the gas phase under typical ambient conditions. These constitute 50–90% of the POA emissions that were traditionally considered as nonvolatile. The POA components that evaporate on dilution are then susceptible to oxidation in the gas phase, forming products with lower volatility that can partition back into the aerosol phase (Robinson et al. 2007; Weitkamp et al. 2007; Sage et al. 2008). An approximate survey of gas- and particle-phase organic emissions over the continental United States as a function of volatility was shown in Figure 14.14, for anthropogenic and biogenic emissions. Total US biogenic emissions are estimated at 34 TgC yr 1, while total anthropogenic emissions are approximately 20 TgC yr 1. Whereas biogenic emissions exceed anthropogenic emissions in the VOC range, anthropogenic emissions greatly exceed biogenic emissions in the LVOC range; the crossover occurs at a volatility of ∼104 μg m 3. This reflects, in part, the fact that high temperature sources (combustion) evaporate lower-volatility compounds than those from biogenic sources. Emissions in the C∗  106 μg m 3 range (shown in the inset of Figure 14.14) total about 2.5 TgC yr 1; about half of these low-volatility emissions were traditionally classified as nonvolatile POA. Anthropogenic does not necessarily mean fossil carbon; more than half of the POA emissions over the continental United States are associated with biomass burning: residential heating, agricultural burning, and wildfires. The volatility distributions of both biogenic and anthropogenic emissions in Figure 14.14 are multimodal. For the biogenics, the major modes correspond to isoprene and the monoterpenes, which collectively dominate biogenic emissions. The anthropogenic emissions show a “dip” in the IVOC range (the right side on the inset in Figure 14.14). Emissions in the lower volatility region contribute only a small fraction of the atmospheric organic mass, but with large carbon numbers and a high degree of substitution, this region may comprise a large number of individual species. Figure 14.19 shows an estimate of the annual organic aerosol budget of the continental United States based on the volatility distribution of emissions. Figure 14.19a shows the cumulative organic (biogenic and anthropogenic) emissions; these are the same data shown in Figure 14.14. Also shown in Figure 14.19a is an estimate of the SOA yield as a function of volatility at typical atmospheric conditions. Precursor structure is important in determining SOA yield; for example, the SOA yield from the oxidation of toluene (C∗ ∼ 108 μg m 3) exceeds that of an n-alkane with a comparable C∗ (n-heptane). Figure 14.19b shows the results of a convolution of the yield estimate with the estimated vapor-phase emissions. The lowest-volatility emissions partition directly into the condensed phase and therefore are classified as POA. At a typical atmospheric COA of 10 μg m 3, ∼0.3 Tg yr 1 of the emissions can be classified as POA, which is about one-quarter of the “nonvolatile” POA emissions in the emission inventory. Oxidation of low-volatility vapors with C∗ < 104 μg m 3 is estimated to produce SOA. For volatility exceeding 104 μg m 3, the emissions dramatically increase, but the production efficiency to oxidized OA drops off. In Figure 14.19b, all VOC species with a C∗ of 106 and higher are classified as traditional SOA precursors. It is estimated that the oxidation of these precursors produces about 4 Tg yr 1 of SOA in the United States. A comparable amount of oxidized OA is estimated to form from the oxidation of low-volatility organic vapors–the large majority of the OA in this simple model comes from emissions with saturation concentrations between 104 and 107 μg m 3. Emissions from the continental United States contribute about 4% of global organic emissions; therefore, a scaling of the results shown in Figure 14.19 suggests annual global oxidized OA production of ∼150 TgC yr 1. Motor vehicles emit both gas-phase organic compounds and primary particulate matter. The majority of particulate matter emissions from motor vehicles is either black carbon or POA. Motor vehicle POA comprises thousands of organic compounds derived from unburned fuel, unburned lubricating oil, and products of incomplete combustion. A substantial fraction of the POA is semivolatile under atmospheric conditions. Because of the historical difficulty in identifying individual species in this region, the collection of compounds in it has been classified as an unresolved complex mixture (UCM) (Fraser et al. 1997, 1998; Schauer et al. 1999). Advances in gas chromatography and highresolution mass spectrometry have, for the first time, revealed the composition of the UCM (Gentner et al. 2012; Chan et al. 2013; Worton et al. 2014). Gasoline is composed mainly of branched alkanes and single-ring aromatic compounds with fewer than 12 carbon atoms and a narrow distribution around C8.

PRIMARY ORGANIC AEROSOL

FIGURE 14.19 (a) Cumulative emissions of anthropogenic and biogenic organics for the continental United States and estimated secondary organic aerosol yield as a function of volatility; (b) estimated cumulative contribution of different classes of organic aerosol as a function of volatility. (Source: Donahue, N. M., et al., Atmos. Environ. 43, 94–106 (2009), Figure 5.)

Diesel fuel contains about 50% branched alkanes and single-ring aromatics and about 50% cyclic compounds with one or more rings; diesel fuel has a broad distribution of carbon numbers (C6–C25) that peaks around C10–C12 (Gentner et al. 2012). Combustion of both gasoline and diesel fuel leads to volatile byproducts of incomplete combustion, such as alkanes, aromatics, and oxygenates. Worton et al. (2014) made measurements of POA in a major traffic tunnel in California. Observed POA mass was dominated (>80%) by branched cycloalkanes with one or more rings and one or more branched alkyl sidechains. The absence of significant combustion products in POA and the similarity in the chemical composition of POA and lubricating oil indicate that unburned lubricating oil is an important source of the vehicular POA. Fossil versus Nonfossil Carbon in Organic Aerosols Radiocarbon analysis (14 C) can unambiguously distinguish fossil and nonfossil sources of EC and OC. C is completely depleted in fossil fuel and its emissions (The halflife of 14 C is 5730 years). Nonfossil carbon sources include biomass burning, cooking, or secondary organic aerosol formed from biogenic emissions. Measurement of 14 C in particulate matter is carried out by analysis of aerosol deposited on filters. Zotter et al. (2014) report time-resolved measurements of 14 C for particulate OC and total particulate carbon (TC) for 7 days in 2010 in Pasadena CA, in the Los Angeles basin, downwind of the heaviest concentration of fossil fuel emissions in the basin. The results show a consistent diurnal pattern. The nonfossil fractions of organic carbon, fOC, and total carbon, fTC, are highest during the 14

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nighttime and early mornings (fOC = 0.71 and fTC = 0.60). In contrast, in the early afternoon the carbona­ ceous aerosol is clearly influenced by fossil sources (fOC = 0.42 and fTC = 0.38), coinciding with the eastward transport of air from the heavy motor vehicle emission areas of the Los Angeles basin. During the 7-day measurement period, the nonfossil OC and TC concentrations stayed nearly constant throughout the day at 2–3 μgC m 3. The afternoon increase of the concentrations is primarily a result of fossil fuel combustion emissions, as fossil OC and TC are observed to increase by ∼3 μgC m 3. Cooking was estimated to contribute at least 25% of the nonfossil OC. In European cities where similar measurements have been made, the fossil fraction of secondary OC (∼25%) is on average ∼20% lower than that in Pasadena.

14.7 THE PHYSICAL STATE OF ORGANIC AEROSOLS Evidence has accumulated from a number of avenues that organic aerosols can often exhibit the properties of highly viscous, even semisolid, particles. This evidence includes observations that (1) some SOA evaporates considerably more gradually than would be consistent with normal liquid behavior (Grieshop et al. 2007; Vaden et al. 2011; Pierce et al. 2011); (2) SOA particles trap more volatile molecules, preventing them from evaporating (Vaden et al. 2011; Kuwata and Martin 2012; Perraud et al. 2012; Abramson et al. 2013); (3) certain organic aerosols bounce off the plates of an impactor (Virtanen et al. 2010; Saukko et al. 2012); (4) 20–50 nm SOA particles composed of water-soluble fractions of α-pinene SOA at RH < 30% are highly viscous (Renbaum-Wolfe et al. 2013a, b); and (5) SOA particles can form stable layered particles with core–shell morphology (Vaden et al. 2010; Loza et al. 2013). Moreover, oligomers and other high molecular weight compounds often prevalent in SOA (see next section) are known to retard diffusion and mixing and to slow the evaporation of smaller molecules (Widmann et al. 1998). Figure 14.20 shows ranges of viscosity, diffusion coefficients, and timescales for diffusion of molecules in the particle phase. The characteristic diffusion time in a 100-nm-diameter liquid particle is ∼10 μs. In this case, gas–particle equilibrium is established rapidly. But in highly viscous SOA, diffusion of particle-phase components becomes very slow, and considerable time is needed for a dissolved species to become uniform throughout the particle. Gas–aerosol models have been developed that explicitly simulate gas- and particlephase mass transport (e.g., Shiraiwa et al. 2012, 2013; Zaveri et al. 2014).

FIGURE 14.20 Characteristic timescale for diffusion in an aerosol particle as a function of diffusion coefficient and particle diameter for a particle of diameter 100 nm. Lines are calculated with equation (12.75). (Source: Shiraiwa, M., et al., Proc. Natl. Acad. Sci. 108, 11003–11008 (2011), Figure 1.)

THE PHYSICAL STATE OF ORGANIC AEROSOLS

Timescale to Achieve Gas–Particle Equilibrium Partitioning The traditional assumption in describing SOA formation is that semivolatile oxidation products achieve instantaneous gas–particle equilibrium. For this assumption to be valid, diffusion of a molecule acquired in the particle phase must be relatively rapid. With the evidence that organic aerosol particles can be highly viscous, the assumption of instantaneous gas–particle partitioning equilibrium must be reconsidered. The computational model KM-GAP that simulates the time-dependent gas-phase diffusion, transport across the air–particle interface, and particle-phase diffusion of a species A can be used, by varying volatility of A and aerosol phase diffusivity over a wide range, to study the time needed to establish gas–particle equilibrium (Shiraiwa et al. 2012; Shiraiwa and Seinfeld 2012). The model is based on numerical solution of the gas and particle transport equations of Chapter 12. To illustrate the effect of volatility of A on the equilibration time τeq, let us consider that a parent VOC is being oxidized to a semivolatile product with a first-order rate constant of 0.1 min 1, that is, a reaction timescale of 10 min. The volatility of a compound is expressed by its effective saturation mass concentration C∗. At gas–particle equilibrium, the partial pressure of A in the vapor phase equals the partial pressure of A just above the particle surface. We vary C∗ over the range from 10 5 to 105 μg m 3. The initial number and mass concentrations of nonvolatile preexisting particles are taken as 104 cm 3 and 1 μg m 3, respectively. The initial particle size distribution is assumed lognormal with geometric mean diameter 50 nm and geometric standard deviation σ = 1.5. The nominal aerosol phase diffusion coefficient is assumed to be 10 8 cm2 s 1, corresponding to that of a viscous liquid. The surface accommodation coefficient for molecules of A is taken to be α = 1. Figure 14.21 shows that τeq increases as C∗ decreases, since the partial pressure gradient between the gas phase and the particle surface is larger for smaller C∗. τeq of ELVOC and SVOC is of order hours while τeq of SVOC and IVOC is the order of seconds, for which instantaneous gas–particle equilibrium partitioning holds.

FIGURE 14.21 Equilibration timescale for SOA gas–particle partitioning for liquid particles as a function of the saturation mass concentration of the condensing product computed using the KM-GAP numerical model for gas– particle interactions (Shiraiwa et al. 2012). The equilibration timescale is eval­ uated as the e 1 point in the approach to equilibrium. The parent VOC is assumed to be reacting to produce the condensable product with a first-order rate constant of 0.1 min 1. The surface accommodation coefficient α = 1.0. The gas-phase and particle-phase diffusion coefficients of the condensing species are 0.1 and 10 8 cm2 s 1, respectively. The molecular weight of the condensing species is 100 g mol 1, and the density of the particle is 1 g cm 3. (Source: Shiraiwa, M., and Seinfeld, J. H., Geophys. Res. Lett. 39, L24801, (2012) Figure 2.)

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FIGURE 14.22 Equilibration timescale for SOA gas–particle partitioning of a semivolatile organic compound (SVOC) with C∗ = 10 μg m 3 in liquid, semisolid, and solid particles as a function of the surface accommodation coefficient α and bulk diffusion coefficient Db computed using the KM-GAP numerical model for gas–particle interactions (Shiraiwa et al. 2012). The equilibration timescale is evaluated as the e 1 point in the approach to equilibrium. (Source: Shiraiwa, M., and Seinfeld, J. H., Geophys. Res. Lett. 39, L24801 (2012), Figure 3.)

Figure 14.22 shows the further effect on τeq of the surface accommodation coefficient α and the bulk diffusivity of A in the particle, Db. Here we consider monodisperse particles of initial diameter 200 nm and a condensing oxidation product of C∗ = 10 μg m 3. Db is varied from 10 21 to 10 5 cm2 s 1. Typical values of Db for organics are 10 10 to 10 5 cm2 s 1 for liquid, 10 20 to 10 10 cm2 s 1 for semisolid, and Qscat in this regime.

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INTERACTION OF AEROSOLS WITH RADIATION

In summary, for sufficiently small particles, if (m2 1)/(m2 + 2) is a weak function of wavelength, the wavelength dependences of the scattering and absorption efficiencies are Qscat  λ

4

Qabs  λ

1

In both scattering and absorption, shorter wavelengths are extinguished more effectively than longer wavelengths. This is the “reddening” of the spectrum referred to above.

15.1.2 Geometric Scattering Regime Particles for which α  1 fall into the so-called geometric scattering regime. In this case the scattering can be determined on the basis of the geometrical optics of reflection, refraction, and diffraction. Scattering is strongly dependent on particle shape and orientation relative to the incoming beam. Let us consider a large, weakly absorbing sphere. We do so because water droplets are the most important class of “large” particles in the atmosphere and they are, for all practical purposes, nonabsorbing. Using geometric optics it can be shown for a weakly absorbing sphere (4αk  1) that (Bohren and Huffman 1983) 8 k Qabs …m; α† ˆ α n3 3 n

…n2

1†3=2

(15.22)

The extinction efficiency of a water droplet as a function of the size parameter α is shown in Figure 15.3. We note that Qext approaches the limiting value 2 as the size parameter increases: lim Qext …m; α† ˆ 2

α!1

(15.23)

This is twice as large as predicted by geometric optics, which is the so-called extinction paradox (Bohren and Huffman 1983). In qualitative terms, the incident wave is influenced beyond the physical boundaries of the sphere; the edge of the sphere deflects rays in its neighborhood, rays that, from the viewpoint of geometric optics, would have passed unimpeded. Also, all the geometrically incident light that is not externally reflected enters the sphere and is absorbed, as long as the absorptive part of the refractive index is not identically zero.

15.1.3 Scattering Phase Function The scattering phase function describes the angle-dependent scattering of light incident on a particle. Figure 15.4 shows the scattering phase functions for (NH4)2SO4 aerosol at 80% RH for several particle sizes at λ = 550 nm. At small diameter, the phase function is symmetric in the forward and backward directions (Figure 15.2). Note that, for all but the very smallest particles, the scattering is highly peaked in the forward direction. The directional asymmetry becomes increasingly pronounced as the particle size increases. The small scattering lobes for θ > 90° become almost imperceptible compared with the strong forward-scattering lobes for the larger particles. The consequence of strong forward scattering is the reason why driving into a bright setting sun can produce a blinding glare. A similar phenomenon occurs at night in a fog when light from oncoming automobile headlights is scattered in the forward direction.

15.1.4 Extinction by an Ensemble of Particles The foregoing discussion relates to the scattering of light by a single particle. A rigorous treatment of the scattering by an ensemble of particles is very complicated. If, however, the average distance between particles is large compared to the particle size, it can be assumed that the total scattered intensity is the sum of the intensities scattered by individual particles, and single-particle scattering theory can be used.

641

SCATTERING AND ABSORPTION OF LIGHT BY SMALL PARTICLES

FIGURE 15.3 Extinction efficiency Qext for a water droplet: (a) wavelength of the radiation is constant at λ = 0.5 μm and diameter is varied; (b) diameter = 2 μm and wavelength is varied (Bohren and Huffman 1983).

Such an assumption is easily obeyed for even the most concentrated atmospheric conditions. Consider, for example, a total particle concentration of 106 cm 3, consisting of monodisperse 1-μm-diameter spheres, a number density well above that of most ambient conditions. Even at this extreme aerosol loading, the volume fraction occupied by particles per cm3 of air is only (π/6) × 10 6. Consider the solar beam traversing an atmospheric layer containing aerosol particles. Light extinction occurs by the attenuation of the incident light by scattering and absorption as it traverses the layer. Recalling Section 4.3, the fractional reduction in intensity over an incremental depth of the layer dz can be expressed as dF = bextF dz, where bext is the extinction coefficient, having units of inverse length (m 1) bext ˆ Cext N

(15.24)

where N is the total particle number concentration (particles m 3). Equation (15.24) is written, for the moment, for a collection of monodisperse particles. Thus bext is the fractional loss of intensity per unit pathlength, and dF ˆ bext F dz

(15.25)

642

INTERACTION OF AEROSOLS WITH RADIATION

FIGURE 15.4 Aerosol scattering phase function for (NH4)2SO4 particles at 80% relative humidity at λ = 550 nm (Nemesure et al. 1995). Incident beam enters from the left.

If F0 is the incident flux at z = 0, taken to represent, say, the top of the layer, then intensity at any distance z into the layer is F ˆ exp… bext z† F0

(15.26)

643

SCATTERING AND ABSORPTION OF LIGHT BY SMALL PARTICLES

As noted in Chapter 4, the dimensionless product τ = bext z is called the optical depth of the layer, and (15.26) is called the Beer–Lambert law. In the visible portion of the spectrum, the optical depth of tropospheric aerosols can range from less than 0.05 in remote, pristine environments to close to 1.0 near the source of intense particulate emissions such as in the plume of a forest fire. For a population of monodisperse, spherical particles at a number concentration of N, the extinction coefficient is related to the dimensionless extinction efficiency by bext ˆ

πD2p 4

NQext

(15.27)

The extinction coefficient can be expressed as the sum of a scattering coefficient bscat and an absorption coefficient babs bext ˆ bscat ‡ babs

(15.28)

with similar relations to Qscat and Qabs. That extinction is the sum of scattering and absorption, and that either of the two phenomena can dominate extinction, is illustrated nicely by the classroom demonstration described by Bohren and Huffman (1983): Two transparent containers (Petri dishes serve quite well) are filled with water, placed on an overhead projector, and their images focused on a screen. To one container, a few drops of milk are added; to the other, a few drops of India ink. The images can be changed from clear to a reddish hue to black by increasing the amount of milk or ink. Indeed, both images can be adjusted so that they appear equally dark; in this instance it is not possible, judging solely by the light transmitted to the screen, to distinguish one from the other: the amount of extinction is about the same. But the difference between the two suspensions immediately becomes obvious if one looks directly at the containers: the milk is white whereas the ink is black. Milk is a suspension of very weakly absorbing particles which therefore attenuate light primarily by scattering; India ink is a suspension of very small carbon particles which attenuate light primarily by absorption. Although this demonstration is not meant to be quantitative, and its complete interpretation is complicated somewhat by multiple scattering, it clearly shows the difference between extinction by scattering and extinction by absorption. Moreover, it shows that merely by observing transmitted light it is not possible to determine the relative contributions of absorption and scattering to extinction; to do so requires an additional independent observation.

We can decompose babs and bscat into contributions from the gas and particulate components of the atmosphere babs ˆ bag ‡ bap

(15.29)

bscat ˆ bsg ‡ bsp

(15.30)

where bsg bsp bag bap

= = = =

scattering coefficient due to gases (the so-called Rayleigh scattering coefficient) scattering coefficient due to particles absorption coefficient due to gases absorption coefficient due to particles

A ratio bscat/bsg of unity indicates the cleanest possible air (bsp = 0). Thus the higher this ratio, the greater is the contribution of particulate scattering to total light scattering.

644

INTERACTION OF AEROSOLS WITH RADIATION

15.2 VISIBILITY Visibility degradation is probably the most readily perceived impact of air pollution. The term visibility is generally used synonymously with visual range, meaning the farthest distance at which one can see a large, black object against the sky at the horizon. Even if no distant objects are within view, subjective judgments about visual range can be made based on the coloration and light intensity of the sky and nearby objects. For example, one perceives reduced visual range if a distant mountain that is usually visible cannot be seen, if nearby objects look “hazy” or have diminished contrast, or if the sky is white, gray, yellow, or brown, instead of blue. Several factors determine how far one can see through the atmosphere, including optical properties of the atmosphere, amount and distribution of light, characteristics of the objects observed, and properties of the human eye. Visibility is reduced by the absorption and scattering of light by both gases and particles. Absorption of certain wavelengths of light by gas molecules and particles is sometimes responsible for atmospheric colorations. However, light scattering by particles is the most important phenomenon responsible for impairment of visibility. Visibility is reduced when there is significant scattering because particles in the atmosphere between the observer and the object scatter light from the sun and other parts of the sky through the line of sight of the observer. This light decreases the contrast between the object and the background sky, thereby reducing visibility. These effects are depicted in Figure 15.5. To examine the effect of atmospheric constituents on visibility reduction, we consider the case in which a black object is being viewed against a white background. We first define the visual contrast at a distance x from the object CV(x) as the relative difference between the light intensity of the background and the target CV …x† ˆ

FB …x† F…x† FB …x†

(15.31)

where FB(x) and F(x) are the intensities of the background and the object, respectively. At the object (x = 0), F(0) = 0, since the object is assumed to be black and therefore absorbs all light incident on it. Thus CV(0) = 1. Over the distance x between the object and the observer, F(x) will be affected by two phenomena: (1) absorption of light by gases and particles and (2) addition of light that is scattered into the line of sight. Light can reenter the beam by multiple scattering; that is, light scattered by particles outside the beam may ultimately contribute to the irradiance at the target. This is because scattered light, in contrast to absorbed light, is not irretrievably lost from the system—it merely changes direction and is lost from a beam propagating in a particular direction—but contributes to other directions. The greater the individual particle scattering coefficient, number concentration of particles, and depth of the beam,

FIGURE 15.5 Contributions to atmospheric visibility: (1) residual light from target reaching the observer; (2) light from the target scattered out of the observer’s line of sight; (3) airlight from intervening atmosphere scattered into the observer’s line of sight; and (4) airlight constituting the horizon sky.

645

VISIBILITY

the greater will be the multiple scattering contribution to the irradiance at x. Over a distance dx the intensity change dF is the result of these two effects. The fraction of F diminished is assumed to be proportional to dx, since dx is a measure of the amount of suspended gases and particles present. The fractional reduction in F is written dF = bextF dx. In addition, the intensity F can be increased over the distance dx by scattering of light from the background into the line of sight. The increase can be expressed as b´ FB(x)dx, where b´ is a constant. The net change in intensity is given by dF…x† ˆ ‰b´ FB …x†

bext F…x†Šdx

(15.32)

By its definition as background intensity, FB must be independent of x. Thus, along any other line of sight, dFB …x† ˆ 0 ˆ ‰b´ FB …x†

bext FB …x†Šdx

(15.33)

We see that b´ = bext. Thus we find that the contrast CV(x) itself obeys the Beer–Lambert law dCV …x† ˆ bext CV …x† dx

(15.34)

and therefore that the contrast decreases exponentially with distance from the object: CV …x† ˆ exp… bext x†

(15.35)

The lowest visually perceptible brightness contrast is called the liminal contrast or threshold contrast. The threshold contrast has been the object of considerable interest since it determines the maximum distances at which various components of a scene can be discerned. Laboratory experiments indicate that for most daylight viewing conditions, contrast ratios as low as 0.018–0.03 are perceptible. Typical observers can detect a 0.02 or greater contrast between large, dark objects and the horizon sky. A threshold contrast value of 2% (CV = 0.02) is usually employed for visual range calculations. Equation (15.35) can be evaluated at the distance at which a black object has a standard 0.02 contrast ratio against a white background. When the contrast in (15.35) becomes the threshold contrast, the distance becomes the visual range. If CV = 0.02, then xv ˆ

3:912 bext

(15.36)

with xv and bext in similar units (i.e., x in m and bext in m 1). This is called the Koschmeider equation. Thus the visual range can be expressed equivalently in terms of an extinction coefficient (bext) or a distance (xv). If the extinction coefficient is measured along a sight path, then xv is the visual range. If the extinction coefficient is measured at a point, then xv is taken to be the local visual range. The two values of xv are equal in a homogenous atmosphere. At sea level the Rayleigh atmosphere has an extinction coefficient bext of approximately 13.2 × 10 6 m 1 at λ = 520 nm wavelength, limiting visibility in the cleanest possible atmosphere to about 296 km. Rayleigh scattering decreases with altitude and is proportional to air density. At λ = 520 nm wavelength: Altitude above sea level (km)

bsg × 106 (m 1)

0 1.0 2.0 3.0 4.0

13.2 11.4 10.6 9.7 8.8

646

INTERACTION OF AEROSOLS WITH RADIATION

FIGURE 15.6 Median midday visual range (km) in the United States [based on data given in Malm et al. (1994)]. The visibility isopleths are the result of a national visibility and aerosol monitoring network of 36 stations established in 1988 to track spatiotemporal trends of visibility and visibility-reducing parti­ cles in the United States. The major visibility-reducing aerosol species—sul­ fates, nitrates, organics, carbon, and windblown dust—are monitored. Sulfates and organics are responsible for most of the extinction at most locations throughout the United States. In the eastern United States, sulfates contribute about two-thirds of the extinction. In almost all cases, extinction is highest in the summer and lowest in the winter months.

For convenience, a megameter, denoted Mm and equal to 1000 km, is often used as the unit of distance; in this unit the sea-level Rayleigh extinction coefficient is 13 Mm 1. Rayleigh scattering thus represents an irreducible level of extinction against which other extinction components (e.g., anthropogenic pollutants) can be compared. Figure 15.6 shows median midday visual range in the United States in the mid-1990s. The greatest visual range in the United States is present in the western states. Indeed, maintenance of the visual range in this region of the country, where there are many national parks, has been a stated objective in the US Clean Air Acts. Scattering by particles of sizes comparable to the wavelength of visible light (the Mie scattering range) is mostly responsible for visibility reduction in the atmosphere. Scattering by particles accounts for 50–95% of extinction, depending on location, with urban sites in the 50–80% range and nonurban sites in the 80–95% range. Particle absorption is on the order of 5–10% of particle extinction in remote areas; its contribution can rise to 50% in urban areas. Values of 10–25% are typical for suburban and rural locations. As we will see shortly, particles in the range 0.1–1 μm in diameter are the most effective, per unit aerosol mass, in reducing visibility. The scattering coefficient bscat is more or less directly dependent on the atmospheric aerosol concentration in this size range. Scattering by air molecules usually has a minor influence on urban visibility. For visual ranges exceeding 30 km the effect of air molecules must be taken into account. The addition of small amounts of submicrometer particles throughout the viewing distance tends to whiten the horizon sky, making distant dark objects and intervening airlight appear more gray. On the whole, as we have seen, particles generally scatter more light in the forward direction than in other directions (see Figure 15.4); thus haze appears bright in the forward scatter mode and dark in the backscatter mode. Nitrogen dioxide (NO2) is the only light-absorbing atmospheric gas present in optically significant quantities in the troposphere. NO2 absorbs light selectively and is strongly blue-absorbing; as a result, its presence will color plumes red, brown, or yellow. Despite the coloration properties of NO2, nevertheless, the brown haze characteristic of smoggy atmospheres is largely a result of aerosol scattering rather than NO2 absorption (Charlson and Ahlquist 1969). Table 15.3 presents estimated contributions of chemical species to the wintertime extinction coefficient in Denver, Colorado in the early 1980s. We note, for example, that in Denver during winter,

647

SCATTERING, ABSORPTION, AND EXTINCTION COEFFICIENTS FROM MIE THEORY

TABLE 15.3 Contribution of Chemical Species to the Extinction Coefficient bext in Denver Wintertime Fine-particle species

Mean percent contribution

(NH4)2SO4 NH4NO3 Organic C Elemental C (scattering) Elemental C (absorption) Other NO2

20.2 17.2 12.5 6.5 31.2 6.6 5.7

Total

100

Source: Groblicki et al. (1981).

elemental carbon was responsible for about 40% of visibility reduction. The disproportionately large influence of black carbon on light extinction, relative to its concentration, arises in part, as we will see, because black carbon is more effective than nonabsorbing aerosol particles (such as sulfates and nitrates) of the same size in attenuating light (Faxvog and Roessler 1978).

15.3 SCATTERING, ABSORPTION, AND EXTINCTION COEFFICIENTS FROM MIE THEORY Aerosol scattering, absorption, and extinction coefficients are functions of the particle size, the complex refractive index of the particles m, and the wavelength λ of the incident light. The extinction coefficient for a monodisperse ensemble of particles was given in terms of the dimensionless extinction efficiency by (15.27). With a population of different-sized particles of identical refractive index m with a number size distribution function of n(Dp), the extinction coefficient is given by1

bext …λ† ˆ

Dmax p

∫0

πD2p 4

Qext …m; α†n…Dp †dDp

(15.37)

is an upper limit diameter for the particle population. Similar expressions can be written for where Dmax p bscat(λ) and babs(λ) in terms of Qscat and Qabs. It is frequently useful to express bext in terms of the aerosol mass distribution function nM …Dp † ˆ ρp

πD3p 6

n…Dp †

(15.38)

where ρp is the density of the particle. The result is bext …λ† ˆ

1

Dmax p

∫0

3 Q …m; α†nM …Dp †dDp 2ρp Dp ext

(15.39)

Although babs, bscat, and bext can be decomposed into contributions from gases and particles as in (15.29) and (15.30), it is common practice to use these terms to refer to the particulate component only.

648

INTERACTION OF AEROSOLS WITH RADIATION

which can be written as bext …λ† ˆ

Dmax p

∫0

Eext …Dp ; λ; m†nM …Dp †dDp

(15.40)

where Eext(Dp, λ, m) is the mass extinction efficiency: Eext …Dp ; λ; m† ˆ

3 Q …m; α† 2ρp Dp ext

(15.41)

Similarly, the mass scattering and mass absorption efficiencies are Escat …Dp ; λ; m† ˆ

3 Q …m; α† 2ρp Dp scat

(15.42)

Eabs …Dp ; λ; m† ˆ

3 Q …m; α† 2ρp Dp abs

(15.43)

The mass scattering efficiencies of spherical particles of water, ammonium nitrate and sulfate, silica (SiO2), and carbon (expressed in units of m2 g 1) are shown as a function of particle diameter in Figure 15.7 at a wavelength of λ = 550 nm. We see that particles between 0.1 and 1.0 μm diameter scatter light most efficiently. The mass scattering efficiencies of the four substances in Figure 15.7 exhibit peaks as follows: H2O, ∼ 0.85 μm; NH4NO3, ∼ 0.5 μm; (NH4)2SO4, ∼ 0.5 μm; carbon, ∼ 0.2 μm. Figure 15.8 shows how the mass scattering and absorption efficiencies vary with systematic variation of the real and imaginary parts of the refractive index. Figure 15.8a–d shows how the imaginary component k influences the mass scattering efficiency at a constant value of the real part n. We note that, at any given value of n,

FIGURE 15.7 Mass scattering efficiencies of homogeneous spheres of (NH4)2SO4, NH4NO3, carbon, H2O, and silica at λ = 550 nm.

SCATTERING, ABSORPTION, AND EXTINCTION COEFFICIENTS FROM MIE THEORY

FIGURE 15.8 Mass scattering (a–d) and absorption (e–h) efficiencies for materials having refractive indices m = n + ik: n = 2.00, 1.75, 1.53, 1.25, and k = 0, 0.1, 0.25 [Note that m = 1.53 0k is the refractive index of (NH4)2SO4.]

increasing k lowers the peak value of the mass scattering efficiency and decreases the diameter at which the scattering peak occurs. Both reflect the fact that, as k increases, more of the total photon energy is being transformed into absorption rather than scattering. We also note that the larger the value of n, the higher the peak value of Escat and the smaller the diameter at which peak scattering occurs. Figure 15.8e–h shows the corresponding behavior of the mass absorption efficiency, Eabs. At any given value of n, increasing k raises the overall Eabs curve. The peak value of Eabs is weakly affected by n (larger n, larger Eabs at the peak). More importantly, for the lower values of n, Eabs remains close to its peak value well into the sub-0.1 μm range. The mass absorption efficiency remains high for absorbing particles less than 0.1 μm in diameter, and, as a result, overall light extinction by particles with diameters smaller than 0.1 μm is due primarily to absorption. The black plume of soot from an oil burner exemplifies this behavior, where most of the particles are smaller than 0.1 μm in diameter. It is of interest to examine the dependence of the mass scattering and absorption coefficients on particle diameter in the two extremes of α  1 (Dp  λ) and α  1 (Dp  λ). Consider first Escat(Dp, λ, m). In the Rayleigh regime (Dp  λ), from (15.19) Qscat  D4p , whereas in the regime Dp  λ, Qscat becomes independent of particle size. Thus from (15.42) the mass scattering efficiency varies with particle size in

649

650

INTERACTION OF AEROSOLS WITH RADIATION

the Rayleigh and large-particle regimes according to Escat …Dp ; λ; m†  D3p  Dp

Dp  λ 1

Dp  λ

Thus the mass scattering efficiency increases as D3p for the smallest particles and falls off as Dp 1 for large particles. This behavior is most clearly evident from the Escat curve for carbon in Figure 15.7. The mass absorption coefficient behavior for Dp  λ is seen from (15.21), from which Qabs  Dp. There is no straight­ forward way to see the dependence of Qabs on Dp for Dp  λ, but Qabs  D0p in this region. Then, from (15.43), we obtain the dependence of mass absorption efficiency on particle size in the two extremes as Eabs …Dp ; λ; m†  D0p  Dp

Dp  λ 1

Dp  λ

Therefore Eabs is independent of particle size for small particles and decreases as Dp 1 for large particles. Any of the curves for k = 0.25 in Figure 15.8e–h illustrate this behavior. In the case of a multicomponent particle, the index of refraction m reflects the mixture of species in the particle and can be approximated by the volume average of the indices of refraction of the individual components. Doing so assumes implicitly that the particle is a homogeneous mixture. This is not quite accurate if the particle, in fact, contains a core of one type of material surrounded by a shell of another. The volume-average index of refraction m for an aerosol containing n components is calculated from n

mˆ

mi f i

(15.44)

iˆ1

where mi is the index of refraction of component i and fi is the volume fraction of component i. Ångstrom Exponent It has proved to be useful in some cases to represent the wavelength dependence of the aerosol extinction coefficient bext, by bext  λ å. The exponent, å, is called the Ångstrom exponent. The Ångstrom exponent is calculated from measured values of the extinction coefficient as a function of wavelength by a ˆ ≅

d log bext d log λ log…bext1 =bext2 † log…λ1 =λ2 †

(15.45)

If the aerosol number distribution is represented in a certain size range by a relation of the form dN  Dp v dDp

(15.46)

then å = v 3. Figure 15.9 shows the Ångstrom exponent for a lognormally distributed water aerosol (σg = 2.0) with refractive index m = 1.33 0i, as a function of volume mean diameter in the range λ = 550–700 nm. In the Rayleigh regime (Dp  0.1 μm), the extinction coefficient varies with wavelength to a power between 3 and 4, reflecting the contributions of both scattering and absorption. In the large-particle regime, the Ångstrom exponent ranges between 1 and 0, again reflecting the wavelength dependence of Qabs and Qscat in this region.

651

CALCULATED VISIBILITY REDUCTION BASED ON ATMOSPHERIC DATA

FIGURE 15.9 Ångstrom exponent for a lognormally distributed water aerosol (σg = 2.0) with refractive index m = 1.33 0i in the wavelength range λ = 550 to 700 nm. (Courtesy of J. A. Ogren.)

15.4 CALCULATED VISIBILITY REDUCTION BASED ON ATMOSPHERIC DATA Table 15.4 presents typical aerosol size distributions and concentrations based on measurements in many locations, including mass median diameters Dpg, geometric standard deviations σg, and volume concentrations V(μm3 cm 3) of the nuclei, accumulation, and coarse-particle modes of eight classifica­ tions of aerosol. We note only a small variability in the accumulation and coarse-mode size distributions for the variety of aerosol classes (excluding the marine aerosol). The average specifications for the accumulation and coarse modes are as follows:

Mode

Accumulation Coarse

Dpg, μm

σg

0.29 ± 0.06 6.3 ± 2.3

2.0 ± 0.1 2.3 ± 0.2

TABLE 15.4 Size Distributions (Lognormal) of Different Classes of Atmospheric Aerosol Nuclei mode Aerosol type

Dpg (μm)

Marine background Clean continental background Average background Background and aged urban plume Background and local sources Urban average Urban and freeway Labadie plume (1976)b

0.019 0.03 0.034 0.028 0.021 0.038 0.032 0.015

a

Accumulation mode

σg

V (μm3 cm 3)

Dpg (μm)

1.6a 1.6 1.7 1.6 1.7 1.8 1.74 1.5

0.0005 0.006 0.037 0.029 0.62 0.63 9.2 0.1

0.3 0.35 0.32 0.36 0.25 0.32 0.25 0.18

σg

V (μm3 cm 3)

Dpg (μm)

σg

V (μm3 cm 3)

2.0 2.1 2.0 1.84 2.11 2.16 1.98 1.96

0.10 1.5 4.45 44.0 3.02 38.4 37.5 12.0

12.0 6.2 6.04 4.51 5.6 5.7 6.0 5.5

2.7 2.2 2.16 2.12 2.09 2.21 2.13 2.5

12.0 5.0 25.9 27.4 39.1 30.8 42.7 24.0

Assumed. Typical distribution observed in the plume from the Labadie coal-fired power plant near St. Louis, Missouri.

b

Coarse-particle mode

652

INTERACTION OF AEROSOLS WITH RADIATION

The contribution to the total scattering coefficient bscat can be calculated for each of the modes of the different aerosol classes in Table 15.4 from the theory in Section 15.3. The accumulation mode is found to be the dominant scattering mode, with the coarse mode contributing a small amount and the nuclei mode a negligible amount. For the two background aerosol cases: Fractional modal contribution to bscat from bscat (m 1)

Type of aerosol

Clean continental background Average background

0.23 × 10 0.57 × 10

4 4

Nuclei

Accumulation

Coarse

Rayleigh

0.01 0.01

0.39 0.46

0.17 0.36

0.43 0.17

These calculations indicate that for background conditions the accumulation mode is a larger contributor to light scattering than the coarse mode but that the coarse mode is a nonnegligible component of the scattering coefficient. This situation contrasts with that of polluted urban atmo­ spheres in which the accumulation mode is responsible for more than 90% of the total scattering coefficient. Larson et al. (1988) analyzed atmospheric composition and visibility in Los Angeles on 2 days in 1983: the first day, April 7, representing relatively clean conditions, and the second, August 25, characterized by heavy smog. Table 15.5 gives the ambient measured concentrations of the species sampled on the two days. In preparing Table 15.5 Larson et al. (1988) used the procedure of Stelson and Seinfeld (1981) for developing a material balance on the chemical composition of the aerosol samples. Stelson and Seinfeld (1981) showed that urban aerosol mass can generally be accounted for from measurements of SO24 , Cl , Br , NO3 , NH‡4 , Na+, K+, Ca2+, Fe, Mg, Al, Si, Pb, carbonaceous

TABLE 15.5 Atmospheric Concentrations Measured on 2 Days in Los Angeles Component (concentration) 3

(NH4)2SO4 (μg m ) NH4NO3 (μg m 3) NaNO3 (μg m 3) Na2SO4 (μg m 3) Elemental carbon (μg m 3) Organic carbon (μg m 3) Al2O3 (μg m 3) SiO2 (μg m 3) K2O (μg m 3) CaO (μg m 3) Fe2O3 (μg m 3) PbO (μg m 3) NO2 (ppm) O3 (ppm) CO (ppm) T (°C) RH (%) bsp (m 1) bsg (m 1) (Rayleigh) bag (m 1) (NO2) bap (m 1) (carbon) Calculated bext (m 1) Source: Larson et al. (1988).

Clear day April 7, 1983 3.54 1.23 — — 0.99 6.78 2.82 4.53 0.36 0.45 0.91 0.06 0.05 0.05 1.6 23.0 23.6 0.259 × 10 0.111 × 10 0.012 × 10 0.093 × 10 0.475 × 10

4 4 4 4 4

Heavy smog Aug. 25, 1983 14.90 1.79 12.80 — 6.37 32.74 9.34 15.54 1.45 1.83 4.58 0.72 0.10 0.21 3.39 29.8 50.5 4.08 × 10 4 0.107 × 10 4 0.030 × 10 4 0.787 × 10 4 5.00 × 10 4

CALCULATED VISIBILITY REDUCTION BASED ON ATMOSPHERIC DATA

material, and aerosol water. Their method assumes that trace metals are present in the form of common oxides: Element

Oxide form

Al Ca Fe Si Mg Pb Na K

Al2O3 CaO Fe2O3 SiO2 MgO PbO Na2O K 2O

In Table 15.5 this procedure was followed with the exception of Na, which was assumed to be in the form of an ionic solid. Mg was present at negligible levels, and thus its chemical form is unimportant to the aerosol mass balance. To account for hydrogen and oxygen present in the hydrocarbons, the mass of organic carbonaceous material was taken to be 1.2 times the organic carbon mass measured (Countess et al. 1980). The ionic material was assumed to be distributed as follows: Na+ was associated with Cl . NH‡4 was associated with SO24 . NH4‡ remaining, if any, was associated with NO3 .

FIGURE 15.10 Aerosol volume distributions measured on 2 days in Los Angeles (Larson et al. 1988).

653

654

INTERACTION OF AEROSOLS WITH RADIATION

Na+ remaining, if any, was associated with remaining NO3 , if any. Na+ remaining, if any, was associated with remaining SO24 , if any. Figure 15.10 shows the aerosol volume distributions and gives the total extinction coefficients on the two days at λ = 550 nm. The estimated contributions to bext on the two days are given at the bottom of Table 15.5. Note that the total calculated bext on the smoggy day was somewhat smaller than that measured since not all atmospheric constituents were included in the calculation. For a comprehensive review of visibility, we refer the reader to Watson (2002).

APPENDIX 15 CALCULATION OF SCATTERING AND EXTINCTION COEFFICIENTS BY MIE THEORY This appendix outlines the evaluation of the coefficients (Pilinis 1989) ak and bk in (15.13) and (15.14). If we define (15A.1) Ak …y† ˆ ψ´k …y†=ψk …y† then (Wickramasinghe 1973) ak ˆ

Ak …y† k ‡ Re ζk …α† m α Ak …y† k ‡ ζk …α† m α k Re ζk …α† α k Ak …y†m ‡ ζk …α† α

Ak …y†m ‡ bk ˆ

Re ζk 1 …α† (15A.2) ζk 1 …α† Re ζk 1 …α† (15A.3) ζk 1 …α†

To generate Ak(y), we use the recurrence relation k k ‡ y y

Ak …y† ˆ

1

Ak 1 …y†

(15A.4)

with A0(y) = cos y/sin y. For ζk(α), we use the relation ζk …α† ˆ

2k

1 p

ζk 1 …α†

ζk 2 …α†

(15A.5)

with ζ 1 …α† ˆ cos α i sin α ζ0 …α† ˆ sin α ‡ i cos α

(15A.6) (15A.7)

Using (15A.1)–(15A.7), one may calculate the sums in (15.13) and (15.14) by adding terms until a desired accuracy is achieved. Convergence is rapid and, for most cases, the number of terms required is less than 20.

PROBLEMS 15.1A Rayleigh scattering is the irreducible minimum of light scattering owing to air molecules along an atmospheric sight path. The amount of scattered light varies as λ 4. The Rayleigh scattering

655

PROBLEMS

coefficient at three wavelengths is bsg ˆ 25:9

293 p T

λ ˆ 450 nm

ˆ 11:4

293 p T

λ ˆ 550 nm

ˆ 5:8

293 p T

λ ˆ 650 nm

where T is in K and p is in atmospheres. Calculate bsg at 550 nm at sea level, 1 km, 2 km, and 3 km altitude. 15.2A Light extinction can be divided into the sum of its scattering and absorption components as follows bext ˆ bsg ‡ bag ‡ bsp ‡ bap where bext = light extinction coefficient, bsg = Rayleigh scattering (light scattering by molecules of air), bag = light absorption because of gases (mainly NO2), bsp = light scattering by particles, and bap = light absorption by particles. The scattering and absorption by gases (bsg and bag) can be calculated knowing the air pressure (altitude) and temperature, and the concentration of NO2, respectively. To deal with the particle-related extinction (bsp and bap), a standard approach is to allocate portions of the extinction to each species of the mixture and then summarize the contributions to arrive at the total particle-related extinction. With this approach, one sums up the individual contributions bsp ˆ esulfate ‰sulfateŠ ‡ enitrate ‰nitrateŠ ‡ eOC ‰organic carbonŠ ‡ esoil ‰soilŠ ‡ ecoarse ‰coarseŠ and, for absorption of light by particles bap ˆ eBC ‰black carbonŠ where the e values are the extinction efficiencies. The units of e are M m 1 per μg m 3; hence m2 g 1. Extinction efficiencies depend on the size distribution and the molecular composition of the sulfates, nitrates, and organic carbon. The extinction efficiencies for hygroscopic substances (sulfate, nitrate, and organic carbon) are dependent on the relative humidity. Values are usually reported for dry particles; the uptake of water can multiply the given sulfate and nitrate efficiencies manyfold at high relative humidities. The effect of water uptake on the organic carbon efficiency is not as well established as that for the inorganic salts. Ranges of dry extinction efficiencies are esulfate ˆ 1:5

4 m2 g

1

enitrate ˆ 2:5

3 m2 g

1

eOC ˆ 1:8 esoil ˆ 1 ecoarse ˆ 0:3 eBC ˆ 8

4:7 m2 g 1:25 m2 g 0:6 m2 g 12 m2 g

1

1 1 1

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INTERACTION OF AEROSOLS WITH RADIATION

Calculate the visual range of an atmosphere at a 0.02 contrast ratio for which esulfate = 3 m2 g 1, enitrate = 3 m2 g 1, eOC = 4 m2 g 1, and eBC = 10 m2 g 1, and for which ‰SulfateŠ

ˆ 20 μg m

‰NitrateŠ

ˆ 5 μg m

3

‰Organic carbonŠ ˆ 25 μg m ‰Black carbonŠ

3

ˆ 7:5 μg m

3 3

15.3B Consider a droplet size distribution given by the modified gamma distribution n…r† ˆ

r N0 Γ…β†rn rn

β 1

exp

r rn

where N0 is the total number concentration of particles; Γ(β) is the gamma function, with β as integer; and rn is related to the mean radius rm of the population by rm = (β + 1)rn. The transmission of a light beam of wavelength λ through the droplet population is T…λ† ˆ exp‰ τ…λ†Š where the optical thickness for pathlength ℓ is given by τ…λ† ˆ ℓ πr2 Qext …r; λ†n…r†dr ∫ N0 typically ranges between 50 and 500 cm 3. rm ranges between 5 and 25 μm, and values of β = 2 and 5 have been used for tropospheric clouds. Consider rm = 20 μm and an optical pathlength of 1 m. Calculate and plot (a) the size distribution and (b) the transmission of light through a droplet population with this modified gamma distribution for both β = 2 and 5. For the plot of T(λ), consider N0 ranging from 0 to 800 cm 3. Identify the sensitivity of the number concentration at which T = 0.5 to the two values of β. [The size distribution can be plotted as n(r)/N0.] 15.4D Calculate the scattering coefficient bscat for a lognormally distributed carbon aerosol of mass concentration 20 μg m 3, with Dpg ˆ 0:02 μm and σg = 3.0 at λ = 550 nm. Note that it will be necessary to numerically evaluate the appropriate integral based on the data in Figure 15.7. What is the visibility under these conditions?

REFERENCES Bohren, C. F., and Huffman, D. R. (1983), Absorption and Scattering of Light by Small Particles, John Wiley & Sons, Inc., New York. Bond, T. C., and Bergstrom, R. W. (2006), Light absorption by carbonaceous particles: An investigative review, Aerosol Sci. Technol. 40, 27–67. Charlson, R. J., and Ahlquist, N. C. (1969), Brown haze: NO2 or aerosol? Atmos. Environ. 3, 653–656. Countess, R. J., et al. (1980), The Denver winter aerosol: A comprehensive chemical characterization, J. Air Pollut. Control Assoc. 30, 1194–1200. Faxvog, F. R., and Roessler, D. M. (1978), Carbon aerosol visibility versus particle size distribution, Appl. Opt. 17, 2612–2616. Groblicki, P. J., et al. (1981), Visibility-reducing species in the Denver “brown cloud”—I. Relationships between extinction and chemical composition, Atmos. Environ. 15, 2473–2484.

REFERENCES Hale, G. M., and Querry, M. R. (1973), Optical constants of water in the 200 nm to 200 μm wavelength region, Appl. Opt. 12, 555–563. Kerker, M. (1969), The Scattering of Light and Other Electromagnetic Radiation, Academic Press, New York. Larson, S. M., et al. (1988), Verification of image processing–based visibility models, Environ. Sci. Technol. 22, 629–637. Malm, W. C., et al. (1994), Spatial and seasonal trends in particle concentration and optical extinction in the United States, J. Geophys. Res. 99, 1347–1370. Marshall, S. F., et al. (1995), Relationship between asymmetry parameter and hemispheric backscatter ratio: Implications for climate forcing by aerosols, Appl. Opt. 34, 6306–6311. Nemesure, S., et al. (1995), Direct shortwave forcing of climate by the anthropogenic sulfate aerosol: Sensitivity to particle size, composition, and relative humidity, J. Geophys. Res. 100, 26105–26116. Pilinis, C. (1989), Numerical simulation of visibility degradation due to particulate matter: Model development and evaluation, J. Geophys. Res. 94, 9937–9946. Stelson, A. W. (1990), Urban aerosol refractive index prediction by partial molar refraction approach, Environ. Sci. Technol. 24, 1676–1679. Stelson, A. W., and Seinfeld, J. H. (1981), Chemical mass accounting of urban aerosol, Environ. Sci. Technol. 15, 671–679. Tegen, I., et al. (1996), The influence of climate forcing of mineral aerosols from disturbed soils, Nature 380, 419–423. Watson, J. G. (2002), Visibility: Science and regulation, J. Air Waste Manage. Assoc. 52, 628–713. Weast, R. C. (1987), Physical constants of organic compounds, in CRC Handbook of Chemistry and Physics, 68th ed., R. C. Weast (ed.), CRC Press, Boca Raton, FL, pp. B67–B146. Wickramasinghe, N. C. (1973), Light Scattering Functions for Small Particles with Applications in Astronomy, John Wiley & Sons, Inc., New York. Wiscombe, W., and Grams, G. (1976), The backscattered fraction in two-stream approximations, J. Atmos. Sci. 33, 2440–2451.

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P A R T I V

Physical and Dynamic Meteorology, Cloud Physics, and Atmospheric Diffusion

We begin this part of the book with an explanation of the notation used in Chapters 16 and 17. The customary notation in physical meteorology and cloud physics differs somewhat from that used up to this point (in Chapters 1–15). To be consistent with the customary notation used in meteorology and cloud physics, the notation in Chapters 16 and 17 is as follows: Gas constant on a molar basis Gas constant for dry air on a mass basis Gas constant for water vapor on a mass basis Molecular mass of dry air (kg molecule 1) Molecular mass of water (kg molecule 1) Latent heat of vaporization of water Specific volume (1/ρ) Specific heat capacity of air at constant volume Specific heat capacity of air at constant pressure Specific heat capacity of dry air at constant pressure Specific heat capacity of water vapor at constant pressure Specific heat capacity of liquid water at constant pressure Specific heat capacity of water vapor at constant volume Partial pressure of water vapor Saturation vapor pressure of water vapor (implied to be over a flat water surface)

R Rd Rv md mv lv α cv cp cpd cpv cpl cvv e es

CHAPTER

16

Physical and Dynamic Meteorology

Meteorology is the study of the atmosphere, its motion, and its phenomena. The word meteorology comes from the Greek word meteoros, meaning “suspended in the air.” The term itself was introduced by Aristotle, who wrote Meteorologica in 340 BC, summarizing what was known at the time about weather and climate, including clouds, rain, snow, wind, hail, and thunder. This first treatment of atmospheric phenomena by Aristotle was speculative and based on qualitative observations and philosophy. There was little progress in the next 2000 years until the invention of the thermometer (end of sixteenth century), the barometer (in 1643), and the hygrometer (eighteenth century) that allowed progress in the science of meteorology, especially in the nineteenth and twentieth centuries. Weather is a result of the atmosphere’s transfer of heat from places that absorb a large amount of sunlight to those that absorb less. The transport of heat occurs both horizontally and vertically, as heat flows from the tropics to the poles and from Earth’s surface to the air aloft. The latent heat released or absorbed as water changes phase plays a key role in energy transfer in the atmosphere. Parcels of air in the troposphere are continually ascending and descending. Air parcels rise when they are more buoyant than their surroundings. A rising air parcel experiences a drop in the surrounding atmospheric pressure and cools by expanding adiabatically. As a moist parcel of air rises and cools, it eventually reaches a point at which the water vapor in it saturates and condenses to form cloud droplets, releasing latent heat. The air in our atmosphere is continuously in motion, with a complicated three-dimensional flow field that varies in scales ranging from a few meters (around a building or a small hill) to 1000s of kilometers (a major storm). At the same time, the temperature and relative humidity of the atmosphere are highly variable. The importance of meteorology for air quality in a given area is clear from Figure 16.1, which shows the PM2.5 concentrations in a large urban area. Both local and regional anthropogenic emissions of the major pollutants in the eastern United States are relatively constant from day to day, varying by a factor of 0, and the parcel does work on its surroundings. We can express the work in terms of the specific volume of the parcel, α, as dw = pdα, where α is the inverse of the density (α = 1/ρ). du can be written as cvdT, where cv is the specific heat capacity of air at constant volume (for dry air, cv = 717.5 J K 1 kg 1). Then (16.1) becomes cv dT ˆ dq

p dα

(16.2)

663

TEMPERATURE IN THE LOWER ATMOSPHERE

We assume that the parcel rises or falls adiabatically (without transfer of heat across its boundaries). Thus, dq = 0, and cv dT ˆ p dα

(16.3)

It is useful to express (16.3) in terms of dT and dp. To do so, we use the ideal-gas law, which is usually stated in terms of the universal gas constant, R = 8.314 J K 1 mol 1. For applications in physical meteorology, the gas constant is defined on a mass basis, in which case the value of R depends on the molecular weight of the specific gas involved. The gas constant for dry air expressed on a mass basis is Rd = 287 J K 1 kg 1, where the molecular weight of dry air is 28.97 g mol 1. The ideal gas law for dry air is pα ˆ Rd T

(16.4)

or, equivalently, p ˆ ρRd T Noting that d(pα) = pdα + αdp, and that d(pα) = RddT, (16.3) becomes …cv ‡ Rd † dT ˆ α dp

(16.5)

The specific heat capacity at constant pressure, cp, is defined in terms of cv by cv + Rd (cp = 1005 J K for dry air). Thus, (16.5) becomes cp dT ˆ α dp

1

kg

1

(16.6)

which is the basic relationship between the changes of T and p for an adiabatic parcel. Using (16.4), (16.6) becomes cp

dT dp ˆ Rd T p

(16.7)

Say that the parcel starts out at an altitude where the pressure is p and the temperature is T and is brought adiabatically down to Earth’s surface at pressure po. For the adiabatic change from (T, p) to (To, po), by integrating (16.7), we obtain T o po

Rd =cp

ˆ Tp

Rd =cp

(16.8)

or To ˆ T

p po

Rd =cp

(16.9)

To is the temperature of the air parcel brought adiabatically from (T, p) to the surface at pressure po. To is given the symbol θ and termed the potential temperature:

θˆT

p po

Rd =cp

(16.10)

Consider an atmosphere in which the vertical temperature structure is such that θ is constant, i.e. the same at any altitude z. In an atmosphere in which θ is constant, an air parcel brought from any altitude adiabatically to the surface pressure po will have the same temperature on arrival at the surface. In this atmosphere, the temperature decreases with increasing altitude at a constant rate. The negative of the rate of change of the atmospheric temperature with height, dT/dz, is called the lapse rate. For a parcel of dry air moving up and down adiabatically, dT/dz is called the dry adiabatic lapse rate and is denoted by

664

PHYSICAL AND DYNAMIC METEOROLOGY

the symbol, Γd. The “dry” means that no condensation of water occurs as the air parcel moves upward and cools. The air parcel can contain water vapor, just not enough for it to condense. We can obtain the actual expression for dT/dz in the dry adiabatic atmosphere from (16.10). Since in this atmosphere θ is a constant with respect to z, dθ/dz = 0. Differentiating (16.10) with respect to z and using the hydrostatic relation for pressure, dp=dz ˆ ρg and the ideal-gas law, we obtain dθ ˆ dz Since Γd is defined as

p po

Rd =cp

dT g ‡ dz cp

ˆ0

(16.11)

dT/dz, the dry adiabatic lapse rate is Γd ˆ

g cp

(16.12)

and the temperature at any height z on the so-called dry adiabat is T…z† ˆ T o

Γd z

(16.13)

For perfectly dry air, Γd ˆ 9:77 K km 1 . This value is considerably larger than that typically observed in the troposphere of ∼6 ---7 K km 1. The difference is mainly due to the presence of condensing water vapor, as will be discussed later. Dry Adiabatic Lapse Rate of an Air Parcel Containing Noncondensing Water Vapor We noted above that the “dry” parcel usually contains some water vapor, just not enough for it to condense. The value of Γd of 9.77 K km 1 is based on parameters for completely dry air. The question arises – how does the value of Γd change as a function of the water vapor content of the air. Since Γd ˆ g=cp ; the effect of the presence of water vapor enters through the value of cp. If wv is the water vapor mass mixing ratio, (mass density of the water vapor divided by the mass density of dry air only) then cp ˆ …1

wv † cpd ‡ wv cpv

(16.14)

where cpd = 1005 J K 1 kg 1 and the heat capacity of water vapor, cpv = 1952 J K 1 kg 1. The higher the water vapor content, the larger the heat capacity of the air parcel, and the smaller the adiabatic lapse rate. The atmospheric concentration of water vapor (see, e.g., Figure 7.1) ranges from a few g m 3 at low temperatures to as much as 40 g m 3 at 40 °C. The mass concentration of air at this latter condition (p = 1 atm, T = 313 K) is 1.1 × 103 g m 3. The water vapor mass mixing ratio is then wv = 40/ (40 + 1100) = 0.035, and cp = 1038 J K 1 kg 1. In this rather extreme case, Γd ˆ 9:45 K km 1 , as compared with 9.77 K km 1 for perfectly dry air, a difference of 3.3%. The effect of the water vapor content of air on the dry adiabatic lapse rate is generally Γd ;that is, if the atmospheric temperature decreases faster than that of the adiabatic rising parcel, then, since Ta(z) < To and (Γd =Λ) 1 < 0, the term in brackets in (16.20) is positive, and B > 0. As the parcel rises, its vertical motion is enhanced by the surrounding atmosphere. If Λ < Γd ;then one can show (considering the cases Λ > 0 and Λ < 0) that the acceleration given by (16.20) is negative, or B < 0.

16.2 ATMOSPHERIC STABILITY Air parcels in the atmosphere are continually displaced upward and downward owing to turbulence and overall atmospheric motion. If a parcel is displaced, say, upward, it can continue to rise, return back to its initial position, or remain at its new altitude. If the parcel continues to rise, then the atmosphere is

666

PHYSICAL AND DYNAMIC METEOROLOGY

termed unstable; if the parcel returns to its original position, the atmosphere is termed stable, and if the parcel simply remains at its new position, the atmosphere is called neutral. Atmospheric stability thus refers to the tendency of the atmosphere to resist or enhance an initial displacement of a parcel of air. The nature of the stability of the atmosphere is crucial when considering, for example, how material released into the atmosphere disperses or simply how the atmosphere itself mixes. Here we develop a criterion that indicates the stability state of the atmosphere. Consider the displacement of an air parcel of volume V from an initial height zo to zo + Δz. The air density in the parcel is ρp, and that of the surrounding atmosphere is ρa. As we have just seen, the difference in density between the parcel and the surrounding air gives rise to a buoyancy force. If we apply Newton’s second law to the parcel, we get ρp V

d2 Δz ˆ g…ρa dt2

ρp †V

(16.21)

where the right-hand side (RHS) of (16.21) is the buoyancy force acting on the parcel and the left-hand side (LHS) is the mass of the parcel times its acceleration. Equation (16.21) becomes ρ d2 Δz ˆg a 2 ρp dt

1

(16.22)

where the RHS of (16.22) is just B. Since the pressure inside the parcel is always equal to that of its surroundings: ρa T p θ p ˆ ˆ ρp T a θ a

(16.23)

The parcel is displaced adiabatically, so θp = θa(zo); thus, the potential temperature of the parcel remains that at zo. The potential temperature of the atmosphere is not necessarily adiabatic, that is, dθ/dz = 0 in (16.11) only if dT/dz = g/cp. For small Δz, θa(z) can be approximated by a first-order Taylor expansion: θa …z† ˆ θa …zo † ‡

dθa dz

Δz

(16.24)

zo

From (16.23), we perform a Taylor expansion of θp(z)/θa(z): θp …z† θp …zo † d θp …z† ˆ ‡ θa …z† θa …zo † dz θa …z†

Δz

(16.25)

zo

Now d θp … z † dz θa …z†

ˆ

dθp θa dz

zo

θp θ2a

dθa dz

(16.26) zo

Since θp does not change, dθp/dz = 0, so the RHS of (16.26) is θp …zo † dθa θ2a …zo † dz

(16.27) zo

Thus, (16.25) becomes (since θp(zo) = θa(zo)) θp …z† ˆ θa …z†

1

1 dθa θa …zo † dz

Δz zo

(16.28)

667

ATMOSPHERIC STABILITY

Substituting (16.23) and (16.28) into (16.22), one finds that the equation governing the parcel’s motion is d2 Δz ‡ N 2 Δz ˆ 0 dt2

(16.29)

where Nˆ

g dθa θa dz

1

=2

(16.30)

is called the Brunt-Väisälä frequency. The nature of the solution of (16.29) depends on the sign of dθa/dz. The general solution of (16.29) is z…t† ˆ A1 exp…iNt† ‡ B1 exp… iNt†

(16.31)

where A1 and B1 are arbitrary constants. If N2 < 0, i.e. dθa/dz < 0, N is imaginary and the solution is z…t† ˆ A1 exp…jNjt† ‡ B1 exp… jNjt†

(16.32)

The first term on the RHS of (16.32) increases exponentially with t; thus, for any upward or downward displacement, the parcel will accelerate away from its initial position, and the atmosphere is unstable. If N2 > 0, that is, dθa/dz > 0, then the solution of (16.29) is z…t† ˆ A1 cos …Nt† ‡ B1 sin …Nt†

(16.33)

and the parcel oscillates vertically with frequency N about its initial position. Since the perturbed parcel remains about its initial position, this situation is stable. Finally, if dθa/dz = 0, the atmospheric potential temperature is a constant, and the atmosphere is neutral. To summarize: dθa ˆ dz

>0 0 Γd , the displaced parcel continues to move in the direction of its displacement. In this case, the atmosphere is unstable.

balloon-borne radiosonde.) Consider the two cases shown in Figure 16.2. In the left panel, a parcel displaced upward becomes cooler than its surroundings and therefore denser and will tend to return to its initial position. In this case, Γd > Λ, and the atmosphere is stable. When Λ > Γd, as in the right panel, the air parcel wants to keep moving in the direction of the displacement, and the atmosphere is unstable. This is exactly what we deduced by analysis of (16.20). The radiative cooling of the ground during the night and its heating by solar radiation during the day cause diurnal changes in the stability of the lower atmosphere. During the night (especially if there are no clouds aloft and the winds are light), the radiative cooling of the ground surface can lead to surface air that is actually colder than the air above it. This leads to a situation in which the atmospheric temperature increases with altitude, a so-called inversion. In this case, the atmosphere is extremely stable. Pollutants emitted into an inversion layer get trapped and can reach relatively high concentra­ tions (Figure 16.3a). As the sun rises, the ground and the air next to it start warming up and a temperature profile corresponding to an unstable atmosphere is established. This change occurs over a period of a few hours in the morning and results in breaking of the inversion usually before noon. The changing atmospheric stability, from stable in the early morning to unstable in the afternoon, has a significant effect on pollutant concentrations. Early morning emissions can produce much higher concentrations than in the afternoon, even if the emission rates are the same. An unstable layer in direct contact with the surface is called the mixed layer and its height the mixing height (or mixing depth). The mixing height is typically of the order of 1 km from a few 100 meters to 3 km (Figure 16.3b). The daytime mixed layer is characterized by vigorous turbulent mixing. A subsidence inversion forms as an airmass aloft, of regional scale and having high pressure at its center, spreads horizontally in response to the high pressure, sinks in conjunction with the divergent motion, compresses as it moves downward, and thus heats. This dynamic process can produce a sharp temperature contrast between the air near the surface and the air aloft. A typical temperature profile for such an inversion is shown in Figure 16.3b. Usually the air below the inversion is unstable, and emissions mix relatively rapidly up to the inversion. The stable inversion layer acts as a lid and prevents penetration. Some of the most extreme air pollution episodes are related to subsidence inversions that persist for several days. The mixing height can be as low as a few hundred meters in such episodes.

ATMOSPHERIC STABILITY

FIGURE 16.3 Temperature profile and pollutant mixing for (a) nighttime radiation inversion and (b) subsidence inversion.

Monthly average diurnal and seasonal variations of the vertical temperature profiles near St. Louis in 1976 are shown in Figure 16.4. The winter nights are characterized by a weak radiation inversion layer extending up to 200 m, while during the day a strong inversion layer is formed at 500 m above the ground extending up to 1000 m. During the summer nights a much stronger radiation inversion layer is present in the lower 100 m, while the atmosphere is unstable during the day. An elevated inversion layer is present with a base close to 1700 m.

FIGURE 16.4 Monthly average diurnal and seasonal variations of the vertical thermal structure of the planetary boundary layer at a rural site near St. Louis, MO, based on 1976 data for January and July.

669

670

PHYSICAL AND DYNAMIC METEOROLOGY

16.3 THE MOIST ATMOSPHERE The presence of water in the atmosphere has a profound effect on Earth’s weather and climate. Because water vapor condenses at temperatures of Earth’s atmosphere, releasing latent heat and forming clouds, virtually every atmospheric process is affected by its presence thermodynamics, dynamical motion, and radiative transfer. Let us consider a parcel of moist air of volume V containing Nd molecules of air and Nv molecules of water. The ideal gas law applied to this mixture is pˆ

…N d ‡ N v †kT V

(16.34)

where the Boltzmann constant, k = 1.381 × 10 23 J K 1 molecule 1, is essentially the gas constant for a single molecule. The mean molecular mass of the mixture is hmi ˆ

N d md ‡ N v mv Nd ‡ Nv

(16.35)

where md and mv are the molecular masses of air and water in units of kg molecule 1. Using (16.35), the ideal-gas law for the mixture, (16.34), can be written as pˆ

…N d md ‡ N v mv †kT ρRT Vhmi

(16.36)

where the mixture of the two gases behaves just like a single ideal gas with the mass-based gas constant, R = k/. The mass-based gas constant for dry air that we introduced in (16.4) is just Rd = k/md, where md is the molecular mass of air (28.97). Likewise, a gas constant for water vapor can be defined as Rv = k/mv, where mv is the molecular mass of H2O (18). Recalling Section 1.6, we summarize the various ways of expressing the water vapor content of air (with typical units indicated):

Mole fraction

fv ˆ

Mass density Specific humidity Mass mixing ratio

(molecules m 3) (hPa)

nv e = nvkT

Number density Partial pressure

nv nv ‡ nd

ρ v = nv m v qv = ρv/(ρv + ρd) = ρv/ρ wv = ρv/ρd

…kg m 3 † …g kg 1 † …g kg 1 †

Note the distinction between specific humidity (mass density of water divided by the total density of water + dry air) and mass mixing ratio (mass density of water divided by the mass density of dry air only). The specific humidity qv and mass mixing ratio wv are dimensionless quantities (mass per mass); their numerical values are customarily expressed in terms of grams per kilogram. These quantities can be interrelated, as shown here: qv ˆ wv ˆ εˆ

ρv wv ˆ ρv ‡ ρd wv ‡ 1 …e=kT†mv e mv ρv nv mv ˆ ˆ ρd nd md pd =kT md p e md

mv 18 ˆ 0:622 ˆ md 28:9

(16.37) ˆ

εe p

e

ˆε

e=p 1 e=p

(16.38) (16.39)

671

THE MOIST ATMOSPHERE

fv ˆ wv ˆ

e wv ˆ p wv ‡ ε qv 1

qv

ˆ

(16.40) qv …ε 1†qv

ε

(16.41)

16.3.1 The Gas Constant for Moist Air In terms of molar, rather than molecular, units, the gas law for a mixture is R = kNA/, where NA is Avogadro’s number. For a mixture of dry air and water vapor, we obtain kN A Rd kN A …k=md †N A ˆ ˆ ˆ ˆ hmi fd md ‡ fv mv fd ‡ fv ε fd ‡ fv ε 1 Rd ˆ 1 ‡ …ε 1†fv

R ˆ

Using fv ˆ qv = ε

…ε

Rd fv ‡ fv ε

(16.42)

1†qv ; we have R ˆ Rd 1 ‡

ε

1

qv

(16.43)

R ˆ Rd 1 ‡ 0:608 qv

(16.44)

ε

Since ε = 0.622, we obtain

The value of qv (expressed as g of water vapor per g air) in the atmosphere seldom exceeds ∼ 0.04, so with an error of less than ∼2%, Rd can be used as the gas constant, even in moist air. A similar analysis was already carried out for the heat capacity in (16.14). Equation (16.14) can be expressed as cp ˆ cpd

cpv ρd ‡ qv ρv ‡ ρd cpd

(16.45)

ˆ cpd 1 ‡ 0:94 qv The exponent in the definition of the potential temperature, (16.10), is Rd/cp. In the derivation of the expression for θ, we assumed completely dry air, and therefore the appropriate gas law constant is Rd. In the derivation of (16.10), the heat capacity cp is considered to be that of dry air. Air always contains some amount of water vapor, so the ratio of the gas law constant to the heat capacity should actually be that appropriate to a mixture of air and water vapor. Using (16.44) and (16.45), we get Rd 1 ‡ 0:608 qv R ˆ cp cpd 1 ‡ 0:94 qv Rd ≅ 1 0:33 qv cpd

(16.46)

Again, since qv < 0:04; R=cp is within 1% or so of Rd/cpd, the dry air value of R/cp can be used even in ∼

moist air.

16.3.2 Level of Cloud Formation: The Lifting Condensation Level The level at which a moist air parcel lifted adiabatically from near the surface first reaches saturation is called the lifting condensation level (LCL). In a rising air parcel the volume mixing ratio of water vapor fv

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is constant, so the partial pressure of water vapor is related to the pressure of the air parcel by e…z† ˆ fv p…z†

(16.47)

≅ fv po exp… z=H†

where H is the scale height for the decrease of pressure with altitude (1.5), and fv po = eo is the partial pressure of water vapor at the surface. Thus the partial pressure of water vapor in an air parcel decreases with altitude in accord with the decrease of total pressure. The saturation vapor pressure of water es decreases with height for the usual tropospheric temperature profile that decreases with height. We can estimate the rate at which es decreases with z by using the Clausius–Clapeyron equation des 1 lv ˆ dT T αv αl

(16.48)

where αv αl is the change in specific volume of water upon vaporization and lv is the latent heat of vaporization. Since αv  αl , (16.48) is approximated by [recall (10.68)] es l v des 1 lv ˆ ˆ dT T αv Rv T2

(16.49)

where Rv is the gas constant for water vapor, Rv = 461.5 J K 1 kg 1. lv is a rather weak function of temperature; it is 2.5 × 106 J kg 1 at 0 °C and 2.25 × 106 J kg 1 at 100 °C. For relatively small temperature changes, lv can be assumed constant, and (16.49) can be integrated as es …T† ˆ eso exp

lv exp Rv T o

lv Rv T

(16.50)

To is a reference temperature at which eso is determined. For example, at To = 273 K, eso = 6.11 hPa. Here we take To as the surface temperature. Let us assume that the atmosphere has a dry adiabatic lapse rate. Then es …z† ˆ eso exp

lv exp Rv T o

lv Rv …To Γd z†

(16.51)

We define the scale height for water vapor Hv by es …z† ˆ eso exp

z Hv

(16.52)

where eso is the saturation vapor pressure of water at the surface, and Hv ˆ

Rv T o2 Rv T o cp T o ˆ g lv lv Γd

(16.53)

Hv can be expressed in terms of the scale height for pressure, H ˆ Rd T o =g; Hv ˆ

cp T o H εlv

For To = 298 K, ε = 0.622, lv = 2.5 × 106 J kg 1, and using cp for dry air (1005 J K Hv < H. For typical tropospheric conditions, a good estimate is Hv ∼ H/5.

(16.54) 1

kg 1), cpTo/εlv = 0.19, and

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At the altitude zLCL where e(z) = es(z), we obtain eso exp… zLCL =H v † ˆ f v po exp… zLCL =H†

(16.55)

Solving for zLCL, we get HHv eso ln H H v f v po HHv 1 ˆ ln H H v RH o

zLCL ˆ

(16.56)

where RHo is the relative humidity at the surface (expressed as a fraction). Note that if RHo = 0, zLCL = 1, which makes sense because there is no water vapor in the parcel. If RHo = 1, zLCL = 0, that is, the parcel is already saturated. For H ≅ 8 km [see (1.5)], Hv ≅ 1:6 km, and for RHo = 0.8, zLCL  440 m. To calculate the temperature TLCL at the lifting condensation level, we recall from (16.8) that the temperature of an air parcel undergoing adiabatic cooling varies as a function of pressure according to T ˆ b pRd =cp

(16.57)

where b is a constant. Because both the air and water vapor masses are conserved during the rise of the air parcel, the water vapor mass mixing ratio wv remains constant, and wv = nv mv/nd md. At the LCL the air is saturated, so e ˆ es …T LCL †: The pressure at this height is pˆ

mv es …TLCL † md w v

(16.58)

Substituting (16.58) into (16.57), we obtain TLCL ˆ b

mv es …T LCL † md w v

Rd =cp

(16.59)

The constant b can be eliminated, noting that initially, T o ˆ b po Rd =cp ; where po is the initial pressure of the air parcel. We get TLCL ˆ T o

mv es …T LCL † m d w v po

Rd =cp

(16.60)

TLCL can be calculated by solving (16.60) numerically. Once TLCL is determined, the height at the LCL, zLCL, can be found from zLCL ˆ

To

T LCL Γd

(16.61)

The LCL, zLCL, is shown in Figure 16.5 as a function of the initial temperature and relative humidity of the air parcel.

16.3.3 Dew-point and Wet-Bulb Temperatures For a volume of moist air that is cooled isobarically, at constant total pressure p and mole fraction fv, the water vapor partial pressure e is constant. A temperature decrease from To to T will lead to a decrease of the saturation vapor pressure from es (To) to es(T). If the parcel is initially at To and relative humidity RH

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FIGURE 16.5 Lifting condensation level as a function of initial temperature and relative humidity of the air parcel assuming that the air parcel starts initially at the ground at p = 1 atm.

(expressed as a fraction), the temperature at which the parcel becomes saturated is called the dew-point temperature Tdp, which can be calculated as follows. Relative humidity RH (expressed as a fraction) is defined as the ratio of the actual mass mixing ratio, wv, to the saturation mixing ratio, ws, RH = wv/ws. From (16.38), the following approximate expression is often used, wv =ws ˆ …e=es † p es = p e ≅ e=es : [Lawrence (2005) has shown that where exponentials are taken the difference between these two expressions for RH can be nonnegligible.] The dew-point temperature is that at which e = es (Tdp). The dependence of the saturation vapor pressure on T is given by the Clausius–Clapeyron equation (16.49). Integrating (16.49) between To ; es …T o † and Tdp , e, gives ln

e es …T o †

ˆ

lv Tdp To Rv To T dp

(16.62)

Noting that the ratio on the LHS is equal to the initial relative humidity, and assuming that the temperature change is small enough so that To T dp ≅ T o 2 ; we get the approximate relation T dp ≅ T o ‡

Rv To 2 ln…RH† lv

(16.63)

The dew-point temperature of a subsaturated air parcel is always lower than its actual temperature. The two become equal only when the relative humidity reaches 100%. Figure 16.6 shows the dew-point temperature of an air parcel as a function of its temperature and relative humidity. For example, for an air parcel initially at 283 K with RH = 0.8, a temperature reduction of 3.3 K is required to bring the parcel to saturation. The wet-bulb temperature Tw is the temperature a volume of air would have if cooled adiabatically to saturation by evaporating water into it, with the latent heat of evaporation being supplied by the volume of air itself. The water being evaporated is assumed to already be at temperature Tw. Let Md and Mv be the masses of dry air and water vapor in the parcel in the initial volume of air. The amount of energy needed to raise the temperature of the volume of air from Tw to T at constant pressure is …Md cpd ‡ Mv cpv † …T

Tw †

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FIGURE 16.6 Dew-point temperature of an air parcel as a function of its temperature and relative humidity.

and the enthalpy change needed to evaporate enough water to saturate the parcel is lv …Mvs

Mv †

where Mvs is the mass of water vapor when the volume of air is saturated. Equating these two quantities, we obtain cpd ‡ wv cpv …T

T w † ˆ lv …ws

wv †

(16.64)

Neglecting the wvcpv term and using wv ≅ εe=p, the wet-bulb temperature is found by solving e ˆ es …T w †

pcpd …T εlv

Tw †

(16.65)

16.3.4 The Moist Adiabatic Lapse Rate Up to now we have considered an atmosphere in which neither condensation nor evaporation of water droplets is occurring. Condensation of water vapor releases large amounts of heat to the atmosphere, and evaporation, likewise, consumes large amounts of heat. The latent heat of vaporization of water lv at 0 °C is 2.5 × 106 J kg 1; about 6 times as much heat is needed to evaporate 1 kg of water as is needed to raise its temperature from 0 °C to 100 °C. If adiabatic cooling of a rising air parcel is sufficient to bring the air to saturation and, if the parcel continues to rise, the air will continue to cool but the release of latent heat from the condensation of water vapor will offset the temperature decrease to some extent. As a result, the rate of decrease of temperature of a rising air parcel in which water vapor is condensing will be less than that of a dry parcel. (We reiterate that “dry” in the present context refers to a parcel that may contain water vapor but in which water vapor is not condensing). The lapse rate of a saturated air parcel is called the moist adiabatic lapse rate and is given the designation Γs.

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The dry adiabatic lapse rate is the temperature profile of an adiabatically rising air parcel that can contain water vapor but for which the water vapor is not condensing. We now derive the analogous lapse rate for an air parcel in which water vapor is condensing. Whereas the dry adiabatic lapse rate is constant (9.77 °C km 1), the moist adiabatic lapse rate depends on the moisture content of the air, and it can vary from ∼ 4 °C km 1 to nearly the dry value. The greater the moisture content of the air, the more latent heat that can be released when condensation occurs. The dry and wet adiabats become nearly the same in the upper troposphere because very cold air does not hold much water vapor, and, therefore, only a small amount of latent heat can be released. To derive the expression for the moist (saturated) adiabatic lapse rate, we apply the first law of thermodynamics to an air parcel saturated with water vapor. As in the dry case, in deriving the equation for the moist adiabat, we assume that the atmospheric temperature profile is the same as that in the adiabatically lifted saturated air parcel. (This need not be the case, but it simplifies the derivation.) For the dry air parcel undergoing an adiabatic change, dq = 0. In the saturated parcel, dq is nonzero and is equal to the source of heat from the condensing water vapor. Then1 dq ˆ cp dT

αdp

(16.66)

Since dp ˆ ρgdz and dq ˆ lv dws (at constant pressure), where ws is the mass mixing ratio of saturated water vapor, (16.66) becomes lv dws ˆ cp dT ‡ gdz

(16.67)

Dividing by cp dz, we obtain dT ˆ dz

g cp

lv dws cp dz

Now we need to determine dws/dz. Using ws ˆ εes = p

(16.68)

es ≅ εes =p;

1 des es dp dws ˆ ε dz p dz p2 dz 1 des dT es dp ˆ ε p dT dz p2 dz

(16.69)

Using the Clausius–Clapeyron equation, (16.49), in (16.69), we obtain 1 lv es dT dws ˆε dz p Rv T 2 dz

es dp p2 dz

(16.70)

The second term in the brackets on the RHS of (16.70) is es dp ˆ p2 dz

1 ws g ε RT

(16.71)

1 To be strictly correct, the temperature in (16.66) should be the so-called virtual temperature Tv, the temperature a moist parcel of air would need to have in order for it to follow the ideal-gas law with the dry air gas constant, Rd. Tv is obtained by expressing the total density of the moist parcel as the sum of the partial densities of dry air and water vapor, doing so as well to the total pressure and the partial pressures:

Tv ˆ T

p

p …1 ε†e

With ε = 0.622 and e=p < 0:03; Tv < 1:01 T: As long as temperatures occur in ratios, the correction between and T and Tv can be ∼ ∼ neglected.

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THE MOIST ATMOSPHERE

and (16.70) becomes dws ε lv es dT ws ˆ ‡ g dz p Rv T2 dz RT

(16.72)

Returning to (16.68) and solving for dT/dz, we obtain the moist (saturated) adiabatic lapse rate as

Γs ˆ

dT g ˆ dz cp

ˆ Γd

lv w s RT l2v ws 1‡ cp Rv T 2 1‡

(16.73)

lv w s 1‡ RT l2v ws 1‡ cp Rv T 2

Whereas Γd is a constant, Γs depends in a complex manner on temperature and varies with altitude depending on the particular conditions. Since the concentration of water vapor in a parcel decreases with height as the water vapor condenses, the release of latent heat is maximum at the surface and decreases with height. As a result, the local temperature gradient rate changes with altitude. As a saturated parcel rises, its lapse rate gradually increases because at cooler temperatures, each increment of adiabatic cooling leads to less condensation and less conversion of latent heat into sensible heat. Moist adiabats have slopes ranging from as small as 2 K km 1 in warm air near the surface to a slope approaching that of the dry adiabat (9.8 K km 1) in cold air aloft. We can estimate Γs for a set of conditions at the surface: lv = 2.5 × 106 J kg 1, R = 287 J K 1 kg 1, Rv = 461.5 J K 1 kg 1, cp = 1005 J K 1 kg 1, es = 6.11 hPa, p = 1013 hPa, and T = 273 K. For these values, the quantity in brackets on the RHS of (16.73) is 0.66. With Γd = 9.77 K km 1, we obtain Γs = 6.45 K km 1. In this estimate we have used the heat capacity for dry air. We now present a more rigorous calculation of the moist adiabatic lapse rate as a function of altitude. Numerical Evaluation of the Moist Adiabatic Lapse Rate To compute T(z) for the moist adiabatic lapse rate of a rising air parcel, it is necessary to integrate (16.73)

dT ˆ dz

g cp

lv w s RT l2v ws 1‡ cp Rv T2 1‡

starting from a given altitude, pressure, and temperature. This cannot be done analytically, because of the complex dependence of the RHS on T and z. For ws, we can use the approximation w s …T † ≅

ε e s …T † p… z †

where es(T) is as given in (16.50). An alternative expression for es(T) is given in Table 17.1. With the approximation that the air density can be taken as that of dry air, p(z) satisfies the hydrostatic equation d ln p ˆ dz

g Rd T … z †

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The coupled differential equations for T(z) and p(z) are solved simultaneously by an appropriate numerical integration routine. lv is a function of T, but this dependence is sufficiently mild that for the range of temperatures of interest, lv can be taken as constant. One final technicality concerns the heat capacity cp. An air parcel in which water vapor is condensing will contain a mixture of dry air, water vapor, and liquid water, so that the heat capacity of the parcel is a weighted sum of these three constituents cp ˆ cpd ‡ ws cpv ‡ wl cpl wl is the liquid water mixing ratio, cpv = 1870 J K 1 kg 1, and cpl is the heat capacity of liquid water, cpl = 4181 J K 1 kg 1. If none of the condensed water precipitates out of the parcel, then ws ‡ wl is a constant equal to the initial mass mixing ratio of water vapor in the parcel. At the other extreme, if all the liquid water falls out, wl ˆ 0; and the calculated moist adiabatic lapse rate is called a pseudoadiabat (since entropy is not precisely conserved when the water is removed from the parcel). In general, the pseudoadiabat differs from the moist adiabat only very slightly, until a parcel rises high enough for the liquid water removal to be appreciable, and even then the difference is not large. We will calculate the moist adiabatic lapse rate for a saturated parcel of air that is at 293 K at the surface, p = 1013 hPa. This requires that we numerically integrate (16.73) coupled to the hydrostatic equation governing p(z) from the surface (z = 0) up to a predetermined altitude. We will perform the integration up to 5000 m. The result is shown in Figure 16.7. We also show in Figure 16.7 the dry adiabat that intersects the moist adiabat at 5000 m. This dry adiabat intersects the surface at 317 K. The pressure at 5000 m is 551.89 hPa. The value of es at the surface is 23.674 hPa, and that at 5000 m is 4.30 hPa. At the surface ws = 0.01454 and at 5000 m, ws = 0.00484. The liquid water mixing ratio wl = 0 at the surface and that at 5000 m is 0.009695. We can evaluate the error incurred by using a constant value of cp in (16.73) as opposed to the more exact equation given above. The value of cp at the surface is 1032 J K 1 kg 1 and that at 5000 m is 1054 J K 1 kg 1, an error of about 2%. (Note that even at the surface the value of cp reflects the weighted contributions from dry air and water vapor.)

FIGURE 16.7 Moist adiabat starting at the surface (z = 0) at temperature 293 K. Also shown is the dry adiabat that intersects the moist adiabat at 5000 m altitude. The moist parcel contains air saturated with water vapor at the surface. As the parcel ascends and cools, water condenses but remains in the parcel. At 5000 m, 67% of the water initially present as vapor has condensed into liquid water. For this parcel the average moist adiabatic lapse rate from the surface to 5000 m is 4.96 K km 1.

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THE MOIST ATMOSPHERE

In summary, consider a parcel initially at some height above the surface at which it has temperature T, pressure p, and water vapor mixing ratio w < ws (the parcel is unsaturated). If the parcel is lowered adiabatically, it will follow a dry adiabat. At the surface, its temperature is its potential temperature θ. If the parcel is raised adiabatically, it will follow a dry adiabat up to the LCL, at which point the parcel is saturated. On further lifting above the LCL, the parcel’s temperature follows a moist adiabat. If all the condensed water is removed from the parcel as it ascends above the LCL, it follows a pseudoadiabat. Following a pseudoadiabat all the way to the top of the atmosphere results in complete drying of the parcel. If that parcel is brought back down, it will follow a dry adiabat. If, when the parcel reaches the LCL on ascending, just enough water vapor is added to the parcel to keep it saturated as it as brought back down, when it reaches its original altitude, its temperature is the wet-bulb temperature of the original parcel.

16.3.5 Stability of Moist Air Imagine that the actual atmospheric lapse rate Λ lies between Γd and Γs (Figure 16.8). If the air is sufficiently dry that no water vapor condensation occurs, a parcel lifted from point A undergoes dry adiabatic cooling to point B. At this elevation, the air in the parcel is colder (and denser) than the surrounding air, so the parcel will tend to sink back to its original level. If the air is saturated and lifting from A to B causes condensation, the parcel temperature follows the moist adiabat Γs to point C, at which point it will be warmer (and less dense) than the surrounding air. Thus, the parcel will continue to rise, and in this situation the atmosphere is unstable. This situation, in which the atmospheric stability depends on the amount of moisture in the air, is termed conditional instability. Thus, an actual atmospheric temperature profile Λ = dT/dz can be classified as follows: Λ > Γd Γd > Λ > Γs Λ < Γs

absolutely unstable conditionally unstable absolutely stable

If Λ < 0, i.e. temperature increases with height, an inversion, the atmosphere is extremely stable.

FIGURE 16.8 Displacement of a saturated (moist) air parcel. The parcel starts out at a point A. The atmospheric lapse rate is Λ. The dry adiabat for the parcel is Γd , and the moist adiabat is Γs . Γs < Γd . If the parcel displaced upwards follows Γd , it arrives at point B, whereas if it follows Γs , it arrives at point C. The stability of the atmosphere depends on the extent to which the air is saturated. In the situation shown, Γd > Λ but Λ > Γs , a conditionally unstable state.

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16.3.6 Convective Available Potential Energy (CAPE) The strength of an unstable atmospheric condition is related to the amount of energy available for upward acceleration (16.18) of an air parcel. This energy is called the convective available potential energy (CAPE). The amount of CAPE of a parcel lifted from an altitude zo to an altitude z1 at which it is no longer positively buoyant is

CAPE ˆ

z1

∫ z0

g

ρa

ρp ρp

dz

(16.74)

The units of CAPE are J kg 1. Assuming the atmosphere is in hydrostatic equilibrium and using the ideal gas law, (16.74) becomes CAPE ˆ Rd

p0

∫p

Tp

T a d …ln p†

(16.75)

1

Since CAPE is proportional to the area between the ambient temperature profile and the parcel’s temperature profile, CAPE can be either positive or negative. The negative area, that where parcels are negatively buoyant, is termed the convective inhibition energy. CAPE is an acceleration. If it is assumed that CAPE is converted into the vertical acceleration of the parcel, then the maximum vertical velocity, vz, a parcel could achieve can be obtained from dvz dvz dvz dz ˆ ˆ vz ˆB dt dz dt dz where B is as given by (16.19), B ˆ g Tp at zLFC yields

(16.76)

T a =T a : Integrating (16.74) from zLFC to zLNB ; and taking vz = 0 v2z;max ˆ 2 CAPE

or vz;max ˆ …2 CAPE†1=2

(16.77)

Equation (16.77) is just an estimate of the maximum updraft velocity corresponding to a particular value of CAPE, as it neglects effects that retard a parcel’s motion, such as mixing between the parcel air and the surrounding air and the loading of liquid water. As a buoyant parcel ascends, mixing across the parcel boundary occurs as a result of turbulence, which is termed entrainment. Since the air outside the parcel is typically cooler and drier than that in a rising saturated parcel, entrainment lowers the buoyancy of the parcel and reduces the amount of condensed water. When these effects are accounted for, vz;max from (16.77) can be overestimated by as much as a factor of 2. Convective available potential energy can be illustrated for the hypothetical temperature–pressure diagram in Figure 16.9. The dashed line represents the atmospheric temperature profile, and the solid line represents the parcel lapse rate. A parcel of air containing noncondensing water vapor is forced to ascend from the surface at 1000 hPa. The parcel follows a dry adiabatic lapse rate. Initially, the parcel is colder than its environment, and it is subject to a downward buoyancy force. The parcel reaches its lifting condensation level (LCL) at 900 hPa, where it becomes saturated with water vapor. Beyond that point the parcel follows its moist adiabatic lapse rate. When the parcel reaches 880 hPa, it becomes warmer than the environment and experiences an upward buoyancy force. This point is termed the level of free convection. The parcel will continue to accelerate upward until it eventually reaches the point where its temperature and the ambient temperature are again equal, the so-called level of neutral buoyancy, at 480 hPa. The area between the two lapse rates in Figure 16.9 is the CAPE for this situation. For the profiles in Figure 16.9, the CAPE value is 612 J kg 1.

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FIGURE 16.9 Illustration of convective available potential energy (CAPE) for a hypotheti­ cal atmospheric temperature profile and parcel lapse rate. The parcel (solid line) starts at the surface and rises according to the dry adiabatic lapse rate up to the lifting condensa­ tion level (LCL), at which point it rises according to the moist adiabatic lapse rate. The atmospheric lapse rate is given by the dashed line. Below a pressure of 880 hPa, the atmosphere is stable with respect to the parcel. Above 880 hPa, the parcel accelerates freely upward. At 480 hPa, the two lapse rates reintersect, and the parcel is no longer buoyant. The area between the two lapse rates represents the CAPE.

Thus, CAPE is a measure of the difference in temperature between an ascending parcel and the surrounding air assuming that no mixing occurs between the parcel and the surrounding air. Integrating this temperature difference produces a number equal to the positive area between the two curves. The larger the value of CAPE, the more likely the occurrence of convective storms. Values of CAPE generally fall in the following ranges: 0 – 1000 J kg 1000 – 2500 2500 – 3500 >3500

1

Marginally unstable Moderately unstable Very unstable Extremely unstable

CAPE for convective storms is often in the range of 1000 – 2000 J kg 1. In the deadly tornado that hit near Birmingham, Alabama on April 27, 2011, CAPE was estimated at 3458 J kg 1 (Markowski and Richard­ son 2014). CAPE is directly related to the maximum potential vertical velocity of updrafts, as in (16.77); 3458 J kg 1 equates to a maximum updraft speed of 83 m s 1. For the two temperature profiles in Figure 16.9, CAPE was evaluated by numerically evaluating the integral in (16.75). We find that CAPE ≅ 612 J kg 1 is a condition of marginal instability.

16.3.7 Thermodynamic Diagrams The atmospheric state can be represented by thermodynamic diagrams. These diagrams allow meteo­ rologists to forecast atmospheric temperature and humidity relationships without lengthy calculations. Figure 16.10 shows such a diagram, in which the horizontal axis is temperature, and the vertical axis is altitude/pressure. There are three types of lines on this diagram. The solid lines are dry adiabats.

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FIGURE 16.10 Thermodynamic diagram. Solid lines are dry adiabats, dashed lines are moist adiabats, and large dashed–small dashed lines are lines of constant saturation mixing ratio. Labels on the moist adiabats indicate the value of the moist adiabat at that altitude; these increase from 4 to 9. Labels on the saturation mixing ratio lines are values of the ratio, which increase from 1 to 48.

These lines show the temperature of a dry air parcel as it rises or descends (RH < 100%). For example, the temperature of a dry air parcel that is 20 °C at 1000 hPa will be 11.5 °C at 900 hPa and 20 °C at 600 hPa. Or, conversely, a dry air parcel at 600 hPa and 20 °C, when brought down to the surface, will be 20 °C. The dashed lines represent moist adiabats. Whereas the dry adiabat lines are straight, those for moist adiabats are curved. As we have noted, moist adiabats have smaller rates of decrease than dry adiabats owing to the conversion of latent heat into sensible heat as water vapor condenses in the rising parcel. But, as the moist parcel rises, its lapse rate gradually increases because at cooler temperatures, the adiabatic cooling causes less condensation to occur. The dry and moist lapse rates can be compared for a parcel at 0 °C at the surface. At 700 mbar, the dry adiabat for the parcel is at 27 °C, while the moist adiabat is at 21 °C. The dashed–dotted lines are lines of constant saturation mixing ratio. (Such lines are called isohomes.) These lines show the saturation mixing ratio of water vapor ws in g water vapor per kg of air at 100% RH, at a particular temperature and pressure. For any particular pressure and temperature, that is, any particular location on the diagram, the isohome passing through that location gives the maximum amount (mass) of water vapor that could be present in each kilogram of dry air at that temperature and pressure. For example, every point on the rightmost dashed–dotted line labeled 30 g kg 1 has ws = 30 g kg 1. Thus, a parcel at 500 hPa altitude and 20 °C has ws of ∼30 g kg 1. Figure 16.10 can be replotted on somewhat different axes (Figure 16.11). The horizontal axis is still temperature, but the vertical axis is now the logarithm of pressure. Lines of constant temperature on Figure 16.10 are vertical; on Figure 16.11, the lines of constant temperature are sloped at a 45° angle to the right. Because of the slanted temperature lines, this plot, shown in Figure 16.11, is called a skew-T log-p diagram, or “skew-T” for short. In this configuration, the angle between the adiabats and isotherms is large enough so that lines do not get bunched together. On this diagram, most of the important isopleths are closer to being straight rather than curved. The logarithmic vertical coordinate approximates the actual atmosphere in which the pressure decreases nearly logarithmically with height. In this way, an entire temperature sounding up to the stratosphere can be plotted on one chart. The skew-T diagram contains the same five sets of lines or curves as those in Figure 16.10: isobars (lines of constant pressure), isotherms (lines of constant temperature), dry adiabats, moist adiabats, and lines of constant saturation mixing ratio. On the skew-T diagram, the dry adiabats are slightly curved lines sloping upward to the

BASIC CONSERVATION EQUATIONS FOR THE ATMOSPHERIC SURFACE LAYER

FIGURE 16.11 Skew-T diagram. The x axis shows both intersection of isotherms (°C) and lines of constant saturation mixing ratio (g kg 1). The y axis is pressure (hPa).

left. The moist adiabats slope from the lower right to the upper left, except those on the extreme right. The lines have slopes ranging from 2 °C km 1 to nearly the same as dry adiabats, 9.8 °C km 1 in cold air aloft. Note that the moist adiabats become more nearly parallel to dry adiabats at lower pressures and temperatures than in Figure 6.10. The dashed–dotted curves, lines of constant saturation mixing ratio, slope from lower left to upper right on the skew-T diagram. Finally, these thermodynamic diagrams show relationships only between temperature, pressure, and saturation mixing ratio. If we know either a parcel’s dewpoint temperature, its water vapor mixing ratio, or its RH, then we can determine the other properties indirectly from the diagram using the relationships developed in this chapter. Figure 16.12 shows a sample environmental temperature profile (sounding) and the corresponding dewpoint temperature on the skew-T diagram. The dewpoint curve is a measure of how much water vapor is actually in the air. For the sounding in Figure 16.12, Tdp at 700 hPa is 13 °C and the saturation mixing ratio line passing through 13 °C is 2.0 g kg 1. Thus, 2.0 g kg 1 is the mixing ratio of water vapor in the air at 700 hPa. The saturation mixing ratio ws of the parcel at 700 hPa can be found by interpolating between the 3.0 and 4.0 saturation mixing ratio lines; the value is 3.8 g kg 1. The relative humidity of the air parcel at 700 hPa is the ratio of the actual mixing ratio of water vapor to the saturation mixing ratio, RH = w/ws = 2.0/3.8 = 0.526. Finally, the potential temperature of the parcel is found from the dry adiabat through the parcel, which is 24 °C at the surface.

16.4 BASIC CONSERVATION EQUATIONS FOR THE ATMOSPHERIC SURFACE LAYER We now turn our attention to air motion in the lowest layers of the atmosphere, which is virtually always turbulent. Atmospheric turbulence is responsible for the transfer of momentum, heat, water vapor, and gases and aerosols in this layer. Our objective is to understand the basic phenomena that influence

683

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PHYSICAL AND DYNAMIC METEOROLOGY

FIGURE 16.12 Hypothetical temperature sounding labeled T and the corresponding profile of dewpoint temperature, Tdp. The temperature of the air parcel at 700 hPa is 5 °C. The dewpoint temperature of the air parcel is 13 °C. The saturation mixing ratio at the dewpoint temperature is 2 g kg 1, and the saturation mixing ratio of the parcel is 3.8 g kg 1. The relative humidity of the air parcel at 700 hPa is RH = 2.0/3.8 = 0.526. The potential temperature of the air parcel is 24 °C.

atmospheric turbulence and to derive equations that govern atmospheric transport phenomena. The interested reader may pursue this area in greater depth in Monin and Yaglom (1971, 1975), Plate (1971, 1982), Nieuwstadt and van Dop (1982), and Panofsky and Dutton (1984). We will derive first the equations that govern the fluid density, temperature, and velocities in the lowest layer of the atmosphere. These equations will form the basis from which we can subsequently explore the processes that influence atmospheric turbulence. In our discussion we shall consider only a shallow layer adjacent to the surface, in which case we can make some rather important simplifications in the equations of continuity, motion, and energy. The equation of continuity for a compressible fluid is @ρ @ @ @ ‡ …ρux † ‡ …ρuy † ‡ …ρuz † ˆ 0 @t @x @y @z

(16.78)

where ρ is the fluid density and ux, uy, uz are the fluid velocity components in the directions x, y, and z, respectively. To make the subsequent equations a little more compact, we will use first the index notation, that is, denote the coordinate axes as x1, x2, and x3 and the corresponding velocities as u1, u2, and u3. With this notation the continuity equation becomes @ρ @ @ @ @ρ ‡ …ρu1 † ‡ …ρu2 † ‡ …ρu3 † ˆ ‡ @t @x1 @x2 @x3 @t

3 iˆ1

@ …ρui † ˆ 0 @xi

(16.79)

Equation (16.79) can be simplified further if we use the summation convention, that is, if we omit the summation symbol from the equation, assuming that repeated symbols are summed from 1 to 3. As a result, the continuity equation becomes @ρ @ ‡ …ρui † ˆ 0 @t @xi

(16.80)

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Because we are interested only in processes taking place on limited spatiotemporal scales over which the air motion is not influenced by Earth’s rotation, we will neglect the Coriolis acceleration (see Chapter 21) and write the equation of motion of a compressible, Newtonian fluid in a gravitational field as ρ

@ui @ui ‡ uj @t @xj

ˆ

2 @uj p‡ μ δik 3 @xj

@ @ui @uk μ ‡ @xk @xi @xk

ρgδ3i

(16.81)

where μ is the fluid viscosity and δij is the Krönecker delta, defined by δij = 1 if i = j, and δij = 0 if i ˆ 6 j. For example, in the x direction (16.81) becomes ρ

@ux @ux @ux @ux ‡ uy ‡ uz ‡ ux @t @x @y @z

@p @ @ux 2 @ux @uy @uz ‡ 2μ μ ‡ ‡ @x @x @x 3 @x @y @z @u @ @ux @uz @ux @ y μ ‡ μ ‡ ‡ ‡ @y @y @x @z @z @x ˆ

(16.82)

Finally the energy equation, assuming that the contribution of viscous dissipation to the energy balance of the atmosphere is negligible, is ρ cv

@T @T ‡ uj @t @xj

ˆk

@2T @xj @xj

p

@uj ‡Q @xj

(16.83)

where cv is the heat capacity of air at constant volume per unit mass, k is the thermal conductivity (assumed constant), and Q represents the heat generated by any sources in the fluid. Equations (16.80), (16.81), and (16.83) represent five equations for the six unknowns u1, u2, u3, p, ρ, and T. The sixth equation necessary for closure is the ideal-gas law: pˆ

ρRT Mair

(16.84)

These six equations can therefore be solved, in principle, subject to appropriate boundary and initial conditions to yield velocity, pressure, density, and temperature profiles in the atmosphere. Atmosphere at Equilibrium Solve the system of the equations given above to calculate the pressure, density, and temperature profiles in the atmosphere for ui = 0, assuming that there are no heat sources and that the temperature is zero at x3 = H. When the atmosphere is at rest, its density, pressure, and temperature are constant with time, and (16.81) and (16.83) become @pe ˆ0 @x1

@pe ˆ0 @x2

@pe ˆ ρe g @x3

@2T @2T @2T ‡ ‡ ˆ0 @x21 @x22 @x32

(16.85) (16.86)

where the subscript e denotes equilibrium. From (16.85) the pressure and density depend only on altitude: pe = p(x3), ρe = ρ(x3). This means, given the ideal-gas law, that the temperature is only a function of altitude and (16.86) becomes @2T ˆ0 @x23

(16.87)

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To solve (16.87) we need two boundary conditions. If T0 is the ground temperature and zero at x3 = H, then, integrating, we obtain x3 Te ˆ T0 1 (16.88) H Substituting (16.84) and (16.88) into (16.85) and integrating, we find that p e ˆ p0 1

…gHMair =RT 0 †

x3 H

(16.89)

where p0 is the surface pressure. Finally, from (16.88), (16.89), and the ideal-gas law, we obtain ρ e ˆ ρ0 1

x3 H

…gHMair =RT 0 † 1

(16.90)

where ρ0 = p0Mair/RT is the air density at the surface. The lapse rate for the atmosphere is in this case equal to Λ = T0/H. Equations (16.80), (16.81) and (16.83) can be simplified for a shallow atmospheric layer next to the ground using the Boussinesq approximations. The conditions of their validity have been examined by Spiegel and Veronis (1960), Calder (1968), and Dutton (1976). The fundamental idea of these approxi­ mations is to first express the equilibrium profiles of pressure, density, and temperature as functions of x3 only as follows: pe ˆ p0 ‡ pm …x3 † ρe ˆ ρ0 ‡ ρm …x3 † T e ˆ T0 ‡ T m …x3 †

(16.91)

We consider only a shallow layer, so that pm/p0, ρm/ρ0, and Tm/T0 are all smaller than unity. When there is motion, we can express the actual pressure, density, and temperature as the sum of the equilibrium values and a small correction due to the motion (denoted by a tilde). Thus we write p…x1 ; x2 ; x3 ; t† ˆ p0 ‡ pm …x3 † ‡ ~p…x1 ; x2 ; x3 ; t† ρ…x1 ; x2 ; x3 ; t† ˆ ρ0 ‡ ρm …x3 † ‡ ~ρ…x1 ; x2 ; x3 ; t† ~ x1 ; x2 ; x3 ; t† T…x1 ; x2 ; x3 ; t† ˆ T 0 ‡ T m …x3 † ‡ T…

(16.92)

where we assume that the deviations induced by the motion are sufficiently small that the quantities ~ 0 are also small compared with unity. Substituting (16.92) into (16.80), (16.81), and ~p=p0 , ~ρ=ρ0 , and T=T (16.83) and simplifying using the assumptions presented above, (see also Appendix 16.2), one can derive the basic equations of atmospheric fluid mechanics that are applicable for a shallow atmospheric layer @ui ˆ0 @xi ~ @ui @ui 1 @~p μ @ 2 ui gT ‡ uj ˆ ‡ ‡ δi3 @t @xj ρ0 @xi ρ0 @xj @xj T0 ρ0 c p

@θ @θ ‡ uj @t @xj

ˆk

@2θ @xj @xj

(16.93)

‡Q

where θ is the potential temperature. The Boussinesq approximations lead to a considerable simplifica­ tion of the original equations. The first is the continuity equation for an incompressible fluid. The

BASIC CONSERVATION EQUATIONS FOR THE ATMOSPHERIC SURFACE LAYER

equation of motion is identical to the incompressible form of the equation with the exception of the last term, which accounts for the acceleration due to buoyancy forces. Finally, the energy equation is just the usual heat conduction equation with T replaced by θ. The complete set of equations consists now of the five equations in (16.93) for the five unknowns, u1, u2, u3, ρ, and θ. The ideal-gas equation of state is no longer required as it has been incorporated into the equations. Although ρ0 and T0 in (16.93) refer to the constant surface values, equations of precisely the same form can be derived in which ρ0 and T0 are replaced by ρe and Te, the reference profiles. The equations written in this form will be useful later when we consider the dynamics of potential temperature in the atmosphere.

16.4.1 Turbulence Equations (16.93) govern the fluid velocity and temperature in the lower atmosphere. Although these equations are at all times valid, their solution is impeded by the fact that the atmospheric flow is turbulent (as opposed to laminar). Turbulence is a characteristic of flows and not of fluids themselves. It is difficult to define turbulence; instead we can cite a number of characteristics of turbulent flows. Turbulent flows are irregular and random, so that the velocity components at any location vary randomly with time. Since the velocities are random variables, their exact values can never be predicted precisely. Thus (16.93) become partial differential equations, the dependent variables of which are random functions. We cannot expect to solve any of these equations exactly; rather, we must be content to determine some statistical properties of the velocities and temperature. The random fluctuations in the velocities result in rates of momentum, heat, and mass transfer in turbulence that are many orders of magnitude greater than the corresponding rates due to pure molecular transport. Turbulent flows are dissipative in the sense that there is a continuous conversion of kinetic to internal energy. Thus, unless energy is continuously supplied, turbulence will decay. The usual source of energy for turbulence is shear in the flow field, although in the atmosphere buoyancy can also be a source of energy. Let us consider a situation of turbulent pipe flow. If the same pipe and pressure drop are used each time the experiment is repeated, the velocity field would always be different no matter how carefully the conditions of the experiment are reproduced. A particular turbulent flow in the pipe can be envisioned as one of an infinite ensemble of flows from identical macroscopic boundary conditions. The mean or average velocity, for instance, as a function of radial position in the pipe could be determined, in principle, by averaging the readings made over an infinite ensemble of identical experiments. Figure 16.13 shows a hypothetical record of the ith velocity component at a certain point in a turbulent flow. The specific features of a second velocity record taken under the same conditions would be different but there might well be decided resemblance in some of the characteristics of the record. In practice, it is rarely possible to repeat measurements, particularly in the atmosphere. To compute the mean value of ui at location x and time t, we would need to average the values of ui at x and time t for all

FIGURE 16.13 Typical record of the velocity in direction i at a point in a turbulent flow.

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the similar records. This ensemble mean is denoted by < ui(t, x) >. If the ensemble mean does not change with time t, we can substitute a time average for the ensemble average. The time-average velocity is defined by 1 t0 ‡τ ui …t†dt τ!1 τ ∫ t 0

ui ˆ lim

In practice, the statistical properties of ui depend on time. For example, the flow may change with time. However, we still wish to define a mean velocity: ui …t† ˆ

1 t‡τ=2 ´ ´ ui …t †dt τ ∫ t τ=2

Clearly, ūi(t) will depend on the averaging interval τ. We need to choose τ large enough so that an adequate number of fluctuations are included, but yet not so large that important macroscopic features of the flow would be masked. For example, if τ1 and τ2 are timescales associated with fluctuations and macroscopic changes in the flow, respectively, we would want τ2  τ  τ1. It is customary to represent the instantaneous value of the wind velocity ui as the sum of a mean and a fluctuating component, ui ‡ u´i . The mean values of the velocities tend to be smooth and slowly varying. The fluctuations u´i ˆ ui ui are characterized by extreme spatiotemporal variations. In spite of the severity of fluctuations, it is observed experimentally that turbulent spatiotemporal inhomogeneities still have considerably greater sizes than molecular scales. The viscosity of the fluid prevents the turbulent fluctuations from becoming too small. Because the smaller scales (or eddies) are still many orders of magnitude larger than molecular dimensions, the turbulent flow of a fluid is described by the basic equations of continuum mechanics. In general, the largest scales of motion in turbulence (the so-called large eddies) are comparable to the major dimensions of the flow and are responsible for most of the transport of momentum, heat, and mass. Large scales of motion have comparatively long timescales, where the small scales have short timescales and are often statistically independent of the large-scale flow. A physical picture that is often used to describe turbulence involves the transfer of energy from the larger to the smaller eddies, which ultimately dissipate the energy as heat.

16.4.2 Equations for the Mean Quantities In the equations of motion and energy (16.93), the dependent variables ui, p, and θ are random variables, making the equations virtually impossible to solve. We modify our goal from trying to calculate the actual variables to trying to estimate their mean values. We decompose the velocities, temperature, and pressure into a mean and a fluctuating component: ui ˆ ui ‡ u´ θ ˆ θ ‡ θ´ p ˆ p ‡ p´

(16.94)

By definition, the mean of a fluctuating quantity is zero: u´i ˆ θ´ ˆ p´ ˆ 0

(16.95)

Our objective is to determine equations for ui , θ, and p. To obtain these equations, we first substitute (16.94) into (16.93). We then average each term in the resulting equations with respect to time. The result, employing (16.95), is @ui ˆ0 @xi

(16.96)

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BASIC CONSERVATION EQUATIONS FOR THE ATMOSPHERIC SURFACE LAYER

@u´ @ui @ui ‡ uj ‡ u´j i ˆ @xj @t @xj ρ0 cp

gθ 1 @p μ @ 2 ui ‡ ‡ δi3 ρ0 @xi ρ0 @xj @xj T 0

(16.97)

@θ @θ @θ´ @2θ ‡ uj ‡ u´j †ˆk @xj @xj @xj @t @xj

It is customary to employ the relation

(16.98)

@u´i ˆ0 @xi

(16.99)

obtained by subtracting (16.96) from the first equation (16.93) to transform the third terms on the LHS of (16.97) and (16.98) to @u´i u´j =@xj and @uj´ θ´ =@xj . Then (16.97) and (16.98) are written in the form @ @ …ρ ui † ‡ …ρ ui uj † ˆ @t 0 @xj 0 ρ0 c p

@p @ ‡ @xi @xj

@ @θ @θ ‡ uj ˆ @xj @t @xj

k

@θ @xj

μ

@ui @xj

ρ0 cp u´j θ´

ρ0 u´i u´j

‡

gθ δi3 T0 (16.100)

The good news is that these equations, now time-averaged, contain only smoothly time-varying average quantities, so that the difficulties associated with the stochastic nature of the original equations have been alleviated. However, there is also some bad news. We note the emergence of new dependent variables, u´i u´j and u´j θ´ for i,j = 1,2,3. When the equations are written in the form of (16.100), we can see that ρ0 u´i u´j represents a new contribution to the total stress tensor and that ρ0 cp u´j θ´ is a new contribution to the heat flux vector. The terms ρ0 u´i u´j , called the Reynolds stresses, indicate that the velocity fluctuations lead to a transport of momentum from one volume of fluid to another. Let us consider the physical interpretation of the Reynolds stresses by using the Cartesian notation for simplicity. We envision the situation of a steady mean wind in the x direction near the ground. Let the y direction be the horizontal direction perpendicular to the mean wind and z the vertical direction. A sudden increase or gust in the mean wind would result in a positive u´x , whereas a lull would lead to a negative u´x . Left- and right-hand swings of the wind direction from its mean direction can be described by positive and negative uy´ , respectively, and upward and downward vertical gusts by positive and negative u´z . The air needed to sustain a gust in the x direction must come from somewhere and usually it comes from more rapidly moving air from above. Therefore, we expect positive values of u´x to be correlated with negative values of uz´ . Similarly, a lull will result when air is transported upward rather than forward, so that we would expect negative ux´ to be associated with positive uz´ . As a result of both effects, u´x u´z is not zero and the corresponding Reynolds stress will play an important role in the transport of momentum. Equations (16.96) and (16.100) have, as dependent variables, ūi, p, θ, u´i uj´ , and uj´ θ´ . We thus have 14 dependent variables (note that u´i uj´ ˆ u´j u´i , so there are six Reynolds stresses plus three u´j θ´ variables, three velocities, pressure, and temperature), but we have only five equations from (16.100). We need eight additional equations to close the system. We could attempt to write conservation equations for the new dependent variables. For example, we can derive such an equation for the variables u´i u´j by first subtracting (16.97) from the second equation in (16.93), leaving an equation for u´i . We then multiply by u´j and average all terms. Although we can arrive at an equation for u´i u´j this way, we unfortunately have at the same time generated still more dependent variables u´i u´j u´j . This problem, arising in the description of turbulence, is called the closure problem, for which no general solution has yet been found. At present we must rely on models and estimates based on intuition and experience to obtain a closed system of equations. Since mathematics by itself will not provide a solution, we will resort to quasiphysical models to obtain additional equations for the Reynolds stresses and the turbulent heat fluxes. The next section is devoted to the most popular semiempirical models for the turbulent momentum and energy fluxes.

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16.4.3 Mixing-Length Models for Turbulent Transport The simplest approaching to closing the equations (16.100) is based on an appeal to the physical picture of the actual nature of turbulent momentum transport. We can envision the turbulent fluid as comprising lumps of fluid, which, for a short time, retain their integrity before being destroyed. These lumps or eddies transfer momentum, heat, and mass from one location to another. Thus it is possible to imagine an eddy, originally at one level in the fluid, breaking away and conserving some or all of its momentum until it mixes with the mean flow at another level. Let us assume a steady turbulent shear flow in which ū1 = ū1(x2) and ū2 = u3̄ = 0. We first consider turbulent momentum transport, that is, the Reynolds stresses. The mean flux of x1 momentum in the x2 direction due to turbulence is ρu´1 u´2 . Let us see if we can derive an estimate for this flux. We can assume that the fluctuation in u1 at any level x2 is due to the arrival at that level of a fluid lump or eddy that originated at some other location where the mean velocity was different from that at x2. We illustrate this idea in Figure 16.14, in which a fluid lump that is at x2 = x2 + la at t τa with an original velocity equal to the mean velocity at that level, namely, u1̄ (x2 + la), arrives at x2 at t. Let u´1a be the fluctuation in u1 at x2 at time t due to the αth eddy. If the eddy maintains its x1 momentum during its sojourn, this fluctuation can be written u´1α ˆ u1α …x2 ; t†

u1 …x2 †

(16.101)

As long as the x1 momentum of the eddy is conserved, then u1α …x2 ; t† ˆ u1 …x2 ‡ lα ; t Substituting (16.102) into (16.101) and expanding ū1(x2 + lα, t we get u´1α ˆ lα

@u1 @x2

τα

τα †

(16.102)

τα) in a Taylor series about the point (x2, t),

@u1 1 2 @ 2 u1 1 2 @ 2 u1 ‡ l ‡ τ ‡ ∙∙∙ @t 2 α @x22 2 α @t2

(16.103)

First, we note that, since the flow has been assumed to be steady, ū1 does not vary with time and the corresponding derivatives are zero. Thus (16.103) becomes ´ ˆ lα u1α

@u1 1 2 @ 2 u1 ‡ l ‡ ∙∙∙ @x2 2 α @x22

(16.104)

The second- and higher-order terms in (16.104) can be truncated if the distance lα over which the eddy maintains its integrity is small compared to the characteristic lengthscale of the ū1 field, leaving ´ u1α ˆ lα

@u1 @x2

FIGURE 16.14 Eddy transfer in a turbulent shear flow.

(16.105)

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BASIC CONSERVATION EQUATIONS FOR THE ATMOSPHERIC SURFACE LAYER

Multiplying (16.105) by u´2α , the turbulent fluctuation in the x2 direction associated with the same αth eddy is ´ ´ ´ u1α u2α ˆ u2α lα

@u1 @x2

(16.106)

To obtain the turbulent flux u1´ u2´ , we need to consider an ensemble of eddies that pass the point x2. If we average (16.106) over all these eddies, we find that u1´ u2´ ˆ u2´ l

@u1 @x2

(16.107)

The term u´2 l represents the correlation between the fluctuating x2 velocity at x2 and the distance of travel of the eddy. Let us try to estimate the order of the term u´2 l. First note that if @ū1/@x2 > 0, then from (16.107) u´2 l will have the same sign as u1´ u2´ , that is, it will be negative (see discussion in Section 16.4.2). If Le is the maximum distance over which an eddy maintains its integrity and û2 is the turbulent intensity 1=2 , then the term u´2 l will be proportional to Leû2 or if c is a positive constant of proportionality, then …u´2 2 † ^2 u´2 l ˆ cLe u

(16.108)

Substituting (16.108) into (16.107), we find that ^2 u´1 u´2 ˆ cLe u

@u1 @x2

(16.109)

The mixing length Le is a measure of the maximum distance in the fluid over which the velocity fluctuations are correlated, or in some sense, of the eddy size. The experimental determination of Le simply involves measuring the velocities at two points separated by larger and larger distances until their correlation approaches zero. From (16.109), we define the eddy viscosity or turbulent momentum diffusivity KM as KM = cLeû2, so that (16.109) becomes u´1 u´2 ˆ KM

@u1 @x2

(16.110)

We can extend the mixing-length concept to the turbulent heat flux. We consider the same shear flow as above, in which buoyancy effects are for the moment neglected. By analogy to the definition of the eddy viscosity, we can define an eddy viscosity for heat transfer by u´2 θ´ ˆ KT

@θ @x2

(16.111)

Equations (16.110) and (16.111) provide a solution to the closure problem inasmuch as the turbulent fluxes have been related directly to the mean velocity and potential temperature. However, we have essentially exchanged our lack of knowledge of u´1 u´2 and u´2 θ´ for KM and KT, respectively. In general, both KM and KT are functions of location in the flow field and are different for transport in different coordinate directions. The variation of these coefficients, as we will see later, is determined by a combination of scaling arguments and experimental data. The result of the mixing-length idea used to derive the expressions (16.110) and (16.111) is that the turbulent momentum and energy fluxes are related to the gradients of the mean quantities. Substitution

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PHYSICAL AND DYNAMIC METEOROLOGY

of these relations into (16.100) leads to closed equations for the mean quantities. Thus, except for the fact that KM and KT vary with position and direction, these models for turbulent transport are analogous to those for molecular transport of momentum and energy. The use of a diffusion equation model implies that the lengthscale of the transport processes is much smaller than the characteristic length over which the mean velocity and temperature profiles are changing. In molecular diffusion in a gas at normal densities, the lengthscale of the diffusion process, the mean free path of a molecule, is many orders of magnitude smaller than the distances over which the mean properties of the gas (e.g., velocity or temperature) vary. In turbulence, on the other hand, the motions responsible for the transport of momentum, heat, and material are usually roughly the same size as the characteristic lengthscale for changes in the mean fields of velocity, temperature, and concentration. In the atmosphere, for example, the characteristic vertical dimension of eddies is of the same order as is the characteristic scale for changes of the velocity profile. Thus, in general, a diffusion model for transport like the one given by (16.110) and (16.111) has some serious weaknesses for the description of atmospheric turbulence. The success in using these equations depends on two factors: (1) they should ideally be employed in situations in which the lengthscale for changes in the mean properties is considerably greater than that of the eddies responsible for transport; and (2) the values and functional forms of KM and KT should be determined empirically for situations similar to those in which (16.110) and (16.111) are applied.

16.5 VARIATION OF WIND WITH HEIGHT IN THE ATMOSPHERE The atmosphere near Earth’s surface can be divided into four layers; proceeding down from the free atmosphere, these are the Ekman layer; the surface layer; and immediately adjacent to the ground surface, the laminar sublayer. The thickness of the laminar sublayer is typically less than a centimeter, and this layer can be neglected for all practical purposes in the present discussion. (The fluid viscosity becomes important in this thin layer; we will examine the processes in this layer in our discussion of dry deposition in Chapter 19.) The surface layer extends typically from the surface to a height of 30–50 m. Within this layer, the vertical turbulent fluxes of momentum and heat are assumed constant with respect to height, and indeed they define the extent of this region. The Ekman layer extends to a height of 300–500 m depending on the terrain type, with the greatest thickness corresponding to the more uneven terrain. In this layer, the wind direction is affected by Earth’s rotation (see Chapter 21). The windspeed in the Ekman layer generally increases rapidly with height; however, the rate lessens near the free atmosphere. In this section we consider the variation of wind with height in the surface and Ekman layers, which constitute the so-called planetary boundary layer. Most of our attention will be devoted to the surface layer, the region in which pollutants are usually first released. The exact vertical distribution of wind velocity depends on a number of parameters, including the surface roughness and the atmospheric stability. In meteorological applications, the surface roughness is usually characterized by the height of the roughness elements (buildings, trees, bushes, grass, etc.) ε. These elements are usually so closely distributed that only their height and spacing are important. In general, smooth surfaces allow the establishment of a laminar sublayer in which they are submerged. On the other hand, a rough surface is one in which the roughness elements are high enough to prevent the formation of a laminar sublayer, so that the flow is turbulent down to the roughness elements. The depth of the laminar sublayer, and hence the classification of the surface as smooth or rough, depends on the Reynolds number of the flow. Knowledge of the governing physics of flows in the surface layer is not sufficiently complete to derive the vertical mean velocity profiles based on first principles. Useful empirical relationships have been developed using approximate theories and experimental measurements. Similarity theories, based on dimensional analysis, provide a convenient method for grouping of the system variables into dimensionless parameters and then the derivation of universal similarity relationships.

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VARIATION OF WIND WITH HEIGHT IN THE ATMOSPHERE

16.5.1 Mean Velocity in the Adiabatic Surface Layer over a Smooth Surface Consider first the steady, ground-parallel flow of air over a flat homogeneous surface. We assume that the wind flow is in the x direction (ūy = 0) and that ūx = ux̄ (z). Our goal is to determine ūx(z), assuming that the vertical temperature profile is adiabatic. Let us see what can be determined about the functional dependence of ūx(z) employing dimensional analysis. For this analysis, we need to define a characteristic velocity for the flow u∗. This characteristic velocity depends on the turbulence flux u´x u´z and is called the friction velocity u∗. It is actually equal to p τ0 =ρ, where τ0 is the shear stress at the surface. The friction velocity can be calculated from an actual measurement of the velocity at a given height. With the addition of the friction velocity, we have five variables in our problem, ūx, ν, ρ, z, and u∗, involving three dimensions (mass, length, and time). We can facilitate the following steps by replacing in this set of variables the velocity ūx with its gradient, so that the variable set becomes dūx/dz, ν, ρ, z, and u∗. The Buckingham π theorem states that in this system there are only 5 3 = 2 independent dimensionless groups, π1 and π2, relating the five variables, so that F(π1, π2) = 0. The π theorem does not tell us what the groups are, only how many exist. As the first group we select zu∗ (16.112) π1 ˆ ν which is essentially a Reynolds number for the flow. For the second dimensionless parameter a convenient choice is a dimensionless velocity gradient π2 ˆ

dux z dz u∗

(16.113)

and according to the π theorem, F(π1, π2) = 0 or equivalently π2 = G(π1). Using (16.112) and (16.113), we obtain dux u∗ zu∗ ˆ G dz z ν

(16.114)

This is as far as dimensional analysis will bring us. We now need some physical insight to determine the unknown function G. The kinematic viscosity ν is important only in the laminar sublayer, and dūx/dz should not depend on ν in the region of interest above the sublayer. Thus we can set the function G equal to a constant (1/κ) obtaining dux 1 u∗ ˆ dz κ z

(16.115)

On integration, we obtain ux …z† ˆ

u∗ ln z ‡ const κ

This equation is usually written in the dimensionless form: ux …z† 1 u∗ z ˆ ln ‡ const κ ν u∗

(16.116)

The integration constant in (16.116) has been evaluated experimentally for smooth surfaces and has been found to be equal to 5.5. The constant κ is known as the von Karman constant with a value of ∼0.4. In spite of an uncertainty of the order of 5% in empirical estimates of κ for different surfaces, it is considered a universal constant. The mean velocity profile can therefore be calculated by ux …z† ˆ u∗

1 u∗ z ln ‡ 5:5 κ ν

(16.117)

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PHYSICAL AND DYNAMIC METEOROLOGY

and the velocity ūx is predicted to increase according to the logarithm of z for a perfectly smooth surface. Inspecting (16.117), we see that ūx becomes zero when ln(u∗z/ν) = 5.5κ = 2.2, that is, when z = z∗ = 0.11 ν/u∗. Typical values of u∗ and ν for the atmosphere are 100 cm s 1 and 0.1 cm2 s 1, so ūx vanishes at around 10 4 cm and becomes negative for even smaller z. Equation (16.117) holds only for values of z greater than z∗. Equation (16.117) has limited utility in the real atmosphere because all actual surfaces have some roughness.

16.5.2 Mean Velocity in the Adiabatic Surface Layer over a Rough Surface For the case of a surface with roughness elements of height ε, one can repeat the analysis used in the previous section for a smooth surface. In this case there is no laminar sublayer, so we do not need to use the kinematic viscosity in our analysis. On the other hand, we do need to add the height of the roughness elements. Thus the five parameters for the dimensional analysis are now dūx/dz, ε, ρ, z, and u∗. One can then show that ux …z† 1 z ˆ ln ‡ Q κ ε u∗

(16.118)

where Q is an integration constant. One can incorporate the integration constant in the logarithm by defining the roughness length z0 so that 1 z 1 z ln ˆ ln ‡ Q κ z0 κ ε

(16.119)

Using (16.119), the velocity profile becomes ux …z† ˆ

u∗ z ln κ z0

z > z0

(16.120)

The roughness length z0 is related to the height of the roughness elements according to (16.119) or, after some algebra, z0 = ε exp( κQ). It has been found experimentally that z0  ε/30. Values of the roughness length for typical surfaces are given in Table 16.1. Note that according to (16.120), ūx = 0 for z = z0, and that the equation is valid only for heights significantly greater than the roughness length.

TABLE 16.1 Roughness Lengths for Various Surfaces Surface Very smooth (ice, mud flats) Snow Smooth sea Level desert Lawn Uncut grass Full-grown root crops Tree covered Low-density residential Central business district Source: McRae et al. (1982).

z0 (m) 10 5 10 3 10 3 10 3 10 2 0.05 0.1 1 2 5–10

695

VARIATION OF WIND WITH HEIGHT IN THE ATMOSPHERE

Commonly, the friction velocity u∗ is obtained from a measurement of the velocity at some reference height hr, often equal to 10 m. Then, if ūx(hr) is known, we obtain u∗ ˆ

κux …hr † ln…hr =z0 †

(16.121)

A better way to obtain u∗ would be a direct measurement of the surface shear stress, but this requires elaborate experimental measurements and is not routinely available. Equation (16.121) works satisfac­ torily in adiabatic boundary layers (Plate 1971). Calculation of the Mean Velocity Profile The mean wind velocity at 4 m over a surface with z0 = 0.0015 m was measured at 7.8 m s 1 during a period of the Wangara experiment (Deardoff 1978). The atmosphere was adiabatic. Calculate the velocity at 0.5 and 16 m and compare with the observed values of 5.7 and 9.1 m s 1. Using (16.121), u∗ = 0.4 m s 1. Using this value and (16.120), we estimate that ūx(0.5 m) = 5.8 m s 1 and ūx(16 m) = 9.3 m s 1. Both of these agree within 2% of the measured values in the field study.

16.5.3 Mean Velocity Profiles in the Nonadiabatic Surface Layer The basic logarithmic velocity profile (16.120) is applicable only to adiabatic conditions. However, the atmosphere is seldom adiabatic, and the velocity profiles for stable and unstable conditions deviate from this logarithmic law. For the more frequently encountered nonadiabatic atmosphere (also called stratified), the Monin–Obukhov similarity theory is usually employed (Monin and Obukhov 1954). According to this theory, the turbulence characteristics in the surface layer are governed in general by the following seven variables: dūx/dz, z, z0, u∗, ρ, (g/T0), and qz ˆ ρcp uz´ θ´ , where the first five were also used in the previous section, (g/T0) is a parameter related to buoyancy, and qz is the vertical mean turbulent flux. Assuming that variations in the roughness length z0 do not affect the form of the velocity profiles but only shift them, we can neglect this parameter and reduce the list to six variables. There are four dimensions in the problem (mass, length, time, and temperature), so, according to the Buckingham π theorem, there are two dimensionless groups governing the behavior of the system. The first group is called the flux Richardson number (Rf) and is defined as Rf ˆ

κgzqz ρcp T 0 u3∗

(16.122)

where κ = 0.4 is the von Karman constant. One can show that the flux Richardson number is equal to the ratio of the production of turbulent kinetic energy by buoyancy to its production by shear stresses. Rf can be positive or negative depending on the sign of the vertical mean turbulent flux qz ˆ ρcp uz´ θ´ . There are three cases: Case 1. If u´z θ´ > 0, then qz > 0 and Rf < 0. For this case, positive values of uz´ tend to be associated with positive values of θ´ and vice versa. This case corresponds to an unstable atmosphere (decreasing potential temperature with height). If an air parcel moves upward because of a positive fluctuation in its velocity u´z , it rises to a region of lower potential temperature. However, the air parcel temperature changes adiabatically during this small rapid fluctuation, and its potential temperature remains constant. As a result, the air parcel potential temperature will exceed the potential temperature of its surroundings and will cause a positive potential temperature fluctuation θ´ in its new position. Case 2. If u´z θ´ < 0, then qz < 0 and Rf > 0. In this case, positive values of u´z occur with negative values of θ´ , and the atmosphere (using the same arguments as above) is stable. Case 3. If u´z θ´ ˆ 0 then qz ˆ 0 and Rf = 0. This case corresponds to an adiabatic (neutral) atmosphere.

696

PHYSICAL AND DYNAMIC METEOROLOGY

TABLE 16.2 Monin–Obukhov Length L with Respect to Atmospheric Stability L Very large negative Large negative Small negative Small positive Large positive Very large positive

Stability condition L < 105 m 105 m  L  100 m 100 m < L < 0 0 < L < 100 m 100 m  L  105 m L > 105 m

Neutral Unstable Very unstable Very stable Stable Neutral

The flux Richardson number according to (16.122) is a function of the distance from the ground. Because it is dimensionless, it can actually be viewed as a dimensionless length Rf ˆ

z L

(16.123)

where L is the Monin–Obukhov length and according to (16.122) and (16.123) is given by Lˆ

ρcp T0 u3∗ κ gqz

(16.124)

By definition the Monin–Obukhov length is the height at which the production of turbulence by both mechanical and buoyancy forces is equal. The parameter L, like the flux Richardson number, provides a measure of the stability of the surface layer. As we discussed, when Rf > 0 and therefore according to (16.123) L > 0, the atmosphere is stable. On the other hand, when the atmosphere is unstable, Rf < 0 and then L < 0. Because of the inverse relationship between Rf and L, an adiabatic atmosphere corresponds to very small (positive or negative) values of Rf and to very large (positive or negative) values of L. Typical values of L for different atmospheric stability conditions are given in Table 16.2. The second dimensionless group is the dimensionless velocity gradient κ L @ux u∗ @z Let us now redefine the first dimensionless group as ζˆ

z L

(16.125)

The dimensionless length ζ is equal to the flux Richardson number, ζ = Rf, only in the surface layer. Using once more the π theorem, we can write the second dimensionless group as a function of the first or κ L @ux ˆ g…ζ† u∗ @z

(16.126)

Our next step is to find the function g(ζ). Replacing L from (16.125) into (16.126) and rearranging, we obtain @ux u∗ ˆ ϕ…ζ† @z κ z

(16.127)

697

VARIATION OF WIND WITH HEIGHT IN THE ATMOSPHERE

where we have defined ϕ(ζ) = ζg(ζ). For ζ → 0 the atmosphere becomes neutral, therefore (16.127) should become (16.114) at this limit. Comparing these two equations, we find that lim ϕ…ζ† ˆ 1

(16.128)

ζ!0

Generally accepted forms of the universal function ϕ(ζ) are those of Businger et al. (1971): Stable ζ > 0

ϕ…ζ† ˆ 1 ‡ 4:7 ζ

Neutral ζ ˆ 0 ϕ…ζ† ˆ 1 Unstable ζ < 0 ϕ…ζ† ˆ …1

(16.129) 15 ζ†

1=4

The velocity ūx(z) can be determined by integrating (16.127) from z = z0 and ux̄ = 0 using (16.129): ux …z† ˆ

u∗ z=L ϕ…ζ† dζ κ ∫ z0 =L ζ

(16.130)

For neutral conditions, (16.130) results in the logarithmic law consistent with (16.120): ux …z† ˆ

u∗ z ln κ z0

(16.131)

For stable conditions, combining (16.130) and the first equation in (16.120) we find that ux …z† ˆ

u∗ z z0 u∗ z ln ‡ 4:7 κ L κ z0

(16.132)

This profile relation may be viewed as a correction to the logarithmic law. For an almost neutral atmosphere, L is a large positive number, and the relationship between velocity and height is logarithmic. As stability increases, the positive L decreases and the deviation from the logarithmic behavior increases. Finally, for very stable conditions, L approaches zero, the second term in (16.132) dominates, and the velocity profile becomes linear. For unstable conditions, the velocity profile given by (16.130) and the third equation in (16.129) is ux …z† ˆ

u∗ z=L dζ κ ∫ z0 =L ζ…1 15 ζ†1=4

(16.133)

Use of (16.130) or (16.131)–(16.133) requires the calculation of the friction velocity u∗. This requires once more the measurement of the velocity at a height hr (say, 10 m). So, if ūx(hr) is known, we can integrate (16.127) between z0 and hr and solve for u∗ assuming that ux̄ (z0) = 0: u∗ ˆ

κux …hr † hr

∫ z0

…ϕ…z=L†=z†dz

(16.134)

For an adiabatic atmosphere, (16.134) reduces to (16.121). For a stable atmosphere, we find that u∗ ˆ

κux …hr † hr 4:7…hr z0 † ln ‡ z0 L

(16.135)

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PHYSICAL AND DYNAMIC METEOROLOGY

TABLE 16.3 Estimation of Pasquill Stability Classesa

Surface (10 m) windspeed (m s 1) 6

Daytimeb

Nighttimeb

Incoming solar radiationc

Cloudcover fraction

Strong

Moderate

Slight

 48

 38

A A–B B C C

A–B B B–C C–D D

B C C D D

--­ E D D D

--­ F E D D

a

Key: A- --extremely unstable; B- --moderately unstable; C- --slightly unstable; D- --neutral; E- --slightly stable; F- ­-moderately stable. The neutral category D should be used, regardless of windspeed, for overcast conditions during day or night. c Solar radiation: strong (>700 W m 2), moderate (350–700 W m 2), slight ( es°, for a pure water droplet to exist at equilibrium in the atmosphere, the air must be supersaturated with water vapor. For the equilibration of a large pure water droplet, a modest supersaturation is necessary, but a large supersaturation is needed for a small droplet. Let us investigate the stability of such an equilibrium state. We assume that a droplet of pure water with diameter Dp is in equilibrium with the atmosphere. At temperature T, the atmosphere will have a water vapor partial pressure es = es(Dp). Say that a few molecules of water vapor collide with the droplet, causing its diameter to increase infinitesimally to Dp´ . This, in turn, will cause a small decrease in the water vapor pressure to es(Dp´ ). The water vapor concentration in the atmosphere has not changed, and therefore es > es(Dp´ ), causing more water molecules to condense on the droplet and the droplet to grow even more. The opposite will happen if a few water molecules leave the droplet. The droplet diameter will decrease, the water vapor pressure at the droplet surface will exceed that of the environment, and the droplet will continue to evaporate. These simple arguments indicate that the equilibrium of a pure water droplet is unstable. A minor perturbation is sufficient either for the complete evaporation of the droplet or for its uncontrollable growth. TABLE 17.1 Saturation Vapor Pressure of Water over a Pure Water or Ice Surface es°…mbar† ˆ a0 ‡ a1 T ‡ a2 T 2 ‡ a3 T 3 ‡ a4 T 4 ‡ a5 T5 ‡ a6 T6 (T is in °C) Water ( 50 to +50 °C)

Ice ( 50 to 0 °C)

a0 = 6.107799961 a1 = 4.436518521 × 10 a2 = 1.428945805 × 10 a3 = 2.650648471 × 10 a4 = 3.031240396 × 10 a5 = 2.034080948 × 10 a6 = 6.136820929 × 10

a0 = 6.109177956 a1 = 5.034698970 × 10 1 a2 = 1.886013408 × 10 2 a3 = 4.176223716 × l0 4 a4 = 5.824720280 × 10 6 a5 = 4.838803174 × 10 8 a6 = 1.838826904 × 10 10

1 2 4 6 8 11

Source: Lowe and Ficke (1974).

710

CLOUD PHYSICS

FIGURE 17.2 Ratio of the equilibrium vapor pressure of water over a droplet of diameter Dp, es(Dp), to that over a flat surface, es , as a function of droplet diameter for 0 °C and 20 °C.

17.1.2 Equilibrium of a Flat Water Solution Droplets in the atmosphere virtually never consist of pure water; they always contain dissolved compounds. Let us consider a water solution (flat surface) at temperature T and pressure p in equilibrium with the atmosphere. Water equilibrium between the gas and aqueous phases requires equality of the corresponding water chemical potentials in the two phases (see Chapter 10): μw g ˆ μw aq

(17.2)

Water vapor behaves in the atmosphere as an ideal gas, so its gas-phase chemical potential is μw g ˆ μw°…T† ‡ RT ln es; soln

(17.3)

where es; soln is the water vapor partial pressure over the solution. The chemical potential of liquid water is given by μw ˆ μ∗w ‡ RT ln γw xw

(17.4)

where γw is the water activity coefficient and xw is the mole fraction of water in solution. Combining (17.2) and (17.4), we obtain  es; soln μ ° μw ˆ exp w γ w xw RT

ˆ K …T †

(17.5)

This equation describes the behavior of the system for any solution composition. Note that the righthand side (RHS) is a function of temperature only and therefore is equal to a constant, K(T). Considering

711

EQUILIBRIUM OF WATER DROPLETS IN THE ATMOSPHERE

the case of pure water (no solute), we note that when xw ˆ 1; γw ! 1 and e°s ˆ K…T †, where e°s is the saturation vapor pressure of pure water. Therefore (17.5) can be rewritten as es;soln ˆ γw xw es°

(17.6)

Equation (17.6) is applicable for any solution and does not assume ideal behavior. Nonideality is accounted for by the activity coefficient γw . The mole fraction of water in a solution consisting of nw water moles and ns solute moles is given by xw ˆ

nw nw ‡ ns

(17.7)

and therefore the vapor pressure of water over its solution is given by es;soln ˆ

nw γ es° nw ‡ ns w

(17.8)

If the solution is dilute, then γw approaches its infinite dilution limit γw ! 1: Therefore, at high dilution, the vapor pressure of water is given by Raoult’s law es;soln ˆ xw es°

(17.9)

and the solute causes a reduction of the water equilibrium vapor pressure over the solution. The vapor pressure of water over NaCl and (NH4)2SO4 solutions as a function of solute mole fraction is shown in Figure 17.3. Also shown is the ideal solution behavior. Note that because NaCl dissociates

FIGURE 17.3 Variation of water vapor pressure ratio es;soln =es as a function of the solute mole fraction at 25 °C for solution of NaCl and (NH4)2SO4 and an ideal solution. The mole fraction of the salts has been calculated taking into account their complete dissociation.

712

CLOUD PHYSICS

FIGURE 17.4 Variation of water vapor pressure ratio es;soln =es as a function of the salt molality (mol salt per kg water) of NaCl and (NH4)2SO4 at 25 °C.

into two ions, the number of equivalents in solution is twice the number of moles of NaCl. For (NH4)2SO4, the number of ions in solution is 3 times the number of dissolved salt moles. In calculating the number of moles in solution ns, a molecule that dissociates into i ions is treated as i molecules, whereas an undissociated molecule is counted only once. A similar diagram is given in Figure 17.4, using now the molality of salt as the independent variable. Solutes that dissociate (e.g., salts) reduce the vapor pressure of water more than do solutes that do not dissociate, and this reduction depends strongly on the type of salt.

17.1.3 Atmospheric Equilibrium of an Aqueous Solution Drop In the previous sections we developed expressions for the saturation vapor pressure of water over a pure water droplet and a flat water solution containing a dissolved solute. Atmospheric droplets virtually always contain dissolved solutes, so we need to combine these two previous results to treat this general case. Let us consider a droplet of diameter Dp containing nw moles of water and ns moles of solute (e.g., a nonvolatile salt). If the water–air interface is flat, the water vapor pressure over it satisfies (17.6). Substituting this expression into (17.1), one obtains es D p 4υw σw ˆ exp es°γw xw RTDp

(17.10)

υw and υs are the partial molar volumes of the two components in the solution, and the total drop volume satisfies 1 3 πD ˆ nw υw ‡ ns υs 6 p

(17.11)

713

EQUILIBRIUM OF WATER DROPLETS IN THE ATMOSPHERE

Note that since we have assumed that the solute is nonvolatile, ns is the same regardless of the value of Dp. Using (17.11) and (17.7), we find that 1 ns ns υw ˆ1‡ ˆ1‡ nw xw …π=6†D3p ns υs

(17.12)

Replacing the water mole fraction appearing in (17.10) by (17.12), we obtain

ln

es Dp es°

ˆ

4υw σw ‡ ln γw RTDp

ln 1 ‡

ns υw …π=6†D3p ns υs

(17.13)

In the derivation of (17.13) we have implicitly assumed that temperature, pressure, and number of solute moles ns remain constant as the droplet diameter changes. This equation relates the water vapor pressure over the droplet solution of diameter Dp to that of water over a flat surface es° at the same temperature. If the solution is dilute, the volume occupied by the solute can be neglected relative to the droplet volume; that is, ns υ s 

π 3 D 6 p

(17.14)

Then (17.13) can be simplified to ln

es Dp es°

ˆ

4Mw σw ‡ ln γw RTρw Dp

ln 1 ‡

6ns υw πD3p

(17.15)

For dilute solutions one can also assume that γw ! 1 and 6ns υw =πD3p ! 0; recalling that ln…1 ‡ x† ≅ x as x ! 0, we find ln

es D p es°

ˆ

4Mw σw RTρw Dp

6ns υw πD3p

(17.16)

Recall that υw is the molar volume of water in the solution, which for a dilute solution is equal to the molar volume of pure water: υw ≅

Mw ρw

(17.17)

Replacing (17.17) in (17.16), we obtain

ln

es D p es°

ˆ

4Mw σw RTρw Dp

6ns Mw πρw D3p

It is customary to define Aˆ

4Mw σw RTρw

Bˆ

6ns Mw πρw

(17.18)

714

CLOUD PHYSICS

and write (17.18) as

ln

es D p es°

ˆ

A Dp

B D3p

(17.19)

Equations (17.13), (17.16), (17.18), and (17.19) are different forms of the Köhler equations (Köhler 1921, 1926). These equations express the two effects that determine the vapor pressure over an aqueous solution droplet: the Kelvin effect that tends to increase vapor pressure and the solute effect that tends to decrease vapor pressure. For a pure water drop there is no solute effect and the Kelvin effect results in higher vapor pressures compared to a flat interface. By contrast, the vapor pressure of an aqueous solution drop can be larger or smaller than the vapor pressure over a pure water surface depending on the magnitude of the solute effect term B=D3p relative to the curvature term A/Dp. Note that both effects increase with decreasing droplet size but the solute effect increases much faster. One should also note that a droplet may be in equilibrium in a subsaturated environment if D2p A < B: Figure 17.5 shows the water vapor pressure over NaCl and (NH4)2SO4 solution drops. The A term in the Köhler equations can be approximated by 4Mw σw 0:66 ≅ …in μm† T RTρw

(17.20)

6ns Mw 3:44  1013 νms ≅ …in μm3 † Ms πρw

(17.21)

Aˆ where T is in K, and the solute term Bˆ

where ms is the solute mass (in g) per particle, Ms the solute molecular weight (in g mol 1), and ν is the number of ions resulting from the dissociation of one solute molecule. For example, ν = 2 for NaCl and NaNO3, while ν = 3 for (NH4)2SO4.

FIGURE 17.5 Köhler curves for NaCl and (NH4)2SO4 solution droplets with dry diameters 0.05, 0.1, and 0.5 μm at 293 K (assuming spherical dry particles). The supersaturation is defined as the saturation minus 1.0. For example, a supersaturation of 1% corresponds to a relative humidity of 101%.

715

EQUILIBRIUM OF WATER DROPLETS IN THE ATMOSPHERE

All the curves in Figure 17.5 pass through a maximum. These maxima occur at the critical droplet diameter Dpc

Dpc ˆ

1=2

3B A

(17.22)

and at this diameter (denoted by the subscript c) ln

es Dp es°

ˆ c

4A3 27B

1=2

(17.23)

The ratio es Dp =es° is the saturation relative to a flat pure water surface required for droplet equilibrium and therefore at the critical diameter, the critical saturation, Sc, is

ln Sc ˆ

4A3 27B

1=2

(17.24)

The steeply rising portion of the Köhler curves represents a region where solute effects dominate. As the droplet diameter increases, the relative importance of the Kelvin effect over the solute effect increases, and finally beyond the critical diameter, the Kelvin effect dominates. In this range, all Köhler curves approach the Kelvin equation, represented by the equilibrium of a pure water droplet. Physically, the solute concentration is so small in this range (recall that each Köhler curve refers to fixed solute amount) that the droplet approaches pure water. The Köhler curves also represent the equilibrium size of a droplet for different ambient water vapor concentrations (or relative humidity values). If the water vapor partial pressure in the atmosphere is e, a droplet containing ns moles of solute and having a diameter Dp satisfying the Köhler equations should be at equilibrium with its surroundings. Realizing that the Köhler curves can be viewed as size–RH equilibrium curves poses some interesting questions: What happens if the atmosphere is supersaturated with water vapor and the supersaturation exceeds the critical supersaturation for a given particle? What happens if for a given atmospheric saturation there are two diameters for which the droplet can satisfy the Köhler equations? Are there two equilibrium states? To answer these questions and understand the cloud and fog creation processes in the atmosphere, we need to investigate the stability of the equilibrium states given by the Köhler equation. 17.1.3.1 Stability of Atmospheric Droplets We have already seen that a pure water droplet cannot be at stable equilibrium with its surroundings. A small perturbation of either the droplet itself or its surroundings causes spontaneous droplet growth or shrinkage. Let us consider first a drop lying on the portion of the Köhler curve for which Dp < Dpc. We assume that the atmospheric saturation is fixed at S. A drop will constantly experience small perturbations caused by the gain or loss of a few molecules of water. Say that the drop grows slightly with the addition of a few molecules of water. At its momentary larger size, its equilibrium vapor pressure is larger than

716

CLOUD PHYSICS

the fixed ambient value and the drop will evaporate water, eventually returning to its original equilibrium state. The same phenomenon will be observed if the droplet loses a few molecules of water. Its equilibrium vapor pressure will decrease, becoming less than the ambient, and water will condense on the droplet, returning it to its original size. Therefore drops in the rising part of the Köhler curve are in stable equilibrium with their environment. Now consider a drop on the portion of the curve for which Dp > Dpc that experiences a slight perturbation, causing it to grow by a few molecules of water. At its slightly larger size its equilibrium vapor pressure is lower than the ambient. Thus water molecules will continue to condense on the drop and it will grow even larger. Conversely, a slight shrinkage leads to a drop that has a higher equilibrium vapor pressure than the ambient so the drop continues to evaporate. If it is a drop of pure water, it will evaporate completely. If it contains a solute, it will diminish in size until it intersects the ascending branch of the Köhler curve that corresponds to stable equilibrium. In conclusion, the descending branches of the curves describe unstable equilibrium states. If the ambient saturation ratio S is lower than the critical saturation Sc for a given particle, then the particle will be in equilibrium described by the ascending part of the curve. If 1 < S < Sc, then there are two equilibrium states (two diameters corresponding to S). One of them is a stable state, and the other is unstable. The particle can reach stable equilibrium only at the state corresponding to the smaller diameter. If the ambient saturation ratio S happens to exceed the particle critical saturation Sc, there is no feasible equilibrium size for the particle. For any particle diameter the ambient saturation will exceed the saturation at the particle surface (equilibrium saturation), and the particle will grow indefinitely. In such a way a droplet can grow to a size much larger than the original size of the dry particle. It is, in fact, through this process that particles as small as 0.01 μm in diameter can grow one billion times in mass to become 10 μm cloud or fog droplets. Moreover, in cloud physics a particle is not considered to be a cloud droplet unless its diameter exceeds its critical diameter Dpc. The critical saturation Sc of a particle is an important property. If the environment has reached a saturation larger than Sc, the particle is said to be activated and starts growing rapidly, becoming a cloud droplet. For a spherical aerosol particle of diameter ds (dry diameter), density ρs ; and molecular weight Ms, the number of moles (after complete dissociation) in the particle is given by ns ˆ

νπd3s ρs 6Ms

(17.25)

and combining this result with (17.24), we find that

ln Sc ˆ

4A3 ρw Ms 27νρs Mw d3s

1=2

(17.26)

This equation gives the critical saturation for a dry particle of diameter ds. Note that the smaller the particle, the higher its critical saturation. When a fixed saturation S exists, all particles whose critical saturation Sc exceeds S come to a stable equilibrium size at the appropriate point on their Köhler curve. All particles whose Sc is below S become activated and grow indefinitely as long as S > Sc. Figure 17.6 shows the critical supersaturation for spherical salt particles as a function of their diameters. One should note that critical saturations are always >1, and one defines sc = Sc 1 as the critical supersaturation. Figure 17.7 presents the Köhler curves for (NH4)2SO4 for complete dissociation and no dissociation. Note that dissociation lowers the critical saturation ratio of the particle (the particle is activated more easily) and increases the drop critical diameter (the particle absorbs more water).

EQUILIBRIUM OF WATER DROPLETS IN THE ATMOSPHERE

FIGURE 17.6 Critical supersaturation for activating aerosol particles com­ posed of NaCl and (NH4)2SO4 as a function of the dry particle diameters (assuming spherical particles) at 293 K.

FIGURE 17.7 Köhler curves for (NH4)2SO4 assuming complete dissociation and no dissociation of the salt in solution for dry radii of 0.02, 0.04, and 0.1 μm.

17.1.4 Atmospheric Equilibrium of an Aqueous Solution Drop Containing an Insoluble Substance Our analysis so far has assumed that the aerosol particle consists of a soluble salt that dissociates completely as the RH exceeds 100%. Most atmospheric particles contain both water-soluble and waterinsoluble substances (dust, elemental carbon, etc.). Here we extend the results of the previous section to

717

718

CLOUD PHYSICS

account for the existence of insoluble material. Our assumption is that the original particle contains soluble material with mass fraction εm, and the rest is insoluble. We also assume that the insoluble portion does not interact at all with water or the salt ions. Our analysis follows exactly Section 17.1.3, until the derivation of the mole fraction of water xw in (17.11) and (17.12). The existence of the insoluble material needs to be included in these equations. If the insoluble particle fraction is equivalent to a sphere of diameter du, then the droplet volume will be 1 3 1 πD ˆ nw υw ‡ ns υs ‡ πdu3 6 p 6

(17.27)

and the mole fraction of water xw in the solution is given by 1 ns ns υw ˆ1‡ ˆ1‡ 3 nw xw …π=6† Dp du3

(17.28)

ns υs

Substituting this expression into (17.10), we find

ln

es Dp es°

ˆ

4Mw σw ‡ ln γw RTρw Dp

ln 1 ‡

ns υw …π=6†

D3p

d3u

ns υ s

(17.29)

For a dilute solution, we may simplify (17.29) as before and obtain expressions analogous to (17.18) and (17.19) with the same definitions of A and B. For example, in this case (17.19) becomes

ln

es Dp es°

ˆ

A Dp

B D3p

d3u

(17.30)

The effect of the insoluble material is to increase in absolute terms the solute effect. Physically, the insoluble material is responsible for part of the droplet volume, displacing the equivalent water. Therefore, for the same overall droplet diameter, the solution concentration will be higher and the solute effect more significant. An alternative expression can be developed replacing the equivalent diameter of the insoluble material du with the mass fraction of soluble material εm, assuming that the insoluble material has a density ρu : Then for a dry particle of diameter ds the following relationship exists between the insoluble core diameter du and the mass fraction εm d3u ˆ

d3s ρu ε m ‡1 ρs 1 ε m

(17.31)

and (17.30) can be rewritten as a function of the particle soluble fraction and the initial particle diameter provided that the densities of soluble and insoluble material are known. The number of moles of solute is now given not by (17.25) but by the following expression: ns ˆ

εm Ms

νπd3s εm 1 εm 6 ‡ ρu ρs

(17.32)

Köhler curves for a particle consisting of various combinations of (NH4)2SO4 and insoluble material are given in Figure 17.8. We see that the smaller the water-soluble fraction, the higher the

CLOUD AND FOG FORMATION

FIGURE 17.8 Variation of the equilibrium vapor pressure of an aqueous solu­ tion drop containing (NH4)2SO4 and insoluble material for an initial dry particle diameter of 0.1 μm for soluble mass fractions 0.2, 0.4, 0.6, and 1.0 at 293 K.

FIGURE 17.9 Critical supersaturation as a function of the particle dry diameter for different contents of insoluble material. The soluble material is (NH4)2SO4.

supersaturation needed for activation of the same particle, and the lower the critical diameter. Critical supersaturation as a function of the dry particle diameter is given in Figure 17.9.

17.2 CLOUD AND FOG FORMATION The ability of a given particle to become activated depends on its size and chemical composition and on the maximum supersaturation experienced by the particle. We will see later on that particle size tends to have the strongest effect on whether a given particle becomes activated; particle composition

719

720

CLOUD PHYSICS

has a lesser influence. As a particle is lifted in a cooling air parcel, it encounters an increasing relative humidity and grows via the uptake of water. (Even though other soluble atmospheric vapor compounds can condense into the growing droplet, since water vapor is, by far, the most prevalent gas-phase species, condensation of liquid water causes the overall growth of the particle.) The water vapor concentration in the rising parcel is depleted by its uptake into growing particles, but the decreasing water saturation vapor pressure with cooling causes the parcel to become supersaturated. As the parcel eventually becomes supersaturated, each growing droplet approaches its activation point. Those particles that activate are termed cloud condensation nuclei (CCN). Once activated, the droplets exhibit runaway growth until the population of growing droplets depletes the water vapor in the parcel so that a steady state condition is reached. Classic Köhler theory does not consider the cases of solutes that are not completely soluble or soluble gases. Extensions of the Köhler theory for soluble trace gases, slightly soluble substances and surface active solutes are presented in Appendix 17 at the end of this chapter. In this section we will examine the mechanisms by which clouds are created in the atmosphere. A necessary condition for cloud formation is the increase of the RH of an air parcel to a value exceeding 100%. From Chapter 16 we know that this RH increase is usually the result of cooling of a moist air parcel. Even if the water mass inside the air parcel does not change, its saturation water vapor concentration decreases as its temperature decreases, and therefore its RH increases. There are several mechanisms by which air parcels can cool in the atmosphere. It is useful to separate these mechanisms into two groups: isobaric cooling and adiabatic cooling. Isobaric cooling is the cooling of an air parcel under constant pressure. It usually is the result of radiative losses of energy (fog and low stratus formation) or horizontal movement of an airmass over a colder land or water surface or colder airmass.

17.2.1 Isobaric Cooling Let us consider a volume of moist air that is cooled at constant pressure. Assuming that we can neglect mass exchange between the air parcel and its surroundings, the water vapor partial pressure (e) will remain constant. A temperature decrease from an initial value T0 to Td will lead to a decrease of the saturation vapor pressure from es°…T 0 † to es°…T d †. We would like to calculate the point at which the parcel becomes saturated. In other words, if a parcel initially has a temperature T0 and relative humidity RH (from 0 to 1), what is the temperature Td at which it will become saturated (RH = 1)? As we saw in Chapter 16, this temperature is called the dew-point temperature, Tdp, and is given by (16.63).

17.2.2

Adiabatic Cooling

The level at which a moist air parcel lifted adiabatically from near the surface reaches saturation is the lifting condensation level (LCL), given by (16.61). So far we have treated a rising air parcel as a closed system. This is rarely the case in real clouds, where air from the rising parcel is mixed with the surrounding air. If m is the mass of the air parcel, one defines the entrainment rate E as Eˆ

1 dm m dz

(17.33)

If the water vapor mass mixing ratio of the environment around the rising parcel is w´v and its temperature T ´ , the lapse rate in the cloud Γc is (Pruppacher and Klett 1997) Γc ˆ Γd ‡

lv dwvs E ‡ lv w v cp dz cp

w´v ‡ cp …T

T ´†

Γc exceeds the moist adiabatic lapse rate Γs because E > 0, wv > w´v , and T > T ´ :

(17.34)

721

CLOUD AND FOG FORMATION

17.2.3 A Simplified Mathematical Description of Cloud Formation Let us revisit the rising moist air parcel assuming that we are in a Lagrangian reference frame, moving with the air parcel. The air parcel is characterized by its temperature T, water vapor mixing ratio wv, liquid water mixing ratio wl , and vertical velocity W. At the same time we need to know the temperature T ´ ; pressure p, and water vapor mixing ratio w´v of the air around it (Figure 17.10). The pressure of the air parcel is (always) assumed to be equal to its environment. Let us assume that the air parcel has mass m and air density ρ (excluding the liquid water). The velocity of the air parcel will be the result of buoyancy forces and the gravitational force due to liquid water. The buoyancy force is proportional to the volume of the air parcel m/ρ and the density difference between the air parcel and its surroundings, ρ´ ρ: The liquid water mass is mwl and the corresponding gravitational force gmwl . The equation for conservation of momentum is ρ´ ρ d …mW† ˆ gm dt ρ

wl

(17.35)

where ρ´ is the density of the surrounding air. As the air parcel is rising, the surrounding airmasses exert a decelerating force on the parcel. The deceleration force is proportional to the mass of the displaced air, m´ , and the corresponding decelera­ tion, dW/dt. Pruppacher and Klett (1997) show that this effect is actually equivalent to an acceleration of an “induced” mass m/2 and therefore a term 12 mdW=dt should be added on the RHS of (17.35). Using the ideal-gas law …ρ´ ρ†=ρ ˆ …T T´ †=T ´ ; and the modified (17.35) can be rewritten as T T´ 3 dW W dm ‡ ˆg 2 dt m dt T´

wl

(17.36)

and by employing the definition of the entrainment rate E, we obtain EjW j ˆ

1 dm m dt

FIGURE 17.10 Schematic description of the cloud formation mathematical framework.

(17.37)

722

CLOUD PHYSICS

Therefore the velocity of the air parcel is described by

dW 2 T T ´ ˆ g dt 3 T´

2 eW 2 3

wl

(17.38)

The rate of change of temperature can be calculated using (17.34), noting that dT/dt = WdT/dz, and also that wvs should be replaced by wv to allow the creation of supersaturations. The final result is

dT gW lυ dwv lυ ‡E ‡ wv ˆ cp dt cp dt cp

w´v ‡ …T

T ´† W

(17.39)

The condensed water is related to wv through the water mass balance for the entraining parcel. If airmass dm enters the parcel from the outside, then the water vapor and liquid water mixing ratios will change according to m…wv ‡ wl † ‡ w´v dm ˆ …wv ‡ dwv ‡ wl ‡ dwl †…m ‡ dm† Neglecting products of differentials and dividing by dt, we find that

dwv ˆ dt

dwl dt

EW wv ‡ wl

w´v

(17.40)

Let us assume that the environmental profiles of temperature and water vapor are constant with time at T ´ …z† and w´v …z† and are known. Then because the air parcel is moving with speed W, it will appear to an observer moving with the air parcel that the surrounding conditions are changing according to

dT ´ dT ´ ˆW dt dz

(17.41)

dw´v dw´ ˆW v dt dz

(17.42)

and

For a given environment and given entrainment rate E, we need one more equation to close the system, namely, an equation describing the liquid water mixing ratio wl of the drop population. This liquid water content can be calculated if the droplet size distribution is known, by an integral over the distribution. We thus need to derive differential equations for the droplet diameter rate of change dDp/ dt. These equations will link the cloud dynamics discussed here with the cloud microphysics discussed in the following sections.

723

GROWTH RATE OF INDIVIDUAL CLOUD DROPLETS

17.3 GROWTH RATE OF INDIVIDUAL CLOUD DROPLETS When cloud and fog droplets have diameters significantly larger than 1 μm, mass transfer of water to a droplet can be expressed by the mass transfer equation for the continuum regime (see Chapter 12) dmd ˆ 2πDp Dυ cw;1 dt

eq cw

(17.43)

where md is the droplet mass, Dp its diameter, Dυ the water vapor diffusivity, cw;1 (in mass per volume of eq air) the concentration of water vapor far from the droplet, and cw (in mass per volume of air) the equilibrium water vapor concentration of the droplet. The diffusivity of water vapor in air is given as a function of temperature and pressure Dυ ˆ

0:211 T p 273

1:94

(17.44)

where Dυ is in cm2 s 1, T is in K, and p is in atm. Equation (17.43) neglects noncontinuum effects that may influence very small cloud droplets. These effects can be included in this equation by introducing a modified diffusivity D´υ , where Dυ´ ˆ

Dυ 2Dυ 2πMw 1‡ αc Dp RT

1=2

(17.45)

where αc is the water accommodation coefficient (also called the condensation coefficient). The corrected diffusivity is plotted as a function of droplet diameter in Figure 17.11. The magnitude of the correction depends strongly on the value of the water accommodation coefficient. For a value equal to unity, the correction should be less than 25% for particles larger than 1 μm and less than 5% for droplet diameters larger than 5 μm.

FIGURE 17.11 Water vapor diffusivity corrected for noncontinuum effects and imperfect accommodation as a function of the droplet diameter at T = 283 K and p = 1 atm.

724

CLOUD PHYSICS

The Water Vapor Condensation Coefficient αc Cloud properties are intimately tied to the activation of individual aerosol particles and subsequent growth of the newly formed droplets by accretion of water vapor. This process depends on the rate of transfer of water vapor to droplets and the fraction, αc, of water molecules impinging on the surface of droplets that is incorporated in the droplet. This fraction, which bears the same definition as the accommodation coefficient discussed previously, is called the condensation coefficient for water vapor. The value of αc has been a subject of research for decades; its value is difficult to predict ab initio since it depends on the nature of the water solution, and is thus usually determined as an adjustable parameter to match growth rate measurements of droplets containing a known concen­ tration of solute. The prevailing view is that αc for a pure water surface is close to unity. Actual cloud droplets, however, may contain solutes or surface films that affect growth kinetics even at low concentrations because slow solute dissolution or glassy or highly viscous aerosol phases may suppress droplet growth rates. Droplet formation can be very sensitive to variations in αc, owing to the dependence of cloud droplet number concentration (Nd) on the maximum supersaturation that develops in clouds. The latter is controlled by a balance between supersaturation generation and depletion from condensation of water vapor on existing droplets. A smaller value of αc results in slower water vapor condensation, allowing supersaturation to develop more fully and increasing Nd before reaching its maximum. The implication is that the extent to which αc varies over space and time (especially if αc < 0.1) is critical in understanding its contribution to Nd variability, hence, global cloud properties and climate. Measurements of the droplet size distribution resulting from exposure of aerosol particles to a given water vapor supersaturation, that is, cloud condensation nuclei (CCN) data, can provide the fundamental information on which values of αc can be inferred. Raatikainen et al. (2013) analyzed eight global datasets, including urban outflows, boreal forests, Arctic air masses, fresh and aged biomass burning plumes, and continental air under anthropogenic and biogenic influence. They also included two previous studies on the activation kinetics of aged marine air in the eastern Mediter­ ranean and near a strong hydrocarbon source at the April 20, 2010 Deepwater Horizon oil spill in the Gulf of Mexico. Because activated droplet size is a function of aerosol size distribution, hygroscopicity, CCN instrument operating conditions (accurately known), and αc (not known), one can simulate the activation and growth of CCN in the CCN counter using a prescribed (constant) αc. If the difference between observed and predicted droplet size is essentially constant for all of the data, then a constant αc is assumed valid. Indeed, this was found to be the case for the globally relevant datasets. Prescribing αc = 0.2 (which is close to perfect accommodation) captures the temporal variability of measured droplet size to within a constant bias and a 0.3 μm variance, which characterizes normal instrument variability and the essentially unresolvable range of αc between 0.1 and 1.0. With this result and the fact that the data suggest that the majority of particles activate as rapidly as (NH4)2SO4, one concludes that αc > 0.1 is a reasonable globally representative constraint. Importantly, αc is effectively constant for all the data analyzed by Raatikainen et al. (2013), even for particles composed largely of organics with very low oxygen content. Limiting αc to the 0.1–1.0 range considerably reduces the uncertainty in cloud droplet number prediction in climate models. The equilibrium vapor concentration in (17.43) corresponds to the concentration at the droplet surface and has been derived in Section 17.1 as a function of temperature. However, during water condensation, heat is released at the droplet surface, and the droplet temperature is expected to be higher than the ambient temperature. Let us calculate the droplet temperature by deriving an appropriate energy balance. If Ta is the temperature at the drop surface and T1 the temperature of the environment, an energy balance gives [see also (12.21)] 2πDp k´a …T 1

T a † ˆ lυ

dmd dt

(17.46)

725

GROWTH RATE OF INDIVIDUAL CLOUD DROPLETS

where k´a is the effective thermal conductivity of air corrected for noncontinuum effects. Equation (17.46) indicates that at steady state the heat released during water condensation is equal to the heat conducted to the droplet surroundings. The temperature at the droplet surface is then Ta ˆ T1 ‡

dDp l υ ρw ˆ T1 …1 ‡ δ† Dp dt 4k´a

(17.47)

where we have used the mass balance dDp dmd 1 ˆ πρw D2p dt 2 dt

(17.48)

dDp lυ ρw Dp ´ dt 4ka T 1

(17.49)

and we have defined δˆ

For atmospheric cloud droplet growth δ  1; however, let us continue the derivation without assuming that Ta ˆ T1 : Combining (17.43) and (17.48) and using the ideal-gas law, we find that Dp

dDp 4D´υ Mw es°…T1 † ˆ Sυ;1 dt ρw RT

es D p ; T a es°…T1 †

(17.50)

where Sυ;1 ˆ es°=es°…T 1 † is the environmental saturation ratio. Recall that for a relative humidity equal to 100% the partial pressure of water in the atmosphere, e, is equal to the saturation vapor pressure es°…T 1 † and Sυ;1 ˆ 1. The ratio of the water saturation pressures at T a and T 1 is given by the Clausius–Clapeyron equation as lυ Mw T a T1 es°…Ta † ˆ exp R Ta T1 es°…T 1 †

(17.51)

Combining (17.47), (17.50), (17.30), and (17.51) we finally get Dp

dDp 4D´υ Mw es …T 1 † ˆ dt ρw RT 1

 Sυ;1

1 lυ Mw δ 4Mw σw ‡ exp 1‡δ RT 1 1 ‡ δ RTρw Dp …1 ‡ δ†

(17.52)

6ns Mw πρw D3p

d3u

This result can be simplified since δ  1 and exp lυ Mw δ=RT 1 …1 ‡ δ† ≅ 1 ‡ lυ Mw δ=RT 1 After some algebra the implicit dependence on δ can be resolved to obtain

Sυ;1 dDp Dp ˆ dt

exp

4Mw σw RT 1 ρw Dp

6ns Mw πρw D3p

lυ ρw lυ Mw ρw RT 1 ‡ 4es°…T 1 †D´υ Mw 4k´a T1 T1 R

d3u 1

(17.53)

726

CLOUD PHYSICS

This equation describes the growth/evaporation rate of an atmospheric droplet. The numerator is the driving force for the mass transfer of water, namely, the difference between the ambient saturation Sυ;1 and the equilibrium saturation for the droplet (or equivalently the water vapor saturation at the droplet surface). The equilibrium saturation includes, as we saw in Section 17.1.4, the contributions of the Kelvin effect (first term in the exponential) and the solute effect (second term in the exponential). When the ambient saturation exceeds the equilibrium saturation, the cloud droplets grow and vice versa. The eq numerator is qualitatively equivalent to the term cw;1 cw in (17.43). The first term in the denominator corresponds to the diffusivity of water vapor [compare with (17.43)], while the second accounts for the temperature difference between the droplet and its surroundings. Note that if no heat were released during condensation, lυ ˆ 0, and this term is zero. The thermal conductivity of air ka is given by ka ˆ 10 3 …4:39 ‡ 0:071T†

(17.54)

where ka is in J m 1 s 1 K 1 and T is in K. The modified form for the thermal conductivity k´a accounting for noncontinuum effects is given by k´a ˆ ka

1‡

2ka 2πMa αT Dp ρcp RT a

1=2

(17.55)

where αT is the thermal accommodation coefficient. The value of αT is also uncertain and it is often set equal to the value of the mass accommodation (condensation) coefficient αc . For a cloud droplet larger than ∼10 μm, using αc ˆ αT ˆ 1; at 283 K, we obtain ρw RT 1 ˆ 4:85  105 s cm es°…T 1 †D´υ Mw l υ ρw l υ M w ka´ T1 T 1 R

1

ˆ 6:4  105 s cm

2

2

and defining Sυ;eq as the equilibrium saturation of the droplet, (17.53) can be rewritten as Dp

dDp ˆ 3:5  10 dt

6

Sυ;1

Sυ;eq

where Dp is in cm and t in s. The growth of an aerosol size distribution under constant supersaturation of 1% is shown in Figure 17.12. The rate of growth of droplets is inversely proportional to their diameters, so smaller droplets grow faster than larger ones. As a result, small droplets catch up in size with larger ones during the growth stage of the cloud.

17.4 GROWTH OF A DROPLET POPULATION The growth of an aerosol population to cloud droplets can be investigated using the growth equation derived in the previous section. In general, one would need to integrate simultaneously the differential equations derived in Section 17.2.4 for the air parcel updraft velocity, temperature, water vapor mixing ratio, and environmental temperature and water vapor mixing ratio, coupled with a set of droplet growth equations, one for each droplet size class. The liquid water mixing ratio of the population consisting of Ni droplets per volume of air of diameters Dpi will then be wl ˆ

ρw π ρa 6

n 3 N i Dpi iˆ1

(17.56)

727

GROWTH OF A DROPLET POPULATION

FIGURE 17.12 Diffusional growth of individual drops with different dry masses as a function of time. The drops are initially at equilibrium at 80% RH.

where we have assumed that there are n groups of droplets. It is instructive, before examining the interactions between cloud dynamics and microphysics, to focus our attention on the microphysics. Figure 17.13 presents the results of the integration of these equations for a polluted urban aerosol population (Pandis et al. 1990a) for an aerosol distribution consisting of seven size sections. At time zero the relative humidity is assumed to be 100%. At this time the particles have grown several times from their dry size as a result of water absorption and are assumed to be in equilibrium with the surrounding environment. The temperature of the air parcel is assumed to decrease at a constant rate of 2 K h 1. As the temperature decreases, the saturation of the air parcel increases. The particles absorb water vapor, but the cooling rate is too rapid in comparison to mass transfer, and the air parcel becomes supersaturated. After a few minutes the particles start to activate. The larger particles become activated first (Section 17.1), and the smaller ones soon follow. As particles become activated, they are able to grow much faster. Note that based on the Köhler curves as a particle eq eq grows the driving force for growth cw;1 cw becomes larger for almost constant cw;1 ; because cw decreases rapidly with size. Therefore, as more and more particles activate, the rate of transport of water from the vapor to the particle phase increases, while the rate of supersaturation increase due to the cooling remains approximately constant. The result is that the supersaturation increase slows down and after 6 min reaches a maximum value of 0.1%. Let us describe this situation quantitatively, by deriving the equation for the rate of change of the supersaturation sυ : The water vapor mixing ratio wv is related to the water vapor partial pressure e, while by definition 1 ‡ sυ ˆ

e es°

One gets sυ ˆ

M a pa wv Mw es°

1

(17.57)

728

CLOUD PHYSICS

FIGURE 17.13 Simulated evolution of temperature, liquid water content, supersaturation, and particle diameters during the lifetime of a cloud (Pandis et al. 1990a,b). The sections denote seven sizes of initial particles.

Differentiating this expression with respect to time and rearranging the terms, we obtain dsυ Ma pa dwv ˆ dt Mw es° dt

…1 ‡ sυ †

1 des° es° dt

1 dpa pa dt

(17.58)

The change of the air pressure with time can be calculated assuming that the environment is in hydrostatic equilibrium so that dpa ˆ dt

gpa Ma W RT

(17.59)

729

GROWTH OF A DROPLET POPULATION

where we have assumed that T ´ ≅ T. The change of the parcel saturation pressure with time can be calculated using the chain rule and the Clausius–Clapeyron equation: des° des° dT lυ Mw es° dT ˆ ˆ dt dT dt RT 2 dt

(17.60)

Substituting (17.59) and (17.60) into (17.58), we get dsυ Ma pa dwv ˆ dt Mw es° dt

…1 ‡ S υ †

lυ Mw dT gMa ‡ W RT RT 2 dt

(17.61)

If we assume that there is no entrainment (E = 0) and substitute (17.39) and (17.40) into (17.61), we obtain dsυ ˆ dt

l υ Mw g cp RT 2

pa Ma l2υ Mw dwl ‡ es°Mw cp RT 2 dt

gMa W RT

(17.62)

where we have assumed that 1 ‡ sυ ≅ 1 as sυ ≅ 0:01 in cloud. Equation (17.62) reveals that in the absence of condensation, the saturation varies linearly with the updraft velocity and is decreased by water condensation. One could replace in (17.62) the updraft velocity with the cooling rate W≅

cp g

dT dt

(17.63)

where we have assumed that the updraft velocity is almost constant, and therefore the air parcel cooling rate is also constant. Equation (17.62) is the mathematical embodiment of our previous qualitative theoretical arguments. It suggests that the supersaturation inside the cloud is the result of a balance between the cooling rate and the liquid water increase. The latter is limited by the mass transport to particles, which, in turn, depends on the particle size distribution and on their state of activation. The maximum supersaturation reached inside a cloud/fog is an important parameter. Particles with critical supersaturations lower than this value will become activated and become cloud droplets. The rest remain close to equilibrium but never grow enough to be considered droplets and are called interstitial aerosol. The aerosol population inside a cloud is therefore separated into two groups: (1) interstitial aerosols that contain significant amounts of water but are not activated (their sizes are usually 0.3) does the relative sensitivity of Nd to changes in κ exceed ∼0.2. This means that in these two regimes at least a 50% change in κ would be required to change Nd by more than 10%. These findings are consistent with other studies investigating the influence of aerosol chemical composition on CCN activation in cloud parcel models (e.g., Rissman et al. 2004). Similar regimes of CCN activation were found in simulations with varying types of size distributions (polluted and pristine continental and marine aerosols with different proportions of nucleation, accumulation, and coarse mode particles). In summary, variability in the initial cloud droplet number concentration in convective clouds is dominated mostly by the variability of updraft velocity and aerosol number concentration in the accumulation mode. Coarse-mode particles and the variability of particle hygroscopicity are estimated to play significant roles only at low supersaturation in the updraft-limited regime of CCN activation.

17.6 CLOUD PROCESSING OF AEROSOLS In a nonraining cloud the aerosol size–composition distribution in a rising parcel of air is transformed by a variety of processes. First, based on the supersaturation in the cloud, a fraction of the aerosol distribution is activated to become cloud droplets while the nonactivated fraction remains as interstitial particles. This process, described as nucleation scavenging of aerosols, determines the initial composition of the cloud droplets. If this is the only process taking place in the cloud, on cloud evaporation the aerosol distribution would return to its original form. However, additional processes occur in the cloud, including collisions between interstitial aerosols and cloud droplets and coalescence among cloud droplets. If the cloud is producing rain, interactions occur between the raindrops and the aerosols both in and below the cloud, leading to removal of aerosols from the atmosphere.

17.6.1 Nucleation Scavenging of Aerosols by Clouds Nucleation scavenging of aerosols in clouds refers to activation and subsequent growth of a fraction of the aerosol population to cloud droplets. This process is described by (17.53) and has been discussed in Section 17.4. If Ci;0 is the concentration (in mass per volume of air) of an aerosol species in clear air before cloud formation (e.g., at the cloud base), and Ci;cloud and Ci;int are its concentrations again in mass per volume of air in the aqueous phase and in the interstitial aerosol, respectively, one can define the cloud mass scavenging ratio for species i (Fi) as Fi ˆ

Ci;0

Ci;int Ci;0

(17.75)

737

CLOUD PROCESSING OF AEROSOLS

Note that if there is no production or removal of i in the cloud, then Ci;0 ˆ Ci;int ‡ Ci;cloud : The mass scavenging ratio defined above may vary from zero to unity. The number scavenging ratio FN can be defined as FN ˆ

N0

N int N0

(17.76)

where N0 is the aerosol number concentration before cloud formation and Nint is the number concentration of interstitial aerosol. Theoretically, as particles larger than ∼0.5 μm become cloud droplets in a typical cloud, and these particles represent most of the aerosol mass, one would expect mass activation efficiencies close to unity. Junge (1963) predicted sulfate scavenging ratios from nucleation scavenging alone to range from 0.5 to 1.0. Theoretical studies that followed have predicted high-mass nucleation scavenging efficiencies for all aerosol species. For example, Flossmann et al. (1985, 1987) calculated aerosol scavenging efficiencies exceeding 0.9 in typical cloud environments. Pandis et al. (1990a) estimated scavenging efficiencies of 0.7 for sulfate and 0.8 for nitrate and ammonia in polluted clouds. In other numerical studies, Flossmann (1991) reported mass scavenging efficiencies of 0.9 or higher for warm clouds over the Atlantic. These theoretical estimates are in good agreement with measured mass scavenging efficiencies in the atmosphere. Ten Brink et al. (1987) observed nearly complete scavenging of aerosol sulfate in clouds. The data of Daum et al. (1984) also showed that the bulk of the sulfate mass is incorporated into cloud droplets. Hegg and Hobbs (1988) reported scavenging ratios for sulfate of 0.5 ± 0.2. On the contrary, low number scavenging efficiencies are expected in clouds influenced by anthropogenic sources because of the prevalence of fine aerosol particles; number scavenging efficien­ cies of a few percent or less are expected in most such situations. Only in clouds in the remote marine atmosphere does the total number scavenging efficiency exceed 0.1.

17.6.2 Chemical Composition of Cloud Droplets During the droplet growth stage of a cloud, droplets of different sizes dilute at different rates; in particular, smaller droplets grow faster and therefore dilute faster than the larger ones. This can be shown by the following argument (Noone et al. 1988). Assume that the aerosol population consists of i particle groups of dry diameters Ds;i : These particles grow and become aqueous droplets of diameters Di : Assuming for simplicity that the density of the dry particles is 1 g cm 3, the solute mass fraction of 3

droplets in section i is Xi ˆ Ds;i =Di : Defining the dilution rate DRi, of section i as the normalized rate of change of the solute mass fraction in the droplet, we obtain DRi ˆ

1 dXi Xi dt

(17.77)

and assuming that the mass of scavenged aerosol in droplets of group i remains constant with time (i.e., neglecting processes like scavenging of gases, coagulation), then, for two acrosol groups with D1 < D2, we obtain DR1 D2 dD1 =dt ˆ DR2 D1 dD2 =dt

(17.78)

For sufficiently large droplets, the growth rate is approximately proportional to the inverse of the droplet diameter [see (17.53)] and K dDi ≅ dt Di

(17.79)

738

CLOUD PHYSICS

where K is a constant that is only a weak function of Di. Then, combining (17.78) and (17.79), one finds that DR1 D2 ≅ D1 DR2

2

>1

(17.80)

and the smaller droplets, D1, are diluted at a faster rate than the larger D2 ones if growth by water diffusion is the dominant process occurring. The above argument suggests that, with time, the total solute concentrations of droplets for an aerosol population would tend to increase with particle size. These arguments are supported by calculations like those by Pandis et al. (1990a) shown in Figure 17.17. Note that initially, before cloud creation, solute concentration is high across the size spectrum and larger particles have slightly lower concentrations as they contain hydrophilic components such as NaCl. As aerosols become activated, their solute concentrations decrease. For a mature cloud, solute concentration shows a minimum at around 10 μm. This minimum is a result of existence of nonactivated particles for which solute concentration decreases with increasing size, and droplets for which the solute concentration increases with increasing size. Note that mature cloud droplets of diameter ∼10 μm have a solute concentration of roughly 100 mg L 1, whereas drops of diameter 24 μm have a solute concentration that is 3.4 times higher. During the evaporation stage of a cloud, smaller droplets get deactivated and evaporate first (Figure 17.17b). Therefore the minimum gradually disappears and the system returns close to its original state.

FIGURE 17.17 Predicted dependence of the total solute concentration on droplet diameter during the lifetime of a cloud (Pandis et al. 1990a).

CLOUD PROCESSING OF AEROSOLS

FIGURE 17.18 Measured composition of the small and large cloud droplets collected in coastal stratus clouds at La Jolla Peak, California, in July 1993 (Collett et al. 1994).

The discussion above concerns total solute concentration. For individual species, because their aerosol concentrations are also generally size-dependent, there is an additional reason for size-depen­ dent droplet concentrations. Concentrations of some major aerosol species measured in small and large droplets in a cloud are shown in Figure 17.18. The droplet population in these measurements was separated into only two samples with significant overlap and therefore the concentration deviations shown probably underestimate the actual differences. The differences in solute concentrations are accompanied by differences in acidity among droplets of different sizes. Figure 17.19 summarizes measurements of Collett et al. (1994) in a variety of environ­ ments. Significant pH differences are observed. Once more, because of significant mixing among droplets of different sizes during sampling, the actual differences are probably even greater than those depicted in Figure 17.19.

17.6.3 Nonraining Cloud Effects on Aerosol Concentrations Figure 17.20 depicts the physical processes that occur involving wet aerosol particles and activated cloud droplets. Cloud droplet chemistry plays an important role in the atmosphere. Significant production of sulfate has been detected and/or predicted in clouds and fogs in different environments (Hegg and Hobbs 1987, 1988; Pandis and Seinfeld 1989; Husain et al. 1991; Pandis et al. 1992; Develk 1994; Liu et al. 1994). Detection of sulfate-producing reactions is often hindered by variability of cloud liquid water content and temporal instability and spatial variability in concentrations of reagents and product species (Kelly et al. 1989). During cloud formation, aerosols that serve as cloud condensation nuclei (CCN) become activated and grow freely by vapor diffusion. Soluble gases such as nitric acid, ammonia, and sulfur dioxide dissolve into the droplets. The cloudwater serves as the reacting medium for aqueous-phase reactions, importantly the transformation of dissolved SO2, S(IV), to sulfate, S(VI). The sulfate formed is not volatile and remains in the particle phase. Other reactions–for example, the oxidation of formaldehyde to formic acid–result in volatile products that return to the gas phase. During the cloud evaporation stage, several species that were dissolved in the cloudwater evaporate. Others, such as sulfate, remain in the aerosol phase. Ammonia often accompanies the sulfate formed as the neutralizing cation. Species such as nitrate or chloride that may have existed in the original particle can be displaced by the sulfate produced and forced to return to the gas phase. The result of these aqueous-phase processes is usually an overall increase in particle mass and size. Chemical composition of the particles may also change, with sulfate

739

740

CLOUD PHYSICS

FIGURE 17.19 Measured pH of small and large droplets in a series of clouds and fogs in typical environments (Collett et al. 1994).

and ammonium concentrations generally increasing and nitrate and chloride decreasing (Pandis et al. 1990b). Cloud processing is a major source of sulfate and aerosol mass in general on regional and global scales. Walcek et al. (1990) calculated that, during passage of a midlatitude storm system, over 65% of tropospheric sulfate over the northeastern United States was formed in cloud droplets via aqueousphase reactions. The same authors estimated that, during a 3-day springtime period, chemical reactions in clouds occupying 1–2% of the tropospheric volume were responsible for sulfate production compa­ rable to the gas-phase reactions throughout the entire tropospheric volume under consideration. McHenry and Dennis (1994) proposed that annually more than 60% of the ambient sulfate in central and eastern United States is produced in mostly nonprecipitating clouds. Similar conclusions were reached by Dennis et al. (1993) and Karamachandani and Venkatram (1992). Aqueous-phase SO2 oxidation in clouds is predicted to be the most important pathway for the conversion of SO2 to sulfate on a global scale (Hegg 1985; Liao et al., 2003). The effect of cloud processing of aerosols in the remote marine atmosphere has been demonstrated in a series of field studies (Hoppel et al. 1986; Frick and Hoppel 1993). Hoppel et al. (1986) proposed that cloud processing of aerosol is an efficient mechanism for accumulating mass in the 0.08–0.5 μm size range in the marine atmosphere. The gap observed in most remote marine aerosol number distributions (Figure 17.21) is, as a result, often referred to as the “Hoppel gap.” These effects on the aerosol number distribution are expected to be important only in the remote atmosphere, where the number of CCN is a significant fraction of the total aerosol number. However, significant effects on the aerosol mass distribution are expected under all circumstances. Simulations of the aerosol–cloud–aerosol cycle have shown that sulfate formed during cloud/fog processing of an airmass favors aerosol particles that have access to most of the cloud liquid water content, which are those with diameters in the 0.5–1.0 μm range (Pandis et al. 1990a). Figure 17.22 depicts

CLOUD PROCESSING OF AEROSOLS

FIGURE 17.20 Cloud processing of an aerosol particle. The y axis (ordinate) represents the mass of solute; as the solute mass increases, the size of the growing wet aerosol particle increases. The x axis (abscissa) is the spectrum of droplet size. Wet aerosol particles activate according to the ambient supersaturation in the cloud. Once activated, the droplet continues to grow in the cloud owing to condensation of water vapor. Activated droplets in the cloud may collide and coalesce, so-called collision–coalescence, to produce larger droplets. It is this process that leads to precipitation-size drops. In the absence of in-cloud chemistry that alters the amount of dissolved solute, if a droplet evaporates, it returns to its original unactivated state.

FIGURE 17.21 Typical remote marine aerosol size distributions. (Reprinted from Atmos. Environ. 24A, Hoppel, W. A., and Frick, G. M., 645–649. Copyright 1990, with kind permission from Elsevier Science, Ltd., The Boulevard, Langford Lane, Kidlington OX5 1GB, UK.)

741

742

CLOUD PHYSICS

FIGURE 17.22 Predicted aerosol size–composition distributions before and after a fog episode (Pandis et al. 1990b).

simulation of fog processing of an urban aerosol population. Note that the shape of the aerosol distribution changes with the creation of an extra mode, resulting mainly from the formation of (NH4)2SO4. Note also that there are significant changes in the aerosol chemical composition before and after the fog with sulfates replacing nitrate and chloride salts.

17.6.4 Interstitial Aerosol Scavenging by Cloud Droplets Interstitial aerosol particles collide with cloud droplets and are removed from cloud interstitial air. The coagulation theory of Chapter 13 can be used to quantify the rate and effects of such removal. If n(Dp,t) is the aerosol number distribution and nd(Dp,t) the droplet number distribution at time t, the loss rate of aerosol particles per unit volume of air due to scavenging by cloud drops is governed by 1 @n Dp ; t K Dp ; x nd …x; t†dx ˆ n Dp ; t ∫0 @t

(17.81)

where K(Dp,x) is the collection coefficient for collisions between an interstitial aerosol particle of diameter Dp and a droplet of diameter x. One can define the scavenging coefficient Λ(Dp, t) for the full droplet population: Λ Dp ; t ˆ

1 @n Dp ; t 1 ˆ K Dp ; x nd …x; t†dx ∫0 @t n Dp ; t

(17.82)

If the scavenging coefficient does not vary with time and is equal to Λ(Dp), then the evolution of the number distribution is given by n Dp ; t ˆ n Dp ; 0 exp

Λ Dp t

(17.83)

Assuming for the time being that cloud droplets are stationary, then particles are captured by Brownian diffusion. The collection of particles by a falling drop will be discussed when we consider wet deposition in Chapter 20. The collection coefficient K Dp ; x can then be estimated by (13.54). Let us

743

OTHER FORMS OF WATER IN THE ATMOSPHERE

TABLE 17.4 Estimated Lifetimes of Aerosols in a Nonraining Cloud (N = 955 cm − 3, Dp = 10 μm, wL = 0.5 g cm − 3) at Standard Conditions Dp …μm† 0.002 0.01 0.05 0.1 1.0 10.0

K(Dp, 10 μm) (cm

3

s 1)

4 × 10 5 1.6 × 10 6 1.1 × 10 7 2.2 × 10 8 1.03 × 10 9 3 × 10 10

Λ Dp …s 1 † 0.038 1.5 × 10 1 × 10 4 2.1 × 10 9.8 × 10 2.9 × 10

3

5 7 7

Lifetime (=1/Λ) 0.4 min 11 min 2.8 h 13 h 11.8 days 40 days

estimate this collection rate assuming that the cloud has a liquid water content of 0.5 g m 3 and that all drops have diameter of 10 μm, resulting in a number concentration of N = 955 cm 3. For such a monodisperse droplet population (17.82) simplifies to Λ(Dp) = Nd K(Dp, 10 μm), and K(Dp,10 μm) can be calculated from (13.54) (see also Figure 13.5 and Table 13.3). Corresponding particle collision efficiencies and lifetimes are shown in Table 17.4. Nuclei smaller than 10 nm will be scavenged in a few minutes in such a cloud whereas particles larger than 0.1 μm will not be collected by droplets during the cloud lifetime. The above results indicate that for average residence times of air parcels in clouds (on the order of an hour) only the very fine aerosol particles will be collected by cloud droplets. These particles often represent a significant fraction of the aerosol number, so the total number may be reduced significantly, and the shape of the aerosol number distribution may change dramatically. However, these particles contain little of the mass, and therefore the mass distribution effectively does not change. These conclusions are in agreement with detailed simulations of aerosol processing by clouds (Flossmann and Pruppacher 1988; Flossmann 1991).

17.7 OTHER FORMS OF WATER IN THE ATMOSPHERE Up to this point we have focused on “warm” nonraining tropospheric clouds. Water in the atmosphere also exists as ice, rain, snow, and so on. We summarize here aspects of the formation and removal of these water forms that are most associated with atmospheric chemistry. The interested reader is referred for more in-depth treatments to Pruppacher and Klett (1997) and Lamb and Verlinde (2011).

17.7.1 Ice Clouds Atmospheric observations indicate that water readily supercools, and water clouds are frequently found in the atmosphere at temperatures below 0 °C. Figure 17.23 shows that supercooled clouds are quite common in the atmosphere, especially if cloudtop temperature is warmer than 10 °C. However, the likelihood of ice increases with decreasing temperature, and at 20 °C only about 10% of clouds consist entirely of water drops. At these low temperatures, ice particles coexist with water drops in the same cloud. The temperature dependence of the equilibrium between water vapor and ice can be described by the Clausius–Clapeyron equation des;i ls ˆ dT T…υυ υi †

(17.84)

744

CLOUD PHYSICS

FIGURE 17.23 Approximate average frequencies of appearance of supercooled water, mixed phase, and ice clouds as a function of temperature in layer clouds.

where es;i is the saturation vapor pressure of water over ice, ls is the molar enthalpy for ice sublimation, and υi and υυ are the molar volumes of ice and water vapor. If we assume that υυ  υi and that water vapor behaves as an ideal gas, then ls d ln es;i ˆ dT RT 2

(17.85)

Note that (17.85) is similar to the Clausius–Clapeyron equation (16.49) for water vapor–water equili­ brium, with the enthalpy of sublimation replacing the enthalpy of evaporation. Finally, if the Clausius–Clapeyron equation is applied to the equilibrium between ice and water, we get des;m lm ˆ dT T…υw υi †

(17.86)

where es;m is the melting pressure of ice, lm the molar enthalpy of melting, and υw the molar volume of water. Equations (10.66), (17.85), and (17.86) can be integrated and plotted to produce the p T phase diagram for pure water shown in Figure 17.24. Note that there is only one point (T = 0 °C, p = 6.1 hPa), the water triple point at which all phases coexist. Another interesting observation is that for temperatures below 0 °C the water vapor pressure over liquid water is higher than the water vapor pressure over ice: es > es;i …T < 0 °C† So, air that is saturated with respect to ice is subsaturated with respect to water. As a result, supercooled water droplets cannot coexist in equilibrium with ice crystals. Figure 17.24 refers to the equilibrium of bulk water (curvature is neglected), without any impurities (zero solute concentration). Curvature and solute effects cause the behavior of water in the atmosphere to deviate significantly from Figure 17.24. We discuss these effects briefly below. 17.7.1.1 Freezing-Point Depression Dissolution of a salt in water lowers the vapor pressure over the solution. A direct result of this is the depression of the freezing point of water. In order to quantify this change, assume that our system has constant pressure and contains air and an aqueous salt solution in equilibrium with ice. According to

745

OTHER FORMS OF WATER IN THE ATMOSPHERE

FIGURE 17.24 Pressure–temperature phase diagram for water. The dashed line corre­ sponds to supercooled water and its metastable equilibrium with water vapor.

thermodynamic equilibrium, the chemical potential of water in the aqueous and ice phases will be the same, μi ˆ μw . If aw is the activity of water in solution, then μi ˆ μw° …T† ‡ RT ln aw Dividing by T and differentiating with respect to T under constant pressure p, we have @ μi =T @T

ˆ p

@ μw°=T @T

‡R p

@ ln aw @T

(17.87) p

But the chemical potential is related to the enthalpy by @ μk =T @T

ˆ p

Hk T2

(17.88)

and therefore @ ln aw @T

ˆ p

Hw Hi lm ˆ 2 RT RT 2

(17.89)

This equation describes the equilibrium between a salt solution and pure ice. This can be integrated as follows, noting that for pure water aw = 1, and denoting the pure water freezing temperature by T0, yields aw

∫ aw ˆ1

d ln aw ˆ

Te

lm dT ∫ T0 RT 2

(17.90)

assuming that lm is approximately constant in the temperature interval from T0 to Te T0

Te ˆ

RT 0 T e … ln aw † lm

(17.91)

746

CLOUD PHYSICS

because aw  1; T 0 T e  0 and the new freezing temperature Te is lower than the pure water freezing temperature. The difference ΔT f ˆ T 0 Te is the equilibrium freezing point depression. To get an estimate of this depression, we can assume that the solution is ideal, so that aw = xw, and that T 0 T e ≅ T20 : Noting that ln aw ˆ ln xw ≅ ns =nw ; we obtain the estimate ΔTf ˆ

RT 20 Mw m 1000 lm

(17.92)

where m is the solution molarity. For ideal solutions this results in a depression of 1.86 °C M 1 of solute, and for a salt that dissociates into two ions this results in a depression of 3.72 °C M 1 of the salt. As a result of the nonideality of real salts, actual depression at 1 M concentration is 3.35 °C for NaCl. 17.7.1.2 Curvature Effects Ice crystals in the atmosphere have a variety of shapes, with hexagonal prismatic being the basic one (Pruppacher and Klett 1997). For illustration purposes, let us ignore this complexity and concentrate on the behavior of a spherical ice particle of diameter Dp. Our analysis for water droplets and the Kelvin effect is directly applicable here and the vapor pressure of water over the ice particle surface is es ˆ es;i exp

4Mw σia RTρi Dp

(17.93)

where es;i is the vapor pressure over a flat ice surface, σia is the surface energy of ice in air, and ρi is the ice density. Pruppacher and Klett (1997) review theoretical and experimental values for σia and suggest that it varies from 0.10 to 0.11 N m 1. This surface tension that is higher than the water/air value results in relative vapor increases higher than those for a water droplet of the same size. Pruppacher and Klett (1997) show that the freezing temperature decreases with decreasing size of the ice particle. This decrease becomes particularly pronounced for crystal diameters smaller than 20 nm and is further enhanced by the solute effect for solute concentrations larger than 0.1 M. 17.7.1.3 Ice Nucleating Particles (INP) Ice nucleation generally requires a preexisting particle. Ice forms via two pathways, homogeneous and heterogeneous freezing. Homogeneous freezing refers to the spontaneous nucleation of ice within a sufficiently cooled liquid droplet. Heterogeneous freezing can involve the freezing of water vapor on impact on a particle surface. Specifically, four heterogeneous ice nucleation modes have been distin­ guished: immersion freezing (initiated from within a liquid cloud droplet), condensation freezing (freezing during droplet formation), contact freezing (freezing due to collision with an INP), and deposition nucleation, which refers to the direct deposition of vapor onto INP. Observations reveal that liquid cloud droplets are generally present before ice crystals form via heterogeneous freezing mecha­ nisms, ruling out deposition nucleation as important for mixed-phase clouds. Laboratory experiments confirm that mineral dust, metallic particles, some biological particles, and anhydrous salts can serve as ice nuclei. Owing to their altitude of formation, cirrus clouds are composed entirely of ice crystals. Cziczo et al. (2013), based on high-altitude aircraft measurements in cirrus clouds, found that the predominant particle category on which freezing took place was mineral dust and metallic particles. Sulfate and organic particles can act as ice nuclei when present as glasses or anhydrous salts. INP can either be bare or mixed with other substances. As bare particles age in the atmosphere, they acquire liquid surface coatings by condensing soluble species and water vapor or by coagulating with soluble particles. A change from contact to immersion freezing implies activation at colder temperatures. Atmospheric concentrations of INP are difficult to determine because of the variation

747

OTHER FORMS OF WATER IN THE ATMOSPHERE

in freezing mechanisms and the challenge of measuring INP in the upper troposphere. The anthropo­ genic fraction is also somewhat uncertain because of a lack of knowledge about the anthropogenic contributions to BC, mineral dust, and other aerosols acting as INP. INP concentrations in the atmosphere are quite variable. A proposed empirical relation for their concentration as a function of temperature is (Pruppacher and Klett 1997) INP L

1

ˆ exp‰0:6…253

T†Š

(17.94)

This relation suggests that ice nucleation in the atmosphere is a very selective process. For example, at temperatures as low as 20 °C the atmosphere typically contains 1 INP L 1 and 106 particles L 1. So, at most, one out of a million particles can serve as an INP. Measurements of ice particle concentrations for cloudtop temperatures below 10 °C have given concentrations varying from 0.1 to 200 L 1. For temperatures below 20 °C, the concentrations vary from 10 to 300 L 1. Mineral dust and some biological particles show the highest degree of ice nucleation activity. Biomass burning particles can act as INP. Marine organic material may be an important INP source in remote ocean environments. See DeMott (2009), Richardson et al. (2010), and DeMott et al. (2009, 2010, 2011).

17.7.2 Rain For a cloud to generate precipitation, some drops need to grow to precipitable size, around 1 mm. Growth of droplets proceeds via several mechanisms: (1) water vapor condensation, (2) droplet coalescence, and (3) ice processes. While the growth of droplets by water vapor condensation is a very effective mechanism for the rapid initial growth of aerosol particles from a fraction of a micrometer to 10 μm, growth by condensation alone cannot account for droplets sufficiently large to initiate rain. Figure 17.12 demon­ strates, for example, that ∼1 h is necessary under a constant supersaturation of 1% for droplet growth to 100 μm. After an entire hour, the drop has collected only 0.1% of the water mass of an average raindrop. Collision–coalescence is the mechanism required to generate precipitation-size droplets. Even so, collision–coalescence occurs too slowly for monodisperse-size droplets to form rain on the necessary timescales. The primary mechanism of collision–coalescence for liquid clouds is that in which large drops collide with and collect smaller droplets that lie in their fall path. Therefore the largest drops in the cloud drop spectrum are of particular importance because they are the ones that initiate precipitation. The typical concentration of large drops required to initiate precipitation is one per liter of air, or only one out of ∼106 cloud drops. These large drops form on giant particles, which are again about one in ∼106 nuclei. The dramatic dependence of precipitation formation on one out of ∼106 droplets is indicative of the difficulty of quantitatively describing the overall process. A broad cloud drop size distribution is more conducive to precipitation development than is a narrow distribution since the rate of collection by falling drops is low if the drops are all about the same size. Remote marine clouds tend to have broad droplet size distributions and generally produce precipitation more effectively than similar continental clouds. Larger drops fall faster than smaller drops, resulting in the larger drops overtaking the smaller drops, colliding and coalescing with them. This process is growth by accretion, sometimes also called gravitational coagulation, coalescence, or collisional growth. The collision process is illustrated in Figure 17.25 for viscous flow around a sphere of diameter Dp. As the large droplet approaches small drops of diameter dp, the viscous forces exerted by the flow field around the large droplet push the droplets away from the center of flow, modifying their trajectories. In Figure 17.25 droplet b is collected by the raindrop and droplet a lies on the trajectory where it just grazes the drop. Owing to fluid mechanical forces the falling raindrop will in general collect fewer drops than those existing in the cylinder of diameter Dp below it. Droplets in the cylinder of diameter y will be collected. The distance y is

748

CLOUD PHYSICS

FIGURE 17.25 Schematic of the flow around a falling drop. The dashed lines are the tra­ jectories of small drops considered as mass points. Trajectory a is a grazing trajectory, while b is a collision trajectory.

defined by the grazing trajectory a and is a function of the raindrop size Dp and drop size dp. One defines the collision efficiency E as the ratio of the actual collision cross section to the geometric cross section, or Eˆ

y2 Dp ‡ d p

(17.95)

2

Note that all drops of diameter dp in the cylinder with diameter y, below the falling drop with diameter Dp, will be collected by it. Because small drops tend to move away from the falling raindrop, E is expected to be smaller than unity for most cases. Figure 17.26 shows theoretically estimated collision efficiencies among drops as functions of the radii of the small and large drops. There is a rapid increase in collision efficiency as the two drops approach equal size. This is a result of fluid mechanical interactions that accelerate the upper drop more than the lower one but have little importance for the atmosphere where the probability of collision of equally sized drops is extremely small. Figure 17.26 shows that drops with diameters below 50 μm have a small collision efficiency and growth of drops to larger size is required for the acceleration of accretional growth. Calculations for larger drops are complicated by phenomena such as shape deformation, wake oscillations, and eddy shedding, making theoretical estimates of E difficult. The overall process of rain formation is further complicated by the fact that drops on collision trajectories may not coalesce but bounce off each other. The principal barrier to coalescence is the cushion of air between the two drops that must be drained before they can come into contact. An empirical coalescence efficiency Ec suggested by Whelpdale and List (1971) to address droplet bounce-off is Ec ˆ

Dp D p ‡ dp

2

(17.96)

which is applicable for Dp > 400 μm. Equation (17.96) can probably be viewed as an upper limit for Ec. Note that while (17.95) describes the area swept by a falling drop, (17.96) quantifies the probability of coalescence of two colliding droplets.

749

OTHER FORMS OF WATER IN THE ATMOSPHERE

FIGURE 17.26 Theoretically estimated collision efficiencies between cloud droplets as func­ tions of their radii. The radius of the smaller drop is on the x axis, and the radius of the larger drop is on the curve. (From H. G. Houghton, Physical Meteorology, The MIT Press, Cambridge, MA, 1985, p. 265.)

The growth of a falling drop of mass m as a result of accretion of drops can be described by dm π 2 ˆ Et Dp ‡ dp wL …υD dt 4

υd †

(17.97)

where Et is the overall accretion efficiency (collision efficiency times coalescence efficiency, Et ˆ EEc ), wL is the liquid water content of the small drops, and υD and υd are, respectively, the fall speeds of the larger and smaller particles. Equation (17.97) is called the continuous-accretion equation and assumes that the smaller drops are uniformly distributed in space. The description of the size change of droplets falling through a cloud by (17.97) implicitly assumes that all droplets of the same size will grow in the same way. According to the continuous equation, if several drops with the same initial diameter fall through a cloud, they would maintain the same size at all times. In reality, each droplet–droplet collision is a discrete event. Droplets that collide first with others grow more rapidly than do droplets of initially the same size that do not. As a result, the simple continuous equation can seriously underestimate the collision frequency and the growth rate of falling drops, especially when the collector and collected drops have the same size. The coagulation equation [see (13.59)] is a better mathematical description of these discrete collisions. Equation (13.59) (often called the stochastic accretion equation in cloud micro­ physics) describes implicitly individual collisions and predicts that even if all falling droplets initially have the same size, some will grow more than others. The role of ice in rain formation was first addressed by Bergeron in 1933, based on the calculations of Wegener. Wegener showed in 1911 that at temperatures below 0 °C supercooled water drops and ice crystals cannot exist in equilibrium. Using this result, Bergeron proposed that in cold clouds, the ice crystals grow by vapor diffusion at the expense of the water droplets until either all drops have been consumed or all ice crystals have fallen out of the cloud as precipitation. Findeisen later produced

750

CLOUD PHYSICS

additional observations supporting the mechanism described above, which is often called the Wegener– Bergeron–Findeisen mechanism. Mathematically, the description of the mechanism requires solution of the growth equations by vapor diffusion for both ice crystals and water drops in a supersaturated environment. 17.7.2.1 Raindrop Distributions A number of empirical formulas have been proposed for the raindrop spectrum. The distribution proposed by Best is often used to describe the fraction of rainwater comprised of raindrops smaller than Dp, F(DP) F Dp ˆ 1

exp

Dp 1:3p0:232 0

2:25

(17.98)

where p0 is the rainfall intensity in (mm h 1) and Dp is in mm. A widely used distribution is the Marshall–Palmer (MP) distribution n Dp ˆ n0 exp

ψDp

(17.99)

where n(Dp) = dn/dDp is the number distribution in drops m 3 mm 1, n0 = 8000 m 3 mm 1, and ψ ˆ 4:1p0 0:21 mm 1 : The MP distribution is often not sufficiently general to describe observed rain spectra. Sekhon and Srivastava (1971), among others, noted that n0 is not a constant but rather depends on the rainfall intensity. They suggested use of n0 ˆ 7000p0:37 m 3 mm 1 and ψ ˆ 3:8p0 0:14 mm 1 : Raindrop 0 spectra and a fitted MP distribution with variable n0 are shown in Figure 17.27. The MP distribution may overestimate by as much as 50% the number of small droplets in the 0.02–0.12 cm range and it is strictly applicable for Dp  0.12 cm.

FIGURE 17.27 Raindrop spectra and fitted MP distributions. (Reprinted from Micro­ physics of Clouds and Precipitation, Pruppacher, H. R., and Klett, J. D., 1997, with kind permission from Kluwer Academic Publishers.)

751

EXTENDED KÖHLER THEORY

APPENDIX 17

EXTENDED KÖHLER THEORY

Derivation of the Köhler equation is based on a combination of two expressions: the Kelvin equation, which governs the increase in water vapor pressure over a curved surface; and modified Raoult’s law, which describes the water equilibrium over a flat solution: es D p 4Mw σw ˆ exp xw γw es° RTρw Dp

(17A.1)

The total volume of the droplet is given by (17.11) π 3 D ˆ nw vw ‡ νs ns vs 6 p

(17A.2)

where νs is the number of ions generated by the dissociation of a molecule of salt that forms the cloud condensation nucleus. Using νs ns vw 1 νs ns ˆ1‡ ˆ1‡π nw xw D3 νs ns vs 6 p

(17A.3)

we get (17.18) ln

es Dp es°

ˆ

4Mw σw RTρw Dp

6νs ns Mw πρw D3p

(17A.4)

We now define the saturation ratio S and the two constants Aw and Bs as Sˆ

es D p es°

Aw ˆ

4Mw σw RTρw

Bs ˆ

6νs ns Mw πρw

and then arrive at the Köhler equation (17.19): ln S ˆ

Aw Dp

Bs D3p

(17A.5)

The subscripts w and s will come in handy as we add more phenomena.

17A.1 Modified Form of Köhler Theory for a Soluble Trace Gas Let us consider how droplet activation changes when a soluble trace gas is present (Laaksonen et al. 1998). We can first rewrite (17A.1) as Sw ˆ xw γw exp

Aw Dp

(17A.6)

When a soluble gas is present, an equation analogous to (17A.6) governs the gas equilibrium Sa ˆ xa γa exp

Aa Dp

(17A.7)

752

CLOUD PHYSICS

where Aa is defined analogously as Aw, and the subscript a refers to the soluble gas: Aa ˆ

4Ma σw RTρa

(17A.8)

To approximate the right-hand side (RHS) of (17A.7), the activity coefficient of a completely dissociated gas is given by x a γ

xa γa ˆ

…ν‡ ‡ν † ν‡ ν ν‡ ν sat pa H a

(17A.9)

where γ± is the mean activity coefficient; ν+ and ν are the number of cations and anions produced by a molecule of dissolved gas, respectively; and Ha is the Henry’s law constant of the gas. We again make the assumption of dilute solution, for which γ± is unity. We also specify that the soluble gas dissociates into exactly one cation and one anion, such as in the case of HNO3 and HC1. Now, to write the left-hand side (LHS) of (17A.7), we express Sa explicitly as the ratio of the partial pressure of the soluble gas to its saturation pressure: Sa ˆ

pa psat a

The partial pressure of the soluble gas can be expressed in terms of its molar concentration, Na (moles per volume of air) by pa ˆ N a RT The amount of soluble gas that is dissolved in droplets per unit volume of air can be expressed as Cna, where C is the number of droplets per unit volume of air and na is the moles of soluble gas per droplet. If the total amount of available soluble gas per unit volume of air is expressed as ntotC, where ntot is the total moles of soluble gas available per droplet, then the soluble gas remaining in the gas phase at any time is (ntot na)C, and Sa ˆ

…ntot

na †CRT psat a

(17A.10)

Combining (17A.7), (17A.9), and (17A.10), we get …ntot

na †CRT psat a

ˆ

xa2 Aa exp sat pa H a Dp

or x2a ˆ H a CRT …ntot

Aa Dp

na † exp

(17A.11)

The exponential term is approximately unity for Dp > 0.2 μm; since cloud droplets exceed this size easily, we will take this term as unity. For a dilute drop, we can approximate the aqueous-phase mole fraction of soluble gas as xa ≅na =nw ; giving n2a ˆ n2w H a CRT …ntot

na †

(17A.12)

753

EXTENDED KÖHLER THEORY

We can then write the number of moles of water in the drop as 3 mw πDp ρw nw ˆ ˆ 6Mw Mw

νs ns Dp3

ˆ

Bs

ˆ

νa na D3p

(17A.13)

Ba

where Ba is defined analogously to Bs as 6νa na Mw πρw

Ba ˆ

(17A.14)

In order to simplify (17A.12), we will define β as πρw 6Mw

⠈ H a RT

2

(17A.15)

and (17A.12) can then be rewritten in terms of this β: n2a ˆ CβD6p …ntot

na †

0 ˆ na2 ‡ CβD6p na

CβDp6 ntot

(17A.16)

The positive root of this quadratic equation is na ˆ

CβD6p

1‡

2

1‡

4ntot CβD6p

(17A.17)

or 2ntot

na ˆ

4ntot 1‡ CβD6p

1‡

(17A.18)

We now turn to (17A.6). We write the mole fraction of water as xw ˆ

nw nw ‡ νs ns ‡ νa na νs ns ‡ νa na nw

≅1

(17A.19)

Recalling the expression for nw from (17A.13), we rewrite xw as xw ˆ 1

Bs D3p

Ba Dp3

(17A.20)

Substituting this back into (17A.6), and again assuming that γw ≅1, we obtain Sw ˆ

1

Bs D3p

Ba Aw exp 3 Dp Dp

(17A.21)

754

CLOUD PHYSICS

The exponential term can then be expanded in a Taylor series. If we collect the lowest-order terms of Dp, we then have Bs D3p

Aw Dp

Sw ˆ 1 ‡

Ba Dp3

(17A.22)

In this expression for saturation ratio, the second term describes the Kelvin effect, while the third term describes the solute effect as in the classical form of the Köhler equation. The fourth term is the added one that accounts for the dissolved gas. In evaluating Sw, we keep in mind that Ba is defined in terms of na, which is given by (17A.18) such that Ba is also a function of Dp. If the soluble gas concentration in the vapor phase is maintained at a constant value, that is, is not depleted as a result of absorption into the droplets, then the foregoing result simplifies. Equation (17A.11) becomes Aa Dp

xa2 ˆ pa H a exp

(17A.23)

where, as above, we can neglect the exponential term for Dp greater than ∼0.2 μm. Approximating xa as na/nw, we get na p (17A.24) ˆ pa H a nw Using (17A.24), the form of the Köhler equation for a constant soluble gas concentration is Sw ˆ 1 ‡

Aw Dp

p ν a pa H a

Bs D3p

(17A.25)

p In summary, the gaseous solute effect term, Ba =D3p ; in (17A.22) is replaced by νa pa H a when the soluble gas concentration in the vapor is constant.

17A.2 Modified Form of Köhler Theory for a Slightly Soluble Substance We take an approach similar to that above; here nw nw ‡ νs ns ‡ νa na ‡ νss nss νs ns ‡ νa na ‡ νss nss ≅1 nw

xw ˆ

(17A.26)

where the subscript ss refers to the slightly soluble substance. Since now the number of moles of water in the drop is written to exclude the volume of the insoluble substance, we obtain nw ˆ ˆ

D3ss ρw

π D3p

6Mw νs ns D3p

(17A.27) 3 Dss

Bs

ˆ

νa na Dp3

3 Dss

Ba

The corresponding water mole fraction becomes xw ˆ 1

Bs D3p

Ba 3 Dss

D3p

D3ss

νss nss nw

(17A.28)

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EXTENDED KÖHLER THEORY

To calculate the number of moles of the slightly soluble species that actually dissolves into the aqueous phase, we use the solubility Γ, expressed in kg of the substance per kg of water: Mw nw Γ Mss

nss ˆ

(17A.29)

Thus, we write the concentration of the slightly soluble substance in the aqueous phase Css as Css ˆ

νss nss νss Mw Γ ˆ Mss nw

(17A.30)

This expression for Css is valid only when a portion of the slightly soluble core has been dissolved. When the entire core has dissolved into the droplet, then Css ˆ

νss Mw Γn0w Mss nw

(17A.31)

0 where nw refers to the number of moles of water required to dissolve the core. Finally, we have

Bs

xw ˆ 1

Dp3

Ba 3 Dss

D3p

D3ss

Css

(17A.32)

and thus we get an equation analogous to (17A.22): Sw ˆ 1 ‡

Aw Dp

Bs D3p

Ba 3 Dss

D3p

D3ss

Css

(17A.33)

This expression is similar to (17A.22) except for the new final term that accounts for the extra solute effect 3 †; from the slightly soluble substance, and all terms involving of D3p in (17A.22) are replaced by …Dp3 Dss reflecting the subtracted volume owing to the undissolved core. This replacement also holds in the 3 2 expression for na as in (17A.18): D6p is replaced by …D3p Dss † : Note that the case of completely insoluble core is included as the limiting case of this equation where Css = 0.

17A.3 Modified Form of Köhler Theory for a Surface-Active Solute Because surface-active compounds are generally organic solutes, we can modify (17A.33) to include the effects of surface-active, slightly soluble material (Shulman et al. 1996). Note that this correction affects only the Kelvin term. First, we assume that the mixture surface tension can be approximated as the mole fraction weighted average of the surface tensions of pure water and pure organic solute: σmix ˆ xw σw ‡ …1 xw †σss ˆ xw …σw σss † ‡ σss

(17A.34)

We can then use this mixture surface tension σmix in place of σw in the Aw term. The resulting expression can be written in shorthand as Sw ˆ where xw is given by (17A.32).

4Mw …xw …σw RTρw Dp

σss † ‡ σss † ‡ xw

(17A.35)

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CLOUD PHYSICS

17A.4 Examples We consider first the effect of a soluble trace gas on droplet activation, in this case HNO3. The Köhler curves are generated with (17A.22); these are shown in Figure 17A.1 for a system in which the HNO3 is depleted and one in which HNO3 is held at constant concentration, both with an initial HNO3 mixing ratio of 10 ppb. The salt assumed to be present is (NH4)2SO4 with a dry diameter of 0.05 μm. We see that in the case of constant HNO3 concentration, the resulting Köhler curve has a shape similar to that of the original curve in the absence of soluble gas. The critical saturation ratio is reduced by about 0.006, translating to easier droplet activation. The case in which the HNO3 is depleted exhibits a similar behavior as the droplet begins to take on water, but as the HNO3 in the gas phase is depleted, the curve approaches the classical curve. The result is a Köhler curve with two maxima and a local minimum. The barrier for activation in this case is set by the second, higher maximum, because after S reaches the first peak, the equilibrium droplet size is given by the point at the same supersaturation on the other ascending branch. As S increases further, the droplet finally activates at the second maximum. Note that if S were to be decreased from some point on the second ascending branch, the change in droplet diameter would be the reverse–it would reach the local minimum, and jump over to the left branch, thereby bypassing the local maximum. In general, the soluble trace gas, regardless of whether its concentration changes or remains constant, helps to accelerate activation by lowering the critical saturation ratio. The higher the soluble gas concentration, the greater the decrease in Sc. Figure 17A.2 shows another set of curves similar to those in Figure 17A.1, but with a larger (NH4)2SO4 dry diameter of 0.1 μm. Here we note that with a sufficiently large dry diameter, a local minimum does not appear in the case in which the HNO3 is depleted, and the particle growth occurs normally. In both cases described above, when the concentration of HNO3 is held constant, the critical diameter is not changed from that in the absence of soluble gas. When depletion occurs, however, a larger critical diameter results. In this case, larger particles can exist in equilibrium without being activated.

FIGURE 17A.1 Köhler curves for dry ammonium sulfate particles of diameter 0.05 μm in the absence and presence of HNO3 at 20 °C and 1 atm. Ammonium sulfate properties are νs ˆ 3 ions molecule 1 and ρs ˆ 1770 kg m 3 : Droplet number concen­ tration C = 1000 cm 3. Surface tension of water σw ˆ 0:073 N m 1 :

EXTENDED KÖHLER THEORY

FIGURE 17A.2 Same conditions as Figure 17A.1 except dry diameter of ammonium sulfate particles is 0.1 μm.

Now we investigate the effect on the Köhler curves of slightly soluble organic matter in the particle core in both presence and absence of HNO3. We choose adipic acid (C6H10O4) as the slightly soluble species because of its moderate solubility and ability to lower surface tension. First, we ignore the effect of adipic acid on the surface tension of the solution and approximate the surface tension as that of pure water. Figure 17A.3 shows the modified curves for the slightly soluble adipic acid at various HNO3 concentrations, as predicted by (17A.33). Again, the general effect is a downward shift of the Köhler curve, which translates to accelerated activation at a lower saturation ratio. A sharp minimum occurs, which is produced by the final dissolution of the slightly soluble core. If a larger core of adipic acid exists initially, say, 0.5 μm, as in Figure 17A.4, the sharp minimum occurs at a later stage, as expected. Since the minimum now occurs at a larger diameter, the effect of the HNO3 depletion that precedes it can be seen in the form of a smooth minimum. For a sufficiently high HNO3 concentration, the effect of HNO3 depletion dominates and masks the smaller sharp minimum. To compare the new critical parameters owing to the slightly soluble organic for the case with no HNO3, we note that from Figure 17A.3 (smaller initial core), Sc = 1.0025 and Dpc = 0.2 μm, while that from Figure 17A.4 (larger initial core), Sc = 1.001 and Dpc = 0.6 μm. Thus, a larger amount of the slightly soluble matter decreases the critical saturation ratio, while the critical diameter is increased mainly because the initial particle size is larger. Finally, we take into account the effect of decreased surface tension as a result of the dissolved adipic acid. The effective surface tension for adipic acid is calculated from literature data (Shulman et al. 1996), and we use (17A.35) to generate the curves in Figure 17A.5. Once more, the curve is shifted down (activation is accelerated) as a result of the decreased surface tension. We also note that the surface-active solute effect lowers the critical S appreciably only when there is very little or no soluble gas. This is because in the presence of sufficient soluble gas, the Sc is determined by the second maximum rather than the first, and the downward shift of this point owing to surface tension effects is quite small.

757

758

CLOUD PHYSICS

FIGURE 17A.3 Köhler curves for particles consisting of a 0.1 μm adipic acid core coated with an equivalent of 0.05 μm ammonium sulfate with 0, 5, and 10 ppb of HNO3 at 20 °C and 1 atm. For adipic acid, a slightly soluble C6 dicarboxylic acid, the following parameters are used: νss ˆ 1 ion molecule 1, Mss = 0.146 kg mol 1, ρss ˆ 1360 kg m 3 (Cruz and Pandis 2000), and Γ = 0.018 kg (kg water) 1 (Raymond and Pandis 2002). Surface tension is taken as that of water.

FIGURE 17A.4 Same conditions as Figure 17A.3 except 0.5 μm adipic acid core.

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PROBLEMS

FIGURE 17A.5 Same conditions as Figure 17A.4 except the effect of adipic acid on the surface tension of the solution is accounted for. The surface tension of the aqueous solution of adipic acid is calculated by interpolation using experimental data (Shulman et al. 1996) as a function of the molar concentration of adipic acid in the droplet.

PROBLEMS 17.1A Verify equations (17.31) and (17.32). 17.2A You dissolve 5 g of NaCl in a glass containing 200 cm3 of water. The glass is in a room with temperature 20 °C and relative humidity 80%. Calculate the volume of water that will be left in the glass after several days of residence in this environment. Repeat the calculation for a relative humidity of 95%. 17.3A Your grandmother and grandfather are upset because it has just started raining. After listening to the weather prediction for a cloudy day but without rain, they planned to spend the day working in the garden. They have started criticizing the local weather forecaster who, despite the impressive gadgets (radar maps, 3D animated maps), still cannot reliably predict if it will rain tomorrow. They turn to you and ask why, if we can send people to the moon, we still cannot tell whether it is going to rain. Explain, avoiding scientific terminology. 17.4B What is the activation diameter at 0.3% supersaturation for particles consisting of 50% (NH4)2SO4, 30% NH4NO3, and 20% insoluble material? 17.5C A 100 nm diameter dry (NH4)2SO4 particle is suddenly exposed to an atmosphere with a 0.3% water supersaturation. How long will it take for the particle to reach a diameter of 5 μm? Assume a unity accommodation coefficient. 17.6C Repeat Problem 17.5, neglecting the correction to the diffusion coefficients and thermal conduc­ tivities for noncontinuum effects (assume that D´υ ˆ Dυ ; ka´ ˆ ka ). Compare with Problem 17.5 and discuss your observations.

760

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17.7C Repeat Problem 17.5, assuming that αc = 0.045. Compare with Problem 17.5 and discuss your observations. 17.8A Dry activation diameters of roughly 0.5 μm have been observed in fogs in urban areas. a. If the particles consist entirely of (NH4)2SO4, what is the maximum supersaturation inside the fog layer? b. Assuming that the maximum supersaturation is 0.05% and the particles contain (NH4)2SO4 and insoluble material, calculate the mass fraction of the insoluble material.

REFERENCES Andreae, M. O., and Rosenfeld, D. (2008), Aerosol-cloud-precipitation interactions. Part 1. The nature and sources of cloud-active aerosols, Earth Sci. Rev. 89, 13–41. Collett, J. L. Jr. et al. (1994), Acidity variations across the cloud drop size spectrum and their influence on rates of atmospheric sulfate production, Geophys. Res. Lett. 21, 2393–2396. Conant, W. C., et al. (2004), Aerosol-drop concentration closure in warm cumulus, J. Geophys. Res. 109, D13204 (doi: 10.1029/2003JD004324). Cruz, C. N., and Pandis, S. N. (2000), Deliquescence and hygroscopic growth of mixed inorganic-organic atmospheric aerosol, Environ. Sci. Technol. 34, 4313–4319. Cziczo, D. J., et al. (2013), Clarifying the dominant sources and mechanisms of cirrus cloud formation, Science 340, 1320–1324. Daum, P. H., et al. (1984), Acidic and related constituents in liquid water stratiform clouds, J. Geophys. Res. 89, 1447–1458. DeMott, P. J. (2009), African dust aerosols as atmospheric ice nuclei, Geophys. Res. Lett. 36, L07808. DeMott, P. J., et al. (2009), Ice nucleation behavior of biomass combustion particles at cirrus temperatures, J. Geophys. Res. 114, D16205. DeMott, P. J., et al. (2010), Predicting global atmospheric ice nuclei distributions and their impacts on climate, Proc. Natl. Acad. Sci. USA, 107, 11217–11222. DeMott, P. J., et al. (2011), Resurgence in ice nuclei measurement research, Bull. Amer. Meteorol. Soc. 92, 1623–1635. Dennis, R. L., et al. (1993), Correcting RADM sulfate underprediction–discovery and correction of errors and testing the correction through comparisons with field data, Atmos. Environ. 27, 975–997. Develk, J. P. (1994), A model for cloud chemistry–a comparison between model simulations and observations in stratus and cumulus, Atmos. Environ. 28, 1665–1678. Dusek, U., et al. (2006), Size matters more than chemistry for cloud nucleating ability of aerosol particles, Science 312, 1375–1378. Flossmann, A. I. (1991), The scavenging of two different types of marine aerosol particles calculated using a two dimensional detailed cloud model, Tellus 43B, 301–321. Flossmann, A. I., et al. (1985), A theoretical study of the wet removal of atmospheric pollutants. Part I: The redistribution of aerosol particles captured through nucleation and impaction scavenging by growing cloud drops, J. Atmos. Sci. 42, 583–606. Flossmann, A. I., et al. (1987), A theoretical study of the wet removal of atmospheric pollutants. Part II: The uptake and redistribution of (NH4)2SO4 particles and SO2 gas simultaneously scavenged by growing cloud drops, J. Atmos. Sci. 44, 2912–2923. Flossmann, A. I., and Pruppacher, H. R. (1988), A theoretical study of the wet removal of atmospheric pollutants 3. The uptake, redistribution, and deposition of (NH4)2SO4 particles by a convective cloud using a two-dimensional cloud dynamics model, J. Atmos. Sci. 45, 1857–1871. Frick, G. M., and Hoppel, W. A. (1993), Airship measurements of aerosol size distributions, cloud droplet spectra, and trace gas concentrations in the marine boundary layer, Bull. Am. Meteorol. Soc. 74, 2195–2202. Ghan, S. J., et al. (2006), Use of in situ cloud condensation nuclei, extinction, and aerosol size distribution measurements to test a method for retrieving cloud condensation nuclei profiles from surface measurements, J. Geophys. Res. 111, D05S10 (doi: 10.1029/2004JD005752).

REFERENCES Hegg, D. A. (1985), The importance of liquid-phase oxidation of SO2 in the troposphere, J. Geophys. Res. 90, 3773–3779. Hegg, D. A., and Hobbs, P. V. (1987), Comparisons of sulfate production due to ozone oxidation in clouds with a kinetic rate equation, Geophys. Res. Lett. 14, 719–721. Hegg, D. A., and Hobbs, P. V. (1988), Comparisons of sulfate and nitrate production in clouds on the mid-Atlantic and Pacific Northwest coast of the United States, J. Atmos. Chem. 7, 325–333. Hegg, D. A., and Hobbs, P. V. (1992), Cloud condensation nuclei in the marine atmosphere: A review, in Nucleation and Atmospheric Aerosols, N. Fukuta and P. E. Wagner (eds.), Deepak Publishing, Hampton, VA, pp. 181–192. Hoppel, W. A., et al. (1986), Effects of non–precipitating clouds on the aerosol size distribution in the marine boundary layer, Geophys. Res. Lett. 13, 125–128. Houghton, H. G. (1985), Physical Meterology, MIT Press, Cambridge, MA. Hudson, J. G., and Frisbie, P. R. (1991), Cloud condensation nuclei near marine stratus, J. Geophys. Res. 96, 20795–20808. Husain, L., et al. (1991), A study of heterogeneous oxidation of SO2 in summer clouds, J. Geophys. Res. 96, 18789–18805. Junge, C. E. (1963), Air Chemistry and Radioactivity, Academic Press, New York. Karamachandani, P., and Venkatram, A. (1992), The role of non–precipitating clouds in producing ambient sulfate during summer–results from simulation with the Acid Deposition and Oxidant Model (ADOM), Atmos. Environ. 26, 1041–1052. Kelly, T. J., et al. (1989), Detection of acid producing reactions in natural clouds, Atmos. Environ. 23, 569–583. Köhler, H. (1921), Zur Kondensation des Wasserdampfe in der Atmosphäre, Geofys. Publ. 2, 3–15. Köhler, H. (1926), Zur Thermodynamic der Kondensation an hygroskopischen Kemen und Bemer-kungen iiber das Zusammenfliessen der Tropfen, Medd. Met. Hydr. Anst. Stockholm 3(8). Kreidenweis, S. M., and Asa-Awuku, A. (2014), Aerosol hygroscopicity: Particle water content and its role in atmospheric processes, in Treatise on Geochemistry Second edition, H. D. Holland and K. K. Turekian (eds.), Elsevier, Oxford, UK, 5, pp. 331–361. Laaksonen, A., et al. (1998), Modification of the Köhler equation to include soluble trace gases and slightly soluble substances, J. Atmos. Sci. 55, 853–862. Lamb, D., and Verlinde, J. (2011), Physics and Chemistry of Clouds, Cambridge Univ. Press, Cambridge, UK. Liao, H., et al. (2003), Interactions between tropospheric chemistry and aerosols in a unified general circulation model, J. Geophys. Res. 108, 4001 (doi: 10. 1029/2001JD001260). Liu, P. S. K., et al. (1994), Sulfate production in a summer cloud over Ontario, Canada, Tellus 45B, 368–389. Lowe, P. R., and Ficke, J. M. (1974), Technical Paper 4–74, The computation of saturation vapor pressure, Environmental Prediction Research Facility, Naval Postgraduate School, Monterey, CA. Mason, B. J. (1971), The Physics of Clouds, Oxford Univ. Press, Oxford, UK. McHenry, J. N., and Dennis, R. L. (1994), The relative importance of oxidation pathways and clouds to atmospheric ambient sulfate production as predicted by the regional acid deposition model, J. Appl. Meteorol. 33, 890–905. Nenes, A., et al. (2001), A theoretical analysis of cloud condensation nucleus (CCN) instruments, J. Geophys. Res. 106, 3449–3474. Noone, K. J., et al. (1988), Cloud droplets: Solute concentration is size dependent, J. Geophys. Res. 93, 9477–9482. Pandis, S. N., and Seinfeld, J. H. (1989), Mathematical modeling of acid deposition due to radiation fog, J. Geophys. Res. 94, 12156–12176. Pandis, S. N., et al. (1990a), Chemical composition differences among droplets of different sizes, Atmos. Environ. 24A, 1957–1969. Pandis, S. N., et al. (1990b), The smog–fog–smog cycle and acid deposition, J. Geophys. Res. 95, 18489–18500. Pandis, S. N., et al. (1992), Heterogeneous sulfate production in an urban fog, Atmos. Environ. 26A, 2509–2522. Petters, M. D., and Kreidenweis, S. M. (2007), A single parameter representation of hygroscopic growth and cloud condensation nucleus activity, Atmos. Chem. Phys. 7, 1961–1971. Pruppacher, H. R., and Klett, J. D. (1997), Microphysics of Clouds and Precipitation, Kluwer, Dordrecht, The Netherlands. Raatikainen, T., et al. (2013), Worldwide data sets constrain the water vapor uptake coefficient in cloud formation, Proc. Natl. Acad. Sci. USA 110, 3760–3764.

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CLOUD PHYSICS Raymond, T. M., and Pandis, S. N. (2002), Cloud activation of single-component organic aerosol particles, J. Geophys. Res. 107, 4787 (doi: 10.1029/2002JD002159). Reutter, P., et al. (2009), Aerosol- and updraft-limited regimes of cloud droplet formation: Influence of particle number, size and hygroscopicity on the activation of cloud condensation nuclei (CCN), Atmos. Chem. Phys. 9, 7067–7080. Richardson, M. S., et al. (2010), Observations of ice nucleation by ambient aerosols in the homogeneous freezing regime, Geophys. Res. Lett. 37, L04806. Rissman, T. A., et al. (2004), Chemical amplification (or dampening) of the Twomey effect: Conditions derived from droplet activation theory, J. Atmos. Sci. 61, 919–930. Rissman, T. A., et al. (2006), Characterization of ambient aerosol from measurements of cloud condensation nuclei during the 2003 Atmospheric Radiation Measurement Aerosol Intensive Observational Period at the Southern Great Plains site in Oklahoma, J. Geophys. Res. 111, D05S11 (doi: 10.1029/2004JD005695). Sekhon, R. S., and Srivastava, R. C. (1971), Doppler observations of drop size distributions in a thunderstorm, J. Atmos. Sci. 28, 983–994. Shulman, M. L., et al. (1996), Dissolution behavior and surface tension effects of organic compounds in nucleating cloud droplets, Geophys. Res. Lett. 23, 277–280. Ten Brink, H. M., et al. (1987), Efficient scavenging of aerosol sulfate by liquid water clouds, Atmos. Environ. 21, 2035–2052. Twomey, S. (1959), The nuclei of natural clouds formation. Part II: The supersaturation in natural clouds and the variation of cloud droplet concentration, Geofis. Pura Appl. 43, 243–249. VanReken, T. M., et al. (2003), Toward aerosol/cloud condensation nuclei (CCN) closure during CRYSTAL-FACE, J. Geophys. Res. 108, 4633 (doi: 10.1029/2003JD003582). Walcek, C. J., et al. (1990), Theoretical estimates of the dynamic radiative and chemical effects of clouds on tropospheric gases, Atmos. Res. 25, 53–69. Warner, J. (1968), The supersaturation in natural clouds, J. Appl. Meteorol. 7, 233–237. Whelpdale, D. M., and List, R. (1971), The coalescence process in droplet growth, J. Geophys. Res. 76, 2836–2856.

CHAPTER

18

Atmospheric Diffusion

A major goal of our study of the atmospheric behavior of trace species is to be able to describe mathematically the spatial and temporal distribution of contaminants released into the atmosphere. This chapter is devoted to the subject of atmospheric diffusion and its description. It is common to refer to the behavior of gases and particles in turbulent flow as turbulent “diffusion” or, in this case, as atmospheric “diffusion,” although the processes responsible for the observed spreading or dispersion in turbulence are not the same as those acting in ordinary molecular diffusion. A more precise term would perhaps be “atmospheric dispersion,” but to conform to common terminology we will use “atmospheric diffusion.” This chapter is devoted primarily to developing the two basic ways of describing turbulent diffusion. The first is the Eulerian approach, in which the behavior of species is described relative to a fixed coordinate system. The Eulerian description is the common way of treating heat and mass transfer phenomena. The second approach is the Lagrangian, in which concentration changes are described relative to the moving fluid. As we will see, the two approaches yield different types of mathematical relationships for the species concentrations that can, ultimately, be related. Each of the two modes of expression is a valid description of turbulent diffusion; the choice of which approach to adopt in a given situation will be seen to depend on the specific features of the situation.

18.1 EULERIAN APPROACH Let us consider N species in a fluid. The concentration of each must, at each instant, satisfy a material balance taken over a volume element. Thus any accumulation of material over time, when added to the net amount of material convected out of the volume element, must be balanced by an equivalent amount of material that is produced by chemical reaction in the element, that is emitted into it by sources, and that enters by molecular diffusion. Expressed mathematically, the concentration of each species ci must satisfy the continuity equation @ci @ @ 2 ci ‡ …uj ci † ˆ Di ‡ Ri …c1 ; . . . ; cN ; T† ‡ Si …x; t† @t @xj @xj @xj

…18:1†

i ˆ 1; 2; . . . ; N

Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, Third Edition. John H. Seinfeld and Spyros N. Pandis.  2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

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where uj is the jth component of the fluid velocity, Di is the molecular diffusivity of species i in the carrier fluid, Ri is the rate of generation of species i by chemical reaction (which depends in general on the fluid temperature T ), and Si is the rate of addition of species i at location x ˆ …x1 ; x2 ; x3 ) and time t.1 In addition to the requirement that the ci satisfy (18.1), the fluid velocities uj and the temperature T, in turn, must satisfy the Navier–Stokes and energy equations, and which themselves are coupled through the uj , ci , and T with the total continuity equation and the ideal-gas law (we restrict our attention to gaseous systems). In general, it is necessary to carry out a simultaneous solution of the coupled equations of mass, momentum, and energy conservation to account properly for the changes in uj , T, and ci and the effects of the changes of each of these on each other. In dealing with atmospheric trace gases, occurring at parts per million and smaller mixing ratios, it is quite justifiable to assume that the presence of these species does not affect the meteorology to any detectable extent; thus the equation of continuity (18.1) can be solved independently of the coupled momentum and energy equations.2 Consequently, the fluid velocities uj and the temperature T can be considered independent of the ci . From this point on we will not explicitly indicate the dependence of Ri on T. The complete description of atmospheric gas behavior rests with the solution of (18.1). Unfortunately, because the flows of interest are turbulent, the fluid velocities uj are random functions of space and time. It is customary to represent the wind velocities uj as the sum of a deterministic and j ‡ u´j . stochastic component, u To illustrate the importance of the definition of the deterministic and stochastic velocity components j and u´j , let us suppose a puff of species of known concentration distribution c…x; t0 ) at time t0 . In the u absence of chemical reaction and other sources, and assuming molecular diffusion to be negligible, the concentration distribution at some later time is described by the following advection equation: @c @ ‡ …uj c† ˆ 0 @t @xj

…18:2†

j and compare the solution with observations, we would find in If we solve this equation with uj ˆ u reality that the material spreads out more than predicted. This extra spreading is, in fact, what is referred to as turbulent diffusion and results from the influence of the random component u´j , which we have ignored. Now let us solve this equation with the precise velocity field uj . We should then find that the solution agrees exactly with the observations (assuming, of course, that molecular diffusion is negligible), implying that if we knew the velocity field precisely at all locations and times, there would be no such phenomenon as turbulent diffusion. Thus turbulent diffusion is an artifact of our lack of complete knowledge of the true velocity field. Consequently, one of the fundamental tasks in turbulent diffusion theory is to define the deterministic and stochastic components of the velocity field. j ‡ u´j in (18.1) gives Replacing uj by u @ @ 2 ci @ci j ‡ uj´ †ci Š ˆ Di ‡ ‰…u ‡ Ri …c1 ; . . . ; cN † ‡ Si …x; t† @xj @xj @t @xj

…18:3†

Since the uj´ are random variables, the ci resulting from the solution of (18.3) must also be random variables; that is, because the wind velocities are random functions of space and time, the airborne species concentrations are themselves random variables in space and time. Thus the determination of

We will use the convention of (x1 ,x2 ,x3 ) as coordinate axes and (u1 ,u2 ,u3 ) as fluid velocity components when dealing with the transport equations in their most general form. Eventually it will be easier to use an (x,y,z), (u,v,w) system when dealing with specific problems and we will do so at that time. 2 Two effects could, in principle, serve to invalidate this assumption. Highly unlikely in the troposphere is that sufficient heat would be generated by chemical reactions to influence the temperature. Absorption, reflection, and scattering of radiation by trace gases and particles could result in alterations of the fluid behavior. 1

765

EULERIAN APPROACH

the ci , in the sense of being a specified function of space and time, is not possible, just as it is not possible to determine precisely the value of any random variable in an experiment. We can at best derive the probability that at some location and time the concentration of species i will lie between two closely spaced values. Unfortunately, the specification of the probability density function for a random process as complex as atmospheric diffusion is almost never possible. Instead, we must adopt a less desirable but more feasible approach, the determination of certain statistical properties of the ci , most notably the mean hci i. The mean concentration can be interpreted in the following way. Let us imagine an experiment in which a puff of material is released at a certain time and concentrations are measured downwind at subsequent times. We would measure ci …x; t†, which would exhibit random characteristics because of the wind. If it were possible to repeat this experiment under identical conditions, we would again measure ci …x; t†, but because of the randomness in the wind field we could not reproduce the first ci …x; t†. Theoretically we could repeat this experiment an infinite number of times. We would then have a socalled ensemble of experiments. If at every location x and time t we averaged all the concentration values over the infinite number of experiments, we would have computed the theoretical mean concentration hci …x; t†i.3 Experiments like this cannot, of course, be repeated under identical conditions, and so it is virtually impossible to measure hci i. Thus a measurement of the concentration of species i at a particular location and time is more suitably envisioned as one sample from a hypothetically infinite ensemble of possible concentrations. Clearly, an individual measurement may differ considerably from the mean hci i. It is convenient to express ci as hci i ‡ c´i , where, by definition, hci´ i ˆ 0. Averaging (18.3) over an infinite ensemble of realizations of the turbulence yields the equation governing hci i: @hci i @ @ @ 2 hci i ´ j hci i† ‡ hu´j ci´ i ˆ Di ‡ hRi …hc1 i ‡ c1´ ; . . . ; hcN i ‡ cN †i ‡ Si …x; t† ‡ …u @t @xj @xj @xj @xj

…18:4†

Let us consider the case of a single inert species, that is, R ˆ 0. We note that (18.4) contains dependent variables hci and hu´j c´ i, j ˆ 1, 2, 3. We thus have more dependent variables than equations. Again, this is the closure problem of turbulence. For example, if we were to derive an equation for the hu´j c´ i by subtracting (18.4) from (18.3), multiplying the resulting equation by u´j and then averaging, we would obtain @ @ ´ ´ @ @hci @ k huj´ c´ i† ‡ huj´ uk´ i huj c i ‡ …u ˆ D hu´ c´ i @t @xk @xk @xk @xk j

huj´ uk´ c´ i

…18:5†

j ˆ 1; 2; 3 Although we have derived the desired equations, we have at the same time generated new dependent variables hu´j u´k c´ i; j;k ˆ 1;2;3. If we generate additional equations for these variables, we find that still more dependent variables appear. The closure problem becomes even worse if a nonlinear chemical reaction is occurring. If the single species decays by a second-order reaction, then the term hRi in (18.4) becomes –k…hci2 ‡ hc´ 2 i†, where hc´ 2 i is a new dependent variable. If we were to derive an equation for hc´ 2 i, we would find the emergence of new dependent variables huj´ c´ 2 i, hc´ 3 i, and h@c´ =@xj @c´ =@xj i. It is because of the closure problem that an Eulerian description of turbulent diffusion will not permit exact solution even for the mean concentration hci.

j versus hci i, We have used different notations for the mean values of the velocities and the concentrations, that is, u in order to emphasize the fact that the mean fluid velocities are normally determined by a process involving temporal and spatial averaging, whereas the hci i always represent the theoretical ensemble average.

3

766

ATMOSPHERIC DIFFUSION

18.2 LAGRANGIAN APPROACH The Lagrangian approach to turbulent diffusion is concerned with the behavior of representative fluid particles.4 We therefore begin by considering a single particle that is at location x´ at time t´ in a turbulent fluid. The subsequent motion of the particle can be described by its trajectory, X‰x´ ; t´ ; tŠ, that is, its position at any later time t. Let ψ…x1 ; x2 ; x3 ; t†dx1 dx2 dx3 ˆ ψ…x; t†dx ˆ probability that the particle at time t will be in volume element x1 to x1 ‡ dx1 , x2 to x2 ‡ dx2 , and x3 to x3 ‡ dx3 , that is, that x1  X1 < x1 ‡ dx1 , and so on. Thus ψ…x; t† is the probability density function (pdf) for the particle’s location at time t. By the definition of a probability density function 1

1

1

1

1

1

ψ…x; t†dx ˆ 1

The probability density of finding the particle at x at t can be expressed as the product of two other probability densities: 1. The probability density that if the particle is at x´ at t´ , it will undergo a displacement to x at t. Denote this probability density Q…x; t j x´ ; t´ † and call it the transition probability density for the particle. 2. The probability density that the particle was at x´ at t´ , ψ…x´ ; t´ †, integrated over all possible starting points x´ . Thus ψ…x; t† ˆ

1

1

1

1

1

1

Q…x; t j x´ ; t´ †ψ…x´ ; t´ †dx´

…18:6†

The density function ψ…x; t† has been defined with respect to a single particle. If, however, an arbitrary number m of particles are initially present and the position of the ith particle is given by the density function ψi …x; t†, it can be shown that the ensemble mean concentration at the point x is given by m

ψi …x; t†

hc…x; t†i ˆ

…18:7†

iˆ1

By expressing the probability density function (pdf) ψi …x; t† in (18.7) in terms of the initial particle distribution and the spatiotemporal distribution of particle sources S…x; t†, say, in units of particles per volume per time, and then substituting the resulting expression into (18.6), we obtain the following general formula for the mean concentration: hc…x; t†i ˆ ‡

1

1

1

1

1

1

1 1

1 1

Q…x; tjx0 ; t0 †hc…x0 ; t0 †idx0 1 1

t

Q…x; tjx´ ; t´ †S…x;´ t´ †dt´ dx´

…18:8†

t0

The first term on the right-hand side (RHS) represents those particles present at t0 , and the second term on the RHS accounts for particles added from sources between t0 and t. By a “fluid particle” we mean a volume of fluid large compared with molecular dimensions but small enough to act as a point that exactly follows the fluid. The “particle” may contain fluid of a different composition than the carrier fluid, in which case the particle is referred to as a “marked particle.”

4

767

EQUATIONS GOVERNING THE MEAN CONCENTRATION OF SPECIES IN TURBULENCE

Equation (18.8) is the fundamental Lagrangian relation for the mean concentration of a species in turbulent fluid. The determination of hc…x; t†i, given hc…x0 ; t0 †i and S…x´ ; t†, rests with the evaluation of the transition probability Q…x; tjx´ ; t´ †. If Q were known for x; x´ ; t, and t´ , the mean concentration hc…x; t†i could be computed by simply evaluating (18.8). However, there are two substantial problems with using (18.8): (1) it holds only when the particles are not undergoing chemical reactions, and (2) such complete knowledge of the turbulence properties as would be needed to know Q is generally unavailable except in the simplest of circumstances.

18.3 COMPARISON OF EULERIAN AND LAGRANGIAN APPROACHES The techniques for describing the statistical properties of the concentrations of marked particles, such as trace gases, in a turbulent fluid can be divided into two categories: Eulerian and Lagrangian. The Eulerian methods attempt to formulate the concentration statistics in terms of the statistical properties of the Eulerian fluid velocities, that is, the velocities measured at fixed points in the fluid. A formulation of this type is very useful not only because the Eulerian statistics are readily measurable (as determined from continuous-time recordings of the wind velocities by a fixed network of instruments) but also because the mathematical expressions are directly applicable to situations in which chemical reactions are taking place. Unfortunately, the Eulerian approaches lead to a serious mathematical obstacle known as the “closure problem,” for which no generally valid solution has yet been found. By contrast, the Lagrangian techniques attempt to describe the concentration statistics in terms of the statistical properties of the displacements of groups of particles released in the fluid. The mathematics of this approach is more tractable than that of the Eulerian methods, in that no closure problem is encountered, but the applicability of the resulting equations is limited because of the difficulty of accurately determining the required particle statistics. Moreover, the equations are not directly applicable to problems involving nonlinear chemical reactions. Having demonstrated that exact solution for the mean concentrations hci …x; t†i even of inert species in a turbulent fluid is not possible in general by either the Eulerian or Lagrangian approaches, we now consider what assumptions and approximations can be invoked to obtain practical descriptions of atmospheric diffusion. In Section 18.4 we shall proceed from the two basic equations for hci i, (18.4) and (18.8), to obtain the equations commonly used for atmospheric diffusion. A particularly important aspect is the delineation of the assumptions and limitations inherent in each description.

18.4 EQUATIONS GOVERNING THE MEAN CONCENTRATION OF SPECIES IN TURBULENCE 18.4.1 Eulerian Approaches As we have seen, the Eulerian description of turbulent diffusion leads to the so-called closure problem, as illustrated in (18.4) by the new dependent variables hu´j ci´ i, j ˆ 1, 2, 3, as well as any that might arise in hRi i if nonlinear chemical reactions are occurring. Let us first consider only the case of chemically inert species, that is, Ri ˆ 0. The problem is to deal with the variables hu´j c´ i if we wish not to introduce additional differential equations. The most common means of relating the turbulent fluxes hu´j c´ i to the hci is based on the mixinglength model of Chapter 16. In particular, it is assumed that (summation implied over k) hu´j c´ i ˆ Kjk

@hci @xk

j ˆ 1;2;3

…18:9†

where Kjk is called the eddy diffusivity. Equation (18.9) is called both mixing-length theory and K theory. Since (18.9) is essentially only a definition of the Kjk which are, in general, functions of location and time,

768

ATMOSPHERIC DIFFUSION

we have, by means of (18.9), replaced the three unknowns hu´j c´ i, j ˆ 1,2,3, with the six unknowns Kjk , j; k ˆ 1,2,3 (Kjk ˆ Kkj †. If the coordinate axes coincide with the principal axes of the eddy diffusivity tensor fKjk g, then only the three diagonal elements K11 , K22 , and K33 are nonzero, and (17.9) becomes5

hu´j c´ i ˆ Kjj

@hci @xj

…18:10†

In using (18.4), two other assumptions are ordinarily invoked: 1. Molecular diffusion is negligible compared with turbulent diffusion: D

@ @ 2 hci  hu´ c´ i @xj j @xj @xj

2. The atmosphere is incompressible: j @u ˆ0 @xj With these assumptions and (18.10), (18.4) becomes @hci @hci @ @hci j ˆ Kjj ‡ S…x; t† ‡u @t @xj @xj @xj

…18:11†

This equation is termed the “semiempirical equation of atmospheric diffusion,” or just the atmospheric diffusion equation, and will play an important role in what is to follow. Let us return to the case in which chemical reactions are occurring, for which we refer to (18.4). Since Ri is almost always a nonlinear function of the ci , we have already seen that additional terms of the type hc´i cj´ i will arise from hRi i. The most obvious approximation we can make regarding hRi i is to replace hRi …c1 ; . . . ; cN †i by Ri …hc1 i; . . . ; hcN i†, thereby neglecting the effect of concentration fluctuations on the rate of reaction. Invoking this approximation, as well as those inherent in (18.11), we obtain for each species i, @hci i @hci i @ @hci i j ˆ Kjj ‡ Ri …hc1 i; . . . ; hcN i† ‡ Si …x; t† ‡u @t @xj @xj @xj

5

…18:12†

No summation is implied in this term, for example hu´1 c´ i ˆ K11

@hci @x1

It is beyond our scope to consider the conditions under which fKjk g may be taken as diagonal. For our purposes we note that ordinarily there is insufficient information to assume that fKjk g is not diagonal.

769

EQUATIONS GOVERNING THE MEAN CONCENTRATION

It can be shown that (18.12) is a valid description of turbulent diffusion and chemical reaction as long as the reaction processes are slow compared with turbulent transport and the characteristic lengthscale and timescale for changes in the mean concentration field are large compared with the corresponding scales for turbulent transport.

18.4.2 Lagrangian Approaches We now wish to consider the derivation of usable expressions for hci …x; t†i based on the fundamental Lagrangian expression (18.8). As we have seen, the utility of (18.8) rests on the ability to evaluate the transition probability Q…x; t j x´ ; t´ †. The first question, then, is: “Are there any circumstances under which the form of Q is known?” To attempt to answer this question let us go back to the advection equation, (18.2), for a onedimensional flow with a general source term S…x; t†: @c @ ‡ …uc† ˆ S…x; t† @t @x

…18:13†

Since the velocity u is a random quantity, the concentration c that results from the solution of (18.13) is also random. What we want to do is to solve (18.13) for a particular choice of the velocity u and find the concentration c corresponding to that choice of u. For simplicity, let us assume that u is independent of x and depends only on time t. Thus the velocity u(t) is a random variable depending on time. Since u(t) is a random variable, we need to specify the probability density for u(t). A reasonable assumption for the probability distribution of u(t) is that it is Gaussian pu …u† ˆ

1 …2π†1=2 σu

exp

…u

†2 u

…18:14†

2σ2u

, and the variance is σ2u . In the process of solving (18.13) we will need where the mean value is hu…t†i ˆ u †…u…τ† u †i. We will assume that u…t† is a stationary random process an expression for the term h…u…t† u with the correlation h…u…t†

†…u…τ† u

†i ˆ σ2u exp… b j t u

τj†

…18:15†

This expression states that the maximum correlation between the velocities at two times occurs when those times are equal, and is equal to σ2u . As the time separation between u…t† and u…τ† increases, the correlation decays exponentially with a characteristic decay time of 1=b. Stationarity implies that the statistical properties of u at two different times t and τ depend only on t τ and not on t and τ individually. Let us now solve (18.13) for a time-varying source of strength S…t† at x ˆ 0. The solution, obtained by the method of characteristics, is t

c…x; t† ˆ

δ…x

X…t; τ††S…τ†dτ

…18:16†

0

where X…t; τ† ˆ

t τ

u…t´ †dt´

…18:17†

770

ATMOSPHERIC DIFFUSION

is just the distance a fluid particle travels between times τ and t. Since X…t; τ† is a random variable, so is c…x; t†. We are really interested in the mean, hc…x; t†i. Thus, taking the expected value of (18.16), we obtain t

hc…x; t†i ˆ

hδ…x

X…t; τ††iS…τ†dτ

…18:18†

0

Since the pdf of u…t† is Gaussian, that for X…t; τ† is also Gaussian pX …X; t; τ† ˆ

1 …2π†1=2 σx

 †2 X

…X

exp

…18:19†

2σ2x

 …t; τ† and σ2 …t; τ† are the mean and variance, respectively, of X…t; τ†. The expression for hc…x; t†i where X x can be written in terms of pX as t

hc…x; t†i ˆ

S…t´ †

0 t

ˆ

1 1

X…t; t´ ††pX …X; t; t´ †dX dt´

δ…x

S…t´ †pX …x; t; t´ †dt´

…18:20†

0

By comparing (18.20) with (18.8), we see that pX …x; t; t´ † is precisely Q…x; t j x´ ; t´ †, except that there is  …t; τ† and σ2 …t; τ† based on the definition of X…t; τ† and no dependence on x´ in this case. Let us compute X x the properties of u. We note that  …t; τ† ˆ …t X

 τ†u

…18:21†

and  †2 i X 2 ˆ hX…t; τ†2 i X

σ2x …t; τ† ˆ h…X…t; τ†

…18:22†

Since t

hX…t; τ†2 i ˆ ˆ

τ

u…t´ †dt´

t τ

t τ

t τ

u…t”†dt”

hu…t´ †u…t”†idt´ dt”

…18:23†

using (18.15), (18.21), (18.22), and (18.23), we obtain σ2x …t; τ† ˆ

2σ2u ‰b…t b2

τ† ‡ e

b…t τ†

…18:24†

1Š

so σ2x …t; τ† ˆ σx2 …t τ†. If the source is just a pulse of unit strength at t ˆ 0, that is, S…t† ˆ δ…t†, then (18.20) becomes

hc…x; t†i ˆ

1 …2π†1=2 σx …t†

exp

t†2 …x u 2σ2x …t†

…18:25†

771

SOLUTION OF THE ATMOSPHERIC DIFFUSION EQUATION FOR AN INSTANTANEOUS SOURCE

Thus we have found that the mean concentration of a tracer released in a flow where the velocity is a stationary, Gaussian random process has a distribution that is, itself, Gaussian. This is an important result. t, just the distance a tracer molecule has traveled The mean position of the distribution at time t is u . The variance of the mean concentration distribution, σ2x …t†, is over a time t at the mean fluid velocity u just the variance of X…t;τ†. This result makes sense since X…t;τ† is just the random distance that a fluid particle travels between times τ and t, and this distance is precisely that which a tracer molecule travels. Let us examine the limits of (18.24) for large and small values of t. For large t, that is, t  b 1 , the characteristic time for velocity correlations, (18.24) reduces to 2σ2u t b

…18:26†

σ2x ˆ σu2 t2

…18:27†

σ2x ˆ For t  b 1 , exp… bt†  1

bt ‡ …bt†2 =2, and

Thus we find that the variance of the mean concentration distribution varies with time according to σ2x 

t2 t

small t large t

This example can readily be generalized to three dimensions. If we continue to assume that there is a mean flow only in the x direction, then the expression for the mean concentration resulting from an instantaneous point source of unit strength at the origin is hc…x; y; z; t†i ˆ

1 …2π†

3=2

 exp

σx …t†σy …t†σz …t†  2 …x ut† 2σ2x …t†

y2 2σ2y …t†

z2 2σ2z …t†

…18:28†

where σ2y and σz2 are given by equations analogous to that for σx2 . To summarize again, we have shown through a highly idealized example that the mean concentration in a stationary, homogeneous Gaussian flow field is itself Gaussian. If the turbulence is stationary and homogeneous, the transition probability density Q of a particle depends only on the displacements in time and space and not on where or when the particle was introduced into the flow. Thus, in that case, Q…x; t j x´ ; t´ † ˆ Q…x x´ ; t t´ †. The Gaussian form of Q turns out to play an important role in atmospheric diffusion theory, as we will see.

18.5 SOLUTION OF THE ATMOSPHERIC DIFFUSION EQUATION FOR AN INSTANTANEOUS SOURCE We begin in this section to obtain solutions for atmospheric diffusion problems. Let us consider, as we did in the previous section, an instantaneous point source of strength S at the origin in an infinite fluid  in the x direction. We desire to solve the atmospheric diffusion equation, (18.11), in this with a velocity u situation. Let us assume, for lack of anything better at the moment, that Kxx , Kyy , and Kzz are constant. Then (18.11) becomes @ 2 hci @ 2 hci @hci @hci @ 2 hci  ‡u ˆ Kxx ‡ K ‡ K yy zz @y2 @z2 @t @x @x2

…18:29†

772

ATMOSPHERIC DIFFUSION

to be solved subject to hc…x; y; z; 0†i ˆ Sδ…x†δ…y†δ…z†

…18:30†

hc…x; y; z; t†i ˆ 0

…18:31†

x; y; z ! 1

The solution of (18.29) to (18.31) is given in Appendix 18A.1. It is hc…x; y; z; t† ˆ

S 8…πt†

3=2

…Kxx Kyy Kzz †1=2 t†2 …x u 4Kxx t

 exp

y2 4Kyy t

z2 4Kzz t

…18:32†

Note the similarity of (18.32) and (18.28). In fact, if we define σ2x ˆ 2Kxx t, σy2 ˆ 2Kyy t, and σz2 ˆ 2Kzz t, we note that the two expressions are identical. There is, we conclude, evidently a connection between the Eulerian and Lagrangian approaches embodied in a relation between the variances of spread that arise in a Gaussian distribution and the eddy diffusivities in the atmospheric diffusion equation. We will explore this relationship further as we proceed.

18.6 MEAN CONCENTRATION FROM CONTINUOUS SOURCES We just obtained expression (18.32) for the mean concentration resulting from an instantaneous release of a quantity S of material at the origin in an infinite fluid with stationary, homogeneous turbulence and  in the x direction. We now wish to consider a continuously emitting source under the a mean velocity u same conditions. The source strength is q…g s 1 †.

18.6.1 Lagrangian Approach A continuous source is viewed conceptually as one that began emitting at t ˆ 0 and continues as t ! 1. The mean concentration achieves a steady state, independent of time, and S…x; y; z; t† ˆ qδ…x†δ…y†δ…z†. The basic Lagrangian expression (18.8) becomes t

hc…x; y; z†i ˆ

Q…x; y; z; tj0; 0; 0; t´ †q dt´

…18:33†

0

The steady-state concentration is given by t

hc…x; y; z†i ˆ limhc…x; y; z; t†i ˆ lim t!1

t!1

Q…x; y; z; tj0;0;0; t´ †q dt´

…18:34†

0

The transition probability density Q has the general Gaussian form of (18.28) Q…x; y; z; tj0; 0; 0; t´ † ˆ …2π†

3=2

‰σx …t

 exp

which can be expressed as Q…x; y; z; t

…x

t´ †σy …t

t´ †σz …t

…t t´ ††2 u 2σ2x …t t´ †

t´ †Š

1

y2 2σ2y …t t´ †

z2 2σ2z …t t´ †

…18:35†

t´ j0; 0; 0; 0†. Thus t

hc…x; y; z†i ˆ lim

t!1

0

Q…x; y; z; τj0; 0; 0; 0†q dτ

…18:36†

773

MEAN CONCENTRATION FROM CONTINUOUS SOURCES

Therefore the steady-state concentration resulting from a continuous source is obtained by integrating the unsteady-state concentration over all time from 0 to 1: 1

hc…x; y; z†i ˆ

q …2π†3=2 σx σy σz

0

…x

exp

 2 ut† 2σ2x

y2 2σy2

z2 dt 2σz2

…18:37†

We need to evalulate the integral in (18.37). To do so, we need to specify σx …t†, σy …t†, and σz …t†. For the moment let us assume simply that σx …t† ˆ σy …t† ˆ σz …t† ˆ σ…t†, where we do not specify how σ depends on  2 =2σ2x Š is peaked at t ˆ x=u  and falls off exponentially for longer or t. We note that the term exp ‰…x ut† shorter values of t. Thus the major contribution of this term to the integral comes from values of t close to . Therefore let us perform a Taylor series expansion of x=u  2 ‡ y2 ‡ z2 ut† 2σ2 …t†

…x

G…t† ˆ exp . Using about t ˆ x=u

† ‡ G…t† ˆ G…x=u

dG dt

 tˆx=u

…t

† x=u

we find dG dt

 tˆx=u

ˆ

y2 ‡ z 2 2σ2 …t†

exp

1 dσ 2 …y ‡ z2 † dt

σ3 …t†

 tˆx=u

If we let E ˆ …y2 ‡ z2 †=2σ2 …t†, then dG dt

 tˆx=u

ˆ

exp… E†

2 dσ E σ dt

 tˆx=u

Thus the Taylor series approximation of the exponential is exp

…x

 2 ‡ y2 ‡ z 2 ut† 2σ2 …t†

ˆe

E

1

t

x  u

E

2 dσ σ dt

…18:38†

 tˆx=u

Now assume that the major contribution to the integral of (18.37) comes from values of t in the range x  u

σ x σ a t ‡a    u u u

Substituting (18.38) into (18.37) yields hc…x; y; z†i ˆ

 …x‡aσ†=u

q

3=2 3  …2π† σ …t† …x aσ†=u

e

E

1

t

x  u

2 dσ E σ dt

 tˆx=u

dt

…18:39†

† term as being of order aσ=u  and assume that within the limits of If we now neglect the …t x=u †, we get integration σ…t†  σ…x=u hc…x; y; z†i ˆ

† 2qe E …aσ=u …2π†3=2 σ3

…18:40†

774

ATMOSPHERIC DIFFUSION

To obtain a, if we impose the condition of conservation of mass, that is, that the total flow of material through the plane at any x is q, we obtain a ˆ …π=2†1=2 . Thus (18.40) becomes hc…x; y; z†i ˆ

y2 ‡ z 2 † 2σ2 …x=u

q exp 2 †  2πuσ …x=u

…18:41†

  x=u . Thus the condition for validity of the The assumptions made in deriving (18.41) require that aσ=u result is † σ…x=u 1 x Physically, the mean concentration emanating from a point source is a plume that can be visualized to be composed of many puffs, each of whose concentration distributions is sharply peaked about its centroid at all travel distances. Thus the spread of each puff is small compared to the downwind distance it has traveled. This assumption is called the slender plume approximation. The procedure we have followed to obtain (18.41) can readily be generalized to σx ˆ 6 σy ˆ 6 σz . The result is hc…x; y; z†i ˆ

y2 2σ2y

q exp 2π uσy σz

z2 2σ2z

…18:42†

This equation for the mean concentration from a continuous point source occupies a key position in atmospheric diffusion theory and we will have occasion to refer to it again and again. 18.6.1.1 An Alternate Derivation of (18.42) Our derivation of (18.42) was based on physical reasoning concerning the amount of spreading of a puff compared to its downwind distance. We can obtain (18.42) in a slightly different manner by assuming specific functional forms for σx , σy , and σz . Specifically, let us select the “long-time” form [recall (18.26)]: σ2x ˆ ax t;

σy2 ˆ ay t;

σz2 ˆ az t

…18:43†

Thus it is desired to evaluate 1

hc…x; y; z†i ˆ

q …2π†3=2 …ax ay az †1=2 t

0

…x

exp 3=2

 †2 ut 2ax t

y2 2ay t

z2 dt 2az t

This integral can be expressed as hc…x; y; z†i ˆ

1

q …2π†3=2 …ax ay az †1=2

t

3=2

exp

…r2

0

 2 t2 † 2 uxt ‡ u dt 2ax t

where r2 ˆ x2 ‡ …ax =ay †y2 ‡ …ax =az †z2 . Thus hc…x; y; z†i ˆ Let η ˆ t

1=2

q 3=2

…2π†

1=2

…ax ay az †

1

 x eux=a

t

3=2

0

exp

2 t u r2 ‡ 2ax t 2ax

dt

and hc…x; y; z†i ˆ

2q …2π†3=2 …ax ay az †

 x eux=a 1=2

1

exp 0

2 u r2 η2 ‡ 2ax 2ax η2

dη

775

MEAN CONCENTRATION FROM CONTINUOUS SOURCES

The integral is of the general form 1

e

…aη2 ‡b=η2 †

dη ˆ

0

so finally hc…x; y; z†i ˆ

1 π 2 a

q 2π…ay az †

1=2

r

1=2

e

exp

2…ab†1=2

 u …r ax

x†

…18:44†

is the expression for the mean concentration from a continuous point source of strength q at the origin in an infinite fluid with the variances given by (18.43). If advection dominates plume dispersion so that only the concentration close to the plume centerline is of importance, we will be interested in the solution only for values of x, y, and z that satisfy ax 2 ax 2 y ‡ z az ay 1 x2 which can be viewed as the result of two assumptions, that y2 ‡ z 2 1 x2 and that

ax ˆ O…1† az

ax ˆ O…1†; ay

The latter assumption implies that the variances of the windspeeds are of the same order of magnitude. Since r ˆ xf1 ‡ ‰…ax =ay †y2 ‡ …ax =az †z2 Š=x2 Š1=2

…18:45†

using …1 ‡ ζ†p ˆ 1 ‡ ζ p ‡


, (18.45) can be approximated by6 rˆx 1‡ 6

…ax =ay †y2 ‡ …ax =az †z2 2x2

The expression …1 ‡ ζ†p ˆ 1 ‡

p ζ‡ 1

p 2 ζ ‡


2

is valid for all p if 1 < ζ < 1. Thus 1 1 …1 ‡ ζ†1=2 ˆ 1 ‡ ζ ‡ ζ2 ‡


2 8 for 1 < ζ < 1. Note that p ν where for noninteger p; p! ˆ Γ…p ‡ 1†.

ˆ

p! ν!…p ν†!

…18:46†

776

ATMOSPHERIC DIFFUSION

In (18.44) we approximate r by x and r

x by the expression above. Thus (18.44) becomes q

hc…x; y; z†i ˆ

2π…ay az †

1=2

x

 u 2ax x

exp

ax 2 ax 2 y ‡ z az ay

…18:47†

If we relate time and distance from the source x by t ˆ x= u, then we can use (18.43) to write σ2y ˆ

ay x ;  u

σ2z ˆ

az x  u

…18:48†

Then (18.47) becomes identical to (18.42). 18.6.1.2

Still Another Derivation of (18.42)

We can use the relation 1

lim

σ!0 …2π†1=2 σ

…x

exp

x´ † 2 2σ2

ˆ δ…x

x´ †

…18:49†

t, we get to take the limit of the x term in (18.38) as σx ! 0. Using (18.49) and letting ξ ˆ u  ξ=u

hc…x; y; z†i ˆ lim

ξ!1

0

q exp σy σz 2πu

y2 2σ2y

z2 δ…x 2σ2z

ξ†dξ

. This last formula can be simplified to give (18.42). This where σy and σz are now functions of ξ=u derivation is the shortest, but we delayed it after those that more clearly illustrate the physical assumptions.

18.6.2 Eulerian Approach The problem of determining the concentration distribution resulting from a continuous source of  in the x direction can be formulated strength q at the origin in an infinite isotropic fluid with a velocity u by the Eulerian approach as follows:  u

@hci @ 2 hci @ 2 hci @ 2 hci ˆK ‡ ‡ ‡ qδ…x†δ…y†δ…z† @x @x2 @y2 @z2 hc…x; y; z†i ˆ 0

x; y; z ! 1

…18:50† …18:51†

The solution of (18.50) and (18.51) is given in Appendix 18A.2. It is hc…x; y; z†i ˆ

q exp 4πKr

…r x† u 2K

…18:52†

where r2 ˆ x2 ‡ y2 ‡ z2 , or rˆx 1‡

y2 ‡ z 2 x2

1=2

…18:53†

If we invoke the slender plume approximation, we are interested only in the solution close to the plume centerline. Thus, as in (18.45), (18.53) can be approximated by rx 1‡

y2 ‡ z 2 2x2

…18:54†

777

MEAN CONCENTRATION FROM CONTINUOUS SOURCES

x by …y2 ‡ z2 †=2x, (18.52) becomes

If in (18.52) r is approximated by x and r hc…x; y; z†i ˆ

18.6.2.1

q exp 4πKx

 u …y2 ‡ z2 † 4Kx

…18:55†

An Alternate Derivation of (18.55)

Equation (18.55) is based on the slender plume approximation as expressed by (18.54). We will now show that the slender plume approximation is equivalent to neglecting diffusion in the direction of the mean flow in the atmospheric diffusion equation. Thus hci is governed by  u

@hci @ 2 hci @ 2 hci ‡ qδ…x†δ…y†δ…z† ‡ ˆK @x @y2 @z2

hc…0; y; z†i ˆ 0 hc…x; y; z†i ˆ 0

…18:56† …18:57† …18:58†

y; z ! 1

Before we proceed to solve (18.56), a few comments about the boundary conditions are useful. When the x diffusion term is dropped in the atmospheric diffusion equation, the equation becomes firstorder in x, and the natural point for the single boundary condition on x is at x ˆ 0. Since the source is also at x ˆ 0 we have an option of whether to place the source on the RHS of the equation, as in (18.56), or in the x ˆ 0 boundary condition. If we follow the latter course, then the x ˆ 0 boundary condition is obtained by equating material fluxes across the plane at x ˆ 0. The result is @hci @ 2 hci @ 2 hci ˆK ‡ @y2 @z2 @x q hc…0; y; z†i ˆ δ…y†δ…z†  u hc…x; y; z†i ˆ 0 y; z ! 1  u

…18:59† …18:60† …18:61†

The sets of (18.56) to (18.58) and (18.59) to (18.61) are entirely equivalent. We present the solution of (18.59) in Appendix 18A.3. The solution is hc…x; y; z†i ˆ

q exp 4πKx

 u …y2 ‡ z2 † 4Kx

…18:62†

If we allow Kyy to be different from Kzz , the analogous result is hc…x; y; z†i ˆ

q 4π…Kyy Kzz †

1=2

x

exp

 y2 u z2 ‡ 4x Kyy Kzz

…18:63†

18.6.3 Summary of Continuous Point Source Solutions Table 18.1 presents a summary of the solutions obtained in this section. Of primary interest at this point is a comparison of the forms of the Lagrangian and Eulerian expressions, in particular, the relationships between eddy diffusivities and the plume dispersion variances. For the slender plume cases, for example, the Lagrangian and Eulerian expressions are identical if σ2y ˆ

2Kyy x  u

σ2z ˆ

2Kzz x  u

…18:64†

In most applications of the Lagrangian formulas, the dependences of σ2y and σz2 on x are determined empirically rather than as indicated in (18.64). Thus the main purpose of the formulas in Table 18.1 is to provide a comparison between the two approaches to atmospheric diffusion theory.

778

ATMOSPHERIC DIFFUSION

TABLE 18.1 Expressions for the Mean Concentration from a Continuous Point Source in an Infinite Fluid in Stationary, Homogeneous Turbulence Approach Lagrangian σ2x ˆ ax t σ2y ˆ ay t σ2z ˆ az t

Solution Employing Slender Plume Approximation

Full Solution q 2π…ay az †1=2 r

exp

 u …r ax

q exp  σy σ z 2πu

x†

y2 z2 ‡ 2 2 2σy 2σz

r2 ˆ x2 ‡ …ax =ay †y2 ‡ …ax =az †z2

Eulerian

q ‡ Kxx Kzz

4π…Kyy Kzz

x2

 exp

 u 2Kxx

q y2

‡ Kxx Kyy

z2 †1=2

x2 y2 z2 ‡ ‡ Kxx Kyy Kzz

4π…Kyy Kzz †1=2 x

1=2

x

 exp

 y2 z2 u ‡ 4x Kyy Kzz

18.7 STATISTICAL THEORY OF TURBULENT DIFFUSION Up to this point in this chapter we have developed the common theories of turbulent diffusion in a purely formal manner. We have done this so that the relationship of the approximate models for turbulent diffusion, such as the K theory and the Gaussian formulas, to the basic underlying theory is clearly evident. When such relationships are clear, the limitations inherent in each model can be appreciated. We have in a few cases applied the models obtained to the prediction of the mean concentration resulting from an instantaneous or continuous source in idealized stationary, homogeneous turbulence. In Section 18.7.1 we explore further the physical processes responsible for the dispersion of a puff or plume of material. Section 18.7.2 can be omitted on a first reading of this chapter; that section goes more deeply into the statistical properties of atmospheric dispersion, such as the variances σ2i …t†, which are needed in the actual use of the Gaussian dispersion formulas.

18.7.1 Qualitative Features of Atmospheric Diffusion The two idealized source types commonly used in atmospheric turbulent diffusion are the instantaneous point source and the continuous point source. An instantaneous point source is the conventional approximation to a rapid release of a quantity of material. Obviously, an “instantaneous point” is a mathematical idealization since any rapid release has finite spatial dimensions. As the puff is carried away from its source by the wind, it will disperse under the action of turbulent velocity fluctuations. Figure 18.1 shows the dispersion of a puff under three different turbulence conditions. Figure 18.1a shows a puff embedded in a turbulent field in which all the turbulent eddies are smaller than the puff. The puff will disperse uniformly as the turbulent eddies at its boundary entrain fresh air. In Figure 18.1b, a puff is embedded in a turbulent field all of whose eddies are considerably larger than the puff. In this case the puff will appear to the turbulent field as a small patch of fluid that will be convected through the field with little dilution. Ultimately, molecular diffusion will dissipate the puff. Figure 18.1c shows a puff in a turbulent field of eddies of size comparable to the puff. In this case the puff will be both dispersed and distorted. In the atmosphere, a cloud of material is always dispersed since there are almost always eddies of size smaller than the cloud. From Figure 18.1 we can see that the dispersion of a puff relative to its center of mass depends on the initial size of the puff relative to the lengthscales of the turbulence. In order to describe such relative dispersion, we must consider the

STATISTICAL THEORY OF TURBULENT DIFFUSION

FIGURE 18.1 Dispersion of a puff of material under three turbulence conditions: (a) puff embedded in a field in which the turbulent eddies are smaller than the puff, (b) puff embedded in a field in which the turbulent eddies are larger than the puff, and (c) puff embedded in a field in which the turbulent eddies are comparable in size to the puff.

statistics of the separation of two representative fluid particles in the puff. The analysis of the wandering of a single particle is insufficient to tell us about the dispersion of a cloud (Csanady 1973). A continuous source emits a plume that might be envisioned, as we have noted, as an infinite number of puffs released sequentially with an infinitesimal time interval between them. The quantity of material released is expressed in terms of a rate, say, grams per minute. The dimensions of a plume perpendicular to the plume axis are generally given in terms of the standard deviation of the mean concentration distribution since the mean cross-sectional distributions are often nearly Gaussian. Figure 18.2 shows the plume “boundaries” and concentration distributions as might be seen in an instantaneous snapshot and exposures of a few minutes and several hours. An instantaneous picture of a plume reveals a meandering behavior with the width of the plume gradually growing downwind of the source. Longer-time averages give a more regular appearance to the plume and a smoother concentration distribution. If we were to take a time exposure of the plume at large distances from the source, we would find that the boundaries of the time-averaged plume would begin to meander, because the plume would come under the influences of larger and larger eddies, and the averaging time, say, several hours, would still be too brief to time average adequately the effect of these larger eddies. Eddies larger in size than the plume dimension tend to transport the plume intact, whereas those that are smaller tend to disperse it. As the plume becomes wider, larger and larger eddies become effective in dispersing the plume and the smaller eddies become increasingly ineffective.

779

780

ATMOSPHERIC DIFFUSION

FIGURE 18.2 Plume boundaries and concentration distributions of a plume at different averaging times.

The theoretical analysis of the spread of a plume from a continuous point source can be achieved by considering the statistics of the diffusion of a single fluid particle relative to a fixed axis. The actual plume would then consist of a very large number of such identical particles, the average over the behavior of which yields the ensemble statistics of the plume.

18.7.2 Motion of a Single Particle Relative to a Fixed Axis Let us consider, as shown in Figure 18.3, a single particle that is at position x0 at time t0 and is at position x at some later time t in a turbulent field. The complete statistical properties of the particle’s motion are embodied in the transition probability density Q…x; t j x´ ; t´ †. An analysis of this problem for stationary, homogeneous turbulence was presented by Taylor (1921) in one of the classic papers in the field of turbulence. If the turbulence is stationary and homogeneous, Q…x; t j x´ ; t´ † ˆ Q…x x´ ; t t´ †; that is, Q depends only on the displacements in space and time and not on the initial position or time. The single particle may be envisioned as one of a very large number of particles that are emitted sequentially from a source located at x0 . The distribution of the concentration of marked particles in the fluid is known once the statistical behavior of one representative marked particle is known. For convenience we assume that the particle is released at the origin at t ˆ 0, so that its displacement corresponds to its coordinate location at time t. The most important statistical quantity is the mean-square displacement of the particle from the source after a time t, since the mean displacement from the axis parallel to the flow direction will be zero. If we envision this particle as one being emitted from a continuous source, we can see that the meansquare displacement of the particle from the axis of the plume will tell us the width of the plume and hence the variances σ2i …t†.

FIGURE 18.3 Motion of a single marked particle in a turbulent flow.

781

STATISTICAL THEORY OF TURBULENT DIFFUSION

The mean displacement of the particle along the ith coordinate is defined by t

hXi …t†i ˆ

xi Q…x; t†dx

…18:65†

0

where the braces (hi) indicate an ensemble average over an infinite number of identical marked particles. If the velocity of the particle in the ith direction at any time is vi …t†, the position of the particle at time t is given by [recall (18.17)] t

Xi …t† ˆ

vi …t´ †dt´

…18:66†

0

where the velocity of the particle at any instant is equal, by definition, to the fluid velocity at the spot where the particle happens to be at that instant: vi …t† ˆ ui ‰X…t†; tŠ

…18:67†

Let us consider a situation in which there is no mean velocity, so that vi …t† ˆ u´i ‰X…t†; tŠ. If there is a mean velocity, say, in the xi direction, we will be interested in the dispersion about a point moving with the mean velocity. Therefore the influence of the fluctuating Eulerian velocities on the wanderings of the marked particle from its axis is the key issue here, not its translation in the mean flow. We might also note that, in discussing the statistics of particle motion, all averages are conceptually ensemble averages, carried out over a very large number of similar particle releases. For this reason, we denote mean quantities by braces, as in (18.65), as opposed to overbars, which have been reserved for time averages. It is understood, however, that when discussing mean properties of the velocity field itself, due to the condition of stationarity, the ensemble average and the time average are identical. The mean displacement can also be computed by ensemble averaging of (18.66), t

hXi …t†i ˆ

hvi …t´ †i dt´

…18:68†

0

where the averaging can be taken inside the integral. The variance of the displacements is the expected value of the product of Xi and Xj , which is expressed in terms of Q by Pij …t† ˆ hXi …t†Xj …t†i ˆ

1 1

xi xj Q…x; t†dx

…18:69†

The diagonal elements hXi2 …t†i are of principal interest since they describe the rate of spreading along each axis. Pii …t† is just σ2i …t†. Using (18.66) in (18.69), we obtain [recall (18.23)] t

Pij …t† ˆ 0

t

0

vj …t”†dt”

0

t

ˆ

t

vi …t´ †dt´

hvi …t´ †vj …t”†i dt´ dt”

…18:70†

0

The integrand of (18.70) is defined as the Lagrangian correlation function Rij …t´ be rewritten t

Pij …t† ˆ 0

t 0

t

ˆ 0

0

Rij …t´

t t´

tӠ, so that (18.70) may

t”† dt´ dt”

Rij …ζ† dζ dt´

…18:71†

782

ATMOSPHERIC DIFFUSION

By definition Rij …ζ† ˆ Rji … ζ†, so that t

Pij …t† ˆ

…t

憉Rij …ζ† ‡ Rji …ζ†Šdζ

…18:72†

0

We now consider the form of Pij …t† in the two limiting situations, t ! 0 and t ! 1. First, for t ! 0, we have Rij …ζ†  Rij …0† ˆ hu´i uj´ i ˆ u´i u´j . Thus Pij …t† ˆ u´i u´j t2

t!0

…18:73†

and the dispersion increases as t2 . Next, as t ! 1 we expect the Lagrangian correlation function Rij …t† to approach zero as the motion of the particle becomes uncorrelated with its original velocity. We expect convergence of the following integral 1

‰Rij …ζ† ‡ Rji …ζ†Š dζ ˆ Iij

0

where Iij is proportional to the Lagrangian timescale of the turbulence. Defining 1

ζ‰Rij …ζ† ‡ Rji …ζ†Š dζ ˆ Jij

0

as t ! 1, Pij …t† becomes proportional to t: Pij …t† ˆ Iij t

…18:74†

Jij

We can obtain these results by a slightly different route. Let us, for example, compute the rate of change of the dispersion hXi2 …t†i: d 2 hX …t†i ˆ 2hXi …t†vi …t†i dt i t

ˆ 2 vi …t†

vi …t´ † dt´

0 t

ˆ2

hvi …t†vi …t´ †i dt´

0 t

ˆ2

Rii …t

t´ † dt´

…18:75†

0

Integration of (18.75) with respect to t gives (18.71). In summary, we have found that mean-square dispersion of a particle in stationary, homogeneous turbulence has the following dependence on time Pii …t† ˆ

u2i t2 2Kii t

t!0 t!1

…18:76†

t´ † dt´

…18:77†

as we had anticipated by (18.26) and (18.27), where t

Kii ˆ lim

t!1

Rii …t

0

Since ui´2 is proportional to the total turbulent kinetic energy, the total energy of the turbulence is important in the early dispersion. After long times the largest eddies will contribute to Rii and Rii will

783

SUMMARY OF ATMOSPHERIC DIFFUSION THEORIES

not go to zero until the particle can escape the influence of the largest eddies. From its definition, Kii has the dimensions of a diffusivity, since as t ! 1 1 dhXi2 …t†i ˆ Kii 2 dt

…18:78†

Now, if we return to Section 18.5 we see that this Kii is precisely the constant eddy diffusivity used in the atmospheric diffusion equation for stationary, homogeneous turbulence. Since the Lagrangian timescale is defined by 1

1

TL ˆ max

´

ui2

i

Rii …t† dt

…18:79†

0

it becomes clear that K theory, with constant K values, should apply only when the diffusion time t is much greater than TL . Because of large atmospheric eddies, TL might be quite large. In general, large eddies dominate atmospheric diffusion when diffusion is measured relative to a fixed coordinate system. It has been recognized that the simple exponential function, exp … t=TL †, where t is travel time from the source, appears to approximate Rii …t† rather well (Neumann 1978; Tennekes 1979). If Rii …t† ˆ u2i exp… t=TL †

…18:80†

then the mean-square particle displacement is given by hXi2 …t†i ˆ 2u2i

t

…t

t´ †e

t´ =TL

dt´

0

ˆ 2u2i TL2

t TL

1

e

t=TL

…18:81†

This result is identical to (18.24) if b ˆ TL 1 , which provides a nice connection between the statistical theory of turbulent diffusion and the basic example considered earlier.

18.8 SUMMARY OF ATMOSPHERIC DIFFUSION THEORIES Turbulent diffusion is concerned with the behavior of individual particles that are supposed to faithfully follow the airflow or, in principle, are simply marked minute elements of the air itself. Because of the inherently random character of atmospheric motions, one can never predict with certainty the distribution of concentration of marked particles emitted from a source. Although the basic equations describing turbulent diffusion are available, there does not exist a single mathematical model that can be used as a practical means of computing atmospheric concentrations over all ranges of conditions. There are two basic ways of considering the problem of turbulent diffusion: the so-called Eulerian and Lagrangian approaches. The Eulerian method is based on carrying out a material balance over an infinitesimal region fixed in space, whereas the Lagrangian approach is based on considering the meandering of marked fluid particles in the flow. Each approach can be shown to have certain inherent difficulties that render impossible an exact solution for the mean concentration of particles in turbulent flow. For the purposes of practical computation, approximate theories have been used for calculating mean concentrations of species in turbulence. The two most prominent are the K theory, based on the atmospheric diffusion equation, and the statistical theory, based on the behavior of individual particles in stationary, homogeneous turbulence. The basic issues of interest with respect to the K theory are (1) under what conditions on the source configuration and the turbulent field can this theory be applied and (2) to what extent can the eddy diffusivities be specified in an a priori manner from measured properties of the turbulence. In summary,

784

ATMOSPHERIC DIFFUSION

the spatial and temporal scales of the turbulence should be small in comparison with the corresponding scales of the concentration field. The statistical theory is concerned with the actual velocities of individual particles in stationary, homogeneous turbulence. Under this assumption the statistics of the motion of one typical particle provides a statistical estimate of the behavior of all particles and that of two particles, an estimate of the behavior of a cluster of particles. In the atmosphere one may expect the crosswind component …v† of turbulence to be nearly homogeneous since the variations in the scale and intensity of v with height are often small. On the other hand, the vertical velocity component (w) is decidedly inhomogeneous, since characteristically w increases with height above the ground. Thus the statistical theory should be suitable for describing the spread of a plume in the crosswind direction regardless of the height but for vertical spread only in the early stages of travel from a source considerably elevated above the ground. The deciding factor in judging the validity of a theory for atmospheric diffusion is the comparison of its predictions with actual data. It must be kept in mind, however, that the theories we have discussed are based on predicting the ensemble mean concentration hci, whereas a single experimental observation constitutes only one sample from the hypothetically infinite ensemble of observations from that identical experiment. Thus it is not to be expected that any one realization should agree precisely with the predicted mean concentration even if the theory used is applicable to the set of conditions under which the experiment has been carried out. Nevertheless, because it is practically impossible to repeat an experiment more than a few times under identical conditions in the atmosphere, one must be content with at most a few experimental realizations when testing any available theory.

18.9 ANALYTICAL SOLUTIONS FOR ATMOSPHERIC DIFFUSION: THE GAUSSIAN PLUME EQUATION AND OTHERS 18.9.1 Gaussian Concentration Distributions We have seen that under certain idealized conditions the mean concentration of a species emitted from a point source has a Gaussian distribution. This fact, although strictly true only in the case of stationary, homogeneous turbulence, serves as the basis for a large class of atmospheric diffusion formulas in common use. The collection of Gaussian-based formulas is sufficiently important in practical application that we devote a portion of this chapter to them. The focus of these formulas is the expression for the mean concentration of a species emitted from a continuous, elevated point source, the so-called Gaussian plume equation. The basic Lagrangian expression for the mean concentration is (18.8) hc…x; y; z; t†i ˆ

1

1

1

1

1

1

Q…x; y; z; t j x0 ; y0 ; z0 ; t0 †hc…x0 ; y0 ; z0 ; t0 †i

 dx0 dy0 dz0 ‡

1

1

1

1

1 ´

1 ´

 S…x´ ; y´ ; z´ ; t´ †dt´ dx´ dy dz

t

Q…x; y; z; t j x´ ; y´ ; z´ ; t´ †

t0

…18:82†

The transition probability density Q expresses physically the probability that a tracer particle that is at x´ ;y´ ;z´ at t´ will be at x;y;z at t. We showed that under conditions of stationary, homogenous turbulence Q has a Gaussian form. For example, in the case of a mean wind directed along the x axis, that is,  ˆ 0, and an infinite domain, Q is v ˆ w Q…x; y; z; t j x´ ; y´ ; z´ ; t´ † ˆ

1 …2π†3=2 σx σy σz

exp

…x

x´

…t u 2σx2

t´ ††2

…y

y´ † 2 2σy2

…z

z´ † 2 2σz2

…18:83†

785

ANALYTICAL SOLUTIONS FOR ATMOSPHERIC DIFFUSION

where the variances σ2x , σy2 , and σ2z are functions of the travel time, t t´ . Up to this point we have considered an infinite domain. For atmospheric applications a boundary at z ˆ 0, the earth, is present. Because of the barrier to diffusion at z ˆ 0 it is necessary to modify the z dependence of Q to account for this fact. We can separate out the z dependence in (18.83) by writing Q…x; y; z; t j x´ ; y´ ; z´ ; t´ † ˆ

x´

…x

1 exp 2πσx σy

t´ ††2

…t u 2σ2x

…y

y´ †2 Qz …z; tjz´ ; t´ † 2σ2y

…18:84†

To determine the form of Qz …z; tjz´ ; t´ †, we enumerate the following possibilities: 1. Form of upper boundary (a) 0  z  1 (b) 0  z  H with no diffusion across z ˆ H (i.e., inversion layer) 2. Type of interaction between the diffusing material and the surface (a) Total reflection (b) Total absorption (c) Partial absorption For the moment let us continue to consider the vertical domain to be 0  z  1. 18.9.1.1

Total Reflection at z = 0

We assume that the presence of a reflecting surface at z ˆ 0 can be accounted for by adding the concentration resulting from a hypothetical source at z ˆ z´ to that from the source at z ˆ z´ in the region z  0. Then Qz assumes the form Qz …z; t j z´ ; t´ † ˆ

18.9.1.2

1 …2π†

1=2

σz

…z

exp

z´ † 2 2σz2

‡ exp

…z ‡ z´ †2 2σz2

…18:85†

Total Absorption at z = 0

If the earth is a perfect absorber, the concentration of material at z ˆ 0 is zero. The form of Qz can be obtained by the same method of an image source at z´ , with the change that we now subtract the distribution from the source at z´ from that for the source at ‡z´ . The result is Qz …z; t j z´ ; t´ † ˆ

1 …2π†1=2 σz

…z

exp

z´ † 2 2σz2

exp

…z ‡ z´ †2 2σz2

…18:86†

The case of partial absorption at z ˆ 0 cannot be treated by the same image source approach since some particles are reflected and some are absorbed. We will consider this case shortly. We now turn to the case of a continuous source. The mean concentration from a continuous point source of strength q at height h above the (totally reflecting) earth is given by (it is conventional to let h denote the source height, and we do so henceforth) t

hc…x; y; z†i ˆ lim

t!1

0

 exp

q …2π†

3=2

…z

σx σy σz h†2

2σ2z

exp

‡ exp

 ´ †2 ut 2σ2x

y2 2σ2y

…z ‡ h†2 2σ2z

dt´

…x

…18:87†

786

ATMOSPHERIC DIFFUSION

As usual, we will be interested in the slender plume case, so we evaluate the integral in the limit of σx ! 0. [Recall (18.49).] The result is

hc…x; y; z†i ˆ

y2 2σ2y

q exp 2π uσy σz  exp

h†2

…z

2σ2z

‡ exp

…z ‡ h†2 2σ2z

exp

…z ‡ h†2 2σ2z

…18:88†

the Gaussian plume equation. For a totally absorbing surface at z ˆ 0, we obtain

hc…x; y; z†i ˆ

y2 2σ2y

q exp σy σz 2πu  exp

h†2

…z

2σz2

…18:89†

18.9.2 Derivation of the Gaussian Plume Equation as a Solution of the Atmospheric Diffusion Equation We saw that by assuming constant eddy diffusivities Kxx , Kyy , and Kzz , the solution of the atmospheric diffusion equation has a Gaussian form. Thus it should be possible to obtain (18.88) or (18.89) as a solution of an appropriate form of the atmospheric diffusion equation. More importantly, because of the facility in specifying different physical situations in the boundary conditions for the atmospheric diffusion equation, we want to include those situations that we were unable to handle easily in Section 18.9.1, namely, the existence of an inversion layer at height H and partial absorption at the surface. Readers not concerned with the details of this solution may skip directly to Section 18.9.3. Let us begin with the atmospheric diffusion equation with eddy diffusivities that are functions of time: @ 2 hci @ 2 hci @ 2 hci @hci @hci  ‡ K ‡ K ‡ S…x;y;z;t† ‡u ˆ Kxx yy zz @y2 @z2 @x2 @t @x hc…x;y;z;0†i ˆ 0 hc…x;y;z;t†i ˆ 0

…18:90†

x; y ! 1

For the boundary conditions on z we assume that an impermeable barrier exists at z ˆ H: @hci ˆ0 @z

zˆH

…18:91†

The case of an unbounded region z  0 is simply obtained by letting H ! 1. To include the case of partial absorption at the surface, we write the z ˆ 0 boundary condition as Kzz

@hci ˆ vd hci @z

zˆ0

…18:92†

787

ANALYTICAL SOLUTIONS FOR ATMOSPHERIC DIFFUSION

where vd is a parameter that is proportional to the degree of absorptivity of the surface, the so-called deposition velocity. We will study the properties and specification of vd in Chapter 19; for now let us treat it merely as a parameter. For total reflection, vd ˆ 0, and for total absorption, vd ˆ 1. 18.9.2.1

Solution of (18.90) to (18.92)

The solution of (18.90) can be expressed in terms of the Green’s function G…x;y;z;t j x´ ;y´ ;z´ ;t´ † as H

hc…x;y;z;t†i ˆ 0

1

1

1

1

t

G…x;y;z;t j x´ ;y´ ;z´ ;t´ †

0

 S…x´ ;y´ ;z´ ;t´ † dt´ dx´ dy´ dz´

…18:93†

where (18.93) is identical to (18.82) and where G satisfies @G @G @2G @2G @2G  ‡u ˆ Kxx 2 ‡ Kyy 2 ‡ Kzz 2 @y @z @t @x @x G…x;y;z;t´ j x´ ;y´ ;z´ ;t´ † ˆ δ…x Gˆ0 @G ˆ βG @z @G ˆ0 @z

x´ †δ…y

y´ †δ…z

…18:94†

z´ †

x; y ! 1 zˆ0 zˆH

where ⠈ vd =Kzz . Physically, G represents the mean concentration at …x;y;z† at time t resulting from a unit source at …x´ ;y´ ;z´ † at time t´ . First we remove the convection term by the coordinate transformation, …t t´ †, which converts (18.94) to ξˆx u @2G @2G @G @2G ˆ Kxx 2 ‡ Kyy 2 ‡ Kzz 2 @t @y @z @ξ To obtain G, we let G…ξ;y;z;t j ξ´ ;y´ ;z´ ;t´ † ˆ A…ξ;y;t j ξ´ ;y´ ;t´ †B…z;t j z´ ;t´ † where @A @2A @2A ˆ Kxx 2 ‡ Kyy 2 @y @t @ξ A…ξ;y;t´ j ξ´ ;y´ ;t´ † ˆ δ…ξ

ξ´ †δ…y

A…ξ;y;t j ξ´ ;y´ ;t´ † ˆ 0

…18:95†

y´ †

ξ; y ! 1

@B @ B ˆ Kzz 2 @t @z B…z;t´ j z´ ;t´ † ˆ δ…z z´ † @B ˆ βB zˆ0 @z @B ˆ0 zˆH @z 2

…18:96†

788

ATMOSPHERIC DIFFUSION

We begin with the solution of (18.95). Using separation of variables, A…ξ;y;t j ξ´ ;y´ ;t´ † ˆ Ax …ξ;t j ξ´ ;t´ †Ay …y;t j y´ ;t´ †. The solutions are symmetric Ax …ξ;t j ξ´ ;t´ † ˆ Ay …y;t j y´ ;t´ † ˆ

exp

…ξ ξ´ †2  xx 4K

exp 1=2

…y y´ †2  yy 4K

a  xx † …K b  yy † …K

1=2

where  ii ˆ K

t t´

Kii …τ

t´ †dτ

Thus A…ξ;y;t j ξ´ ;y´ ;t´ † ˆ

…ξ ξ´ †2  xx 4K

a´ exp  xx K  yy †1=2 …K

…y y´ †2  yy 4K

We determine a´ from the initial condition A…ξ;y;t´ j ξ´ ;y´ ;t´ † ˆ δ…ξ

ξ´ †δ…y

y´ †

or 1

a´  xx K  yy †1=2 …K

1

1

ˆ

1

exp

1

1

1

1

δ…ξ

…ξ ξ´ †2 exp  xx 4K ξ´ †δ…y

…y y´ †2 dξ dy  yy 4K

y´ †dξ dy

which reduces to a´ ˆ 1=4π. Thus A…ξ;y;t j ξ´ ;y´ ;t´ † ˆ

1 exp  xx K  yy †1=2 4π…K

…ξ ξ´ †2  xx 4K

…y y´ †2  yy 4K

…18:97†

Now we proceed to solve (18.96) to obtain B…z;t j z´ ;t´ † ˆ 2

1

…λ2n ‡ β2 † cos‰λn …H z´ †Š cos‰λn …H H…λn2 ‡ β2 † ‡ β nˆ1

z†Š

 zz † exp… λ2n K

where the λn are the roots of λn tan …λn H† ˆ β In the case of a perfectly reflecting surface, ⠈ 0, and B…z;t j z´ ;t´ † ˆ

2 1‡ H

1 nˆ2

cos‰λn …H

z´ †Š cos‰λn …H

 zz Š z†Š exp‰ λ2n K

…18:98†

789

ANALYTICAL SOLUTIONS FOR ATMOSPHERIC DIFFUSION

where sin (λn H† ˆ 0. This result can be simplified somewhat to B…z;t j z´ ;t´ † ˆ

2 H

1

1‡

nπz nπz´ cos exp H H

cos nˆ1

nπ 2  Kzz H

…18:99†

The desired solution for G…x;y;z;t j x´ ;y´ ;z´ ;t´ † is obtained by combining the expressions for A…ξ;y;t j ξ´ ;y´ ;t´ † and B…z;t j z´ ;t´ †. In the case of a totally reflecting earth, we have 1 G…ξ;y;z;t j ξ´ ;y´ ;z´ ;t´ † ˆ   yy †1=2 2πH…Kxx K 

1‡

1

nπz nπz´ cos exp H H

cos nˆ1

…ξ ξ´ †2  xx 4K

 exp

nπ 2  Kzz H

…y y´ †2  yy 4K

…18:100†

As usual, we are interested in neglecting diffusion in the x direction as compared with convection, that is, the slender plume approximation. We could return to the original problem, neglecting the term  xx ! 0. To do so @ 2 hci=@x2 in (18.90) and repeat the solution. We can also work with (18.100) and let K 2  return to (18.97), and let σx ˆ 2Kxx . Using (18.49) to take the limit of the x term as σx ! 0: G…x;y;z;t j x´ ;y´ ;z´ ;t´ † ˆ

1

1‡

cos nˆ1



nπz nπz´ cos exp H H …y y´ †2 δ…x  yy 4K

1 exp  2…πKyy †1=2

nπ 2  Kzz H x´

…t u

t´ ††

…18:101†

Now we consider a continuous point source of strength q at (0,0,h). The continuous source solution is obtained from the unsteady solution from t!1

1

H

hc…x;y;z†i ˆ lim

1

1

0

1

 qδ…x´ †δ…y´ †δ…z´

t

G…x;y;z;t j x´ ;y´ ;z´ ;t´ †

0

h†dt´ dx´ dy´ dz´

The solution is illustrated for the case of a totally reflecting earth. Using (18.101) and carrying out the integration, we obtain t

hc…x;y;z†i ˆ q

1‡

0

1

cos nˆ1

1 exp  yy †1=2 2…πK



nπz nπh exp cos H H y2  yy δ…x 4K

…t u

nπ 2  Kzz H

t´ ††dt´

Evaluating the integral yields hc…x;y;z†i ˆ

q  yy †1=2 Hu  …πK  exp

y2  yy 4K

1‡

1

cos nˆ1

nπz nπh exp cos H H

nπ 2  Kzz H …18:102†

790

ATMOSPHERIC DIFFUSION

This is the expression for the steady-state concentration resulting from a continuous point source located at (0,0, h) between impermeable, nonabsorbing boundaries separated by a distance H when diffusion in the direction of the mean flow is neglected. Finally, we wish to obtain the result when H ! 1, that is, only one bounding surface at a distance h from the source. Let 2

Kyy ˆ

1 dσy 2 dt

1 dσ2z 2 dt

Kzz ˆ

and (18.102) can be expressed as hc…x;y;z†i ˆ

2q  σy H …2π†1=2 u

1‡

1

cos nˆ1

nπ 2 σ2z H 2

nπh exp H

 cos

nπz H exp

y2 2σ2y

Now let H ! 1: hc…x;y;z†i ˆ

2q  σy …2π†1=2 u

exp

1



y2 2σ2y

πz πh cos exp H H

cos 0

…πσx †2 d…1=H† 2H 2

Now cos

πz πh cos H H

ˆ

1 π…z ‡ h† π…z h† ‡ cos cos 2 H H

and hc…x;y;z†i ˆ

q …2π†

1=2

 σy u

1



cos 0 1

‡

cos 0

exp

y2 2σ2y

π…z ‡ h† exp H π…z h† exp H

…πσz †2 d…1=H† 2H 2 …πσz †2 d…1=H† 2H2

The integrals can be evaluated7 to produce (18.88). 7

Note that 1

e 0

αx2

cos βx dx ˆ

1 π 2 α

1=2

exp

β2 4α

791

DISPERSION PARAMETERS IN GAUSSIAN MODELS

18.9.3 Summary of Gaussian Point Source Diffusion Formulas The various point source diffusion formulas we have derived are summarized in Table 18.2.

18.10 DISPERSION PARAMETERS IN GAUSSIAN MODELS We have derived several Gaussian-based models for estimating the mean concentration resulting from point source releases of material. We have noted that the conditions under which the equation is valid are highly idealized. Because of its simplicity, however, the Gaussian plume equation has been applied widely (U.S. Environmental Protection Agency 1980). The justification for these applications is that the dispersion parameters σy and σz used have been derived from concentrations measured in actual atmospheric diffusion experiments under conditions approximating those of the application. This section is devoted to a summary of several results available for estimating Gaussian dispersion coefficients.

18.10.1 Correlations for σy and σz Based on Similarity Theory As we noted in Section 18.7, the variances of the mean plume dimensions can be expressed in terms of the motion of single particles released from the source. (At a particular instant the plume outline is defined by the statistics of the trajectories of two particles released simultaneously at the source. We have not considered the two-particle problem here.) In an effort to overcome the practical difficulties associated with using (18.72) to obtain results for σy and σz , Pasquill (1971) suggested an alternate definition that retained the essential features of Taylor’s statistical theory but that is more amenable to parametrization in terms of readily measured Eulerian quantities. As adopted by Draxler (1976), the American Meteorological Society (1977), and Irwin (1979), the Pasquill representation leads to σy ˆ σv tFy σz ˆ σw tFz

…18:103† …18:104†

where σv and σw are the standard deviations of the wind velocity fluctuations in the y and z directions, respectively, and Fy and Fz are universal functions of a set of parameters that specify the characteristics of the atmospheric boundary layer. The exact forms of Fy and Fz are to be determined from data. The variables on which Fy and Fz are assumed to depend are the friction velocity u∗ , the Monin– Obukhov length L, the Coriolis parameter f, the mixed-layer depth zi , the convective velocity scale w∗ , the surface roughness z0 , and the height of pollutant release above the ground h.8 The variances σ2y and σ2z are therefore treated as empirical dispersion coefficients, the functional forms of which are determined by matching the Gaussian solution to data. In that way, σy and σz actually compensate for deviations from stationary, homogeneous conditions that are inherent in the assumed Gaussian distribution. Of the two standard deviations, σy and σz , more is known about σy . First, most of the experiments from which σy and σz values are inferred involve ground-level measurements. Such measurements provide an adequate indication of σy , whereas vertical concentration distributions are needed to determine σz . Also, the Gaussian expression for vertical concentration distribution is known not to be obeyed for ground-level releases, so the fitting of a measured vertical distribution to a Gaussian form is considerably more difficult than that for the horizontal distribution where lateral symmetry and an approximate Gaussian form are good assumptions.

8

It is conventional when referring to unstable conditions to represent the depth of the unstable, or mixed, layer by zi . From a mathematical point of view in terms of the equations for mean concentration, zi is identical to H, the height of an elevated layer impermeable to diffusion. The Coriolis parameter is discussed in Chapter 21.

794

ATMOSPHERIC DIFFUSION

For lateral dispersion, Irwin (1979) developed expressions for σv based on the work of Deardorff and Willis (1975), Draxler (1976), and Nieuwstadt and van Duuren (1979): σv ˆ

1:78 u ‰1 ‡ 0:059… zi =L†Š1=3 1:78u

zi =L < 0 zi =L  0

…18:105†

Irwin (1979), based on the work of Panofsky et al. (1977) and Nieuwstadt (1980), proposed the following form for Fy :

Fy ˆ

t 1‡ Ti 1 ‡ 0:9

1

1=2

; Ti 1 ˆ

2:5u 1 ‡ 0:0013 zi

zi L

1=3

1

t T0

zi 0 L zi >0 L

; T0 1 ˆ 1:001

…18:106†

We note that (18.103) when combined with (18.105)–(18.106) possesses the same limiting behavior as the statistical theory; that is, σy  t as t ! 0 and σy  t1=2 as t ! 1. The case of unstable or convective conditions is a special one in determining atmospheric dispersion. Since under convective conditions the most energetic eddies in the mixed layer scale with zi , the timescale relevant to dispersion is zi =w∗ . This timescale is therefore roughly the time needed after release for material to become well mixed through the depth of the mixed layer. A wide range of field and laboratory measurements on vertical and velocity fluctuations under unstable conditions can be represented by σw ˆ w∗ G…z=zi † (Irwin 1979), where 1:342…z=zi †0:333

z=zi < 0:03

0:763…z=zi †0:175

G…z=zi † ˆ

0:722…1

z=zi †

0:03 < z=zi < 0:40 0:207

0:40 < z=zi < 0:96

…18:107†

z=zi > 0:96

0:37

Under neutral and stable conditions the formulation for σw developed by Binkowski (1979) can be used σw ˆ u∗

ϕm …z=L† z=L 3κfm

1=3

z=L  0

…18:108†

where from (16.129) ϕm …z=L† ˆ 1 ‡ 4:7z=L

…18:109†

and where fm ˆ

0:4‰1 ‡ 3:9z=L 0:25…z=L†2 Š 0:4‰6:78 ‡ 2:39…z=L 2†Š

z=L  2 z=L > 2

…18:110†

The next step to complete parametrization of the vertical dispersion coefficients is to specify Fz . Between neutral and very stable conditions, Irwin (1979) gives an interpolation formula that we do not reproduce here. Draxler (1976) developed the following results for Fz under neutral and stable conditions: Fz ˆ

‰1 ‡ 0:9…t=T0 †1=2 Š 0:8

1

‰1 ‡ 0:945…t=T0 † Š

1

z < 50 m z  50 m

…18:111†

795

DISPERSION PARAMETERS IN GAUSSIAN MODELS

Both expressions require specification of the characteristic time T0 . While an initial estimate of 50 s was given by Draxler, Irwin (1979) proposed additional values of T0 .

18.10.2 Correlations for σy and σz Based on Pasquill Stability Classes The correlations for σy and σz in the previous subsection require knowledge of atmospheric variables that may not be available. In that case, one needs correlations for σy and σz based on readily available ambient data. The Pasquill stability categories A through F introduced in Chapter 16 provide a basis for such correlations. The most widely used σy and σz correlations based on the Pasquill stability classes are those developed by Gifford (1961). The correlations, commonly referred to as the “Pasquill–Gifford curves,” appear in Figures 18.4 and 18.5. For use in dispersion formulas it is convenient to have analytical expressions for σy and σz as functions of x. Many of the empirically determined forms can be represented by the power-law expressions σy ˆ Ry xry

…18:112†

σ z ˆ R z xr z

…18:113†

where Ry , Rz , ry , and rz depend on the stability class and the averaging time. Some commonly used dispersion coefficients, including those of Pasquill–Gifford (PG), are summarized in Table 18.3. Both the ASME and Klug dispersion coefficients are expressible by (18.112) and (18.113). Although the σy

FIGURE 18.4 Correlations for σy based on the Pasquill stability classes A to F (Gifford 1961). These are the so-called Pasquill–Gifford curves.

796

ATMOSPHERIC DIFFUSION

FIGURE 18.5 Correlations for σz based on the Pasquill stability classes A to F (Gifford 1961). These are the so-called Pasquill–Gifford curves.

correlation for the PG coefficients is expressible in the form (18.112), that for σz requires a three-parameter form:9 σz ˆ exp‰Iz ‡ Jz ln x ‡ Kz …ln x†2 Š

…18:114†

For completeness we also give σy in this form in Table 18.3 for the PG correlations. The three sets of coefficients in Table 18.3 are based on different data. In choosing a set for a particular application, one should attempt to use that set most representative of the conditions of interest [see Gifford (1976), Weber (1976), AMS Workshop (1977), Doran et al. (1978), Sedefian and Bennett (1980), and Hanna et al. (1982)].

18.11 PLUME RISE In order to ensure that waste gases emitted from stacks will rise above the top of the chimney, the gases are usually released at temperatures above that of the ambient air and are emitted with considerable initial momentum. Since the maximum mean ground-level concentration of effluents

9

There are points of uncertainty surrounding the PG coefficients, including the stack heights at which they apply and the downwind distances to which they extend. Regardless of their technical merits, the PG coefficients have been incorporated (with and without modifications) into computer programs developed and endorsed by the U.S. Environmental Protection Agency. Note that some of these programs conservatively assume that the PG coefficients are for an averaging time of 1 h despite the fact that the true PG averaging time is about 10 min.

798

ATMOSPHERIC DIFFUSION

from an elevated point source depends roughly on the inverse square of the effective stack height, the amount of plume rise obtained is an important factor in reducing ground-level concentrations. The effective stack height is taken to be the sum of the actual stack height hs and the plume rise Δh, defined as the height at which the plume becomes passive and subsequently follows the ambient air motion: h ˆ hs ‡ Δh

…18:115†

The behavior of a plume is affected by a number of parameters, including the initial source conditions (exit velocity and difference between the plume temperature and that of the air), the stratification of the atmosphere, and the wind speed. Based on the initial source conditions, plumes can be categorized in the following manner: Buoyant plume Forced plume Jet

Initial buoyancy  initial momentum Initial buoyancy  initial momentum Initial buoyancy  initial momentum

We shall deal here with buoyant and forced plumes only. Our interest is in predicting the rise of both buoyant and forced plumes in calm and windy, thermally stratified atmospheres. Chracterization of plume rise in terms of the exhaust gas properties and the ambient atmospheric state is a complex problem. The most detailed approach involves solving the coupled mass, momentum, and energy conservation equations. This approach is generally not used in routine calculations because of its complexity. An alternate approach, introduced by Morton et al. (1956), is to consider the integrated form of the conservation equations across a section normal to the plume trajectory [e.g., see Fisher et al. (1979) and Schatzmann (1979)]. Table 18.4 (see also Table 18.5) presents a summary of several available plume rise formulas expressed in the form Δh ˆ

Exb a u

…18:116†

18.12 FUNCTIONAL FORMS OF MEAN WINDSPEED AND EDDY DIFFUSIVITIES While the Gaussian equations have been widely used for atmospheric diffusion calculations, the lack of ability to include variation of windspeed with height and nonlinear chemical reactions limits the situations in which they may be used. The atmospheric diffusion equation provides a more general approach to atmospheric diffusion calculations than do the Gaussian models, since the Gaussian models have been shown to be special cases of that equation when the windspeed is uniform and the eddy diffusivities are constant. The atmospheric diffusion equation in the absence of chemical reaction is @hci @hci @hci @hci   ‡u ‡ v ‡w @t @x @y @z @ @hci @ @hci @ @hci ‡ ‡ Kxx Kyy Kzz ˆ @x @x @y @y @z @z

…18:117†

, v, and w , The key problem in the use of (18.117) is to specify the functional forms of the windspeeds, u and the eddy diffusivities, Kxx , Kyy , and Kzz , for the particular situation of interest.

799

FUNCTIONAL FORMS OF MEAN WINDSPEED AND EDDY DIFFUSIVITIES

a TABLE 18.4 Summary of Several Plume Rise Formulas Expressed in the Forma Δh ˆ Ex b =u Atmospheric Stability

a

b

E

Conditions

Reference

Plumes Dominated by Buoyancy Forces Neutral and unstable Stableb Neutral and unstable

b;c

Stable

1 1 3

1 1 1 1 1 3

0 1

0 0 2 3

0 2 3

0 0 0 2 3

7:4…Fh2s †1=3 29…F=S1 †1=3 1:6F1=3 21:4 F3=4 1:6 F1=3 38:7 F3=5 2:4…F=S2 †1=3 5F1=4 S2 1:6F1=3

ASME (1973) F < 55, x < 49F5=8 F < 55, x  F5=8 F  55, x < 119F2=5 F  55, x  119 F2=5

Briggs (1969, 1971, 1974)

3=8

Plumes Dominated by Momentum Forces All

1.4

0

dVs1:4

Neutrald

2 3

1 3

1

0

1:44…dVs †2=3 3dVs

Vs > 10 m s  Vs > u ΔT < 50 K 4 Vs =u 4 Vs =u

1

ASME (1973)

Briggs (1969)

Nomenclature for Table 18.4: d = stack diameter, m F = buoyancy flux parameter, gd2 Vs …Ts Ta †=4Ts ; m4 s 3 g = acceleration due to gravity, 9:807 m s 2 p = atmospheric pressure, kPa p0 ˆ 101:3 kPa S1 ˆ …g@θ=@z=Ta †…p=p0 †0:29 ; s 2 S2 ˆ …g@θ=@z†=Ta ; s 2 Ta ˆ ambient temperature at stack height; K Ts ˆ stack exit temperature at stack height; K ΔT ˆ Ts Ta Vs ˆ stack exit velocity, m s 1 a For further information we refer the reader to Hanna et al. (1982). b If the appropriate field data are not available to estimate S1 and S2 . Table 18.5 can be used. c Of these formulas for stable conditions, use the one that predicts the least plume rise. d Of the two formulas for neutral conditions, use the one that predicts the least plume rise.

TABLE 18.5 Relationship between Pasquill–Gifford Stability Classes and Temperature Stratification Stability Class A (extremely unstable) B (moderately unstable) C (slightly unstable) D (neutral) E (slightly stable) F (moderately stable)

Ambient Temperature Gradient @T=@z, °C 100 m 1 < 1:9 1:9 to 1:7 1:7 to 1:5 1:5 to 0:5 0:5 to 1.5 > 1:5

Potential Temperature Gradienta @θ=@z, °C 100 m 1 < 0:9 0:9 to 0:7 0:7 to 0:5 0:5 to 0.5 0.5 to 2.5 > 2:5

Calculated by assuming @θ=@z ˆ @T=@z ‡ Γ, where Γ is the dry adiabatic lapse rate, 0.986°C 100 m 1 .

a

800

ATMOSPHERIC DIFFUSION

18.12.1 Mean Windspeed The mean windspeed, usually taken as that coinciding with the x direction, is often represented as a power-law function of height by (16.138)  u ˆ  ur

z zr

p

…18:118†

where p depends on atmospheric stability and surface roughness (Figure 16.16).

18.12.2 Vertical Eddy Diffusion Coefficient Kzz The expressions available for Kzz are based on Monin–Obukhov similarity theory coupled with observational or computationally generated data. It is best to organize the expressions according to the type of stability. In the surface layer Kzz can be expressed as Kzz ˆ

κu∗ z ϕ…z=L†

…18:119†

where ϕ…z=L† is given by (16.129) 1 ‡ 4:7 z=L 1 …1 15 z=L†

ϕ…z=L† ˆ

1=2

z=L > 0 stable z=L ˆ 0 neutral z=L < 0 unstable

…18:120†

We note that for stable and neutral conditions ϕ…z=L† is identical to that for momentum transfer, ϕm …z=L†. For unstable conditions, ϕ…z=L† ˆ …ϕm …z=L††2 (Galbally 1971; Crane et al. 1977). Since we generally need expressions for Kzz that extend vertically beyond the surface layer, we now consider some available correlations for the entire Ekman layer. 18.12.2.1

Unstable Conditions

In unstable conditions there is usually an inversion base height at z ˆ zi that defines the extent of the mixed layer. The two parameters that are key in determining Kzz are the convective velocity scale w∗ and ~ zz ˆ Kzz =w∗ zi , which is a function only of z=zi , should be zi . We expect that a dimensionless profile K ~ zz is independent of the nature of the source applicable. This form should be valid as long as K ~ zz does depend on the source height. With the distribution. Lamb and Duran (1977) determined that K proviso that the result be applied when emissions are at or near ground level, Lamb et al. (1975) and ~ zz under unstable conditions, using the Lamb and Duran (1977) derived an empirical expression for K numerical turbulence model of Deardorff (1970): 2:5 κ

z zi

4=3

0:021 ‡ 0:408 Kzz ˆ w ∗ zi

4:096

z zi

0:2 exp 6 0:0013

15 z=L†1=4

…1 z zi

‡ 1:351

3

‡2:560 10

z zi

z zi

0 z zi

z < 0:05 zi

2

4

0:05  0:6
1:1 zi

…18:121†

801

FUNCTIONAL FORMS OF MEAN WINDSPEED AND EDDY DIFFUSIVITIES

FIGURE 18.6 Vertical turbulent eddy diffusivity Kzz under unstable conditions derived by Lamb and Duran (1977).

Equation (18.121) is shown in Figure 18.6. The maximum value of Kzz occurs at z=zi  0:5 and has a magnitude  0:21w∗ zi . For typical meteorological conditions this corresponds to a magnitude of O…100 m2 s 1 † and a characteristic vertical diffusion time, z2i =Kzz , of O…5zi =w∗ †. Other expressions for Kzz in unstable conditions exist, notably those of O‘Brien (1970) and Myrup and Ranzieri (1976), that are similar in nature to (18.121). 18.12.2.2

Neutral Conditions

Under neutral conditions Kzz ˆ κu∗ z in the surface layer. Since Kzz will not continue to increase without limit above the surface layer, it is necessary to specify its behavior at the higher elevations. Myrup and Ranzieri (1976) proposed the following empirical form for Kzz under neutral conditions: Kzz ˆ

κu∗ z κu∗ z…1:1 0

z=zi †

z=zi < 0:1 0:1  z=zi  1:1 z=zi > 1:1

…18:122†

Shir (1973) developed the following relationship for Kzz under neutral conditions from a onedimensional turbulent transport model: Kzz ˆ κu∗ z exp

8fz u∗

…18:123†

We note that Myrup and Ranzieri chose the mixed-layer depth zi as the characteristic vertical lengthscale, whereas Shir uses the Ekman layer height, u∗ =f . (Since L ˆ 1 under neutral conditions, the Monin–Obukhov length cannot be used as a characteristic length scale.)

802

ATMOSPHERIC DIFFUSION

Lamb et al. (1975) calculated Kzz under neutral conditions from a numerical turbulence model and obtained u2 7:396  10 f Kzz ˆ

12:72

zf u∗

4

‡ 6:082  10

3

‡15:17

zf u∗

2

zf u∗

‡ 2:532

4

;

0

0

zf u∗

2

zf u∗

 0:45

zf u∗

> 0:45

…18:124†

The predictions of the three expressions (18.122) to (18.124) are compared in Figure 18.7. Note that the scale height H in the figure differs for each equation, in particular Hˆ

zi 0:5 u∗ =f u∗ =f

for …18:122† for …18:123† for …18:124†

It is clear that substantial differences exist in the magnitude of Kzz predicted by the three expressions. This is due to the lack of knowledge of the form of Kzz above the surface layer.

FIGURE 18.7

Comparison of vertical profiles of vertical turbulent eddy diffusivity Kzz .

803

SOLUTIONS OF THE STEADY-STATE ATMOSPHERIC DIFFUSION EQUATION

18.12.2.3

Stable Conditions

Under stable conditions the appropriate characteristic vertical length scale is L. Businger and Arya (1974) proposed a modification of surface layer similarity theory to extend its vertical range of applicability: Kzz ˆ

κu∗ z exp 0:74 ‡ 4:7…z=L†

8fz u∗

…18:125†

18.12.3 Horizontal Eddy Diffusion Coefficients Kxx and Kyy We have seen that when σ2 varies as t the crosswind eddy diffusion coefficient Kyy is related to the variance of plume spread by 2

Kyy ˆ

2

1 dσy 1 dσy  ˆ u 2 dt 2 dx

…18:126†

for t  TL , where TL is the Lagrangian timescale. Measurements of TL in the atmosphere are challenging to perform, and it is difficult to establish whether the condition t  TL holds for urban scale flows. Csanady (1973) indicates that a typical eddy that is generated by shear flow near the ground has a Lagrangian timescale on the order of 100 s. Lamb and Neiburger (1971), in a series of measurements in the Los Angeles basin, estimated the Eulerian timescale TE to be about 50 s. In a discussion of some field experiments, Lumley and Panofsky (1964) suggested that TL < 4TE . If the averaging interval is selected to be equal to the travel time, then an approximate value for Kyy can be deduced from the measurements of Willis and Deardorff (1976). Their data indicate that for unstable conditions (L > 0) and a travel time t ˆ 3zi =w∗ σ2y z2i

 0:64

…18:127†

Employing the previous travel time estimate and combining this result with (18.126) gives 1 σy w∗ zi  0:1 w∗ zi 6 z2i 2

Kyy ˆ

…18:128†

This latter result can be expressed in terms of the friction velocity u∗ and the Monin–Obukhov length L as 3=4

Kyy  0:1 zi … κL†

1=3

u∗

…18:129†

For a range of typical meteorological conditions this formulation results in diffusivities under unstable conditions of O…50--100 m2 s 1 †. In practical applications it is usually assumed that Kxx ˆ Kyy .

18.13 SOLUTIONS OF THE STEADY-STATE ATMOSPHERIC DIFFUSION EQUATION The Gaussian expressions are not expected to be valid descriptions of turbulent diffusion close to the surface because of spatial inhomogeneities in the mean wind and the turbulence. To deal with

804

ATMOSPHERIC DIFFUSION

diffusion in layers near the surface, recourse is generally made to the atmospheric diffusion equation, in which, as we have noted, the key problem is proper specification of the spatial dependence of the mean velocity and eddy diffusivities. Under steady-state conditions, turbulent diffusion in the direction of the mean wind is usually neglected (the slender plume approximation), and if the wind direction coincides with the x axis, then Kxx ˆ 0. Thus it is necessary to specify only the lateral, Kyy ,  and Kyy and vertical, Kzz , coefficients. It is generally assumed that horizontal homogeneity exists so that u are independent of y. Hence (18.117) becomes  u

@hci @ @hci @hci @ ˆ ‡ Kzz Kyy @x @y @y @z @z

…18:130†

, Kzz , and Kyy based on atmospheric boundarySection 18.12 has been devoted to expressions for u  and Kzz on z, (18.130) must generally be layer theory. Because of the rather complicated dependence of u solved numerically. However, if they can be found, analytical solutions are advantageous for studying the behavior of the predicted mean concentration. For solutions beyond those presented here we refer the reader to Lin and Hildemann (1996, 1997).

18.13.1 Diffusion from a Point Source A solution of (18.130) has been obtained by Huang (1979) in the case when the mean windspeed and vertical eddy diffusivity can be represented by the power-law expressions …z† ˆ az p u

…18:131†

Kzz …z† ˆ bz n

…18:132†

and when the horizontal eddy diffusivity is related to σ2y by (18.126). For a point source of strength q at height h above the ground, the solution of (18.130) subject to (18.126), (18.131), and (18.132) is hc…x; y; z†i ˆ

q …2π†1=2 σy

exp

y2 …zh†…1 n†=2 bαx 2σ2y

a…zα ‡ hα † I bα2 x

 exp

ν

2a…zh†α=2 bα2 x

…18:133†

where α ˆ 2 ‡ p n, ν ˆ …1 n†=α, and I ν is the modified Bessel function of the first kind of order ν. Equation (18.133) can be used to obtain some special cases of interest. If it is assumed that p ˆ n ˆ 0, then (18.133) reduces to hc…x; y; z†i ˆ

q …2π†

1=2

σy

exp

y2 …zh†1=2 exp  2σ2y σ2z u

z 2 ‡ h2 I 2σ2z

1=2

zh σ2z

…18:134†

where  σ2y ˆ 2Kyy x=u

 σ2z ˆ 2Kzz x=u

…18:135†

Using the asymptotic form I

1=2 …x†

ˆ

2 πx

1=2

cosh x

x!1

…18:136†

805

SOLUTIONS OF THE STEADY-STATE ATMOSPHERIC DIFFUSION EQUATION

(18.134) reduces to the Gaussian plume equation. Note that the asymptotic condition in (18.136) corresponds to zh  σ2z . The case of a point source at or near the ground can also be examined. We can take the limit of (18.133) as h ! 0 using the asymptotic form of Iν …x† as x ! 0 Iν …x† ˆ

xν 2ν Γ…1

‡ ν†

x!0

…18:137†

to obtain hc…x; y; z†i ˆ

α

q …2π†

1=2

σy

x…1‡p†=α aν …bα2 †…1‡p†=α Γ……1 y2 exp 2σ2y

 exp

‡ p†=α† …18:138†

a…zα ‡ hα † bα2 x

18.13.2 Diffusion from a Line Source The mean concentration downwind of a continuous, crosswind line source at a height h emitting at a rate ql …g m 1 s 1 † is governed by  u

@hci @ @hci ˆ Kzz @x @z @z

…18:139†

…h††δ…z hc…0; z†i ˆ …ql =u Kzz …0†

@hci @z

ˆ0

h†

zˆ0

zˆ0

hc…x; z†i ˆ 0

z !1

The solution of (18.139) for the power-law profiles (18.131) and (18.132) is hc…x; z†i ˆ

ql …zh†…1 n†=2 exp bαx

a…zα ‡ hα † I bα2 x

ν

2a…zh†α=2 bα2 x

…18:140†

azα α2 bx

…18:141†

For a ground-level line source, h ˆ 0, and hc…x; z†i ˆ

αql a 2 aΓ……p ‡ 1†=α† α bx

…p‡1†=α

exp

The General Structure of Multiple-Source Plume Models Multiple-source plume (and particularly Gaussian plume) models (Calder 1977) are commonly used for predicting concentrations of inert pollutants over urban areas. Although there are many specialpurpose computational algorithms in use, the basic element that is common to most is the single­ point-source release. The spatial concentration distribution from such a source is the underlying component and the multiple-source model is then developed by simple superposition of the individual plumes from each of the sources.

806

ATMOSPHERIC DIFFUSION

The starting point for predicting the mean concentration resulting from a single point source is the assumption of a quasi-steady state. In spite of the obvious long-term variability of emissions, and of the meteorological conditions affecting transport and dispersion, it is assumed as a working approximation that the pollutant concentrations can be treated as though they resulted from a time sequence of different steady states. In urban modeling the time interval is normally relatively short and on the order of 1 h. The time sequence of steady-state concentrations is regarded as leading to a random series of (1 h) concentration values, from which the frequency distribution and long-term average can be calculated as a function of receptor location. An assumption normally made is that the dispersion of material is “horizontally homogeneous” and independent of the horizontal location of the source—that is, the source–receptor relation is invariant under arbitrary horizontal translation of the source–receptor pair. Furthermore, the dispersion in a simple point source plume is assumed to be “isotropic” with respect to direction, and independent of the actual direction of the wind transport over the urban area. In principle, the short-term multiple-source plume model then only involves simple summation or integration over all point, line, and area sources and for the different meteorological conditions of plume dispersion. In most short-term plume models, it is assumed that for each time interval of the quasi-steady­ state sequence, transport by the wind over the urban area can be characterized in terms of a single “wind direction.” Also, for each time interval the three meteorological variables that influence the transport and dispersion are taken to be the mean wind speed, the atmospheric stability category, and the mixing depth. Although the wind direction θ is evidently a continuous variable that may assume any value (0 < θ < 360 ), for reasons of practical simplicity the three dispersion variables are usually considered as discrete, with a relatively small number of possible values for each. The short-term, steady-state, three-dimensional spatial distribution of mean concentration for an individual point source plume must then be expressed in terms of a plume dispersion equation of fixed functional form, such as the Gaussian plume equation, although the dispersion parameters that appear in its arguments will, of course, depend on actual conditions. Many models assume the well-known Gaussian form, and the horizontal and vertical standard deviation functions, σy and σz , are expressed as functions of the downwind distance in the plume and of the stability category only. To illustrate these ideas we consider a fixed rectangular coordinate system, with the plane z ˆ 0 at ground level, and also assume that the pollutant transport by the wind over the urban area can be characterized in terms of a single horizontal direction that makes an angle, say, θ, with the positive x axis of the coordinate system. We thus explicitly separate the wind direction variable from all other meteorological variables that influence transport and dispersion, for example, windspeed, atmospheric stability category, and mixing depth. These latter variables may be regarded as defining a meteorological dispersion index j … j ˆ 1; 2; . . .† that will be a function of tn , the interval of time over which meteorological conditions are constant; that is, j ˆ j…tn †. The index number j simply designates a given meteorological state out of the totality of possible combinations, that is, a specified combination of windspeed, stability, and mixing depth. Then from the additive property of concentration it follows that the mean concentration hc…x; y; z; tn †i for the time interval tn that results from the superposition can be written as a summation (actually representing a three-dimensional integral) Qfx; y; z; x´ ; y´ ; z´ ; θn ; j…tn †gS…x´ ; y´ ; z´ ; tn †ΔV

hc…x; y; z; tn †i ˆ …x´ ;y´ ;z´ †

where S…x´ ;y´ ;z´ ; tn † is the steady emission rate per unit volume and unit time at position …x´ ;y´ ;z´ † for time interval t ˆ tn ; Qfx;y;z; x´ ;y´ ;z´ ; θn ; j…tn †g is the mean concentration at (x;y;z) produced by a steady point source of unit strength located at …x´ ;y´ ;z´ †, and for time interval tn when θ ˆ θn ; and the summation is taken over all the elemental source locations of the entire region. The function Q is a meteorological dispersion function that is explicitly dependent on the wind direction θn and implicitly on the other meteorological variables that will be determined by the time sequence tn . In

807

FURTHER SOLUTIONS OF ATMOSPHERIC DIFFUSION PROBLEMS

simple urban models Q is taken to be a deterministic function, although its variables θn and j…tn † may be regarded as random. It is usually assumed that the dispersion function Q is independent of the horizontal location of the source, so that Q is invariant under arbitrary horizontal translations of the source–receptor pair, reducing Q to a function of only two independent horizontal spatial variables, so that the preceding equation may be rewritten as QfX; Y; z; z´ ; θn ; j…tn †gS…x

hc…x; y; z; tn †i ˆ

X; y

Y; z´ ; tn †ΔV

…X;Y;z´ †

where X ˆ x x´ and Y ˆ y y´ , and the summation extends over the entire source distribution. Finally, it is normally assumed that the dispersion function Q depends on θ only through a simple rotation of the horizontal axes, and that it is independent of the actual value of the single horizontal wind direction that is assumed to characterize the pollutant transport over the urban area, provided the x axis of the coordinate system is taken along this direction. If this is done then the dispersion function is directionally isotropic. The long-term average concentration, which we denote by an uppercase C, is given by C…x; y; z† ˆ

1 N

N

hc…x; y; z; tn †i nˆ1

When the emission intensities S and the dispersion functions Q, that is, the meteorological conditions, can be assumed to be independent or uncorrelated, the average value of the product of S and Q is equal to the product of their average values, and we thus have …x  …X; Y; z; z´ †S Q

C…x; y; z† ˆ

X; y

Y; z´ †ΔV

…X;Y;z´ †

 denotes the time-average source strength and Q  the average value of the meteorological where S dispersion function. Thus if P…θ; j† denotes the joint frequency function for wind direction θ and dispersion index j, so that P…θ; j† ˆ 1 θ

j

then  …X; Y; z; z´ † ˆ Q

QfX; Y; z; z´ ; θ; jgP…θ; j† θ

j

APPENDIX 18.1 FURTHER SOLUTIONS OF ATMOSPHERIC DIFFUSION PROBLEMS 18A.1 Solution of (18.29)–(18.31) To solve this problem we let hc…x; y; z; t†i ˆ cx …x; t†cy …y; t†cz …z; t† with cx …x; 0† ˆ S1=3 δ…x†, cy …y; 0† ˆ S1=3 δ…y†, and cz …z; 0† ˆ S1=3 δ…z†. Then (18.29) becomes @c @2c @cx  x ˆ Kxx 2x ‡u @t @x @x

…18A:1†

808

ATMOSPHERIC DIFFUSION

@cy @ 2 cy ˆ Kyy 2 @t @y

…18A:2†

@cz @ 2 cz ˆ Kzz 2 @t @z

…18A:3†

Each of these equations may be solved by the Fourier transform. We illustrate with (18A.1). The Fourier transform of cx …x; t† is C…α; t† ˆ Ffcx …x; t†g ˆ

1

1

…2π†1=2

1

cx …x; t† e

iαx

dx

and thus transforming, we obtain @C ‡ iα uC ˆ α2 Kxx C @t C…α; 0† ˆ

…18A:4†

S1=3 …2π†1=2

The solution of (18A.4) is C…α; t† ˆ

S1=3 …2π†1=2

†tŠ exp‰ …α2 Kxx ‡ iαu

The inverse transform is cx …x; t† ˆ

1

1

…2π†1=2

1

C…α; t† eiαx dα

Thus cx …x; t† ˆ

S1=3 2π

1 1

expf ‰α2 Kxx t

iα…x

tŠgdα u

Completing the square in the exponent, we obtain α2 Kxx t

ˆ

Let η ˆ α…Kxx t†1=2

i…x

 ut†

iα…x

α…Kxx t†

1=2

i…x

t†2  †2 …x u …x ut ‡ 4Kxx t 4Kxx t  ut†

2…Kxx t†1=2

2

 2 …x ut† 4Kxx t

t†=2…Kxx t†1=2 and dη ˆ …Kxx t†1=2 dα. Then u cx …x; t† ˆ

S1=3 2π…Kxx t†

exp 1=2

t† …x u 4Kxx t

1 1

e

η2

dη

809

FURTHER SOLUTIONS OF ATMOSPHERIC DIFFUSION PROBLEMS

the integral equals π1=2 , so cx …x; t† ˆ

S1=3 2…πKxx t†

 2 …x ut† 4Kxx t

exp 1=2

By the same method cy …y; t† ˆ cz …z; t† ˆ

S1=3

exp

y2 4Kyy t

exp 2…πKzz t†1=2

z2 4Kzz t

2…πKyy t†

1=2

S1=3

and thus the mean concentration is given by hc…x; y; z; t†i ˆ

S 8…πt†

3=2

…Kxx Kyy Kzz †1=2

 exp

 2 …x ut† 4Kxx t

y2 4Kyy t

18A.2 Solution of (18.50) and (18.51) To solve (18.50) we begin with the transformation f …x; y; z† ˆ hc…x; y; z†ie

kx

=2K, and (18.50) becomes where k ˆ u q kx e δ…x† δ…y† δ…z† K f …x; y; z† ˆ 0 x; y; z ! 1

r2 f

k2 f ˆ

First we will solve r2r f

k2 f ˆ 0

where r2 ˆ x2 ‡ y2 ‡ z2 . To do so, let f …r† ˆ g…r†=r and r2r f ˆ

1 d 2 df r r2 dr dr

ˆ

1 d2 g r dr2

so that the equation to be solved is d2 g dr2

k2 g ˆ 0

The solution is g…r† ˆ A1 ekr ‡ A2 e

kr

z2 4Kzz t

…18A:5†

810

ATMOSPHERIC DIFFUSION

Then A1 ˆ 0 satisfies the condition that g is finite as r ! 1, and A e r

f …r† ˆ

kr

To determine A, we will evaluate

V

…r2r f

q e VK

k2 f †dV ˆ

kx

δ…x† δ…y† δ…z† dV

on the sphere with unit radius. The right-hand side is simply hand side is

V

k2 f †dV ˆ

…r2r f

S

@f dS @n

q=K. Using Green’s theorem, the left-

k2

f dV V

On a sphere of unit radius

S

@f dS ˆ @n

2π 0

π π

cos θ

@f @r

dθ dϕ rˆ1

ˆ 4π Ae k …k ‡ 1† and 1

f dV ˆ V

4πr2 f dr ˆ 4πA‰e k …k ‡ 1†

1Š

0

Thus we obtain 4πA ˆ q=K. Finally hc…x; y; z†i ˆ

…r x† u 2K

q exp 4πKr

18A.3 Solution of (18.59)–(18.61) The solution can be carried out by Fourier transform, first with respect to the y direction and then with respect to the z direction. If C…x; α; z† ˆ Fy fhc…x; y; z†ig and C´ …x; α; β† ˆ Fz fC…x; α; z†g, then (18.59) becomes  u

@C´ ˆ K…α2 ‡ β2 †C´ …x; α; β† @x

…18A:6†

The x ˆ 0 boundary condition, when transformed doubly, is C´ …0; α; β† ˆ

q  2πu

…18A:7†

The solution of (18A.6) subject to (18A.7) is C´ …x; α; β† ˆ

q exp  2πu

Kx 2 …α ‡ β2 †  u

…18A:8†

811

ANALYTICAL PROPERTIES OF THE GAUSSIAN PLUME EQUATION

We must now invert (18A.8) twice to return to hc…x; y; z†i. First C…x; α; z† ˆ

1

1

…2π†1=2

1

C´ …x; α; β†eiβz dβ

Thus C…x; α; z† ˆ

q  …2π†3=2 u

e

 α2 Kx=u

1 1

Kxβ2 dβ  u

exp iβz

…18A:9†

It is now necessary to express the exponential in the integrand as  …Kxβ2 =u †1=2 β and let η ˆ …Kx=u

†1=2 β iβz† ˆ ‰…Kx=u

=Kx†1=2 Š2 …iz=2†…u

=4Kx z2 u

=Kx†1=2 . Then (18A.9) becomes …iz=2†…u C…x; α; z† ˆ

 u 3=2  Kx …2π† u q

1=2

exp

α2 Kx  u

 z2 u 4Kx

1 1

e

η2

dη

Proceeding through identical steps to invert C…x; α; z† to hc…x; y; z†i, we obtain hc…x; y; z†i ˆ

q exp 4πKx

 u …y2 ‡ z2 † 4Kx

…18A:10†

APPENDIX 18.2 ANALYTICAL PROPERTIES OF THE GAUSSIAN PLUME EQUATION When material is emitted from a single elevated stack, the resulting ground-level concentration exhibits maxima with respect to both downwind distance and windspeed. Both directly below the stack, where the plume has not yet touched the ground, and far downwind, where the plume has become very dilute, the concentrations approach zero; therefore a maximum ground-level concentration occurs at some intermediate distance. Both at very high wind speeds, when the plume is rapidly diluted, and at very low windspeeds, when plume rise proceeds relatively unimpeded, the contribution to the ground-level concentration is essentially zero; therefore a maximum occurs at some intermediate wind speed. To investigate the properties of the maximum ground-level concentration with respect to distance from the source and windspeed, we begin with the Gaussian plume equation evaluated along the plume centerline …y ˆ 0† at the ground (z ˆ 0): hc…x; 0; 0†i ˆ

q exp π uσy σz

h2 2σ2z

…18A:11†

The maximum in hc…x; 0; 0†i arises because of the x dependence of σy , σz , and h, as parametrized, for example, by (18.112), (18.113), (18.115), and (18.116). As a buoyant plume rises from its source, the effect  of the wind is to bend it over until it becomes level at some distance xf from the source. Windspeed u enters the expression for the ground-level concentration through two terms having opposite effects: (1)  1 , such that the lighter the wind, the greater is the ground-level hc…x; 0; 0†i is proportional to u a , such that the lighter the wind, concentration; and (2) the plume rise Δh is inversely proportional to u the higher the plume rise and the smaller is the ground-level concentration. Thus a “worst” windspeed

812

ATMOSPHERIC DIFFUSION

exists at which the ground-level concentration is a maximum at any downwind location. This maximum is different from the largest ground-level concentration that is achieved as a function of downwind  suggests that one distance for any given wind speed. The fact that hc…x; 0; 0†i depends on both x and u may find simultaneously the windspeed and distance at which the highest possible ground-level concentration may occur, the so-called critical concentration. In computing these various maxima, there are two regimes of interest. The first is the region of x < xf , where the plume has not reached its final height, and the second is x  xf , when the plume has reached its final height. The latter case corresponds to b ˆ 0 in the formula for the plume rise. We want to calculate the location xm of the maximum ground-level concentration for any given wind speed. By differentiating (18A.11) with respect to x and setting the resulting equation equal to zero, we find 1 dσz h2 dσz ‡ σz dx σ3z dx

1 dσy σy dx

h dh σ2z dx

ˆ0

…18A:12†

Now using the expressions (18.112), (18.113), (18.115), and (18.116) for h and σy and σz as a function of x, we obtain the following implicit equation for xm hbΔh ‡ h2 rz ry ‡ rz

σ2z …xm † ˆ

…18A:13†

where if (18.114) and its analogous relation for σy are used, ry and rz in (18A.13) are replaced by Jy ‡ 2Ky ln xm and Jz ‡ 2Kz ln xm , respectively. In the special case in which Δh ˆ 0 and σy and σz are given by (18.112) and (18.113), the expression for xm reduces to (Ragland 1976) xm ˆ

h2s rz 2 Rz …ry ‡ rz †

1=2rz

…18A:14†

The value of the maximum ground-level concentration at xm in the case of σy and σz given by (18.112) and (18.113) and when b ˆ 0 is hcim ˆ

qγ γ=2 Rγz 1  πeγ=2 Ry hγ u

…18A:15†

where 㠈 1 ‡ ry =rz .  on the maximum ground-level concentration. The Now we consider the effect of wind speed u . Differentiating highest concentration at any downwind distance can be determined as a function of u , we obtain (18A.11) with respect to u  2a u

hs aExb a  u σz2

aE2 x2b ˆ0 σ2z

…18A:16†

w , is from which we find that the “worst” windspeed, u

w ˆ u where ζ ˆ …1 ‡ 4σ2z =ah2s †1=2 .

2Exb hs …ζ 1†

1=a

…18A:17†

813

ANALYTICAL PROPERTIES OF THE GAUSSIAN PLUME EQUATION

The value of the ground-level concentration at this windspeed is hc…x†iw ˆ

q2

1=a 1=a hs …ζ 1†1=a πσy σz E1=a xb=a

hs2 …ζ ‡ 1†2 8σ2z

exp

…18A:18†

When the plume has reached its final height, h becomes independent of x, and the effective plume height can be written as h ˆ hs ‡

E a u

…18A:19†

In that case the “worst” windspeed is (Roberts 1980; Bowman 1983) w ˆ u

1=a

2E hs …ζ 1†

…18A:20†

Finally, we can find the combination of location and windspeed that produces the highest possible ground-level concentration, the so-called critical concentration. Considering σy and σz given by (18.112) and (18.113), the critical downwind distance xc is given by xc ˆ

hs Rz

r r 1=a

1 r

1=2 1=rz

…18A:21†

c is where r ˆ …ry ‡ rz ‡ b=a†=rz . The critical windspeed u c ˆ …aE†1=a u

r1=2 Rz

b=arz

r

1=a hs

…1=a b=arz †

…18A:22†

and the critical concentration hcic is hcic ˆ

qRrz 1 …r

1=a†r

1=a

r 1=a r=2 r=2 e r

π…aE†1=a Ry hs

…18A:23†

All terms in (18A.23), other than exponents, are positive except for r 1=a. If this term is zero or negative, a critical concentration does not exist. Thus for a critical concentration to exist, it is necessary that r > 1=a, or a>

rz b ry ‡ rz

…18A:24†

Once the plume has reached its final height, b ˆ 0, and (18A.24) becomes simply a > rz =…ry ‡ rz †. Let us now apply these general results to some specific cases. Figures 18.A.1 and 18.A.2 show the critical distance xc as a function of the source height hs and stability class for the cases in which the plume has reached its final height and before it has reached its final height, respectively. For the level plume, that is, one that has reached its final height, the critical downwind distance is xc ˆ

a2 rz …ry ‡ rz † R2z ‰a…ry ‡ rz †

rz Š2

1=2rz z h1=r s

…18A:25†

814

ATMOSPHERIC DIFFUSION

FIGURE 18A.1 Critical downwind distance xc as a function of source height hs, and stability class for a plume that has reached its final height.

For a plume that has not yet reached its final height, a ˆ 1, and xc ˆ

3rz …3rz ‡ 3ry ‡ 2† R2z …3ry ‡ 2†2

1=2rz z h1=r s

…18A:26†

For the level plume, a ˆ 1 in neutral and unstable conditions and a ˆ 13 under stable conditions. Coefficients for different stability conditions are given in Table 18.3.

FIGURE 18A.2 Critical downwind distance xc as a function of source height hs , and stability class for a plume that has not yet reached its final height.

815

PROBLEMS

We see that as hs increases, the downwind distance at which the critical concentration occurs increases. The distance also increases as the atmosphere becomes more stable. When a ˆ 1, (18A.24) holds regardless of ry and rz (neutral and unstable conditions); for stability classes E and F, however, a ˆ 13, and from the ry and rz values in Table 18.3, we see that (18A.24) is not satisfied. Thus for a level plume, a critical concentration does not exist for stability classes E and F.

PROBLEMS 18.1A A power plant burns 104 kg h 1 of coal containing 2.5% sulfur. The effluent is released from a single stack of height 70 m. The plume rise is normally about 30 m, so that the effective height of emission is 100 m. The wind on the day of interest, which is a sunny summer day, is blowing at 4 m s 1 . There is no inversion layer. Use the Pasquill–Gifford dispersion parameters from Table 18.3. a. Plot the ground-level SO2 concentration at the plume centerline over distances from 100 m to 10 km. (Use log–log coordinates.) b. Plot the ground-level SO2 concentration versus crosswind distance at down-wind distances of 200 m and 1 km. c. Plot the vertical centerline SO2 concentration profile from ground level to 500 m at distances of 200 m, 1 km, and 5 km. 18.2A Repeat the calculation of Problem 18.1 for an overcast day with the same windspeed. 18.3A Repeat the calculation of Problem 18.1 assuming that an inversion layer is present at a height of 300 m. 18.4A A power plant continuously releases from a 70-m stack a plume into the ambient atmosphere of temperature 298 K. The stack has diameter 5 m, the exit velocity is 25 m s 1 , and the exit temperature is 398 K. a. For neutral conditions and a 20 km h 1 wind, how high above the stack is the plume 200 m downwind? b. If a ground-based inversion layer 150 m thick is present through which the temperature increases 0.3°C and if the wind is blowing at 10 km h 1 , will the plume penetrate the inversion layer? 18.5C Mathematical models for atmospheric diffusion are generally derived for either instantaneous or continuous releases. Short-term releases of material sometimes occur and it is of interest to develop the appropriate diffusion formulas to deal with them (Palazzi et al. 1982). The mean concentration resulting from a release of strength q at height h that commenced at t ˆ 0 is given by t

hc…x; y; z; t†i ˆ 0

q …2π†3=2 σx σy σz

 exp

exp

h†2

…z

2σ2z

‰x

‡ exp

…t t´ †Š2 u 2σ2x

y2 2σ2y

…z ‡ h†2 2σ2z

dt´

…A†

The duration of the release is tr . a. Show that if the slender plume approximation is invoked, the mean concentration resulting from the release is t

hcG …x; y; z†i 0

hc…x; y; z; t†i ˆ hcG …x; y; z†i

0

 u

…2π†1=2 σx tr  u …2π†1=2 σx

exp exp

‰x

…t t´ †Š2 u dt´ 2σ2x

t  tr

‰x

…t t´ †Š2 u dt´ 2σ2x

t > tr

816

ATMOSPHERIC DIFFUSION

where hcG …x; y; z†i is the steady-state Gaussian plume result. Show that this reduces to

hc…x; y; z; t†i ˆ

1 x hcG …x; y; z†i erf p 2 2σx

 x ut erf p 2σx

…t tr † 1 x u p hcG …x; y; z†i erf 2 2σx

t  tr

 x ut erf p 2σx

t > tr

b. For t < tr show that hci reaches its maximum at t ˆ tr . After t ˆ tr , the puff continues to flow  r =2, the concentration decreases with downwind. Show that at locations where x  ut tr =2, show increasing t and the maximum concentration is given by that at t ˆ tr . For x > u  and is that the maximum concentration occurs when @hci=@t ˆ 0, which gives t ˆ tr =2 ‡ x=u  ut t > tr pr tr =2 x>u 2 2σx p  r =2 2σx . What do you conclude? and hcG i as a function of ut

hc…x; y; z; t†imax ˆ hcG …x; y; z†ierf c. Compare hcimax

d. We can define the dosage over a time interval from t to t0 ‡ te as c…x; y; z† ˆ

1 te

t0 ‡te

hc…x; y; z; t†idt

…B†

t0

Three different situations can be visualized, according to when the release stops: tr  t0 t0  tr  t0 ‡ te tr  t0 ‡ te

release stops before exposure period begins release stops during exposure period release stops after the exposure period

Show that substituting the appropriate form of hc…x; y; z; t†i into (B) leads to the following formulas for the concentration dosage: p …t x u t† hcG …x; y; z†i 2σx p 0 r F  2te u 2σx F

…t0 ‡ te u p 2σx

tr †

…t ‡ t †  t0 u x u p 0 e F p 2σx 2σx p x x hcG …x; y; z†i 2σx F p F  2te u 2σx ‡F

c…x; y; z† ˆ

x

‡F

x

x

…tr u p

…t ‡ t † u p 0 e 2σx t0 †

2σx

erfc p

F x 2σx

tr  t0 …t0 ‡ te u p 2σx

tr †

t x u p 0 2σx t0  tr  t0 ‡ te

p …t ‡ t † x u hcG …x; y; z†i 2σx p 0 e F  2te u 2σx F where F…x† ˆ

1 x

erfc (ξ) dξ:

t x u p 0 2σx

 ut x p e erfc p 2σx 2σx

tr  t0 ‡ te

817

PROBLEMS

e. The dosage reaches its maximum value in the interval defined by t0 > tr …tr ‡ te †=2, the maximum dosage corresponds to xu t0 ˆ

te . Show that when

x tr te ‡  u 2

and that it can be found from p …tr ‡ te † u jte tr j hcG …x;y;z†i 2σx tr ‡ te p p p u cmax ˆ u ‡ 2F  2te u 2σx 2 2σx 2σx

2F

jte tr j u p 2 2σx

18.6B Chlorine is released at a rate of 30 kg s 1 from an emergency valve at a height of 20 m. We need to evaluate the maximum ground-level dosage of Cl2 for different durations of release and exposure. Assume a 5 m s 1 windspeed and neutral stability. We may use the dispersion parameters for a continuous emission (they are a good approximation for exposure times exceeding about 2 min). Plot the Cl2 dosage in ppm versus exposure time in minutes on a log– log scale over a range of 1 to 100 min exposure time for release durations of 2, 10, and 30 min. (The results needed to solve this problem can be taken from the statement of Problem 18.5.) 18.7B SO2 is emitted from a stack under the following conditions: Stack diameter ˆ 3 m Exit velocity ˆ 10 m s

1

Exit temperature ˆ 430 K Emission rate ˆ 103 g s

1

Stack height ˆ 100 m For Pasquill stability categories A, B, C, and D, calculate (Appendix 18.2 is needed): a. The distance from the source at which the maximum ground-level concentration is achieved as a function of windspeed. b. The windspeed at which the ground-level concentration is a maximum as a function of downwind distance. c. The plume rise at the wind speed in part (b). d. The critical downwind distance, windspeed, and concentration. 18.8C Gifford (1968, 1980) has discussed the determination of lateral and vertical dispersion parameters σy and σz from smoke plume photographs. Let us consider a single plume emanating from a continuous point source that will be assumed to be at ground level. Thus the mean concentration divided by the source strength is given by the Gaussian plume equation with h ˆ 0: χ…x; y; z† ˆ

hc…x; y; z†i † ˆ …πσy σz u q

1

exp

y2 2σ2y

z2 2σ2z

…A†

A view of the plume from either above or the side is sketched in Figure 18.P.1, where yE and zE represent the coordinates of the visible edge of the plume, and ym and zm denote the maximum values of yE and zE with respect to downwind distance x. Let us consider the visible edge in the

818

ATMOSPHERIC DIFFUSION

FIGURE 18P.1 Plume boundaries.

y direction, that is, the plume as observed from above. Integrating (A) over z from 0 to 1 and evaluating y at yE gives χ E …x† ˆ

1

χ…x; yE ; z†dz

0

† ˆ …πσy u ˆ …2π†

1

1=2

yE2 2σ2y

exp

† …σy u

1

exp

1 0

1 e σz

z2 =2σ2z

dz

…B†

y2E 2σ2y

χE …x† represents the mean concentration normalized by the source strength, evaluated at the value of y corresponding to the visible edge of the plume and integrated through the depth of the plume. a. Because the visible edge of the plume is presumably characterized by a constant integrated concentration, χE …x† is a constant χE . Show that, at y ˆ ym y2m ˆ σ2z …xm † ˆ σ2 ym

…C†

b. Since χE is a constant, (B) is invariant whether evaluated at any x or at xm . Thus show that at any x χE ˆ …2π†

1=2

† …σy u

1

y2E 2σ2y

exp

and at xm χE ˆ …2πe†

1=2

† …ym u

1

Thus show that σ2y

ˆ

y2E

2 eym ln σ2y

1

18.9C Derive (18.89) as a solution of the atmospheric diffusion equation.

819

PROBLEMS

18.10C Murphy and Nelson (1983) have shown that the Gaussian plume equation can be modified to include dry deposition with a deposition velocity vd by replacing the source strength q, usually taken as constant, with the depleted source strength q…t† as a function of travel time t, where q…t† ˆ q0 exp

2 π

1=2

t

vd 0

1 exp σz …t´ †

h2 dt´ 2σ2z …t´ †

Verify this result. Show that, if σz ˆ …2Kzz t†1=2 , then q…t† ˆ q0 exp

2vd t1=2 …πKzz †1=2

18.11A An eight-lane freeway is oriented so that the prevailing wind direction is usually normal to the freeway. During a typical day the average traffic flow rate per lane is 30 cars per minute, and the average speed of vehicles in both directions is 80 km h 1 . The emission of CO from an average vehicle is 90 g km 1 traveled. a. Assuming a 5 km h 1 wind and conditions of neutral stability, with no elevated inversion layer present, determine the average ground-level CO concentration as a function of downwind distance, using the appropriate Gaussian plume formula. b. A more accurate estimate of downwind concentrations can be obtained by taking into account the variation of wind velocity and turbulent mixing with height. For neutral conditions we have seen that …z† ˆ u0 z1=7 u and Kzz ˆ κu∗ z ˆ 0:4 u∗ z should be used. Assume that the 5 km h 1 wind reading was taken at a 10 m height and that for the surface downwind of the freeway u∗ ˆ 0:6 km h 1 (grassy field). Repeat the calculation of (a). Discuss your result. 18.12B It is desired to estimate the ground-level, centerline concentration of a contaminant down­ wind of a continous, elevated point source. The source height including plume rise is 100 m. The wind speed at a reference height of 10 m is 4 m s 1 . The atmospheric diffusion equation with the power-law correlations (18.131) and (18.132) is to be used. The roughness length is 0.1 m. a. Calculate the ground-level, centerline mean concentration under the following conditions: (1) Stable (Pasquill stability class F) (2) Neutral (Pasquill stability class D) (3) Unstable (Pasquill stability class B) b. Compare your predictions to those of the Gaussian plume equation assuming that  ˆ 4 m s 1 at all heights. Discuss the reasons for any differences in the two results. u 18.13B To account properly for terrain variations in a region over which the transport and diffusion of pollutants are to be predicted, the following dimensionless coordinate transformation is used

820

ATMOSPHERIC DIFFUSION

ρˆ

z

h…x; y† Z

ZˆH

h…x; y†

where h…x; y† is the ground elevation at point …x; y† and H is the assumed extent of vertical mixing. Likewise, a similar change of variables for the horizontal coordinates may be performed xW X y yS ηˆ Y

ξˆ

x

X ˆ xE

xW

Y ˆ yN

yS

where xE , xW , yN , and yS are the coordinates of the horizontal boundaries of the region. Show that the form of (18.117) in ξ; η; ρ; t coordinates is  @hci v @hci W @hci @hci u 1 @ @hci ‡ ‡ ‡ ˆ 2 Kxx @t X @ξ Y @η Z @ρ X @ξ @ξ

‡

‡  where W ˆ w

 …u=X†Λ ξZ Λξ ˆ

Λξ

@hci @ρ

Λξ @ Kxx @hci X @ρ X @ξ

Λξ

@hci @ρ

1 @ @hci Kyy 2 Y @η @η

Λη

@hci @ρ

Λη @ Kyy @hci Y @ρ Y @η

Λη

@hci @ρ

1 @ @hci Kzz Z2 @ρ @ρ

v=Y†Λη Z and where …

1 @h @Z ‡ρ Z @ξ @ξ

Λη ˆ

1 @h @Z ‡ρ Z @η @η

18.14C You have been asked to formulate the problem of determining the best location for a new power plant from the standpoint of minimizing the average yearly exposure of the population to its emissions of SO2. Assume that the long-term average concentration of SO2 downwind of the plant can be fairly accurately represented by the Gaussian plume equation. Let (X´ ; Y´ ; Z´ ) be the location of the source in the (x´ ; y´ ; z´ ) coordinate system. Assume that α …x´ σ2y ˆ au

X ´ †β

α …x´ σ2z ˆ bu

X ´ †β

…A†

 is assumed to be in the x´ where a, b, α, and β depend on the atmospheric stability. The wind u direction. A conventional fixed (x; y; z) coordinate system with x and y pointing in the east and north directions, respectively, will not necessarily coincide with the (x´ ; y´ ; z´ ) system, which is always chosen so that the wind direction is parallel to x´ . The vertical coordinate is the same in each system, that is, z ˆ z´ . Given that the (x´ ; y´ ) coordinate system is at an angle θ to the fixed (x; y) system, show that x´ ˆ x cos θ ‡ y sin θ y´ ˆ x sin θ ‡ y cos θ

…B†

821

PROBLEMS

It is necessary to convert the ground-level concentration hc…x´ ; y´ ; 0†i to hc…x; y; 0†i, the average concentration at …x; y; 0† from a point source at …X; Y; Z† with the wind in a direction at an angle of θ to the x axis. Show that the result is hc…x; y; 0†i ˆ

q 1‡α ‰…x π…ab†1=2 u  exp

X† cos θ ‡ …y

Y† sin θŠβ

b‰…X x† sin θ ‡ …y Y† cos θŠ2 ‡ aZ2  α ab‰…x X† cos θ ‡ …y Y† sin θŠβ 2u

…C†

and that this equation is valid as long as …x

X† cos θ ‡ …y

Y† sin θ > 0

…D†

 j ; θk † be the fraction of time over a year that the wind blows with speed u j in Now let P…u direction θk . Thus, if you consider J discrete windspeed classes and K directions: J

K

 j ; θk † ˆ 1 P…u

…E†

jˆ1 kˆ1

The yearly average concentration at location …x; y; 0† from the source at …X; Y; Z† is given by J

K

c…x; y; 0† ˆ

 j ; θk † hc…x; y; 0†iP…u

…F†

jˆ1 kˆ1

where hc…x; y; 0†i is given by (C). The total exposure of the region is defined by Eˆ

…G†

C…x; y; 0†dx dy

The optimal source location problem is then to choose …X; Y† (assuming that the stack height Z is fixed) to minimize E subject to the constraint that …X; Y† lies in the region. Carry through the solution for the optimal location X of a ground-level cross-wind line source (parallel to the y j † and P1 …u j † be the fractions of time that the wind blows in axis) on a region 0  x  L. Let P0 …u the ‡x and x directions, respectively. Show that the value of X to minimize E subject to 0  X  L is 0 α0 < α1 Xopt ˆ L α0 > α1 αi ˆ

L…1 β=2† π1=2 a 1 β=2 q

J

j 1 u

α=2

j † Pi …u

i ˆ 0; 1

jˆ1

18.15B When a cloud of pollutant is deep enough to occupy a substantial fraction of the Ekman layer, its lateral spread will be influenced or possibly dominated by variations in the direction of the mean velocity with height. The mean vertical position of a cloud released at ground level increases at a velocity proportional to the friction velocity u∗ . If the thickness of the Ekman layer can be estimated as 0:2u∗ =f , where f is the Coriolis parameter, estimate the distance that a cloud must travel from its source in midlatitudes for crosswind shear effects to become important.

822

ATMOSPHERIC DIFFUSION

18.16B Consider a continuous, ground-level crosswind line source of finite length b and strength S…g km 1 s 1 †. Assume that conditions are such that the slender plume approximation is applicable. a. Taking the origin of the coordinate system as the center of the line, show that the mean ground-level concentration of material at any point downwind of the source is given by hc…x; y; 0†i ˆ

S …2π†

1=2

 σz u

b=2 y p 2σ y

erf

b=2 ‡ y p 2σy

‡ erf

b. For large distances from the source, show that the ground-level concentration along the axis of the plume …y ˆ 0† may be approximated by hc…x; 0; 0†i ˆ

Sb  πσy σz u

c. The width w of a diffusing plume is often defined as the distance between the two points where the concentration drops to 10% of the axial value. For the finite line source, at large enough distances from the source, the line source can be considered a point source. Show that under these conditions σy may be determined from a measurement of w from σy ˆ

w 4:3

18.17C In this problem we wish to examine two aspects of atmospheric diffusion theory: (1) the slender plume approximation and (2) surface deposition. To do so, consider an infinitely long, continuously emitting, ground-level crosswind line source of strength ql . We will assume that the mean concentration is described by the atmospheric diffusion equation,  u

@hci @ 2 hci @ 2 hci ˆK ‡ ‡ ql δ…x†δ…z† @x @z2 @x2

K

@hci ˆ vd hci @z

hc…x; z†i ˆ 0

zˆ0 z ! ‡1

x ˆ L

 and eddy diffusivity K are independent of the height, vd is a where the mean velocity u parameter that is proportional to the degree of absorptivity of the surface (the so-called deposition velocity), and L is an arbitrary distance from the source at which the concentration may be assumed to be at background levels. (L is a convenience in the solution and can be made as large as desired to approximate the condition hci ˆ 0 as x ! 1.) a. It is convenient to place this problem in dimensionless form by defining X ˆ x=L, Z ˆ z=L, L=K, and Sh ˆ vd L=K. C…X; Z† ˆ hc…x; z†iK=ql , Pe ˆ u Show that the dimensionless solution is C…X; Z† ˆ ePe…x=2†

1

…θn ‡ Sh† 1 ‡

nˆ1

where θ2n ˆ λn2 ‡ …Pe†2 =4 and λn ˆ ‰…2n

sin 2λn 2λn

1†=2Šπ.

1

e

θn Z

cos λn X

823

REFERENCES

b. To explore the slender plume approximation, let us now consider @hci @ 2 hci ˆK ‡ ql δ…x†δ…z† @x @z2 @hci ˆ vd hci zˆ0 K @z hc…x; z†i ˆ 0 z ! ‡1  u

hc…0; z†i ˆ 0 First, express this problem with the source ql in the x ˆ 0 boundary condition and then in the z ˆ 0 boundary condition. Solve any one of the three formulations to obtain C…X; Z† ˆ

1 Pe  erfc

Pe πX

1=2

X Pe

exp

…Pe†Z2 4X

1=2

Sh ‡

Sh exp …Sh†Z ‡

…Sh†2 X Pe

…Pe†Z 2X

(Note that even though the length L does not enter into this problem, we can use it merely to obtain the same dimensionless variables as in the previous problem.) c. We want to compare the two solutions numerically for typical parameters to explore (1) the effect of diffusion in the X direction and (2) the effect of surface deposition. Thus evaluate the two solutions for the following parameter values:  ˆ 1 and 5 m s u

1

K ˆ 1 and 10 m2 s vd ˆ 0 and 1 cm s

1 1

L ˆ 1000 m Plot C…X; Z† against Z at a couple of values of X to examine the two effects. Discuss your results.

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ATMOSPHERIC DIFFUSION Calder, K. L. (1977), Multiple-source plume models of urban air pollution—their general structure, Atmos. Environ. 11, 403–414. Crane, G., Panofsky, H., and Zeman, O. (1977), A model for dispersion from area sources in convective turbulence, Atmos. Environ. 11, 893–900. Csanady, G. I. (1973), Turbulent Diffusion in the Environment, Reidel, Dordrecht, The Netherlands. Deardorff, J. W. (1970), A three-dimensional numerical investigation of the idealized planetary boundary layer, Geophys. Fluid Dyn. 1, 377–410. Deardorff, J. W., and Willis, G. E. (1975), A parameterization of diffusion into the mixed layer, J. Appl. Meteorol. 14, 1451–1458. Doran, J. C., Horst, T. W., and Nickola, P. W. (1978), Experimental observations of the dependence of lateral and vertical dispersion characteristics on source height, Atmos. Environ. 12, 2259–2263. Draxler, R. R. (1976), Determination of atmospheric diffusion parameters, Atmos. Environ. 10, 99–105. Fisher, H. B., List, E. J., Koh, R. C. Y., Imberger, J., and Brooks, N. H. (1979), Mixing in Inland and Coastal Waters, Academic Press, New York. Galbally, I. E. (1971), Ozone profiles and ozone fluxes in the atmospheric surface layer, Quart. J. Roy. Meteorol. Soc. 97, 18–29. Gifford, F. A. (1961), Use of routine meteorological observations for estimating atmospheric dispersion, Nucl. Safety 2, 47–51. Gifford, F. A. (1968), An outline of theories of diffusion in the lower layers of the atmosphere, in Meteorology and Atomic Energy, D. Slade, ed., USAEC TID-24190, Chapter 3, U.S. Atomic Energy Commission, Oak Ridge, TN. Gifford, F. A. (1976), Turbulent diffusion-typing schemes: A review, Nucl. Safety 17, 68–86. Gifford, F. A. (1980), Smoke as a quantitative atmospheric diffusion tracer, Atmos. Environ. 14, 1119–1121. Hanna, S. R., Briggs, G. A., and Hosker, R. P. Jr. (1982), Handbook on Atmospheric Diffusion, U.S. Department of Energy Report DOE/TIC-11223, Washington, DC. Huang, C. H. (1979), Theory of dispersion in turbulent shear flow, Atmos. Environ. 13, 453–463. Irwin, J. S. (1979), Scheme for Estimating Dispersion Parameters as a Function of Release Height, EPA-600/4-79-062, U.S. Environmental Protection Agency, Washington, DC. Klug, W. (1969), A method for determining diffusion conditions from synoptic observations, Staub-Reinhalt. Luft 29, 14–20. Lamb, R. G., Chen, W. H., and Seinfeld, J. H. (1975), Numerico-empirical analyses of atmospheric diffusion theories, J. Atmos. Sci. 32, 1794–1807. Lamb, R. G., and Duran, D. R. (1977), Eddy diffusivities derived from a numerical model of the convective boundary layer, Nuovo Cimento 1C, 1–17. Lamb, R. G., and Neiburger, M. (1971), An interim version of a generalized air pollution model, Atmos. Environ. 5, 239–264. Lin, J. S., and Hildemann, L. M. (1996), Analytical solutions of the atmospheric diffusion equation with multiple sources and height-dependent wind speed and eddy diffusivities, Atmos. Environ. 30, 239–254. Lin, J. S., and Hildemann, L. M. (1997), A generalized mathematical scheme to analytically solve the atmospheric diffusion equation with dry deposition, Atmos. Environ. 31, 59–72. Lumley, J. L., and Panofsky, H. A. (1964), The Structure of Atmospheric Turbulence, Wiley, New York. Martin, D. O. (1976), Comment on the change of concentration standard deviations with distance, J. Air Pollut. Control Assoc. 26, 145–146. Morton, B. R., Taylor, G. I., and Turner, J. S. (1956), Turbulent gravitational convection from maintained and instantaneous sources, Proc. Roy. Soc. Lond. Ser. A, 234, 1–23. Murphy, B. D., and Nelson, C. B. (1983), The treatment of ground deposition, species decay and growth and source height effects in a Lagrangian trajectory model, Atmos. Environ. 17, 2545–2547. Myrup, L. O., and Ranzieri, A. J. (1976), A Consistent Scheme for Estimating Diffusivities to Be Used in Air Quality Models, Report CA-DOT-TL-7169-3-76-32, California Department of Transportation, Sacramento. Neumann, J. (1978), Some observations on the simple exponential function as a Lagrangian velocity correlation function in turbulent diffusion, Atmos. Environ. 12, 1965–1968. Nieuwstadt, F. T. M. (1980), Application of mixed layer similarity to the observed dispersion from a ground level source, J. Appl. Meteorol. 19, 157–162.

REFERENCES Nieuwstadt, F. T. M., and van Duuren, H. (1979), Dispersion experiments with SF6 from the 213 m high meteorological mast at Cabau in The Netherlands, Proc. 4th Symp. Turbulence, Diffusion and Air Pollution, Reno, NV, American Meteorological Society, Boston, pp. 34–40. O‘Brien, J. (1970), On the vertical structure of the eddy exchange coefficient in the planetary boundary layer, J. Atmos. Sci. 27, 1213–1215. Palazzi, E., DeFaveri, M., Fumarola, G., and Ferraiolo, G. (1982), Diffusion from a steady source of short duration, Atmos. Environ. 16, 2785–2790. Panofsky, H. A., Tennekes, H., Lenschow, D. H., and Wyngaard, J. C. (1977), The characteristics of turbulent velocity components in the surface layer under convective conditions, Boundary Layer Meteorol. 11, 355–361. Pasquill, F. (1971), Atmospheric diffusion of pollution, Quart. J. Roy. Meteorol. Soc. 97, 369–395. Ragland, K. W. (1976), Worst case ambient air concentrations from point sources using the Gaussian plume model, Atmos. Environ. 10, 371–374. Roberts, E. M. (1980), Conditions for maximum concentration, J. Air Pollut. Control Assoc. 30, 274–275. Schatzmann, M. (1979), An integral model of plume rise, Atmos. Environ. 13, 721–731. Sedefian, L., and Bennett, E. (1980), A comparison of turbulence classification schemes, Atmos. Environ. 14, 741–750. Shir, C. C. (1973), A preliminary numerical study of atmospheric turbulent flows in the idealized planetary boundary layer, J. Atmos. Sci. 30, 1327–1339. Taylor, G. I. (1921), Diffusion by continuous movements, Proc. Lond. Math. Soc. Ser. 2 20, 196. Tennekes, H. (1979), The exponential Lagrangian correlation function and turbulent diffusion in the inertial subrange, Atmos. Environ. 13, 1565–1567. Turner, D. B. (1969), Workbook of Atmospheric Diffusion Estimates, USEPA 999-AP-26, U.S. Environmental Protection Agency, Washington, DC. U.S. Environmental Protection Agency (1980), OAQPS Guideline Series, Guidelines on Air Quality Models, Research Triangle Park, NC. Weber, A. H. (1976), Atmospheric Dispersion Parameters in Gaussian Plume Modeling, EPA-600/4-76-030A, U.S. Environmental Protection Agency, Washington, DC. Willis, G. E., and Deardorff, J. W. (1976), A laboratory model of diffusion into the convective boundary layer, Quart. J. Roy. Meteorol. Soc. 102, 427–447.

825

P A R T V

Dry and Wet Deposition

CHAPTER

19

Dry Deposition

Wet deposition and dry deposition are the ultimate paths by which trace gases and particles are removed from the atmosphere. The relative importance of dry deposition, as compared with wet deposition, for removal of a particular species depends on the following factors: • • • •

The The The The

extent to which the substance is present in the gaseous or particulate form solubility of the species in water meteorological conditions (amount of precipitation, windspeed, etc.) terrain and type of surface cover

Dry deposition is, broadly speaking, the transport of gaseous and particulate species from the atmosphere onto surfaces in the absence of precipitation. The factors that govern the dry deposition of a gaseous species or a particle are the level of atmospheric turbulence, the chemical properties of the depositing species, and the nature of the surface itself. The level of turbulence in the atmosphere, especially in the layer nearest the ground, governs the rate at which species are delivered down to the surface. For gases, solubility and chemical reactivity may affect uptake at the surface. For particles, size, density, and shape may determine whether capture by the surface occurs. The surface itself is a factor in dry deposition. A nonreactive surface may not permit absorption or adsorption of certain gases; a smooth surface may lead to particle bounce-off. Natural surfaces, such as vegetation, whereas highly variable and often difficult to describe theoretically, generally promote dry deposition.

19.1 DEPOSITION VELOCITY It is generally impractical, in terms of the atmospheric models within which such a description is to be embedded, to simulate, in explicit detail, the microphysical pathways by which gases and particles travel from the bulk atmosphere to individual surface elements where they adhere. In the universally used formulation for dry deposition, it is assumed that the dry deposition flux is directly proportional to the local concentration C of the depositing species, at some reference height above the surface (e.g., 10 m) F ˆ υd C

(19.1)

Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, Third Edition. John H. Seinfeld and Spyros N. Pandis.  2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

829

830

DRY DEPOSITION

where F represents the vertical dry deposition flux, the amount of material depositing to a unit surface area per unit time. The proportionality constant between flux and concentration υd has units of length per unit time and is known as the deposition velocity. Because C is a function of height z above the ground, υd is also a function of z and must be related to a reference height at which C is specified. By convention, a downward flux is negative, so that υd is positive for a depositing substance. The advantage of the deposition velocity representation is that all the complexities of the dry deposition process are bundled in a single parameter, υd. The disadvantage is that, because υd contains a variety of physical and chemical processes, it may be difficult to specify properly. The flux F is assumed to be constant up to the reference height at which C is specified. Equation (19.1) can be readily adapted in atmospheric models to account for dry deposition and is usually incorporated as a surface boundary condition to the atmospheric diffusion equation. The process of dry deposition of gases and particles is generally represented as consisting of three steps: (1) aerodynamic transport down through the atmospheric surface layer to a very thin layer of stagnant air just adjacent to the surface; (2) molecular (for gases) or Brownian (for particles) transport across this thin stagnant layer of air, called the quasilaminar sublayer, to the surface itself; and (3) uptake at the surface. Each of these steps contributes to the value of the deposition velocity υd. Transport through the atmospheric surface layer down to the quasilaminar sublayer occurs by turbulent diffusion. Both gases and particles are subject to the same eddy transport in the surface layer. Sedimentation of particles may also contribute to the downward flux for larger particles. As a consequence of the no-slip boundary condition for airflow at a surface, the air at an infinitesimal distance above a stationary surface will also be stationary. Earth’s surface consists generally of a large number of irregularly shaped obstacles, such as leaves and buildings. Adjacent to each surface is a boundary layer, of thickness on the order of millimeters, in which the air is more or less stationary. As noted above, this layer is called the quasilaminar sublayer. Transport across the quasilaminar sublayer occurs by diffusion (gases and particles) and sedimentation (particles). Particle removal can actually occur within the quasilaminar sublayer by interception, when particles moving with the mean air motion pass sufficiently close to an obstacle to collide with it. Particle deposition by impaction may also occur when particles cannot follow rapid changes of direction of the mean flow, and their inertia carries them across the sublayer to the surface. Like interception, impaction occurs when there are changes in the direction of airflow; unlike interception, a particle subject to impaction leaves its air streamline and crosses the sublayer with inertia derived from the mean flow (recall Figure 9.12). The final step in the dry deposition process is actual uptake of the vapor molecules or particles by the surface. Gaseous species may absorb irreversibly or reversibly into the surface; particles simply adhere. The amount of moisture on the surface and its stickiness are important factors at this step. For moderately soluble gases, such as SO2, the presence of surface moisture can have a marked effect on whether the molecule is actually removed. For highly soluble and chemically reactive gases, such as HNO3, deposition is rapid and irreversible on almost any surface. Solid particles may bounce off a smooth surface; liquid particles are more likely to adhere on contact.

19.2 RESISTANCE MODEL FOR DRY DEPOSITION It has proved useful to interpret the deposition process in terms of an electrical resistance analogy, in which the transport of material to the surface is assumed to be governed by three resistances in series: the aerodynamic resistance ra, the quasilaminar layer resistance rb, and the surface or canopy (where the term canopy refers to the vegetation canopy) resistance rc. The total resistance, rt, to deposition of a gaseous species is the sum of the three individual resistances and is, by definition, the inverse of the deposition velocity: υd 1 ˆ rt ˆ ra ‡ rb ‡ rc

(19.2)

831

RESISTANCE MODEL FOR DRY DEPOSITION

FIGURE 19.1 Resistance model for dry deposition.

(The overall resistance for particles includes the effect of sedimentation and will be discussed shortly.) This basic conceptual model is depicted schematically in Figure 19.1. Equation (19.2) is derived as follows. By reference to Figure 19.1, let C3 be the concentration at the top of the surface layer [the reference height referred to in (19.1)]; C2, that at the top of the quasilaminar sublayer; C1, at the bottom of the quasilaminar sublayer, and C0 = 0, in the surface itself. The difference between C1 and C0 accounts for the resistance offered by the surface itself. At steady state the overall flux of a vapor species is related to the concentration differences and resistances across the layers by Fˆ

C3

C2 ra

ˆ

C2

C1 rb

ˆ

C1

C0 rc

ˆ

C3

C0 rt

(19.3)

where C0 = 0. Thus C3 C3 C2 C2 C1 C1 ˆ ˆ ˆ rt ra rb rc

(19.4)

By solving the three independent equations for the three unknowns, C1, C2, C3, one obtains (19.2). For particles, the model is identical to that for gases except that particle settling operates in parallel with the three resistances in series. The sedimentation flux is equal to the particle settling velocity, υs, multiplied by the particle concentration. It is usually assumed that particles adhere to the surface on contact so that the surface or canopy resistance rc = 0. In this case the vertical flux is Fˆ

C2

C3 ra

‡ υs C 3 ˆ

C2

C1 rb

‡ υs C 2 ˆ

C3

C1 rt

(19.5)

832

DRY DEPOSITION

where, because of assumed perfect removal, C1 = 0. Thus C2 C3 C3 C2 ˆ ‡ υs C 3 ˆ ‡ υs C2 ra rb rt

(19.6)

Solving the two independent equations to eliminate C2 and C3 yields the resistance relation analogous to (19.2) for particle dry deposition: υd ˆ

1 1 ˆ ‡ υs rt ra ‡ rb ‡ ra rb υs

(19.7)

Therefore the deposition velocity of particles may be viewed as the reciprocal of three resistances in series (ra, rb, and rarbυs) and one in parallel (1/υs). The third resistance in series is a virtual resistance since it is a mathematical artifact of the equation manipulation and not an actual physical resistance. The aerodynamic and quasilaminar resistances are affected by windspeed, vegetation height, leaf size, and atmospheric stability. In general, ra + rb decreases with increasing windspeed and vegetation height. Thus smaller resistances and hence higher deposition rates are expected over tall forests rather than over short grass. Also, smaller resistances are expected under unstable than under stable and neutral conditions. Typical aerodynamic layer resistances for a 4 m s 1 windspeed are as follows: z0 ˆ 0:1 m z0 ˆ 1 m z0 ˆ 10 m

Grass Crop Conifer forest

ra ≅ 0:6 s cm ra ≅ 0:2 s cm ra ≅ 0:1 s cm

1 1 1

In daytime, values of ra tend to be relatively small,1 and hence the precise value is seldom important (an exception is very reactive species such as HNO3, HC1, and NH3 that deposit efficiently onto any surface with which they come into contact). At night, ra is considerably higher over land and is often the controlling factor in the overall rate of deposition. Values of ra up to 1.5 s cm l can occur at night over land when turbulent mixing is reduced. When ra is high, it is usually difficult to evaluate. Fortunately, in this case, the resulting deposition fluxes are low, and hence the error in predicting the dry deposition rate tends to be relatively less important. Over water, ra lacks the strong diurnal cycle typical of that over land. The larger the body of water and the farther from shore, the smaller the diurnal cycle in ra. The absolute magnitude of rb does not change significantly as atmospheric conditions vary. Using perfect-sink surfaces to provide zero surface resistance, Wu et al. (1992a) showed that rb for SO2 in the ambient atmosphere reached a maximum value of about 100 s cm 1 for a range of windspeeds for smooth anthropogenic (human-made) surfaces. Much smaller values would be expected for rough natural vegetation. This suggests that rb can be of comparable value to ra during the day and is probably much smaller than ra at night. Thus rb is not often rate-limiting for gases. Under stable conditions, especially with low winds, ra dominates dry deposition, whereas under neutral and unstable conditions, ra is seldom the controlling resistance for dry deposition. Over some surfaces (especially more barren landscapes), rc can be quite large, thereby dominating the dry deposition process. The greatest deposition velocities are achieved under highly unstable conditions over transpiring vegetation, when the three resistances are all small and of approximately the same order of magnitude. Table 19.1 summarizes typical dry deposition velocities for some atmospheric trace gases. For gaseous species, solubility and reactivity are the major factors affecting surface resistance and overall deposition velocity. For particles, the factor most strongly influencing the deposition velocity is the particle size. Particles are transported by turbulent diffusion toward the surface identically to gaseous species, a process enhanced for larger particles by gravitational settling. Across the quasila­ minar layer ultrafine particles (< 0.05 μm diameter) are transported primarily by Brownian diffusion, 1

Remember that a small resistance equates to a large deposition velocity.

833

RESISTANCE MODEL FOR DRY DEPOSITION

TABLE 19.1 Typical Dry Deposition Velocities over Various Surfaces υd cm s

1

Species

Continent

Ocean

Ice/snow

CO N2 O NO NO2 HNO3 O3 H2O2

0.03 0 0.016 0.1 4 0.4 0.5

0 0 0.003 0.02 1 0.07 1

0 0 0.002 0.01 0.5 0.07 0.32

Source: Hauglustaine et al. (1994).

analogous to the molecular diffusion of gases. Larger particles possess inertia, which may enhance the flux through the quasilaminar sublayer (Davidson and Wu 1990). Figure 19.2 shows deposition velocity data for particles as a function of particle size for a water surface in a wind tunnel. Zufall et al. (1999) have shown that deposition to bodies of water exposed to the atmosphere can be twice as fast as deposition to a flat water surface owing to the presence of waves.

FIGURE 19.2 Particle dry deposition velocity data for deposition on a water surface in a wind tunnel (Slinn et al. 1978).

834

DRY DEPOSITION

Furthermore, growth of hygroscopic particles in the humid layer above the air–water interface can increase deposition velocities compared with deposition to dry surfaces (Zufall et al. 1998). The data in Figure 19.2 indicate a characteristic minimum in deposition velocity in the particle diameter range between 0.1 and 1.0 μm. The reason for this behavior is that small particles behave much like gases and are efficiently transported across the quasilaminar layer by Brownian diffusion. Brownian diffusion ceases to be an effective transport mechanism for particles with diameters above ∼0.05 μm. Moderately large particles in the 2–20 μm diameter range are efficiently transported across the laminar sublayer by inertial impaction (see Figure 9.12); the deposition of even larger particles (Dp > 20 μm) is governed principally by gravitational settling since the settling velocity increases with the square of the particle diameter. The transport mechanisms for particles in the 0.05–2-μm-diameter range are less effective than those of smaller or larger particles. The ambient particle size distribution thus has a marked effect on the overall predicted particle dry deposition velocity (Ruijgrok et al. 1995).

19.3 AERODYNAMIC RESISTANCE Turbulent transport is the mechanism that brings material from the bulk atmosphere down to the surface and therefore determines the aerodynamic resistance. The turbulence intensity is dependent principally on the lower atmospheric stability and the surface roughness and can be determined from micrometeorological measurements and surface characteristics such as windspeed, temperature, and radia­ tion and the surface roughness length (see Chapter 16). During daytime conditions, the turbulence intensity is typically large over a reasonably thick layer (i.e., the well-mixed layer), thus exposing a correspondingly ample reservoir of material to potential surface deposition. During nighttime, stable stratification of the atmosphere near the surface often reduces the intensity and vertical extent of the turbulence, effectively diminishing the overall dry deposition flux. The aerodynamic resistance is independent of species or whether a gas or particle is involved. The aerodynamic component of the overall dry deposition resistance is typically based on gradient transport theory and mass transfer/momentum transfer similarity (or mass transfer/heat transfer similarity). It is presumed that turbulent transport of species through the surface layer (i.e., constant-flux layer) is expressible in terms of an eddy diffusivity multiplied by a concentration gradient, that turbulent transport of material occurs by mechanisms that are similar to those for turbulent heat and/or momentum transport, and that measurements obtained for one of these entities thus can be applied, using scaling parameters, to calculate the corresponding behavior of another (see Chapter 16). Expressions for the aerodynamic resistance are most easily obtained by integrating the micrometeorological flux–gradient relationships. Applications of similarity theory to turbulent transfer through the surface layer suggest that the eddy diffusivity should be proportional to the friction velocity and the height above the ground. Under diabatic conditions the eddy diffusivity is modified from its neutral form by a function dependent on the dimensionless height scale ζ = z/L, where L is the Monin–Obukhov length. The vertical turbulent flux of a quantity, say, C, through the (constant-flux) surface layer is expressed as Fa ˆ K

@C @z

(19.8)

where K is the appropriate eddy diffusivity and Fa is, by definition, constant across the layer. From dimensional analysis and micrometeorological measurements, the eddy momentum (KM) and heat diffusivities (KT) can be expressed by κu∗ z (19.9) KM ˆ ϕM … ζ † KT ˆ

κu∗ z ϕT … ζ †

(19.10)

835

QUASILAMINAR RESISTANCE

where κ is the von Karman constant, u∗ is the friction velocity, and ϕM and ϕT, respectively, are empirically determined dimensionless momentum and temperature profile functions. If (19.8) is integrated across the depth of the constant-flux (i.e., surface) layer from z0 (the roughness less length) to zr (the reference height implicit in the definition of vd in (19.1)), the flux Fa may be written as Fa ˆ …C3

C2 †

zr

ϕ… ζ † dz ∫ z0 κu∗ z

1

(19.11)

where, as above, C3 and C2 refer to concentrations at the top and bottom of the constant- flux layer and ϕ(ζ) denotes either ϕM(ζ) or ϕT(ζ), whichever is deemed analogous to the species profile function. The aerodynamic resistance is thus given by ra ˆ

zr

ϕ… ζ † dz ∫ z0 κu∗ z

(19.12)

Explicit Expressions for ra If the stability-dependent temperature profile function is given by (16.129), then 1 ‡ 4:7ζ ϕT … ζ † ˆ

1 …1

15ζ†

1=4

for 0 < ζ < 1

…stable†

for ζ ˆ 0

…neutral†

for

1H

Thus the source is taken at a height h and with strength q. Horizontal inhomogeneity is neglected, and neutral stability is assumed up to a layer at height H, thereafter remaining constant. The object of the model is to be able to study the effect of υd on the vertical concentration profiles and thereby to assess the degree of importance of dry deposition. a. Assuming that the source height is in the constant diffusivity layer, that is, h > H, show that 1‡ c … z† ˆ

υd 1‡ κu∗ 1

υd z ln zυ κu∗ h H H ‡ ln zυ H

zυ  z  H

υd …h z† υd h H H κu∗ H 1 ‡ ‡ ln zυ κu∗ H

b. Given a source height h < H, show that υd ln κu∗ c…z† ˆ υd 1‡ ln κu∗ 1‡

z zυ h zυ

Hz Ceq), and vice versa (when Cg < Ceq). However, in the limiting case when Cg  Ceq ; one can neglect the flux from the aqueous to the gas phase and assume that W t …z; t† ≅ Kc Cg …z; t†

(20.13)

This simplification frees us from the need to estimate the aqueous-phase concentration Caq …z; t†.This will be a rather good approximation for a very soluble species, that is, a species with sufficiently high effective Henry’s law constant H∗ . Nitric acid, with a Henry’s law constant of 2.1 × 105 M atm 1, is a good example of such a species. Recalling that dissolved nitric acid dissociates to produce nitrate, we obtain HNO3 aq „ NO3 ‡ H‡ and the effective Henry’s law coefficient H ∗HNO3 for the equilibrium between nitric acid vapor and total dissolved nitrate N…V† ˆ ‰HNO3 Š ‡ NO3 ; is given by (7.59) H ∗HNO3 ˆ HHNO3 1 ‡

Kn H‡

(20.14)

where H HNO3 ˆ 2:1  105 M atm 1 at 298 K, and Kn is the dissociation constant of HNO3 (aq) equal to 15.4 M. Therefore the effective Henry’s law coefficient is H ∗HNO3 ˆ 3:23  106 =‰H∗ Š and for a reasonable ambient pH range from 2 to 6 it varies from 3.2 × 108 to 3.2 × 1012 M atm 1. If the gas-phase HNO3 mixing ratio is 10 ppb (10 8 atm), then the corresponding equilibrium aqueous-phase concentration is 3–30,000 M. This is an extremely high concentration (rainwater concentrations of nitrate are several orders of magnitude less), and the assumption Cg  Caq =H ∗HNO3 RT appears reasonable. To view it from a different perspective, nitrate rainwater concentrations Caq are in the 10–300 μM range (US NAPAP 1991), so the corresponding equilibrium concentrations Ceq ˆ Caq =H ∗HNO3 RT < 0.001 ppb and as Cg  10 ppb the assumption Cg  Ceq should hold under most ambient conditions. Let us calculate the scavenging rate of HNO3(g) during a rain event. There are two ways to approach this problem: 1. To follow a falling raindrop calculating the amount of HNO3(g) removed by it 2. To simulate the evolution of the gas-phase concentration of HNO3 as a function of height Let us use the falling-drop approach. The rate of increase of the concentration Caq of an irreversibly scavenged gas in a droplet can be estimated by a mass balance between the rate of increase of the mass of species in the droplet and the rate of transport of species to the drop 1 3 dCaq ˆ πD2p W t πD 6 p dt

(20.15)

or, using the irreversible flux given by (20.13): dCaq 6Kc ˆ Cg dt Dp

(20.16)

862

WET DEPOSITION

For a falling raindrop with a terminal velocity U t , we can change the independent variable in (20.16) from time to height noting that, from the chain rule dCaq dCaq dz dCaq ˆ ˆ Ut dt dz dt dz

(20.17)

where z is the fall distance. Combining (20.16) and (20.17), we obtain dCaq 6Kc ˆ Cg dz Dp U t

(20.18)

Assuming that the droplet diameter remains constant as the drop falls and that its initial concentration is C0aq , we get the following, by integrating (20.18): Caq ˆ C0aq ‡

6Kc z Cg y dy U t Dp ∫ 0

(20.19)

If the atmosphere is homogeneous, and Cg is constant with height, then Caq ˆ C0aq ‡

6Kc Cg z U t Dp

(20.20)

and the concentration of the irreversibly scavenged species in the drop varies linearly with the fall distance. Note that z = 0 corresponds to cloud base level. If the drop falls a distance h corresponding to the cloud base, the amount scavenged below cloud per droplet will be mscav ˆ

1 3 πD 6 p

C0aq ˆ

Caq

πD2p Kc Cg h Ut

(20.21)

Let us assume that the rain intensity is p0 (mm h 1) and that all rain droplets have diameter Dp (m). The volume of water deposited per unit surface area will then be 10 3p0 (m3 m 2 h 1). The number of droplets falling per surface area per hour is 6  10 3 p0 =πD3p and the rate of wet removal of the material below cloud Fbc is [using (20.21)] Fbc ˆ

6  10 3 p0 Kc h Cg U t Dp

(20.22)

A mass balance on the species in a 1 m2 column of air below the cloud suggests that Fbc ˆ h

dCg dt

(20.23)

and after integration Cg ˆ C0g e

Λt

(20.24)

where the scavenging coefficient is ˈ

6  10 3 p0 Kc U t Dp

(20.25)

863

BELOW-CLOUD SCAVENGING OF GASES

TABLE 20.1 Estimation of the Scavenging Coefficient Λ for Irreversible Scavenging in a Homogeneous Atmosphere (p0 = 1 mm h − 1) Dp ; cm

Ut ; cm s

0.001 0.01 0.1 1.0

0.3 26 300 1000

1

Kc ; cm s 220 32 13 6

1

Λ; h

1

4.4 × 105 73.8 0.26 0.0036

Λ 1; h 2.3 × 10 0.01 3.8 278

6

The parameters Kc and U t are given in Table 20.1 as a function of the droplet diameter. The scavenging coefficients in Table 20.1 have been estimated for p0 = 1 mm h 1 assuming that all the raindrops have the same size Dp. As the scavenging coefficient according to (20.25) depends linearly on p0, one just needs to multiply the Λ values in Table 20.1 by the appropriate rainfall intensity. Table 20.1 demonstrates that the scavenging coefficient depends dramatically on raindrop diameter. Small drops are very efficient in scavenging soluble gases for two reasons: (1) they fall more slowly, so they have more time in their transit to “clean” the atmosphere; and (2) mass transfer is more efficient for these drops (large Kc). Note that the scavenging rate varies over eight orders of magnitude when the drop diameter increases by three orders. Also note that the scavenging timescale (1/Λ) can vary from less than a second to several hours depending on the raindrop size distribution. Droplets smaller than 2 mm are responsible for most of the scavenging in general. Typical scavenging rates are in the range of 1–3% min 1 for irreversibly soluble gases such as HNO3. These rates indicate that HNO3 and other very soluble gases are significantly depleted during a typical 30-min rainfall. An assumption inherent in our preceding analysis is that the diameter of a raindrop remains approximately constant during its fall. Raindrops often evaporate during their fall, as they pass through a subsaturated environment. Finally, we can now revisit our initial assumption that HNO3 scavenging is practically irreversible. Figure 20.3 shows the HNO3 concentration in the aqueous phase for droplets of

FIGURE 20.3 Concentration of dissolved HNO3 and corresponding equilibrium vapor pressure over drops of different diameters as a function of fall distance for pHNO3 ˆ 10

8

atm (10 ppb) (Levine and Schwartz 1982).

864

WET DEPOSITION

different sizes, falling through an atmosphere containing 10 ppb of HNO3. The equilibrium gas-phase mixing ratio (plotted on the upper x axis) remains below 10 ppb (10 8 atm) even for the smaller droplets, providing more support for the validity of our assumption.

20.2.2 Below-Cloud Scavenging of a Reversibly Soluble Gas For a reversibly soluble gas, one needs to account for both the flux from the gas to the aqueous phase and the inverse flux. The mass balance of (20.15) is still valid, so combining it with (20.11), we get dCaq …z; t† 6Kc ˆ Cg …z; t† dt Dp

Caq …z; t† H ∗ RT

(20.26)

where the effective Henry’s law coefficient H∗ can be a function of the pH, and Caq is the net aqueousphase species concentration (including both the associated and dissociated forms). Equation (20.26) can be rewritten using height as an independent variable as dCaq …z; t† 6Kc ˆ Cg …z; t† dz Dp U t

Caq …z; t† H ∗ RT

(20.27)

This equation indicates that the driving force for scavenging is the difference between the gas-phase concentration of the species and the concentration at the drop surface (droplet equilibrium concentra­ tion). As the aqueous-phase concentration increases during the droplet fall, the driving force will tend to decrease. Let us assume that the rain droplet size and droplet pH remain constant during its fall and that the atmosphere is homogeneous so that Cg …z; t† ˆ Cg …t†: Then (20.27) can be integrated using Caq …0; t† ˆ C0aq to give Caq …z; t† ˆ Cg …t†H ∗ RT

Cg …t†H ∗ RT

C0aq exp

6Kc z Dp Ut H ∗ RT

(20.28)

This solution exhibits several interesting limiting cases. If the droplet starts at equilibrium with the below-cloud atmosphere …C0aq ˆ H ∗ RTCg …t††; then its aqueous-phase concentration will remain constant and the saturated droplet will not scavenge any of the species from the below-cloud region. We note also that for z  Dp U t H ∗ RT=…6Kc †; Caq ≅ H ∗ RTCg …t† and the drop after a certain fall distance reaches equilibrium with the environment and stops scavenging additional material from the gas phase. The scavenging rate s(t) for one droplet of diameter Dp can be calculated by dCaq 1 3 dCaq 1 s…t† ˆ πD3p ˆ πDp U t 6 dt 6 dz

(20.29)

Differentiating (20.28) and substituting into (20.29), we obtain s…t† ˆ

πD2p Kc H ∗ RT

Cg …t†H ∗ RT

C0aq exp

6Kc z Dp U t H ∗ RT

(20.30)

The overall below-cloud scavenging rate, W bc ; can be calculated once more by multiplying the per droplet scavenging rate by the number of droplets per volume of air …6  10 3 p0 =πD3p U t †: W bc ˆ

6  10 3 p0 Kc Cg …t†H ∗ RT H ∗ RTDp U t

C0aq exp

6Kc z Dp U t H ∗ RT

(20.31)

865

BELOW-CLOUD SCAVENGING OF GASES

Because of the reversibility of the scavenging process, one can define a scavenging coefficient only if

Cg H ∗ RT  C0aq . This definition is valid when the initial raindrop species concentration is much lower

than the equilibrium concentration corresponding to the below-cloud atmospheric conditions. If so, then Λ … z† ˆ

6  10 3 p0 Kc exp Dp U t

6Kc z Dp U t H ∗ RT

(20.32)

and, as expected, the local scavenging coefficient is a function of fall distance. As the droplet continues to fall, its aqueous-phase concentration approaches equilibrium and the local scavenging rate approaches zero. When the drop reaches the ground after a fall through an atmospheric layer of thickness h, it will have a concentration according to (20.28) equal to Caq …h; t† ˆ Cg …t†H∗ RT

Cg …t†H ∗ RT

6Kc h Dp U t H ∗ RT

C0aq exp

(20.33)

and will have scavenged a mass of species equal to 1 3

πD Caq …h; t† C0aq 6 p 1 ˆ πD3p Cg …t†H ∗ RT C0aq 6

mscav ˆ

1

exp

6Kc h Dp Ut H ∗ RT

(20.34)

The total scavenging rate by the rain from the below-cloud atmosphere (equal to the wet deposition rate due to below-cloud scavenging) is Fbc ˆ 10 3 p0 Cg …t†H ∗ RT

C0aq

1

exp

6Kc h Dp U t H ∗ RT

(20.35)

It is interesting to note that this overall scavenging rate can be negative if Cg …t†H ∗ RT < C0aq . This case corresponds to a droplet falling through an atmosphere that is below saturation compared to the species concentration in the droplet. In this case, the species will evaporate from the droplet during its fall and rain will result in a net gain of species mass for the below-cloud atmosphere. Note also that the maximum scavenging rate is reached for 6Kc h=Dp U t H ∗ RT  1 and is equal to …Fbc †max ˆ 10 3 p0 Cg …t†H∗ RT

C0aq

(20.36)

Our analysis above applies to an idealized scavenging scenario. In general, during a drop’s fall, the dissolution of species changes the pH and therefore the effective Henry’s law constant of dissociating species. At the same time, aqueous-phase reactions may take place inside the raindrops, usually resulting in an acceleration of the scavenging process. Study of the interaction of these processes requires solution of a system of differential equations numerically. A representative example is given next. Fall of a Droplet through a Layer Containing CO2, SO2, HNO3, O3, and H2O2 We want to calculate the evolution of the aqueous-phase concentrations in a drop of diameter Dp (Dp = 5 mm), falling through an h = 1000-m-deep atmospheric layer, as a function of its fall distance. Assume that the atmospheric gas-phase concentrations of all species remain constant with values given in Table 20.2.

866

WET DEPOSITION

TABLE 20.2 Species Mixing Ratios (ppb) for Example Case A B C D

ξH2 O2

ξO3

ξHNO3

ξNH3

ξSO2

ξCO2 a

10 10 0 10

50 50 0 50

10 0 10 10

5 5 5 0

20 20 20 20

300 300 300 300

a

In ppm.

Case A represents the base-case scenario. In case B, no HNO3 is present, so by comparing cases A and B, we can determine the HNO3 effect on droplet pH. In case C we remove the oxidants (H2O2 and O3), to study the effect of S(IV) on the drop composition. Finally, case D has no NH3, so comparing cases A and D we can see the effect of the neutralization by NH3. We assume that initally the drop is in equilibrium with the ambient CO2 and NH3. The droplet composition will be characterized by the concentrations of the following species: CO2
H2O, HCO3 , CO23 ; SO2
H2O, HSO3 , SO23 , NH4OH, NH‡4 , HNO3 (aq), NO3 , O3, H2O2, HO2 , H+, OH , H2SO4, HSO4 , and SO24 . As we saw in our discussion of aqueous-phase chemistry, instead of describing all the individual concentrations, we can define the total species concentrations as COT2

ˆ ‰CO2 ? H2 OŠ ‡ HCO3 ‡ CO23

‰S…IV†Š

ˆ ‰SO2 ? H2 OŠ ‡ HSO3 ‡ SO23

HNOT3 ˆ HNO3 aq NHT3 H2 O2T ‰S…VI†Š

‡ NO3

(20.37)

ˆ ‰NH3 ? H2 OŠ ‡ NH‡4 ˆ ‰H2 O2 Š ‡ HO2 ˆ ‰H2 SO4 Š ‡ HSO4 ‡ SO24

and adding O3 we have reduced the problem size from 18 variables to 7, plus the pH that can be calculated from the electroneutrality equation. Note that sulfate will be the product of the S(IV) oxidation. The mass balance for total aqueous-phase nitric acid (assuming reversible transfer between the droplets and the surrounding atmosphere) will then be pHNO3 d HNOT3 1 3 ˆ πD2p Kc;HNO3 πDp U t dz RT 6

HNOT3 H ∗HNO3 RT

(20.38)

where H ∗HNO3 is the effective Henry’s law coefficient for nitric acid given by (20.14) H ∗HNO3 ˆ H HNO3 1 ‡

Kn H‡

(20.39)

with H HNO3 and Kn the nitric acid Henry’s law coefficient and the first dissociation constant, respectively. Equation (20.38) indicates that the total dissolved nitric acid changes due to mass transfer, as described by the product of the mass transfer coefficient for HNO3 transport to a falling sphere of diameter Dp in air Kc;HNO3 and the driving force, which is the difference between the ambient HNO3 concentration pHNO3 =RT and the vapor pressure of HNO3 just above the drop surface. Using the definition of the effective Henry’s law coefficient in (20.39), we can rewrite (20.38) as d HNOT3 6Kc;HNO3 ˆ dz U t Dp RT

pHNO3

HNOT3 H‡ H HNO3 H‡ ‡ Kn

(20.40)

867

BELOW-CLOUD SCAVENGING OF GASES

Note that only HNOT3 and H‡ appear on the RHS of the equation. Equation (20.40) is an example of a material balance obtained on a drop for a gas that dissociates on absorption but does not participate in further aqueous-phase chemistry. Consider now O3, a gas that does not dissociate on absorption but that participates in aqueousphase chemistry reacting with S(IV). The material balance in this case is 1 d‰O3 Š ‰O 3 Š ‡ H O3 Ut dt

d‰ O 3 Š 6Kc;O3 ˆ p dz U t Dp RT O3

(20.41) react

The aqueous-phase reaction rate between O3 and S(IV) can be calculated using the kinetic expressions given in Chapter 7. Dissolved SO2 both dissociates and reacts with O3 and H2O2. The mass balance in this case is d‰S…IV†Š 6Kc;SO2 ˆ dz U t Dp RT

pSO2

1 d‰S…IV†Š ‡ Ut dz

‰S…IV†Š H‡ H SO2

H‡

2

2

‡ Ks1 H‡ ‡ Ks1 Ks2

(20.42)

react

Sulfate, [S(VI)], is an example of a species that is produced in the droplet but is not transferred between the gas and aqueous phases. Its material balance is simply d‰S…IV†Š ˆ dz

1 d‰S…IV†Š dz Ut

(20.43) react

The mass balances for the remaining species can be written similarly, recalling that COT2 and NHT3 are species that are transferred between the two phases, dissociate, but do not participate in aqueous-phase reactions, while H2O2 has a behavior similar to that of S(IV).

All these differential equations are coupled through the electroneutrality equation H‡ ‡ NH‡4 ˆ ‰OH Š ‡ HCO3 ‡ 2 CO23 ‡ NO3 ‡ 2 SO24

‡ HSO3 ‡ 2 SO23

‡ HSO4 ‡ HO2

(20.44)

Each term in (20.44) can be related to NHT3 , COT2 , ‰S…IV†Š, HNOT3 , H2 OT2 , and [S(VI)]. Mathematically the problem is equivalent to the solution of seven differential equations (for COT2 ; ‰S…IV†Š; HNOT3 ; NHT3 ; H2 O2T ; ‰S…IV†Š; and [O3]) coupled with one algebraic equation, (20.44). The problem could be simplified depending on the conditions by using appropriate equilibrium assumptions. For example, if one assumes that CO2 is in equilibrium between the gas and aqueous phases and its gas-phase concentration remains constant, one of the differential equations is eliminated. The reader is referred back to Chapter 12 for a discussion of the validity of similar assumptions for the other species. In the solutions presented below we have assumed that CO2, NH3, and O3 are in equilibrium between the two phases. Figure 20.4 shows [H+] as a function of drop fall distance in cases A, B, C, and D. As expected, the pH is lowest in case D in the absence of NH3. The removal of HNO3 in case B relative to A leads to only a slight increase in pH, primarily because the large drop does not absorb an appreciable amount of HNO3 during its fall. The corresponding sulfate profiles are given in Figure 20.5. Sulfate concentrations increase to 300 μM in cases A and B, since the absence of HNO3 for the 5 mm drop was seen in the previous figure to have only a negligible effect on pH. Absence of NH3 in case D leads to a much lower sulfate concentration of a little more than 100 μM, due to the significant pH reduction.

868

WET DEPOSITION

FIGURE 20.4 Fall of a drop through a layer containing CO2, SO2, HNO3, NH3, O3, and H2O2. [H+] as a function of fall distance in cases A–D in text and Table 20.2.

FIGURE 20.5 Fall of a drop through a layer containing CO2, SO2, HNO3, NH3, O3, and H2O2. SO24 as a function of fall distance in cases A to D in text.

20.3 PRECIPITATION SCAVENGING OF PARTICLES As a raindrop falls through air, it collides with airborne particles and collects them. As the droplet falls, it sweeps per unit time the volume of a cylinder equal to πD2p U t =4; where U t is its fall velocity. As a first approximation, one would be tempted to conclude that the droplet would collect all the particles that are in this volume. Actually, if the aerosol particles have a diameter dp, a collision will occur if the center of the particle is inside the cylinder with diameter Dp + dp. Also, the particles are themselves settling with a 2

ut dp =4. The major velocity ut dp . So the collision volume per time is actually π Dp ‡ dp U t Dp complication associated with this simple picture, as we saw in Chapter 17, arises because the falling drop perturbs the air around it, creating the flow field depicted in Figure 17.25. The flow streamlines diverge

869

PRECIPITATION SCAVENGING OF PARTICLES

around the drop. As the raindrop approaches the particle, it exerts a force on it via the air as a medium and modifies its trajectory. Whether a collision will in fact occur depends on the sizes of the drop and the particle and their relative locations. Prediction of the actual trajectory of the particle is a complicated fluid mechanics problem. Solutions are expressed as the collision efficiency E Dp ; dp ; which is defined in exactly the same manner as is the drop-to-drop collision efficiency. Therefore E Dp ; dp is the fraction of particles of diameter dp contained within the collision volume of a drop of diameter Dp that are collected. E Dp ; dp can be viewed as a correction factor accounting for the interaction between the falling raindrop and the aerosol particle. If the aerosol size distribution is n dp ; the number of collisions between particles of diameters in the range dp ; dp ‡ ddp and a drop of diameter Dp is π D p ‡ dp 4

2

ut dp E Dp ; dp n dp ddp

U t Dp

(20.45)

The rate of mass accumulation of these particles by a single falling drop can be calculated by simply replacing the number distribution n dp by the mass distribution nM dp to obtain π D p ‡ dp 4

2

ut dp E Dp ; dp nM dp ddp

U t Dp

(20.46)

The total rate of collection of mass of all particles of diameter dp is obtained by integrating (20.46) over the size distribution of collector drops 1

π D ‡ dp ∫0 4 p  E Dp ; dp N Dp dDp

W bc dp ˆ nM dp ddp

2

U t Dp

ut d p

(20.47)

where N Dp is the raindrop number distribution.

Two approximations can generally be made in (20.47):

1. U t Dp  ut dp 2 2. Dp ‡ dp ≅ D2p Using these approximations, (20.47) becomes W bc dp ˆ nM dp ddp

1

π 2 D U D E Dp ; dp N Dp dDp ∫0 4 p t p

(20.48)

Therefore the below-cloud scavenging (rainout) rate of aerosol particles of diameter dp can be written as dnM dp ˆ Λ d p nM d p dt

(20.49)

where the scavenging coefficient Λ dp is given by Λ dp ˆ

1

π 2 D U D E Dp ; dp N Dp dDp ∫0 4 p t p

(20.50)

Calculation of the aerosol scavenging rate, for a given aerosol diameter dp, therefore requires knowledge of the droplet size distribution N Dp and the scavenging efficiency E Dp ; dp :

870

WET DEPOSITION

The total aerosol mass scavenging rate can be calculated by integrating over the aerosol size distribution and using (20.49) to get 1 dMaer d 1 ˆ nM dp ddp ˆ Λ dp nM dp ddp ∫0 dt dt ∫ 0

Finally, the rainfall rate p0 mm h

1

(20.51)

is related to the raindrop size distribution by p0 ˆ

1

π 3 D U D N Dp dDp ∫0 6 p t p

(20.52)

While routine measurements of p0 are available, knowledge of reliable raindrop size distributions remains a problem because of variability from event to event as well as during the same rainstorm.

20.3.1 Raindrop–Aerosol Collision Efficiency The collision efficiency E Dp ; dp is by definition equal to the ratio of the total number of collisions occurring between droplets and particles to the total number of particles in an area equal to the droplet’s effective cross-sectional area. A value of E = 1 implies that all particles in the geometric volume swept out by a falling drop will be collected. Usually E  1; although E can exceed unity under certain conditions (charged particles). Experimental data suggest that all particles that hit a hydrometeor stick; therefore, a sticking efficiency of unity is assumed. Theoretical solution of the Navier–Stokes equation for prediction of the collision efficiency, E Dp ; dp , for the general raindrop–aerosol interaction case is a difficult undertaking. Complications arise because the aerosol size varies over orders of magnitude, and also because the large raindrop size results in complicated flow patterns (drop oscillations, wake creation, eddy shedding, etc.) Pruppacher and Klett (1997) present a critical overview of the theoretical attempts for the solution of the problem. A detailed discussion of these efforts is outside our scope. However, it is important to understand at least qualitatively the various processes involved. Particles can be collected by a falling drop as a result of their Brownian diffusion. This random motion of the particles will bring some of them in contact with the drop, and will tend to increase the collection efficiency E. Because the Brownian diffusion of particles decreases rapidly as particle size increases, this removal mechanism will be most important for the smaller particles (dp < 0.2 μm). Inertial impaction occurs when a particle is unable to follow the rapidly curving streamlines around the falling spherical drop and, because of its inertia, continues to move toward the drop and is eventually captured by it. Inertial impaction increases in importance as the aerosol size increases and accelerates scavenging of particles with diameters larger than 1 μm. The preceding arguments indicate that whereas the scavenging of small and large particles is expected to be efficient, scavenging of particles in the 0.1–1 μm size range is expected to be relatively inefficient. This minimum is sometimes referred to as the Greenfield gap, after S. Greenfield, who first identified it. Finally, interception takes place when a particle, following the streamlines of flow around an obstacle, comes into contact with the raindrop, because of its size. If the streamline on which the particle center lies is within a distance dp/2 or less from the drop surface, interception will occur. Interception and inertial impaction are closely related, but interception occurs as a result of particle size neglecting its mass, while inertial impaction is a result of its mass neglecting its size. An alternative to an exact solution of the Navier–Stokes equation is the use of dimensional analysis coupled with experimental data. To formulate a correlation for E based on dimensional analysis, we must identify first the parameters that influence E. There are eight such variables: aerosol diameter (dp), raindrop diameter (Dp), velocities of the aerosol and raindrop …ut ; U t †; viscosities of water and air μw ; μair ; aerosol diffusivity (D), and air density ρa . We assume here that the aerosol has density ρp equal to 1 g cm 3, so this density is not included in the list. Water viscosity appears in the list because internal circulations may be established in the drop, affecting the flow field around it and thus the

871

PRECIPITATION SCAVENGING OF PARTICLES

capture efficiency. These eight variables have three fundamental dimensions (mass, length, and time).

Thus, by the Buckingham π theorem, there are five (eight minus three) independent dimensionless

groups. These groups can be obtained by nondimensionalizing the equations of motion for air and for

the particles. They are as follows (Slinn 1983):

Re ˆ Dp U t ρa =2μa Sc ˆ μa =ρa D St ˆ 2τ…U t ut †=Dp ϕ ˆ dp =Dp ω ˆ μw =μa

(Reynolds number of raindrop based on its radius)

(Schmidt number of collected particle)

(Stokes number of collected particle, where τ is its characteristic relaxation time)

(ratio of diameters)

(viscosity ratio)

On the basis of these equations, Slinn (1983) proposed the following correlation for E that fits experimental data Eˆ

4 1 ‡ 0:4 Re1=2 Sc1=3 ‡ 0:16 Re1=2 Sc1=2 Re Sc 3=2

‡4ϕ ω

1

‡ 1 ‡ 2 Re1=2 ϕ ‡

St St

S∗ S∗ ‡

(20.53)

2 3

where S∗ ˆ

1 ln…1 ‡ Re† 1:2 ‡ 12 1 ‡ ln…1 ‡ Re†

(20.54)

For particles of density different from 1 g cm 3, the last term in (20.53) should be scaled by …ρw =ρp †1=2 . The first term in (20.53) is the contribution from Brownian diffusion, the second is due to interception, and the third represents impaction. The third term on the RHS of (20.53) requires some comment; this term represents the effect of impaction. If S∗ St < 2/3 , the term inside the parentheses is 20 μm) and extremely small ones, the collection efficiency approaches unity. However, for particles close to the minimum (dp  1 μm) the drops collect only particles that are close to the center of the volume swept by the raindrop.

20.3.2 Scavenging Rates Using expressions obtained for the collision efficiency for E Dp ; dp in (20.50) and (20.51), one can estimate the scavenging coefficient, and the scavenging rate for a rain event. The calculation requires knowledge of the size distributions of the raindrops and the below-cloud aerosols. The scavenging coefficient Λ(dp) given by (20.50) describes the rate of removal of particles of diameter dp by rain with a raindrop size distribution N(Dp). If one assumes that all raindrops have the

872

WET DEPOSITION

FIGURE 20.6 Semiempirical correlation for the collection efficiency E of two drops (Slinn 1983) as a function of the collected particle size. The collected particle is assumed to have unit density.

same diameter Dp, and a number concentration ND, then (20.50) simplifies to π Λ dp ˆ D2p U t Dp E Dp ; dp N D 4

(20.55)

The only term depending on the aerosol size dp is the collection efficiency E. The number concentration of drops can be estimated using (20.52) as a function of the rainfall intensity by π p0 ˆ Dp3 U t Dp N D 6

(20.56)

and for monodisperse aerosols and raindrops, (20.55) becomes Λ dp ˆ

3 E D p ; d p p0 2 Dp

(20.57)

The scavenging coefficient for this simplified scenario is shown in Figure 20.7 for p0 = 1 mm h 1 for drops of diameters Dp = 0.2 and 2 mm. This figure indicates the sensitivity of the scavenging coefficient to sizes of both aerosols and raindrops, suggesting the need for realistic size distributions for both aerosol and drops in order to obtain accurate estimates for ambient scavenging rates. The total aerosol mass scavenging rate is given by (20.51). Replacing the aerosol mass distribution with the number distribution, the overall mass scavenging rate is given by 1 dMaer π 3 ˆ d n dp ρp Λ dp ddp ∫0 6 p dt

(20.58)

873

IN-CLOUD SCAVENGING

FIGURE 20.7 Scavenging coefficient for monodisperse particles as a function of their diameter collected by monodisperse raindrops with diameters 0.2 and 2 mm assuming a rainfall intensity of 1 mm h 1.

Our calculations can be simplified by defining a mean mass scavenging coefficient Λm so that 1 1 π π ρp d3p n dp Λ dp ddp ˆ Λm ρp d3p n dp ddp 6 ∫0 6 ∫0

(20.59)

or equivalently by 1

Λm ˆ

∫0

Λ dp d3p n dp ddp 1

∫0

(20.60) n dp d3p ddp

Note that the denominator is equal to 6V aer =π, where V aer is the particle volume concentration and assuming a particle density ρp the denominator is equal to 6Maer =πρp : Therefore, by definition, we have dMaer ˆ Λm Maer dt

(20.61)

and all the scavenging information has effectively been incorporated into one parameter.

20.4 IN-CLOUD SCAVENGING Species can be incorporated into cloud and raindrops inside the raining cloud. These processes determine the initial concentration C0aq of the raindrops, before they start falling below the cloud base, and have been discussed previously. Let us summarize the rates of in-cloud scavenging for gases and aerosols. Gases like HNO3, NH3, and SO2 can be removed from interstitial cloud air by dissolution into clouddrops. For a droplet size distribution N(Dp), the local rate of removal of an irreversibly soluble gas

874

WET DEPOSITION

like HNO3 with a concentration Cg is given by W ic ˆ Cg

1

∫0

Kc πD2p N Dp dDp ˆ ΛCg

(20.62)

where Λ is the in-cloud scavenging coefficient. Levine and Schwartz (1982) used the cloud droplet distribution of Battan and Reitan (1957) N Dp ˆ ae

bDp

(20.63)

with the parameters a = 2.87 cm 4, b = 2.65 cm 1, and Dp = 5–40 μm, to calculate Λ = 0.2 s 1 for a cloud with 288 drops cm 3 and L = 0.17 g m 3. This value of Λ corresponds to a characteristic time of approximately 5 s for the scavenging of a highly soluble gas such as HNO3 in a typical cloud. In this case the uptake process in a cloud is rapid compared with the characteristic times of air movement through the cloud and the change of cloud liquid water content amounts by condensation and precipitation. For less soluble gases such as SO2, uptake is a complex function not only of species properties and cloud droplet distribution but also of other species that may participate in aqueous-phase reactions (e.g., H2O2) or control the cloud droplet pH (e.g., NH3). Finally, aerosol in-cloud scavenging is the result of two processes. First, nucleation scavenging (growth of CCN to clouddrops) and then collection of a fraction of the remaining aerosols by cloud or rain droplets. Nucleation scavenging as we have seen is an efficient process, often incorporating in clouddrops most of the aerosol mass. On the other hand, as we saw in Chapter 17, interstitial aerosol collection by cloud droplets is a rather slow process and often scavenges a negligible fraction of the interstitial aerosol mass.

20.5 ACID DEPOSITION The atmosphere is a potent oxidizing medium. In Chapters 6 and 7 we have seen that, once emitted to the atmosphere, SO2 and NOx become oxidized to sulfate and nitrate through both gas- and aqueous-phase processes. Organic acids can be produced during the oxidation of organic compounds. The result of these reactions is the existence in the atmosphere of acids in the gas phase (HNO3, HCl, HCOOH, CH3COOH, etc.), in the aerosol phase (sulfate, nitrate, chloride, organic acids, etc.), and in the aqueous phase. These acidic species are removed from the atmosphere by both wet and dry deposition, and the overall process is termed acid deposition. The removal of acidic material by rain has traditionally been termed acid rain. Therefore acid deposition includes acid rain, dry deposition of both acid vapor and acid particles, and other forms of wet removal (acid fogs, cloud interception, etc.). Because of the historical focus on the composition of rainwater, this entire process is often called simply acid rain. Even though the term acid rain implies removal only by wet deposition, it is important to keep in mind that the effects attributable to acid rain are in fact a result of the combination of wet and dry deposition.

20.5.1 Acid Rain Overview A raindrop in a pollutant-free atmosphere has, as we have seen, a pH of 5.6 as a result of the dissolution of CO2. However, emissions of SO2 and NOx lead to conversion of these species during transport from their sources to acidic sulfate and nitrate, and their incorporation into cloud and rainwater. Addition of these acids lowers the rainwater pH, and the rain reaching the ground is acidic. 20.5.1.1 Historical Perspective The phenomenon of acid rain appears to have been recognized at least three centuries ago. In 1692 Robert Boyle published A General History of the Air, recognizing the presence of sulfur compounds and acids in air and rain. Boyle referred to them as “nitrous or salino-sulfurous spirits.” In 1853 Robert Angus Smith, an English chemist, published a report on the chemistry of rain in and around the city of Manchester, “discovering” the phenomenon. Smith published Air and Rain: The Beginnings of Chemical Climatology in

ACID DEPOSITION

1872, introducing the term acid rain. Little attention was paid to acid rain for almost a century thereafter. In 1961, Svante Odin, a Swedish chemist, established a Scandinavian network to monitor surface water chemistry. On the basis of his measurements, Odin showed that acid rain was a large-scale regional phenomenon in much of Europe with well-defined source and sink regions, that precipitation and surface waters were becoming more acidic, and that there were marked seasonal trends in the deposition of major ions and acidity. Odin also hypothesized long-term ecological effects of acid rain, including decline of fish population, leaching of toxic metals from soils into surface waters, and decreased forest growth. The major foundations for our present understanding of acid rain and its effects were laid by Eville Gorham. On the basis of his research in England and Canada, Gorham showed as early as 1955 that much of the acidity of precipitation near industrial regions can be attributed to combustion emissions, that progressive acidification of surface waters can be traced to precipitation, and that the acidity in soils receiving acid precipitation results primarily from sulfuric acid. Scandinavian and European studies, including the Norwegian Interdisciplinary Research Pro­ gramme “Acid Precipitation–Effects on Forest and Fish,” and the study by the European Organization for Economic Cooperation and Development, elucidated the effects of acid rain on fish and forests and long-distance transport of pollutants in Europe. Studies of acid rain in North America by individual researchers started as early as 1923. Concern about acid rain developed first in Canada and later in the United States with the true scope of the problem not being appreciated until the 1970s. In 1978 the United States and Canada established a Bilateral Research Consultation Group on the Long Range Transport of Pollutants and in 1980 signed a memorandum of intent to develop bilateral agreements on transboundary air pollution, including acid deposition. Both countries have instituted long-term programs for the chemical analysis of precipitation. The National Acid Precipitation Assessment Program (NAPAP) served from 1980 to 1990 as the umbrella of one of the most comprehensive acid rain studies (US NAPAP 1991). Brimblecombe (1977) and Cowling (1982) have provided excellent historical perspectives associated with the discovery of and attempts to deal with acid rain. 20.5.1.2 Definition of the Problem The global pattern of precipitation acidity in the late 1980s as determined by the Background Air Pollution Monitoring Program of the World Meteorological Organization (Whelpdale and Miller 1989) is shown in Figure 20.8. Average pH measured in background sites varied from 3.8 to 6.3. All of these values are below the absolutely neutral precipitation pH, which has a value of 7. The “natural” acidity of rainwater is often taken to be pH 5.6, which is that of pure water in equilibrium with the global atmospheric concentration of CO2. (This was about 350 ppm at the time this calculation was made; now the CO2 mixing ratio is ∼400 ppm.) Thus a pH value of 5.6 has been considered as the threshold of acidic precipitation. Figure 20.8 demonstrates that the answers to “What is acid rain?” and “Who is responsible for it– ourselves or nature?” are complicated. Note first that rainwater in several areas has average pH values higher than 6 (China, SE Australia), indicating an alkaline tendency. High pH values observed in several regions of the world are the result of alkaline dust emissions, usually from deserts. These alkaline particles have significant buffering capacity, and even after the addition of anthropogenic acidity, pH of the resulting rain may exceed 6. A more puzzling observation is that pH values measured in precipitation over remote oceans that are often acidic and sometimes very acidic (Figure 20.8). Galloway et al. (1982) reported precipitation chemistry data from remote areas of the globe, with average values around 5.0. Analysis of eastern US precipitation data by Stensland and Semonin (1982) also indicated background values of around 5.0. Keene et al. (1983) showed that organic acids (mainly formic and acetic) can play a significant role in lowering rainwater pH in some remote areas. Their measurements in Katherine, Australia showed that these acids accounted for more than half the rainwater acidity. While it is undeniable that human activities contribute, sometimes overwhelmingly, to rain acidity, it is often difficult to quantify this contribution from rain pH alone. It is reasonable to conclude that rain of pH higher than 5.6 has not been influenced by humans, or if it has, it has sufficient buffering capacity so that acidification did not occur. Rain with pH between 5.0 and 5.6 may have been influenced by anthropogenic emissions, but not to an extent exceeding that of natural background compounds.

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WET DEPOSITION

FIGURE 20.8 Map of the global pattern of precipitation acidity in the late 1980s as determined by the Background Air Pollution Monitoring Program of the World Meteorological Organization (after Whelpdale and Miller (1989)).

Precipitation measurements are usually reported as volume-weighted means. The volume-weighted mean is computed by first multiplying the concentration of a species by either the depth of rainfall or the liquid equivalent depth of snowfall. Summing over all precipitation events and dividing by the total liquid depth for all events gives the volume-weighted mean. The volume-weighted mean, assuming that the species is conserved during mixing, is the concentration that would result if the samples from all events were physically mixed and the resulting concentration measured. The volume-weighted pH is computed by first converting the individual pH measurement to H+ concentration, averaging using the sample volume as a weighting factor and reconverting to pH.

20.5.2 Surface Water Acidification Lakes, rivers, and other surface waters acidify when they lose alkalinity (Roth et al. 1985). The total alkalinity or acid-neutralizing capacity is the reservoir of bases in solution. The acid-neutralizing capacity of a lake buffers it against large changes in pH. In natural clean waters, most of the acidneutralizing capacity consists of bicarbonate ion, HCO3 . Carbonate alkalinity is defined by Alkalinity ˆ HCO3 ‡ 2 CO23

‡ ‰OH Š

H‡

When strong acids such as sulfuric acid enter bicarbonate waters, the additional acidity can be neutralized by interacting with the bicarbonate. In other terms, addition of acidity can be balanced by a shifting of the equilibrium, H‡ ‡ HCO3 ƒ CO2 ? H2 O toward CO2
H2O. If input of hydrogen ion continues, the supply of buffering ions is eventually exhausted and the lake pH decreases. Watersheds vary in their ability to neutralize acidity, so some are more susceptible than others. Soils around the surface water help buffer the pH of the water. Many soils take up hydrogen ions from acidic deposition by cation exchange with Ca2+, Mg2+, Na+, or K+.

ACID DEPOSITION

Surface water acidification can thus be avoided if the supply of base cations is sufficient and if precipitation contacts soil long enough before flowing into surface waters. The hydrology and chemistry of lakes and streams are highly individualistic. Lakes surrounded by poorly buffered soil and underlain by granitic bedrock appear to be more susceptible to acidification when exposed to acid rain (Havas et al. 1984). Other lake characteristics such as dominance of atmospheric input or surface/subsurface runoff as the major source of water, type and depth of soil, bedrock characteristics, lake size and depth, area of the drainage basin, and residence time of water in the lake are all features that influence the response of a lake to acid rain. As the pH of a lake decreases, concentrations of several potentially toxic metals, such as aluminum, iron, manganese, copper, nickel, zinc, lead, cadmium, and mercury increase (Dickson 1980; Havas et al. 1984). Quantification of the relationship between atmospheric acidity input and surface water acidification is difficult. First, certain lakes, known as dystrophic lakes, are naturally acidic. They usually have brown to yellow water caused by humic-derived organic acids. However, organic acids in these naturally acidic lakes reduce metal toxicity over that in clear water lakes (Baker and Schofield 1980). Because measurements of only the precipitation component of the atmospheric acid deposition input are widely available, measurements of acid deposition tend to be correlated with the acid rain component only. Moreover, often the only measurements available include the pH alone. Damage is likely occurring with wet sulfate loadings exceeding 20 kg ha 1 yr 1, and almost certainly occurs for loadings over 30 kg ha 1 yr 1. Lakes with alkalinities less than 200 μeq L 1 are sensitive to current levels of acid precipitation.

20.5.3 Cloudwater Deposition Materials scavenged by cloud droplets (in-cloud scavenging) can be removed by wet deposition without rain formation if the cloud comes in contact with Earth’s surface. This wet deposition of intercepted cloudwater can be locally important, especially for slopes of mountains that regularly intercept clouds. In general, the concentrations of SO24 , NO3 , H+, and NH‡4 in cloudwater are 5–10 times greater than in precipitation at the same sites in both the eastern and western United States. This process is relevant to high-altitude ecosystems and will be distinguished here from in situ fog formation, which can as easily occur at low altitudes. The deposition of cloudwater is dependent on the height of the site. For example, the site at 1500 m in Whiteface Mountain, New York spends roughly 4 times more time immersed in clouds than a location at the same mountain at 1000 m (Mohnen 1988). Most cloud droplets have diameters over 5 μm, and their main removal mechanism for flat terrain at low windspeeds is gravitational settling. However, surfaces of interest for cloudwater deposition are usually forest canopies on complex terrain with relatively high turbulence. Wet removal rate is therefore a function not only of the cloud droplet size distribution, but also the size, shape, and distribution of elements protruding into the airflow. Droplet deposition velocities of 2–5 cm s 1 have been measured on grass, while 10–20 cm s 1 have been estimated for forests (US NAPAP 1991). These values can be contrasted with the settling velocity of a 10 μm diameter droplet of 0.3 cm s 1, of a 20 μm droplet of 1.2 cm s 1, and of a 50 μm droplet of 7.5 cm s 1. Volume mean diameters are on the order of 10 μm, indicating that turbulence has the potential to significantly increase the deposition velocity of clouddrops even over a flat surface in mountaintops characterized by high windspeeds and high roughness lengths. Cloud droplets are typically far more acidic than precipitation droplets collected at the ground. In essence, clouddrops are small and have been subjected to neither the dilution associated with growth to the size of raindrops, snowflakes, and so on nor the neutralization associated with the capture of surfacederived NH3 and alkaline particles held in layers at lower altitudes. Interception of these droplets therefore provides a route by which concentrated solutions of sulfate and nitrate can be transferred to foliage in high-elevation areas that are exposed to clouds.

20.5.4 Fogs and Wet Deposition Fogs can be viewed as clouds that are in contact with Earth’s surface. Fogs are created during cooling of air next to Earth’s surface either by radiation to space (radiation fogs) or by contact with a surface (advection fogs) and in general can be distinguished from clouds that are just intercepted by a mountain

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WET DEPOSITION

slope. Urban fogs scavenge acidic particles and gases by the same mechanisms as clouds and are often characterized by low pH values. An extreme event observed in southern California was a relatively light fog at Corona del Mar during which the fog pH reached a low of 1.7. The nitrate level was roughly 6 times the sulfate level in that particular fog (Jacob et al. 1985). Composition and pH of fogs are a function of time. At the beginning of a fog, the concentrations of its major constituents are high; as the fog develops, liquid water content rises, droplets are diluted, and acidity drops, and then as the air is heated, evaporation takes place, relative humidity decreases, and the pH is lowered again. While fogs have liquid water contents (0.1–0.2 g m 3) and droplet sizes (∼10 μm) similar to clouds, they are usually much more concentrated. This should be expected because they are in general created in areas of higher particulate and gaseous concentrations than those in clouds.

20.6 ACID DEPOSITION PROCESS SYNTHESIS The nature of acid deposition depends on competition between gas-phase and liquid-phase chemistry, competition between airborne transport and removal, and competition between dry and wet deposition. The key questions in acid deposition are related to identifying the essential processes involved and then understanding their interactions and quantifying their contributions.

20.6.1 Chemical Species Involved in Acid Deposition The gas/particulate/aqueous forms of sulfuric and nitric acid are the major contributors to acid deposition in polluted areas. Smaller contributions in these areas are made by hydrochloric acid and organic acids (mainly formic and acetic). Atmospheric bases, mainly NH3 and Ca2+, and other crustal elements play as important a role as do the acids, neutralizing to various extents the atmospheric acidity. The net acidity is the overall result of the acid and basic contributions.

20.6.2 Dry versus Wet Deposition Dry deposition operates slowly, but continuously, whereas wet deposition occurs episodically in precipitation events. Let us follow the simple calculation of Schwartz (1989) to compare the magnitude of the various processes. Average emission fluxes of SO2 and NOx in the northeastern United States in the late 1980s were 130 mmol m 2 per year and 120 mmol m 2 per year, respectively. For comparison, total annual emissions in Ohio were 360 mmol m 2 yr 1 and 210 mmol m 2 yr 1. Let us for the purposes of this calculation assume a dry deposition velocity of SO2 of 1 cm s 1 as a representative value and use available SO2 concentration measurements to get a range of annual average SO2 concentrations over the United States for that time period. Schwartz (1989) estimated an annual average value of SO2 concentration equal to 0.16–1.25 μmol m 3, leading to dry deposition fluxes of 16–125 mmol m 2 yr 1. Measured annual average sulfate wet deposition fluxes ranged from 10 to 40 mmol m 2 yr 1 in this region. Such estimates indicate that, on an annual basis, dry deposition of sulfur exceeded wet deposition in near-source regions (corresponding to high SO2 concentrations) and was comparable to wet deposi­ tion farther from source regions. These conclusions are in agreement with the results of threedimensional acid deposition models and the available deposition measurements (US NAPAP 1991).

20.6.3 Chemical Pathways for Sulfate and Nitrate Production Sulfur dioxide can be converted to sulfate by reactions in gas, aerosol, and aqueous phases. As we noted in Chapter 17, the aqueous-phase pathway is estimated to be responsible for more than half of the ambient atmospheric sulfate concentrations, with the remainder produced by the gas-phase oxidation of SO2 by OH (Walcek et al. 1990; Karamachandani and Venkatram 1992; Dennis et al. 1993; McHenry and Dennis 1994). These results are in agreement with box model calculations suggesting that gas-phase daytime SO2 oxidation rates are ∼l–5% per hour, while a representative in-cloud oxidation rate is ∼10% per minute for 1 ppb of H2O2.

ACID DEPOSITION PROCESS SYNTHESIS

Fogs in polluted environments have the potential to increase aerosol concentrations by dropletphase reactions but, at the same time, to cause reductions because of the rapid deposition of larger fog droplets compared to smaller particles (Pandis et al. 1990). Pandis et al. (1992) estimated that more than half of the sulfate in a typical aerosol air pollution episode was produced inside a fog layer the previous night. The low amount of liquid water associated with particles (volume fraction 10 10, compared to clouds, for which the volume fraction is on the order of 10 7) precludes significant aqueous-phase conversion of SO2 in such particles. These particles can contribute to sulfate formation only for very high relative humidities (RH  90%) and in areas close to emissions of NH3 or alkaline dust. Seasalt particles can also serve as the sites of limited sulfate production (Sievering et al. 1992), as they are buffered by the alkalinity of seawater. The rate of such a reaction as a result of the high pH of fresh seasalt particles is quite rapid, 60 μM min 1, corresponding to 8% h 1 for the remote oceans (SO2 = 0.05 ppb). Despite this initial high rate of the reaction, the extent of such production may be quite limited. For a seasalt concentration of 100 nmol m 3, the alkalinity of seasalt is around 0.5 nmol m 3. Consequently, after 0.25 nmol m 3 of SO2 is taken up in solution and oxidized, the initial alkalinity would be exhausted and the reaction rapidly quenched as the pH would immediately decrease. Gas-phase production of HNO3 by reaction of OH with NO2 is approximately 10 times faster than the OH reaction with SO2. The peak daytime conversion rate of NO2 to HNO3 in the gas phase is ∼10–50% per hour. A second pathway for formation of nitrate is the reaction sequence (see Chapter 6) NO2 ‡ O3 ! NO3 ‡ O2 NO3 ‡ NO2 ƒ N2 O5 N2 O5 ‡ H2 O aq ! 2HNO3 aq Note that this reaction pathway operates only at night, as during the daytime NO3 photolyzes rapidly. The NO3 radical formed during the nighttime also reacts with a series of organic compounds, producing organic nitrates. Aqueous-phase production of nitrate by NO and NO2 reactions is negligible under most conditions, including reactions of NO2 with O2, O3, H2O2, and H2O (see Chapter 7). This leaves the gasphase NO2 + OH reaction as the dominant daytime nitrate production pathway and the NO3 heteroge­ neous reaction as the main nighttime production pathway. Both NO and NO2 have low solubility in water. Virtually no NOx is removed from fresh plumes. HNO3 formed by gas-phase oxidation of NO2 is very soluble in water and the principal source of nitrate in precipitation. Since the secondary products are much more easily scavenged than NOx, scavenging increases with plume dilution and oxidation. About one-third of the anthropogenic NOx emissions in the United States are estimated to be removed by wet deposition. The distinct seasonal character of sulfur wet deposition is absent in the case of nitrogen wet deposition for a series of reasons: HNO3 has a strong affinity for ice as well as liquid water; its formation has no direct dependence on hydrogen peroxide, which peaks in summer; and nitrate can be formed in low winter sunlight.

20.6.4 Source–Receptor Relationships In developing control strategies to deal with acid deposition, one seeks to establish relationships between emissions and depositional fluxes, so-called source–receptor relationships. For sulfur, for example, it is desired to relate deposition at site j, as mmol m 2 per year, to the strength of source i, as measured, say, also in mmol m 2 per year. Clearly, the amount of deposited sulfur will vary from day to day depending on the prevailing meteorological conditions, such as the windspeed and direction, the presence of clouds and precipitation, mixing state and chemical reactivity of the atmosphere, and so forth. Such source–receptor relationships on a day-to-day basis or averaging over a year can be derived only from mathematical models that include transport, chemical reactions, and removal processes. Sulfur and nitrogen oxides emitted into the atmosphere are necessarily returned to Earth’s surface. A key measure of the source–receptor relation for acid deposition is the mean transport distance x, the distance of travel between source and receptor averaged over all emitted molecules (Schwartz 1989).

879

880

WET DEPOSITION

This quantity defines the region of influence of a specific source and is a measure of the decay of the deposition flux as one moves away from the source. One approach to getting a rough estimate for x is to multiply the transport velocity by the mean residence time of the emitted material in the atmosphere. Climatologically averaged transport distances in the atmospheric mixed layer indicate a mean transport velocity of around 400 km per day. Atmospheric lifetimes of SO2, NOx, and their oxidation products are 1–3 days (Schwartz 1979; Levine and Schwartz 1982), suggesting mean transport distances of 400–1200 km. Detailed chemical transport model simulations are consistent with these rough estimates, suggesting, for example, that SO2 sources located 100–600 km from receptor areas were the dominant contributors to the corresponding local surface levels in the eastern United States (Wagstrom and Pandis 2011). If the receptor region is, for example, the Adirondack Mountains region of New York State, a possible specific question of the overall source–receptor problem would be which states contribute to acid deposition in the area. One could make the question even more specific, by asking what fraction of the sulfate deposition in the Adirondacks is emitted as SO2 in the state of Ohio. The development of source–receptor relationships is a key policy question associated with acid deposition.

20.6.5 Linearity The source–receptor relationships reveal the fraction of acid deposition at a receptor that results from emissions of a particular source over a given averaging time. Then, what one needs to calculate is the extent to which deposition of, say, sulfate at a receptor site will be reduced if SO2 emissions at a certain source are reduced by a certain amount. Assume that the utility SO2 emissions in the Lower Ohio Valley are reduced by 50% (cut in half). This area as of about three decades ago (according to chemical transport model results) appeared to contribute on average 1.8 kg (S) ha 1 yr 1 to the sulfur deposition on the Adirondacks. What would be the contribution after the emission reductions? One could argue that, if the emissions are reduced by 50%, and because “Whatever goes up must eventually come down,” the sum of sulfate acid deposition (wet and dry) averaged over a year would also decrease by 50%. Such an answer implies that sulfate deposition varies linearly with changes in SO2 emissions. We know enough about the processes and pathways connecting the SO2 emissions with the sulfate deposition (Figure 20.9) to critically evaluate the validity of such an assumption. Let us take a step back and synthesize what we know about the behavior of the system and its response to an SO2 emission change. A reasonable assumption for our discussion is that this local change of emissions will not affect the meteorological component of the acid deposition process (windspeed and direction, mixing, cloud occurrence and pollutant processing, rainfall, etc.). This leaves us free to

FIGURE 20.9 Conceptual depiction of atmospheric sulfur source–receptor relationship.

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PROBLEMS

concentrate on the changes of the chemical component of acid deposition. Simplifying the problem this way, we can now focus on the two components of the acid deposition–the clean-air and the cloud-related pathways. The clean-air processes include emissions of SO2, atmospheric transport, conversion to sulfate by reaction with the OH radical, and dry deposition of SO2 and sulfate. If we follow an air parcel moving from the source to the receptor, then all the clean-air process rates depend for all practical purposes linearly on the SO2 concentration. For example, the gas-phase conversion rate of SO2 to sulfate is k‰OHŠ‰SO2 Š. The OH radical concentration is not sensitive to the levels of SO2 present as it is determined by the available organic and NOx concentrations present. Therefore a doubling of the SO2 gas-phase concentration would lead to a doubling of the conversion rate. The same is true for the dry removal rate, which for a well-mixed box boundary layer is equal to υd ‰SO2 Š=H m , where υd is the dry deposition velocity of SO2 and Hm is the characteristic mixing height. Again, because the SO2 concentration does not affect the meteorological conditions and because the deposition velocity and mixing height will remain constant even if SO2 changes, a doubling of SO2 concentrations will lead to a doubling of the dry removal rate. These arguments show that the clean air component of the sulfur deposition process is practically linear. This leaves us with the aqueous-phase chemistry component. Let us first consider the conversion of SO2 to sulfate in clouds by H2O2. As we saw, this reaction often dominates the overall process as its rate can exceed 10% SO2 per minute. Let us assume a scenario where an air parcel contains 0.8 ppb of SO2 and 0.5 ppb of H2O2. One would expect that, in this case, all the available H2O2 will react with 0.5 ppb SO2, producing roughly 2 μg m 3 of sulfate. Let us assume that the cloud pH is low enough that no other reaction is taking place with an appreciable rate. Then assume that SO2 emissions increase by a factor of 2 and the SO2 concentration entering the cloud doubles to 1.6 ppb. Let us assume that this emission increase does not significantly affect the H2O2 concentration, which will remain close to 0.5 ppb. Again, the H2O2 will control the reaction and again roughly 2 μg m 3 of sulfate will be produced. This example shows that if the sulfate production in the aqueous phase is controlled by the availability of oxidants, then the sulfate concentrations may respond nonlinearly to SO2 emission and concentration will change. This can be the case in areas very close to sources, where SO2 concentrations are high, and during the winter, when H2O2 concentrations are lower. In these scenarios, there may not be adequate oxidants to permit all the sulfur dioxide to be transformed to sulfate. Note that in this case a 50% reduction in sulfur emissions will result in a lesser reduction of acid deposition than 50%. In our simple example, a 50% reduction of emissions would lead to a reduction of the SO2 concentration to 0.4 ppb, and the sulfate produced would be 1.6 μg m 3, corresponding to a reduction by only 20%.

PROBLEMS 20.1A Calculate the aerosol scavenging coefficient Λ as a function of the diameter dp of the scavenged particle over a size range of 0.01–0.1 μm diameter for drops of diameter Dp = 5 mm falling at their terminal velocity in air at 25 °C. Assume that p0 = 1 mm h 1. 20.2A An industrial accident has resulted in a significant release of HCl to the atmosphere. The HCl mixing ratio downwind of the release is 40 ppb, at which the HCl is well-mixed in the belowcloud layer. Luckily, at this point it starts raining, so part of the HCl becomes dissolved in the rainwater and is removed from the atmosphere. The atmospheric conditions are as follows: Temperature = 10°C Rain intensity = 5 mm h 1 Rain drop diameter = 2 mm

Rain duration = 2 h Cloud base height = 2 km

a. Using the information in Chapter 7, calculate the effective Henry’s law coefficient for HCl for the conditions of the problem and pH = 2, 4, and 6. What do you conclude about the partitioning of the HCl between the gas and aqueous phases for the atmospheric pH range of 2–6?

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WET DEPOSITION

b. What is the fraction of HCl that is removed by the rain? c. In the same area there is a lake with surface area of 1 km2. How much HCl (in kg) will be deposited in the lake during the rain event? 20.3B Calculate the mean aerosol number scavenging coefficient for the same conditions as in Problem 20.1, where the aerosol being scavenged has a lognormal size distribution with dpg ˆ 0:01 μm and 1.0 μm, each case having σg ˆ 2:0 20.4B For a more accurate determination of the scavenging rate of a gas, we need to include in the calculation the full droplet size spectrum. Show that if the droplet size distribution is given by the function N(Dp), the rate of removal of an irreversibly soluble gas is given by W bc ˆ

1

∫0

Kc πD2p Cg N Dp dDp

Using this, calculate the scavenging coefficient for the Marshall–Palmer distribution N Dp ˆ 0:08 exp for p0 = 1 mm h

1

41Dp p0 0:21

and 25 mm h 1.

20.5B Prepare a plot analogous to Figure 20.3 for the scavenging of NH3 for an ammonia mixing ratio of 10 ppb. Will any droplet be saturated in ammonia after falling 3 km? 20.6B Calculate the scavenging coefficient for below-cloud scavenging of NH3 based on the Marshall– Palmer raindrop size distribution for rainfall intensities of p0 = 1, 5, 15, and 25 mm h 1. Assume a minimum raindrop diameter of 0.02 cm. 20.7B Consider a raindrop falling through a layer containing a uniform concentration of SO2. Show how to calculate the rate of approach of the dissolved S(IV) to its equilibrium value as a function of time of fall. Apply your result to a drop with 5 mm diameter, T = 25 °C, ξSO2 ˆ 10 ppb to compute how long it takes for the drop to reach equilibrium with the surrounding gas phase. Repeat the calculation for H2O2 at a background mixing ratio of 1 ppb. 20.8B Sulfate can be formed not only in cloud but also in the falling raindrops. Assume a raining cloud with cloud base at 2 km and monodisperse raindrops with diameter equal to 1 mm. The rain intensity is 4 mm h 1. a. What is the concentration of rainwater (in g H2O m 3 of air) in the below-cloud atmosphere? b. The atmosphere below the cloud has the following pollutant mixing ratios at the beginning of the rain event: 20 ppb NOx 250 ppbC VOCs 50 ppb O3

c. d.

e. f.

5 ppb SO2 1 ppb H2O2

What is the sulfate production rate in the rainwater? Assume that all species are in equilibrium between the gas and rainwater phases and that the initial pH of the rainwater (at cloud base) is equal to 4 throughout the rain event. If the sulfate concentration in the rainwater at cloud base is 500 μM, what is the sulfate concentration when the rain arrives at the ground? Write the appropriate mass balance in the below-cloud air for SO2, assuming that its concentration does not depend on height. Solve the resulting differential equation to calculate the SO2 concentration as a function of time. Repeat part d. for H2O2 and the other species. Calculate the sulfate deposition rate as a function of time.

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PROBLEMS

20.9C Assuming a rainfall with intensity p0 = 10 mm h 1 and a monodisperse raindrop distribution with Dp = 4 mm, calculate the mass scavenging coefficients for the model continental, marine, and urban distributions in Table 8.3. 20.10C Calculate the number scavenging efficiencies of the aerosol populations for the conditions of Problem 20.1. Compare with the mass scavenging efficiencies and discuss your results. 20.11A Calculate the effect on the pH of atmospheric liquid water of dissolved aerosol SO24 and the liquid water content L, assuming that the mixing ratios of CO2 and SO2 are constant and equal to 350 ppm and 100 ppt, respectively. Ammonia is assumed to be absent. In particular, fill in the pH values in the table below: L gm SO24 μg m

3

0.1

3

0.5

2.5

0.4 0.2 1.0

20.12B The probability of an SO2 molecule being transformed to sulfate and wet-deposited on land can be expressed as the product of the probabilities of the molecule being processed through precipitating air parcels over land and of being absorbed and deposited (Oppenheimer 1983a). Show that if tSO2 is the regional mean lifetime of SO2 and f(t) is the probability of a dry period lasting a time of length t, that is, of a wet event occurring at time t, then the probability of an SO2 molecule being processed through precipitation is equal to 1

∫0

f …t†e

t=tSO2

dt

Assuming that the frequency distribution of rain events is given by f … t† ˆ

1 e TD

t=T D

show that this probability is equal to tSO2 = tSO2 ‡ T D . If we identify tSO2 with the characteristic time for SO2 removal by dry deposition, we obtain an upper limit on tSO2 because some SO2 will be lost by gas-phase conversion into material that is not wet-deposited. Let us use tSO2 ˆ 60 h. A value of TD typical of the eastern United States is TD = 138 h. Thus we see that the probability of an emitted SO2 molecule being processed through precipitation before dry deposition is about 0.3. Considering the region of interest to be the eastern United States and Canada, Oppenheimer (1983a) estimated the following sulfur emissions and wet deposition for the 1977–1979 period: Emissions

ˆ 10:26  109 kg…S†yr

Wet depositon ˆ 2:25  109 kg…S†yr

1

1

Calculate the probability of an SO2 molecule being absorbed and deposited in this region. Discuss your result. 20.13B In order to evaluate potential emission control strategies for acid rain, one would like to understand the relationship of decreases in SO2 emissions to changes in precipitation sulfate levels. A lower limit on the change in precipitation sulfate can be obtained by assuming that all

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WET DEPOSITION

sulfate formation occurs in the liquid phase under oxidant-limited circumstances. Assume that airborne SO2 occurs in two streams, a part that is processed through precipitating air parcels during its airborne lifetime and a part that is not, labeled A and B, respectively. The part that is passed through precipitating air parcels (A) is then subdivided into two streams, a portion that is absorbed, oxidized, and deposited as sulfate and a part that is not oxidized, labeled A1 and A2, respectively. Assume that an emissions reduction is not reflected in A1, but only in A2 and B, until A2 is exhausted. Thereafter, the emissions reduction reduces A1 on a proportional basis. Show that the response of wet deposition to a fractional emissions reduction X/(A + B) is 1 X A2 β A‡B A‡B where β is the probability of an SO2 molecule being absorbed and deposited. [Note that if β = 1, A2 = 0, and δ = X/(A + B).] Using the value of β determined in Problem 20.12, evaluate δ for a 50% reduction in emissions. δˆ

20.14C In this problem, following Oppenheimer (1983b), we wish to explore the scenario that the atmospheric path leading to precipitation sulfate occurs only by gas-phase oxidation of SO2 to sulfate aerosol, followed by either incorporation of the aerosol in droplets or dry deposition. Let g6(t) = probability density that an air parcel is subjected to a dry period of length t, followed by precipitation beginning in the interval (t, t + dt) p6 = probability that aerosol processed through a precipitating air parcel is scavenged and deposited p1 ; p5 = probabilities that SO2 or sulfate aerosol, respectively, are not lost from the regional boundary layer before wet or dry deposition f 2 …t†; f 7 …t† ˆ probabilities that SO2 or sulfate aerosol, respectively, escape dry deposition for time t g4 …t† = probability density that an SO2 molecule remains unoxidized for a period of length t, followed by oxidation in the interval …t; t ‡ dt† Show that the probability of an SO2 molecule being oxidized to sulfate aerosol in the regional boundary layer before passing through a precipitating air parcel is pox ˆ p1

1

∫0

g4 …t†f 2 …t†dt

and the probability of sulfate aerosol being wet-deposited is pwd ˆ p5 p6

1

∫0

g6 …t†f 7 …t†dt

and thus that the probability that an SO2 molecule is both oxidized to sulfate and deposited is pdep ˆ pox pwd Let us assume that f 7 …t† ˆ e t=τ7 , with τ7 ˆ 280 h. Based on υd ˆ 0:1 cm s 1 and a 1 km boundary layer, p1 = p5 = 0.75, g6 …t† ˆ τ6 1 e t=τ6 , with τ6 ˆ 138 h. Also, we take pdep = 0.22. Finally, values of p6 ranging from 1/3 to 1.0 have been assumed. Essentially, p6 is very uncertain because the cloud microphysics is unknown. Using the above values, show that pwd ˆ 0:5p6 and consequently that pox ˆ 0:44p6 1 . Assuming that all SO2 emission reductions reduce the unoxidized SO2 fraction before any of the oxidized part is reduced, and using p6 = 0.5, determine

885

PROBLEMS

the percentage reduction in wet sulfate deposition resulting from a 50% reduction in SO2 emissions. 20.15C To quantify source–receptor relationships, long-range transport models have been developed [e.g., see Gislason and Prahm (1983)]. Many of these models are of the so-called trajectory type in which the concentration changes in a parcel of air are computed as the parcel is advected by the wind field. In some of the trajectory models used in the SO2/sulfate system, all transformations and removal processes are represented as first-order. (The use of trajectory models to describe long-range transport presumes that we can estimate the location of an air parcel as a function of time. It is clearly an approximation to assume that an air parcel maintains its integrity over a period of many hours to days. Nevertheless, the concept of an integral air parcel has been useful in allowing the formulation of the trajectory model.) The first-order dry deposition rate constants for SO2 and SO24 for use in a trajectory model are kdSO2 ˆ vdSO2 =H and kdSO4 ˆ vdSO4 =H, where vd are the deposition velocities and H is the height of the vertical extent of the model. The height of the mixed layer H varies both diurnally and seasonally, although it is common for regional-scale, single-layer trajectory models to consider only the seasonal variation. Using the following parameter values, characteristic of noontime, summertime conditions in the northeastern United States (Samson and Small 1982) H ˆ 1500 m kc ˆ 0:03 h 1 vdSO2 ˆ 1 cm s

1

vdSO4 ˆ 0:4 cm s

1

kwSO2 ˆ 5  104 p0 =H

p0 in mm h

=H kwSO4 ˆ 2:32  105 p0:625 0

H in mm

1

carry out a trajectory model simulation of long-range transport of the sulfur species in that area. Note that the model does not include cloud removal of SO2. Start a trajectory at the Indiana/ Ohio/Kentucky three-state intersection and run it on a line to Albany, New York, at a speed of 10 km h 1. Continue the calculation for 96 h and record the complete sulfur material balance for the parcel, including airborne SO2 and SO24 , dry-deposited SO2 and SO24 , and wet-deposited sulfur. The SO2 emission rates can be taken as those for 1980: Ohio Pennsylvania New York

2602 × 103 Mt yr 1 1971 × 103 Mt yr 1 901 × 103 Mt yr 1

Assume that the emissions are distributed uniformly over each state. The areas of the states are Ohio Pennsylvania New York

115,750 km2 119,316 km2 137,976 km2

and the distance as the crow flies from the Indiana/Ohio/Kentucky intersection to Albany, New York, is 966 km. Assume that it rains during the last hour of each 24 h period at a rate of p0 = 10 mm h 1. Calculate the total quantity SO2 emitted along the trajectory, together with the quantity remaining at the end, the quantities lost by wet and dry deposition, and the quantity converted to sulfate. Perform the same material balance on sulfate. For the purpose of calculation, assume the moving air parcel to have a base of 104 m2.

886

WET DEPOSITION

20.16D Clouds act like giant pumps through which large quantities of air are processed. Aside from transporting boundary-layer air into the free troposphere, clouds act as filters and remove soluble gases and aerosols from that air. Although the physics of the interaction among cloud droplets, gases, and aerosols is very complex, insights can be gained by representing in-cloud processes as a flowthrough reactor in which boundary-layer air containing SO2, HNO3, NOx, NH3, O3, H2O2, CO2, and aerosol flows through the well-mixed region (reactor) representing the cloud (Hong and Carmichael 1983). We say that the cloud consists of cloud droplets of radius Rc and number concentration Nc. Gases are absorbed and aerosol particles are captured, and inside the droplets chemical reactions occur. Show that, subject to these assumptions, within the gas phase the concentration of a species i obeys V

dci ˆ q ci0 dt

ci

Vkni ci

kgi V4πR2c N c ci

cli =H i RT

where V is cloud volume, q is volumetric flow rate of air through the cloud, c0i is the concentration of species i in the entrained air, knl is the first-order rate constant for gas-phase conversion of species i, kgi is the mass transfer coefficient, and cli is the liquid-phase concentration of species i. Then show that the liquid-phase concentration of species i is governed by Vl

dcli ˆ kgi V4πRc2 N c ci dt

cli =H i RT

qi cli ‡ V l Ri

where Vl is the liquid water volume, ql is the volumetric flow rate of water out of the cloud by rainfall, and Ri is the rate of generation of species i by liquid-phase reactions. The microphysics of the cloud will be represented in terms of the liquid water content wL, rainfall intensity p0, updraft velocity w, cloud depth hc, and cloud cross-sectional area A. We wish to use this model to investigate sulfate production in clouds. The conditions to be simulated are as follows (take T = 273 K): The in-cloud S(IV) oxidation reactions to be considered are those involving O3 and H2O2. The necessary equilibrium and kinetic data can be obtained from Chapter 7. The dynamic calculations should be done by numerically integrating the differential equations for ci and cli together with the relevant electroneutrality relation. The results of the computa­ tion should be presented in terms of the ratio of ci at t = 25 min to ci at t = 0. Discuss your results.

Case

[SO2] (ppb)

[O3] (ppb)

[NH3] (ppb)

[HNO3] (ppb)

[H2O2] (ppb)

Rc (cm)

p0 (mm h 1)

w (m s 1)

hc (km)

wL (cm3 m 3)

50 50 50 50

100 100 100 100

1 1 1 1

10 10 10 10

10 50 50 50

0.003 0.003 0.003 0.03

1 1 10 1

2.5 2.5 2.5 2.5

5 5 5 5

0.3 0.3 0.3 0.3

1 2 3 4

REFERENCES Baker, J. P., and Schofield, C. L. (1980), Proc. Int. Conf. Ecological Impact of Acid Precipitation, Sandefjord, Norway, D. Drablos and A. Tollau (eds.), SNSF Project, Oslo. pp. 292–293. Battan, L. J., and Reitan, C. H. (1957), Droplet size measurements in convective clouds, in Artificial Stimulation of Rain, Pergamon Press, New York, pp. 184–191. Bird, R. B., et al. (2002), Transport Phenomena, Second edition, John Wiley & Sons, Inc., New York.

REFERENCES Brimblecombe, P. (1977), London air pollution, 1500–1900, Atmos. Environ. 11, 1157–1162. Chamberlain, A. C. (1953), Aspects of Travel and Deposition of Aerosols and Vapor Clouds, AERE Harwell, Report R1261 HMSO, London. Cowling, E. B. (1982), Acid precipitation in historical perspective, Environ. Sci. Technol. 16, 110A–123A. Dennis, R. L., et al. (1993), Correcting RADM’s sulfate underprediction: Discovery and correction of model errors and testing the corrections through comparisons against field data, Atmos. Environ. 27A, 975– 997. Dickson, W. (1980), Proc. Int. Conf. Ecological Impact of Acid Precipitation, Sandefjord, Norway, D. Drablos and A. Tollau (eds.), SNSF Project, Oslo. pp. 75–83. Galloway, J. N., et al. (1982), The composition of precipitation in remote areas of the world, J. Geophys. Res. Oceans 87, 8771–8776. Gislason, K. B., and Prahm, L. P. (1983), Sensitivity study of air trajectory long-range transport modeling, Atmos. Environ. 17, 2463–2472. Havas, M., et al. (1984), Red herrings in acid rain research, Environ. Sci. Technol. 18, 176A–186A. Hong, M. S., and Carmichael, G. R. (1983), An investigation of sulfate production in clouds using a flow through chemical reactor model approach, J. Geophys. Res. 88, 733–743. Jacob, D. J., et al. (1985), Chemical composition of fogwater collected along the California coast, Environ. Sci. Technol. 19, 730–736. Karamachandani, P., and Venkatram, A. (1992), The role of non-precipitating clouds in producing ambient sulfate during summer: results from simulations with the Acid Deposition and Oxidant Model (ADOM), Atmos. Environ. 26A, 1041–1052. Keene, W. C., et al. (1983), Measurements of weak organic acidity in precipitation from remote areas of the world, J. Geophys. Res. 88, 5122–5130. Levine, S. Z., and Schwartz, S. E. (1982), In-cloud and below-cloud scavenging of nitric acid vapor, Atmos. Environ. 16, 1725–1734. McHenry, J. N., and Dennis, R. L. (1994), The relative importance of oxidation pathways and clouds to atmospheric ambient sulfate production as predicted by the Regional Acid Deposition Model, J. Appl. Meteorol. 33, 890–905. Mohnen, V. A. (1988), Mountain Cloud Chemistry Project: Wet, Dry, and Cloud Water Deposition, Report to EPA Contract CR-813934-01-2, US EPA, Research Triangle Park, NC. Oppenheimer, M. (1983a), The relationship of sulfur emissions to sulfate in precipitation, Atmos. Environ. 17, 451–460. Oppenheimer, M. (1983b), The relationship of sulfur emissions to sulfate in precipitation–II. Gas phase processes, Atmos. Environ. 17, 1489–1495. Pandis, S. N., et al. (1990), The smog–fog–smog cycle and acid deposition, J. Geophys. Res. 95, 18489–18500. Pandis, S. N., et al. (1992), Heterogeneous sulfate production in an urban fog, Atmos. Environ. 26A, 2509–2522. Pruppacher, H. R., and Klett, J. D. (1997), Microphysics of Clouds and Precipitation, Kluwer, Dordrecht, The Netherlands. Roth, R., et al. (1985), The American West’s Acid Rain Test, World Resources Institute, Washington, DC. Samson, P. J., and Small, M. J. (1982), The use of atmospheric trajectory models for diagnosing the sources of acid precipitation, American Chemical Society, 183rd National Meeting, Las Vegas. Schwartz, S. E. (1979), Residence times in reservoirs under non-steady-state conditions: Application to atmospheric SO2 and aerosol sulfate, Tellus 31, 520–547. Schwartz, S. E. (1989), Acid deposition: Unraveling a regional phenomenon, Science 243, 753–763. Sievering, H., et al. (1992), Removal of sulfur from the marine boundary layer by ozone oxidation in sea-salt, Nature 360, 571–573. Slinn, W. G. N. (1983), Precipitation scavenging, in Atmospheric Sciences and Power Production–1979, Chapter 11, Division of Biomedical Environmental Research, US Dept. Energy, Washington, DC. Stensland, G. J., and Semonin, R. G. (1982), Another interpretation of the pH trend in the United States, Bull. Am. Meterol. Soc. 63, 1277–1284. US NAPAP (US National Acid Precipitation Assessment Program) (1991), The U.S. National Acid Precipitation Assessment Program, Vol. I, US Government Printing Office, Washington, DC.

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WET DEPOSITION Wagstrom, K. M., and Pandis, S. N. (2011), Contribution of long range transport to local fine particulate matter concerns, Atmos. Environ. 45, 2730–2735. Walcek, C. J., et al. (1990), Theoretical estimates of the dynamic, radiative and chemical effects of clouds on tropospheric gases, Atmos. Res. 25, 53–69. Whelpdale, D. M., and Miller, J. W. (1989), GAW and Precipitation Chemistry Measurement Activities, Fact Sheet 5, Background Air Pollution Monitoring Program, World Meteorological Organization, Geneva, Switzerland.

P A R T V I

The Global Atmosphere, Biogeochemical Cycles, and Climate

CHAPTER

21

General Circulation of the Atmosphere

The fundamental process driving the global-scale circulation of Earth’s atmosphere is the uneven heating of Earth’s surface by solar radiation. Although the total energy received by Earth from the sun is balanced by the total energy radiated back to space, this balance does not hold for every location on Earth. The tropics, for example, receive more energy from the sun than is radiated back to space; the polar regions receive less energy than they emit. Figure 21.1 shows the zonally annual averaged absorbed solar and emitted infrared fluxes, as observed from satellites. We note a net gain of radiative energy between about 40°N and 40°S, and a net loss of energy in the polar regions. This pattern results largely from the decrease in insolation to the polar regions in winter and from the high surface albedo in the polar regions. The outgoing infrared flux displays only a modest latitudinal dependence. As a result of the net gain of radiative energy in the tropics and the net loss in the polar regions, an equator-to-pole temperature gradient is generated. Earth’s atmospheric circulation is one mechanism (about 60%) for redistributing energy from areas of the globe with an energy excess to those with an energy deficit; Earth’s ocean circulation is the other mechanism (about 40%). The circulation of the global atmosphere is one of the most complex, and theoretically deep, areas in fluid dynamics. It is the subject of several texts [e.g., Holton (1992), Salby (1996)]. To understand atmospheric flows at a truly fundamental level requires an advanced fluid mechanics background. The approach taken here is to describe the general nature of large-scale flow in the atmosphere at a level corresponding to a basic course in fluid mechanics. Imagine, for a moment, that Earth is not rotating and that the atmosphere can be represented as being confined to a huge rectangular enclosure that is heated at one end (tropics) and cooled at the other end (poles). Air at the warm end rises because of its decreased density, and air at the cool end falls because of its increased density. The buoyant rise of air along the heated end creates a decreased pressure at the surface, while the sinking of cold air at the other end leads to a higher pressure. This pressure difference creates a flow along the bottom of the enclosure from the cold end to the warm end. To conserve mass, rising air at the warm end must travel along the top of the enclosure to the cold end. The result is a single large circulation. Applied to the atmosphere, this single-cell model leads to a single circulation between the equator and the poles, in which air rises in the equatorial regions, spreads poleward at the tropopause, converges, and sinks in the polar regions, finally moving back toward the equator at Earth’s surface. Such a circulation, proposed in the early eighteenth century by the British meteorologist George Hadley (1685–1768), is called a Hadley cell (also the “tropical cell”) (Hadley 1735). Although the Hadley cell captures the concept of rising and falling masses of air resulting from differential heating and cooling, it fails to describe the large-scale circulation of the atmosphere because it does not take into account the rotation of Earth.

Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, Third Edition. John H. Seinfeld and Spyros N. Pandis.  2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

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GENERAL CIRCULATION OF THE ATMOSPHERE

FIGURE 21.1 Zonally averaged components of the absorbed solar flux and emitted thermal infrared flux at the top of the atmosphere. The + and signs denote energy gain and loss, respectively. (From Radiation and Cloud Processes in the Atmosphere: Theory Observation and Model­ ing by Kuo-Nan Liou. Copyright  1992 by Oxford University Press, Inc. Used by permission of Oxford University Press, Inc.)

The fact that Earth is rotating gives rise to another force, besides pressure gradients, that profoundly influences circulation of the atmosphere, the Coriolis force. Because of this force, air cannot flow in one unhindered cell from pole to equator and back. It turns out that a good approximation is that of a threecell model (Figure 21.2). (There must be an odd number of cells so that air rising and falling in adjacent cells is oriented in the proper direction.) The cell in the equatorial region retains the name of the Hadley cell. The reverse cell in midlatitudes is called the Ferrel cell (also called the midlatitude cell). The third cell at high latitudes, called the polar cell, contains the sinking motion over the poles. The three-cell model, while highly oversimplified, is a reasonable qualitative description of the large-scale pattern of atmospheric circulation, especially that associated with surface flow.

FIGURE 21.2 Three-cell representation of global circulation of the atmosphere.

FERRELL CELL AND POLAR CELL

21.1 HADLEY CELL The Hadley cell in the Northern Hemisphere is characterized by ascent at the equator, northward movement along the tropopause, convergence, and descent at about 30°N and southward movement toward the equator along the surface. In the Southern Hemisphere, flow is southward along the tropopause, with convergence and descent at about 30°S, and northward movement toward the equator. Air over the equator is warm, and horizontal pressure gradients are weak. Little wind exists in this region, which has become known as the doldrums. The excess radiative heating in the tropics is compensated partially by evaporation, but even with evaporation, the air is still warmer than that to the north and south. The warm, moist air over the tropics is unstable; as soon as a small uplift occurs, the air saturates with water vapor and continues to move upward. Large amounts of latent heat are released as towering cumulus clouds form, providing even more energy for the upward motion. At the tropical tropopause (∼15 km), the atmosphere is stable and the upward motion is largely brought to a halt. Air that does not enter the stratosphere is deflected to the north and south. The Hadley cell accomplishes a large fraction of global heat transport toward the higher latitudes. While the Hadley cell also transports latent heat, in the form of water vapor, from higher latitudes toward the equator, on net balance the transport of sensible heat (the warm air itself) poleward is larger. In the Northern Hemisphere, air converges as it moves northward and cools. At latitudes of about 30°N, cooling and convergence produce a large, heavy mass of air above Earth’s surface. This large, heavy mass of air sinks and produces a widespread, semipermanent high-pressure system known as a subtropical high. While sinking, the air warms by compression, producing clear skies, warm, dry climates, and weak surface winds.1 These zones are where the great deserts of the world are found. At the surface, some of the descending air moves back toward the equator and is deflected to the right in the Northern Hemisphere as a result of the Coriolis force (which we will discuss shortly) as it moves. On both sides of the equator, there exists a wide region where winds blow from east to west (easterlies) with a slight equatorward tilt (see Figure 21.2).2 This region is called the trade wind belt. Such winds were well known to early mariners, taking sailing ships from east to west. Near the equator, northeasterly winds from the Northern Hemisphere converge with southeasterly winds from the Southern Hemisphere. This region is known as the intertropical convergence zone (ITCZ). The convergence aids the ascent of air resulting in a widespread area of low pressure known as the equatorial low. The predominant rising motion in the ITCZ leads to a band of extensive convective clouds and high rainfall. Major tropical rainforests are found in this zone.

21.2 FERRELL CELL AND POLAR CELL At 30°N, the air descending to the surface that does not move south toward the equator in the Hadley cell begins to move north. As it does so, it is deflected toward the right by the Coriolis force. In these regions north and south of the trade wind belt (in the Northern and Southern Hemispheres, respectively), winds tend to blow from west to east (westerlies); these are referred to as westerly wind belts. When the temperature gradient and upper-level winds are sufficiently large, the westerly flow breaks down into large-scale eddies. As a result, day-to-day weather patterns do not exhibit the zonal symmetry of the large-scale climatological Ferrell cell. The breakdown into eddies is termed baroclinic instability (baros means “weight” and clini means “sloping”). The stronger the flow, the more likely it is to be unstable. Baroclinic eddies, the eddies resulting from baroclinic instability, are the source of midlatitude weather. They are characterized by trains of alternating low- and high-pressure systems moving eastward, which are the source of frontal systems that produce the clouds and storminess of the 1

In the days of sailing, winding up in these regions meant serious delays in the voyage. They were referred to as horse latitudes, because it is said that the horses that were carried on board had to be thrown off to lighten the load or to conserve drinking water. 2 Recall that winds are identified by the direction they are coming from, not heading to.

893

894

GENERAL CIRCULATION OF THE ATMOSPHERE

midlatitudes. Baroclinic eddies push warm air poleward and cold air southward, cooling the subtropics and warming the polar latitudes. Baroclinic eddy heat fluxes are so important that their description lies at the heart of any theory of the general circulation. Figure 21.3 sketches four conditions of midlatitude flow in the Northern Hemisphere. Panel (a) is the typical westerly flow under baroclinically stable conditions, when the energy imbalance between tropics and poles is not excessively large. Panel (a) shows a distinct band of relatively strong winds (usually exceeding 30 m s 1) called jetstreams. They are located above areas of especially strong temperature gradients. Since westerly flow parallel to latitude circles does not transport much energy poleward, the equator-to-pole temperature gradient will continue to increase as long as the atmo­ sphere is in this state. When a critical north–south temperature gradient is reached, meanders begin to form in the jetstream and the atmosphere becomes baroclinically unstable [panel (b)]. Strong waves form in the upper airflow [panel (c)] and relative highs and lows are formed along each latitude band.

FIGURE 21.3 Baroclinic stability and instability: (a) gently undulating upper airflow; (b) meanders form in jetstream; (c) strong waves form in upper airflow; (d) return to a period of flatter flow aloft.

895

CORIOLIS FORCE

Finally, the flow returns to a pattern of flatter flow aloft [panel (d)]. In short, baroclinic instability arises when inevitable small perturbations in the velocities grow rather than decay. The overall direction of winds between 30° and 60° is from the west. Above ∼500 mbar, these winds actually flow in wavelike patterns, with troughs and ridges. Troughs occur where the flow dips equatorward, and ridges occur where the flow moves poleward [panels (b) and (c) in Figure 21.3]. The flow in these upper-level waves leads to storms that move warm air poleward and cold air equatorward. The Northern Hemisphere is typically encircled by several of these waves at any given time. These long-wavelength waves, which drift slowly eastward, are referred to as Rossby waves, after Carl-Gustav Rossby, the Swedish meteorologist who first identified them. When the waves have large amplitude with deep troughs and peaked ridges, cold air flows equatorward and warm air flows poleward. The high-latitude end of the Ferrel cell is characterized by rising warm, moist air. Some of the rising air, on reaching the tropopause, moves back southward toward the region of convergence above 30°N. This circulation, northward along the surface, upward at the polar front, back to the south along the tropopause, and sinking toward the surface at the horse latitudes, constitutes the Ferrel cell. The portion of the air aloft that does not move equatorward moves toward the poles where it converges and sinks. This circulation, which results in high pressure at the pole (the polar high), is known as the polar cell; it is the weakest of the three cells (Figure 21.2). As the cold air sinks over the poles, it must flow equatorward at the surface. The Coriolis force turns these winds toward the west, forming the polar easterlies at 60° latitude and above. Mild air moving north and east along Earth’s surface in the midlatitude Northern Hemisphere encounters much cooler air moving south and west in the polar easterlies. These airmasses of differing temperature and density result in the formation of the polar front along the boundary where they meet. The polar front is a zone of strong convergence and rapidly rising air that results in a widespread surface low-pressure system known as the subpolar low. This region of rapid, widespread ascent produces conditions conducive to storm formation. Polar air, which is cooler and more dense than the air moving up from the south, can push the location of the polar front farther south, especially in the wintertime, producing a cold polar outbreak.

21.3 CORIOLIS FORCE Because Earth is a rotating sphere, points on its surface move at different speeds depending on their latitude. Consider a point completing a circle of radius R in time t. Its tangential speed is v = 2πR/t. The angular velocity (Ω) in radians per second, is 2π/t, since 2π radians is the angle covered in t seconds. Thus, v = ΩR. The angular velocity of Earth is 2π/(24 × 60 × 60) = 7.27 × 10 5 rad s 1. Since Earth’s radius at the equator is about 6370 km, the tangential speed of a point on the surface at the equator is v = 463 m s 1. At latitude ϕ, the radius of the circle described by a point on the surface is R cos ϕ, so that the tangential speed is v ˆ ΩR cos ϕ

(21.1)

which is just cos ϕ times the speed at the equator. At latitude 30°, for example, the tangential speed is 401 m s 1. Thus, at 30 °N, a parcel of upper-level air transported from the equator has a “surplus” momentum corresponding to a velocity of 62 m s 1. In a simplistic view, this surplus momentum forms the westerly subtropical jetstream, the average velocity of which is about 60 m s 1. The difference in tangential speeds between points on Earth’s surface at different latitudes gives rise to the Coriolis force, named for the French engineer who first described it. Consider in Figure 21.4 a parcel of air moving northward from point X to point Y. As the parcel heads north toward Y, it carries with it the eastward momentum from Earth’s rotation at X. Point X has a greater tangential speed than Y. As a result, as the parcel moves slowly northward, it appears to an observer on Earth to gain speed toward the east. The combination of the northward and eastward velocities results in a bending path that deflects to the right of its destination. Now imagine the air parcel moving southward from point Y to X. Since the parcel has the lower tangential velocity of point Y as it

896

GENERAL CIRCULATION OF THE ATMOSPHERE

FIGURE 21.4 Illustration of the effect of Coriolis force on air parcels in the Northern Hemisphere (Ackerman and Knox 2003).

heads southward, it appears to an observer on the surface to lag behind the spinning Earth, and its path also deflects to the right of its direction of travel. We conclude that the Coriolis force deflects moving air to the right (in its direction of travel) in the Northern Hemisphere. If point Y is in the Southern Hemisphere, the air parcel, in going from X to Y, deflects toward the left. Air, traveling from Y to X, also deflects to the left. Thus, in the Southern Hemisphere, the Coriolis force deflects moving air to the left (in its direction of travel). The magnitude of the Coriolis force per unit mass of air is proportional to the distance from the equator and the windspeed. This force is expressed as fv, where f is the Coriolis parameter: f ˆ 2Ω sin ϕ

(21.2)

The Coriolis parameter f > 0 in the Northern Hemisphere (ϕ > 0) and f < 0 in the Southern Hemisphere (ϕ < 0). Note that the Coriolis parameter f equals zero at the equator and increases in magnitude as sin ϕ toward the poles; thus, winds at higher latitudes are more strongly affected by the Coriolis force. The momentum of an object moving in a straight line is equal to the product of its mass and its velocity. A rotating body has angular momentum, defined as the product of its mass, its rotation velocity, and the perpendicular distance from the axis of rotation. If undisturbed, a rotating object conserves its angular momentum. Thus, if the distance from the axis of rotation decreases, then its velocity must accordingly increase. (This is why a figure skater spins faster as his or her arms are pulled closer to the body.) As an air parcel moves north or south from the equator, its distance from Earth’s axis of rotation decreases, and therefore its angular velocity must increase to conserve angular momentum. As air flows northward in the upper level of the Hadley cell, the Coriolis force turns it to the right (eastward). The magnitude of the Coriolis force increases as the air flows toward the poles, and by the time the air reaches about 30°N latitude, the Coriolis force has turned the flow entirely from west to east (see Figure 21.2). To conserve angular momentum, the speed of the wind also increases as the air moves toward the pole.

897

GEOSTROPHIC WINDSPEED

Order of Magnitude of Atmospheric Quantities It is useful to estimate the order of magnitude of quantities in the equations of motion on the synoptic scale (systems having a spatial scale of ∼ 1000 km). Typical orders of magnitude of quantities are as follows (McIlveen 1992): L H t Δp P ρ Ω g

Horizontal lengthscale Vertical lengthscale Timescale Horizontal pressure Vertical pressure change Air density Earth’s angular velocity Gravitational constant

1000 km 10 km 1 day 10 mbar 1000 mbar

106 m 104 m 105 s 103 Pa 105 Pa 1 kg m 3 10 4 rad s 10 m s 2

1

L is the horizontal distance over which there is a substantial change in the pressure or wind field. Δp is a typical horizontal pressure gradient for synoptic weather systems. The vertical pressure change is just the hydrostatic head of air from the surface to the tropopause. (In the spirit of order of magnitude, we estimate this as 1000 mbar rather than the accurate 900 mbar.) The timescale of 1 day is set by Earth’s rate of rotation. Moreover, the observed timescale of synoptic scale weather systems is about one day; such systems take days rather than hours to form and dissipate. From the orders of magnitude above, we can estimate the following characteristic values: Horizontal windspeed Vertical windspeed Horizontal acceleration Vertical acceleration Coriolis acceleration Horizontal pressure gradient

u ∼ L/t w ∼ H/t u/t (or u2/L) w/t (or w2/H) Ωu Δp/L

10 m s 1 10 1 m s 1 10 4 m s 2 10 6 m s 2 10 3 m s 2 10 3 Pa m 1

21.4 GEOSTROPHIC WINDSPEED At an altitude sufficiently far above Earth’s surface (about 500 m) where the effect of the surface can be neglected, the airflow is influenced only by horizontal pressure gradients and the Coriolis force. This layer is called the geostrophic layer, and the windspeed attained is the geostrophic windspeed. In the geostrophic layer the flow may be assumed to be inviscid (molecular viscosity effects are negligible). We can compute the geostrophic windspeed and direction at any latitude as a function of the prevailing pressure gradient from the equations of continuity and motion. It can be shown that the acceleration experienced by an object on the surface of the Earth (or in the atmosphere) moving with a velocity vector u consists of two components, Ω × (Ω × r) and 2(Ω × u), where Ω is the angular rotation vector for Earth and r is the radius vector from the center of Earth to the point in question. The first term is simply the centrifugal force, in a direction that acts normal to Earth’s surface and is counterbalanced by gravity. The second term, Ω × u, is the Coriolis force. This force arises only when an object, such as an air parcel, is moving; that is, u ˆ 6 0. Even though the Coriolis force is of much smaller magnitude than the centrifugal force, only the Coriolis force has a horizontal component. Since the winds are horizontal in the geostrophic layer, the Coriolis acceleration is given by the horizontal component of the Coriolis term, namely, 2ugΩ sin ϕ. The direction of the Coriolis force is perpendicular to the wind velocity. Windspeed ug at latitude ϕ lies in the horizontal plane. In the geostrophic layer it may be assumed that the atmosphere is inviscid (frictionless). The equations of continuity and motion for such a fluid are @u @v @w ‡ ‡ ˆ0 @x @y @z

(21.3)

898

GENERAL CIRCULATION OF THE ATMOSPHERE

and @u @u @u @u ‡u ‡v ‡w ˆ @t @x @y @z @v @v @v @v ‡u ‡v ‡w ˆ @t @x @y @z @w @w @w @w ‡u ‡v ‡w ˆ @t @x @y @z

1 @p ‡ Fcx ρ @x 1 @p ‡ Fcy ρ @y 1 @p ‡ Fcz ρ @z

(21.4)

where u, v, and w are the three components of the velocity and Fcx, Fcy, and Fcz are the three components of the Coriolis force. Let the axes be fixed in Earth, with the x axis horizontal and extending to the east, the y axis horizontal and extending to the north, and the z axis normal to Earth’s surface. As before, Ω is the angular velocity of rotation of Earth and ϕ is the latitude. The components of the Coriolis force in the x, y, and z directions on an air parcel are the components of Fc = 2(Ω × u): Fcx ˆ

2Ω…w cos ϕ

Fcy ˆ

2Ωu sin ϕ

v sin ϕ† (21.5)

Fcz ˆ 2Ωu cos ϕ In the geostrophic layer, the vertical velocity component w can usually be neglected relative to the horizontal components u and v. Therefore, substituting (21.5) into (21.4), we obtain, for steady motion @u @u 1 @p ‡v ˆ 2Ωv sin ϕ @x @y ρ @x @v @v 1 @p ˆ 2Ωu sin ϕ u ‡v @x @y ρ @y u

(21.6)

We see that the air moves so that a balance is achieved between the pressure gradient and the Coriolis force. Let us consider the situation in which the velocity vector is oriented in the x direction, and so v = 0; then u

@u ˆ @x

1 @p ρ @x

0 ˆ 2Ωu sin ϕ

(21.7) 1 @p ρ @y

(21.8)

From the continuity equation (21.3), we see that @u/@x = 0, since v = w = 0. Thus from (21.7) we obtain @p/@x = 0, and we are left with only (21.8). From (21.8), we see that the component of the Coriolis force, fu, is exactly balanced by the pressure gradient, (1/ρ) @p/@y, and the direction of flow is perpendicular to the pressure gradient @p/@y. Therefore the geostrophic windspeed ug is given by ug ˆ

@p=@y ˆ 2ρΩ sin ϕ

1 @p ρf @y

(21.9)

The approach to the geostrophic equilibrium for an air parcel starting from rest, accelerated by the pressure gradient and then affected by the Coriolis force, is shown in Figure 21.5.

899

GEOSTROPHIC WINDSPEED

FIGURE 21.5 Approach to geostrophic equilibrium.

21.4.1 Buys Ballot’s Law In the geostrophic equilibrium, the pressure gradient must be parallel to the y axis, with pressure decreasing in the positive y direction. The fact that, for a flow in the easterly direction, the isobars (lines of constant pressure) must themselves lie east–west was noted empirically by the nineteenth-century Dutch meteorologist Buys Ballot, and is called Buys Ballot’s law. The law states that “Low pressure in the Northern Hemisphere is on the left when facing downwind” (on the right in the Southern Hemisphere). Because the pressure gradient is directly proportional to the geostrophic windspeed, the spacing of the isobars must decrease as the windspeed increases and vice versa. Instead of flowing in the direction of decreasing pressure gradient, the geostrophic wind flows perpendicular to the pressure gradient. For an area of low pressure in the Northern Hemisphere that is completely surrounded by areas of high pressure, the isobars are concentric circles with a low pressure at the center. The pressure gradient force causes the air to flow toward the center of this low-pressure system, but because of the Coriolis force the air is forced to deflect to the right. As a result, the Northern Hemisphere low or cyclone is characterized by a counterclockwise flow that spirals inward with rising air at the center [see Figure 21.6 as well as panel (c) of Figure 21.3]. In the case of an area of high pressure completely surrounded by areas of low pressure, the flow outward from the center will also be deflected to the right. This high or anticyclone exhibits a clockwise flow that spirals outward with descending air at its center (Figure 21.6). In the Southern Hemisphere, while the pressure gradient force acts in the direction from high pressure to low pressure, the Coriolis force now causes the deflection to be directed to the left of the path of motion. Thus a cyclone in the Southern Hemisphere is characterized by a clockwise flow of air that spirals inward with rising air at the center, while an anticyclone exhibits a counterclockwise flow that spirals outward

FIGURE 21.6 Northern Hemisphere cyclones and anticyclones.

900

GENERAL CIRCULATION OF THE ATMOSPHERE

with descending air in the center. Note that the only difference between the two hemispheres is the direction of rotation.

21.4.2 Ekman Spiral The geostrophic balance determines the wind direction at altitudes above about 500 m. In order to describe the air motions at lower levels, we must take into account the friction of Earth’s surface. The presence of the surface induces a shear in the wind profile, as in a turbulent boundary layer over a flat plate generated in a laboratory wind tunnel. In analyzing the geostrophic windspeed, we found that for steady flow a balance exists between the pressure force and the Coriolis force. Consequently, steady flow of air at levels near the ground leads to a balance of three forces: pressure force, Coriolis force, and friction force due to Earth’s surface. Thus, as shown in Figure 21.7, the net result of these three forces must be zero for a nonaccelerating air parcel. Since the pressure gradient force Fp must be directed from high to low pressure, and the frictional force Ff must be directed opposite to the velocity u, a balance can be achieved only if the wind is directed at some angle toward the region of low pressure. This angle between the wind direction and the isobars increases as the ground is approached since the frictional force increases. At the ground, over open terrain, the angle of the wind to the isobars is usually between 10° and 20°. Because of the relatively smooth boundary existing over this type of terrain, the windspeed at a 10 m height (the height at which the so-called surface wind is usually measured) is already almost 90% of the geostrophic windspeed. Over built-up areas, on the other hand, the speed at a 10 m height may be only 50% of the geostrophic windspeed, owing to the mixing induced by the surface roughness.

FIGURE 21.7 Variation of wind direction with altitude: (a) balance of forces among pressure gradient, Coriolis force, and friction; (b) the Ekman spiral.

901

GEOSTROPHIC WINDSPEED

In this case the surface wind may be at an angle of 45° to the isobars. As a result of these frictional effects, the wind direction commonly turns with height, as shown in Figure 21.7. The variation of wind direction with altitude is known as the Ekman spiral. Derivation of the expression for the Ekman spiral is the subject of Problem 21.4. Horizontal and Vertical Flows The relationship between horizontal and vertical velocities is embodied in the mass continuity equation @ρ @ @ @ ‡ …ρu† ‡ …ρv† ‡ …ρw† ˆ 0 @t @x @y @z

(21.10)

Atmospheric flows are such that the horizontal mass flux divergence, the second and third terms on the LHS of (21.10) are essentially balanced by the fourth term, the vertical mass flux divergence. Thus @ @ @ …ρu† ‡ …ρv† ‡ …ρw† ≅ 0 @x @y @z

(21.11)

When flow converges at lower levels, it flows vertically upward in such a way so as to satisfy (21.11). If we consider a column of air and integrate the last term on the LHS of (21.11) from the bottom (b) of the column to its top (t), we get t

|

t @ …ρw†dz ˆ ρw ˆ …ρw†t b ∫ b @z

(21.12)

since there is no mass flux of air into Earth’s surface. Thus, from (21.11), we have wt ˆ

@ 1 t @ …ρu† ‡ …ρv† dz ρt ∫ b @x @y

(21.13)

Atmospheric measurements in a variety of circumstances show that the RHS of (21.13) generally has a magnitude of order 10 cm s 1 in the midtroposphere. (Updraft rates in cumulonimbus clouds can, however, reach values of ∼5 m s 1.) A simple depiction of a tropospheric air column with surface-level convergent flow is given in Figure 21.8, wherein horizontal convergence at the surface is matched by horizontal divergence at upper levels. The situation sketched in Figure 21.8 especially represents that in cloudy weather systems.

FIGURE 21.8 Tropospheric air column with vertical convection.

902

GENERAL CIRCULATION OF THE ATMOSPHERE

21.5 THE THERMAL WIND RELATION We have seen that the horizontal wind components in the atmosphere obey the geostrophic equations: 1 @p ρf @y

ug ˆ

vg ˆ

1 @p ρf @x

(21.14)

A basic question is how the geostrophic windspeed varies with height, the so-called vertical wind shear. If one can obtain an expression for the gradients of ug and vg with altitude, this relation is the key to understanding how the flow in the atmosphere behaves at all levels. The two fundamental relations that govern the large-scale flow in the atmosphere are the geostrophic and hydrostatic balances, and these two relations can be combined to produce the desired relation for the vertical gradient of the windspeed. We desire the expressions for the variation of ug and vg with height; it turns out to be easier to use pressure as the vertical coordinate rather than height. So we seek expressions for @ug/@p and @vg/@p, where the minus sign allows the derivative with respect to the vertical coordinate to be taken upward (since pressure decreases with altitude). The quantities @ug @p

@vg @p

and

are called the thermal wind, and the relation we will obtain is called the thermal wind relation. We note, despite the name, that these quantities are vertical gradients of the windspeed, not windspeeds. The hydrostatic relation is usually expressed as the gradient of pressure with respect to height @p ˆ gρ @z

(21.15)

but it can be written with pressure as the coordinate and z as the dependent variable by simply inverting: @z ˆ @p

1 gρ

(21.16)

We can substitute for ρ using the ideal-gas law (ρ = p/RT), and we get @z RT ˆ @p pg

(21.17)

This equation can be nicely interpreted in finite difference form. Consider two pressure surfaces, ρ0 + Δρ/2 and ρ0 Δρ/2. The two pressure surfaces, at any given location, must occur at different heights, with a separation Δz. This distance between the two pressure surfaces is called its thickness. Thus Δz ≅

Δρ RT ρ g

(21.18)

Then, combining (21.14) and (21.16) gives ug ˆ

g @z f @y

vg ˆ

g @z f @x

(21.19)

903

THE THERMAL WIND RELATION

Taking the p derivative of the x component of the geostrophic wind in pressure coordinates, we have @ug ˆ @p

g @2z f @p@y g @ @z f @y @p

ˆ ˆ

1 @ 1 f @y ρ

(21.20) p

p

Since 1/ρ = RT/p, it follows that @ 1 @y ρ

ˆ p

R @T p @y

(21.21) p

Thus, we obtain the thermal wind relations: @ug R @T ˆ @p f p @y

p

@vg ˆ @p

R @T f p @x

(21.22) p

This relation shows that horizontal temperature gradients must be accompanied by vertical gradients of geostrophic windspeed. Imagine that in the Northern Hemisphere temperature decreases in the northward direction, that is, @T/@y < 0. At a certain height z at lower latitude, say, at point A, the air is warm and therefore has relatively low density, while beneath, say, point B, at higher latitude, the air is colder and more dense. From the hydrostatic balance, pA > pB, since the air density is less at point A than at point B and it must be p0 at the surface at both places. The x component of the geostrophic wind (in the Northern Hemisphere) is westerly since @p/@y < 0 (lower pressure on its left). Because @T/@y < 0, it follows that @ug/@p < 0, and thus ug increases with height. In summary, in the Northern Hemisphere, a northward decrease of temperature leads to westerly winds that increase in speed with altitude. In the Northern Hemisphere, (@T/@y)p is negative, and in the Southern Hemisphere, (@T/@y)p is positive. The Coriolis parameter f is positive in the NH and negative in the SH. The negative signs cancel, and ug flows in the same direction (west to east) in each hemisphere. As we noted earlier, the jetstreams are bands of relatively strong winds (usually exceeding 30 m s 1) located above areas of especially strong temperature gradients. The occurrence of jetstreams is a consequence of the thermal wind relation. In such areas, the pressure gradients and resulting windspeeds increase with height all the way up to the tropopause. Jetstreams occur in the upper troposphere, at altitudes between 9 and 18 km. Earlier, we mentioned the polar front jetstream. The polar front is the boundary between polar and midlatitude air. In winter this boundary can extend all the way down to 30°, while in summer it tends to reside at 50°–60°. The temperature gradient (@T/@y)p is stronger in winter than in summer. Thus, the polar front jetstream is located closer to the equator in winter and is stronger during that time. The polar front jets move west to east but meander with the general upper-air waves, serving to affect the movement of major low-level airmasses. Another important jetstream is located at the boundary between the Hadley and Ferrel cells, the poleward limit of the equatorial tropical air above the transition zone between tropical and midlatitude air (about 30°–40° latitude). This jet is not necessarily characterized by especially large surface temperature gradients but rather by strong temperature gradients in the midtroposphere. The sub­ tropical jet is generally less intense than the polar jet and is found at higher altitudes. Finally, a high-level jet called the tropical easterly jetstream is found in the tropical latitudes of the Northern Hemisphere (NH). Such jets are located about 15°N over continental regions; they result from latitudinal heating contrasts over tropical land masses that do not exist over the tropical oceans. Jetstreams in the Southern

904

GENERAL CIRCULATION OF THE ATMOSPHERE

Hemisphere (SH) generally resemble those in the NH, although they exhibit less day-to-day variability because of the presence of less landmass. The thermal wind relation also describes the prevailing flows in the stratosphere. In contrast to the troposphere, the midlatitude winds in the stratosphere blow from west to east in the winter and from east to west in the summer. The explanation of the differing large-scale flows between the troposphere and stratosphere lies in the temperature structure of the stratosphere. Above ∼30 mbar, stratospheric temper­ atures have their maximum at the summer pole, not the equator, and decrease uniformly from that point to the winter pole. Thus, (@T/@y)p > 0 in the summer NH. When the NH is in summer, the SH is in winter, and (@T/@y)p > 0 in the SH. Since f changes sign in the two hemispheres, but (@T/@y)p does not, @ug/@p has different signs in the two hemispheres. In the summer hemisphere, @ug/@p > 0 [in the NH, both (@T/@y)p and f are positive; in the SH both are negative]. In the winter hemisphere, @ug/@p < 0. Thus, in the summer hemisphere of the stratosphere, winds become more easterly as altitude increases; in the winter hemi­ sphere, they become more westerly. As the stratospheric temperature structure changes over the course of a year, the direction of the jets changes. In contrast to the troposphere, stratospheric jets, therefore, change direction seasonally and flow in opposite directions in the two hemispheres. Application of the Thermal Wind Relation The mean temperature in the layer between 65 and 60 kPa decreases in the eastward direction by 4°C per 100 km. If the geostrophic wind at 65 kPa is from the southeast at 20 m s 1, what are the geostrophic windspeed and direction at 60 kPa. Assume f = 10 4 s 1. Winds from the southeast are midway between easterly and southerly, so the geostrophic windspeed components at 65 kPa altitude are ug ˆ … 20 m s 1 †…cos 45 † ˆ 14:14 m s vg ˆ …20 m s 1 †…sin 45 † ˆ ‡14:14 m s

1

1

Now we need to determine the thickness Δz of the layer between 65 and 60 kPa. To do so we need the mean density of air in the layer. Taking T = 248 K at the average pressure of 62.5 kPa, we obtain ρˆ

p 6:25  104 Pa ˆ ˆ 0:88 kg m RT …287 J kg 1 K 1 †…248 K†

3

Then the thickness of the layer between 65 and 60 kPa is found from the hydrostatic relation Δz ˆ

Δp 5000 Pa ˆ ˆ 581 m ρg …0:88 kg m 3 †…9:8 m s 2 †

Now the thermal wind relation will allow us to calculate the change in geostrophic wind over this layer. The only temperature gradient is in the eastward (x) direction. From (21.22), there is no change ug with altitude. From (21.22), we can express @vg/@p as @vg g @T ˆ @z f T @x

p

In finite difference form, this is Δvg ˆ Δz

g ΔT f T Δx

ˆ …581 m†

9:8 m s 2 …10 4 s 1 †…248 K†

4K 105 m

ˆ 9:2 m s

1

905

THE HYDROLOGIC CYCLE

With vg = 14.14 m s 1 at 65 kPa, the meridional wind component at 60 kPa is 14.14 – 9.2 = 4.9 m s 1. The zonal component is 14.14 m s 1, so the net windspeed at 60 kPa = [(4.9)2 + ( 14.14)2]1/2 = 15 m s 1. The wind direction at 60 kPa is out of the southeast at a direction of tan θ = 4.9/14.14. Thus, θ = 19°, and the wind is from the southeast at 19° north of west.

21.6 STRATOSPHERIC DYNAMICS The classic depiction of stratospheric transport is that material enters the stratosphere in the tropics, is transported poleward and downward, and finally exits the stratosphere at middle and high latitudes. The mean meridional stratospheric circulation, known as the Brewer–Dobson circulation, is generated by stratospheric wave forcing, with the circulation at any level being controlled by the wave forcing above that level (recall Figure 5.34). This process is also called the extratropical pump. The composition of the lowermost stratosphere varies with season, suggesting a seasonal dependence in the balance between the downward transport of stratospheric air and the horizontal transport of air of upper tropospheric character. The stratosphere and troposphere are actually coupled by more dynamically complex mechanisms than the traditional model of large-scale circulation-driven exchange. Waves generated in the troposphere propagate into the stratosphere, where they can exert forces on circulation, which then can extend back down into the troposphere. A coupling exists between the variability of the stratosphere and troposphere through what are termed the northern and southern annular modes (NAM and SAM), also called the Arctic and Antarctic oscillations (AO and AAO). The extreme states of this mode of variability correspond to strong and weak polar vortices.

21.7 THE HYDROLOGIC CYCLE The term hydrologic cycle refers to the fluxes of water, in all its states and forms, over Earth. Table 21.1 gives the estimated volume and percentage of total water in all the major global reservoirs, and Table 21.2 shows the estimated annual fluxes between major reservoirs. We note that 97% of Earth’s water resides in the oceans. Of the remainder, about 3/4 is sequestered in the ice caps of Greenland and Antarctica, and ¼ is in underground aquifers and lakes and rivers. Only a fraction of 10 5 is in the atmosphere, almost all of which is in the form of water vapor. If the total water vapor content of the atmosphere were suddenly precipitated, it would cover Earth with a rainfall of only 3 cm (see Problem 21.2). The hydrosphere is likely to have been formed by outgassing of steam from volcanoes during the early life of Earth. Although the mass of the current hydrosphere is effectively unchanging, water rapidly cycles between reservoirs (Table 21.2). More water is evaporated from the oceans than falls on them as precipitation, and more water falls as precipitation on the landmasses than is evaporated. The system is

TABLE 21.1 Water in Major Global Reservoirs Reservoir Oceans Glaciers and ice sheets Underground aquifers Lakes Rivers Atmosphere Biosphere Total

Volume of water (km3) 1,370,000,000 29,000,000 9,565,000 125,000 1,700 13,000 600 1,408,705,300

Percentage of total 97.25 2.05 0.69 0.01 0.0001 0.001 0.00001 100

906

GENERAL CIRCULATION OF THE ATMOSPHERE

TABLE 21.2 Water Fluxes between Reservoirs Flux (km3 yr 1)

Reservoirs

Process

Ocean–atmosphere

Evaporation Precipitation Evaporation Precipitation Runoff

Landmasses–atmosphere Landmasses–ocean

400,000 370,000 60,000 90,000 30,000

balanced by river runoff. Most of the evaporation occurs from the tropical oceans, where the water vapor is carried to the ITCZ. Here it is uplifted in rising convective systems, where it precipitates in intense downpours. This intense convection over tropical landmasses accounts for much of the world’s rainfall and much of the flux of water from the oceans to landmasses. The annual flux of water through the atmosphere is about 460,000 km3 yr 1, about 35 times greater than the amount of water stored in the atmosphere at any one time. Thus, the average residence time of a molecule of water in the atmosphere can be estimated as 13,000 km3/460,000 km 3 yr 1, or about 10 days. By contrast, the ocean reservoir is over 3000 times larger than the annual flux to the atmosphere, so the average residence time of a molecule of water in the oceans is very long (about 3400 years).

PROBLEMS 21.1A Compute the magnitude of the geostrophic windspeed (westerly) at 40°N latitude and 5000 m altitude assuming that the pressure at this latitude and altitude decreases from south to north by 400 Pa over a distance of 200 km. 21.2A The total mass of water in the atmosphere is 1.3 × 1016 kg, and the global mean rate of precipitation is estimated to be 0.2 cm day 1. Estimate the global mean residence time of a molecule of water in the atmosphere. 21.3A Accounting for the order of magnitude of atmospheric quantities, estimate the time needed to mix a species throughout a hemisphere. 21.4C The Ekman spiral describes the variation of wind direction with altitude in the planetary boundary layer. The analytical form of the Ekman spiral can be derived by considering a two-dimensional wind field (no vertical component), the two components of which satisfy @ @u KM @z @z

1 @p ‡f v ˆ 0 ρ @x

@ @v KM @z @z

1 @p ρ @y

fu ˆ0

where KM is a constant eddy viscosity and f is the Coriolis parameter. At some height the first terms in each of these equations are expected to become negligible, leading to the geostrophic wind field. Take the x axis to be oriented in the direction of the geostrophic wind, in which case f ug ˆ

1 @p ρ @y

907

REFERENCES

Show that the solutions for ū(z) and v(z) are eαz cos αz†

u…z† ˆ u g …1 v…z† ˆ u g e where α ˆ

αz

sin αz

p f =2KM . In what way is this solution oversimplified?

21.5B The outer boundary of the planetary boundary layer can be defined as the point at which the component in the Ekman spiral disappears. From the solution of Problem 21.4, it is seen that this occurs when αzg = π. a. Using representative values of f and KM, estimate the depth of the planetary boundary layer. b. The magnitude of the total turbulent stress at the surface is given by ρKM @u τ0 ˆ p 2 @z

‡ zˆ0

@v @z

zˆ0

Show that for a planetary boundary layer zg τ0 ˆ p f ρug 2π Estimate the magnitude of τ0. c. The surface layer can be estimated as that layer in which τ0 changes by only 10% of its value at the surface. In the planetary boundary layer the turbulent stress terms in the equations of motion are of the same order as the Coriolis acceleration terms. Estimate @τ/@z, and from this estimate the thickness of the surface layer.

REFERENCES Ackerman, S. A., and Knox, J. A. (2003), Meteorology: Understanding the Atmosphere, Brooks/Cole, Pacific Grove, CA. Hadley, G. (1735), Concerning the cause of the general trade winds, Philos. Trans. 39. Holton, J. R. (1992), An Introduction to Dynamic Meterology, 3rd ed., Academic Press, San Diego. McIlveen, R. (1992), Fundamentals of Weather and Climate, Chapman & Hall, London. Salby, M. L. (1996), Fundamentals of Atmospheric Physics, Academic Press, San Diego.

CHAPTER

22

Global Cycles: Sulfur and Carbon

Biogeochemical cycles describe the exchange of molecules containing a given atom between various reservoirs within the Earth system. Important biogeochemical cycles in the Earth system include oxygen, nitrogen, sulfur, carbon, and phosphorus. Here we consider the global biogeochemical cycles of sulfur and carbon, two of the most important elements from the standpoint of atmospheric chemistry and climate.

22.1 THE ATMOSPHERIC SULFUR CYCLE Figure 22.1 depicts the major reservoirs in the biogeochemical cycle of sulfur, with estimated quantities [in Tg(S)] in each reservoir. Arrows indicate the direction of fluxes between the reservoirs. The major pathways of sulfur compounds in the atmosphere are depicted in Figure 22.2. The numbers on each arrow refer to the description of the process given in the caption to the figure (not to fluxes). Note the small amount of sulfur in the atmosphere relative to that in the other reservoirs. Also note the significant amount of sulfur in the marine atmosphere; this is the result of dimethyl sulfide (DMS) emissions from the sea. We wish now to analyze that portion of the global sulfur cycle involving atmospheric SO2 and sulfate shown in Figure 22.2. We will denote the natural and anthropogenic emissions of SO2 as PnSO2 and PaSO2 , respectively. PnSO2 includes a contribution from the oxidation of reduced sulfur species to SO2. SO2 is removed by dry and wet deposition and oxidized to sulfate by chemical reaction. Sulfate is also removed from the atmosphere by dry and wet deposition. Our goal is to obtain estimates for the lifetimes of SO2 and SO24 . Writing (2.15) for both SO2 and SO24 , we obtain PnSO2 ‡ PaSO2 kcSO2 QSO2

c kdSO2 ‡ kw SO2 ‡ kSO2 QSO2 ˆ 0

kdSO4 ‡ kw SO4 QSO4 ˆ 0

(22.1) (22.2)

Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, Third Edition. John H. Seinfeld and Spyros N. Pandis.  2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

908

909

THE ATMOSPHERIC SULFUR CYCLE

FIGURE 22.1 Major reservoirs and burdens of sulfur, in Tg(S) (Charlson et al. 1992). (Reprinted by permission of Academic Press.)

The mean residence time of SO2 is τSO2 ˆ ˆ

QSO2 n ‡ PaSO2 PSO 2 1

(22.3)

c kdSO2 ‡ kw SO2 ‡ kSO2

whereas that for sulfate is τSO4 ˆ ˆ

QSO4 kcSO2 QSO2 1

(22.4)

d kSO ‡ kw SO4 4

The mean residence time of a sulfur atom is τS ˆ

QSO2 ‡ QSO4 PnSO2 ‡ PaSO2

(22.5)

which can be expressed in terms of the two previous mean residence times as τS ˆ τSO2 ‡ bτSO4

(22.6)

where bˆ

kcSO2 QSO2 PnSO2 ‡ PaSO2

the fraction of S converted to SO24 before being removed.

(22.7)

910

GLOBAL CYCLES: SULFUR AND CARBON

FIGURE 22.2 Major pathways of sulfur compounds in the atmosphere (Berresheim et al. 1995). The paths are labeled according to the processes: (1) emission of DMS, H2S, CS2, and OCS; (2) emission of S(+4) and S(+6); (3) oxidation of DMS, H2S, and CS2 by OH, and DMS, by NO3 in the troposphere; (4) transport of OCS into the strato­ sphere; (5) photolysis of OCS or reaction with O atoms to form SO2 in the strato­ sphere; (6) oxidation of SO2 in the stratosphere; (7) transport of stratospheric OCS, SO2, and sulfate back into the troposphere; (8) oxidation of SO2 and other S(+4) products by OH in the troposphere; (9) absorption of S(+4), mainly SO2, into hydrosols (cloud/fog/rain droplets, moist aerosol particles); (10) liquid-phase oxidation of S(+4) by H2O2(aq) in hydrosols (and by O2 in the presence of elevated levels of catalytic metal ions); (11) absorption/growth of S(+6) aerosol—mainly sulfate—into hydrosols; (12) evaporation of cloudwater leaving residual S(+6) aerosol; (13) deposition of OCS, S(+4), and S(+6).

We can also define individual characteristic times such as τdSO2 ˆ kdSO2

1

w τw SO2 ˆ k SO2

;

1

τdSO4 ˆ kdSO4

;

1

so that 1 τSO2

ˆ

1 τdSO2

‡

1 τw SO2

‡

1 τcSO2

1 1 1 ˆ ‡ τSO4 τdSO4 τw SO4

(22.8) (22.9)

We can also define the mean residence time of a sulfur atom before surface removal or precipitation scavenging by τd;w S ˆ

QSO2 d;w kSO2 QSO2

‡ QSO4 ‡ kd;w SO4 QSO4

(22.10)

911

THE ATMOSPHERIC SULFUR CYCLE

so that using PnSO2 ‡ PaSO2 QSO2 QSO4 ˆ kdSO2 ‡ kw ‡ kdSO4 ‡ kw SO2 SO4 QSO2 ‡ QSO4 QSO2 ‡ QSO4 QSO2 ‡ QSO4

(22.11)

1 1 1 ˆ d‡ w τS τS τS

(22.12)

we get

The mean residence times for a sulfur atom before surface removal (or precipitation scavenging) can be related to the mean surface removal residence times for SO2 and SO24 as follows. Noting that τdS ˆ ˆ

ˆ

QSO2 d kSO2 QSO2

‡ QSO4 ‡ kdSO4 QSO4

QSO2 ‡ QSO4 QSO2 QSO4 ‡ d d τSO τSO4 2

(22.13)

1 QSO2 QSO4 1 1 ‡ d d QSO2 ‡ QSO4 τSO2 QSO2 ‡ QSO4 τSO4

we have 1 c …1 c† ˆ d ‡ d d τS τSO2 τSO4

(22.14)

where

cˆ 1‡

1

c b kdSO2 ‡ kw SO2 ‡ k SO2

(22.15)

w kdSO4 ‡ kSO 4

By virtue of (22.14), c is the fraction of the total sulfur that is SO2. Rodhe (1978) has estimated values of the sulfur residence times (in hours): τdSO2

τw SO2

τcSO2

τSO2

τdSO4

τw SO4

τSO4

60

100

80

25

> 400

80

80

Assuming c = 0.5, the sulfur atom residence times are (in hours): τSd 120

τw S 90

τS 50

Because of the uneven spatial distribution of anthropogenic sources and the relatively brief residence time of sulfur in the atmosphere, global averages do not provide an accurate description of human influence on the sulfur cycle in populated parts of the world.

912

GLOBAL CYCLES: SULFUR AND CARBON

22.2 THE GLOBAL CARBON CYCLE 22.2.1 Carbon Dioxide In terms of total mass of emissions, carbon dioxide is the single most important waste product of our industrialized society. The primary effect of CO2 in the atmosphere is climatic (see Chapter 23). Charles D. Keeling of the Scripps Institution of Oceanography began measurement of atmospheric CO2 levels at Mauna Loa, Hawaii in 1958 (Keeling 1983; Keeling and Whorf 2003). Figure 22.3 (top panel) shows monthly mean CO2 mixing ratios at Mauna Loa from 1958 to 2013. The bottom panel of Figure 22.3 shows the CO2 mixing ratio from 2009 to mid-2013. The monthly average CO2 mixing ratio in August 2012 was 392.41 ppm; that in August 2013 was 395.15 ppm. April 2014 was the first month in human history with an average CO2 level in Earth’s atmosphere exceeding 400 ppm.

FIGURE 22.3 CO2 mixing ratio at Mauna Loa, Hawaii. The curve with periodic oscillation is the monthly mean mixing ratio, centered on the middle of each month. The line through this curve represents the same, after correction for the seasonal average cycle. Top panel: The complete data record since 1958. Bottom panel: The data record for 2009–2013. Source: Scripps Institution of Oceanography (scrippsco2.ucsd.edu/) and National Oceanic and Atmospheric Administration (www.esrl.noaa.gov/gmd/ccgg/trends/).

THE GLOBAL CARBON CYCLE

FIGURE 22.4 CO2 mixing ratio over the past 1000 years from the recent ice core record and (since 1958) from the Mauna Loa measurement site. The inset shows the period from 1850 in more detail, including CO2 emissions from fossil fuel. The smooth curve is based on a 100-year running mean. All ice core measurements were taken in Antarctica.

The CO2 record exhibits a seasonal cycle. Each year during the Northern Hemisphere growing season in spring and summer, atmospheric CO2 mixing ratios decrease as carbon is incorporated into leafy plants. From October through January, photosynthesis is largely confined to the tropics and the relatively small landmass of the Southern Hemisphere. At this time of year, plant respiration and decay dominate, and CO2 levels increase. At Mauna Loa, the peak-to-trough CO2 seasonal cycle has remained between 5 and 6 ppm since 1958. The amplitude of the seasonal cycle increases to about 15 ppm in the boreal forest zone (55°–65°N). In the Southern Hemisphere, the mean amplitude of the CO2 seasonal cycle is only about 1 ppm. The preindustrial (pre–Industrial Revolution) level of CO2 over the last 1000 years, as determined from ice core records, was about 280 pm (Figure 22.4).1 Over the period 1750–2011, it is estimated that 365 Pg C (1 Pg = 1 Gt = 1015 g) were released to the atmosphere by fossil fuel combustion and cement manufacture (Figure 22.5). In addition, land-use changes are estimated to have resulted in a net transfer of 180 Pg C to the atmosphere since 1750. This totals 545 Pg C (1 Pg C = 3.67 Pg CO2). As seen in Figure 22.3, the atmospheric CO2 mixing ratio rose from 280 to 391 ppm in 2011. Each ppm of CO2 in the atmosphere corresponds to 2.1 Pg C (see Problem 1.6), so the increase in the atmospheric burden of carbon from 1750 to 2011 was 233 Pg C. Thus, about 43% (233/545) of the carbon added to the atmosphere has remained in the atmosphere; the other 57% has been absorbed by the oceans and the terrestrial biosphere. The 391 ppm of CO2 translates into 821 Pg C, of which 233 Pg C has been added since preindustrial time. Figure 22.6 shows the estimated percentage of an initial atmospheric CO2 pulse remaining in the atmosphere, as computed by a range of climate–carbon cycle models. The left panel shows the first 100 years; and the right panel, the first 1000 years. Scenarios of initial pulses of 100 and 5000 Pg C are shown. Figure 22.7 shows the estimated fate of an initial pulse of 5000 Pg C in terms of redistribution among atmosphere, land, and ocean (IPCC 2013). 1

Estimates of the preindustrial CO2 mixing ratio vary from 280 to 290 ppm. Records generated both by proxy techniques and recovery of ancient air trapped in glacial ice show the CO2 abundance cycling between 200 and 300 ppm as the climate cycled from glacial to interglacial conditions (Barnola et al. 1987; Genthon et al. 1987).

913

914

GLOBAL CYCLES: SULFUR AND CARBON

FIGURE 22.5 Annual global fossil fuel carbon emissions. Emissions from flaring and cement manufacture are included in the total but not shown owing to their relatively small magnitude. (Carbon Dioxide Information Analysis Center, Oak Ridge National Laboratory; http://cdiac.ornl.gov/trends/emis/glo_2010.html).

22.2.2 Compartmental Model of the Global Carbon Cycle Of the values for all the quantities of interest in the global carbon cycle, such as the preindustrial CO2 mixing ratio, the total fossil fuel emissions of CO2, the transfer of CO2 to the atmosphere as a result of deforestation, and the amounts of carbon in the preindustrial reservoirs, only the fossil fuel emissions are known with significant accuracy. As a result, different studies employ somewhat different procedures or databases to estimate these quantities, leading to a range of estimates in the literature. For example, the parameters in the global carbon cycle model that we will employ here have been determined on the basis of a value for preindustrial atmospheric carbon loading somewhat different from the 588 Pg C corresponding to a CO2 mixing ratio of 280 ppm. Nonetheless, the important features of the carbon cycle, such as the percentage of carbon emissions remaining in the atmosphere, are reflected in most models.

FIGURE 22.6 Percentage of initial atmospheric CO2 perturbation remaining in the atmosphere in response to an idealized instantaneous pulse in year 0 as calculated by a range of coupled climate–carbon cycle models. Solid line is the multimodel mean and the shading is the 2-standard deviation range. Left panel, 100 years; right panel, 1000 years. (IPCC 2013.)

THE GLOBAL CARBON CYCLE

FIGURE 22.7 Decay of a CO2 excess amount of 5000 PgC emitted into the atmosphere at time zero and its subsequent redistribution into land and ocean as a function of time, computed by coupled carbon-cycle climate models. The size of the bands indicates the carbon uptake by the respective reservoir. The first two panels show the multimodel mean from a model intercomparison project (Joos et al. 2013). The last panel shows the longer-term redistribution, including ocean dissolution of carbonaceous sediments as com­ puted by an Earth System model. (IPCC 2013.)

Carbon is naturally contained in all of Earth’s compartments; the global carbon cycle is a description of how carbon moves among those compartments in response to perturbations, such as emissions from fossil fuel burning and deforestation. Compartmental models of the global carbon cycle provide a means to simulate the flux of carbon between the various reservoirs and to estimate future levels corresponding to different CO2 emission scenarios. Figure 22.8 shows a six-compartment model of the carbon cycle due to Schmitz (2002). The quantities shown in parentheses in each compartment are estimates of the preindustrial (∼1850) amount of carbon, indicated by the Mi symbols, measured in petagrams (Pg C) in each reservoir. Since the amount of carbon in the aquatic biosphere and in rivers, streams, and lakes is negligible compared with those in the other reservoirs in shown Figure 22.8, these are omitted. The fossil fuel reservoir affects the global carbon cycle only as a source of carbon. Sediments are actually the largest carbon reservoir of all, but the fluxes of carbon into and out of sediments are so small that sediments can be neglected as a compartment over any realistic timescale.2 Estimates of the preindustrial flux of carbon from reservoir i to reservoir j (Fij) are given in parentheses next to the arrow indicating the flux; at the preindustrial steady state, the net flow of carbon into or out of any reservoir was zero. Some of the fluxes shown in Figure 22.8 can be elaborated on: 1. F12, F21, F13 and F31 are rates of transfer of CO2 between the atmosphere and surface waters of the ocean. 2. F23 is the flux of carbon from warm to cool surface ocean reservoirs. This flow, which accounts for much of mixing in the oceans, results from sinking of cool surface water at high latitudes and upwelling at low latitudes. 3. F15 is the uptake of atmospheric CO2 by terrestrial vegetation (photosynthesis). 4. F56 is the flux of carbon in litter fall, such as dead leaves. 5. F51 and F61 are the fluxes of carbon to the atmosphere by respiration of biota.

2 Certain carbon sequestration approaches under consideration are aimed at transferring some of the CO2 that would otherwise be released to the atmosphere to the sediments or underground deep-water reservoirs by direct injection of CO2 emitted by power plants.

915

916

GLOBAL CYCLES: SULFUR AND CARBON

FIGURE 22.8 Six-compartment model of the global carbon cycle (Schmitz 2002). Values shown for reservoir masses (Mi, in Pg C) and fluxes (Fij, in Pg C yr 1) represent those for the preindustrial steady state.

The carbon content of the oceans is more than 50 times greater than that of the atmosphere. Over 95% of the oceanic carbon is in the form of inorganic dissolved species, bicarbonate HCO3 and carbonate …CO23 † ions; the remainder exists in various forms of organic carbon (Druffel et al. 1992). Oceanic uptake of CO2 involves three steps: (1) transfer of CO2 across the air–sea interface, (2) chemical interaction of dissolved CO2 with seawater constituents, and (3) transport to the deeper ocean by vertical mixing processes. Steps 2 and 3 are the rate-determining processes in the overall transfer of CO2 from the atmosphere to the ocean, and oceanic transport and mixing processes are the primary uncertainties in predicting the rate of oceanic uptake of CO2. The ocean is both a source and sink for atmospheric CO2. Two fundamental effects control the ocean distribution of CO2. First, the cold waters, which fill the deep ocean from the high latitudes, are rich in CO2; this results because the solubility of CO2 increases as temperature decreases. In the sunlit uppermost 100 m of the ocean, photosynthesis is a sink for CO2 and a source of O2. Dying marine organisms decay as they settle into the deeper layers of the ocean, consuming dissolved O2 and giving off CO2. Thus, these layers have higher CO2 concentrations and lower O2 concentrations than the surface waters. This process of transporting CO2 to deeper water is referred to as the biological pump. Since conditions during preindustrial times, the rise in atmospheric CO2 has led to an increased air­ to-sea CO2 flux everywhere on Earth. From 10s of 1000s of individual measurements made in every corner of the world’s ocean in every season of the year, a global map of the present-day flux of CO2 across the sea surface has emerged (Takahashi et al. 1997). While the net global flux of CO2 is into the ocean, there are regions in which CO2 is outgassed. The equatorial Pacific is the strongest continuous global natural source of CO2 to the atmosphere, the cause of which is vigorous upwelling along the equator, driven by divergence of surface currents. The upwelling water comes from only a few 100 meters’ depth

917

THE GLOBAL CARBON CYCLE

but is warmed by several degrees as it ascends, decreasing the solubility of CO2. Even though the upwelling water also brings abundant nutrients to the surface, the plankton are not able to make full use of these because of the lack of sufficient amounts of the nutrient iron. The absence of biological production in these surface waters means that the CO2 brought to the surface is released to the atmosphere rather than taken up by photosynthesis. The most intense oceanic sink region for CO2 is the North Atlantic. The Gulf Stream and North Atlantic drift transport warm water rapidly northward. As the water cools, CO2 solubility increases. In addition, the North Atlantic is biologically the most productive ocean region on Earth, because of an abundant supply of nutrients, including iron from Saharan dust. Other important ocean sink areas are all associated with zones where the ocean is transferring heat to the atmosphere and cooling. Carbon in the form of CO2 is removed from the atmosphere and incorporated into the living tissue of green plants by photosynthesis. Respiration returns carbon to the atmosphere (Schimel 1995). Photo­ synthesis responds essentially instantaneously to an increase in atmospheric CO2. Respiration does not respond directly to atmospheric CO2 concentrations but exponentially increases with increasing temperature. Carbon is stored in terrestrial ecosystems in two main forms: carbon in biomass …∼1=3†, principally as wood in forests; and carbon in soil …∼2=3†. The lifetime of carbon in these reservoirs can be quite different. If elevated CO2 leads to increased growth of a tree, this results in a CO2 sink. The tree might typically live for a century before it dies and decomposes, releasing CO2 back into the atmosphere via heterotrophic respiration. Large terrestrial carbon reservoirs in the form of trees or forest floor litter are susceptible to rapid return to the atmosphere through fire. Figure 22.8 shows three fluxes denoted by dashed lines: Ff emissions of carbon from fossil-fuel burning Fd flux of carbon from renewable terrestrial biota to the atmosphere (deforestation) Fr flux of carbon from the atmosphere to terrestrial biota (reforestation) These three fluxes are the direct anthropogenic perturbations to the carbon cycle and are assumed to have been negligibly small in preindustrial times. This is an excellent assumption for Ff and Fr, but it also presumes that natural forest fires (Fd) were responsible for small emissions of CO2. The preindustrial compartmental masses of carbon in Figure 22.8 are those assumed by Schmitz (2002). The preindustrial atmospheric mass of CO2 is taken to be 612 Pg C, which corresponds to a CO2 mixing ratio of 291 ppm, toward the high end of the range of 280–290 ppm usually assumed as the preindustrial value. The carbon mass balances for each of the six compartments follow the development in Chapter 2. For example, for the carbon in the atmosphere, we have dM1 ˆ F21 dt

F12 ‡ F31

F13 ‡ F51

F15 ‡ F61 ‡ Ff ‡ Fd

Fr

(22.16)

At the preindustrial steady state, the perturbation fluxes, Ff = Fd = Fr = 0, and F21 F12 + F31 F13 + F51 F15 + F61 = 0, so that dM1/dt = 0. The total amount of carbon stored in fossil fuels affects the carbon cycle only through the amount actually released to the atmosphere. Falkowski et al. (2001) have estimated that the fossil fuel reservoir contains 4130 Pg C. (The ratio of the estimated fossil fuel reservoir to the current atmospheric reservoir, 4130/788 = 5.2. Thus, if the entire estimated fossil fuel reservoir were added to the atmosphere with no carbon taken up by other reservoirs, the atmospheric concen­ tration of CO2 would increase by a factor of 6.2. This can be considered as an upper-limit estimate for increase of atmospheric CO2.) As a first approximation, the intercompartmental fluxes can be assumed to be proportional to the mass in the compartment from which the flux is taking place Fij ˆ kij Mi

(22.17)

918

GLOBAL CYCLES: SULFUR AND CARBON

where the kij values are first-order exchange coefficients. These kij values can be determined from the fluxes and masses in Figure 22.8 using (22.17), assuming that the exchange coefficients between compartments have remained the same from preindustrial times to the present. There are three exceptions: F21 (warm ocean waters to atmosphere), F31 (cold ocean waters to atmosphere), and F15 (atmosphere to terrestrial biota). Once CO2 dissolves in water, an equilibrium is established among dissolved CO2 (CO2
H2O), bicarbonate ions …HCO3 †, and carbonate ions …CO23 † (see Chapter 7). Thus, while the sea-to-air fluxes F21 and F31 can be linearly related to dissolved CO2 (CO2
H2O), they are not linearly related to the total dissolved carbon (M2 and M3), which is the sum of CO2
H2O, HCO3 , and CO23 . This relationship depends on seawater temperature and pH, the latter of which depends in a complex way on salinity and concentrations of dissolved salts. Seawater pH varies between ∼ 7.5 and 8.4, and therefore most of the CO2 that dissolves in the ocean is not in the CO2
H2O form (Chapter 7). For the present global carbon cycle model we will avoid the complication of an explicit ocean chemistry model to relate F21 and F31 to M2 and M3, respectively. The following empirical relationship has been developed (Ver et al. 1999) β

F21 ˆ k21 M22

β

F31 ˆ k31 M33

(22.18)

where β1 and β2 are positive empirical constants. Given numerical values of β2 and β3, the values of k21 and k31 can be determined from the preindustrial conditions shown in Figure 22.8. Note that the units of k21 and k31 are no longer simply yr 1, but are Pg C…1 β2 † yr 1 and Pg C…1 β3 † yr 1 , respectively. As M2 and M3 increase with more dissolved carbon in the oceans, the fluxes of carbon back to the atmosphere F21 and F31 increase so that the net flux from atmosphere to ocean decreases. In this way, the fraction of CO2 added to the atmosphere that is taken up by the oceans decreases with increasing CO2 burden because of reduced buffer capacity of the CO2
H2 O=HCO3 =CO23 system. F15 expresses the rate of photosynthetic uptake of atmospheric CO2 by vegetation. Physically, F15 increases with increasing CO2 but eventually saturates. Lenton (2000) has suggested the following empirical form for F15: F15 ˆ

0 M1  γ M1 γ M1 > γ k15 G M1 ‡ Γ

(22.19)

γ is a threshold value for M1, below which no CO2 uptake occurs (γ = 62 Pg C). Γ is a saturation parameter (Γ = 198 Pg C). At sufficiently high CO2 (in the range of 800–1000 ppm), carbon assimilation via photosynthesis would cease to increase. G(t) is a scaling factor that corrects the flux from the atmosphere to the terrestrial biota for the permanent effects of deforestation and reforestation that have taken place from preindustrial times (1850) until time t. For example, if forested areas are cleared and converted to urban use, then those forested areas are forever removed from Earth’s supply of terrestrial biota, and, even in the absence of any fossil fuel burning, the new carbon cycle steady state will be different from the preindustrial state because the CO2 uptake capacity of terrestrial biota will have been permanently altered. The preindustrial value of G is taken to be 1.0. G(t) can be expressed by G…t† ˆ 1 ‡

t

∫0

ar Fr ad Fd dt M 5 … 0†

(22.20)

where time zero is taken as preindustrial (1850), ad is the fraction of forested area that is not available for regrowth after deforestation, and ar is the fraction of reforested area that increases Earth’s terrestrial biota. If no deforestation or reforestation occurs, then the integral is zero, and G = 1. The integral accounts for the permanent effects of changes to terrestrial vegetation. For example, if only deforestation occurs, then G < 1, and F15, the flux of CO2 from the atmosphere to vegetation, as given by (22.19), is less than that if deforestation had not taken place.

919

THE GLOBAL CARBON CYCLE

Equation (22.20) can be replaced by its corresponding differential equation: dG ar Fr ad Fd ˆ ; dt M 5 … 0†

G … 0† ˆ 1

(22.21)

Returning to (22.19), the numerical value of k15 can be calculated from the preindustrial values in Figure 22.8, using G = 1 and the values of γ and Γ. The full compartmental model and numerical values of the coefficients are given in Table 22.1. It is assumed that all reforested land increases the terrestrial biota, so ar = 1.0. The value of ad = 0.23 is that

TABLE 22.1 Compartmental Model for the Global Carbon Cycle Parameters Governing equations

Symbol

Value

Units

k12

0.0931

yr

1

k13

0.0311

yr

1

dM2 ⠈ k12 M1 …k23 ‡ k24 †M2 k21 M22 ‡ k42 M4 dt dM3 ⠈ k13 M1 ‡ k23 M2 k34 M3 k31 M33 ‡ k43 M4 dt

k15

147

yr

1

k21

58…730

dM4 ˆ k24 M2 ‡ k34 M3 …k42 ‡ k43 †M4 dt M1 γ dM5 ˆ k15 G …k51 ‡ k56 †M5 Fd …t† ‡ Fr …t† dt M1 ‡ Γ dM6 ˆ k56 M5 k61 M6 dt

k23

0.0781

yr

1

k24

0.0164

yr

1

k31

18…140

dM7 ˆ Ff …t† dt dG ad Fd …t† ar Fr …t† ˆ dt M5 …0†

k34

0.714

yr

1

k42

0.00189

yr

1

k43

0.00114

yr

1

k51

0.0862

yr

1

k56

0.0862

yr

1

k61

0.0333

yr

1

β2 β3 γ

9.4 10.2 62.0

PgC

Γ ad ar

198 0.230 1.0

dM1 ˆ …k12 ‡ k13 †M1 dt β

k15 G

M1 γ β ‡ k21 M22 M1 ‡ Γ

‡ k31 M33 ‡ k51 M5 ‡ k61 M6 ‡ Ff …t† ‡ Fd …t†

Source: Schmitz (2002).

Fr …t†

β2

β3

†

†

PgC…1

PgC…1

PgC

β2 †

β3 †

yr

1

yr

1

920

GLOBAL CYCLES: SULFUR AND CARBON

FIGURE 22.9 Prediction of atmospheric carbon loading from preindustrial time to the year 2000 from numerical integration of the six-compartment model given in Table 22.1 with initial conditions based on the carbon amounts shown in Figure 22.8. Data points correspond to global average CO2 data.

suggested by Schmitz (2002) [Lenton (2000) proposed 0.27]. Initial values for all Mi are the preindustrial values given in Figure 22.8. The preindustrial fossil fuel reservoir is assumed to have contained 5300 Pg carbon; the actual value is not important, only the emission rate Ff(t). The model requires as input Ff(t), Fd(t), and Fr(t) from preindustrial times to the present. Ff(t) is obtained from the historical record of carbon emissions from fossil fuels; Fd(t) is that for deforestation, expressed also in units of Pg C yr 1. Until very recently, reforestation, Fr, can be assumed to have been negligibly small. The differential equations for M1, M2, M3, M4, M5, M6, M7, and G can be integrated numerically to predict burdens of atmospheric CO2 from 1850 to the present using the emissions from Figure 22.5; the result is shown in Figure 22.9, and agreement with observed mixing ratios over this period is close. Table 22.2 compares the predicted quantities of carbon in each of the seven compartments in 1990 with their preindustrial values. We also note that G is predicted to have decreased from 1.0 to 0.952 over this time period. Of the six reservoirs, only the atmosphere has shown a substantial increase in carbon loading since preindustrial times. Carbon in terrestrial biota (M5) is predicted to have decreased by only 3 Pg, from 580 to 577 Tg; decreases resulting from deforestation and increased uptake owing to higher ambient CO2 levels virtually offset each other.

TABLE 22.2 Preindustrial and 1990 Predicted Compartmental Burdens of Carbon (Pg C) Burden M1 M2 M3 M4 M5 M6 M7 G Source: Schmitz (2002).

Preindustrial

1990

612 730 140 37,000 580 1,500 5,300 1.0

753 744 143 37,071 577 1,489 5,086 0.952

921

THE GLOBAL CARBON CYCLE

22.2.3 Atmospheric Lifetime of CO2 The atmospheric lifetime of CO2 has taken on a particular significance owing to attempts to predict the impact of fossil fuel emissions of CO2 on Earth’s climate (Archer et al. 2009). Two definitions of the lifetime of anthropogenic CO2 are prevalent. One definition is the average time a molecule of CO2 spends in the atmosphere before it is removed. The second definition is the time required for the atmospheric CO2 concentration to return to its preindustrial level following cessation of fossil fuel CO2 emissions. It is important to distinguish between each of these definitions of lifetime. Let us begin with the definition of lifetime for a species at steady state, that is, the ratio of its total abundance (e.g., Tg) to its total rate of removal (e.g., Tg yr 1). Let us apply that definition to CO2 under preindustrial conditions. Carbon dioxide can be presumed to have been in steady state under preindustrial conditions, in which, according to Figure 22.8, the total atmospheric abundance was 612 Pg and the total rate of removal was 176 Pg yr 1. Thus, the overall mean lifetime of a molecule of CO2 in the preindustrial atmosphere was 3.5 years. For the deep-ocean reservoir, the mean carbon residence time was 37,000 Pg/ 112 Pg yr 1 = 330 years. At these steady-state conditions the overall sources and sinks of carbon into and out of any reservoir are equal; for example, the overall rate of removal from the atmosphere of 176 Pg yr 1 was balanced by an equal overall emission rate into the atmosphere. The removal of CO2 from the atmosphere occurs for three major reservoirs: terrestrial biota and warm and cool ocean surface waters; each of these reservoirs exchanges carbon not just with the atmosphere but with additional reservoirs, soils and detritus, and the deep ocean, where the soil and detritus reservoir is also a direct source of carbon to the atmosphere. Under preindustrial steady-state conditions, flows into and out of each of these reservoirs were in balance. Note, however, that the mean carbon lifetime at steady state in each of these reservoirs is different. For terrestrial biota, that lifetime is 11.6 years; for warm ocean surface waters, 5.7 years; for cool ocean surface waters, 1.2 years; and for soils and detritus, 30 years. And, as noted above, the carbon residence time in the deep-ocean reservoir was 330 years. Starting with preindustrial conditions, increasing amounts of CO2 have been emitted year by year to the atmosphere (Figure 22.5), and CO2 was no longer in steady state. The dynamic response of CO2 in the atmospheric compartment is represented in simplified form by (2.14) dQ ˆP dt

Q τ

(22.22)

where Q is the atmospheric CO2 abundance (in Pg C) above the initial preindustrial steady-state value and P is the anthropogenic emission rate (Pg C yr 1). With these definitions of Q and P, the initial condition for (22.22) is Q = 0 at t = 0; because we are starting from a steady state, it suffices to consider the dynamics of the perturbed system. A critical issue is how to interpret the value of τ in (22.22). Equation (22.22) represents only a balance on the amount of atmospheric carbon after the preindustrial steady state. It does not account for how carbon levels in the other reservoirs are changing and how these changes, in turn, affect the dynamics of the atmospheric carbon. In (22.22), then, τ represents essentially a relaxation time for the atmospheric carbon system to reach a new steady state corresponding to P. The mean atmospheric carbon lifetime derived from the preindustrial steady state of Figure 22.8 is not the same as τ in (22.22) because the atmospheric carbon is no longer in steady state once the anthropogenic perturbation occurs. Because Q, dQ/dt, and P are relatively well known, one can actually use these quantities in (22.22) to solve for the relaxation time τ from τˆ

P

Q dQ=dt

(22.23)

Because Q, dQ/dt, and P vary yearly, the relaxation time τ inferred from this equation also varies from year to year. Let us estimate τ, for example, in year 2000 on the basis of (22.23). To determine dQ/dt, consider the period 1995–2000, during which CO2 increased at a rate of 1.8 ppm yr 1 (Keeling and Whorf 2003). This

922

GLOBAL CYCLES: SULFUR AND CARBON

equates to (1.8 ppm yr 1)(2.1 Pg C ppm 1) = 3.78 Pg yr 1. The fossil fuel CO2 emission rate in 2000 has been estimated as 6.6 Pg C yr 1 (Marland et al. 2003) and that from biomass burning as ranging between 1.5 and 2.7 Pg C yr 1 (Jacobson 2004). This produces a CO2 emission range of 8.1–9.3 Pg C yr 1. The global average CO2 mixing ratio in 2000 was 370 ppm. With an assumed preindustrial mixing ratio of 280 ppm, the anthropogenic burden of CO2 in 2000 was (90 ppm) (2.1 Pg C ppm 1) = 189 Pg C. On the basis of the upper and lower estimates of the CO2 emission rate in year 2000, we obtain the following inferred range for the mean atmospheric relaxation time of CO2 in year 2000: τ ∼ 34

44 yr

Similar calculations are presented by Gaffin et al. (1995) and Jacobson (2005). Only about half of the annual CO2 emissions remain in the atmosphere; the remainder partitions primarily into the near-surface ocean layer. The equilibrium ratio is about 0.7: 1.0 for CO2 partitioning between the atmosphere and the near-surface ocean water. At present, the total amount of CO2 in the atmosphere is about 750 Pg C, with approximately 900 Pg C in the near-surface ocean layer. If humans add 700 Pg C by the year 2050, this CO2 will partition by equilibration (with the roughly 35-year relaxation time) between the atmosphere and the surface ocean layer. Neglecting any significant uptake by biomass, the total C in the atmosphere/surface ocean will be 750 + 900 + 700 = 2350 Pg C. At an equilibrium partitioning ratio of 0.7: 1.0, this will lead to ∼950 Pg C in the atmosphere and ∼ 1400 Pg C in the near-surface ocean. The atmospheric CO2 mixing ratio will be ∼ 475 ppm. As CO2 continues to be emitted, the fraction that remains airborne increases owing to the depleted carbon-buffering capacity of the ocean. But the radiative impact of that incremental amount of CO2 decreases because of saturation of the CO2 absorption bands. These two effects largely cancel each other (Caldeira and Kasting 1993), so that the radiative impact of a kg of CO2 is nearly independent of the time that that kg was emitted in the fossil fuel era. Residence Time of Excess CO2 Emitted to the Atmosphere The concept of a single, characteristic atmospheric lifetime does not apply to CO2. Table 22.3 summarizes the principal natural processes that remove CO2 following an emission pulse into the atmosphere, together with the timescales for adjustment, and the main chemical reactions involved. The timescales of atmospheric CO2 adjustment to anthropogenic emissions can be divided into three phases (IPCC 2013): Phase 1. Within several decades, 1/3 –½ of an initial pulse of anthropogenic CO2 is absorbed into the land and ocean. Within a few centuries, most of the pulse of CO2 will be in the form of dissolved inorganic carbon in the ocean. Within 1000 years, the remaining atmospheric fraction is expected to be between 15% and 40%, with the actual fraction depending on the quantity of carbon released. The carbonate buffer capacity of the ocean decreases as CO2 increases, so that the larger the cumulative emissions, the larger the remaining atmospheric fraction.

TABLE 22.3 Processes that Remove CO2 Following a Large Emission Pulse to the Atmosphere Process

Timescale (yr)

Chemistry

Land uptake: photosynthesis respiration Ocean absorption Reaction with CaCO3 on seafloor Silicate weathering in atmosphere

10–100 10–1000 1000–10,000 104–106

6CO2 ‡ 6H2 O ‡ hν ! C6 H12 O6 ‡ 6O2 CO2 ‡ CO23 ‡ H2 O „ 2HCO3 CO2 ‡ CaCO3 ‡ H2 O ! Ca2‡ ‡ 2HCO3 CO2 ‡ CaSiO3 ! CaCO3 ‡ SiO2

Source: IPCC (2013).

923

SOLUTION FOR A STEADY-STATE FOUR-COMPARTMENT MODEL OF THE ATMOSPHERE

Phase 2. Within a few thousand years, the decrease of ocean pH that occurred in phase 1 will be restored by reaction of dissolved CO2 with CaCO3 in seafloor sediments; this leads to further absorption of atmospheric CO2. During phase 2, the atmospheric CO2 fraction is reduced to 10–25% of the original CO2 pulse after about 10,000 years. Phase 3. Within several 100,000 years, the remainder of the CO2 pulse will be removed by silicate weathering through reaction of CO2 with land-based CaSiO3. Ocean Acidification Ocean acidification occurs as seawater absorbs rising atmospheric CO2. It is estimated that to date the oceans have absorbed approximately 155 Pg C as CO2 from the atmosphere (roughly 25% of the total amount of anthropogenic CO2 emitted since preindustrial times). The average pH of ocean surface waters has decreased by ∼0.1 unit from ∼8.2 to ∼8.1 (Zeebe 2012). Absorption of CO2 by water was considered in Section 7.3.2. Dissolved carbon dioxide in the ocean occurs in the form of aqueous carbon dioxide, CO2
H2O, bicarbonate ion, HCO3 , and carbonate ion, CO32 . At typical surface ocean conditions of pH = 8.2, the speciation among CO2
H2O, HCO3 , and CO32 is 0.5%, 89%, and 10.5%, respectively. The sum of dissolved carbonate species is called total dissolved inorganic carbon, [CO2T] and is given by (7.23): ‰CO2T Š ˆ ‰CO2
H2 OŠ ‡ ‰HCO3 Š ‡ ‰CO32 Š As the partial pressure of CO2 in the atmosphere increases, the equilibrium shifts so that the H+ concentration increases, decreasing pH. Also, as the ocean itself warms, the solubility of CO2 in seawater decreases, reducing the amount of CO2 the oceans can absorb from the atmosphere. For a 2 °C temperature increase, for example, seawater will absorb ∼10% less CO2 than its does in the absence of a temperature increase. Table 22.4 gives ocean pH under preindustrial and 2 × CO2 conditions, with and without a 2 °C warming. TABLE 22.4 Ocean pH Parameter pH H+ (mol kg 1) CO2 (aq) (μmol kg 1) HCO3 (μmol kg 1) CO23 (μmol kg 1) Total Carbon (μmol kg 1)

Preindustrial (280 ppm): 20 °C 8.17 6.739 × 10 9.10 1723 228.3 1960.8

9

2 × preindustrial (560 ppm): 20 °C 7.92 1.202 × 10 18.10 1932 143.6 2094.5

8

2 × preindustrial (560 ppm): 22 °C 7.92 1.200 × 10 17.2 1910 152.9 2080.5

8

Source: IPCC (2013).

22.3 SOLUTION FOR A STEADY-STATE FOUR-COMPARTMENT MODEL OF THE ATMOSPHERE Consider Figure 22.10, in which four natural atmospheric reservoirs are depicted, the NH and SH troposphere and the NH and SH stratosphere.3 We use this model of four interconnected atmospheric

3

While the division of the troposphere into two compartments, NH and SH, is reasonable in terms of estimating lifetimes of relatively longlived tropospheric constituents, such a division is less applicable in the stratosphere. In the case of stratospheric transport and mixing, a better compartmental division would be between the tropical and midlatitude stratosphere. With such a division, the stratosphere would actually be represented by three compart­ ments, the tropical stratosphere and the NH and SH midlatitude to polar stratospheres. The development in this section can be extended to such a five-compartment model, if desired.

924

GLOBAL CYCLES: SULFUR AND CARBON

FIGURE 22.10 Four-compartment model of the atmosphere (NH = Northern Hemi­ sphere, SH = Southern Hemisphere, S = stratosphere, T = troposphere). PNH and PSH are source emission rates of the compound in the NH and SH troposphere, respec­ tively. The quantities of the species of interest in the four reservoirs are denoted QTSH , S T QNH , QSH , and QSSH . The fluxes between the reservoirs are proportional to the content of the compound in the reservoir where the flux originates. The flux of material from the NH troposphere to the SH troposphere is kTNH=SH QTNH , and the reverse flux is kTSH=NH QTSH . Other intercompartmental fluxes are defined similarly.

compartments as a vehicle to derive balance equations from which global biogeochemical cycles can be analyzed. We will assume that the substance of interest is removed by different first-order processes in the troposphere and stratosphere, characterized by first-order rate constants kT and kS. Thus the rates of removal of the substance in the two tropospheres are kT QTNH and kT QTSH ; the rates of removal in the two S . kT could represent, for example, OH reaction in the troposphere, and stratospheres are kS QSNH and kS QSH kS could represent photolysis in the stratosphere. kT can also include removal at Earth’s surface. A dynamic balance on the mass of substance in the NH troposphere component is dQTNH T T ˆ kTSH=NH QSH kTNH=SH QNH dt …exchange between NH and SH tropospheres† NH S ‡ kS=T QNH

NH T kT=S QNH

…exchange between NH tropospheres and stratosphere† T kNH T QNH

…removal in troposphere† ‡ PNH …source emission into NH troposphere†

(22.24)

925

SOLUTION FOR A STEADY-STATE FOUR-COMPARTMENT MODEL OF THE ATMOSPHERE

At steady state, the sources and sinks balance: T 0 ˆ kTSH=NH QSH

NH S T kTNH=SH QNH ‡ kS=T QNH

NH T kT=S QNH

kTNH QTNH ‡ PNH

(22.25)

This equation can be rearranged as follows: NH T S QTSH ‡ kNH QTNH ‡ kSH=NH kTNH=SH ‡ kNH T=S ‡ k T S=T QNH ‡ PNH

0ˆ

(22.26)

A similar steady-state balance on the SH troposphere yields SH SH kTSH=NH ‡ kSH QSSH ‡ PSH QTSH ‡ kTNH=SH QTNH ‡ kS=T T=S ‡ k T

0ˆ

(22.27)

Comparable balances on the two stratosphere compartments yield 0ˆ 0ˆ

S NH S T kNH=SH ‡ kNH ‡ kSSH=NH QSSH ‡ kNH QNH T=S QNH S=T ‡ k S

(22.28)

SH T ‡ kSH QSSH ‡ kSNH=SH QSNH ‡ kSH kSSH=NH ‡ kS=T S T=S QSH

(22.29)

T T Equations (22.26)–(22.29) constitute four equations in the four unknowns QNH , QSH , QSNH , and QSSH . It is useful to rewrite these equations as

0 ˆ α1 Q1 ‡ α2 Q2 ‡ α3 Q3 ‡ P1

(22.30)

0 ˆ α4 Q2 ‡ α5 Q1 ‡ α6 Q4 ‡ P2

(22.31)

0 ˆ α7 Q3 ‡ α8 Q4 ‡ α9 Q1

(22.32)

0 ˆ α10 Q4 ‡ α11 Q3 ‡ α12 Q2

(22.33)

S where Q1 ˆ QTNH , Q2 ˆ QTSH , Q3 ˆ QNH , Q4 ˆ QSSH , P1 = PNH, and P2 = PSH. Also NH ‡ kTNH α1 ˆ kTNH=SH ‡ kT=S

T α2 ˆ kSH=NH

α3 ˆ kNH S=T

SH α4 ˆ kTSH=NH ‡ kSH T=S ‡ k T

T α5 ˆ kNH=SH

α6 ˆ kSH S=T

NH α7 ˆ kSNH=SH ‡ kNH S=T ‡ k S

S α8 ˆ kSH=NH

α9 ˆ kNH T=S

S SH α10 ˆ kSH=NH ‡ kS=T ‡ kSH S

α11 ˆ kSNH=SH

SH α12 ˆ kT=S

Equations (22.32) and (22.33) can be solved simultaneously to obtain Q3 and Q4 in terms of Q1 and Q2 Q3 ˆ β1 Q2 ‡ β2 Q1

(22.34)

Q4 ˆ β3 Q2 ‡ β4 Q1

(22.35)

926

GLOBAL CYCLES: SULFUR AND CARBON

where α8 α12 α7 α10 β1 ˆ α8 α11 1 α7 α10 α11 α8 α12 α12 α α α β3 ˆ 10 α78 α10 ‡ 11 α10 1 α7 α10

α9 α7 β2 ˆ α8 α11 1 α7 α10 α9 α11 α7 α10 β4 ˆ α8 α11 1 α7 α10

The resulting equations for Q1 and Q2 are 0 ˆ α3 β2

α1 Q1 ‡ α2 ‡ α3 β1 Q2 ‡ P1

0 ˆ α5 ‡ α6 β4 Q1 ‡ α6 β3

α 4 Q2 ‡ P2

(22.36) (22.37)

the solutions of which are Q1 ˆ Q2 ˆ

α3 β2

α2 ‡ α3 β1 P2 α1 α6 β3 α4

α6 β3 α4 P1 α5 ‡ α6 β4 α2 ‡ α3 β1

(22.38)

α3 β2

α5 ‡ α6 β4 P1 α1 α6 β3 α4

α3 β2 α1 P2 α5 ‡ α6 β4 α2 ‡ α3 β1

(22.39)

Equations (22.38) and (22.39) give the steady-state concentrations of the substance in the NH and SH T ), as a function of the source rates into the two troposphere, respectively (Q1 ˆ QTNH and Q2 ˆ QSH hemispheres and all the transport and removal parameters of the four compartments. Steady-state concentrations in the two stratospheric reservoirs, Q3 ˆ QSNH and Q4 ˆ QSSH , are then obtained from (22.34) and (22.35). These equations provide a general, steady-state analysis of a four-compartment model of a substance that is emitted into the troposphere and removed by separate first-order processes in each of the four compartments. The mass of the substance in the entire atmosphere is the sum of the masses in the four compart­ ments: Qtotal ˆ QTNH ‡ QTSH ‡ QSNH ‡ QSSH

(22.40)

The overall average residence time of a molecule of the substance in the atmosphere can be obtained by dividing its total quantity in the atmosphere by its total rate of introduction from sources: τˆ

Qtotal PNH ‡ PSH

(22.41)

Average residence times in any of the four compartments can be calculated from the quantity in the compartment divided by the net source rate in that compartment. The foregoing analysis can be simplified to fewer than four atmospheric compartments. For a substance that is completely removed in the troposphere, only the two tropospheric hemispheric components need be considered. If the lifetime of such a substance is shorter than the time needed to mix throughout the entire global troposphere, then a two-compartment model, NH troposphere and SH troposphere, would be called for. If the substance’s lifetime is long compared to the interhemispheric mixing time, then the entire troposphere can be considered as a single compartment. Because horizontal mixing in the stratosphere is so much faster than vertical mixing, for many substances that reach the stratosphere, the stratosphere can be considered as a single compartment.

927

PROBLEMS

Application of the Four-Compartment Model to Methyl Chloroform (CH3CCI3) Methyl chloroform is an anthropogenic substance, the total emissions of which to the atmosphere are reasonably well known. Its atmospheric degradation occurs almost entirely by hydroxyl radical reaction. The CH3CCl3 mixing ratio in the atmosphere is well established; thus the global steady-state budget of CH3CCl3 can be used as a means of estimating a global average OH radical concentration. Such estimates are usually accomplished with a three-dimensional atmospheric model (Prinn et al. 1992), but here we apply the simple four-compartment model to analyze the global budget of CH3CCl3. The following emissions data for CH3CCl3 are available (Prinn et al. 1992): PNH ˆ 5:647  1011 g yr PSH ˆ 2:23  10 g yr 10

1

…1978 1990 average†

1

CH3CCl3 is removed by OH reaction CH3 CCl3 ‡ OH ! CH2 CCl3 ‡ H2 O with a rate constant k = 1.6 × 10 12 exp( 1520/T) (see Table B.1). To evaluate average values of the rate constant in the troposphere and stratosphere, we use the tropospheric average temperature of 277 K. An average temperature for the stratosphere is taken as that at 12 km in the US Standard Atmosphere, T = 216.7 K. A global average tropospheric OH concentration is taken as 8.7 × 105 molecules cm 3 (Prinn et al. 1992). An average stratospheric OH mixing ratio in the midlatitude is 1 ppt (see Chapter 5), which translates, at 12 km, into a concentration of 6.48 × 106 molecules cm 3. CH3CCl3 is also degraded by photolysis in the stratosphere, but at a rate almost 4000 times slower than OH reaction, so we will neglect it here. Finally, CH3CCl3 is lost by deposition to Earth’s surface with a first-order loss coefficient of 0.012 yr 1 (Prinn et al. 1992). For the exchange rates among the four atmospheric compartments, we use the following values: T kSH=NH ˆ kTNH=SH ˆ 1:0 yr

1

S kSH=NH ˆ kSNH=SH ˆ 0:25 yr NH SH kS=T ˆ kS=T ˆ 0:4 yr

kNH T=S

ˆ

kSH T=S

…van Velthoven and Kelder 1996†

1

ˆ 0:063 yr

1

1

…van Velthoven and Kelder 1996†

The four-compartment model can be used, with the parameter values and yearly emission rates given above, to estimate the steady-state mixing ratios of CH3CCl3 in the troposphere and stratosphere and the overall atmospheric residence time of CH3CCl3. The results are given below.

Tropospheric mixing ratio (ppt) Stratospheric mixing ratio (ppt) Atmospheric lifetime (yr) a b

Calculated by four-compartment model

Observed mixing ratio

129 75 5.0a

160b

The CH3CCl3 lifetime estimated by IPCC (2001) is 4.8 years based on three-dimensional model calculations. See Figure 2.5 for ∼1990 mixing ratio.

PROBLEMS 22.1A In the simplified calculation of the atmospheric sulfur cycle, if the value of c, the SO2 fraction of the total sulfur, is taken as 0.5, a sulfur atom residence time of 50 h is estimated. What is the value of b, the fraction of sulfur converted to SO24 before being removed, that is consistent with this choice of c?

928

GLOBAL CYCLES: SULFUR AND CARBON

22.2A It is desired to compute the global emissions of carbon that resulted from oil and gas production in 1986. Worldwide oil production in 1986 was 5.518 × 107 barrels day 1, and worldwide gas production in the same year was 6.22 × 1013 ft3 yr 1 (Katz and Lee 1990). For oil, a density of 0.85 g cm 3 can be assumed. Assume that oil has an average composition of C10H22. A barrel has a volume of 0.16 m3. Assume that the gas produced is entirely methane. Calculate global emissions from each in g(C) yr 1. 22.3A The current human population of Earth is estimated to be approximately 7.0 billion. Each person, on average, exhales about 1 kg of CO2 per day. How does this compare with current fossil fuel emissions? This amount is not included in inventories of CO2 emissions. Why? 22.4A A MJ (megajoule) of heat can be generated by combusting ∼19.9 g of methane, assuming a heating value of 50.3 MJ kg 1. a. Assuming complete combustion of CH4, show that 54.6 g of CO2 is released in combusting this quantity of CH4. b. Assuming that 44 g of coal is needed to generate this same MJ of heat (based on a heating value of 22.7 MJ kg 1), and that coal is 85% carbon by mass, show that 137 g of CO2 is released in combusting this coal. 22.5A In 1980 the atmospheric CO2 mixing ratio was ∼340 ppm. By 1988, it had increased to ∼350 ppm. a. How many moles of CO2 were added to the atmosphere during this time period? How many grams of C is this? b. Fossil fuel burning released C at a rate of 5 Gt C yr 1 during this period. What fraction of this carbon was added to the atmosphere? 22.6B Consider the global balance equations for a steady-state substance that is completely generated and removed in the troposphere, but for which the two tropospheric reservoirs, NH and SH, should be considered. Apply the balance to CO using the following source and sink data, derived from Table 2.15 (all in Tg yr 1): Sources:

Sinks:

Methane oxidation Biogenic oxidation Biomass burning Industrial emissions OH reaction Soil uptake

800 380 700 900 400

As a first approximation, assume that industrial sources of CO are totally concentrated in the Northern Hemisphere and that biomass burning sources are totally in the Southern Hemisphere. The CH4 oxidation and biogenic oxidation sources can be apportioned equally between the NH and SH. Soil uptake can be apportioned 2/3 in the NH and 1/3 in the SH. The NH/SH exchange rate of 1 yr 1 can be used. A global mean OH concentration of 8.7 × 105 molecules cm 3 can be assumed, and the CO + OH reaction rate constant is given in Table B.1. The tropospheric average temperature is 277 K. Calculate the CO mixing ratios in the NH and SH. 22.7D a. Use the compartmental model for the global carbon cycle (Table 22.1) to estimate the concentration of CO2 in the atmosphere, assuming that F r …t † ˆ 0 Fd …t† ˆ 0:3 ‡ 0:01 t Ff …t† ˆ

…t ˆ 0 for 1850†

0:014 t

from 1850 to 1950 …t ˆ 0 for 1850†

1:4 ‡ …4:6=40†t

from 1950 to 1990 …t ˆ 0 for 1950†

929

REFERENCES

where the fluxes are in Pg(C) yr 1 and t is in yr. Compare this value to the current measurements. b. Assume that both deforestation and fossil fuel combustion emissions continue to increase at the same rate [0.01 Pg(C) yr 1 for deforestation and 46/40 Pg(C) yr 1] for fossil fuel combus­ tion. When will the concentration of CO2 in the atmosphere double? c. Assume that global emissions of CO2 from fossil fuel combustion and deforestation are stabilized at their 1990 levels. What will be the CO2 concentration in 2050 and 2100? 22.8D One approach to slowing down the increase of the CO2 concentration in the atmosphere and stabilizing it at a lower level is carbon sequestration, that is, the transfer of some of the atmospheric CO2 (or preferably of the CO2 produced by a source such as power plants) to one of the reservoirs in the hydrosphere or lithosphere. Let us investigate the effectiveness of sequestration in the deep-ocean waters as an example using the compartmental model for the global cycle (Table 22.1). Assume that t = 0 corresponds to year 2010, and use the 1990 values of Table 22.2 as the initial conditions for the various reservoirs. a. Calculate the CO2 concentration from 2010 to 2150. assuming that Fr … t † ˆ 0 Fd …t† ˆ 1:8 ‡ 0:01 t …t ˆ 0 for 2010† Ff …t† ˆ 6 ‡ …4:6=40†t …t ˆ 0 for 2010† This is the “business as usual” scenario that assumes that we will continue to increase the CO2 emissions at the present rate. b. Let us assume that starting in 2010, F0 Pg(C)yr 1 are transferred from the atmosphere to the deep ocean. Extend the model to account for this additional flux and repeat the calculations of part (a) for F0 = 1.3 Pg(C)yr 1 (this corresponds to 20% of the current emissions from fossil fuel combustion). c. What should be the value of F0 to avoid reaching a CO2 concentration of 500 ppm by 2150?

REFERENCES Archer, D., et al. (2009), Atmospheric lifetime of fossil fuel carbon dioxide, Annu. Rev. Earth Planet. Sci. 37, 117–134. Barnola, J. M., et al. (1987), Vostok ice core provides 160,000-year record of atmospheric CO2, Nature 329, 408–414. Berresheim, H., et al. (1995), Sulfur in the atmosphere, in Composition, Chemistry, and Climate of the Atmosphere, H. B. Singh (ed.), Van Nostrand Reinhold, New York, pp. 251–307. Caldeira, K., and Kasting, J. K. (1993), Insensitivity of global warming potentials to carbon dioxide emission scenarios, Nature 366, 251–253. Charlson, R. J., et al. (1992), The sulfur cycle, in Global Biogeochemical Cycles, S. S. Butcher et al. (eds.), Academic Press, New York, pp. 285–300. Druffel, E. R. M., et al. (1992), Cycling of dissolved particulate organic matter in the open ocean, J. Geophys. Res. 97, 15639–15659. Falkowski, P., et al. (2001), The global carbon cycle: A test of our knowledge of earth as a system, Science 290, 291–296. Gaffin, S. R., et al. (1995), Comment on “The lifetime of excess atmospheric carbon dioxide,” by Berrien Moore III and B. H. Braswell, Global Biochem. Cycles 9, 167–169. Genthon, C., et al. (1987), Vostok ice core: Climate response to CO2 and orbital forcing changes over the last climatic cycle, Nature 329, 414–418. IPCC (Intergovernmental Panel on Climate Change) (2013), Climate Change 2013: The Physical Science Basis, Cambridge Univ. Press, Cambridge, UK. Jacobson, M. Z. (2004), The short-term cooling but long-term global warming due to biomass burning, J. Climate 17, 2909–2926.

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GLOBAL CYCLES: SULFUR AND CARBON Jacobson, M. Z. (2005), Correction to “Control of fossil-fuel particulate black carbon and organic matter, possibly the most effective method of slowing global warming,” J. Geophys. Res. 110, D14105 (doi: 10.1029/2005JD005888). Joos, F., et al. (2013), Carbon dioxide and climate impulse response functions for the computation of greenhouse gas metrics: A multi-model analysis, Atmos. Chem. Phys. 13, 2793–2825. Katz, D. L., and Lee, R. L. (1990), Natural Gas Engineering, McGraw-Hill, New York. Keeling, C. D. (1983), The global carbon cycle: What we know and could know from atmospheric, biospheric, and oceanic observations, Proc. CO2 Research Conf.: Carbon Dioxide, Science and Consensus, DOE Conf-820970, II.3-II.62, US Dept. Energy, Washington, DC. Keeling, C. D., and Whorf, T. P. (2003), Atmospheric CO2 concentrations (ppmv) derived from in situ air samples collected at Mauna Loa Observatory, Hawaii (http://cdiac.esd.ornl.gov/ftp/maunaloaco2/maunaloa.co2). Lenton, T. M. (2000), Land and ocean carbon cycle feedback effects on global warming in a simple earth system model, Tellus B 52, 1159–1188. Marland, G., et al. (2003), Global CO2 emissions from fossil-fuel burning, cement manufacture, and gas flaring: 1751–2000, in Trends Online: A Compendium of Data on Global Change, Carbon Dioxide Information Analysis Center, Oak Ridge National Laboratory, US Dept. Energy, Oak Ridge, TN. Prinn, R., et al. (1992), Global average concentration and trend for hydroxyl radicals deduced from ALE/GAGE trichloroethane (methyl chloroform) data from 1978–1990, J. Geophys. Res. 97, 2445–2461. Rodhe, H. (1978), Budgets and turn-over times of atmospheric sulfur compounds, Atmos. Environ. 12, 671–680. Schimel, D. S. (1995), Terrestrial ecosystems and the carbon cycle, Global Change Biol. 1, 77–91. Schmitz, R. A. (2002), The Earth’s carbon cycle, Chem. Eng. Educ. 296. Takahashi, T., et al. (1997), Global air-sea flux of CO2: An estimate based on measurements of sea-air pCO2 difference, Proc. Natl. Acad. Sci. USA 94, 8292–8299. van Velthoven, P. F. J., and Kelder, H. (1996), Estimates of stratosphere–troposphere exchange: Sensitivity to model formulation and horizontal resolution, J. Geophys. Res. 101, 1429–1434. Ver, L. M. B., et al. (1999), Biogeochemical responses of the carbon cycle to natural and human perturbations: Past, present, and future, Am. J. Sci. 299, 762–801. Zeebe, R. E. (2012), History of seawater carbonate chemistry, atmospheric CO2, and ocean acidification, Annu. Rev. Earth Planet. Sci. 40, 141–165.

CHAPTER

23

Global Climate

Earth’s climate is the result of a balance between incident solar (shortwave) radiation absorbed and thermal infrared (longwave) radiation emitted. Averaged over the globe and over a sufficiently long period of time, this balance must be zero. Because virtually all the energy for the climate system comes from the sun, globally the amount of incoming solar radiation on average must be equal to the sum of the outgoing reflected solar radiation and the outgoing infrared longwave radiation emitted by the planet. A perturbation of this global radiation balance, be it anthropogenic or natural, is called radiative forcing. Climate refers to the mean behavior of the weather over some appropriate averaging time. The actual averaging time when defining climate is not immediately obvious; times too short are insufficient to average out normal year-to-year fluctuations; times too long may incur on timescales of climate change itself. Timescales of relevance for climate change can range from decades to millennia. A traditional averaging time for defining climate is 30 years. The most fundamental climate variable is the global annual mean surface temperature, although other variables can also be considered.

23.1 EARTH’S ENERGY BALANCE Annually averaged, Earth receives 341 W m 2 of solar radiation at the top of the atmosphere (Trenberth et al. 2009) (Figure 23.1). Of this amount, ∼102 W m 2 is reflected back to space by Earth’s surface and by clouds and particles in the atmosphere. As a result, the net radiant energy absorbed by Earth is 239 W m 2. At thermal equilibrium, 239 W m 2 of energy must be radiated back to space from Earth, establishing the equilibrium temperature of the planet. The equilibrium climate is, more precisely, a quasiequilibrium climate, owing to the fact that changes do occur on millennial and longer timescales, such as glacial–interglacial cycles caused by orbital variations. Global average surface temperature is about 288 K (15 °C), for which the corresponding blackbody irradiance is 396 W m 2. Much of this infrared energy emitted at Earth’s surface is absorbed by molecules in the atmosphere, such as carbon dioxide (CO2), water vapor, methane (CH4), nitrous oxide (N2O), chlorofluorocarbons, and ozone (O3) and reemitted in both upward and downward directions, further heating Earth’s surface and main­ taining the atmospheric temperature gradient. Nitrogen and oxygen, which account for about 99% of the volume of the atmosphere, are essentially transparent to infrared radiation. At equilibrium, the emitted longwave flux at the top of the atmosphere, 239 W m 2, is considerably less than that emitted at the surface, 396 W m 2. Without this natural greenhouse effect, owing primarily to water vapor and carbon dioxide, Earth’s mean surface temperature would be a freezing 18 °C, instead of the habitable 15 °C. Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, Third Edition. John H. Seinfeld and Spyros N. Pandis.  2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

931

932

GLOBAL CLIMATE

FIGURE 23.1 Earth’s energy balance (Trenberth et al. 2009). Incoming and outgoing energy fluxes from Earth on an annual-average basis. The greenhouse effect refers to the absorption and reradiation of energy by atmospheric gases, resulting in a downward flux of infrared radiation from the atmosphere to the surface. At equilibrium, the total rate at which energy leaves Earth (102 W m 2 of reflected sunlight plus 239 W m 2 of infrared radiation) is equal to 341 W m 2 of incident sunlight.

Despite their small amounts, the greenhouse gases profoundly affect Earth’s temperature. As compared with the average annual energy input to the earth from the sun, the amount of energy released annually as a result of humanity’s energy production is about 0.025 W m 2 (Nakicenovic et al. 1998). Internal terrestrial energy generation is ∼0.087 W m 2 (Pollack et al. 1993). Human energy production, therefore, is totally negligible as compared with energy flows associated with solar and terrestrial radiation. The troposphere is characterized by constant vertical motion; in the process of rising, the air cools by expansion, and the temperature decreases with height. Solar energy is absorbed primarily at Earth’s surface and transferred to the atmosphere by surface heat fluxes. Convection and other heat transport mechanisms connect all levels in the troposphere, and, to a good approximation, the troposphere can be considered to warm and cool as a unit. When a greenhouse gas is present, the upwelling infrared radiation from the surface is absorbed and reemitted. As a result, the infrared radiation that actually escapes to space comes from the higher, colder parts of the atmosphere. Since the emission rate of radiation from a blackbody varies with the fourth power of temperature, the flux of radiation from these upper levels is considerably less than that emitted from the surface. By contrast, the downwelling radiation to the surface comes predominantly from the warmer layers nearest the surface. Earth’s clouds, whether liquid or frozen water, act essentially as blackbodies. They emit at the cloudtop temperature, which is cold if the cloud tops are in the mid–upper troposphere. Infrared longwave radiation emitted by Earth has a maximum near 10 μm wavelength. Oxygen and ozone absorb radiation strongly in the ultraviolet region, but their absorption is essentially zero in the visible and infrared regions. Methane absorbs strongly in two narrow regions around 3.5 and 8 μm wavelength, which are in the infrared portion of the spectrum. Nitrous oxide has absorption peaks at about 5 and 8 μm. The CO2 molecule has four main groups of absorption features in the thermal infrared, of which the most important is that at wavelength near 15 μm. As a polar molecule, water has a richer set of vibrational and rotational modes that allows it to absorb infrared photons over a much broader range of wavelengths than CO2. Water vapor is so abundant in the atmosphere that in those regions of the

RADIATIVE FORCING

spectrum where H2O vapor absorbs infrared radiation, the spectrum is saturated. CO2 and CH4 have absorption in some of the “windows” in the infrared spectrum where H2O vapor does not absorb and where terrestrial radiation escapes. The CO2 greenhouse effect is directly visible in satellite observations as the bite taken out of the infrared spectrum near 15 μm wavelength, a feature the details of which agree precisely with first-principles radiative transfer calculations (Pierrehumbert 2011). The radiative forcing change resulting from a change in CO2 abundance is approximately propor­ tional to the logarithm of change in the CO2 mixing ratio. If the CO2 level increases by a factor β, the radiative forcing is given by ΔF = 5.3 ln β. For a relatively small change in CO2 abundance, since ln(1 + ε)  ε, the forcing change due to a change in CO2 is approximately proportional to the fractional change in the CO2 mixing ratio. For a doubling of CO2 from its pre-industrial value, ΔF = 5.3 × ln 2 = 3.67 W m 2. The concept that increasing amounts of CO2 can cause the temperature of the globe to increase was first enunciated by Svante Arrhenius in 1896 (Arrhenius 1896). After Arrhenius’ pioneering paper almost half a century passed before concern was initially expressed about possible global warming. In 1938, British meteorologist Guy Callendar asserted that atmospheric CO2 levels had increased by 10% since the 1890s, but his paper received little attention (Weart 1997). In fact, it was believed that the oceans would absorb the vast majority of any CO2 that was put into the atmosphere. Moreover, it was thought that CO2 infrared absorption lines lie right on top of those for water vapor and thus addition of CO2 would not affect radiation that was already being absorbed by H2O. In 1955, G. Plass accurately calculated the infrared absorption spectrum of CO2 and showed that the CO2 lines do not lie right on top of water vapor absorption lines as had previously been assumed; this result meant that adding CO2 to the atmosphere would lead to interception of more infrared radiation. Then in 1956, Roger Revelle showed that, because of the chemistry of seawater, the ocean surface layer does not have the virtually unlimited capacity for CO2 absorption as previously believed. Finally, in late 1957, Charles Keeling began continuous atmospheric measurements of CO2 on the top of the Mauna Loa volcano in Hawaii; by 1960, he was able to detect a rise after only 2 years of measurements. These data, which became the standard for atmospheric CO2, have chronicled the increase in CO2 ever since (see Figure 22.3).

23.2 RADIATIVE FORCING Earth’s radiation balance can be perturbed in a number of ways, for example: (1) a change in solar output of energy (either positive or negative) alters the solar radiation flux, So; (2) a large volcano injects particles directly into the stratosphere these particles reflect a portion of incoming solar radiation back to space and thereby prevent that radiation from reaching Earth’s surface, increasing Earth’s reflectivity or albedo A; (3) an increase in the concentration of infrared absorbing gases in the atmosphere leads to increased absorption of the Earth’s longwave radiation, which reduces the outgoing infrared energy flux at the top of the atmosphere–in this case, the incoming solar energy flux then exceeds the outgoing energy flux, and Earth warms as a consequence. Radiative forcing (RF) is the net change in Earth’s energy balance due to an imposed perturbation, measured in units of W m 2. RF is a quantitative basis for comparing the potential climate impact of different imposed agents and is generally calculated as the value due to changes between preindustrial to present day. Instantaneous RF refers to the instantaneous change in net (down minus up) radiative flux (shortwave plus longwave, in W m 2) due to an imposed perturbation. The change in flux can be that at the top of the atmosphere (TOA) or at the climatological tropopause. In determining instantaneous RF, any climate changes that occur as the Earth system responds to counteract the change in radiative flux are excluded; that is, implicit in the concept of radiative forcing is that the change in net radiative flux can be separated from all subsequent responses to the forcing. Traditionally, the stratospherically adjusted RF has been used, as the change in net irradiance at the tropopause allowing for stratospheric temperatures to readjust to radiative equilibrium, while holding surface and tropospheric temperatures at their unperturbed values.

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For a number of forcing agents, the change in net radiative flux induces relatively rapid adjustments in the troposphere that act to enhance or reduce the perturbation in radiative flux (Sherwood et al. 2015). These rapid changes can be considered a part of the RF itself (in much the same way as the stratospheric adjustment is incorporated in the calculation of the RF). The effect of changes in aerosol levels on clouds is a particularly relevant example. As will be addressed in more detail in Chapter 24, changes in aerosol levels affect cloud properties that occur on the microphysical scale of the cloud droplets as well as on a larger scale involving cloud systems. A number of these cloud changes take place on the relatively short time scales of the cloud system itself. Such changes can be termed “rapid adjustments”; these changes are not a result of climate feedbacks from surface temperature change. IPCC (2013) terms a forcing that accounts for rapid adjustments the effective radiative forcing (ERF). The ERF represents the change in net TOA downward radiative flux after allowing for atmospheric temperatures, water vapor, and clouds to adjust, but with global mean surface temperature unchanged. The primary means to calculate ERF is to fix sea surface temperature (SST) and sea ice cover at climatological values while allowing other parts of the climate system to reach a new steady state. There is not a definite timescale dividing line that defines these rapid adjustments with the fixed SST. The majority takes place on timescales of seasons or less. Changes in land ice and snow cover may take place over many years. Those forcings that are considered as part of the ERF contribute to the ultimate steady-state climate response. In actuality, the differences between ERF and RF are often very small.

23.2.1 Climate Sensitivity The change in climate that results from a perturbation to Earth’s energy balance is manifested in a number of ways, including temperature, precipitation, and snow and ice cover. The index that is used most commonly as a measure of climate change is the annual and global mean surface air temperature. In assessing the extent to which climate has changed, one usually defines a temperature anomaly, the difference in temperature at a given site relative to a climatological mean temperature at that site. The global mean temperature change that results in response to an imposed and sustained perturbation on Earth’s energy balance after a time sufficiently long for both the atmosphere and the oceans to come to thermal equilibrium is termed Earth’s equilibrium climate sensitivity. Climate sensitivity has units of temperature change per W m 2 (°C W 1 m2 or K W 1 m2) and is given the symbol λ. Two assumptions inherent in this definition of climate sensitivity are that a change in global-mean surface temperature is directly proportional to the imposed forcing and that different forcings are additive in contributing to a total forcing. Both of these assumptions are supported by climate models. There are two approaches to determining climate sensitivity. The empirical approach is based on an observed change in global mean temperature over a given historical time period with respect to an estimated or known forcing. The second way of determining climate sensitivity is based on simulations using climate models. A standard benchmark that is used to assess climate change is a doubling of CO2 from its preindustrial level of about 280 to 560 ppm. (This scenario is referred to in shorthand notation as 2 × CO2.) The perturbation to Earth’s energy balance that would result from 2 × CO2 is ∼3.7 W m 2. The increase in the global mean temperature required to reequilibrate Earth’s energy balance to this change, considering solely Earth’s blackbody Stefan–Boltzmann response (see Section 23.3), is 1.2 °C. A temperature increase of 1.2 °C in response to a forcing of 3.7 W m 2 implies a climate sensitivity of 0.32 °C W 1 m2. IPCC (2013) stated that the climate response to 2 × CO2 is likely in the range 1.5–4.5 °C, without giving a best estimate. For a mean estimate of 3 °C, the corresponding climate sensitivity is 3 °C/3.7 W m2 = 0.81 °C W 1 m2. The estimated global mean temperature increase of 3.0 °C from a doubling of CO2 represents an amplification by a factor of about 2.5 over the pure blackbody temperature change of 1.2 °C. The climate sensitivity is defined as the equilibrium change in global mean surface temperature following a doubling of atmospheric CO2. Implicit in this definition is the time required for Earth to come to equilibrium. For example, continental ice sheets respond slowly to changes in radiative forcing and their feedback on temperature may not be accounted for in climate sensitivity derived from climate model simulations. Climate sensitivities derived from paleoclimate data implicitly include such long timescale responses. A climate sensitivity of 3.0 °C for 2 × CO2 is based on relatively fast feedback processes, such as changes in water vapor, clouds, snow, and sea ice. In contrast, for some warm periods

RADIATIVE FORCING

in Earth’s history, such as the Pliocene epoch (from ∼5.3 to 2.6 million years ago), climate sensitivity has been estimated at 7–10 °C per doubling of CO2. This estimate of climate sensitivity includes feedbacks on all timescales, such as changes in vegetation, dust, aerosols, ice sheets, ocean circulation, marine productivity, and weathering (Zeebe 2011). The Pliocene was a long-term steady state of the Earth system that was established over millions of years. By contrast, the present input of CO2 represents a massive, rapid perturbation of CO2 over a timescale of a couple of centuries. The closest analog to present day is the Palaeocene–Eocene Thermal Maximum (PETM), 55 million years ago, in which a large mass of carbon was released over a relatively short period, and surface temperatures rose by 6 °C in a few thousand years (Zeebe 2011; Zeebe and Zachos 2013).

23.2.2 Climate Feedbacks The explanation of why the estimated temperature change to 2 × CO2 is so much larger than that based purely on the increase of absorbed radiation lies in climate feedbacks. Once Earth starts to warm, in response, say, to an increase in greenhouse gas levels, other changes take place that act to amplify the initial warming. A warmer atmosphere holds more water vapor, which itself is a powerful infrared absorber. Also, as temperature increases, sea ice begins to melt. Because the darker ocean absorbs more sunlight than the sea ice it replaced, this leads to further warming. To illustrate the concept of a positive climate feedback, imagine a climate forcing that causes a unit of warming. Say that a strong positive feedback exists, one that amplifies the initial warming by 50%. The added warming of 0.5 induces more feedback, by 0.5 × 0.5 = 0.25, and so on. The ultimate response to the unit of warming is 1 ‡ 0:5 ‡ 0:25 ‡ 0:125 ‡ ∙ ∙ ∙ ˆ 2. Thus, this strong feedback of 0.5 ultimately leads to a warming twice as large as would have occurred in the absence of the feedback. The water vapor feedback is almost as strong as this example, but it occurs together with other feedbacks, some positive (amplifying) and some negative (retarding). The net effect of these feedbacks is to amplify the global temperature response to greenhouse gas warming by about a factor of 2–3. Of the overall greenhouse effect on Earth, including the effect of climate feedbacks, water vapor accounts for ∼50%, clouds ∼25%, CO2 itself ∼20%, and the minor GHGs ∼5%. Despite its only fractional contribution to the greenhouse effect, CO2 is, in fact, the controller of climate. If CO2 were removed from the atmosphere, the atmosphere would cool sufficiently that much of the water vapor would precipitate out. That loss of water vapor would lead to further cooling, with still more water vapor removal, ultimately spiraling the planet into a globally glaciated state. It is the presence of CO2 that maintains a level of warmth to keep the atmospheric level of water vapor that sustains our climate. Of course, increasing CO2 and warming the atmosphere produces the opposite effect, additional warming, through positive water vapor feedback. In summary, the 25% contribution due to the noncondensing, longlived GHGs supports and sustains the entire greenhouse effect, with the remaining 75% a result of the fast feedbacks involving water vapor and clouds (Lacis et al. 2010). Subsequently, we will discuss climate feedbacks in more detail, but it is important to stress that the presence of the well-mixed GHGs in the atmosphere (especially CO2) provides the basis for our present climate, upon which the fast feedback processes operate. As we noted above when discussing climate sensitivity, on timescales shorter than centuries, only fast feedback processes, such as water vapor, lapse rate, clouds, and snow/sea ice are usually considered. On millennial and longer timescales, feedback processes involving ice sheets, vegetation, ocean circulation, etc. come into play (Zeebe 2013). These long-term feedbacks prolong the warming period and ultimately lead to a larger climate sensitivity.

23.2.3 Timescales of Climate Change If Earth’s radiation balance is perturbed, the global surface temperature will adjust so as to reestablish a balance between incoming and outgoing radiation. The climate response time is that needed for the climate system or its components to reequilibrate to a new state, following a forcing resulting from external and internal processes or feedbacks. The response time of the troposphere is relatively short, from days to weeks, whereas the stratosphere reaches equilibrium on a timescale of typically a few months. Most of the heat capacity of Earth lies in the ocean. The climate responds to increases in atmospheric CO2 by

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warming, but this warming is not realized instantaneously owing to the timescale associated with heat storage in the ocean. This is often referred to as the “inertia” of the climate system. In the usual scenario that assesses this inertia, the atmospheric CO2 concentration is held fixed at, say, its current level. The time scale associated with heat equilibration between the atmosphere and upper layers of the ocean leads to delay in the climate system reaching its ultimate steady state temperature corresponding to the fixed atmospheric level of CO2. The upper 100 m or so of the ocean is efficiently mixed by wind stress and convection. The thermal inertia of this ocean mixed layer, by itself, would produce a surface temperature response time of about a decade. Climate models show that, in response to an instantaneous doubling of CO2, about 40% of the equilibrium response is attained in about a decade, followed by a slow warming on century time scales, owing to exchange between the mixed layer and the deeper ocean. Climate feedbacks, which drive the ultimate temperature change, operate on an initial temperature change, not on the forcing. Thus, the climate response time depends on the climate sensitivity itself, as well as on the rate at which heat is transported into the deeper ocean. The rate of ocean heat uptake determines the planetary energy imbalance, which is the portion of the net climate forcing to which the planet has not yet responded. The Earth’s energy imbalance is currently 0.6–0.7 W m 2. This imbalance is a fundamental characterization of the state of the climate. It determines the amount of additional temperature change “in the pipeline.” In the terminology of climate change, the transient temperature lags behind the equilibrium temperature. Under equilibrium climate conditions, the solar heat flux to the upper layer of the ocean is balanced by a heat flux from the ocean to the atmosphere, and that balance establishes the temperature of the ocean surface layer. Heat transfer between the ocean and atmosphere is controlled by a thin layer at the ocean surface through which heat is transferred by conduction. The temperature gradient across this thin conduction layer determines the heat flux. If the GHG concentration in the atmosphere were to suddenly increase, the increased absorption of infrared radiation by GHGs leads to a downward heat flux that slightly increases the surface temperature, decreasing the flux of heat through the layer to the atmosphere. As a result, more of the energy acquired by the bulk of the ocean surface layer from absorption of solar radiation remains in the ocean, leading to an increased temperature of the upper layer of the ocean. The characteristic time for the near-surface ocean temperature to adjust in response to climate forcing is about a decade. The estimated increase in global energy from 1971 to 2010 was 274 × 1021 J, with a heating rate of 213 × 1012 W over that time period (IPCC 2013). Ocean warming dominates the total heating rate, with warming of the full ocean depth accounting for ∼93% and warming of the upper 700 m accounting for ∼64%. The upper 75 m has warmed by 0.11 °C per decade over 1971–2010. The remainder of the energy increase is: ∼3% continents, ∼3% melting ice, 1% atmosphere. The increase in upper ocean heat content during this time period was 17 × 1022 J. If CO2 emissions decrease sufficiently, the CO2 level in the atmosphere can also decrease over time due to the continued slow uptake of CO2 by the ocean, that is, inertia in the carbon cycle itself. This carbon cycle inertia affects global temperature in the opposite direction from the climate inertia that is based on heat transport (Matthews and Solomon 2013). As a result of these two opposing effects, if CO2 emissions were to cease abruptly, global average temperature would remain roughly constant for several centuries, as the effects of thermal inertia and carbon cycle inertia roughly balance each other.

23.3 THE GREENHOUSE EFFECT Total solar irradiance (TSI) or the solar constant, So, is the amount of solar radiance received outside the Earth’s atmosphere on a surface normal to the incident radiation and at Earth’s mean distance from the sun. The most reliable measurements of solar radiation are made from space, and the precise satellite record extends back only to 1978. The generally accepted value of TSI is 1368 W m 2, with an accuracy of about 0.2%. Since only one-half of Earth is illuminated at any time, the incoming solar energy flux is the product of the solar constant So, the projected area of the sphere πR2, and (1–A), where A is the shortwave albedo of Earth (the fraction of incoming radiation that is immediately reflected back to space). The outgoing

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FIGURE 23.2 Radiative model of a one-layer atmosphere that is transparent to incoming solar radiation but completely absorbing of longwave radiation.

longwave radiation flux is the product of the entire area of the planet, 4πR2, and the Stefan-Boltzmann flux, σT4 (assuming, for the moment, a longwave emissivity ε of unity), where σ is the Stefan–Boltzmann constant (5.67 × 10 8 W m 2 K 4) and Earth’s radius is 6371 km. For Earth, A  0.3 (Bender et al. 2006).1 At equilibrium, So …πR2 †…1 A† ˆ …4πR2 †σT 4e , so the global average effective temperature of emission is Te ˆ

So …1 A† 4σ

1=4

(23.1)

With So = 1368 W m 2 and A = 0.3, equation (23.1) predicts Te = 255 K. If Earth were totally devoid of clouds, the global albedo would be about A = 0.15, for which Te = 268 K. The actual global average surface temperature of Earth is ∼288 K. Since the longwave radiation emitted by Earth originates not just from Earth’s surface but also from the atmosphere itself, which is substantially colder than the surface, Te = 255 K is a measure of the overall equilibrium temperature of the Earth–atmosphere system. This simple energy balance predicts that Te varies ∼0.5 K for a 10 W m 2 (0.7%) variation in the solar constant, or for a reflectance variation around A = 0.3 of ΔA = 0.005. We now show how the greenhouse effect establishes Earth’s temperature. Consider a one-layer atmosphere that is transparent to incoming shortwave (SW) radiation but completely absorbing of longwave (LW) radiation (Figure 23.2). We also assume a LW emissivity ε = 1. At radiative equilibrium at Earth’s surface: SW # ‡ LW #ˆ LW " …So =4†…1

A† ‡ σT a4 ˆ σT 4s

At the top of the atmosphere SW # ˆ …So =4†…1 LW " ˆ σT4a

A†

so at radiative equilibrium SW # ˆ LW " σT 4a 1

ˆ …So =4†…1

A†

In a comprehensive study of satellite data, Kim and Ramanathan (2008) estimated the planetary albedo for a 3-year average over 2000–2002 to be 28.9 ± 1.2%, as compared with CERES and ERBE satellite estimates of 28.6% and 29.6%, respectively. (Without clouds, but including aerosol, the Earth’s albedo is only 15.0 ± 0.6%). For purposes of estimating Earth’s emission temperature, a value of the albedo of 0.3 is commonly used.

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Summarizing, we have σT 4s ˆ …So =4†…1

A† ‡ σT4a

σT 4a

A†

ˆ …So =4†…1

so the equilibrium temperature of the totally absorbing atmospheric layer is Ta ˆ

So …1 A† 4σ

1=4

(23.2)

Note that Ts = (2)1/4 Ta and thus Ts > Ta. In this simple example, Ta is the effective temperature of the Earth system, the same as the result of the overall planetary balance in (23.1). We can extend this idea to the more realistic case of a one-layer atmosphere that is transparent to SW radiation and partially absorbing of LW radiation (Figure 23.3). Let ε be the fraction of LW absorption by the atmosphere, 0 < ε < 1. We continue to assume a LW emissivity of unity, for convenience. At the surface SW # ˆ …1 A†…So =4† LW # ˆ σT4a LW " ˆ σT4s so σT 4s ˆ …So =4†…1

A† ‡ σT4a

At the top of the atmosphere SW # ˆ …So =4†…1

A†

LW " ˆ σT 4a ‡ …1

ε†σT 4s

so σT4a ‡ …1

ε†σTs4 ˆ …So =4†…1

A†

FIGURE 23.3 Radiative model of a one-layer atmosphere that is transparent to incoming solar radiation but partially absorbing of longwave radiation.

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so, together, Ts = (2/ε)1/4 Ta. For example, if ε = 0.8 (80% of the LW radiation emitted by the Earth’s surface is absorbed by the atmosphere): Ta ˆ …0:8=2†1=4 T s ˆ 0:795Ts Note that as ε increases, the atmosphere becomes warmer. In this example, the outgoing LW flux at the top of the atmosphere has two components, one from the layer at Ta and one from the surface at Ts. The planetary emission temperature Te can be obtained by collecting all the outgoing LW radiation at the top of the atmosphere and defining Te based on that flux: σTe4 ˆ σT4a ‡ …1

ε†σT4s

Using Ts = (2/ε)1/4 Ta, we get Ta ˆ Ts ˆ

ε

1=4

ε

2

(23.3)

Te

(23.4)

1=4

2 2

Te

ε

so Ta  Te  Ts, that is, the planetary emission temperature represents an average of radiation from the warm surface and the cold atmosphere. For example, for A = 0.3, ε = 0.78, Te = 255 K, Ta = 230 K and Ts = 288.5 K. At ε = 0.78, 22% of the longwave radiation from the Earth escapes into space. For a “perfect” greenhouse, ε = 1, Ts = 303 K. The planetary emission temperature depends only on the solar constant and the planetary albedo. The presence of greenhouse gases affects the surface temperature and atmospheric temperature profile but does not affect Te or LW ↑ at the top of the atmosphere. If the CO2 concentration in the atmosphere is doubled, the Earth’s surface will become warmer but Te will be unchanged. (Note that this simple analysis assumes that the planetary albedo A remains constant, which may not be the case, such as if global cloudcover changes.) Two Absorbing Atmospheric Layers To represent an atmospheric temperature profile, we can extend the foregoing analysis to two absorbing layers, as shown in Figure 23.4. For simplicity, assume that each layer has 100% LW absorptivity. The radiative balances are Top of the atmosphere : σT41 ˆ …So =4†…1

A†

Surface :

σT4s ˆ σT 42 ‡ …So =4†…1

Layer 2 :

σT41

‡

σT4s

ˆ

A†

2σT42

Combining these three relations, we find T s ˆ …3†1=4 T 1 T 2 ˆ …2†1=4 T 1 and Ts > T2 > T1 This simple model produces an atmospheric temperature profile that decreases with altitude. Because both layers are assumed to be totally absorbing, the planetary emission temperature is T1.

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FIGURE 23.4 Radiative model of a two-layer atmosphere that is transparent to incoming solar radiation but completely absorb­ ing of longwave radiation.

To be more realistic, we now consider the two-layer model in which each of the layers is partially absorbing (ε < 1) (Figure 23.5). While the accounting for upward and downward radiative fluxes is somewhat more involved, the procedure follows from the cases we have already considered. The radiative balance at Earth’s surface is σT4s ˆ …So =4†…1

A† ‡ σT 42 ‡ …1

ε†σT 41

(23.5)

At the top of the atmosphere, the planetary emission temperature Te is defined on the basis of the total upward flux of radiation: σT 4e ˆ σT 41 ‡ …1

ε†σT 42 ‡ …1

ε†2 T s4

FIGURE 23.5 Radiative model of a two-layer atmosphere that is transparent to incoming solar radiation but partially absorbing of longwave radiation.

(23.6)

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The radiative balance on layer 2 is εσT4s ‡ εσT 41 ˆ 2σT42

(23.7)

and finally the balance of net incoming and outgoing radiation is σT 4e ˆ …So =4†…1

A†

(23.8)

As before, Te is specified by the overall radiative balance on the planet. T1, T2, and Ts are determined by simultaneous solution of (23.5)–(23.8). The three equations governing T1, T2, and Ts become T 41 ‡ …1 …1 εT 41

ε†T 42 ‡ …1

ε†2 T s4 ˆ T e4

ε†T 41 T42 ‡ T 4s ˆ T 4e 2T 42 ‡ εT4s ˆ 0

(23.9)

This 3 × 3 set of linear equations can be solved using Cramer’s rule. With a single atmospheric absorbing layer, we can assign the absorptivity as that approximating the actual atmosphere, ε = 0.8. With two layers, we cannot assign an absorptivity of 0.8 to each layer, since that will produce far too much absorption as the upward longwave radiation traverses sequentially each layer. Since this is still an idealized model, we can ask: What value of ε applied to each layer produces a reasonable value of Ts? If we choose ε = 0.5, we obtain T1 ˆ 194 K T2 ˆ 214 K Ts ˆ 291 K The Ts = 291 K is remarkably close to the actual value of 288 K. Figure 23.6 illustrates the effect of an increase in GHG abundance on the atmospheric temperature profile. The mean atmospheric temperature profile is shown starting from the current mean surface temperature of 288 K. If we assume a global average rate of decrease of temperature with height of 5.5 K km 1, the temperature of 255 K, the blackbody temperature corresponding to the mean emitted power of the Earth–atmosphere system, is reached at about 6 km altitude. A doubling of CO2 from the preindustrial level diminishes the mean emitted power of the Earth–atmosphere system. The result is that the mean altitude from which thermal energy escapes to space rises by about 200 m. Because it is colder 200 m higher, the energy emitted to space is temporarily reduced and Earth radiates less energy than it absorbs. Considering only the blackbody response, the surface temperature must rise by 1.2 K to restore the energy balance, if the vertical temperature gradient and other factors are held fixed. Earth’s Albedo The average value of Earth’s albedo is ∼0.3. The albedo ranges of individual types of surfaces is Liquid water Fresh snow Thick clouds Ice Soil (dry) Soil (wet) Desert Forest

0.05–0.1 0.7–0.9 0.4–0.9 0.3–0.4 0.2–0.3 ∼0.1 0.2–0.4 0.1–0.15

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FIGURE 23.6 Greenhouse effect. Doubling CO2 increases infrared absorption, raising by ∼200 m the mean level from which thermal energy escapes to space. Because it is colder, the higher up one goes in the troposphere, the energy emitted to space is temporarily reduced and Earth radiates less energy than it absorbs. The surface temperature must rise by 1.2 K to restore the energy balance, if the temperature gradient and other factors are held constant.

23.4 CLIMATE-FORCING AGENTS 23.4.1 Solar Irradiance The amount of solar radiation reaching Earth and Earth’s changing orientation to the sun are the major causes for climate change throughout its history. Long-term variations in the orbital parameters of Earth have profound effects on climate. Earth’s orbit is elliptical with the sun at one focus of the ellipse. As a result, the amount of sunlight reaching Earth varies throughout the year. The eccentricity of Earth’s orbit is 1.7%. Earth is closest to the sun in January (the perihelion), when the Earth–Sun distance is 147.5 million km, versus July (aphelion), when the distance is 152.6 million km. The eccentricity of Earth’s orbit changes with time, as does the time of year when the Earth is farthest from the sun, while the tilt of Earth’s axis to the plane of the orbit (the obliquity) wobbles back and forth periodically. The eccentricity of Earth’s orbit changes over 100,000- and 413,000-year cycles. The precession of Earth’s axis varies over a 19,000–24,000-year cycle, and the tilt of Earth’s axis changes from 22.0° to 24.6° in a 41,000­ year cycle. The result of the combination of these three orbital parameters is that the amount of solar energy received at different latitudes at different times of the year changes appreciably. During the past million years, Earth has experienced 10 major and 40 minor episodes of glaciation, all of which appear to have been controlled by these orbital elements. The climate cycles caused by these orbital factors are called Milankovitch cycles after the Serbian mathematician Milutin Milankovitch, who first described them in 1920. Orbital changes initiate the vast climate shifts associated with glaciation and deglaciation. Feedbacks, such as changes in Earth’s reflectivity, amount of particles in the atmosphere, and the carbon dioxide and methane content of the atmosphere, act together with the orbital changes to produce the periods of cooling and warming. Although the total radiation received from the sun is called the solar constant, like other stars of similar age, size, and composition, the sun exhibits variability in its output. Most pronounced, and most familiar, is a cycle of about 11 years in the number of dark spots (sunspots) on its surface. More sunspots equate to higher solar output. The so-called Maunder Minimum, or Little Ice Age, which lasted from 1650 to 1800, was characterized by very low sunspot activity and unusually cold

CLIMATE-FORCING AGENTS

FIGURE 23.7 Variations in solar radiation incident on Earth (in W m 2), on different timescales (Lean and Rind 1996). (a) Recorded day-to-day changes for a period of 7 months at a time of high solar activity. The largest dips of 0.3% persist for about a month and are the result of large sunspot groups that are carried across the face of the sun with solar rotation. (b) Observed changes for the 15–year period over which direct measurements have been made, showing the 11–year cycle of amplitude about 0.1%. (c) A reconstruction of variations in solar radiation since about 1600, based on historical records of sunspot numbers and postulated solar surface brightness during the 70-year Maunder Minimum. Estimated variations are of larger amplitude than have yet been observed. (d) A longer record of solar activity based on postulated changes in solar radiation that are derived from measured variations in 14 C and 10 Be.

temperatures over Europe. Spaceborne measurements made since 1979 provide a precise record of solar output (Figure 23.7). The total radiation from the sun is continually changing, with variations of 0.2% from one month to the next. The timing and nature of these shorter-term fluctuations are consistent with the sun’s 27-day period of rotation. They occur because brighter or darker areas on the solar surface alter the amount of sunlight received at Earth. This “rotational” variation of total solar radiation (see top panel in Figure 23.7) is superimposed on the 11-year sunspot cycle (Haigh 1996). The 11-year sunspot cycle is clearly evident in the second panel from the top of Figure 23.7. Variations in satellite records of lower troposphere temperatures since 1978, in upper ocean temperatures since 1955, and in surface

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FIGURE 23.8 Estimate of surface temperature for the last 800,000 years, inferred from measurements of the ratio of 16 O to 18 O in fossil plankton that settled to the seafloor. The use of oxygen isotope ratios is based on the assumption that changes in global temperature approximately track changes in the global ice volume. Detailed studies for the Last Glacial Maximum provide the temperature scale. Shown are changes in temperature (in °C) from the modern value.

temperatures during the past century are approximately in phase with this 11-year solar cycle (Lean and Rind 2001). There are no direct measurements of solar radiation extending back beyond the last century or so. Much longer records of solar activity are derived from the abundance of atomic isotopes that are produced in the atmosphere by the impact of cosmic rays, the rate of incidence of which on Earth is affected by conditions on the sun. When the sun is more active, its own magnetic field deflects some of the cosmic rays that would otherwise impact on the earth; conversely, when the sun is less active, the earth receives more cosmic rays. Isotopes 14 C, found in tree rings, and 10 Be, sequestered in ice deposits, are sensitive to the cosmic ray influx. Records of both of these isotopes have existed for thousands of years. They exhibit cyclic variations of 2300, 210, and 88 years, as well as that of the 11-year sunspot cycle, all of which are ascribed to the sun. Even longer-term changes in orbital features of 19,000, 24,000, 41,000, and 100,000 years exist induced by the changing gravitational pull of the other planets and the moon. The prominent 100,000-year periodicity in the climate record (see Figure 23.8) is associated with the 100,000-year cycle of the eccentricity of Earth’s orbit. The average global variation of surface temperature over the 11-year solar cycle is ∼0.2 °C, corresponding to a change in radiative forcing from solar-min to solar-max of ∼0.18 W m 2 (Camp and Tung 2007). This amount of temperature change is too large to be explainable purely by the direct effect of a 0.18 W m 2 variation in total solar intensity. The actual observed temperature response is the result of the positive feedbacks inherent in the climate system. The inferred climate sensitivity based on the climate response to sunspot cycles is thus λˆ

0:2 °C 0:18 W m

2

∼1

°C Wm

2

Since this climate sensitivity is a transient, rather than an equilibrium, sensitivity (i.e., the climate has not come to equilibrium in response to the varying solar output), and since the equilibrium climate sensitivity always exceeds the transient sensitivity, this method of inferring climate sensitivity suggests a value exceeding 1 °C W 1 m2. Direct radiative forcing owing to increases in total solar irradiance since 1750 is estimated by IPCC (2013) to be +0.05 W m 2. The period of greatest global warming over the past century or so has occurred since 1970, and if the sun is responsible for this warming, a sustained increase in solar irradiance would have to be evident in the satellite record. That record (Figure 23.9) indicates no net increase in solar

CLIMATE-FORCING AGENTS

FIGURE 23.9 Total solar irradiance observations since 1978. Since November 1978, a set of total solar irradiance (TSI) measurements from space has been available. From measurements made by several different instruments a composite record of TSI can be constructed.

irradiance since 1978, and reconstructions of presatellite data show this period of quiescence extends back to 1940. Moreover, comparison of climatic fingerprints of high and low solar forcing derived from model simulations with an ensemble of surface air temperature reconstructions over the past millennium indicate that neither a high magnitude of solar forcing nor a strong climate effect of that forcing agree with temperature reconstructions (Schurer et al. 2013). Volcanic eruptions and greenhouse gas concen­ trations appear to have exerted the most important influence over that period.

23.4.2 Greenhouse Gases Atmospheric concentrations of CO2, CH4, and N2O in 2011 were 391 ppm, 1803 ppb, and 324 ppb, respectively. These exceed preindustrial levels by ∼40%, 150%, and 20%, respectively. Concentrations of these three GHGs now exceed substantially the highest concentrations recorded in ice cores in the last 800,000 years. That the increase in the atmospheric level of CO2 over the past century or so is a result largely of fossil fuel burning is conclusively demonstrated from several lines of evidence. These include (1) records of coal, oil, and natural gas consumption and (2) concomitant decreases in the relative abundance of both the stable (13C) and radioactive (14 C) carbon isotopes and the decrease in atmospheric oxygen. Adding up the human sources of CO2 primarily from fossil fuel burning, cement production, and land use changes (e.g., deforestation), one finds that about 56% of the CO2 emitted as a result of human activities is accumulating in the atmosphere. The remaining 44% of the emitted CO2 is being taken up by oceans and the biosphere. Emissions of CO2 from human activities can account for the increase in atmospheric CO2 concentrations. Concentrations of CO2 in the ocean have increased along with the atmospheric concentrations, showing that the increase in atmospheric CO2 cannot be a result of release from the oceans. All lines of evidence taken together lead to the unambiguous conclusion that the increase in atmospheric CO2 concentration is primarily a result of fossil fuel burning. Volcanic emissions of CO2 arise from erupting magma and from degassing of unerupted magma beneath volcanoes. Over very long timescales, these CO2 emissions act to restore CO2 lost from the atmosphere and oceans by silicate weathering, carbonate deposition, and organic carbon burial. Global estimates of the annual present-day CO2 volcanic emissions range from 0.15 to 0.26 gigatons per year (Gerlach 2011); this is to be compared with anthropogenic CO2 emissions, estimated at 35 gigatons in

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2011 (IPCC 2013). Moreover, if atmospheric CO2 increases over the past century were the result of volcanic missions, not anthropogenic activity, the massive eruption of Mt. Pinatubo in the Philippines in 1991 would have produced a large pulse in CO2. No CO2 pulse appeared in any global measurements, and, in fact, the main effect of the eruption was a temporary cooling from the increased reflection of sunlight by the resulting stratospheric aerosols. Globally averaged CH4 in 1750 is estimated at 722 ± 25 ppb (IPCC 2013). In 2011, the global annual mean CH4 level was 1803 ± 2 ppb. Direct atmosphere measurements of CH4 of sufficient spatial coverage began in 1978. Assuming no long-term trend in hydroxyl radical (OH) concentration,2 the decrease in CH4 growth rate from the early 1980s through 2006 indicates an approach to steady state in which total global emissions have been approximately constant at ∼550 Tg CH4 yr 1. The global methane concen­ tration has increased by a factor of 2.5 since 1750, with an estimated radiative forcing of 0.97 W m 2 (IPCC 2013). This estimate includes the forcing due to CH4 alone of 0.48 W m 2 and effects on other greenhouse gases from methane. Increases in CH4 lead to an increase in stratospheric water vapor (about 7% of CH4 is oxidized in the upper atmosphere) and to an increase in tropospheric O3 through reactions involving oxides of nitrogen, thus having an indirect effect on forcing through stratospheric water vapor and ozone production (Shindell et al. 2005). Accounting for these additional indirect effects, the full contribution of CH4 to radiative forcing is estimated to be 0.97 W m 2. Globally averaged N2O under preindustrial conditions (∼1750) was 270 ± 7 ppb, based on data from ice cores (IPCC 2013). In 2011, N2O mixing ratio was 324 ppb, a 20% increase over preindustrial conditions. The increase since the early 1950s is dominated by emissions from soils treated with synthetic and organic nitrogen fertilizer. Systematic measurements of N2O began in the late 1970s, from which time N2O has been increasing at an average rate of ∼0.75 ppb yr 1. Figure 2.3 shows globally averaged N2O mixing ratio since 1999. Persistent latitudinal gradients exist in N2O concen­ trations that reflect differences in anthropogenic emissions from fertilizer use and natural emissions from soils and ocean upwelling regions of the tropics. Exchange of air between the stratosphere, where N2O is destroyed photochemically (see Chapter 5), and troposphere is the main contributor to observed seasonal cycles, not seasonality in emissions. In summary, shortlived climate pollutants, so-called SLCPs, which include methane, black carbon, tropospheric ozone, and hydrofluorocarbons (HFCs), contribute as much as one-third of the current total greenhouse forcing. These SLCPs have a strong impact on climate, but their lifetime in the atmosphere is short relative to those of the long-lived GHGs; the lifetime of black carbon is days to weeks, that of methane is about a decade, and the HFCs reside in the atmosphere for a few decades. It has been argued that parallel strategies for reducing CO2 and SLCP emissions should be pursued, as reduction of the latter could lead to near-term climate benefits (Velders et al. 2007, 2012; Shoemaker et al. 2013).

23.4.3 Radiative Efficiencies of Greenhouse Gases The radiative efficiency of climate forcing for a greenhouse gas can be expressed as W m 2 of forcing per unit of increase of atmospheric concentration (Table 23.1). Per unit part-per-million increase, CH4 and N2O are 26.5 and 221 times as effective in radiative forcing as CO2; however, CH4 is more than 200 times less abundant than CO2, and N2O is ∼1000 times less abundant, so CO2 still exerts the dominant effect. We return to the relative global warming potentials in Section 23.9.

23.4.4 Aerosols Aerosols scatter and absorb solar radiation and serve as the seed particles on which cloud droplets form. Through scattering and absorption of solar radiation, aerosols reduce solar insolation at the surface, leading to changes in surface fluxes of energy and water, with concomitant changes to the 2

Reaction with OH is the main loss process for CH4 and is the largest term in the CH4 sink budget. Thus, any change in the global average OH concentration would significantly affect the CH4 budget. There was no apparent trend in OH concentration from 1979 to 2011, so this potential change is not a factor in the CH4 budget.

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TABLE 23.1 Lifetimes, Mixing Ratios, and Radiative Efficiencies (REs) of Greenhouse Gases Species CO2 CH4 N2O CFC-11 CFC-12

Lifetime (yr)

Mixing ratio (2011)

9.1 131 45 100

390 ppm 1803 ppb 324 ppb 238 ppt 527 ppt

RE (W m

2

ppb 1)

1.37 × 10 3.63 × 10 3.03 × 10 0.263 0.32

5 4 3

Source: IPCC (2013).

hydrological cycle. Aerosols containing absorbing material, such as black carbon, also heat the atmosphere, altering profiles of temperature and relative humidity and consequently atmospheric stability. Furthermore, black carbon deposited on ice and snow can reduce the surface albedo and accelerate melting. Aerosols have a tropospheric lifetime measured in weeks; because of this, aerosol levels are highly nonuniform over Earth, with highest concentrations occurring in regions of greatest emissions. Aerosols injected into the stratosphere, such as from a volcanic eruption, have a lifetime of a year or more, owing to the absence of removal mechanisms (wet and dry deposition) present in the troposphere. Because aerosols interact with solar radiation, their forcing occurs only during daylight hours. Changes in aerosol levels can change cloud properties through alteration of the number of activated droplets; changes in cloud properties can then lead to changes in cloud reflectivity, cloud lifetime, and precipitation. The role of aerosols in climate is addressed in Chapter 24.

23.4.5 Summary of IPCC (2013) Estimated Forcing Table 23.2 summarizes the IPCC (2013) estimates and uncertainties bars (5–95% confidence range) on the change in radiative forcing from all important climate agents over the period 1750 to present day. Total forcing from the increase in CO2 over the industrial period is estimated as +1.68 W m 2. Increases in CH4, N2O, chlorofluorocarbons, and tropospheric O3, in aggregate, are estimated to have produced a forcing of +1.32 W m 2, a value close to that of CO2 itself. (The decrease in stratosphere O3 due to chloro­ fluocarbons is estimated to produce a slight cooling effect.) Forcings resulting from changes in surface

TABLE 23.2 Radiative Forcings since Preindustrial Time Species Longlived greenhouse gases CO2 CH4 Halocarbons N2O Shortlived gases CO NMVOC NOx Aerosols and precursors Mineral dust, Sulfate, Nitrate, OC, BC Cloud adjustments due to aerosols

Radiative forcing (W m 2) +1.68 [1.33–2.03] +0.97 [0.74–1.20] +0.18 [0.01–0.35] +0.17 [0.13–0.21] +0.23 [0.16–0.30] +0.10 [0.05–0.15] 0.15 [ 0.34 to 0.03] 0.27 [ 0.77 to 0.23] 0.55 [ 1.33 to -0.06]

Albedo change due to land use Solar irradiance

0.15 [ 0.25 to -0.05] +0.05 [0.00–0.10]

Total anthropogenic RF relative to 1750

+2.29 [1.13–3.33]

Source: IPCC (2013).

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albedo are estimated to have produced both warming and cooling effects, with a slight net cooling. The change in aerosols, on the whole, was estimated by IPCC to have exerted a cooling effect on climate of 0.82 W m 2, with an uncertainty range [ 1.9 to 0.1]. Most aerosols exert a negative forcing, but black carbon has a positive contribution from its absorption of solar radiation. IPCC (2013) suggests that the direct radiative forcing of atmospheric black carbon is +0.6 W m 2, which would place black carbon as second only to CO2. On the whole, aerosols contribute the largest uncertainty to the total radiative forcing estimate.

23.4.6 The Preindustrial Atmosphere Estimates of radiative forcing are based on the difference between the level of forcing agent in the present-day atmosphere and that in the preindustrial atmosphere, generally taken as 1750. For most gases, like CO2, the preindustrial level is accurately known from a variety of data sources, such as ice cores, etc. This is not the case for atmospheric aerosols, for which we need to know the aerosol content of the atmosphere in the absence of human industrial activities. Knowledge of the preindustrial aerosol burden in the atmosphere is of particular importance in assessing how Earth’s clouds have changed as a result of the evolution of the atmosphere’s aerosol level from preindustrial time to present day (Carslaw et al. 2013). It turns out that information about aerosol levels in the absence of human industrial activity is difficult to obtain. One might have presumed that areas of the Earth beyond the immediate vicinity of pollution sources can be considered as pristine. Owing to large anthropogenic emissions of aerosols and their gaseous precursors, together with efficient long-range transport, however, it is difficult to identify today areas on Earth, especially on the continents, that are not impacted by anthropogenic aerosols (Hamilton et al. 2014). The atmospheric lifetime of aerosols is usually considered to be the order of a week or two, depending on their size, water solubility, and location. The concept of a lifetime is, as we know, the time when the concentration has decreased to 1/e (∼37%) of its initial value. So, even after one lifetime, ∼37% remains, and after 3 lifetimes ∼5% still remains. Since airmasses can easily travel thousands of km in a week, it is difficult to identify places truly as pristine, especially in the Northern Hemisphere. Aerosol concentrations approaching pristine conditions are found, if anywhere, over the oceans, especially in the Southern Hemisphere. The natural aerosol over the remote ocean is a mixture of sea salt particles, sulfates from the oxidation of dimethyl sulfide (DMS), and primary and secondary organics from the sea. Mineral dust and biomass smoke may be present in the air over the remote ocean, and these would likely have been present in the preindustrial atmosphere as well. Identifying continental regions that can exhibit pristine conditions is more difficult; regions considered include the high northern latitude boreal forest and the Brazilian rainforest. Even the continental aerosol at remote sites in the Northern Hemisphere has similar composition to pollution aerosols. For example, Putaud et al. (2004) showed that even at the cleanest sites in Europe the aerosol composition is very similar to that at urban and regionally polluted sites, with a large fraction of non-sea salt sulfate and traces of combustionrelated black carbon. In terms of estimating aerosol indirect (cloud) forcing, the relevant aerosol quantity is the CCN concentration. The cleanest regions of the atmosphere over the remote ocean represent the relevant situation of the response of clouds to changes in aerosol levels. Natural CCN concentrations over the oceans, removed from sources of dust and biomass burning, in preindustrial time are probably similar to those over the remote Southern Hemisphere ocean, 50–200 CCN cm 3. These consist mainly of sulfates from DMS, seasalt particles, and some organics. Over the continents, natural CCN concentrations lie in the range, 100–300 cm 3, primarily biogenic SOA. The lowest continental CCN concentrations result when biogenic emissions are at a minimum, such as in wintertime or over desert areas, when CCN concentrations are just a few tens per cm3. [Inherent in these estimates is the definition of CCN. For example, Hamilton et al. (2014) define CCN concentrations as the number concentration of soluble particles with a dry diameter equal to or greater than 50 nm at cloud base, taken to be 915 hPa.] Based on global aerosol modeling, Hamilton et al. (2014) find that, averaged over a full year, approximately one-third of the Southern Hemisphere is predicted to have CCN concentrations similar to preindustrial, whereas only 9% of the Northern Hemisphere can be classified as pristine. The southern Pacific Ocean (∼20–60°S, 90–180°W) is predicted to be a large region close to pristine, especially during

COSMIC RAYS AND CLIMATE

SH summer when monthly mean CCN concentrations are in the range of 53–285 cm 3. In the NH, prolonged pristine periods are predicted to occur only over continental regions above 60°N, such as in boreal Canada and Russia, where particle levels are affected strongly by forest fire emissions. The major marine stratocumulus cloud decks are the most influential climatically in indirect aerosol forcing (see Chapter 24). In order to assess the effect of aerosol perturbations on such clouds, areas that experience both pristine and polluted conditions are most effective. In summary, a substantial uncertainty in predicting aerosol indirect forcing is specifying the aerosol in the preindustrial atmosphere.

23.5 COSMIC RAYS AND CLIMATE Galactic cosmic rays (GCRs) are the high-energy particles that flow into our solar system from far away in the galaxy. GCRs are mostly pieces of atoms: protons, electrons, and atomic nuclei that have had all of the surrounding electrons stripped during their high-speed passage through the galaxy. (Cosmic rays provide one of the few direct samples of matter from outside the solar system.) About 90% of the cosmic ray nuclei are hydrogen (protons), and about 9% are helium (alpha particles). Cosmic rays are the main source of ions in the free atmosphere. When ionizing radiation impacts an air molecule, an ion–electron pair is formed. The positive ion formed can be a molecular ion, N2+ or O2+, or, in the case of highly energetic radiation, N+ or O+. The newly formed positive ion behaves similar to that of the surrounding neutral molecules. The electron formed gains momentum from the interaction and intercepts atmospheric molecules, ionizing them, such as O2 . Positive ions are transformed into hydronium–water clusters and ammonium–water clusters. Negative ions, mainly O2 , cluster with water molecules (Luts et al. 2011). All of these processes occur in t1

d. Show that the expression for the temperature increase that results after a very long time in scenario c is ΔT …t† ˆ

bt1 α

23.9A Evaluate the parameter α in the energy balance model in Problem 23.8, assuming that the equilibrium temperature increase resulting from an instantaneous doubling of CO2 is 3 K. What is the feedback response time in this case?

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23.10B The radiative forcing due to an increase of the atmospheric CO2 mixing ratio by a factor β can be approximated by ΔF ˆ 5:3 ln β

…W m 2 †

a. For a sustained increase in the CO2 mixing ratio of 1% per year, that is, ΔF = bt, evaluate the constant b. b. At this rate of increase, how many years would it take for the CO2 mixing ratio to increase from the level of 400 ppm to 560 ppm, approximately double the preindustrial value?

REFERENCES Allen, M. R., et al. (2009), Warming caused by cumulative carbon emissions towards the trillionth tonne, Nature 458, 1163–1166. Arrhenius, S. (1896), On the influence of carbonic acid in the air upon the temperature of the ground, Philos. Mag. 4, 237–276. Barkstrom, B. R. (1989), Earth radiation budget experiment (ERBE) archival and April 1985 results, Bull. Am. Meteorol. Soc. 70, 1254–1262. Bender, F. A. M., et al. (2006), 22 views of the global albedo-comparison between 20 GCMs and two satellites, Tellus 58A, 320–330. Bony, S., et al. (2006), How well do we understand and evaluate climate change feedback processes? J. Climate 19, 3445–3482. Brasseur, G. P. and Roeckner, E. (2005), Impact of improved air quality on the future evolution of climate, Geophys. Res. Lett. 32, L23704 (doi: 10.1029/2005GL023902). Camp, C. D., and Tung, K. K. (2007), Surface warming by the solar cycle as revealed by the composite mean difference projection, Geophys. Res. Lett. 34, L14703 (doi: 10.1029/2007GL030207). Carslaw, K. S., et al. (2002), Cosmic rays, clouds, and climate, Science 298, 1732–1737. Carslaw, K. S., et al. (2013), Large contribution of natural aerosols to uncertainty in indirect forcing, Nature 503, 67–71. Dessler, A. E. (2010), A determination of the cloud feedback from climate variations over the past decade, Science 330, 1523–1527. Friis-Christensen E., and Lassen, K. (1991), Length of the solar cycle: An indicator of solar activity closely associated with climate, Science 254, 687–700. Gerlach, T. (2011), Volcanic versus anthropogenic carbon dioxide, Eos 92, 201–202. Haigh, J. D. (1996), The impact of solar variability on climate, Science 272, 981–984. Hamilton, D. S., et al. (2014), Occurrence of pristine aerosol environments on a polluted planet, Proc. Natl. Acad. Sci. USA 111, 18466–18471. IPCC (Intergovernmental Panel on Climate Change) (2007), Climate Change 2007: The Physical Science Basis, Cambridge Univ. Press, Cambridge, UK. IPCC (Intergovernmental Panel on Climate Change) (2013), Climate Change 2013: The Physical Science Basis, Cambridge Univ. Press, Cambridge, UK. Isaksen, I. S. A., and Hov. O. (1987), Calculation of trends in the tropospheric concentration of O3, OH, CH4, and NOx, Tellus 39B, 271–285. Isaksen, I. S. A., et al. (2009), Atmospheric composition change: Climate-chemistry interactions, Atmos. Environ. 43, 5138–5192. Kim, D., and Ramanathan, V. (2008), Solar radiation budget and radiative forcing due to aerosols and clouds, J. Geophys. Res. 113, D02203 (doi: 10.1029/2007JD008434). Knutti, R., and Hegerl, G. C. (2008), The equilibrium sensitivity of the Earth’s temperature to radiation changes, Nature Geosci. 1, 735–743.

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GLOBAL CLIMATE Krissansen-Totton, J., and Davies, R. (2013), Investigation of cosmic ray-cloud connections using MISR, Geophys. Res. Lett. 40, 5240–5245. Kulmala, M., et al. (2011), Atmospheric data over a solar cycle: No connection between galactic cosmic rays and new particle formation, Atmos. Chem. Phys. 10, 1885–1898. Lacis, A. A., et al. (2010), Atmospheric CO2: Principal control knob governing Earth’s temperature, Science 330, 356–359. Lean, J. and Rind, D. (2001), Earth’s response to a variable sun, Science 292, 234–236. Luts, A., et al. (2011), Composition of negative air ions as a function of ion age and selected trace gases: Mass and mobility distribution, J. Aerosol Sci. 42, 820–838. Madronich, S., and Grainer, C. (1992), Impact of recent total ozone changes on tropospheric ozone photodissociation, hydroxyl radicals and methane trends, Geophys. Res. Lett. 19, 465–467. Madronich, S., and Grainer, C. (1994), Tropospheric chemistry changes due to increased UV-B radiation, in Stratospheric Ozone Depletion – UV-B Radiation in the Biosphere, R. H. Biggs and M. E. B. Joyner (eds.), Springer, Berlin, pp. 3–10. Matthews, H. D., et al. (2009), The proportionality of global warming to cumulative carbon emissions, Nature 459, 829–832. Matthews, H. D., and Solomon, S. (2013), Irreversible does not mean unavoidable, Science 340, 438–439. Meinshausen, M., et al. (2009), Greenhouse-gas emission targets for limiting global warming to 2 °C, Nature 458, 1158–1162. Nakicenovic, N., et al. (1998), Global Energy Perspectives, Cambridge Univ. Press, Cambridge, UK. Pierce, J. R., and Adams, P. J. (2007), Efficiency of cloud condensation nuclei formation from ultrafine particles, Atmos. Chem. Phys. 7, 1367–1379. Pierce, J. R., and Adams, P. J. (2009), Can cosmic rays affect cloud condensation nuclei by altering new particle formation rates? Geophys. Res. Lett. 36, L09820 (doi: 10.1029/2006GL037946). Pierrehumbert, R. T., et al. (2007), On the relative humidity of the atmosphere, in The Global Circulation of the Atmosphere, T. Schneider and A. H. Sobel (eds.), Princeton Univ. Press, Princeton, NJ. Pierrehumbert, R. T. (2011), Infrared radiation and planetary temperature, Physics Today 64, 33–38. Pistone, K., et al. (2014), Observational determination of albedo decrease caused by vanishing Arctic sea ice, Proc. Natl. Acad. Sci. USA 111, 3322–3326. Pollack, H., et al. (1993), Heat flow from the Earth’s interior: Analysis of the global data set, Rev. Geophys. 31, 267–280. Prather, M. J. (1994), Lifetimes and eigenstates in atmospheric chemistry, Geophys. Res. Lett. 21, 801–804. Prather, M. J., and Holmes, C. D. (2013), A perspective on time: Loss frequencies, time scales and lifetimes, Environ. Chem. 10, 73–79. Putaud, J. P., et al. (2004), European aerosol phenomenology – 2: Chemical characteristics of particulate matter at kerbside, urban, rural and background sites in Europe, Atmos. Environ. 38, 2579–2595. Raes, F. and Seinfeld, J. H. (2009), New directions: Climate change and air pollution abatement: A bumpy road, Atmos. Environ. 43, 5132–5133. Rohde, R., et al. (2013a), A new estimate of the average earth surface land temperature spanning 1753 to 2011, Geoinform. Geostat. Overview 1: 1. Rohde, R., et al. (2013b), Berkeley earth temperature averaging process, Geoinform. Geostat. Overview 1: 2. Roe, G. H., and Baker, M. B. (2007), Why is climate sensitivity so unpredictable? Science 318, 629–632. Roe, G. (2009), Feedbacks, timescales, and seeing red, Annu. Rev. Earth Planet Sci. 37, 93–115. Schmittner, A. et al. (2011), Climate sensitivity estimated from temperature reconstructions of the Last Glacial Maximum, Science, 334, 1385–1388. Schurer, A. P., et al. (2013), Small influence of solar variability on climate over the past millennium, Nature Geosci. 7, 104–108. Sherwood, S. C., et al. (2015), Adjustments in the forcing-feedback framework for understanding climate change, Bull. Am. Meteorol. Soc. 96, 217–228. Shindell, D. T., et al. (2005), An emissions-based view of climate forcing by methane and tropospheric ozone, Geophys. Res. Lett. 32, L04803 (doi: 10.1029/2004GL021900). Shoemaker, J. K., et al. (2013), What role for short-lived climate pollutants in mitigation policy? Science 342, 1323–1324.

REFERENCES Sloan, T., and Wolfendale, A. W. (2008), Testing the proposed causal link between cosmic rays and cloud cover, Environ. Res. Lett. 3 (doi: 10.1088/1748-9326-3-2-0234001:1-6). Soden, B. J., et al. (2002), Global cooling after the eruption of Mount Pinatubo: A test of climate feedback by water vapor, Science 296, 727–730. Soden, B. J., and Held, I. M. (2006), An assessment of climate feedbacks in coupled ocean-atmosphere models, J. Climate 19, 3354–3360. Stocker, T. F. (2013), The closing door of climate targets, Science 339, 280–282. Sun, B., and Bradley, R. S. (2002), Solar influences on cosmic rays and cloud formation, J. Geophys. Res. 107, D14 (doi: 10.1029/2001JD000560). Svensmark, H. (1998), Influence of cosmic rays on Earth’s climate, Phys. Rev. Lett. 81, 5027–5030. Trenberth, K. E., et al. (2009), Earth’s global energy budget, Bull. Am. Meteorol. Soc. 90, 311–323. Velders, G. J. M., et al. (2007), The importance of the Montreal Protocol in protecting climate, Proc. Natl. Acad. Sci. 104, 4814–4819. Velders, G. J. M., et al. (2012), Preserving Montreal Protocol climate benefits by limiting HFCs, Science 335, 922–923. Weart, S. R. (1997), The discovery of the risk of global warming, Physics Today 34–40. Zeebe, R. E. (2011), Where are you heading Earth? Nature Geosci. 4, 416–417. Zeebe, R. E. (2013), Time-dependent climate sensitivity and the legacy of anthropogenic greenhouse gas emissions, Proc. Natl. Acad. Sci. USA 110, 13739–13744. Zeebe, R. E., and Zachos, J. C. (2013), Long-term legacy of massive carbon input to the Earth system: Anthropocene versus Eocene, Phil. Trans. Royal Soc. 371 (http//dx.doi.org/10.1098/rsta.2012.0006).

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24

Aerosols and Climate

Aerosols influence climate directly by the scattering and absorption of solar radiation and indirectly through their role as cloud condensation nuclei (CCN). The magnitude of the direct forcing of aerosols (measured in W m 2) at a particular time and location depends on the amount of radiation scattered back to space, which itself depends on the size, abundance, and optical properties of the particles and the solar zenith angle. The so-called indirect effect refers to the effect of changes in aerosol levels on cloud properties. For example, increases in aerosol number concentrations from anthropogenic sources lead to increased concentrations of cloud condensation nuclei, which, in turn, lead to clouds with larger number concentrations of droplets with smaller radii, which, in turn, lead to altered cloud albedos. Particles can both scatter and absorb radiation; as particles become increasingly absorbing versus scattering, a point is reached, depending on their size and the albedo of the underlying surface, where the overall effect of the particle layer changes from one of cooling to heating. In addition, if the particles consist of a mixture of purely scattering material, such as ammonium sulfate, and partially absorbing material, such as soot, the cooling–heating effect depends on the manner in which the two substances are mixed throughout the particle population. The two extremes in this regard are when every particle contains some absorbing material and when the absorbing material is distinct from the scattering particles. The effects are further complicated when a cloud is present. Particles exist both above and below clouds and, to some extent, even in the cloud itself. The amount of light scattered back to space depends on the properties of both the aerosols and the cloud. The direct effect can be observed visually as the sunlight reflected upward from haze when viewed from above (e.g., from a mountain or an airplane). The result of the process of scattering of sunlight is an increase in the amount of light reflected by the planet and hence a decrease in the amount of solar radiation reaching the surface. The amount of light reflected upward by aerosol is roughly proportional to the total column mass burden of particles (typically reported in grams per square meter). The direct effect of aerosols on climate is a result of the same physics responsible for the reduction of visibility that occurs in airmasses laden with particles. The major difference is that, whereas visibility reduction is attributable to aerosol scattering in all directions, the direct climatic effect of aerosols results only from radiation that is scattered in the upward direction, that is, back to space. Indirect climate effects of aerosols are more complex and more difficult to assess than direct effects because they involve a chain of phenomena that connect aerosol levels to concentrations of cloud condensation nuclei, cloud condensation nuclei concentrations to cloud droplet number concentrations (and size), and these, in turn, to cloud albedo and cloud lifetime. Changes in the number concentration of aerosols are observed to cause variations in the population and sizes of cloud droplets, which are

Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, Third Edition. John H. Seinfeld and Spyros N. Pandis.  2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

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expected to cause changes in cloud albedo and areal extent. Other meteorological influences might occur as a result of perturbations in the number concentration of aerosols, such as changes in precipitation. In contrast to GHGs, which act only on outgoing, infrared radiation, aerosol particles can influence both sides of the energy balance (Table 24.1). Particles of diameters less than 1 μm are highly effective at scattering incoming solar radiation, sending a portion of that scattered radiation back to space. In so

TABLE 24.1 Comparison of Climate Forcing by Aerosols with Forcing by Greenhouse Gases (GHGs) Factor

Longlived GHGs (CO2, CH4, CFCs)

Shortlived GHGs (O3, HCFCs, VOCs)

Optical properties

Infrared absorption is well quantified for all major and minor GHGs

Infrared absorption is reasonably well quantified

Important electromagnetic spectrum

Almost entirely longwave (λ > 1 μm)

For O3, solar and longwave are important

Amounts of material

Well mixed; nearly uniform within the troposphere

Determination of forcing

Well-posed problem in radiative transfer

Highly variable in space and time; concentrations may be estimated by chemical transport models, but with some uncertainty Radiative aspects well posed; global networks provide some data to test model predictions of geographic and altitudinal distributions

Dependence of forcing on loading (at present)

Varies as weak function (square root or logarithm) of concentration Forcing is exerted at the surface and in the troposphere, operates night and day

Nature of forcing

Source: National Research Council (1996).

Generally nonlinear with concentration; for halocarbons, a linear dependence Strongly dependent on geographic, altitudinal, and temporal variation (e.g., O3); forcing is exerted at the surface and troposphere; operates night and day

Aerosols Refractive indices of pure substances are known, but size-dependent mixing of numerous species and the nature of mixing have optical effects difficult to quantify For tropospheric aerosol, mainly solar; for stratospheric aerosol, solar and longwave contributions lead to stratospheric warming Pronounced spatioemporal variations

Direct: Relatively well-posed problem, but dependent on empirical values for several key aerosol properties; dependent on models for geographical/temporal variations of forcing Indirect: Depends on aerosol number distribution; inadequacy of descriptions of aerosols and clouds restricts abilities to predict indirect forcing Direct: Almost linear in the concentration of the particles Indirect: Nonlinear Tropospheric forcing varies strongly with location and season and occurs only during daytime; maxima of forcing occur near sources and at Earth’s surface; stratospheric forcing includes some longwave effect but is dominated by shortwave radiation (daytime only); following major volcanic events, stratospheric mixing yields a forcing that is substantially global in nature For nonabsorbing tropospheric aerosols, forcing is almost entirely at surface; for stratospheric aerosols, there is a small heating resulting in a transient warming of stratosphere

972

AEROSOLS AND CLIMATE

doing, these particles reduce the amount of incoming solar energy as compared with that in their absence and consequently cool the earth. Over industrialized parts of the world, sulfates, organics, and black carbon constitute much of this light-scattering aerosol. In the tropics, biomass burning of forests and savannas is a dominant source of airborne particles, which consist mainly of organic matter and soot. Mineral dust from wind acting on soils is always present in the atmosphere to some degree, although human activities, such as disruption of soils by changing use of land in arid and subarid regions, can increase the loading of dust over that present “naturally.” Because of their size and composition, mineral dust particles can scatter and absorb both incoming and outgoing radiation. In the visible part of the spectrum, the light-scattering effect dominates, and mineral dust exerts an overall cooling effect; in the infrared region, mineral dust is an absorber and acts like a greenhouse gas. Greenhouse gases such as CO2, CH4, N2O, and the CFCs are virtually uniform globally; aerosol concentrations, on the other hand, are highly variable in space and time. With lifetimes of about 2 weeks, aerosols are most abundant close to their sources in the industrialized areas of the Northern Hemisphere. Biomass aerosols are emitted predominantly during the dry season in tropical areas. Mineral dust is at highest concentrations downwind of large arid regions. Moreover, greenhouse gas forcing operates day and night; aerosol forcing operates only during daytime. Aerosol radiative effects depend in a complicated way on the solar angle, relative humidity, particle size and composition, as well as the albedo of the underlying surface. When superimposed on each other, the spatial distribution of GHG warming and aerosol cooling do not occur at the same locations. Black carbon (BC) is the strongest absorber of solar radiation of all the constituents of the atmosphere (Bond et al. 2013). In addition to BC the atmosphere contains light-absorbing organic or brown carbon (BrC), which can account for as much as 20% of light absorption in the atmosphere. Aerosol residence times in the troposphere are roughly 1–2 weeks; if all anthropogenic sources were shut off today, anthropogenic aerosols would disappear from the planet in a month. By contrast, not only are GHG residence times measured in decades to centuries, but because of the great inertia of the climate system, as noted in the previous chapter, the effect of GHG forcing takes decades to be fully transformed into equilibrium climate warming. As a result, if both CO2 and aerosol emissions were to cease today, Earth would continue to warm as the climate system continues to respond to the accumulated amount of CO2 already in the atmosphere.

24.1 SCATTERING–ABSORBING MODEL OF AN AEROSOL LAYER Consider a direct solar beam impinging on the layer shown in Figure 24.1. Assume, for the moment, that the beam is directly overhead, at a solar zenith angle of θ0 = 0°. The fraction of the incident beam transmitted through the layer is e τ , where τ is the optical depth of the layer. The fraction reflected back

FIGURE 24.1 Scattering model of an aerosol layer above Earth’s surface. Total downward transmitted fraction is t = e τ + ω(1 β) (1 e τ). Total reflected off surface = Rst (Rs = surface albedo).

973

SCATTERING–ABSORBING MODEL OF AN AEROSOL LAYER

in the direction on the beam is r = (1 e τ )ωβ, where ω is the single-scattering albedo of the aerosol, and β is the upscatter fraction, the fraction of light that is scattered by a particle into the upward hemisphere. The fraction of light absorbed within the layer is (1 ω)(1 e τ ). The fraction scattered downward is ω(1 β)(1 e τ ). The total fraction of radiation incident on the layer that is transmitted downward is tˆe

τ

‡ ω…1

e τ†

β†…1

(24.1)

If the albedo of the underlying Earth surface is Rs, then the fraction of the radiation incident on the surface that is reflected is Rst. Of the intensity RstF0 reflected upward by Earth back into the aerosol layer, some is backscattered back to Earth, some is absorbed by the layer, and some is scattered upward. On the first downward pass through the layer the fraction of the intensity transmitted is t. Thus, on the first upward pass of the reflected beam, the fraction transmitted is also t. By turning the layer upside down in our minds, the fraction of the beam from Earth reflected back downward to Earth from the layer is just rRst. That beam reflected downward is itself reflected off the surface. So, for two complete passes, the total upward flux is Fr ˆ r ‡ Rs tF0 t ‡ Rs rRs tF0 t ˆ …r ‡ t2 Rs ‡ t2 Rs2 r†F0 The process can be continued, giving Fr ˆ …r ‡ t2 Rs ‡ t2 R2s r ‡ ∙ ∙ ∙ †F0 ˆ ‰r ‡ t2 Rs …1 ‡ Rs r ‡ ∙ ∙ ∙ †ŠF0 The next term in the series is R2s r2 , so Fr ˆ ‰r ‡ t2 Rs …1 ‡ Rs r ‡ R2s r2 ‡ R3s r3 ‡ ∙ ∙ ∙ †ŠF0

(24.2)

Since both Rs and r are 0, then the planetary albedo is increased in the presence of aerosols, and more radiation is reflected back to space than in their absence. In this case, the change in outgoing radiative flux ΔF > 0. In terms of climate forcing, one is concerned with the net change in incoming radiative flux, so the net change in forcing is ΔF. To summarize, ΔF depends on the following parameters: F0 = incident solar flux, W m 2 Ac = fraction of the surface covered by clouds Ta = fractional transmittance of the atmosphere Rs = albedo of the underlying Earth surface ω = single-scattering albedo of the aerosol β = upscatter fraction of the aerosol τ = aerosol optical depth The single-scattering albedo ω depends on the aerosol size distribution and chemical composition and is wavelength-dependent. The upscatter fraction β depends on aerosol size and composition, as well as on the solar zenith angle θ0. Aerosol optical depth depends largely on the mass concentration of aerosol.

975

COOLING VERSUS HEATING OF AN AEROSOL LAYER

As a first approximation, it was assumed that direct aerosol forcing occurs only in cloud-free regions. Actually, some amount of aerosol forcing does occur when clouds are present. Imagine a relatively uniform layer of particles extending from the surface up to a height of, say, 5 km. Consider first a cloud layer above the aerosol layer. Some solar radiation does penetrate the cloud layer to be intercepted by the aerosol layer beneath it. However, because the aerosols receive only a portion of the incoming radiation and because any radiation they do scatter upward is, in turn, scattered by the clouds, the aerosol forcing in this situation turns out not to be terribly significant. If the cloud layer exists within the aerosol layer, then those particles above the clouds scatter radiation back to space in the same manner as in the cloudfree case. A somewhat mitigating factor is that the radiation scattered upward by the cloud layer is intercepted by the aerosol layer and some of that radiation is scattered back down toward the cloud layer. In an early calculation, Boucher and Anderson (1995) found, for example, in their global simulation that the ratio of aerosol forcing in the presence of clouds to that in the absence of clouds is ΔFcloud/ΔFclear = 0.25. When aerosol scattering within and below clouds was removed from the simulation, the resulting decrease in forcing was 7%. Thus aerosols above the cloud layer are responsible for most of the direct effect in cloudy regions.

24.2 COOLING VERSUS HEATING OF AN AEROSOL LAYER The sign of the change in planetary albedo as the result of an aerosol layer ΔRp determines whether the forcing is negative (cooling effect) or positive (heating effect). The key parameter governing the amount of cooling versus heating is the single-scattering albedo ω. The value of ω at which ΔRp = 0 defines the boundary between cooling and heating. The scattering versus absorption properties of an aerosol are measured by the value of the single-scattering albedo. Whether particle absorption actually results in an increase or decrease of the planetary albedo depends on the particle single-scattering albedo as well as on the albedo of the underlying surface. Over dark surfaces such as the ocean, virtually all particles tend to increase the planetary albedo because upscattering of incident solar radiation by the particle layer exceeds that from the dark surface regardless of how much absorption is occurring. On the other hand, over high-albedo surfaces, such as snow or bright deserts, absorption by particles can reduce the solar flux reflected from the surface and result in a net reduction of radiation to space. Let us derive an expression for the value of ω at which ΔRp = 0. To do so it is useful to employ the fact that for global tropospheric conditions optical depth values are usually about 0.1, so that the mathe­ matical approximation τ1 is a reasonable one. Thus r ˆ …1

e τ †ωβ

(24.11)

≅ τωβ and tˆ e

τ

≅1

‡ ω…1 τ ‡ ω…1

β†…1

e τ†

(24.12)

β†τ

Thus, from (24.6), we obtain ΔRp ≅ τωβ ‡

‰1

τ ‡ ω…1 β†τŠ2 Rs 1 Rs τωβ

Rs

Rearranging the RHS of (24.13) and neglecting terms involving τ2 yield ΔRp ≅ τωβ ‡ …1

Rs †2

2Rs 1 β ω

1

(24.13)

976

AEROSOLS AND CLIMATE

We can also neglect τωβ relative to the other term, giving ΔRp ˆ …1

Rs †2

2Rs 1 β ω

1

(24.14)

The boundary between cooling and heating, that is, at ΔRp = 0, occurs for values of ω satisfying ωcrit ˆ

2Rs 2Rs ‡ β…1

R s †2

(24.15)

Values of ω > ωcrit lead to cooling (ΔRp > 0). Figure 24.2 shows the critical single-scattering albedo that defines the boundary between negative (cooling) and positive (heating) forcing regions as a function of surface albedo Rs and upscatter fraction β. The critical value of ω is practically independent of the actual value of τ. A typical global mean surface albedo Rs is about 0.15, and a representative value of the spectrally and solar zenith angle averaged β is about 0.29. For these values of Rs and β, the critical value of ω that defines the crossover point between heating and cooling is about 0.6. At values of surface albedo approaching 1.0, the critical value of ω itself approaches 1.0, independently of the value of β. At high values of Rs the total reflectance of the aerosol– surface system is large to begin with and even a small amount of aerosol absorption leads to a heating effect. At values of Rs approaching zero, only a small amount of aerosol scattering is required to produce a cooling effect of the aerosol layer.

FIGURE 24.2 Critical single-scattering albedo that defines the boundary between cooling and heating regions.

977

SCATTERING MODEL OF AN AEROSOL LAYER FOR A NONABSORBING AEROSOL

24.3 SCATTERING MODEL OF AN AEROSOL LAYER FOR A NONABSORBING AEROSOL The aerosol layer scattering model can be simplified if a nonabsorbing aerosol, that is, ω = 1, is considered and account is taken of the fact that the optical depth for most tropospheric conditions is usually about 0.1 or smaller, thereby allowing one to assume τ  1. We note that hydrated ammonium/sulfate particles ranging from pure (NH4)2SO4 to pure H2SO4, a mixture frequently used to represent background global aerosol, are essentially nonabsorbing at wavelengths below 2 μm (Figure 24.3). This, coupled with the fact that the small amount (∼5%) of solar intensity above λ = 2 μm, is mostly absorbed by atmospheric gases, means that the approximation ω = 1 is reasonable for global sulfate aerosols. The simplification occurs in the expression for the change in planetary reflectance (24.6): ΔRp ˆ Ras ˆ r‡

Rs t2 Rs 1 Rs r

Rs

For a nonabsorbing aerosol, ω = 1 and r + t = 1. Thus …1 r†2 Rs Rs 1 Rs r ≅ …r ‡ Rs 2rRs †…1 ‡ Rs r†

ΔRp ˆ r ‡

(24.16) Rs

where we have used the condition that 1  R2s r2 . Using this condition yet again, we find ΔRp ≅ r…1

Rs †2

(24.17)

Note that ΔRp > 0. This means that for a nonabsorbing aerosol layer, the change in planetary reflectance is always positive; that is, the presence of the aerosol layer leads to an increased planetary albedo over that in its absence and, consequently, a cooling effect.

FIGURE 24.3 Single-scattering albedo for (NH4)2SO4 particles as a function of wave­ length (RH = 50%).

978

AEROSOLS AND CLIMATE

Since with ω = 1, and τ  1, it follows that

r ˆ …1

e τ †β

(24.18)

≅ τβ so Rs †2 τβ

ΔRp ≅ …1 Adding the factors of (1

(24.19)

Ac) and T 2a as in (24.10), we obtain the change in outgoing radiative flux: ΔF ≅ F0 T 2a …1

Ac †…1

Rs †2 βτ

(24.20)

For radiation at solar zenith angle θ0 (see Chapter 4), the pathlength relative to a unit depth of the layer is (neglecting for the moment the correction given in Table 4.1) sec θ0, so at any solar zenith angle θ0, the flux along the path of the sun is F (24.21) ˆ exp… τ sec θ0 † F0 where τ sec θ0 can be thought of as the slant path optical depth, which replaces τ in (24.20). But the incident solar flux F0 at solar zenith angle θ0 has an intensity, normal to the surface, of F0 cos θ0, which replaces F0 in (24.20). Thus the two factors, sec θ0 and cos θ0, accounting for the solar zenith angle just cancel each other. The upscatter fraction β depends on aerosol particle size and solar zenith angle θ0. For use of (24.20) in estimating radiative forcing, β can be replaced by its spectrally and solar zenith angle averaged value, β. Equation (24.20) can be used to estimate the globally and annually averaged forcing, ΔF, as a result of an aerosol layer. First F0 is replaced by F0/2 to reflect the fact that any point on the globe is illuminated by sunlight only one-half of the time over the course of a year. Finally, the optical depth, which is a function of wavelength λ, is replaced by its spectrally weighted value, τ. The result for the change in incoming solar flux (the negative of the outgoing flux) is 1 F0 T 2a …1 2

ΔF ˆ

Ac †…1

Rs †2 βτ

(24.22)

It is useful to reiterate the approximations inherent in (24.22): (1) the aerosol is assumed to be nonabsorbing, that is, ω = 1; (2) the optical depth is considerably smaller than 1.0; (3) annual mean conditions are assumed; (4) a single, globally averaged value of the surface albedo Rs is used; (5) β is a solar zenith angle and spectrally averaged; (6) a single, globally averaged value of atmospheric transmittance Ta is used; and (7) direct aerosol forcing occurs only in cloud-free regions. The expression for ΔF combines geophysical variables, aerosol microphysics, and optical depth: ΔF ˆ

1 F0 T2a …1 2

Rs †2 βτ

Ac †…1

optical depth

geophysical variables aerosol microphysics

The general result for the net radiative forcing of an aerosol that both scatters and absorbs (ω < 1) radiation is

ΔF ˆ

1 F0 T 2a …1 2

Ac †ωβτ …1

R s †2

2Rs 1 β ω

1

(24.23)

This equation was first derived by Haywood and Shine (1995). We will now consider explicitly how upscatter fraction and optical depth may be calculated.

UPSCATTER FRACTION

24.4 UPSCATTER FRACTION The fraction of light that is scattered by a particle into the upward hemisphere relative to the local horizon is the fraction of the integral over the angular distribution of scattered radiation that is in the upward hemisphere. This upscatter fraction β therefore depends on particle size and solar zenith angle. Note that it is the upward hemisphere that is important in the effect of aerosols on the Earth’s radiative balance, not the back hemisphere relative to the direction of direct of incident radiation (scattering angles of 90°–270° with respect to the incident ray). The two coincide exactly only when the solar zenith angle is directly overhead, θ0 = 0°. At any given solar zenith angle θ0, the shift toward forward scattering associated with increasing particle size decreases the upscatter fraction (see Figure 24.4). Radiative transfer theory to calculate the upscatter fraction β as a function of particle size and solar zenith angle θ0 was developed by Wiscombe and Grams (1976). Figure 24.5 gives β as a function of

FIGURE 24.4 Schematic of the relationship between upscatter fraction and hemispheric backscatter ratio as a function of solar zenith angle for two particle sizes.

FIGURE 24.5 Dependence of upscatter fraction β on the cosine of the solar zenith angle, μ0 = cos θ0, and particle radius (in μm) at λ = 550 nm and m = 1.4 0i (Nemesure et al. 1995).

979

980

AEROSOLS AND CLIMATE

μ0 = cos θ0 and particle radius. For Sun at the horizon (θ0 = 90°, μ0 = 0) β = 0.5, independent of particle size because of the symmetry of the phase function (see Figure 24.4). For solar zenith angles decreasing from 90°, forward scatter leads to a decrease in the value of β, although the decrease is slight for the smallest particles because of the relative symmetry of the scattering function in the forward and backward directions for small particles. For models used to assess the effect of aerosols on radiative forcing, it is desirable to average the forcing over solar zenith angle. In doing so, it is necessary to average β over solar zenith angle. Figure 24.6 shows the global annual average β (β) averaged over both diurnal solar zenith angle and the solar spectrum as calculated for a (NH4)2SO4 aerosol. The global average β approaches 0.5 for the smallest particles and decreases to a value of about 0.2 for the largest particles.

FIGURE 24.6 Diurnal (24-h) average upscatter fraction for (NH4)2SO4 particles averaged over the solar spectrum, for three seasons and several latitudes (°N) as a function of particle radius (Nemesure et al. 1995). Here winter = December 21, spring = March 21, and summer = June 21. The bottom panel represents the global annual average β.

981

OPTICAL DEPTH AND COLUMN FORCING

24.5 OPTICAL DEPTH AND COLUMN FORCING It is useful to define the aerosol optical depth as the product of coefficients per unit mass of aerosol and the aerosol mass concentration. For sulfate aerosol, for example, we can express the optical depth as (recall that the absorption component for sulfate aerosols is zero) τ ˆ αSO2 mSO2 H 4

(24.24)

4

where αSO2 is the light-scattering mass efficiency of the aerosol, expressed in units of m2 …g SO24 † 1 , 4 mSO2 is the sulfate mass concentration (g m 3), and H is the pathlength through the aerosol layer (m). 4 Equation (24.24) defines αSO2 , a quantity that, when multiplied by the mass concentration of sulfate, 4 produces the sulfate aerosol scattering coefficient bsp. By reference to Chapter 15, we see that αSO2 is just the mass scattering efficiency Escat(m, Dp, λ), 4 expressed in units of m2 …g SO24 † 1 , for sulfate-containing particles. Figure 15.7 showed the mass scattering efficiency of a dry (NH4)2SO4 aerosol at λ = 550 nm. In that figure, the units of Escat are m2 g 1, referring to the total mass of (NH4)2SO4. Expressed in units of m2 …g SO24 † 1 , the mass scattering efficiency increases over that shown in Figure 15.7 by the ratio 132/96. Thus αSO2 for (NH4)2SO4 aerosol 4

reaches a maximum of about 9 m2 …gSO24 † 1 at a diameter just about equal to the wavelength of the radiation. Because particle size and therefore αSO2 depend strongly on the ambient relative humidity (RH), it is 4 customary to define αSO2 at a reference, low RH (RHr) and then multiply it by a factor f(RH) that is the 4 ratio of the scattering cross section at any RH to that at RHr r αSO2 ˆ αRH f …RH† SO2 4

(24.25)

4

represents the light-scattering efficiency of sulfate aerosol, including its associated cations, at where αRHr SO2 4

RHr, typically 30%. Its value depends on the aerosol size distribution and the chemical compounds associated with sulfate in the particles. An alternative to including all the associated cations and other is to have the quantity refer only to the sulfate ion substances along with sulfate in the definition of αRHr SO2 4

itself and treat the mass scattering efficiency of all substances separately and additively. The dependence of scattering efficiency on RH is accounted for in (24.25) by the factor f(RH). Most inorganic aerosols are hygroscopic. Sulfuric acid particles are always hydrated, but salts such as (NH4)2SO4 and NH4HSO4 exist as dry particles at sufficiently low RH and experience an abrupt uptake of water at the relative humidity of deliquescence (DRH) (see Chapter 10). The DRH for these two ammonium sulfate salts are, for example, 80% for (NH4)2SO4 (Shaw and Rood 1990; Tang and Munkelwitz 1994) and 39% for NH4HSO4 (Tang and Munkelwitz 1994). Figure 10.4 showed the particle size change that occurs with increasing RH for (NH4)2SO4, NH4HSO4, and H2SO4. Sulfuric acid remains a liquid throughout the entire range of RH. As we saw in Chapter 10, starting at an RH > DRH and decreasing RH, a particle does not crystallize when the DRH is reached but remains in a metastable equilibrium until a much lower humidity (RHC) at which crystallization finally occurs. This behavior gives rise to a hysteresis phenomenon in which a particle of a particular compound can be a different size at RHs between the RHC and DRH depending on whether it reached that RH through increasing or decreasing relative humidity (Figure 10.4). Crystallization relative humidities for the two ammonium sulfate salts are 37% for (NH4)2SO4 and 0.46). This occurs because the decrease in Escat that takes place when soot is mixed into every particle is proportionately more than the fraction of soot that has been added. (Figure 15.8 shows this behavior in terms of the refractive index.) Now consider the behavior of the absorption coefficients in the internal and external mixtures. The absorption coefficient increases with increasing soot content of the overall aerosol, regardless of whether the soot and sulfate are internally or externally mixed, but, contrary to the scattering coefficient, the absorption coefficient of the internal mixture always exceeds that of the external mixture. In the external mixture, some particles are pure soot and the others are pure (NH4)2SO4, and all the absorption is concentrated in the pure soot particles. In the internal mixture, the absorbing component is spread equally over all particles. The absorption cross section of a given amount of soot in a particle that consists also of nonabsorbing material is greater than that of the pure soot particle in air. Thus, in the internal mixture the soot exerts its influence in the absorption of every single particle, which exceeds that if all the soot is concentrated exclusively in pure soot particles (frequently the case) and that the soot particles are dispersed uniformly in the sulfate droplets. Simply from geometric optics, more light is incident on a small sphere when it is at the center of a much larger sphere than when it is in air by itself. On the whole, the increase in scattering coefficient with sulfate content outweighs the increase in absorption coefficient with soot content and the overall extinction coefficient increases with increasing sulfate content. The single-scattering albedo ω for the two mixtures is shown in Figure 24.11. The fact that ω for the external

FIGURE 24.11 Single-scattering albedo for internal versus external mixture of soot and (NH4)2SO4.

TOP-OF-THE-ATMOSPHERE VERSUS SURFACE FORCING

FIGURE 24.12 Real (n) and imaginary (k) parts of the refractive indices of internally mixed soot and (NH4)2SO4 particles.

mixture exceeds that for the internal mixture at all compositions reflects the dominance of the scattering of pure (NH4)2SO4 particles because of the absence of an absorbing component (recall Figure 15.8). The volume-average refractive indices of the internally mixed particles are shown in Figure 24.12.

24.7 TOP-OF-THE-ATMOSPHERE VERSUS SURFACE FORCING Aerosol direct radiative forcing measured at the top of the atmosphere (TOA) is called TOA aerosol forcing; measured at the surface, it is called surface aerosol forcing. If the aerosol does not absorb radiation, the surface forcing is approximately equal to that at TOA. With black carbon and certain mineral dusts, which absorb solar radiation, the surface forcing can be a factor of 2–3 greater than the magnitude of the TOA forcing. Of all particulate species, black carbon (BC) exerts the most complex effect on climate. Like all aerosols, BC scatters a portion of the direct solar beam back to space, which leads to a reduction in solar radiation reaching the Earth’s surface. This reduction is manifest as an increase in solar radiation reflected back to space at the top of the atmosphere, that is, a negative radiative forcing (cooling). A portion of the incoming solar radiation is absorbed by BC-containing particles in the air. This absorption leads to a further reduction in solar radiation reaching the surface. At the surface, the result of this absorption is cooling because solar radiation that would otherwise reach the surface is prevented from doing so. However, the absorption of radiation by BC-containing particles leads to a heating in the atmosphere itself. Thus, absorption by BC leads to a negative radiative forcing at the surface and a positive radiative forcing in the atmosphere. Finally, the BC-containing aerosol absorbs radiation from the diffuse upward beam of scattered radiation. This reduces the solar radiation that is reflected back to space, leading to a positive radiative forcing at the top of the atmosphere. This effect is particularly accentuated when BC aerosol lies above clouds. A portion of organic aerosols is light-absorbing. This fraction has been termed brown carbon. Brown carbon is found predominantly in urban and biomass burning regions and is most absorbing at UV wavelengths. On an analysis of the wavelength dependence of aerosol absorption optical depth, Chung et al. (2012) estimate that brown carbon contributes globally 20% of the total absorption at 550 nm from all carbonaceous aerosols (BC + OA). Laboratory experiments show that light-absorbing organic aerosol is especially associated with aromatic carbonyls.

987

988

AEROSOLS AND CLIMATE

TABLE 24.3 Scattering and Absorption Properties of East Asian Aerosola Type Water-soluble (used for sulfate and organic carbon) Black carbon Seasalt (accumulation) Seasalt (coarse) Mineral (fine) Mineral (intermediate) Mineral (coarse) Externally mixed model (sulfate + OC + BC) Internally mixed model (sulfate + OC + BC)

Dm (nm)

σg

α (m2/g) (500 nm; 80% RH)

αe …500 nm;80% RH† ω αe …500 nm; dry† (500 nm; 80% RH)

42

2.24

8.13

0.987

2.25

24 418 3500 140 780 3800

2.0 2.03 2.03 1.95 2.0 2.15

10.66 4.60 0.97 2.79 0.56 0.23 8.28

0.226 1.000 1.000 0.986 0.939 0.864 0.928

1 3.63 3.87 1 1 1 2.15

9.52

0.905

2.36

70/150

1.4/1.6

a

The external and internal mixtures correspond only to sulfate, organic carbon (OC), and black carbon (BC) for average concentrations predicted over land. Dm is geometric mean diameter, σg is geometric variance, α is mass scattering efficiency, αe is mass extinction efficiency, ω is single-scattering albedo. Source: Conant et al. (2003).

Top-of-atmosphere (TOA) forcing for BC is the sum of the negative forcing at the surface due to scattering of incoming radiation and the positive forcing from absorption of upward diffuse radiation. The absorption of incoming solar radiation by BC does not contribute to TOA forcing because it adds heat to the atmosphere and reduces solar heating at the surface by the same amount. The net effect of BC is to add heat to the atmosphere and decrease radiative heating of the surface. Regions of the world have been studied for their significant amounts of absorbing aerosol. As discovered in the Indian Ocean Experiment (INDOEX), a large plume of biomass and industrial emissions rich in organic and BC aerosol is carried from the Indian subcontinent by the northeast monsoon over the northern Indian Ocean each winter (Satheesh and Ramanathan 2000; Ramanathan et al. 2001; Collins et al. 2002). This plume is responsible for a large (from 15 to 30 W m 2) surface radiative forcing over a substantial fraction of the northern Indian Ocean. The Asian Pacific Regional Aerosol Characterization Experiment (ACE-Asia), carried out in spring 2001, characterized the aerosol that is carried from East Asia over the western Pacific (Huebert et al. 2003; Seinfeld et al. 2004). East Asia is one of the most concentrated aerosol source regions on the globe, with emissions from industrial sources, biomass burning, and dust storms. Conant et al. (2003) constructed an optical model of the East Asian aerosol (size, chemistry, single-scatter albedo, hygroscopicity, and mixing state) measured during ACE-Asia and predicted radiative forcing and forcing efficiency for all the species and their combinations (Table 24.3) during April 5–15, 2001, when a powerful dust storm carried significant levels of mineral dust mixed with anthropogenic sulfur, organic carbon, and BC over the North Pacific. Figure 24.13 shows predicted column optical depth, surface and TOA forcing efficiency, and surface and TOA forcing for each of the aerosol types averaged over April 5–15, 2001 over the area 20°N–50°N, 100°E–150°E. Mineral dust was the strongest single contributor to the aerosol optical depth and forcing over this period, contributing 10 W m 2 to the surface forcing and 6 W m 2 to the TOA forcing. Sulfate and organic carbon (treated together as a water-soluble component) exerted a combined forcing of 8 W m 2 at the surface and 7 W m 2 at TOA. Black carbon is estimated to have had a slight (0.25 W m 2) warming effect at the TOA, yet exerted surface cooling of 4 W m 2. Seasalt forcing contributed only 0.5 W m 2 at the surface and TOA. Mean predicted forcing efficiency per unit optical depth by dust is 65 W m 2τ 1. Sulfate and organic carbon are predicted to have comparable surface and TOA forcing efficiencies per unit optical depth of 30 W m 2τ 1, whereas

TOP-OF-THE-ATMOSPHERE VERSUS SURFACE FORCING

FIGURE 24.13 Radiative effects of East Asian aerosol during the period April 5–15, 2001 (Conant et al. 2003): Optical depth; surface forcing efficiency (W m 2 τ 1); TOA forcing efficiency (W m 2 τ 1); surface forcing (W m 2); and TOA forcing (W m 2). Values are for clear skies assuming that the aerosol components are externally mixed.

BC is far more efficient at reducing surface radiation with a surface forcing efficiency per unit optical depth of 220 W m 2τ 1. The predicted forcing efficiency is sensitive to the extent to which the aerosol is externally or internally mixed and whether clouds are present. Figure 24.14 shows the overall predicted regional forcing for the externally mixed case (clear sky), the internally mixed case (clear sky), and the internally mixed all-sky case (clear and cloudy skies). As noted earlier, clouds generally decrease the magnitude of TOA forcing relative to clear-sky conditions because of multiple scattering effects. The effect of clouds on atmospheric absorption, however, depends on the vertical distribution of the cloud and aerosol layers. Aerosol absorption is reduced by high clouds, which shade the aerosol, but is enhanced by low clouds, which reflect radiation back up through the absorbing aerosol layers.

989

990

AEROSOLS AND CLIMATE

FIGURE 24.14 Direct aerosol forcing for the East Asian aerosol during the period April 5–15, 2001 at the TOA, the atmosphere, and at the surface for an externally mixed aerosol (clear sky), an internally mixed aerosol (clear sky), and the all-sky case (clear and cloudy skies) (Conant et al. 2003). The range bars for the all-sky cases reflect the sensitivity to cloud fraction.

24.8 INDIRECT EFFECTS OF AEROSOLS ON CLIMATE Clouds cover vast regions of the globe and play a central role in Earth’s radiation budget. If aerosol number concentrations Na increase, the number concentration of cloud droplets Nd that result from activation of these particles will, in principle, increase. The direct link between increasing CCN levels and cloud properties was proposed by Twomey (1977). Twomey argued that an increase in CCN entering shallow, warm clouds, and for a constant liquid water content, will lead to more numerous, smaller droplets. Smaller droplets lead to an increase in the surface area of cloud droplets available to interact with radiation, thereby reflecting more sunlight back to space. Aerosol indirect forcing is so named because the radiative forcing is indirectly mediated through the response of clouds rather than by the particles themselves directly interacting with sunlight. The considerable challenge in describing quantitatively the effect of aerosol perturbations on clouds arises because clouds are influenced heavily by meteorology. Changes in aerosol concentration and, less so, composition can alter cloud microphysics to initiate a cascade of feedbacks that may amplify or dampen the original perturbation, and these effects are influenced strongly by the meteorological tapestry within which the cloud exists. For example, Albrecht (1989) argued that a reduction in size of cloud droplets would retard and reduce rain formation in shallow marine clouds, perhaps resulting in longer-lived clouds and increased cloud cover. It is these responses, often subtle, resulting from a change in aerosol amount, that must be unraveled to predict the climatic aerosol indirect effect. Formally, aerosol indirect forcing is a measure of the radiative perturbation due to changes in Earth’s clouds from preindustrial to present-day aerosol levels. Another challenge in assessing the aerosol indirect effect on climate is the vast range of spatio­ temporal scales associated with clouds. The aerosols that activate to form droplets can range from 10s to 1000s of nanometers in size. The droplets formed span the scale of micrometers, whereas the scale of clouds themselves can range from 100 m to 1000s of meters. CCN activation occurs in ci , this entrainment term is positive and vice versa. The entrainment term given by (25.12) should be included in the box model only if the mixing height is increasing. Summarizing, the entraining Eulerian box model equations are q dci ˆ i ‡ Ri dt H…t†

vd;i c0 ci ci ‡ i H…t† τr

q dci ˆ i ‡ Ri dt H…t†

vd;i c0 ci cai ci dH ci ‡ i ‡ H…t† τr H…t† dt

for

dH 0 dt

(25.13)

for

dH >0 dt

(25.14)

Equations (25.13) and (25.14) describe mathematically the concentration of species above a given area assuming that the corresponding airshed is well mixed, accounting for emissions, chemical reactions, removal, advection of material in and out of the airshed, and entrainment of material during growth of the mixed layer. These equations cannot be solved analytically if one uses a realistic gas-phase chemical mechanism for the calculation of the Ri terms. Numerical solutions will be discussed subsequently.

25.2.2 A Lagrangian Box Model The Eulerian box model developed in the previous section does not have any spatial resolution, in that the entire airshed is assumed to be well mixed. This weakness can be circumvented by selecting as the modeling framework a much smaller box that is allowed to move with the wind in the airshed (Figure 25.2). This box extends vertically up to the mixing height, while its horizontal dimensions can be selected arbitrarily. This model simulates advection of an air parcel over the airshed, its subsequent movement and collection of emissions, the buildup of primary and secondary species, and finally its exit from the airshed. The major assumption of the Lagrangian box model is that there is no horizontal dispersion; that is, material in the box is not removed by mixing and dilution with the surrounding air. The Lagrangian box is assumed to behave like a point identically following the wind patterns. Knowing the position of the box at every moment s(t), one can calculate the corresponding emission fluxes Ei(t). For example, if such a box is located in the eastern suburbs of a city at 3 p.m. and in the downtown area at midnight, then the emissions into the box at 3 p.m. are those from the eastern suburbs at this time, and at midnight those from downtown. The mass balance for the Lagrangian box is identical to (25.13) and (25.14) with the exception that the advection terms are absent: q dci ˆ i ‡ Ri dt H…t†

vd;i ci H…t†

q dci ˆ i ‡ Ri dt H…t†

ca ci dH vd;i ci ‡ i H…t† H…t† dt

for

dH 0 dt for

(25.15) dH >0 dt

(25.16)

The initial condition for the solution of (25.15) and (25.16) is ci …0† ˆ c0i perhaps reflecting concentrations upwind of the area of interest.

(25.17)

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ATMOSPHERIC CHEMICAL TRANSPORT MODELS

While (25.13) and (25.14) and (25.15) and (25.16) appear to be similar, they are fundamentally different; these differences are demonstrated in the following example. Urban SO2 SO2 is emitted in an urban area with a flux of 2000 μg m 2 h 1. The mixing height over the area is 1000 m, the atmospheric residence time 20 h, and SO2 reacts with an average rate of 3% h 1. Rural areas around the city are characterized by a SO2 concentration equal to 2 μg m 3. What is the average SO2 concentration in the urban airshed for these conditions? Assume an SO2 dry deposition velocity of 1 cm s 1 and a cloud/fog-free atmosphere. Let us first represent this situation with a Eulerian box model enclosing the entire airshed. The concentration of SO 2 over this area cs will satisfy (25.13), or in the case of a first-order reaction dcs qs ˆ dt H

kcs

vs c0 cs cs ‡ s H τr

(25.18)

with solution cs …t† ˆ

A ‡ cs …0† B

A e B

Bt

(25.19)

where Aˆ

qs cs0 ‡ H τr

B ˆ k‡

(25.20)

vs 1 ‡ H τr

(25.21)

The term B corresponds to the various sinks of SO2, namely, chemical reaction (k = 0.03 h 1), dry removal (vs/H = 0.036 h 1), and advection out of the urban airshed (l/τr = 0.05 h 1). All sinks contribute significantly to SO2 removal and B = 0.116 h 1. The term A corresponds to the sources of SO2, that is, emissions (qs/H = 2 μg m 3 h 1) and advection of background SO2 c0s =τr ˆ 0:1 μg m 3 h 1 . The urban emissions dominate in this case and A = 2.1 μg m 3 h 1. The concentration cs(0) is the initial SO2 concentration for the airshed. Independently of our choice of this value, the predicted concentration cs(t) approaches a steady-state value for t  1/B, or t  12.5 h, equal to A/B or 18.1 μg m 3. The fundamental assumption of the Eulerian box model is that the airshed is homogeneous, so all areas even if they are close to the airshed boundary will have, according to this modeling approach, the same concentration of 18.1 μg m 3. Let us solve the same problem using a Lagrangian box advected from an area outside the urban center into the airshed and out of it. Equation (25.15) is applicable in this case and therefore dcs qs ˆ dt H

kcs

vs cs H

(25.22)

with initial condition cs …0† ˆ c0s

(25.23)

1019

BOX MODELS

The solution of (25.22) is cs …t† ˆ

D ‡ c0s E

D e E

Et

(25.24)

where Dˆ

qs H

Eˆk‡

vd H

(25.25)

Note that the SO2 advection term does not appear explicitly in the D and E terms in this moving framework. Under the conditions of the example, D = 2 μg m 3 h 1 and E  0.066 h 1. Once more, the predicted concentration eventually reaches a steady-state value equal to D/E = 30.3 μg m 3. How­ ever, cs approaches this steady-state value for t  33.3 h and the air parcel stays in the urban airshed only τr = 20 h. After this period, (25.22) is no longer valid since qs = 0. Therefore the steady state arising from (25.24) is irrelevant as it is not reached during the available residence time of the air parcel in the airshed. Equation (25.24) predicts that as the air parcel enters the urban area its SO2 concentration increases rapidly from the background value of 2 μg m 3 to 15.7 μg m 3 in the center of the airshed, and to 22.7 μg m 3 at the opposite end (Figure 25.5). The average concentration over the area is then 14.6 μg m 3.

FIGURE 25.5 Concentration as a function of travel time across an urban area predicted by a Lagrangian box model assuming length 200 km and a constant windspeed of 2.8 m s 1.

The results described above illustrate the strengths and weaknesses of the two box models. The Eulerian box model is easy to apply but oversimplifies everything by assuming a homogeneous airshed. The Lagrangian model can provide more information, for example, a spatial distribution of concentra­ tions, but by neglecting horizontal dispersion, it may predict higher concentrations downwind of emission sources.

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ATMOSPHERIC CHEMICAL TRANSPORT MODELS

25.3 THREE-DIMENSIONAL ATMOSPHERIC CHEMICAL TRANSPORT MODELS The starting point of atmospheric chemical transport models is the mass balance equation (18.1) for a chemical species i1 @ci ‡ r
…uci † ˆ Ri …c1 ; c2 ; . . . ; cn † ‡ Ei @t

Si

(25.26)

where ci(x, t) is the concentration of i as a function of location x and time t, u(x, t) is the velocity vector (ux, uy, uz), Ri is the chemical generation term for i, and Ei(x, t) and Si(x, t) are its emission and removal fluxes, respectively. This equation leads directly to the atmospheric diffusion equation (18.12) by splitting the atmospheric transport term into an advection and turbulent transport contribution2 @ci @ci @ci @ci ‡ ux ‡ uy ‡ uz @t @x @y @z ˆ

@ @ci Kxx @x @x

‡

@ @ci Kyy @y @y

‡

‡ Ri …c1 ; c2 ; . . . ; cn † ‡ Ei …x; y; z; t†

@ @ci Kzz @z @z

(25.27)

Si …x; y; z; t†

where ux(x, y, z, t), uy(x, y, z, t), and uz(x, y, z, t) are the x, y, and z components of the wind velocity and Kxx(x, y, z, t), Kyy(x, y, z, t), and Kzz(x, y, z, t) are the corresponding eddy diffusivities. The turbulent fluctuations u´ and c´i of the velocity and concentration fields relative to their average values u and ci have been approximated using the K theory (or mixing length or gradient transport theory) in (25.27) by hu´ c´ i ˆ K
rc

(25.28)

Equation (25.28) is the simplest solution to the closure problem and is currently used in the majority of chemical transport models. Higher-order closure approximations have been developed but are com­ putationally expensive.

25.3.1 Coordinate System—Uneven Terrain In our discussion so far we have implicitly assumed that the terrain is flat. Obviously, this is rarely the case. For example, severe air pollution problems often occur in regions that are at least partially bounded by mountains that restrict air movement. Earth’s surface can be characterized by its topography: Z ˆ h…x; y†

(25.29)

The upper boundary of the modeling domain can be characterized by the mixing height Hm(x, y, t) or an appropriately chosen constant height Ht. The presence of the topographic relief complicates the numerical solution of the atmospheric diffusion equation. Instead of using the actual height of a site

1

We showed in Chapter 18 that the molecular diffusion term on the RHS of (18.1) can be neglected in atmospheric flows. In Chapter 18 we used index notation, for example, @(ujci)/@xj. Here we use both vector and index notation. The three wind velocity components can be denoted (u1, u2, u3), (ux, uy, uz), or (u, v, w) in Cartesian coordinates, and each of these notations has been used at various points in the book. 2 For simplicity we drop the bracket notation on the concentration (ci), which has been used in Chapter 18 to denote the ensemble mean concentration. All concentrations in this chapter are, however, understood to represent theoretical ensemble mean values. Likewise, we drop the overbars on the wind velocity components, but they are also understood to represent mean values.

1021

THREE-DIMENSIONAL ATMOSPHERIC CHEMICAL TRANSPORT MODELS

(i.e., measured using sea level as a reference height), one usually transforms the domain into one of simpler geometry (Figure 25.6). This can be accomplished by a mapping that transforms points (x, y, z) from the physical domain to points (x, y, ζ) of the computational domain. The terrain-following coordinate transformation is commonly used, where ζˆ

z h…x; y† z h…x; y† ˆ H t h…x; y† ΔH…x; y; t†

(25.30)

that scales the vertical extent of the modeling domain into a new domain ζ that varies from 0 to 1. One may also simply subtract the surface height and define the new vertical coordinate by z´ ˆ z

h…x; y†

(25.31)

and now z´ has units of length and is not bounded between 0 and 1. The above transformations result in changes of the components of the wind field from (ux, uy, uz) to (ux, uy, w), where, for the terrain-following coordinate system (McRae et al. 1982a), we obtain wˆ

1 uz ΔH

ux

@h @ΔH ‡z @x @x

uy

@h @ΔH ‡z @y @y

z

@ΔH @t

(25.32)

Note that even if the original vertical windspeed uz is small, the new windspeed w may be significant if the terrain is rough (large derivatives @h/@x and @h/@y). For the simple coordinate system transforma­ tion of (25.31), the vertical velocity is w ´ ˆ uz

ux

@ΔH @x

uy

@ΔH @y

FIGURE 25.6 Coordinate transformation for uneven terrain: (a) two-dimensional terrain in x z space; (b) same as (a) but with contours of constant ζ superimposed; (c) same as (a) but with contours of constant z´ superimposed; (d) two-dimensional terrain in x ζ computa­ tional space (the terrain is indicated by the shaded region).

(25.33)

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ATMOSPHERIC CHEMICAL TRANSPORT MODELS

The coordinate transformations discussed above result in changes of the eddy diffusivities. Initially, the eddy diffusivity tensor K was diagonal, but the transformed form is no longer diagonal. However, the off-diagonal terms contribution to turbulent transport in most urban flows is negligible (McRae et al. 1982a). If the terrain is extremely rugged, the off-diagonal terms must be maintained, significantly complicating the overall problem. Note that even if these terms are neglected, if the terrain-following coordinate system is used, then Kζζ ˆ

Kzz ΔH 2

(25.34)

No change is necessary for the simple transformation of (25.31). The changes required for the three coordinate systems are summarized in Table 25.2. One should note that for the physical coordinate system boundary conditions must be applied at z = h(x, y), while for the terrain-following systems application is at ζ = 0 and z´ = 0 (Figure 25.6). In the following discussion we will use the simple terrain-following coordinate transformation (25.31), which effectively “flattens out” the terrain, leading to the flat modeling domain shown in Figure 25.6d.

25.3.2 Initial Conditions The solution of the full atmospheric diffusion equation requires the specification of the initial concen­ tration field of all species: ci …x; y; z; 0† ˆ c∗i …x; y; z† A typical modeling domain at the urban or global scales contains on the order of 105 computational cells, and as a result one needs to specify the concentrations of all simulated species at all these points. There are practically never sufficient measurements (especially for the upper-lever computational cells), so one needs to extrapolate from the few available data to the rest of the modeling domain. Use of an inaccurate concentration field may introduce significant errors during the early part of the simulation. The sensitivity of the model to the specified initial conditions is qualitatively similar to that of (25.19). Initially, the model prediction is equal to c∗i , the specified initial concentration, and will obviously be only as good as the specified value. As the simulation proceeds, the initial condition term decays exponen­ tially, while those due to emissions and advection increase in relative magnitude. Even if the specified initial conditions contain gross errors, their effect is eventually lost and the solution is dominated by the emissions qi and the boundary conditions c0i . It is therefore common practice to start atmospheric simulations some period of time before that which is actually of interest. At the end of this “startup” period, the model should have established concentration fields that do not seriously reflect the initial conditions, and the actual simulation and comparison with observations may commence. The startup period for any atmospheric chemical transport model is determined by the residence time τr of an air parcel in the modeling domain.

TABLE 25.2 Coordinate Systems for Solution of the Atmospheric Diffusion Equation Coordinate system

Coordinates

Physical Terrain-following

x, y, z x, y, ζ

Simple terrain-following

x, y, z´

a

Given by (25.32). Given by (25.33).

b

Definitions z h x; y ζˆ ΔH x; y; t z´ = z h(x, y)

Vertical windspeed

Vertical eddy diffusivity

uz wa

Kzz Kzz =ΔH 2

w´ b

Kzz

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THREE-DIMENSIONAL ATMOSPHERIC CHEMICAL TRANSPORT MODELS

25.3.3 Boundary Conditions The atmospheric diffusion equation in three dimensions requires horizontal boundary conditions, two for each of the x, y, and z directions. The only exception are global-scale models simulating the whole Earth’s atmosphere. One usually specifies the concentrations at the horizontal boundaries of the modeling domain as a function of time: ˆ ˆ ˆ ˆ

c…x; 0; z; t† c…x; ΔY; z; t† c…0; y; z; t† c…ΔX; y; z; t†

cx0 …x; z; t† cx1 …x; z; t† cy0 …y; z; t† cy1 …y; z; t†

(25.35)

Unfortunately, species concentration fields are practically never known at all points of the boundary of a modeling domain. Unlike initial conditions, boundary conditions, especially at the upwind boundaries, continue to affect predictions throughout the simulation. The simple box model solution given by (25.19) and (25.20) once more provides valuable insights as c0i represents the corresponding boundary conditions. After the startup period the solution is the sum of the emission term and a term proportional to the concentration at the boundary. If c0i  qi τr =H, errors in c0i will have a small effect on the model predictions, which will be dominated by emissions. Therefore one should try to place the limits of the modeling domain in relatively clean areas (low c0i ), where boundary conditions are relatively well known and have a relatively small effect on model predictions. A rule of thumb is to try to include all sources that have any effect on the air quality of a given region inside the modeling domain. If this cannot be done because of computational constraints, then the source effect has to be included implicitly in the boundary conditions. Uncertainty in urban air pollution model predictions as a result of uncertain side boundary conditions may be reduced by use of larger-scale models (e.g., regional models) to provide the boundary conditions to the urban-scale model. This technique is called nesting (the urbanscale model is “nested” inside the regional-scale model). Treatment of upper and lower boundary conditions is different from that of the side conditions. One usually chooses a total reflection condition at the upper boundary of the computational domain (e.g., the top of the planetary boundary layer): Kzz

@c @z

ˆ0

(25.36)

zˆHt

An alternative boundary condition can be formed using the vertical velocity uz at the top of the modeling region (Reynolds et al. 1973) juz jzˆHt cai

ci …z ˆ H t † ˆ Kzz Kzz

@c @z

@c @z

for

uz  0

zˆHt

ˆ0

(25.37) for uz > 0

zˆHt

where cai is the concentration above the modeling region. The two conditions in (25.37) correspond to the case where material is transported into the region from above (uz  0) and out of the region (uz > 0). In desirable cases the top of the domain is a stable layer for which vertical transport is small, and either (25.36) or (25.37) can be used with little effect on model predictions. The boundary condition used at Earth’s surface accounts for surface sources and sinks of material vd;i ci

Kzz

@ci @z

ˆ Ei zˆ0

(25.38)

1024

ATMOSPHERIC CHEMICAL TRANSPORT MODELS

where vd,i is the deposition velocity and Ei is the ground-level emission rate of the species. Note that ground-level emissions can be included either as part of the z = 0 boundary condition or directly in the differential equation as a source term in the ground-level cells.

25.4 ONE-DIMENSIONAL LAGRANGIAN MODELS Whereas the atmospheric diffusion equation (25.27) is ideally suited for predicting the concentration distribution over extended areas, in many situations the atmosphere needs to be simulated only at a particular location. A Lagrangian model that simulates air parcels that eventually reach the receptor can often be used in this case. The air parcels are vertical columns of air extending from the ground up to a height H. Assume that our goal is to simulate air quality in a given location, say, Claremont, California on a given day, for example, August 28, 1987. The first step using a Lagrangian modeling approach is to calculate, based on the wind field, the paths for the 24 air parcels that arrived at Claremont at 0:00, 1:00, . . . , and 23:00 of the specific date. These backward trajectories can be calculated if we know the three-dimensional wind field ux(x, y, z, t), uy(x, y, z, t), and uz(x, y, z, t) from ds…t† ˆ u…t† dt

(25.39)

where s(t) is the location of the air parcel at time t and u is the wind velocity vector. Integrating from t to t0, we obtain s0

s…t† ˆ

t0

∫t

u…τ†dτ

(25.40)

or

s…t† ˆ s0

t0

∫t

u…τ†dτ

(25.41)

where we have assumed that at time t0 the trajectory ends at the desired location s0. The location of the air parcel s(t) for a given moment t on this backward trajectory can be calculated by a straightforward integration according to (25.41). This integration is carried backward a couple of days until the air parcel is in a relatively clean area or in one with well-characterized atmospheric composition. In Figure 25.7 we show a calculated air parcel trajectory arriving at Claremont during August 28, 1987. The air parcel started over the Pacific Ocean, traversed the Los Angeles air basin picking up emissions from the various sources along their way, and eventually arrived at the receptor (Claremont). After the trajectory path s(t) has been calculated from (25.41), the second step is calculation of the emission fluxes corresponding to the trajectory path by interpolation of the emission field E(x, y, z, t). For example, at 22:00 on August 27, 1987 the air parcel arriving in Claremont at 14:00 of the next day is over Hawthorne and therefore will pick up emissions from that area. Thus emission fluxes along the trajectory Et(t) are given by Et …t† ˆ E…s…t†; t†

(25.42)

Let us now derive the simplified form of (25.27) corresponding to a moving coordinate system. Let x´ , y´ , and z´ be the coordinates in the new system and u´x , u´y , and let u´z be the wind velocities with respect to

1025

ONE-DIMENSIONAL LAGRANGIAN MODELS

FIGURE 25.7 Air parcel trajectory arriving at Claremont, California, at 2 p.m. on August 28, 1987 (Pandis et al. 1992). SCAQS monitoring stations refer to those established in the 1987 Southern California Air Quality Study (SCAQS).

this new moving coordinate system. The coordinate system moves horizontally with velocity equal to the windspeed and therefore u´x ˆ uy´ ˆ 0, while u´z ˆ uz . Therefore the second and third terms on the left­ hand side (LHS) of (25.27) are zero in this case. Physically, the air parcel is moving with velocity equal to the windspeed, so there is no exchange of material with its surroundings by advection. The atmospheric diffusion equation then simplifies to @ci @ @ci @ci ‡ uz ˆ Kxx @t @z @x @x

‡

@ @ci Kyy @y @y

‡ Ri …c1 ; c2 ; . . . ; Cn † ‡ Et;i …t†

‡

@ @ci Kzz @z @z

(25.43)

Si …t†

A number of additional simplifying assumptions can be invoked at this stage. The first is that vertical advective transport is generally small compared to the vertical turbulent dispersion uz

@ci @ @ci  Kzz @z @z @z

(25.44)

so that the former term can be neglected. This assumption can easily be relaxed, and the term may be retained if the vertical component of the wind field is large. According to the second assumption, the horizontal turbulent dispersion terms can be neglected: @ @ci @ @ci Kxx ≅ 0; ≅0 Kyy @x @x @y @y

(25.45)

This assumption implies that horizontal concentration gradients are relatively small so that these terms make a negligible contribution to the overall mass balance. The error introduced by this assumption is small in areas with spatially homogeneous emissions but becomes significant in areas dominated by a few strong point sources. Finally, use of a Lagrangian trajectory model implies that the column of air retains its integrity during its transport. This assumption is equivalent to neglecting the wind shear: ux …x; y; z; t† ≅ ux …x; y; t† uy …x; y; z; t† ≅ uy …x; y; t†

(25.46)

1026

ATMOSPHERIC CHEMICAL TRANSPORT MODELS

This is a critical assumption and a major source of error in some trajectory model calculations, especially those that involve long transport times (Liu and Seinfeld 1975). After employing the preceding three assumptions, the Lagrangian trajectory model equation is simplified to @ci @ @ci ˆ Kzz @t @z @z

‡ Ri …c1 ; c2 ; . . . ; cn † ‡ Et;i …t†

Si …t†

(25.47)

Our previous discussion regarding the treatment of uneven terrain and boundary conditions is directly applicable to one-dimensional (1D) atmospheric models. For example, it is often difficult to specify appropriate initial conditions for a 1D column model. In this modeling framework the characteristic species decay time is much longer than the residence time τr, because the only loss mechanism of inert chemical species is dry deposition [see (25.24) and (25.25)]. The characteristic time for a Lagrangian model can then be defined as τL ˆ H=vd;i where H is the height of the column and vd,i is the pollutant dry deposition velocity. For example, for H  1000 m and vd,i  0.3 cm s 1, τL  4 days. This order-of-magnitude calculation suggests that column models will be relatively sensitive to the specified initial conditions in that an incorrectly specified value may persist for days after initiation of the simulation. The solution to this problem is to start the trajectory simulation far upwind of the source region in a relatively clean area with relatively well-known background concentrations. One advantage of 1D column models is that there are no side boundaries, and therefore no horizontal boundary conditions are necessary. One could argue, however, that the tradeoff is increased sensitivity to the initial conditions. Equations (25.36)–(25.38) can be used as the upper and lower boundary conditions in this case also.

25.5 OTHER FORMS OF CHEMICAL TRANSPORT MODELS Our discussion in the previous sections has been based on a Cartesian coordinate system using x, y, and z as the coordinates and the molar concentration of species as the dependent variable. Other coordinate systems and concentration units used in chemical transport models are outlined below.

25.5.1 Atmospheric Diffusion Equation Expressed in Terms of Mixing Ratio Atmospheric trace gas levels are frequently expressed in terms of mixing ratios. The volume mixing ratio of a species i (ξi) is identical to its mole fraction. We have developed forms of the atmospheric diffusion equation using the concentration ci as the dependent variable. Let us take ci as the molar concentration, expressed in units of mol i m 3. Since the mass concentration mi and the molar concentration ci are related by mi = ci Mi, where Mi is the molecular weight of species i, the atmospheric diffusion equation applies equally well to the mass concentration. Recall that the volume mixing ratio of species i at any point in the atmosphere is its mole fraction ξι ˆ

ci cair

(25.48)

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OTHER FORMS OF CHEMICAL TRANSPORT MODELS

where cair is the total molar concentration of air: cair = p/RT. The atmospheric mass density of air is ρˆ

pMair RT

(25.49)

To emphasize the altitude dependence of ρ, p, and T, we write (25.49) as ρ…z† ˆ

p…z†Mair RT…z†

(25.50)

Thus we can express the molar concentration of i in terms of its volume mixing ratio at any height z as ci …z† ˆ ξi …z†

ρ…z† Mair

(25.51)

The basic question we address is: What is the proper form of the atmospheric diffusion equation when written in terms of the mixing ratio? Let us assume initially a horizontally homogeneous atmosphere. In this case, because only the vertical coordinate is affected by a changing density, we need consider only the form of equation for vertical transport. In what ensues we follow the development by Venkatram (1993). The mixing-length argument in Chapter 16 that led to the gradient transport equations applies to a variable that is conserved as an air parcel moves from one level to another. If, for some variable q, its substantial derivative satisfies Dq ˆ0 Dt

(25.52)

then the gradient transport relation in the z direction is u´z q´ ˆ K

@ q @z

(25.53)

To investigate whether ci satisfies (25.52), we begin with the conservation equation using index notation @ci @ ‡ …uj ci † ˆ 0 @t @xj

(25.54)

where uj is the instantaneous velocity in direction j. The substantial derivative @ci Dci @ci  ‡ uj Dt @t @xj

(25.55)

is then as follows, combining (25.54) and (25.55): @uj Dci ˆ ci Dt @xj

(25.56)

Previously we assumed an incompressible atmosphere, for which @uj/@xj = 0. This is a reasonable approximation for layers close to Earth’s surface and therefore for urban-scale atmospheric chemical transport models. The continuity equation for the compressible atmosphere is @ρ @ ‡ …ρuj † ˆ 0 @t @xj

(25.57)

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ATMOSPHERIC CHEMICAL TRANSPORT MODELS

Then @uj ˆ @xj

1 @ρ @ρ uj ‡ ρ @xj @t

(25.58)

which is not zero in the vertical direction. Thus Dci/Dt is not zero in the vertical direction when vertical variations of density are taken into account. To test if the mixing ratio ξi = ciMair/ρ is conserved, we write (25.54) in terms of the mixing ratio as @ ξi ρ @t Mair

‡

@ ξρ uj i @xj Mair

ˆ0

(25.59)

which gives ρ

@ξi @ρ @ξ @ ‡ ξi ‡ ρuj i ‡ ξi …uj ρ† ˆ 0 @t @t @xj @xj

(25.60)

From (25.57), this reduces to @ξi @ξ ‡ uj i ˆ 0 @t @xj

(25.61)

or Dξi ˆ0 Dt

(25.62)

Thus the mixing ratio is a conserved quantity in the sense of the mixing-length arguments. This suggests that the proper mixing-length form is u´z ξi´ ˆ Kzz

@ ξi @z

(25.63)

The continuity equation for the mean concentration of species i, with a zero mean vertical velocity, is @ hci i ˆ @t

@ ´ ´ uc @z z i

(25.64)

Expressing ci in terms of mixing ratio using (25.51), ci = ξiρ/Mair, the mean values and fluctuations of each quantity are hci i ‡ ci´ ˆ ξi ρ ‡ ξ´i ρ ‡ ξi ρ´ ‡ ξ´i ρ´ (25.65) Taking the mean of both sides of the equation, we obtain hci i ˆ ξi ρ

(25.66)

since ξ´i ρ´ , which is proportional to ρ ´ 2 =ρ 2 , can be neglected. The vertical turbulent flux term in (25.64) becomes u´z ci ˆ u´z …ξ´i ρ ‡ ξ´i ρ´ ‡ ξ´i ρ´ †

(25.67)

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OTHER FORMS OF CHEMICAL TRANSPORT MODELS

which, on averaging and neglecting u´z ξ´i ρ´ , becomes u´z c´i ˆ ρ u´z ξ´i ‡ ξi ρ´ u´z

(25.68)

Substituting (25.66) and (25.68) into (25.64), we obtain @ ξi 1 @ ˆ @t ρ @z

ρ uz´ ξ´i

@ ξi ρ´ u´z @z ρ

(25.69)

where we have used the conservation equation for air density: @ρ @ ´ ´ ‡ …ρ uz † ˆ 0 @t @z

(25.70)

On the basis of the earlier analysis showing that the mixing ratio is conserved along a vertical fluctuation, we adopt (25.63), and (25.69) becomes @ ξi @ ξi 1@ ˆ ρKzz @t ρ @z @z

@ ξi ρ´ uz´ @z ρ

(25.71)

which states that turbulent transport will not change the mean mixing ratio of a species hξii in an atmosphere in which hξii does not already vary with height. Venkatram (1993) has shown that the second term on the RHS of (25.71) involving the flux of air can generally be neglected. Therefore the form of the atmospheric diffusion equation for vertical transport when vertical variations of density are taken into account is @ ξi @ ξi 1 @ ˆ ρKzz @t ρ @z @z

(25.72)

If the atmosphere is not horizontally homogeneous, the same arguments can be used for the x and y directions to obtain 1 @ @ξ @ξi @ξ @ξ @ξ 1 @ @ξ ‡ ux i ‡ uy i ‡ uz i ˆ ρKyy i ρKxx i ‡ @y @t @x @y @z ρ @x @x ρ @y 1@ @ξi ‡ ‡ Ri Si ρKzz @z ρ @z

(25.73)

where ξi and ρ are the mean mixing ratio and air density, respectively.

25.5.2 Pressure-Based Coordinate System For modeling domains that cover the full troposphere and not only the lowest 1–2 km, it is useful to replace the height z, based coordinate system with the σ vertical coordinate system, defined as σˆ

p ps

pt pt

(25.74)

where ps is the surface pressure, pt is the pressure that defines the top of the modeling domain (usually around 0.1 atm for the troposphere), and p is the pressure at the point where σ is evaluated. At the

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ATMOSPHERIC CHEMICAL TRANSPORT MODELS

TABLE 25.3 Vertical Levels, σ Coordinates, Pressures, and Altitudes Level

Coordinate (σ)

Pressure (atm)

z (km)

0.995 0.97 0.90 0.75 0.60 0.45 0.30 0.15 0

0.995 0.972 0.905 0.763 0.620 0.478 0.355 0.193 0.050

0.04 0.22 0.75 2.0 3.6 5.5 8.2 12.4 22.5

1 2 3 4 5 6 7 8 9

surface, even if the terrain is not flat, σ = 1, while at the top of the modeling domain σ = 0. An example of a σ-coordinate system is given in Table 25.3. The σ-coordinate system is similar to the height-based ζ system defined by (25.30). The corresponding form of the diffusion equation can then be developed, noting that from (25.74) pt †dσ

dp ˆ …ps

(25.75)

and recalling that dp = ρg dz. Thus dz ˆ

p∗ dσ ρg

(25.76)

where p∗ ˆ p s

pt

(25.77)

Substituting (25.76) into (25.73), we obtain @ξi @ξ @ξ @ξ ‡ ux i ‡ uy i ‡ uσ i @t @x @y @σ

ˆ

1 @ @ξ 1 @ @ξ ρKxx i ‡ ρKyy i @x @y ρ @x ρ @y 2 g @ @ξ ρKzz i ‡ Ri Si ‡ @σ …p∗ †2 @σ

(25.78)

where uσ is the vertical velocity in the σ-coordinate system. The velocity uσ can be computed using the mass continuity equation in σ coordinates @p∗ ˆ @t

@ …ux p∗ † @x

@ …uy p∗ † @y

@ …uσ p∗ † @σ

2uz p∗ Re

(25.79)

where Re is Earth’s radius. The last term is much smaller than the other terms and can be ignored. The velocity uσ can be calculated by integrating (25.79): uσ ˆ

@ 1 σ @ …ux p∗ † ‡ …uy p∗ † dσ ∗ ∫ @y p 0 @x

σ @p∗ p∗ @t

(25.80)

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NUMERICAL SOLUTION OF CHEMICAL TRANSPORT MODELS

25.5.3 Spherical Coordinates For global atmospheric chemical transport models with a domain representing the entire atmosphere of the planet, spherical coordinate systems offer a series of advantages. The coordinates chosen are the latitude ϕ, the longitude l, and the σ-vertical coordinate described in the previous section. The coordinate-system-independent atmospheric diffusion equation is @ξi 1 ˆ u
rξi ‡ r
…ρKrξi † @t ρ

Si ‡ R i

(25.81)

This equation can be written in spherical coordinates 1 @ 1 @ 1 @ @ …p ξ † ‡ …p ul ξ † ‡ …p uϕ ξi cos ϕ† ‡ …uσ ξi † ps @t s i ps @l s i ps cos ϕ @ϕ s @σ ˆ ‡

1 @ @ξ 1 @ @ξ ps Kll i ‡ ps Kϕϕ cos ϕ i @l @ϕ ps @l ps cos ϕ @ϕ g ps

2

@ @ξ ρ2 Kzz i @σ @σ

(25.82)

Si ‡ Ri

where ps is the atmospheric surface pressure at the corresponding point; Kll, Kϕϕ, and Kzz are the components of the diffusion tensor. The components of the wind velocity are given by

1 ux Re cos ϕ 1 uϕ ˆ uy Re @p σ @ps @ps ‡ ux ‡ s uy uσ ˆ @x @y ps @t ul ˆ

(25.83) ρg uz ps

and Kll ˆ

1 Kxx R2e cos2 ϕ

Kϕϕ ˆ

1 Kyy R2e

(25.84)

25.6 NUMERICAL SOLUTION OF CHEMICAL TRANSPORT MODELS In this section we will discuss some aspects of numerical solution of chemical transport models. We do not attempt to specify which numerical method is best; rather, we point out some considerations in assessing the adequacy and appropriateness of numerical methods for chemical transport models. Our treatment is, by necessity, brief. We refer the reader to Peyret and Taylor (1983) and Oran and Boris (1987) for more study in this area.

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ATMOSPHERIC CHEMICAL TRANSPORT MODELS

Chemical transport models solve chemical species equations of the general form @ci ˆ @t

@ci @ci ‡ @t adv @t diff @ci @ci ‡ ‡ @t cloud @t

‡ dry

@ci @t

(25.85) ‡ Rgi ‡ Ei aeros

where ci is the concentration of species i; (@ci/@t)adv, (@ci/@t)diff, (@ci/@t)cloud, (@ci/@t)dry, and (@ci/@t)aeros are the rates of change of ci due to advection, diffusion, cloud processes (cloud scavenging, evaporation of cloud droplets, aqueous-phase reactions, wet deposition, etc.), dry deposition, and aerosol processes (transport between gas and aerosol phases, aerosol dynamics, etc), respectively; Rgi is the net production from gas-phase reactions; and Ei is the emission rate. Species simulated can be in the gas, aqueous, or aerosol phases. Therefore chemical transport models are characterized by the following operators: A D C G P S

Advection operator Diffusion operator Cloud operator Gas-phase chemistry operator Aerosol operator Source/sink operator

For example, the advection operator is u
▿ci. If only the advection operator is applied to all species, then they will be advected following the wind patterns. If c(x, y, z; t) is the concentration vector of all species at time t, then its value at the next timestep t + Δt will be the net result of the simultaneous application of all operators c…x; y; z; t ‡ Δt† ˆ c…x; y; z; t† ‡ ‰A…Δt† ‡ D…Δt† ‡ C…Δt† ‡ G…Δt† ‡ P…Δt† ‡ S…Δt†Šc…x; y; z; t†

(25.86)

Each of these operators is distinctly different in basic character, each usually requiring quite different numerical techniques to obtain numerical solutions. We should note that no single numerical method is uniformly best for all chemical transport models. The relative individual contributions of all the operators to the overall solution as well as other considerations, such as boundary conditions and wind fields, can easily change from application to application, leading to different numerical requirements.

25.6.1 Coupling Problem—Operator Splitting Equation (25.85), the basis of every atmospheric model, is a set of time-dependent, nonlinear, coupled partial differential equations. Several methods have been proposed for their solution, including global finite differences, operator splitting, finite element methods, spectral methods, and the method of lines (Oran and Boris 1987). Operator splitting, also called the fractional step method or timestep splitting, allows significant flexibility and is used in most atmospheric chemical transport models. 25.6.1.1 Finite Difference Methods Finite difference methods approximate and solve the full equation (25.85). To illustrate the basic idea let us focus on the one-dimensional diffusion equation @c @ 2 c ˆ @t @x2

(25.87)

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NUMERICAL SOLUTION OF CHEMICAL TRANSPORT MODELS

FIGURE 25.8 Example of a discrete approximation of the continuous concentration function c(x).

which is one of the simplest subcases of (25.85). First, a spatial grid x1, x2, . . . , xk is defined over the spatial domain of interest 0  x  L. The concentration function c(x, t) is approximated by its values c1, c2, . . . , ck at the corresponding grid points (Figure 25.8). Partial spatial derivatives of concentration can be approximated by divided difference quotients; for example, if the grid spacing is uniform and equal to Δx [e.g., xi = x1 + (i 1)Δx], then @c ci ci ˆ @x Δx

1

@ 2 c ci‡1 2ci ‡ ci ˆ @x2 …Δx†2

(25.88) 1

(25.89)

As the concentration c(x, t) changes with time, at each moment there will be a different set of values ci. Time can be discretized similarly to space by using a time interval Δt. So we are interested in the values of ci at t0, t0 + Δt, . . . Defining tn ˆ t0 ‡ nΔt

(25.90)

we would like to calculate the concentration values cni corresponding to the grid point i at the time tn. Time derivatives can be approximated similarly to space derivatives by cni @c cn‡1 ˆ i @t Δt

(25.91)

By combining (25.89) and (25.91), a finite difference approximation of (25.87) is n 2cni ‡ cni 1 cn‡1 cin ci‡1 i ˆ Δt …Δx2 †

(25.92)

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ATMOSPHERIC CHEMICAL TRANSPORT MODELS

or solving for cn‡1 , we obtain i cn‡1 ˆ cin ‡ i

cni‡1

2cni ‡ cni‡1 …Δx†2

Δt

(25.93)

Initially (time zero) the values of ci0 are known and therefore the values of ci1 (concentrations at t = Δt) can be calculated by direct application of (25.93). This approach can be repeated, and the values of cn‡1 can be calculated from the previously calculated values of cni . This is an example of an i explicit finite difference method, where, if approximate solution values are known at time tn = nΔt, then approximate values at time tn+1 = (n + 1)Δt may be explicitly and immediately calculated using (25.93). Typically, explicit techniques require that constraints be placed on the size of Δt that may be used to avoid significant numerical errors and for stable operation. In a stable method, unavoidable small errors in the solution are suppressed with time; in an unstable method a small initial error may increase significantly, leading to erroneous results or a complete failure of the method. Equation (25.93) is stable only if Δt < (Δx)2/2, and therefore one is obliged to use small integration timesteps. The derivative @ 2c/@x2 can also be approximated by using concentration values at time tn+1 or @ 2 c cn‡1 ˆ i‡1 @x2

2cn‡1 ‡ cin‡1 i 1 …Δx†2

(25.94)

leading to the following finite difference approximation of (25.87): cin cn‡1 cn‡1 i ˆ i‡1 Δt

n‡1 2cin‡1 ‡ ci‡1

…Δx†2

(25.95)

Now, even if values of cni …i ˆ 1; 2; . . . ; k† are known, (25.95) cannot be solved explicitly, as cn‡1 is also a i n‡1 function of the unknown cn‡1 and c . However, all k equations of the form (25.95) for i = 1, 2, ...,k i‡1 i 1 n‡1 n‡1 , c ; . . . c . This system can form a system of linear algebraic equations with k unknowns, namely, cn‡1 1 2 k be solved and the solution can be advanced from tn to tn+1. This is an example of an implicit finite difference method. In general, implicit techniques have better stability properties than do explicit methods. They are often unconditionally stable, and any choice of Δt and Δx may be used (the choice is ultimately based on accuracy considerations alone). Similar global implicit formalisms can be developed to treat any form of partial differential equations in an atmospheric model. Accuracy of the solutions can be tested by increasing the temporal and spatial resolution (decreasing Δx, Δy, Δz, and Δt) and repeating the calculation. However, because the full problem is solved as a whole, implicit formalisms require solution of very large systems, are slow, and require huge computational resources. Even if finite difference methods are easy to apply, they are rarely used for solution of the full (25.85) in atmospheric chemical transport models. 25.6.1.2 Finite Element Methods In using a finite element method, one typically divides the spatial domain in zones or “elements” and then requires that the approximate solutions have the form of a specified polynomial over each of the elements. Discrete equations are then derived by requiring that the error in the piecewise polynomial approximate solution (i.e., the residual when the polynomial is substituted into the basic partial differential equation) be minimum. The Galerkin approach is one of the most popular, requiring that the error be orthogonal to the piecewise polynomial space itself. Characteristically, finite element methods lead to implicit approximation equations to be solved, which are usually more complicated than analogous finite difference methods. Their computational expense is comparable to, or can exceed, global implicit methods.

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NUMERICAL SOLUTION OF CHEMICAL TRANSPORT MODELS

25.6.1.3 Operator Splitting Operator splitting is the most popular technique for the solution of (25.85). The basic idea is, instead of solving the full equation at once, to solve independently the pieces of the problem corresponding to the various processes and then couple the various changes resulting from the separate partial calculations (Yanenko 1971). Considering (25.86), each process contributes a part of the overall change in the concentration vector c. Let us define the change over the timestep Δt as Δc ˆ c…t ‡ Δt†

c…t†

(25.96)

A…Δt†c…t† D…Δt†c…t† C…Δt†c…t† G…Δt†c…t† P…Δt†c…t† S…Δt†c…t†

(25.97)

Then, decoupling the operators, we obtain ΔcA ΔcD ΔcC ΔcG ΔcP ΔcS

ˆ ˆ ˆ ˆ ˆ ˆ

where ΔcA is the change to the concentration vector because of advection, ΔcD is the change because of diffusion, and so on. The overall change for the timestep can then be found by summing all changes Δc ˆ ΔcA ‡ ΔcD ‡ ΔcC ‡ ΔcG ‡ ΔcP ‡ ΔcS

(25.98)

and the new concentration vector is c…t ‡ Δt† ˆ c…t† ‡ Δc

(25.99)

Equations (25.97)–(25.99) suggest that each process can be simulated individually and then the results can be combined. The approach of (25.97)–(25.99) can work quite well, but other alternatives exist and are often used. Instead of applying the operators in parallel to c, one can apply them in series, or c1 …t ‡ Δt† c2 …t ‡ Δt† c3 …t ‡ Δt† c4 …t ‡ Δt† c5 …t ‡ Δt† c…t ‡ Δt†

ˆ ˆ ˆ ˆ ˆ ˆ

A…Δt†c…t† D…Δt†c1 …t ‡ Δt† C…Δt†c2 …t ‡ Δt† G…Δt†c3 …t ‡ Δt† P…Δt†c4 …t ‡ Δt† S…Δt†c5 …t ‡ Δt†

(25.100)

For example, c3(t + Δt) includes changes in concentration because of advection, diffusion, and cloud processing. Operators can be combined; for example, diffusion and advection can be applied together as a transport operator T, or they may be split. One may apply first the transport operator in the x direction, Tx, then Ty, and finally the vertical transport operator Tz. Order of operator application is another issue. McRae et al. (1982) recommended using a symmetric operator splitting scheme for the solution of the atmospheric diffusion equation. They used the scheme Δt Δt Δt Ty Tz G…Δt† 2 2 2 Δt Δt Δt  Tz Ty Tx c…t† 2 2 2

c…t ‡ Δt† ˆ T x

(25.101)

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ATMOSPHERIC CHEMICAL TRANSPORT MODELS

where Tx is the horizontal transport (advection and diffusion) operator in the east–west direction, Ty is the north–south transport operator, Tz is the vertical transport operator, and G is the gas-phase chemistry operator. Note that the transport operators are applied twice for Δt/2 and the gas-phase chemistry operator once for Δt during each cycle, to advance the solution for Δt. The qualitative criterion for the validity of (25.97) or (25.100) is that the concentration values must not change too quickly over the applied timestep from any of the individual processes. The error because of the splitting becomes zero as Δt → 0, but unfortunately the maximum allowable value of Δt cannot be easily determined a priori. The allowable Δt depends strongly on the simulated processes and scenario. For example, for the application of (25.101) in an urban airshed, McRae et al. (1982a) recommended a value of Δt smaller than 10 min. To illustrate the advantages of operator splitting, consider the solution of the three-dimensional diffusion equation @c @ 2 c @ 2 c @ 2 c ˆ ‡ ‡ @t @x2 @y2 @z2

(25.102)

on a region having the shape of a cube. We divide the cube up with a uniform grid with N divisions in each direction; that is, we divide it up into N3 smaller cubes. A global implicit finite difference method would require the solution of a linear system of equations with N3 unknowns. Because the resulting matrix will have a bandwidth of about N2 at each timestep, the cost of solving the linear system using conventional banded Gaussian elimination would be proportional to (N2)2N3 = N7. Using a splitting technique for each step, one would solve discrete versions of the following three one-dimensional diffusion problems @c @2c ˆ @t @x2 @c @2c ˆ @t @y2 @c @2c ˆ @t @z2

(25.103)

or using finite differences n‡1=3

ci

Δt

n‡2=3

cin

n‡1=3

cj

cj Δt

cn‡1 k

n‡1=3

ˆ

ˆ

n‡1=3

2ci

n‡1=3 1

‡ ci

…Δx†2 n‡2=3

ˆ

n‡2=3

ck Δt

ci‡1

cj‡1

n‡2=3

2cj

Δy n‡1 ck‡1

n‡2=3 1

‡ cj

2

(25.104)

2ckn‡1 ‡ cn‡1 k 1 …Δz†2

where cn are the values at tn = n Δt and cn+1 are the values at tn+1. To accomplish the solution of (25.104) on the N × N × N grid displayed above would require N2 solutions of each of the three equations above, and the cost of each solution would be proportional to N. So the total cost to advance over one timestep using splitting would be proportional to 2N3, which is considerably less than the N7 cost for the fully implicit case. If each one-dimensional method is stable, then the overall splitting technique is usually stable. Splitting does not require exclusive use of finite difference methods; for example, finite element techniques could be used to solve the above one-dimensional problems. Operator splitting methods require less computational resources but demand more thought. In general, stability is not guaranteed. However, operator splitting encourages modular models and allows the use of the best available numerical technique for each module. Thus advection, kinetics, cloud

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NUMERICAL SOLUTION OF CHEMICAL TRANSPORT MODELS

processes, and aerosol processes each reside in different modules. Techniques used for these subpro­ blems will be outlined subsequently. For applications in other problems, the reader is referred to Oran and Boris (1987).

25.6.2 Chemical Kinetics The gas-phase chemistry operator involves solution of a system of ordinary differential equations of the form dci ˆ Pi dt

Li

(25.105)

where ci are the species concentrations and Pi and ci Li are production and loss terms, respectively. Equation (25.105) is a system of coupled nonlinear differential equations. The Pi and Li terms are functions of the concentrations ci and provide the coupling among the equations. The simplest method for the solution of (25.105) is based on the finite difference approach discussed in the previous section. Writing cni dci cn‡1 ˆ i dt Δt yields and solving for cn‡1 i cn‡1 ˆ cni ‡ Pni i

cni Lni Δt

(25.106)

where Pni and cni Lin are the production and loss terms for t = tn. This method is known as the Euler method and is explicit. Starting with the initial conditions c0i one can apply (25.106) to easily find the concentrations to integrate (25.105), finding cni for each timestep. However, integration of chemical kinetic equations poses a major difficulty as indicated by the following example. Stiff ODEs Consider the following two ordinary differential equations (ODEs) dc1 ˆ 1001 c1 ‡ 999 c2 ‡ 2 dt dc2 ˆ 999 c1 1001 c2 ‡ 2 dt

(25.107)

with initial conditions c1(0) = 3 and c2(0) = 1. Let us try to integrate this system numerically from t = 0 to t = 1 min using the Euler method. Then ˆ cn1 ‡ 2 1001 cn1 ‡ 999 cn2 Δt cn‡1 1 ˆ cn2 ‡ 2 ‡ 999 cn1 1001 cn2 Δt cn‡1 2

(25.108)

If we select a step Δt = 0.1 min, then using c10 ˆ 3 and c20 ˆ 1 and (25.108), we calculate that c11 ˆ 197 and c12 ˆ 201. If we reapply the algorithm for t = 0.2 min, then c12 ˆ 4  104 and c22 ˆ 4  104 , and both solutions approach infinity. The numerical solution of the problem has failed. Let us try once more using Δt = 0.01 min. Applying (25.108), we find that for t = 0.01 min, c11 ˆ 17, c21 ˆ 21. In the next step c21 ˆ 362 and c22 ˆ 359, and the algorithm once more fails. One may suspect that there is something

1038

ATMOSPHERIC CHEMICAL TRANSPORT MODELS

FIGURE 25.9 Numerical solution of the stiff ODE example using the explicit Euler method and Δt = 0.001. Also shown is the true solution.

peculiar going on with the solution of the problem. The system of differential equations has the solution c1 …t† ˆ 1 ‡ exp… 2t† ‡ exp… 2000t† (25.109) c2 …t† ˆ 1 ‡ exp… 2t† exp… 2000t†

FIGURE 25.10 Numerical solution of the stiff ODE example using the explicit Euler method and Δt = 0.0001. Also shown is the true solution.

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NUMERICAL SOLUTION OF CHEMICAL TRANSPORT MODELS

and each function is simply the sum of two exponential terms and a constant. The true c1(t) starts from its initial value c1(0) = 3 and, as the two exponentials decay, approaches a steady-state value of c1 = 1. The second function reaches a maximum and then decays back to 1.0. If the solution that we are seeking is so simple, then why is the Euler method having so much trouble reproducing it? Figure 25.9 and (25.109) suggest that the desired solution consists of two transients. The fast one, which could be due to a fast reaction, decays by t = 0.002 min. The slow transient affects the solution up to t = 10 min or so. As a result of the fast transient, the concentration c1 decays in less than 0.005 min by 33% from c1 = 3 to c1 = 2. Extremely small timesteps are necessary to resolve this rapid decay. If we use a timestep Δt = 0.001 min, our solution has significant errors throughout the integration interval but at least remains positive and does not explode (Figure 25.9). Once more the fast transient during the first 0.002 min is the cause of the numerical error. Finally, if Δt = 0.0001 min is used, the algorithm is able to resolve both transients and reproduce the correct solution (Figure 25.10). Small errors exist only in the first few steps. Note that application of (25.109) 10,000 times is necessary to integrate the system of ordinary differential equations (ODEs). In the example above, a term in the solution important for only 10 3 min dictated the choice of the timestep. If a step larger than 10 3 min is used, the solution is unstable. Similar problems almost always accompany the numerical solution of systems of ODEs describing a set of chemical reactions. These systems are characterized by timescales that vary over several orders of magnitude and are character­ ized as numerically stiff. For example, the differential equations (25.107) have timescales on the order of 10 3 min and 10 min. Stiffness can be defined rigorously for a system of linear ODEs that can be expressed in the vector form dc ˆ Ac (25.110) dt with c the unknown concentration vector and A an N × N matrix based on the real part of the eigenvalues of A, Re(λi). The system (25.110) is said to be stiff if the following two conditions are satisfied: 1. The real part of at least one of the eigenvalues is negative

Re…λi † < 0

2. There are significant differences between the maximum and minimum eigenvalues maxjRe…λi †j  1 minjRe…λi †j

(25.111)

dc ˆ f…c† dt

(25.112)

For a more general nonlinear system

the Jacobian matrix of the system is defined as Jˆ

@f @c

(25.113)

J ik ˆ

@f i @ck

(25.114)

and has elements

Note that if the system is linear, then J = A. For a nonlinear system the stiffness definition is applied to its Jacobian and its eigenvalues. For the previous example the corresponding Jacobian is Jˆ

1001 999

and the corresponding eigenvalues are λ1 = 2 min

999 1001 1

(25.115)

and λ2 = 2000 min 1, with a ratio of λ2/λ1 = 103.

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ATMOSPHERIC CHEMICAL TRANSPORT MODELS

McRae et al. (1982a) calculated the eigenvalues and characteristic reaction times for a typical tropospheric chemistry mechanism. The 24 eigenvalues of the corresponding Jacobian span 12 orders of magnitude (ranging from 10 5 to 107 min 1). The inverse of these eigenvalues (1/|λi|) corresponds roughly to the characteristic reaction times of the reactive species. For example, the rapidly reacting OH radical with a lifetime of 10 4 min corresponds to an eigenvalue of approximately 104 min 1. The system of the corresponding 24 differential equations is extremely stiff and its integration a formidable task. One approach commonly used in the integration of such chemical kinetics problems is the pseudo­ steady-state approximation (PSSA) (see Chapter 3). For example, instead of solving a differential equation for short-lived species like O, OH, and NO3, one calculates and solves the corresponding PSSA algebraic equations. For example, McRae et al. (1982a) estimated that nine species (O, RO, OH, RO2, NO3, RCO, HO2, HNO4, and N2O5) with characteristic lifetimes less than 0.1 min in the environment of interest could be assumed to be in pseudosteady state. This assumption resulted in the replacement of the 24-ODE system by one consisting of 15 ODEs and 9 algebraic equations. The ratio of the larger to the smaller eigenvalues and the corresponding stiffness were reduced to 106. The choice of the species in pseudosteady state is difficult, as it is an approximation and therefore introduces numerical errors. As a rule of thumb, the longer the lifetime of a species set in pseudo-steady state the larger the numerical error introduced. Even after the use of the PSSA, the remaining problem is stiff and its integration cannot be performed efficiently with an explicit Euler method. Fully implicit, stiffly stable integration techniques have been developed and are routinely used for such problems. 25.6.2.1 Backward Differentiation Methods If the rate of change of ci on the RHS of (25.105) is approximated not by its value at tn but with the value at tn+1, then cn‡1 ˆ cni ‡ Pin‡1 i

Δt cn‡1 Ln‡1 i i

(25.116)

This algorithm is the backward Euler or fully implicit Euler method. Because this formula is implicit, a system of algebraic equations must be solved to calculate cn‡1 from cni for i = 1, 2, . . . , N. This step is i expensive because the Jacobian of the ODE system needs to be inverted, a process that requires on the order of N3 operations. This inversion should in principle be repeated in each step. A number of other backward differentiation methods and numerical routines for their use can be found in Gear (1971). These algorithms have been rewritten and incorporated in the package ODEPACK (Hindmarsh 1983). 25.6.2.2 Asymptotic Methods Equation (25.105) can be rewritten in the form dci ˆ Pi dt

ci τi

(25.117)

where τi = 1/Li is the characteristic time for the approach of ci to equilibrium. Dropping the subscript i, (25.117) can be approximated numerically by

Solving for cn+1, we obtain cn‡1 ˆ

cn‡1 cn Pn‡1 ‡ Pn ˆ Δt 2 cn τn‡1 ‡ τn

Δt

cn‡1 ‡ cn τn‡1 ‡ τn

‡ 0:5τ Pn‡1 ‡ Pn τn‡1 ‡ τn …τn‡1 ‡ τn ‡ Δt†

(25.118)

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NUMERICAL SOLUTION OF CHEMICAL TRANSPORT MODELS

Equation (25.118) cannot be used directly as cn+1 is a function of the unknown τn+1 and Pn+1. However, if we assume that the solution varies slowly so that τn+1  τn and Pn+1  Pn, then cn‡1 ≅ ∗

cn …2τn

Δt† ‡ 2Δtτn Pn 2τn ‡ Δt

(25.119)

and the concentrations cn‡1 can now be calculated explicitly. To improve the accuracy of the approxi­ ∗ mation (25.119) and (25.118) are combined in a predictor–corrector algorithm. First, the predictor (25.119) and P∗n‡1 are calculated using the cn‡1 is used. Then, τn‡1 ∗ ∗ . Finally, the corrector (25.118) is applied using n+1 n+1 n‡1 n‡1 τ∗ and P∗ instead of τ and P . This asymptotic method is stable and does not require the solution of algebraic systems (no expensive Jacobian inversions) for the advancement of the solution to the next timestep. As a result it is very fast but moderately accurate. The described above asymptotic method does not necessarily conserve mass. This weakness provides a convenient check on accuracy. Deviation of the mass balance corresponds roughly to the numerical error that has been introduced by the method. Algorithms (CHEMEQ) have been developed for the asymptotic method (Young and Boris 1977) and are routinely used in atmospheric chemistry models. In these algorithms, equations are often split into stiff ones (integrated by the asymptotic method) and nonstiff ones (integrated with an inexpensive explicit Euler method). Oran and Boris (1987) and Dabdub and Seinfeld (1995) present additional methods for the integration of stiff ODEs in chemical kinetics.

25.6.3 Diffusion From a numerical perspective the diffusion operator that solves @c @ @c @ @c @ @c ˆ Kxx Kyy Kzz ‡ ‡ @t @x @x @y @y @z @z

(25.120)

is probably the simplest to deal with of those involved in the full atmospheric diffusion equation. Because of its physical nature, diffusion tends to smooth out gradients and lend overall stability to the physical processes. Most typical finite difference or finite element procedures are quite adequate for solving diffusion-type operator problems. In Section 25.6.1 we discussed finite difference schemes for the solution of the one-dimensional diffusion equation. This explicit scheme of (25.93) is stable only if Δt < (Δx)2/2. If K is not equal to unity, the corresponding stability criterion is KΔt < (Δx)2/2. Therefore, explicit schemes cannot be used efficiently because stability considerations dictate relatively small timesteps. Thus implicit methods are used for the diffusion aspect of a problem. In higher dimensions this requirement implies that splitting will almost certainly have to be used. Implicit algorithms that can be used include the globally implicit algorithm of (25.95) or the popular Crank–Nicholson algorithm that can be derived as follows. For the one-dimensional problem with constant diffusivity K, we obtain @c @2c ˆK 2 @x @t

(25.121)

If we approximate the spatial derivative by 1 @2c ˆ @x2 2Δx2

cni 1 ‡ cn‡1 i 1

2 cni ‡ cn‡1 ‡ cni‡1 ‡ cn‡1 i i‡1

(25.122)

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ATMOSPHERIC CHEMICAL TRANSPORT MODELS

and @c=@t ˆ cn‡1 i

cni =Δt, then (25.121) becomes KΔt n‡1 KΔt n‡1 KΔt n‡1 c ‡ 1‡ c c 2Δx2 i 1 Δx2 i 2Δx2 i‡1 KΔt n KΔt n KΔt n ˆ c ‡ 1 c ‡ c 2Δx2 i 1 Δx2 i 2Δx2 i‡1

(25.123)

The algebraic system of equations resulting from the Crank–Nicholson scheme in (25.123) is tridiagonal and can therefore be solved efficiently with specialized routines.

25.6.4 Advection Advection problems tend to be more difficult to solve numerically than are diffusion problems. The advection operator included in @c @c @c @c ‡ uy ‡ uz ˆ0 ‡ ux @t @x @y @z

(25.124)

does not smooth or damp out gradients or solutions but simply transports them about intact. Consider, for example, the one-dimensional advection equation with constant velocity u @c @c ‡u ˆ0 @t @x

(25.125)

and let us assume that it is used to describe the movement of a homogeneous plume with concentration c (x, t) initially given by c…x; 0† ˆ 1 c…x; 0† ˆ 0

1x2 x < 1 and 2 < x

(25.126)

This is the classic square-wave problem, in which the wave moves along the x axis with velocity u without changing shape. The center of the wave will be centered at x = ut + 1.5 as shown in Figure 25.11. Let us attempt to solve this problem numerically using finite differences. First, we need to discretize the time and space domains; let us select an equally spaced grid Δx and timesteps Δt. Then, using finite differences, both derivatives in (25.125) can be written as cni @c cn‡1 ˆ i @t Δt

@c cni cni 1 ˆ @x Δx

and (25.125) can be written, solving for cn‡1 , as i

FIGURE 25.11 Solution of the square-wave test problem for u = 1.

(25.127)

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NUMERICAL SOLUTION OF CHEMICAL TRANSPORT MODELS

cn‡1 ˆ cni ‡ i

uΔt n c Δx i 1

cni

(25.128)

The algorithm presented above uses the solution information for the grid point i 1 to estimate the solution at i and is known as the upwind, or “one-sided,” or “donor cell” algorithm. Consider the numerical solution of the square-wave problem assuming that the concentration initially satisfies (25.126) and using Δx = 0.1 m, Δt = 0.01 s, and u = 1 m s 1. The results of (25.128) after one step are shown in Figure 25.12. Note that for the 21st grid point corresponding to x = 2.1 m, we have c121 ˆ c021 ‡ 0:1…1 0† ˆ 0:1, and in the first step the concentration increases significantly. In reality, during this 0.01 s the wave advances 1 × 0.01 = 0.01 m and occupies only 10% of the corresponding grid cell. Our grid structure is not sufficiently detailed to accurately describe this wave advancement, and the concentration is artificially spread over the whole grid cell. This numerical error is known as numerical diffusion because the solution “diffuses” artificially into the next grid cell. This diffusion takes place in every timestep of the upwind scheme and the error accumulates (Figure 25.13). After 100 steps (t = 1 s) the predicted concentration peak is 10% less than the true one, and after 500 steps 50% less. Similar numerical diffusion is introduced by most advection algorithms. The upwind scheme is characterized by significant numerical diffusion and therefore has severe limitations. If the spatial derivative is expressed using the values around the point of interest @c cni‡1 cni 1 ˆ @x 2Δx

(25.129)

then after substitution into (25.125) the explicit algorithm ˆ cni ‡ cn‡1 i

uΔt n c 2Δx i 1

cni‡1

FIGURE 25.12 Solution of the square-wave problem using Δx = 0.1 m, Δt = 0.01 s, and u = 1 m s 1 with the upwind finite difference scheme. Numerical results and true solution after one timestep.

(25.130)

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ATMOSPHERIC CHEMICAL TRANSPORT MODELS

FIGURE 25.13 Solution of the square-wave problem using Δx = 0.1 m, Δt = 0.01 s, and u = 1 m s 1 with the upwind finite difference scheme. Numerical results and true solution for t = 1 s (100 steps) and t = 5 s (500 steps).

is obtained. For this algorithm information from both upwind and downwind grid points is used to advance the solution for each grid point. Solution of the same square-wave problem using Δx = 0.1 m, Δt = 0.02 s, and u = 1 m s 1 results in the terrible solution shown in Figure 25.14. Oscillations that get larger with time are evident, and errors have even propagated backward to x = 0. This explicit algorithm is unstable for all timesteps Δt and the solution obtained has little resemblance to the true solution.

FIGURE 25.14 Solution of the square-wave problem using Δx = 0.1 m, Δt = 0.02 s, and u = 1 m s 1 with the explicit finite difference scheme. Numerical results and true solution for t = 1 s (50 steps).

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NUMERICAL SOLUTION OF CHEMICAL TRANSPORT MODELS

Stability is a problem associated with most explicit advection algorithms. The explicit algorithm of (25.130) has the worst behavior. The upwind scheme is stable if uΔt 1 Δx

(25.131)

The preceding criterion is known as the Courant limit. For example, for our application of the upwind scheme discussed above, Δt < 0.1 s must be used to ensure stability of the solution. A third problem plaguing numerical solution of advection problems can be seen if the upwind scheme is modified to include the updated information in the cell i 1. Replacing cni 1 with cn‡1 i 1 in (25.128), we get ˆ cni ‡ cn‡1 i

uΔt n‡1 c Δx i 1

cni

(25.132)

This algorithm is still explicit, because, as applied sequentially for i = 1, . . . , N, the value of cn‡1 i 1 is known when we need to calculate cn‡1 . Results from the application of the algorithm to the same squarei wave problem are depicted in Figure 25.15. The numerical solution in this case not only diffuses but also moves more rapidly than the actual solution. Use of the updated information resulted in an artificial acceleration of the solution and the creation of a phase difference. All the problems encountered during application of the algorithms presented above illustrate the difficulty of solution of the advection problem in atmospheric transport models. A number of techniques have been developed to treat advection accurately, including flux-corrected transport (FCT) algorithms (Boris and Book, 1973), spectral and finite element methods [for reviews, see Oran and Boris (1987), Rood (1987), and Dabdub and Seinfeld (1994)]. Bott (1989, 1992), Prather (1986), Yamartino (1992), Park and Liggett (1991), and others have developed schemes specifically for atmospheric transport models.

FIGURE 25.15 Solution of the square-wave problem using Δx = 0.1 m, Δt = 0.02 s, and u = 1 m s 1 with the finite difference scheme of (25.132). Numerical results and true solution for t = 1 s (100 steps) and t = 5 s (500 steps).

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ATMOSPHERIC CHEMICAL TRANSPORT MODELS

25.7 MODEL EVALUATION Atmospheric chemical transport models are evaluated by comparison of their predictions against ambient measurements. Various statistical measures of the agreement or disagreement between predicted and observed values can be used. Although raw statistical analysis may not reveal the cause of the discrepancy, it can offer valuable insights about the nature of the mismatch. Three different classes of performance measures have been used for urban ozone models. These tests are heavily weighted toward the ability of the model to reproduce the peak ozone concentrations and include 1. Analysis of predictions and observations paired in space and time. This is the most straightfor­ ward and demanding test. 2. Comparison of predicted and observed maximum concentrations. 3. Comparisons paired in space but not in time. These comparisons allow timing errors of model predictions. Several numerical and graphical procedures are used routinely to quantify the performance of atmospheric models. Assume that there are n measurement stations each performing m measurements during the evaluation period. Let PREDi,j and OBSi,j be, respectively, the predicted and observed concentrations of a species for the station i during interval j. Tests that appear to be most helpful in determining the adequacy of a simulation include 1. The paired peak prediction accuracy 2. The unpaired peak prediction accuracy 3. The mean normalized bias (MNB) defined by MNB ˆ

1 nm

n

m

iˆ1 jˆ1

PREDi;j OBSi;j OBSi;j

(25.133)

4. The mean bias (MB) defined by MB ˆ

1 nm

n

m

PREDi;j

OBSi;j

(25.134)

iˆ1 jˆ1

5. The mean absolute normalized gross error (MANGE) defined by MANGE ˆ

1 nm

n

m

iˆ1 jˆ1

jPREDi;j OBSi;j j OBSi;j

(25.135)

6. The mean error (ME) defined by ME ˆ

1 nm

n

m

jPREDi;j

OBSi;j j

(25.136)

iˆ1 jˆ1

A series of diagnostic tests are used to gain insights into the operation of complex atmospheric chemical transport models. Similar tests can often reveal model flaws or diagnose model problems. These tests for urban-scale models include use of zero emissions, zero initial conditions, zero boundary conditions, zero surface deposition, and mixing-height variations. Similar tests can be performed for regional- and global-scale models.

RESPONSE OF ORGANIC AND INORGANIC AEROSOLS TO CHANGES IN EMISSION

The zero emission test is intended to quantify the sensitivity of the model to emissions. The model should produce concentrations close to background or at least representative of upwind concentra­ tions. Insensitivity to emissions raises questions about the utility of the simulated scenario for design of control strategies. The zero (or as close to zero as possible) initial and boundary condition tests reveal the effect of these often uncertain conditions on predicted concentrations. This initial condition effect should be small for the second or third day of an urban-scale simulation. The expected results of the zero deposition tests are increases of the primary pollutant concentrations downwind of their sources. However, concentrations of secondary pollutants may increase or decrease during this test because of nonlinear chemistry. The objective of the mixing height test is to quantify the sensitivity of predictions to the mixing height used in the simulations. Additional tests that are useful for atmospheric chemical transport models include overall mass balances and sensitivity analysis to model parameters and inputs.

25.8 RESPONSE OF ORGANIC AND INORGANIC AEROSOLS TO CHANGES IN EMISSION Ambient aerosols are a combination of organic and inorganic constituents. The VOCs that, on atmospheric oxidation, contribute to secondary organic aerosol arise from both anthropogenic and biogenic sources. The dominant inorganic components of atmospheric aerosols, sulfates, and nitrates, arise largely from anthropogenic sources. Reduction of ambient aerosol levels is a major abatement goal. Given the complex composition of the atmospheric aerosol, a key question is: What are the relative benefits of reducing emissions of SO2, NOx, NH3, VOC, black carbon, and primary organic aerosol in reducing PM mass? Table 25.4 summarizes the direction of responses of PM concentrations to changes in emission. The magnitudes of the effects, and even the direction of changes, will depend on the overall mix of atmospheric species, most especially whether the region is dominated by anthropogenic or biogenic emissions. For example, analysis of data in Pittsburgh, PA, a region influenced largely by anthropogenic emissions, indicates that in that area in wintertime nitrate can be reduced either with reductions in NOx or NH3, while in the summer nitrate levels are controlled by the availability of NH3, since gas-phase nitric acid is relatively plentiful from gas-phase photochemistry. Ammonia is the predominant gasphase base in the atmosphere and provides much of the neutralizing potential for acidic gases. While sulfuric acid transfers directly to the PM phase after its formation (even in the absence of ammonia), ammonium sulfate, ammonium bisulfate, and ammonium nitrate require ammonia to form from their gas-phase precursors, sulfuric acid and nitric acid. However, sulfuric acid reacts more readily with ammonia to form the ammonium salt than does nitric acid. Thus, for ammonium nitrate to form requires that sufficient ammonia is available to first neutralize the existing sulfuric acid. A consequence of this competition for ammonia is that SO2 reduction may result in less than a one-to-one reduction in PM mass since some of the sulfate not formed may be replaced by additional nitrate as more ammonia is then available for reaction with nitric acid. In regions where the dominant source of organic aerosol is oxidation of biogenic VOCs, such as the southeastern United States, the effect of anthropogenic emission reductions is particularly interesting. Emissions of anthropogenic SO2 provide a source of sulfate aerosol as well as organosulfates in SOA. Sulfate aerosol is especially hygroscopic, and the anthropogenic sulfate component will cause the ambient aerosol to grow markedly in response to increases in humidity (which is prevalent in such warm, biogenically dominated regions). The resulting increase in the volume of aerosol water owing to its presence from anthropogenic inorganic PM species has been referred to as anthropogenic water. This larger volume of aerosol water facilitates increased partitioning to the particle phase of gas-phase oxidation products of biogenic VOCs, including small water-soluble species such as glyoxal, which, once in the particle phase, react to form SOA. Owing to this so-called anthropogenic water effect, one might anticipate that reductions in anthropogenic sulfates and nitrates could lead to more than proportionate reductions in organic aerosol in such regions. Reduction of black carbon does not have a particularly

1047

1048

ATMOSPHERIC CHEMICAL TRANSPORT MODELS

TABLE 25.4 Interrelationships between Reductions in Primary Pollutant Emissions and Changes in Ambient Ozone and Particulate Matter Levels

a Arrow direction indicates increase (↑) or decrease (↓). Arrow size signifies magnitude of change. A blank entry indicates a

negligible response.

b In and downwind of some urban areas in which O3 formation is VOC-limited.

c Decrease of BC leading to an increase in solar flux.

d Due to effect of NOx on oxidant levels (OH, H2O2, O3).

e Due to effect of NH3 on cloud/fog pH.

f Decrease in sulfate may make more NH3 available for reaction with HNO3 to form NH4NO3. More important when NH4NO3

formation is NH3-limited.

g Decrease except in special cases; decrease in NOx may lead to increase in O3 with associated increase in HNO3 formation.

h Increase due to less organic nitrate formation and more OH available for reaction with NOx; decrease due to decrease in oxidant

levels.

i Related to effect of NOx on oxidant levels (OH, O3, NO3).

j Decrease of SOA, depends on fraction of SOA that is anthropogenic.

k Reduction of OC adsorbed or emitted with black carbon.

strong effect chemically on other aerosol constituents; its reduction alone has a one-to-one effect on total aerosol mass. Reduction of primary organic aerosol leads directly to reduction of OA, as well as reduction of SOA that forms as a result of atmospheric oxidation of evaporated POA.

PROBLEMS 25.1B SO2 is emitted in an urban area at a rate of 3 mg m varies diurnally according to

2

‰OHŠ ˆ ‰OHŠmax sin π…t

h 1. The hydroxyl radical concentration 6†=12

where t is the time in hours using midnight as a starting point. Assume that [OH] is zero from 0  t  6 and 18  t  24. The mixing height changes at sunrise from 300 m during the night to 1000 m during the day. Assuming that, for this area, [OH]max = 107 molecules cm 3, residence

1049

PROBLEMS

time = 24 h, background SO2 = 0.2 μg m 3, and background sulfate = 1.0 μg m 3, and using an Eulerian box model, calculate the following: a. The average and maximum SO2 and sulfate concentrations. b. The time of occurrence of the maximum concentrations. Explain the results. c. Assume that during an air pollution episode the residence time increases to 36 h. Calculate the maximum SO2 and sulfate concentration levels during the episode. 25.2B Repeat the calculations of Problem 25.1 using a Lagrangian box model. Discuss and explain the differences in predictions between the two modeling approaches. 25.3B Derive the atmospheric diffusion equations (25.82)–(25.84) in spherical coordinates. What happens when cos ϕ = 0? Assume constant surface pressure. 25.4D The following simplified reaction mechanism describes the main chemical processes taking place in an irradiated mixture of ethene/NOx at a constant temperature of 298 K and a pressure of 1 atm. (1) NO2 + hν → NO + O k1 = 0.5 (2) O + O2 + M → O3 + M k2 = 2.183 × 10 5 (3) O3 + NO → NO2 + O2 k3 = 26.6 (4) HO2 + NO → OH + NO2 k4 = 1.2 × 104 (5) OH + NO2 → HNO3 k5 = 1.6 × 104 (6) C2H4 + OH → HOCH2CH2O2 k6 = 1.3 × 104 (7) HOCH2CH2O2 + NO → NO2 + 0.72 HCHO + 0.72 CH2OH + 0.28 HOCH2CHO + 0.28 HO2 k7 = 1.1 × 104 (8) CH2OH + O2 → HCHO + OH k8 = 2.1 × 103 (9) C2H4 + O3 → HCHO + 0.4 H2COO + 0.18 CO2 + 0.42 CO + 0.12 H2 + 0.42 H2O + 0.12 HO2 k9 = 2.6 × 10 2 (10) H2COO + HO2 → HCOOH + H2O k10 = 0.034 The units of all the reaction constants are in the ppm-min unit system. The initial mixture contains C2H4, NO, NO2, and air. Note that several additional reactions like photolysis of ozone, CO reactions, and carbonyl reactions have not been included in the preceding list to avoid unnecessary complications. a. Simulate the system above without using the pseudo-steady-state assumption (PSSA) for a period of 12 h. Use the following values as initial conditions: ‰NOx Š0 ˆ 0:5 ppm;

‰NO2 Š0 =‰NOx Š0 ˆ 0:25;

‰C2 H4 Š0 ˆ 3:0 ppm

To account for the varying light intensity during the day, assume that the reaction constant k1 is k1 min 1 ˆ 0:2053t 0:02053t2 for 0  t  10 k1 ˆ 0 for 10  t

b. c. d. e.

where t is the time from the beginning of the simulation in hours. Therefore t = 0 corresponds to sunrise, t = 10 h corresponds to sunset, and the last 2 h cover the beginning of the night. Prepare a list of the species participating in the chemical processes listed above and decide which will be treated as active and constant and which can be treated by using the PSSA in a typical polluted atmosphere. Explain your choices. Use the PSSA to derive expressions for the concentrations of the steady-state species as function of the concentrations of the active and constant species. Repeat part a using the PSSA. Compare the solution with a. Discuss the computational requirements and timesteps for the two approaches. Discuss the effect of the hydrocarbon on the maximum ozone value by repeating the preceding simulation for initial ethene values of 1, 2, and 4 ppm.

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ATMOSPHERIC CHEMICAL TRANSPORT MODELS

25.5D Integrate numerically the two differential equations of the stiff ODE (ordinary differential equation) using (a) the backward differentiation method given by (25.116) and (b) the asymptotic method given by (25.118)–(25.119). Compare your numerical solutions with the true solution of the system and discuss the necessary timesteps for accurate solution of the problem. 25.6D Solve the square-wave problem discussed in Section 25.6.4 using the Lax–Wendroff scheme given by cn‡1 ˆ cni ‡ i

1 2

uΔt=Δx

cni 1

cni‡1 ‡ 12…uΔt=Δx†2 cni 1

n 2cin ‡ ci‡1

Compare your solution with the true solution and the results from other methods discussed in Section 25.6.4.

REFERENCES Boris, J. P., and Book, D. L. (1973), Flux corrected transport I: SHASTA a fluid transport algorithm that works, J. Comput. Phys. 66, 1–20. Bott, A. (1989), A positive definite advection scheme obtained by non-linear renormalization of the advective fluxes, Mon. Weather Rev. 117, 1006–1015. Bott, A. (1992), Monotone flux limitation in the area-preserving flux-norm advection algorithm, Mon. Weather Rev. 120, 2592–2602. Dabdub, D., and Seinfeld, J. H. (1994), Numerical advective schemes used in air-quality models-sequential and parallel implementation, Atmos. Environ. 28, 3369–3385. Dabdub, D., and Seinfeld, J. H. (1995), Extrapolation techniques used in the solution of stiff ODEs associated with chemical kinetics of air quality models, Atmos. Environ. 29, 403–410. Gear, C. W. (1971), Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, NJ. Hindmarsh, A. C. (1983), ODEPACK, a systematic collection of ODE solvers, in Numerical Methods for Scientific Computation, R. S. Stepleman (ed.), North Holland, New York, pp. 55–64. Liu, M. K., and Seinfeld, J. H. (1975), On the validity of grid and trajectory models of urban air pollution, Atmos. Environ. 9, 553–574. McRae, G. J., et al. (1982), Numerical solution of the atmospheric diffusion equation for chemically reacting flows, J. Comput. Phys. 45, 1–42. Oran, E. S., and Boris, J. P. (1987), Numerical Simulation of Reactive Flow, Elsevier, New York. Pai, P., and Tsang, T. H. (1993), On parallelization of time dependent three-dimensional transport equations in air pollution modeling, Atmos. Environ. 27A, 2009–2015. Pandis, S. N., et al. (1992), Secondary organic aerosol formation and transport, Atmos. Environ. 26A, 2269–2282. Park, N. S., and Liggett, J. A. (1991), Application of a Taylor least–squares finite element to three dimensional advection–diffusion equation, Int. J. Numer. Meth. Fluids 13, 769–773. Peyret, R., and Taylor, T. D. (1983), Computational Methods for Fluid Flow, Springer-Verlag, New York. Prather, M. J. (1986), Numerical advection by conservation of second-order moments, J. Geophys. Res. 91, 6671–6681. Reynolds, S. D., et al. (1973), Mathematical modeling of photochemical air pollution. I. Formulation of the model, Atmos. Environ. 7, 1033–1061. Rood, R. B. (1987), Numerical advection algorithms and their role in atmospheric chemistry and transport models, Rev. Geophys. 25, 71–100. Venkatram, A. (1993), The parameterization of the vertical dispersion of a scalar in the atmospheric boundary layer, Atmos. Environ. 27A, 1963–1966. Yamartino, R. J. (1992), Non-negative, conserved scalar transport using grid-cell-centered, spectrally constrained Blackman cubics for applications on a variable thickness mesh, Mon. Weather Rev. 120, 753–763. Yanenko, N. N. (1971), The Method of Fractional Steps, Springer-Verlag, New York. Young, T. R., and Boris, J. P. (1977), A numerical technique for solving stiff ordinary differential equations associated with chemical kinetics for reactive-flow problems, J. Phys. Chem. 81, 2424–2427.

CHAPTER

26

Statistical Models

Mathematical models based on fundamental atmospheric chemistry and physics are essential tools in tracking emissions from sources, their atmospheric transport and transformation, and their contribu­ tion to concentrations at a given location (receptor). A number of factors may limit the application of these mathematical models, including need for spatially resolved, time-dependent emission invento­ ries and meteorological fields. In some cases an alternative to the use of atmospheric chemical transport models is available. It is possible to attack the source contribution identification problem in reverse order, proceeding from concentrations at a receptor site backward to responsible emission sources. The corresponding tools, named receptor models, attempt to relate measured concentrations at a given site to their sources without reconstructing the atmospheric transport of the material. Sometimes, receptor and atmospheric chemical transport models are used synergistically. For example, receptor models can refine the input emission information used by atmospheric chemical transport models. In the first part of this chapter a number of receptor modeling approaches will be discussed. These models are used for apportionment of the contributions of each source, identification of sources and their emission composition, and determination of the spatial distribution of emission fluxes from a group of sources. In the second part, we will develop the tools needed to analyze the statistical character of air quality data.

26.1 RECEPTOR MODELING METHODS Receptor models are based on measured mass concentrations and the use of appropriate mass balances. For example, assume that the total concentration of particulate iron measured at a site can be considered to be the sum of contributions from a number of independent sources Fetotal ˆ Fesoil ‡ Feauto ‡ Fecoal ‡ ? ? ?

(26.1)

where Fetotal is the measured iron concentration, Fesoil and Feauto are the concentrations contributed by soil emissions and automobiles, and so on. Let us start from a rather simple scenario illustrating the major concepts used in receptor modeling.

Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, Third Edition. John H. Seinfeld and Spyros N. Pandis.  2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

1051

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STATISTICAL MODELS

Source Apportionment Assume that for a rural site the measured PM10 concentration is 32 μg m 3 containing 2.58 μg m 3 Si and 3.84 μg m 3 Fe. The two major sources contributing to the location’s particulate concentration are a coal-fired power plant and soil-related dust. Analysis of the emissions of these sources indicates that the soil contains 200 mg(Si) g 1 (20% of the total soil emissions) and 32 mg(Fe) g 1 (3.2% of the total emissions), while the particles emitted by the power plant contain 10 mg(Si) g 1 (1%) and 150 mg(Fe) g 1 (15%). Neglecting Si and Fe contributions from other sources: Sitotal ˆ Sisoil ‡ Sipower

(26.2)

Fetotal ˆ Fesoil ‡ Fepower

(26.3)

If S and P are the total aerosol contributions (in μg m3) from soil and the power plant to the PM10 concentration in the receptor, then PM10 ˆ S ‡ P ‡ E

(26.4)

where E is the contribution from any additional sources. If the composition of the particles does not change during their transport from the sources to the receptor, then, using the initial composition of the emissions, we obtain Sisoil Fesoil Sipower Fepower

ˆ ˆ ˆ ˆ

0:2S 0:032S 0:01P 0:15P

(26.5)

Substituting (26.5) into (26.2) and (26.3) yields Sitotal ˆ 0:2S ‡ 0:01P Fetotal ˆ 0:032S ‡ 0:15P

(26.6)

The preceding leads to an algebraic system of two equations with two unknowns, the contributions of the two sources, S and P, to the receptor aerosol concentration. The solution of the system using the measured Si and Fe concentrations is S = 12 μg m 3 and P = 18 μg m 3. Using (26.4), we also find that E = 2 μg m 3, and therefore the power plant is contributing 56.2%, the dust 37.5%, and the unknown sources 6.3% to the PM10 of the specific location. Recall that we have implicitly assumed that the unknown sources contribute negligible Si or Fe to the levels measured at the location. This example describes a simple scenario but demonstrates the utility of receptor modeling. One can calculate the contribution of several sources to the atmospheric concentrations at a given location with knowledge of only source and receptor compositions. No information regarding meteorology, topog­ raphy, location, and magnitude of sources is necessary. However, both the measurements and the source compositions may be somewhat uncertain, and the number of sources influencing the receptor and their characteristics may not be known exactly. On the other hand, there may be multiple measurements at different time periods or receptors available allowing the use of more complex receptor models. A general mathematical framework can be developed for solution of problems similar to that in the example above. Suppose that there are m sources affecting a receptor in which the concentrations of n species are measured. If fij is the fraction of chemical species i in the emissions from source j, then the composition of sources can be described by a matrix F with dimensions n x m. For the sources and species of the previous example, we have Fˆ

fSi;soil fFe;soil

fSi;power fFe;power

ˆ

0:2 0:01 0:032 0:15

1053

RECEPTOR MODELING METHODS

Let ci be the concentration (in μg m 3) of element i (i = 1, 2, . . ., n) measured at the receptor and gj the total contribution (in μg m 3) of the emissions from source j to the total particulate concentration at the receptor site. Assuming that the source signature fij is not modified by processes (reactions, removal, etc.) occurring during atmospheric transport between sources and receptor, we can express the concentration of element i at the site of interest as follows: m

ci ˆ

f ij gj

i ˆ 1; 2; . . . ; n

(26.7)

jˆ1

The corresponding equations for the example are cFe ˆ f Fe;soil gsoil ‡ f Fe;power gpower cSi ˆ f Si;soil gsoil ‡ f Si;power gpower

(26.8)

Thus the concentration of each chemical element at a receptor site becomes a linear combination of the contributions of each source to the particulate matter at that site. If there are s ambient aerosol samples, then the source contributions will in general be different from sampling period to sampling period depending on wind direction, emission strength, and so on. If cik is the concentration (in μg m 3) of element i in the sample k, and gjk the contribution (in μg m 3) of source j to the particulate matter of during the kth sampling period, then Equation (26.7) can be written in a more general form as m

cik ˆ

f ij gjk

i ˆ 1; 2; . . . ; n

and

k ˆ 1; 2; . . . s

(26.9)

jˆ1

If no errors exist in the measurements cik and the source characteristics fij, then the mass balances expressed by Equation (26.9) would be satisfied exactly. However, in reality these errors are always nonzero. If the corresponding random errors are denoted by eik, then the source–receptor equation becomes m

cik ˆ

f ij gjk ‡ eik

i ˆ 1; 2; . . . ; n

and

k ˆ 1; 2; . . . s

(26.10)

jˆ1

or in a more compact vector form C ˆ FG‡E

(26.11)

In this formulation, C is the n × s matrix of measurements, F is the n × m matrix of source fingerprints, G is the m × s matrix of source contributions to the measurements for each sampling period, and E is the n × s error matrix. In all source–receptor models C is known and we are trying to determine the source contributions G. This is also known as a bilinear model. A number of approaches based on (26.10) and (26.11) have been developed to quantify the source–receptor relationships for nonreactive species in an airshed. These methods can be divided in two groups: (1) chemical mass balance (CMB) and other methods, in which the source fingerprints F are known; and (2) factor analysis methods, in which the characteristics of the sources are not known and F needs to be found together with G. The CMB is therefore used strictly for source apportionment while the factor analysis approaches are used for both source identification and source apportionment. There are additional approaches, including the empirical orthogonal function (EOF) method, which can also be used for identification of the location and emission strengths of sources. These methods are discussed in the following sections.

1054

STATISTICAL MODELS

26.2 CHEMICAL MASS BALANCE (CMB) The chemical mass balance (CMB) model combines the chemical and physical characteristics of particles or gases measured at the sources and the receptors to quantify the source contributions to the receptor (Winchester and Nifong 1971; Miller et al. 1972). CMB is a method for the solution of the set of equations (26.10) to determine the unknown gjk. The source profiles fij, that is, the fractional amount of the species in the emissions from each source type, and the receptor concentrations cik, with appropriate uncertainty estimates, serve as input to the CMB model. The analysis can be done for each sample separately, so we can use Equation (26.7) without loss of generality. The first assumption of CMB is that all sources contributing to the measured concentrations ci in the receptor have been identified. Each measured concentration ci can then be expressed as the sum of the true value ~ci , and a random error ei: (26.12) ci ˆ ~ci ‡ ei It is also assumed that the measurement errors ei are random, uncorrelated, and normally distributed about a mean value of zero. These errors can be characterized statistically by the standard deviation σi of their normal distributions. For an initial guess of source contributions gj, the predicted concentrations pi for all elements are given by m

pi ˆ

f ij gj

(26.13)

jˆ1

We would like to minimize the “distance” between the measurements ci and the predictions pi. This distance can be expressed by the sum n

…ci

pi †2

iˆ1

Because of the measurement uncertainty, no choice of gj values will result in perfect agreement between predictions and observations. The measurement uncertainty depends on the element i. To account for these different degrees of uncertainties, 1=σ2i are used as weighting factors. Summarizing, one needs to minimize n 1 2 ξ2 ˆ c i pi (26.14) 2 σ i iˆ1 by choosing appropriate values of the contributions gj. Note that by using the weighting factors 1=σ2i , elements with large uncertainties contribute less to the ξ2 function compared to elements with smaller uncertainties. Combining (26.13) and (26.14), we obtain n

ξ2 ˆ iˆ1

1 σ2i

2

m

ci

f ij gj

(26.15)

jˆ1

where n is the number of species and m is the number of sources. The solution approach is to minimize the value of ξ2 with respect to each of the m coefficients gj, yielding a set of m simultaneous equations with m unknowns (g1, g2, . . ., gm). This is the common multiple regression analysis problem. It can be shown (Watson 1979; Hopke 1985) that the solution is the vector g of source contributions given by g ˆ ‰FT W FŠ 1 FT W c

(26.16)

1055

CHEMICAL MASS BALANCE (CMB)

where F is the n × m source matrix with the source compositions fij, W is the n × n diagonal matrix with elements of the weighting factors, wii ˆ 1=σ2i , FT is the m × n transpose1 of A, c is the vector with the measurements of the n elements, and g is the vector with the m source contributions. Note that [FT W F] is an m × m square matrix so it can be inverted. The solution of the receptor problem using (26.16) considers uncertainties in the measurements ci but neglects the inherent uncertainty in the source contributions fij. Let us denote by σf ij the standard deviation of a determination of the fraction fij of element i in the emissions of source j. The solution can then be calculated by an expression analogous to (26.16) (Watson 1979; Hopke 1985, 1991): g ˆ ‰FT V FŠ 1 FT V c

(26.17)

Here V is the diagonal matrix with elements m

vii 1 ˆ σ2i ‡ jˆ1

σ2f ij g2j

(26.18)

The unknown source contributions gj are included in the elements of the V matrix, and therefore an iterative solution of (26.17) is necessary. The first step is to assume that σf ij ˆ 0, solve (26.17) directly, and calculate the first approximation of gj. Then vii can be calculated from (26.18), and a second approxima­ tion is found. If this approach converges, the solution is found. This approach using (26.17) and (26.18) is known as the effective variance method. Summarizing, the following major assumptions are employed by the CMB model: 1. 2. 3. 4. 5. 6.

Compositions of source emissions are constant. Species included are not reactive. All sources contributing significantly to the receptor have been included in the calculations. There is no relationship among the source uncertainties. The number of sources is less than or equal to the number of species. Measurement uncertainties are random, uncorrelated, and normally distributed.

These assumptions are fairly restrictive and may be difficult to satisfy for many CMB applications. When they are not satisfied, the CMB predictions may be unrealistic (e.g., negative contributions) or may include significant uncertainties. The application of CMB to a given area poses a number of challenges in addition to the assumptions of the method. Let us assume that a particulate sample has been collected in the area of interest and its elemental composition has been determined. The first issue is which sources should be included in the model. If an emission inventory exists for the region, it can be used to determine the major sources. The second issue is which source profiles should be used. Profiles used by studies in other areas may be applicable only to that specific area. For example, the emission fingerprint of a power plant in Ohio may not be representative of a power plant in Texas. Local sources of road and soil dust are usually different from location to location. To complicate things even further, emission profiles often change with time. For example, motor vehicle emission composition has changed dramatically since the 1970s with the introduction of new fuels (unleaded gasoline), new engines, and control technologies. Uncertainties or errors in the CMB results can be reduced noticeably by obtaining source profile measurements that correspond to the period of the ambient measurements (Glover et al. 1991). It is clearly essential for the CMB application to know the area that is to be modeled (Hopke 1985).

The transpose of an n × m matrix F denoted by FT is simply the m × n matrix obtained by interchanging all the rows and columns.

1

1056

STATISTICAL MODELS

When multiple samples are available, CMB should be applied to each sample separately, and then the results can be averaged (Hopke 1985). This time-consuming approach is more accurate than the CMB application to the averaged measurements. Information is generally lost during averaging of sample composition data and cannot be recovered later by CMB. CMB Application to Central California PM Chow et al. (1992) apportioned source contributions to aerosol concentrations in the San Joaquin Valley of California. The source profiles used for CMB application are shown in Table 26.1. The standard deviations σf ij of the profiles (three or more samples were taken) are also included. To account for secondary aerosol components in the CMB calculations, ammonium sulfate, ammonium nitrate, sodium nitrate, and organic carbon were expressed as secondary source profiles using the stoichiometry of each compound. The average elemental concentrations observed at one of the receptors—Fresno, California, in 1988–1989—are shown in Table 26.2. The ambient concentrations of some species (e.g., Ga, As, Y, Mo, Ag) included in the source profiles were below the detection limits. These species cannot be used for source apportionment in this specific case. Results of the source apportionment using the CMB method (using CMB for each sample and then averaging the results) are shown in Table 26.3. The major contributors to the annual average PM10 concentrations that exceeded 50 μg m 3 were primary geologic material and ammonium nitrate. For the PM2.5, secondary NH4NO3 and (NH4)2SO4, together with primary motor vehicle emissions and vegetative burning, were the major contributors.

TABLE 26.1 Source Profiles (Percent of Mass Emitted) for Central California Chemical species

Paved road dust

NO3 SO24 NH‡4 Na+ EC OC Al Si P S Cl K Ca Ti V Cr Mn Fe Co Ni Cu Zn Ga As Se Br Sr Y

0 ± 0.47 0.547 ± 1.17 0 ± 0.008 0.181 ± 0.055 2.69 ± 1.44 19.5 ± 4.67 9.34 ± 1.11 23.2 ± 2.62 0.304 ± 0.05 0.520 ± 0.17 0.163 ± 0.031 1.95 ± 0.28 2.98 ± 0.43 0.499 ± 0.067 0.0311 ± 0.008 0.0299 ± 0.003 0.106 ± 0.016 5.41 ± 0.88 0.0059 ± 0.076 0.0111 ± 0.001 0.02 ± 0.002 0.172 ± 0.026 0.0003 ± 0.006 0.0014 ± 0.042 0.0001 ± 0.002 0.0095 ± 0.001 0.0794 ± 0.006 0.0025 ± 0.004

Vegetative burning 0.462 ± 0.123 1.423 ± 0.423 0.0852 ± 0.057 0.143 ± 0.052 15.89 ± 5.80 44.60 ± 7.94 0.0019 ± 0.027 0 ± 0.015 0 ± 0.022 0.521 ± 0.176 1.908 ± 0.64 3.993 ± 1.24 0.0659 ± 0.056 0.0009 ± 0.016 0.0005 ± 0.007 0 ± 0.0016 0.0007 ± 0.001 0.0006 ± 0.001 0.0001 ± 0.001 0.0001 ± 0.001 0.0001 ± 0.001 0.0866 ± 0.036 0 ± 0.0021 0.0002 ± 0.002 0.0004 ± 0.001 0.0096 ± 0.002 0.0007 ± 0.001 0.0001 ± 0.001

Primary crude oil

Motor vehicles

0 ± 0.002 20.32 ± 4.24 0.0076 ± 0.005 0.762 ± 0.399 0 ± 0.072 0.0894 ± 0.118 0 ± 0.009 0.011 ± 0.016 0 ± 0.17 5.45 ± 0.39 0.024 ± 0.021 0.044 ± 0.054 0.062 ± 0.005 0.012 ± 0.002 0.823 ± 0.058 0.007 ± 0.025 0.0056 ± 0.001 0.2134 ± 0.022 0.0185 ± 0.002 0.789 ± 0.093 0.0009 ± 0.003 0.260 ± 0.034 0.0132 ± 0.002 0.0006 ± 0.001 0.0114 ± 0.002 0.0003 ± 0.0002 0.0015 ± 0.0003 0.0008 ± 0.0003

0 ± 0.001 3.11 ± 3.55 0 ± 0.001 0 ± 0.001 54.15 ± 19.78 49.81 ± 24.15 0.077 ± 0.051 0.957 ± 1.39 0.057 ± 0.02 1.037 ± 1.182 0.029 ± 0.02 0.008 ± 0.008 0.072 ± 0.079 0.001 ± 0.003 0.001 ± 0.002 0 ± 0.002 0.028 ± 0.024 0.001 ± 0.005 0 ± 0.001 0 ± 0.002 0.005 ± 0.003 0.053 ± 0.028 0.002 ± 0.002 0.004 ± 0.012 0 ± 0.002 0.264 ± 0.152 0 ± 0.003 0 ± 0.004

Limestone 0 ± 0.001 3.06 ± 0.3 0 ± 0.001 0 ± 0.001 0 ± 0.001 0 ± 0.001 2.11 ± 0.21 6.5 ± 0.65 0 ± 0.001 1.02 ± 0.1 0.46 ± 0.05 0.16 ± 0.04 29.52 ± 2.95 0.08 ± 0.04 0 ± 0.1 0 ± 0.01 0.05 ± 0.03 1.04 ± 0.1 0 ± 0.001 0 ± 0.1 0.02 ± 0.01 0.1 ± 0.01 0 ± 0.001 0 ± 0.001 0 ± 0.001 0.03 ± 0.01 0 ± 0.001 0 ± 0.001

1057

CHEMICAL MASS BALANCE (CMB)

TABLE 26.1 (Continued ) Chemical species

Paved road dust

Zr Mo Ag Cd In Sn Sb Ba La Hg Pb

0.0091 ± 0.002 0.0004 ± 0.006 0 ± 0.016 0.0015 ± 0.017 0.0030 ± 0.02 0.0037 ± 0.027 0.0054 ± 0.03 0.064 ± 0.103 0.0142 ± 0.117 0.0015 ± 0.008 0.265 ± 0.032

Chemical species NO3 SO24 NH‡4 Na+ EC OC S Cl K Ca Br

Vegetative burning

Primary crude oil

Motor vehicles

Limestone

0 ± 0.0019 0 ± 0.0033 0.0003 ± 0.007 0.0007 ± 0.008 0.0001 ± 0.009 0 ± 0.012 0.0022 ± 0.014 0.0095 ± 0.05 0.0016 ± 0.056 0 ± 0.0037 0.004 ± 0.003

0.0006 ± 0.0004 0.0168 ± 0.002 0.0002 ± 0.002 0.0006 ± 0.002 0.0009 ± 0.002 0.0007 ± 0.003 0.0006 ± 0.003 0.0013 ± 0.011 0.0041 ± 0.013 0 ± 0.0009 0 ± 0.0013

0 ± 0.019 0 ± 0.012 0 ± 0.016 0 ± 0.02 0 ± 0.026 0 ± 0.031 0 ± 0.069 0 ± 0.129 0 ± 0.236 0 ± 0.002 0.373 ± 0.207

0 ± 0.001 0 ± 0.001 0 ± 0.001 0 ± 0.001 0 ± 0.001 0 ± 0.001 0 ± 0.001 0 ± 0.001 0 ± 0.001 0 ± 0.001 0.27 ± 0.03

Marine

(NH4)2SO4

NH4NO3

Secondary OC

NaNO3

0 ± 0.001 10.0 ± 4.0 0 ± 0.001 40.0 ± 4.0 0 ± 0.001 0 ± 0.001 3.3 ± 1.3 40.0 ± 10.0 1.4 ± 0.2 1.4 ± 0.2 0.2 ± 0.05

0 72.7 27.3 0 0 0 0 0 0 0 0

77.5 0 22.5 0 0 0 0 0 0 0 0

0 0 0 0 0 100 0 0 0 0 0

72.9 0 0 27.1 0 0 0 0 0 0 0

Source: Chow et al. (1992).

TABLE 26.2 Annual (1988–1989) Aerosol Composition in Fresno, California Chemical species NO3 SO24 NH‡4 EC OC Al Si P S Cl K Ca Ti V Cr Mn Fe Ni Cu

PM2.5(μg m 3) 9.43 ± 11.43 2.75 ± 1.32 4.04 ± 3.89 6.27 ± 5.68 8.05 ± 5.31 0.15 ± 0.18 0.38 ± 0.46 0.013 ± 0.012 1.12 ± 0.54 0.17 ± 0.22 0.28 ± 0.18 0.072 ± 0.068 0.016 ± 0.015 0.0034 ± 0.0019 0.0016 ± 0.002 0.0073 ± 0.0058 0.17 ± 0.16 0.0023 ± 0.0024 0.069 ± 0.064

PM10(μg m 3) 10.26 ± 10.52 3.20 ± 1.51 4.06 ± 3.85 6.73 ± 5.68 12.89 ± 7.66 2.94 ± 2.63 7.49 ± 6.31 0.072 ± 0.067 1.27 ± 0.76 0.34 ± 0.37 0.85 ± 0.53 0.85 ± 0.64 0.14 ± 0.12 0.01 ± 0.008 0.0081 ± 0.0061 0.035 ± 0.029 1.48 ± 1.22 0.0051 ± 0.0029 0.0077 ± 0.095 (continued )

1058

STATISTICAL MODELS

TABLE 26.2 (Continued ) Chemical species Zn Se Br Sr Zr Ba Pb

PM2.5(μg m 3) 0.069 ± 0.052 0.0016 ± 0.003 0.017 ± 0.011 0.0007 ± 0.0006 0.0011 ± 0.0028 0.013 ± 0.014 0.051 ± 0.034

PM10(μg m 3) 0.087 ± 0.066 0.0019 ± 0.004 0.017 ± 0.008 0.0043 ± 0.0035 0.0031 ± 0.0029 0.044 ± 0.034 0.067 ± 0.034

Source: Chow et al. (1993).

TABLE 26.3 Estimated Annual Average Source Contributions (μg m − 3) to PM10 and PM2.5 in Fresno, California, Based on CMB Source

PM10

PM2.5

Geologic Motor vehicles Vegetative burning Primary crude oil (NH4)2SO4—secondary NH4NO3—secondary Seasalt (NaCl–NaNO3) OC—secondary

31.78 6.80 5.10 0.29 3.58 10.39 0.96 0.07

2.26 9.24 5.92 0.25 3.48 12.35 0.45 0.36

Calculated mass

58.97

34.34

Measured mass

71.49

49.30

Source: Chow et al. (1992).

26.2.1 CMB Evaluation A method often used to evaluate the CMB method is use of only selected measurement elements for estimation of source contributions and then use of the remaining measurement elements and predictions as a test of the analysis. For example, Kowalczyk et al. (1982) used the CMB and nine elements (Na, V, Pb, Zn, Ca, Al, Fe, Mn, As) to calculate contributions of seven sources to the Washington, DC, aerosol. Each of the selected elements was characteristic of a source: Na for seasalt, V for fuel oil, Pb for motor vehicles, Zn for refuse incineration, Ca for limestone, Al and Fe for coal and soil, and As for coal. The authors used 130 samples from a network of 10 stations. Cr, Ni, Cu, and Se were significantly underestimated, but the concentrations of the remaining elements were successfully reproduced by CMB. Kowalczyk et al. (1982) repeated the exercise using 9–30 marker elements and found little difference in the results as the key elements (Pb, Na, and V) were included. They also observed that including some elements (namely, Br and Ba) as markers gave erroneous results for several other elements. The absolute accuracy of the CMB cannot be tested easily, because the true results are unknown. However, artificial datasets can be created by assuming a realistic distribution of sources, source strengths, and meteorology, simulating the scenario with a deterministic transport model (see Chapter 25), and using CMB to apportion the source contributions to the modeled concentrations. Gerlach et al. (1983) reported the results of such a test using a typical city plan and 13 known sources. The results of the CMB application indicated that the contributions of nine of the sources were accurately predicted (errors < 20%) while errors as much as a factor of 4 were found for the remaining four sources. The contributions of the six most important sources were accurately predicted by CMB, and the errors were associated with sources of secondary importance.

FACTOR ANALYSIS

26.2.2 CMB Resolution Another issue that may complicate the application of the CMB on ambient datasets is existence of two sources with similar fingerprints or, more generally, a source whose profile is a linear combination of other source profiles. This is called the collinearity problem. If this is the case, then the matrix [FT WF] used in (26.16) has two columns that are almost similar, or a linear combination of several others. This matrix from a mathematical point of view is close to singular, and the result of its inversion is extremely sensitive to small errors. Then, the results of CMB can produce large positive and negative source contributions. The simplest solution to this problem is identification of the “offending” sources and elimination of one of them. Physically, because the sources are too similar, it is difficult for CMB to quantify the contribution of each. Thus there are limits to how far source contributions can be resolved even with almost perfect information; only significantly different sources can be treated by CMB. Similar sources have to be combined into a lumped source. Henry (1983) and Hopke (1985) have proposed algorithms that can be used for the a priori identification of estimable sources and the estimable source combinations that can be determined for a given source matrix. We should note, once more, that during the derivation of (26.16) and (26.17) we have assumed that atmospheric transformations of all elements used in the CMB are negligible. Therefore these equations should not be applied to species that are produced or consumed (e.g., sulfate) during transport from source to receptor. Gravitational settling is often assumed not to modify fij (the elemental fractions of the source emissions) to a first approximation, even if it changes the net concentrations of these elements. This assumption is equivalent to assuming that all elements have the same size distribution. Application of (26.16) to gaseous pollutants that react in the atmosphere is seldom appropriate.

26.2.3 CMB Codes The US Environmental Protection Agency (USEPA) maintains and provides a user-friendly version of CMB (Coulter et al. 2004) currently in version 8.2 (EPA-CMB8.2). The code is available through the EPA website (www.epa.gov/ttn/scram/receptor_cmb.htm).

26.3 FACTOR ANALYSIS If the nature of the major sources influencing a particular receptor is unknown, statistical factor analysis methods can be combined with ambient measurements to estimate the source composition. For a particular location, assuming that several ambient particulate samples are collected and analyzed for several elements, the resulting data will probably include information about the fingerprints of the sources affecting the location. Factor analysis models are mathematically complex, and their results are often subtle to interpret. Let us consider first a simple example given by Hopke (1985). A specific location, unbeknownst to us, is heavily influenced by two sources—automobiles and a coal-fired power plant. All samples are analyzed for aluminum (Al), lead (Pb), and bromine (Br). We assume that Al is emitted only by the power plant, while lead and bromine (mass ratio 3:1) are emitted solely by automobiles. Our samples, depending on the prevailing wind direction, traffic intensity, and so on, will have different Al, Pb, and Br concentrations. A three-dimensional plot of these concentrations (Figure 26.1) reveals little information about the underlying sources. However, all the data points are actually located on the same plane defined by the Al axis and the line Br = 1/3 Pb (Figure 26.2). If the z-Al plane is rotated, all the measurements can be plotted on a two-dimensional graph, with axes defined by Al and z (Figure 26.3). Note that the three-dimensional dataset has collapsed to a two-dimensional dataset and the two axes correspond to the composition of the sources. The Al and Br = 1/3 Pb axes are the principal factors influencing the aerosol concentration at the receptor. A significant issue in the use of factor analysis to quantify atmospheric source–receptor relationships is that in general the problem has multiple solutions. In our simple example the measurements can be reproduced by the combination of the two source profiles [Al=1, Pb=0, Br=0] and [Al=0, Pb=3, Br=1] as

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FIGURE 26.1 Measured aerosol composition of three elements in seven samples for a site influenced by automobiles (emitting Pb and Br) and a coal-fired power plant (emitting Al).

FIGURE 26.2 Plane passing through the data points of Figure 26.1. The z axis is defined by Al = 0 and Br = 1/3 Pb.

FIGURE 26.3 Two-dimensional depiction of the data shown in Figures 26.1 and 26.2.

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FACTOR ANALYSIS

shown in Figures 26.2 and 26.3. However, they can also be reproduced by other source profiles shown in Figure 26.3. For example, all data points can be expressed as linear combinations of the profiles z1 and w1. Actually any profile lying between z and z2 can be combined with any profile lying between Al and w2 in Figure 26.3 to accurately explain all observations. Henry (1987) showed that this is true for more complex problems and that in general the factor analysis receptor models are ill posed and need additional constraints to produce unique solutions. If there are more than three aerosol species, then we need to work with higher than threedimensional spaces, and locating hyperplanes passing through (or close to) all the data points becomes a complicated exercise. Principal-component analysis (PCA) and positive matrix factorization (PMF) are two of the factor analysis approaches that are commonly used in air quality source–receptor analysis.

26.3.1 Principal-Component Analysis (PCA) Principal-component analysis (PCA) is one of the factor analysis methods used to unravel the hidden source information from a rich ambient measurement dataset. It is used in all forms of data analysis to reduce a complex dataset to a lower dimension and is available in most software packages for statistical analysis. The PCA algorithms usually perform an eigenvector analysis of a correlation matrix (Hopke 1985), but PCA can also be performed by a singular value decomposition of the measurement matrix C (Abdi and Williams 2010). We will follow the first path in this section. Additional details and the corresponding mathematical derivations can be found in the references above and in Shlens (2014). Let us assume that the collected dataset includes k samples of n aerosol species. These species measurements are then analyzed for the calculation of the correlation coefficients. If Al and Pb were two of the species measured, one would have available the values of (cAl,1, cAl,2, . . ., cAl,k), and (cPb,1, cPb,2, . . ., cPb,k), where cAl,i and cPb,i are the Al and Pb concentrations in the ith sample. The mean Al and Pb values will be cAl ˆ

1 k

k

cPb ˆ

cAl;i iˆ1

k

1 k

cPb;i

(26.19)

iˆ1

The correlation coefficient around the mean between lead and aluminum rPb,Al is then defined as k

rPb;Al ˆ

cAl;i

iˆ1 k

cAl;i

iˆ1

cAl

cPb

cAl cPb;i

2 1=2

k iˆ1

cPb;i

cPb

2 1=2

(26.20)

and is a measure of the interrelationship between Pb and Al concentrations. If σAl and σPb are the standard deviations of the corresponding samples, the correlation coefficient is given by rPb;Al ˆ

k

1 k

1

cAl;i

iˆ1

cAl cPb;i σAl σPb

cPb

(26.21)

If the two are completely unrelated, rPb,Al = 0. If the two variables are strongly related to each other (positively or negatively), they have a high correlation coefficient (positive or negative). It is important to stress at this point that high correlation coefficients do not necessarily imply a cause-and-effect relationship. Variables can be related to each other indirectly through a common cause. For example, let us assume that for a given receptor, Pb and Al concentrations are highly correlated; high Al values are always accompanied by high Pb values and vice versa. One would be tempted to conclude that they have a common source. However, the same correlation can also be a result of the fact that the lead source (a major traffic artery) is next to the Al source (a coal-fired power plant), and depending on the wind direction, their concentrations in the receptor vary proportionally to each other. Correlation coefficients can be calculated for each pair of elements and a correlation matrix (n × n) can be constructed. Note that because rPb;Al ˆ rAl;Pb

and

rPb;Pb ˆ rAl;Al ˆ 1

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TABLE 26.4 Correlation Matrix for Elements Measured in Whiteface Mountain, New York

Na K Sc Mn Fe Zn As Br Sb

Na

K

1 0.48 0.03 0.22 0.21 0.14 0.13 0.04 0.08

0.48 1 0.46 0.57 0.61 0.03 0.30 0.28 0.15

Sc

Mn

Fe

Zn

As

Br

0.03 0.46 1 0.82 0.72 0.26 0.64 0.06 0.07

0.22 0.57 0.82 1 0.88 0.08 0.65 0.03 0.13

0.21 0.61 0.72 0.88 1 0.03 0.46 0.08 0.09

0.14 0.03 0.26 0.08 0.03 1 0.12 0.04 0.62

0.13 0.30 0.64 0.65 0.46 0.12 1 0.18 0.07

0.04 0.28 0.06 0.03 0.08 0.04 0.18 1 0.27

Sb 0.08 0.15 0.07 0.13 0.09 0.62 0.07 0.27 1

Source: Parekh and Husain (1981).

the matrix will be symmetric, and the elements in the diagonal will be equal to unity. A correlation matrix for nine elements measured at Whiteface Mountain, New York, is shown in Table 26.4. The correlation matrix S is the basis of PCA. Let λ1, λ2, . . ., λn be its eigenvalues and e1, e2, . . ., en the corresponding eigenvectors. Each eigenvector corresponds to a particular aerosol composition influencing the receptor. Each aerosol sample collected at the receptor can be expressed as a linear combination of these eigenvectors. The corresponding eigenvalues can be viewed as a measure of the importance of each eigenvector for the receptor. For example, for the correlation matrix for Whiteface Mountain, the eigenvalues are 3.6, 1.83, 1.16, 1.01, 0.54, 0.32, 0.31, 0.16, and 0.078. Eigenvectors corresponding to eigenvalues close to zero are neglected because they are usually artifacts due to either the numerical or the sampling analysis procedures. The decision as to which eigenvector should be retained is not straightforward. In practice, eigenvectors corresponding to eigenvalues larger than one are usually retained. Following this rule of thumb, the four corresponding eigenvectors are shown in Table 26.5. At this stage one is left with a set of vectors corresponding to the specific elemental combinations influencing the receptor (Table 26.5). Even if the solution to the PCA problem so far is straightforward and unique, these calculated factors often have little physical significance. For example, the factors in Table 26.5 are difficult to interpret, as they have several negative components, and so a further transformation is needed. To solve these problems, the next step involves a rotation of the remaining eigenvectors. Two types of rotation are used: orthogonal when the new axes are also orthogonal to each other and oblique when they are not. The most popular rotation is an orthogonal rotation known as varimax rotation (Kaiser 1959), which rotates the remaining eigenvectors in space in such a way as to maximize the number of values that are TABLE 26.5 Normalized Eigenvectors for Whiteface Mountain, New York Eigenvector Element Na K Sc Mn Fe Zn As Br Sb

1 0.174 0.381 0.455 0.499 0.468 0.05 0.375 0.024 0.081

Source: Parekh and Husain (1981).

2 0.262 0.222 0.182 0.044 0.014 0.591 0.158 0.357 0.588

3 0.403 0.397 0.148 0.104 0.027 0.394 0.309 0.504 0.377

4 0.696 0.083 0.193 0.035 0.095 0.188 0.096 0.617 0.191

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TABLE 26.6 Rotated Factors for Whiteface Mountain, New York Factor Element Na K Sc Mn Fe Zn As Br Sb

1 0.047 0.478 0.901 0.913 0.815 0.152 0.803 0.134 0.756

2 0.086 0.020 0.128 0.006 0.006 0.897 0.021 0.126 0.887

3 0.943 0.628 0.015 0.210 0.251 0.144 0.038 0.141 0.039

4 0.071 0.436 0.104 0.103 0.251 0.068 0.274 0.890 0.227

Source: Parekh and Husain (1981).

close to unity (Hopke 1985). This is a rather controversial step because it is an arbitrary procedure. In any case, the final eigenvectors are usually interpreted as fingerprints of emission sources based on the chemical species with which they are highly correlated. The rotated eigenvectors are shown in Table 26.6, and their physical significance can be discussed even if there are only a few elements available. For example, factor 1, containing elements coming from crustal sources, is probably soil and other crustal materials. Factor 2, characterized by Zn and Sb, apparently results from refuse incineration. Factor 3, containing Na and K, could be marine aerosol with road salt. Bromine on factor 4 indicates a motor vehicle component. One should note that the above solution is not unique. Hopke (1985) reanalyzed the same data using a different rotation and concluded that there were five factors affecting the location with composition different from the one in Table 26.6. The major assumptions of PCA modeling are that: 1. The composition of emission sources is constant. 2. Chemical species used in PCA do not interact with each other, and their concentrations are linearly additive. 3. Measurement errors are random and uncorrelated. 4. The variability of the concentrations from sample to sample is dominated by changes in source contributions and not by the measurement uncertainty or changes in the source composition. 5. The effect of processes that affect all sources equally (e.g., atmospheric dispersion) is much smaller than the effect of processes that influence individual sources (e.g., wind direction, emission rate changes). 6. There are many more samples than source types for a statistically meaningful calculation. 7. Eigenvector rotations (if used) are physically meaningful. Additional examples of PCA application can be found in Henry and Hidy (1979, 1981), Wolff and Korsog (1985), Cheng et al. (1988), Henry and Kim (1989), Koutrakis and Spengler (1987), and Zeng and Hopke (1989). PCA provides a rather qualitative description of source fingerprints, which can be used later as input to a CMB model or a similar source apportionment tool. One of the most popular PCA algorithms/codes for air quality applications is UNMIX by Henry (1997). UNMIX tries to find non-negative solutions for Equation (26.10), assuming that the contribution of each source is negligible for at least a few of the collected samples. These are called edge points, and UNMIX tries to find them and fit a hyperplane (called an edge) through them. The intersections of the edges provide the composition of the sources. For UNMIX to identify N sources, it is necessary to have at least N(N 1) data points that have little or no impact from that source. The use of these geometric constraints by UNMIX eliminates the need for rotations of the vectors. A user-friendly version of the

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STATISTICAL MODELS

code is available by the US Environmental Protection Agency (Norris et al. 2007) through their website (www.epa.gov/heasd/research/unmix.html).

26.3.2 Positive Matrix Factorization (PMF) Positive matrix factorization (PMA) is a newer approach to factor analysis (Paatero and Tapper 1994) that tries to address among others the problem of scaling of data so that the more precise data will have more influence on the solution than will points with higher uncertainties. It addresses the source–receptor problem of Equations (26.10) or (26.11) as a least-squares problem. For a given guess of the matrices F and G, the elements of the residual (or error) matrix E can be calculated according to Equation (26.10): m

eik ˆ cik

f ij gjk

(26.22)

jˆ1

Similarly to least squares, PMF tries to minimize the objective function Q defined as n

s

eik sik

Qˆ iˆ1 kˆ1

2

(26.23)

where sik is an estimate of the uncertainty in element i of sample k. By definition Q depends on the choices of F and G, and the goal of PMF is to find the corresponding nonnegative matrices that lead to the minimum value of Q. If the measurement uncertainties sik are properly estimated, then the errors eik for the various data points will be similar to the corresponding sik, and each term in the sum of (26.23) will be of the order of unity. Paatero (1997) developed a global minimization scheme that determines directly both matrices F and G in PMF. The corresponding algorithm/code is called PMF2 (the 2 signifies two-dimensional arrays). The nonnegativity constraints are imposed by adding two logarithmic penalty terms (one for each matrix) to the objective function, which now becomes n

s

Qˆ iˆ1 kˆ1

eik sik

2

n

s

n

log…gik †

α

s

log…f ik †

β

iˆ1 kˆ1

(26.24)

iˆ1 kˆ1

The two penalty terms above prevent the elements of matrices F and G from becoming negative, and the coefficients α and β control the corresponding magnitude of the penalty. Two more quadratic terms are added to constrain the solution. This is known as regularization and is often used in ill-posed problems to prevent unacceptable solutions: n

s

Qˆ iˆ1 kˆ1

eik sik

2

n

s

α

n

log…gik † iˆ1 kˆ1

s

n

s

iˆ1 kˆ1

n

s

gik2 ‡ δ

log…f ik † ‡ γ

β

iˆ1 kˆ1

f ik2

(26.25)

iˆ1 kˆ1

All coefficients α, β, γ, and δ are given progressively smaller values during the iterations of the algorithm, so their final values are very small but still not zero. 26.3.2.1 Uncertainties and Missing Values Solution of the PMF problem depends critically on the uncertainties of the measured values, sik. At the same time there may be additional problems in the dataset, including missing values for some species (the sample was analyzed, but some of the n concentrations are not available) or values below the detection limit (the value is known to be small but the exact concentration is not known). Values and the corresponding uncertainties are needed for these measurements, too. Let us assume that uik is the analytical uncertainty of element i in sample k and di the detection limit. Suggestions for the uncertainties

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FACTOR ANALYSIS

TABLE 26.7 Recommended uncertainties in PMF Concentration of i

Concentration in PMF

Uncertainty (sik)

cik di/2

uik + (di/3) 5di/6

Measured (cik) Below detection limit Missing

∏skˆ1 cik

1=s

4 ∏skˆ1 cik

1=s

Source: Polissar et al. (1998).

that should be used are given in Table 26.7 (Polissar et al. 1998). Concentrations below the detection limit are assigned a value equal to half of the limit and a significant uncertainty. Missing concentrations are set equal to the geometric mean concentration of all samples with a large uncertainty that is 4 times the estimated value. In this way, the below detection values (which still carry important information) and samples that are missing some values can be still be treated by PMF, taking into account, though, the high corresponding uncertainties. 26.3.2.2 Extreme Values The treatment of extreme values is another important issue in PMF applications. These values can be real or can be outliers due to sample contamination or analytical errors. In either case, these high values can influence significantly the solution and may even distort it. It is possible in PMF to control the influence of these values in a process called robust factorization (Paatero 1997). This is achieved by modifying the objective function Q given by Equation (26.23) with an influence term hik resulting in the robust Qrob: n

s

Qrob ˆ iˆ1 kˆ1

eik hik sik

2

(26.26)

The influence term (Huber 1981) is defined as h2ik ˆ

1 if jeik j=sik  q jeik j=…qsik † otherwise

(26.27)

where q is the outlier distance. A typical choice for this parameter is q = 4. If the absolute value of the error for a concentration during the PMF calculation is less than q times the corresponding uncertainty, the influence term plays no role as it is equal to unity. However, if the error exceeds the above threshold, then the measurement is treated as an outlier. The contribution of this extreme value to the objective function Qrob is not neglected but it is assumed to be equal to q |eij|/sij, thus limiting its influence on the answer. The difference in the PMF results using Q and Qrob illustrates the importance of the extreme values on the solution. It is generally advisable to use the robust mode (Qrob) for environmental data, however examining the solution using Q may provide additional information. 26.3.2.3 Choice of Number of Factors The appropriate choice of the number of factors P that should be used in the solution is one of the major challenges of PMF and of factor analysis in general. Additional factors result in general in reductions of the minimum Q value obtained. A large decrease in the minimum Q after the addition of a factor suggests that the new factor can explain a significant fraction of the data variation unaccounted by the others. After an appropriate number of factors have been used in the analysis, addition of factors does not result in significant further reduction of the minimum Q.

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FIGURE 26.4 PMF results for an aerosol mass spectrometer (AMS) dataset collected in New York in 2009 (Zhang et al. 2011): (a) Q/Qexp as a function of the number of factors used in the solution; (b) Q/Qexp as a function of the rotational parameter Fpeak.

The expected value of Q, Qexp is equal to the degrees of freedom in the dataset (Paatero et al. 2002), that is, the number of points in the data matrix minus the total number of elements in the factor matrices: Qexp ˆ …n  s†

P…n ‡ s†

(26.28)

For typical atmospheric chemistry datasets (e.g., obtained by the aerosol mass spectrometer) n  s  P…n ‡ s†, and therefore Qexp  n  s

(26.29)

the expected minimum value of Q is approximately equal to the number of data points or Q/Qexp  1. If Q/Qexp  1 then either the PMF model is not appropriate for the system or the measurement uncertainties have been underestimated. On the other hand, Q/Qexp  1 indicates overestimation of the data uncertainties. A typical diagram of the Q/Qexp versus the number of factors P is shown in Figure 26.4. The number of factors is usually chosen taking into account not only the Q/Qexp variation but also the correlations of factor time series with the concentrations of other pollutants not used in the PMF analysis, the average diurnal profiles of the factors, meteorological data, and so on. In that application, five factors were chosen (Sun et al. 2011) resulting in a Q/Qexp = 3.5. 26.3.2.4 Rotational Ambiguity One of the major weaknesses of factor analysis methods is that they suffer from rotational ambiguity; different sets of factors provide equally good fits to the measurements. This problem can be seen in Figure 26.3, in which the measurements can be explained equally well by selecting different axes. Mathematically, the pair of matrices G and F in the source–receptor problem of Equation (26.11) can be transformed to other pairs G´ and F´ corresponding to the same Q. Assume that T is a nonsingular square matrix of dimensions m × m and T 1 is its inverse. Then using the unit vector I = T T 1 and F G ˆ F I G ˆ F T T 1 G ˆ …F T†…T

1

G† ˆ F´ G´

The matrices F´ = F T and G´ = T 1 G therefore correspond to the same solution of the factor analysis problem. The above transformation of the pair G and F to G´ and F´ is a rotation. In the case of PMF this rotation is acceptable only if all the elements of the new matrices G´ and F´ are also nonnegative. These nonnegativity constraints for both the source profiles and their contributions limit the domain of possible rotation, but they still do not result in a unique solution. In most cases some rotations are possible, resulting in an infinite number of solutions to the problem. One important characteristics of PMF is that rotations are part of the fitting process and are not applied after the estimation of the factors as done in principal-component analysis and other eigenvector-based methods. If a number of elements of the fingerprint matrix F or the source contribution matrix G are known a priori to be zero, then there is little or no rotational ambiguity in the solution. In these cases the problem is solved by setting these elements to zero during the execution of the PMF2 algorithm (Paatero et al. 2002).

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One of the ways that the effect of rotations can be quantified in the code PMF2 (Paatero 1997) is by using the rotational parameter Fpeak. A specific rotation may be regarded as additions and subtractions of factor vectors (Paatero and Tapper 1994). The negative direction (Fpeak < 0) corresponds to subtracting columns of matrix G from each other while adding the corresponding rows of the matrix F to each other. Fpeak > 0 corresponds to the addition of G columns and the subtraction of F rows. Unfortunately, there is no theoretical basis for choosing a particular value of Fpeak. The rotation result is usually not an exact rotated image of the original because of the need to satisfy the nonnegativity constraints. This leads to an increase of the Q value. If the Q increases too much because of a rotation, then the result is probably not valid any more. Exploring the sensitivity of the solution to both positive and negative Fpeak values allows the quantification of the variability of the solution due to rotations (Figure 26.4). Additional details about the practical use of Fpeak, the choice of factors, and other practical information for aerosol mass spectrometry data analysis can be found in Zhang et al. (2011). 26.3.2.5 The Multilinear Engine (ME) The multilinear engine (ME) is another algorithm with the corresponding code (currently ME-2) for the solution of the positive matrix factorization problem that provides flexibility in the specification of the system model (Paatero 1999). The ME-1 algorithm was described by Paatero (1999) and its successor, the ME-2, by Paatero (2000). While there are similarities in the two PMF approaches (PMF-2 and ME-2) the underlying algorithms are quite different. The PMF-2 algorithm solves a well-defined problem (discussed in the previous paragraphs), and the user can make only limited changes (e.g., rotations through the Fpeak parameter) but cannot change the equations of the model. ME-2 can be viewed as a general equation solver with the user defining all aspects of the problem in the beginning. To reduce the uncertainty of the PMF solutions, additional information such as known source compositions and/or source contributions can be used in ME. To solve the problem with the auxiliary equations corresponding to the various kinds of a priori information, auxiliary terms are added to the objective function Q. These auxiliary terms Qa have the form V

Qa ˆ vˆ1

rv sv

2

(26.30)

where the subscript v denotes the corresponding auxiliary equation (there are V such equations) in arbitrary order, rv is the corresponding residual, and sv is the “softness” of that equation. The sv values play the same role as the uncertainties sik in Equation (26.23) and are selected by the user. The larger the value of sv, the “softer” the equation, resulting in a small contribution to the objective function Q. ME-2 allows four types of modifications of the model given by Equation (26.10) (Paatero and Hopke 2009): 1. Setting some factor elements equal to desired values (zero or nonzero) 2. Specifying upper and/or lower bounds for some factor elements 3. Introducing “pulling equations” into the model for pulling individual factor elements toward specified values 4. Introducing “pulling equations” for functions of factor elements For cases 3 and 4, if f is the factor element (or a function of factor elements) and t the target value, then the corresponding auxiliary term is Qa = (f t)2/s2, where s is the softness of the pull.

26.4 METHODS INCORPORATING WIND INFORMATION The source–receptor methods discussed in the previous sections rely on measurements collected at the receptor site and estimate the contributions of the various sources to these measured pollution levels. Use of the wind field during the measurements and physical models of pollutant transport, together with the measurements, can result in valuable additional information such as the location of the major sources affecting the receptor or receptors. A number of methods have been developed incorporating wind data

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into the analysis. Two of them are discussed below: the relatively simple potential source contribution function (PSCF) and the more complex empirical orthogonal function (EOF) method.

26.4.1 Potential Source Contribution Function (PSCF) This method, developed originally by Ashbaugh et al. (1985), uses the trajectories of the air parcels arriving at the receptor site (known as backward trajectories) during the period of the measurements (see Section 25.4). The central idea of the method is that the trajectories arriving at the receptor during periods of high concentration of a pollutant have passed over major source areas of the pollutant. If many measurements are available, then the common areas of these trajectories will be the locations of the sources. The backward trajectories can be readily calculated with applications such as the HYSPLIT (hybrid single-particle Lagrangian integrated trajectory) model (Draxler 1999), available through the US National Oceanic and Atmospheric Administration (NOAA) website (ready.arl.noaa.gov/HYSPLIT.php). The corresponding calculation requires first the division of the region of interest (e.g., the continental United States) into grid cells. The size of the domain and that of the grid cells depends on the application. Let (i, j) be one of these cells. A time interval Δt (e.g., one hour) is chosen for the period of interest T (e.g., 5-day backward trajectories) and the location of the trajectories every Δt is found and is allocated to the corresponding grid cell. If nij is the number of trajectory points that were allocated to the cell (i, j), then the probability that an air parcel was at this location at a given time P(Aij) is given by P Aij ˆ nij =

nij i

(26.31)

j

This probability also expresses the fraction of the time that air parcels arriving at the receptor spent in that cell during the period T and therefore the product P(Aij)T is its total residence time in that location. The concentrations at the receptor site vary during the measurement period. We are interested in the trajectories arriving at the site during high-concentration periods, say, periods with concentrations above the mean value or some other threshold value. The corresponding high-concentration trajectories are selected and their points are allocated once more to the corresponding grid cells. These are obviously a subset of all the trajectories used in the analysis. If mij is the number of trajectory points that were allocated to the cell (i, j) then the probability that a high-concentration air parcel passed through this location P(Bij) is given by P Bij ˆ mij =

nij i

(26.32)

j

The probability P(Bij) expresses the fraction of the time that “contaminated” air parcels spent in that cell during the period T. The next step in the algorithm is the calculation of the conditional probability P(Bij/Aij). This is the probability that if an air parcel is observed to reside in a given cell, then it will also arrive at the receptor with a high concentration of the pollutant of interest. The conditional probability can be calculated as P Bij =Aij ˆ

P…Bij ∩ Aij † P…Aij †

(26.33)

where P Bij ∩ Aij is the probability of the intersection of the sets Bij and Aij. However, since the set of events Bij is a subset of the set Aij, their intersection is simply Bij, resulting in the following equation: P Bij =Aij ˆ

P…Bij † P…Aij †

(26.34)

The conditional probability is therefore the ratio of the fraction of time spent by contaminated air parcels in the cell (i, j) to the total fraction of time spent in the cell by all air parcels. This conditional probability is defined as the potential source contribution function (PSCF), PSCFij = P(Bij/Aij), and can be calculated replacing Equations (26.31) and (26.32) into (26.34):

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METHODS INCORPORATING WIND INFORMATION

PSCFij ˆ

mij nij

(26.35)

Cells containing major emission sources should have high PSCF values close to one. The PSCF model allows the creation of a map of the source areas affecting the receptor during the period of the analysis. It does not apportion though the measured concentrations to the various source areas. PSCF Application to the 2002 Quebec Fires Begum et al. (2005) used PSCF to identify source areas affecting Philadelphia during July 2002. In the beginning of the month there were major forest fires in central Quebec that affected the eastern United States. Semi-continuous measurements (2-h time resolution) of OC, EC, sulfate, and PM2.5 levels were used in the analysis. The vertically mixed HYSPLIT model starting at 500 m above ground level was used to calculate 5-day backward trajectories arriving at the measurement site every 2 h (at the midpoint of the measurement period). The region covered by the trajectories (North and Central America) was divided into 2664 1° × 1° latitude/longitude cells. There were 360 trajectories, each with 120 points (one every hour), resulting in 43,200 points. This corresponds to an average of approxi­ mately 16 points per cell. Small values of nij can produce erroneous high PSCF values with high uncertainties given the small number statistics. In order to minimize this artifact, the authors multi­ plied the PSCF with an empirical weight function W(nij) that reduced the PSCF values when the number of trajectory points in a cell was less than about 3 times the average values of trajectory points:

W…nij † ˆ

1:0; 48 < nij 0:7; 5 < nij  48; 0:4; 2 < nij  5; 0:2; 2  nij

The PSCF results can be displayed in the form of maps with the PSCF values ranging from 0 to 1 displayed using a grey scale. The map for OC sources (Figure 26.5) shows clearly large source areas southeast of Hudson Bay in Quebec. This was the area of the fires, demonstrating in this way the ability of PSCF to locate major sources.

FIGURE 26.5 Potential source contribution function plot for OC for July 2002 (adapted from Begum et al. (2005)).

1070

STATISTICAL MODELS

26.4.2 Empirical Orthogonal Function (EOF) Receptor models can also be used together with spatial distribution of measurements to estimate the spatial distribution of emission fluxes. The empirical orthogonal function (EOF) method is one of the most popular models for this. Henry et al. (1991) improved the EOF method by using wind direction in addition to spatially distributed concentration measurements as input. We describe this approach below. Let us assume that during a given period the ground-level concentration field of an atmospheric species can be written as N

ai …t†ϕι …x; y†

C…x; y; t† ˆ

(26.36)

iˆ1

where ϕi(x,y) are N orthogonal functions and ai(t) are time-weighting functions. The functions ϕi(x, y) include the information about the sources of the species, while the time-weighting functions represent its atmospheric transport. Neglecting vertical concentration variations and dispersion, the atmospheric diffusion equation for this species is @C @C @C ‡u ‡v ˆ Q…x; y; t† @t @x @y

(26.37)

where u(x, y, t) and v(x, y, t) are the wind components and Q(x, y, t) is the net source term for the species including emissions and removal processes. Using (26.36), we obtain @C ˆ @x

N

ai …t†

@ϕι @x

(26.38)

ai …t†

@ϕι @y

(26.39)

iˆ1

and @C ˆ @y

N iˆ1

Substituting (26.38) and (26.39) into (26.37) and integrating from time zero to the end of the sampling period T, one gets Q…x; y† ˆ

C…x; y; T†

C…x; y; 0† T

N

‡

ui …x; y; t† iˆ1

@ϕi ‡ @x

N

vi …x; y; t† iˆ1

@ϕi @y

(26.40)

where Q is the average source strength spatial distribution and ui …x; y† ˆ

1 T ai …t†u…x; y; t†dt T ∫0

(26.41)

vi …x; y† ˆ

1 T ai …t†v…x; y; t†dt T ∫0

(26.42)

Equation (26.40) suggests that the spatial distribution of the source strength of the species can be found using concentration and wind data if the EOFs ϕi(x, y) and the time-weighting functions ai(t) can be calculated (Henry et al. 1991). The EOFs can be found with the following procedure. Assume that a given species is measured simultaneously at s sites during n sampling periods. The measurements can be used to construct the n × s concentration matrix C. The first column of C contains all the measurements at the first site, the second column the measurements at the second site, and so on. Equation (26.36) can be viewed as the continuous

1071

METHODS INCORPORATING WIND INFORMATION

form of the singular value decomposition of matrix C (26.43) C ˆ UBVT T where U is an n × n orthogonal matrix whose columns are the eigenvectors of CC , V is the matrix of the eigenvectors of CTC, and B is a diagonal matrix of singular values (square roots of the eigenvalues). In this case there are s nonzero singular values. If a singular value is zero or very small, the corresponding eigenvectors can be removed from the matrices U and V, leaving us with N significant eigenvectors. This step is similar to selection of the principal components during PCA. Let U∗ be the n × N matrix with the significant eigenvectors, B∗ the N × N diagonal matrix with the remaining singular values, and V∗ the corresponding s × N matrix. Then let Φ ˆ B∗ V∗T

(26.44)

The columns of Φ are then the discrete EOFs and the columns of U∗ are the discrete-time functions. These discrete EOFs can be interpolated in space to obtain the continuous EOFs using, for example, 1/r2 interpolation. Then (26.40) can be applied to obtain the spatial distribution of the source strength. Note that while PCA is applied to many samples from the same site taken over a number of sampling periods, the EOF operates on many samples from many sites taken over the same period. Assumptions implicit in the use of the EOF are that: 1. 2. 3. 4. 5.

The fluxes of spatially distributed species are linearly additive. Species are homogeneously distributed in the mixed layer. Measurement errors are random and uncorrelated. The number of sampling sites exceeds the number of sources. Measurements are located in areas where there are significant spatial concentration gradients.

Assumptions 4 and 5 are rarely met in practice. The spatial resolution of the EOF is limited by the number of observation sites and the distance between them. Sudden changes of windspeed and direction during a sampling period often result in problems. Applications of the EOF have been presented by Gebhart et al. (1990), Ashbaugh et al. (1984), Wolff et al. (1985), and Henry et al. (1991). Henry et al. (1991) compared simulated two-dimensional data generated by a simple dispersion model and the above-described version of the EOF using simple wind fields. One of the comparisons is shown in Table 26.8. For this comparison, a sampling site was located in each square and the model was able to reproduce the location of the two sources. However, the source strength is underpredicted as a result of numerical diffusion to the neighboring cells. TABLE 26.8 Predicted Emission Location and Strengtha 0.5

0.5

0.8

1.5

1.6

1.3

0.8

2.9

3.7

0.3

0.2

1.0

0.8

0.5

0.1

2.4

3.0

4.2

0.6

0.5

1.9

2.2

9.4

9.7

5.2

2.9

3.3

1.6

2.0

2.6

7.3

28.5

31.3

10.8

0.3

0.9

0.6

1.2

0.6

8.1

25.5

28.4

11.4

1.2

1.5

2.0

4.6

5.7

5.5

6.2

7.0

4.1

0.4

0.2

3.9

14.3

15.5

5.5

1.0

1.4

1.3

1.5

0.4

4.0

13.3

14.2

5.4

0.1

0.4

0.4

0.6

0.4

2.1

3.1

3.6

2.2

1.2

0.7

0.7

0.8

0.7

a

Actual emissions in the shaded cells are 50 (upper square) and 25 (lower square) emission units. For the rest the actual emissions are zero.

Source: Henry et al. (1991).

1072

STATISTICAL MODELS

26.5 PROBABILITY DISTRIBUTIONS FOR AIR POLLUTANT CONCENTRATIONS Air pollutant concentrations are inherently random variables because of their dependence on the fluctuations of meteorological and emission variables. We already have seen from Chapter 18 that the concentration predicted by atmospheric diffusion theories is the mean concentration hci. There are important instances in analyzing air pollution where the ability simply to predict the theoretical mean concentration hci is not enough. Perhaps the most important situation in this regard is in ascertaining compliance with ambient air quality standards. Air quality standards are frequently stated in terms of the number of times per year that a particular concentration level can be exceeded. In order to estimate whether such an exceedance will occur, or how many times it will occur, it is necessary to consider statistical properties of the concentration. One objective of this section is to develop the tools needed to analyze the statistical character of air quality data. Hourly average concentrations are the most common way in which urban air pollutant data are reported. These hourly average concentrations may be obtained from an instrument that actually requires a 1-h sample in order to produce a data point or by averaging data taken by an instrument having a sampling time shorter than 1 h. If we deal with 1-h average concentrations, those concentrations would be denoted by cτ(ti), where τ = 1 h. For convenience we will omit the subscript τ henceforth; however, it should be kept in mind that concentrations are usually based on a fixed averaging time. There are 8760 h in a year, so if we are interested in the statistical distribution of the 1-h average concentrations measured at a particular location in a region, we will deal with a sample of 8760 values of the random variable c. The random variable is characterized by a probability density function p(c), such that p(c) dc is the probability that the concentration c of a particular species at a particular location will lie between c and c + dc. Our first task will be to identify probability density functions (pdf’s) that are appropriate for representing air pollutant concentrations. Once we have determined a form for p(c), we can proceed to calculate the desired statistical properties of c. If we plot the frequency of occurrence of a concentration versus concentration, we would expect to obtain a histogram like that sketched in Figure 26.6a. As the number of data points increases, the histogram should tend to a smooth curve such as that in Figure 26.6b. Note that very low and very high concentrations occur only rarely. We recall that aerosol size distributions exhibited a similar overall behavior; there are no particles of zero size and no particles of infinite size. Thus a probability distribution that is zero for c = 0 and as c → 1 is also desired for atmospheric concentrations. Although there has been speculation as to what probability distribution is optimum in representing atmospheric concentrations from a physical perspective (Bencala and Seinfeld 1976), it is largely agreed that there is no a priori reason to expect that atmospheric concentrations should adhere to a specific probability distribution. Moreover, it has been demonstrated that a number of pdf’s are useful in representing atmospheric data. Each of these distributions has the general features of the curve shown in Figure 26.4b; they represent the distribution of a nonnegative random variable c that has probabilities of

FIGURE 26.6 Hypothetical distributions of atmospheric concentrations: (a) histogram; (b) continuous distribution.

1073

PROBABILITY DISTRIBUTIONS FOR AIR POLLUTANT CONCENTRATIONS

occurrence approaching zero as c → 0 and as c → 1. Georgopoulos and Seinfeld (1982) summarize the functional forms of many of the pdf’s that have been proposed for air pollutant concentrations [see also Tsukatami and Shigemitsu (1980)]. In addition, Holland and Fitz-Simons (1982) have developed procedures for fitting statistical distributions to air quality data. Which probability distribution actually best fits a particular set of data depends on the character­ istics of the set. The two distributions that have been used most widely for representing atmospheric concentrations are the lognormal and the Weibull, and we will focus on these two distributions here.

26.5.1 The Lognormal Distribution We have already dealt extensively with the lognormal distribution in Chapter 8 for representing aerosol size distribution data. If a concentration c is lognormally distributed, its pdf is given by

pL …c† ˆ

1 …2π†1=2 c ln σg

ln c exp

ln μg

2

2 ln2 σg

(26.45)

where μg and σg are the geometric mean and standard deviation of c. Figure 26.7 shows sample points of the distribution of 1-h average SO2 concentrations equal to or in excess of the stated values for Washington, DC, for the 7-year period December 1, 1961–December 1, 1968. A lognormal distribution has been fitted to the high-concentration region of these data. We recall that the geometric mean, or median, is the concentration at which the straight-line plot crosses the 50th percentile. The slope of the line is related to the standard geometric deviation, which can be calculated from the plot by dividing the 16th percentile concentration (which is one standard deviation from the mean) by the 50th percentile concentration (the geometric mean). [This is the 16th percentile of the complementary distribution function F(c); equivalently, it is the 84th percentile of the

FIGURE 26.7 Frequency of 1-h average SO2 concentrations equal to, or in excess of, stated values at Washington, DC, from December 1, 1961 to December 1, 1968 (Larsen 1971).

1074

STATISTICAL MODELS

cumulative distribution function F(c).] F(x) = Prob {c > x} = 1 F(x). For the distribution of Figure 26.7, μg = 0.042 ppm, and σg = 1.96. Plots such as Figure 26.7 are widely used because it is important to know the probability that concentrations will equal or exceed certain values.

26.5.2 The Weibull Distribution The pdf of a Weibull distribution is pw …c† ˆ …λ=σ†…c=σ†λ 1 exp‰ …c=σ†λ Š σ; λ > 0

(26.46)

As was done with the lognormal distribution, it is desired to devise a set of coordinates for a graph on which a set of data that conforms to a Weibull distribution will plot as a straight line. To do so, we can work with the complementary distribution function, FW …c†. The complementary distribution function of the Weibull distribution is FW …c† ˆ exp‰ …c=σ†λ Š

(26.47)

Taking the natural logarithm of (26.47) and changing sign, we obtain 1 FW …c†

ln

ˆ

c σ

λ

(26.48)

Now take the logarithm (base 10) of both sides of (26.48): log ln

1 FW …c†

ˆ λ…log c

log σ†

(26.49)

Therefore, if we plot log[ln(1/FW …c†)] versus log c, data from a Weibull distribution will plot as a straight line. The values log[ln(1/FW …c†)] are the values on the ordinate scale of Figure 26.8, so-called extremevalue probability scale. The scale is constructed so that a computation of log[ln(1/FW …c†)] of each value of FW …c† yields a linear scale. By setting log c = log σ in (26.49), we obtain FW …c† ˆ e 1 ˆ 0:368. This point is shown in Figure 26.8. The parameter λ, which is the slope of the straight line, can be found by using any other point on the line and solving (26.49) for λ: 1 FW …c† log c log σ

log ln λˆ

(26.50)

Suppose that we use, for a second point, that point on the line that crosses FW …c† ˆ 0:01. Then λˆ

0:663 log c0:01 log σ

(26.51)

26.6 ESTIMATION OF PARAMETERS IN THE DISTRIBUTIONS Each of the two distributions that we have presented is characterized by two parameters. The fitting of a set of data to a distribution involves determining the values of the parameters of the distribution so that the distribution fits the data in an optimal manner. Ideally, this fitting is best carried out using a systematic optimization routine that estimates the parameters for several distributions from the given set of data and then selects that distribution that best fits the data, using a criterion of “goodness of fit” (Holland and Fitz-Simons 1982). We present two methods here that do not require a computer

1075

ESTIMATION OF PARAMETERS IN THE DISTRIBUTIONS

FIGURE 26.8 Weibull distribution fits of the 1971 hourly average (1) and daily maximum hourly average (2) ozone mixing ratio at Pasadena, California (Georgopoulos and Seinfeld 1982).

optimization routine. The first method has, in effect, already been described. It is formally called the method of quantiles. The second is the method of moments, which requires the computation of the first (as many as the parameters of the distribution) sample moments of the “raw data.” The procedure suggested should start with the construction of a plot of the available data on the appropriate axes (that gives a straight line for the theoretical distribution), in order to get a preliminary notion of the goodness of fit. Then one would apply one of the following methods to evaluate the parameters of the “best” distribution.

26.6.1 Method of Quantiles We have seen that the parameters μg and σg of the lognormal distribution can be estimated by using the 50th percentile and 16th percentile values. These percentiles are called quantiles. Also, (26.50) and (26.51) indicate how this approach can be used to determine the two parameters of the Weibull distribution. For the lognormal distribution, for example, we have already illustrated how μg and σg are estimated from the concentrations at the 50% and 84% quantiles: ln c0:50

ln μg ˆ 0

ln c0:84

ln μg ˆ ln σg

Choosing as a further illustration the 95% and 99% quantiles, we obtain ln c0:95

ln μg ˆ 1:645 ln σg

ln c0:99

ln μg ˆ 2:326 ln σg

1076

STATISTICAL MODELS

For the Weibull distribution, the quantile concentration is given by 1=λ

1

cqi ˆ σ ln

qi

1

Using the 0.80 and 0.98 quantiles, for example, we obtain the estimates for the parameters λ and σ as 0:88817 ln c0:98 ln c0:80 σ ˆ exp…1:53580 ln c0:80

λ ˆ

0:53580 ln c0:98 †

26.6.2 Method of Moments To estimate the parameters of a distribution by the method of moments, the moments of the distribution are expressed in terms of the parameters. Estimates for the values of the moments are obtained from the data, and the equations for the moments are solved for the parameters. For a two-parameter distribution, values for the first two moments are needed. The rth noncentral moment of a random variable c with a pdf p(c) is defined by 1

μ´r ˆ

∫0

cr p…c†dc

(26.52)

and the rth central moment (moment about the mean) is μr ˆ

1

∫r

μ´r †r p…c†dc

…c

(26.53)

The mean value of c is μ´1 , and the variance is μ2, which is commonly denoted by σ2, or sometimes Var {c}. Consider first the estimation of μg and σg for the lognormal distribution. Let μ = ln μg and σ = ln σg. The first and second non-central moments of the lognormal distribution are μ´1 ˆ exp μ ‡

σ2 2

(26.54)

μ´2 ˆ exp…2μ ‡ 2σ2 †

(26.55)

After solving (26.54) and (26.55) for μ and σ2, we have μ ˆ 2 ln μ´1

1 ln μ´2 2

(26.56)

σ2 ˆ ln μ´2

2 ln μ´1

(26.57)

where μ´1 , μ´2 , and μ2 are related through μ2 ˆ μ´2

μ´2 1 and are estimated from the data by

M´1 ˆ

n

1 n

ci

(26.58)

c2i

(26.59)

iˆ1

and M´2 ˆ M2 ˆ

iˆ1 n

1 n

n

1 n

1

…ci iˆ1

M´1 †2

(26.60)

1077

ESTIMATION OF PARAMETERS IN THE DISTRIBUTIONS

where n is the number of data points. Thus the moment estimates of the parameters of the lognormal distribution are given by μ ˆ 2 ln M´1

1 ln M´2 2

(26.61)

σ2 ˆ ln M´2

2 ln M´1

(26.62)

For the Weibull distribution, the mean and variance are given by μ´1 ˆ σΓ…1 ‡ 1=λ†

(26.63)

where Γ is the gamma function and μ2 ˆ σ2 ‰Γ…1 ‡ 2=λ†

Γ2 …1 ‡ 1=λ†Š

(26.64)

In solving these equations for σ and λ, we can conveniently use the coefficient of variation given by …μ2 †1=2 =μ´1 . Then the moment estimators of the sample correspond to λ so that Γ…1 ‡ 2=λ† Γ2 …1 ‡ 1=λ† σˆ

1=2

1

ˆ

M´1 Γ…1 ‡ 1=λ†

…M2 †1=2 M´1

(26.65)

(26.66)

To aid in the determination of λ in fitting a set of data to the Weibull distribution, one can use Figure 26.9, in which the left-hand side (LHS) of (26.65) is shown as a function of λ for 0 < λ < 9. As an example, we consider the fitting of a Weibull distribution to 1971 hourly average ozone data from Pasadena, California. The data consist of 8303 hourly values (there are 8760 h in a year). The maximum hourly value reported was 53 parts per hundred million (pphm). The arithmetic mean and

FIGURE 26.9 Calculation of parameter λ in fitting a set of data to a Weibull distribution by the method of moments (Georgopoulos and Seinfeld 1982).

1078

STATISTICAL MODELS

TABLE 26.9 1971 Hourly Average Pasadena, California, Ozone Data Fitted to a Weibull Distribution Mixing ratio (pphm) equaled or exceeded by stated percent of observations

Data (hourly average) Weibull distribution Data (daily maximum) Weilbull distribution

1%

2%

3%

4%

5%

10%

25%

50%

75%

24 23.5 34 38.8

20 19.2 33 34.6

18 16.8 32 32

16 15.1 30 30.1

15 13.8 28 28.6

11 10 25 24.8

5 5.3 17 16.6

2 2.3 10 10.2

1 0.76 5 5.5

standard deviation of the data are M´1 ˆ 4:0 pphm and M2 ˆ 5:0 pphm, and the geometric mean and geometric standard deviation are 2.4 and 2.6 pphm, respectively. If one assumes that the hourly average ozone mixing ratios fit a Weibull distribution, the parameters of the distribution can be estimated from (26.65) and (26.66) to give λ = 0.808 and σ = 3.555 pphm. It is interesting also to fit only the daily maximum hourly average ozone values to a Weibull distribution. For these data there exist 365 data points, the maximum value of which is, as already noted, 53 pphm. The arithmetic mean and standard deviation of the data are M´1 ˆ 12:0 pphm and 1=2 M2 ˆ 8:6 pphm; the geometric mean and geometric standard deviation are 9.1 and 2.2 pphm, respec­ tively. Table 26.9 gives a comparison of the data and the Weibull distribution concentration frequencies in the two cases. Both fits are very good, but the fit to the daily maximum values is slightly better. 1=2

26.7 ORDER STATISTICS OF AIR QUALITY DATA One of the major uses of statistical distributions of atmospheric concentrations is to assess the degree of compliance of a region with ambient air quality standards. These standards define acceptable upper limits of pollutant concentrations and acceptable frequencies with which such concentrations can be exceeded. The probability that a particular concentration level x will be exceeded in a single observation is given by the complementary distribution function F(x) = Prob{c > x} = 1 F(x). When treating sets of air quality data, available as successive observations, we may be interested in certain random variables, as, for example: The highest (or, in general, the rth highest) concentration in a finite sample of size m The number of exceedances of a given concentration level in a number of measurements or in a given time period The number of observations (waiting time or “return period”) between exceedances of a given concentration level The distributions and pdf’s, as well as certain statistical properties of these random variables, can be determined by applying the methods and results of order statistics (or statistics of extremes) (Gumbel 1958; Sarhan and Greenberg 1962; David 1981).

26.7.1 Basic Notions and Terminology of Order Statistics Consider the m random unordered variates c(t1), c(t2), . . ., c(tm), which are members of the stochastic process {c(ti)} that generates the time series of available air quality data. If we arrange the data points by order of magnitude, then a “new” random sequence of ordered variates c1;m  c2;m  ∙ ∙ ∙  cm;m is formed. We call ci;m the ith highest-order statistic or ith extreme statistic of this random sequence of size m.

1079

ORDER STATISTICS OF AIR QUALITY DATA

In the exposition that follows we assume in general that The concentration levels measured in successive nonoverlapping periods—and hence the unordered random variates c(ti)—are independent of one another. The random variables c(ti) are identically distributed.

26.7.2 Extreme Values Assume that the distribution function F(x), as well as the pdf p(x), corresponding to the total number of available measurements, are known. They are called the “parent” (or initial) distribution and pdf, respectively. The probability density function pr,m(x) and the distribution function Fr,m(x) of the rth highest concentration out of samples of size m are evaluated directly from the parent pdf p(x) and the parent distribution function F(x) as follows. The probability that cr,m = x equals the probability of m r trials producing concentration levels lower than x, times the probability of r 1 trials producing concentrations above x, times the probability density of attaining a concentration equal to x, multiplied by the total number of combinations of arranging these events (assuming complete independence of the data). In other words, the pdf of the rth highest concentration has the trinomial form m! ‰F…x†Šm r ‰1 F…x†Šr 1 p…x† 1†!…m r†! 1 ˆ ‰F…x†Šm r ‰1 F…x†Šr 1 p…x† B…r; m r ‡ 1†

pr;m …x† ˆ

…r

(26.67)

where B is the beta function (Pearson 1934). In particular, for the highest and second highest concentration values (r = 1, 2), we have p1;m …x† ˆ m‰F…x†Šm 1 p…x†

(26.68)

and p2;m …x† ˆ m…m

1†‰F…x†Šm 2 ‰1

F…x†Šp…x†

(26.69)

The probability Fr,m(x) that cr;m  x is identical to the probability that no more than r 1 measure­ ments out of m result in cr;m > x. Every observation is considered as a Bernoulli trial with probabilities of “success” and “failure” F(x) and 1 F(x), respectively. Thus r 1

Fr;m …x† ˆ kˆ0

m ‰1 k

F…x†Šk ‰F…x†Šm

k

(26.70)

For the particular cases of the highest and the second highest values (r = 1, 2), (26.70) becomes F1;m …x† ˆ ‰F…x†Šm F2;m …x† ˆ m‰F…x†Šm

1

…m

(26.71) 1†‰F…x†Šm

(26.72)

It is worthwhile to note the dependence of the probability of the largest values on the sample size. From (26.71), we obtain F1;n …x† ˆ ‰F1;m …x†Šn=m

(26.73)

Thus, if the distribution of the extreme value is known for one sample size, it is known for all sample sizes.

1080

STATISTICAL MODELS

Let the pdf of the rth highest concentration out of a sample of m values be denoted pr;m(c). Once this pdf is known, all the statistical properties of the random variable cr;m can be determined. However, the integrals involved in the expressions for the expectation and higher-order moments are not always easily evaluated, and thus there arises the need for techniques of approximation. The most important result concerns the evaluation of the expected value of cr;m. In fact, for sufficiently large m, an approximation for E{cr;m} is provided by the value of x satisfying (David 1981) F…x† ˆ

m r‡1 m‡1

(26.74)

In terms of the inverse function of F(x), F 1(x) [i.e., F 1[F(x)] = x], we have the asymptotic relation E cr;m  F

1

m r‡1 ; m‡1

as m ! 1

(26.75)

26.8 EXCEEDANCES OF CRITICAL LEVELS The number of exceedances (episodes) Nx(m) of a given concentration level x in a set of m successive observations c(ti) is itself a random function. Similarly, the number of averaging periods (or observa­ tions) between exceedances of the concentration level x, another random function called the waiting time, passage time, or return period, is of principal interest in the study of pollution episodes. In the case of independent, identically distributed variates, each one of the observations is a Bernoulli trial; therefore, the probability density function of Nx(m) is, in terms of the parent distribution F(x) pN …N x ; m; x† ˆ

m ‰1 Nx

F…x†ŠNx F…x†m

Nx

(26.76)

Thus the expected number of exceedances N x …m† of the level x in a sample of m measurements is N x …m† ˆ m…1

F…x†† ˆ mF…x†

(26.77)

The expected percentage of exceedances of a given concentration level x is just 100 F(x).

26.9 ALTERNATIVE FORMS OF AIR QUALITY STANDARDS In the evaluation of whether ambient air quality standards are satisfied in a given region, aerometric data are used to estimate expected concentrations and their frequency of occurrence. If it is assumed that a certain probability distribution can be used to represent the air quality data, the distribution is fit to the current years’ data by estimating the parameters of the distribution. It is then assumed that the probability distribution will hold for data in future years; only the parameters of the distribution will change as the source emissions change from year to year. If the parameters of the distribution can be estimated for future years, then the expected number of exceedances of given concentration levels, such as the ambient air quality standard, can be assessed. Table 26.10 gives four possible forms for an ambient air quality standard for ozone. (The ozone mixing ratio standard in the table does not correspond to the current value.) The ambient air quality standards involve a concentration level and a frequency of occurrence of that level. In this section we want to examine the implications of the form of the standard on the degree of compliance of a region. The choice of one form of the standard over another can be based on the impact that each form implies for the concentration distribution as a whole (Curran and Hunt 1975; Mage 1980).

1081

ALTERNATIVE FORMS OF AIR QUALITY STANDARDS

TABLE 26.10 Alternative Statistical Forms of the Ozone Air Quality Standarda Number 1 2 3 4

Form 0.12 ppm hourly average with expected number of exceedances per year less than or equal to 1 0.12 ppm hourly average not to be exceeded on the average by more than 0.01% of the hours in 1 year 0.12 ppm annual expected maximum hourly average 0.12 ppm annual expected second highest hourly average

a

For most practical purposes, forms 1 and 3 can be considered equivalent. As noted in the text, the value of 0.12 ppm does not correspond to the current ozone standard.

The first step in the evaluation of an air quality standard is to select the statistical distribution that supposedly best fits the data. We will assume that the frequency distribution that best fits hourly averaged ozone concentration data is the Weibull distribution. Since the standards are expressed in terms of expected events during a 1-year period of 1-h average concentrations, we will always use the number of trials m equal to the number of hours in a year, 8760. We would use m < 8760 only to evaluate the parameters of the distribution if some of the 8760 hourly values are missing from the dataset. Let us now analyze each of the four forms of the ozone air quality standard given in Table 26.10 from the point of view that ozone concentrations can be represented by a Weibull distribution. 1. Expected Number of Exceedances of 0.12 ppm Hourly Average Mixing Ratio. The expected number of exceedances N x …m† of a given concentration level in m measurements is given by (26.77), which, in the case of the Weibull distribution, becomes N x ˆ m exp‰ …x=σ†λ Š

(26.78)

If we desire the expected exceedance to be once out of m hours, that is, N x1 ˆ 1, the concentration corresponding to that choice is x1 ˆ σ…ln m†1=λ

(26.79)

x1 ˆ σ…9:08†1=λ

(26.80)

For m = 8760, (26.79) becomes

2. Hourly Average Mixing Ratio Not to Be Exceeded on Average by >0.01% of the Hours in 1 Year. The expected percentage of exceedance of a given concentration x is given by 100 F(x), which, for the Weibull distribution, is ∏…x† ˆ 100 exp‰ …x=σ†λ Š (26.81) Equation (26.81) can be rearranged to determine the concentration level that is expected to be exceeded Π…x† percent of the time: x ˆ σ‰ln…100=∏…x††Š1=λ

(26.82)

Therefore, we can calculate the mixing ratio that is expected to be exceeded 0.01% of the hours in 1 year as x0:01 ˆ σ…9:21†1=λ

(26.83)

3. Annual Expected Maximum Hourly Average Mixing Ratio and Annual Expected Second Highest Hourly Average Mixing Ratio. The expected value of the rth highest concentration for a Weibull distribution is 1 m! …x=σ†λ Efcr;m g ˆ …r 1†!…m r†! ∫ 0 (26.84) m r r  f1 exp‰ …x=σ†λ Šg fexp‰ …x=σ†λ Šg dx

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STATISTICAL MODELS

We wish to evaluate this equation for r = 1 and r = 2, corresponding to standards 3 and 4, respectively, in Table 26.10, to obtain E{c1;m} and E{c2;m}, the expected highest and second highest hourly concentrations, respectively, in the year, with m = 8760. Unfortunately, the integral in (26.84) cannot be evaluated easily. Even numerical techniques fail to give consistent results, because of the singularity at x = 0. Thus the asymptotic relation for large m, (26.75), must be used in this case. For the Weibull distribution, we have 1

Efcr;m g σ

exp

λ

ˆ

m r‡1 m‡1

(26.85)

For m = 8760 and r = 1,2, we must solve, respectively, the equations exp

Efc1;m g σ

λ

1

exp

Efc2;m g σ

λ

1

ˆ

8760 8761

(26.86)

ˆ

8759 8761

(26.87)

and

to obtain Efc1 ; mg ˆ σ…9:08†1=λ

(26.88)

Efc2 ; mg ˆ σ…8:38†1=λ

(26.89)

and

Evaluation of Alternative Forms of the Ozone Air Quality Standard with 1971 Pasadena, California, Data Earlier, 1971 hourly average and maximum daily hourly average ozone mixing ratios at Pasadena, California, were fit to Weibull distributions. We now wish to evaluate the four forms of the ozone air quality standard with these data. For convenience all mixing ratios values will be given as pphm rather than ppm. 1. Expected Number of Exceedances of 12-pphm Hourly Average Mixing Ratio. The expected number of exceedances of 12 pphm, based on the Weibull fit of the 1971 Pasadena, California, hourly average data, is, from (26.78) N 12 ˆ 8760 exp

12 3:555

0:808

ˆ 605:2

The hourly average ozone mixing ratio that is exceeded at most once per year is from (26.79) x1 ˆ 3:555…ln 8760†1=0:808 ˆ 54:41 pphm which agrees well with the actual measured value of 52 pphm. However, if, instead of the complete hourly average Weibull distribution, we use the distribution of daily maximum hourly average values, the expected number of exceedances of a daily maximum of 12 pphm is N 12 ˆ 365 exp

12 13:189

1:416

ˆ 152:2

RELATING CURRENT AND FUTURE AIR POLLUTANT STATISTICAL DISTRIBUTIONS

and the daily maximum 1 h mixing ratio that is exceeded once per year, at most, is x1 ˆ 13:189…ln 365†1=1:416 ˆ 46:2 pphm It is interesting to note that this value is underpredicted if we use the distribution of daily maxima instead of the distribution based on the complete set of data. 2. Hourly Average Mixing Ratio Not to Be Exceeded on the Average by >0.01% of the Hours in 1 Year. The expected percentage of exceedances of 12 pphm is, from (26.81) ∏…12† ˆ 100 exp

12 3:555

0:808

ˆ 6:91%

The mixing ratio that is expected to be exceeded 0.01% of the hours in the year is, from (26.82) x0:01 ˆ 3:555 ln

100 0:01

1=0:808

ˆ 55:5 pphm

This form of the standard cannot be evaluated from the distribution of daily maxima since it is stated on the basis of a percentage of all the hours of the year. 3,4. Annual Expected Maximum Hourly Average Mixing Ratio and Annual Expected Second Highest Average Mixing Ratio. The annual expected maximum hourly average mixing ratio is obtained from (26.88) for E{c1;m}, and for σ = 3.555, λ = 0.808. We have Efc1 g ˆ 54:41 pphm (whereas the observed (sample) maximum hourly average value was 53 pphm). Similarly, for the annual expected second highest hourly average mixing ratio we have, from (26.89) Efc2 g ˆ 49:40 pphm

26.10 RELATING CURRENT AND FUTURE AIR POLLUTANT STATISTICAL DISTRIBUTIONS The reduction R in current emission source strength to meet an air quality goal can be calculated by the so-called rollback equation Efcg Efcgs (26.90) Rˆ Efcg cb where E{c} is the current annual mean of the pollutant concentration, E{c}s is the annual mean corresponding to the air quality standard cs, and cb is the background concentration assumed to be constant. Since, as we have seen, the air quality standard cs is usually stated in terms of an extreme statistic, such as the concentration level that may be exceeded only once per year, it is necessary to have a probability distribution to relate the extreme concentration cs to the annual mean E{c}s. We assume that if, in the future, the emission level is halved, the annual mean concentration will also be halved. The key question is: If the emission level is halved, what happens to the predicted extreme concentration in the future year; is it correspondingly halved or does it change by more or less than that amount? To address this question, assume that the concentration in question can be represented by a lognormal distribution (under present as well as future conditions). If a current emission rate changes by a factor κ (κ > 0), while the source distribution remains the same (if meteorological conditions are

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STATISTICAL MODELS

unchanged, and if background concentrations are negligible), the expected total quantity of inert pollutants having an impact on a given site over the same time period should also change by the factor κ. The expected concentration level for the future period is therefore given, for a lognormally distributed variable, by (recall μ = ln μg and σ = ln σg) ´

Efc´ g ˆ eμ ‡σ

´2

=2

´

ˆ κeμ ‡σ =2 2

(26.91)

where the primed quantities of c´ , μ´ , and σ´ apply to the future period, and the unprimed quantities apply to the present. On an intuitive basis, if meteorological conditions remain unchanged, the standard ´ geometric deviation of the lognormal pollutant distribution should remain unchanged; that is, eσ ˆ eσ . ´ Thus eμ ˆ κeμ , or (26.92) μ´ ˆ μ ‡ ln κ The probability that future concentration level c´ will exceed a level x is Fc´ …x† ˆ 1

1 …ln x p 2π ∫ 1

μ´ †=σ´

e

η2

dη

(26.93)

ˆ Fc …x=κ† using (26.92). Similarly Fc´ …κx† ˆ 1

1 …ln x p 2π ∫ 1

μ´ †=σ´

e

η2

dη

(26.94)

ˆ Fc …x† Thus the probability that the future level κx will be exceeded just equals the probability that, with current emission sources, the level x will be exceeded. Therefore, with equal σ, all frequency points of the distribution shift according to the factor κ. This results in a parallel translation of the graph of F(x) or F(x), when plotted against ln(x). Figure 26.10 shows two lognormal distributions with the same standard geometric deviation but with different geometric mean values. The geometric mean concentrations of the two distributions are

FIGURE 26.10 Two lognormal distributions with the same standard geometric deviation.

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PROBLEMS

0.05 and 0.10 ppm, and the standard geometric deviations are both 1.4. The mean concentrations can be calculated with the aid of ln Efcg ˆ ln μg ‡ 12 ln2 σg and the material presented above. We find that E1{c} = 0.053 ppm, and E2{c} = 0.106 ppm for the two distributions, respectively. The variances can 2 likewise be calculated with the aid of Varfcg ˆ ‰exp…2μ ‡ σ2 †Š‰eσ 1Š to obtain Var1{c} = 0.00034 ppm2 2 and Var2{c} = 0.00134 ppm . Suppose that distribution 1 represents current conditions, and therefore that the current probability of exceeding a concentration of 0.13 ppm is about 0.0027 (which corresponds to about 1 day per year if the distribution is of 24-h averages). If the emission rate were doubled, the new distribution function would be given by distribution 2. The new distribution has a median value twice that of the old one, since total loadings attributable to emissions have doubled. In the new case, a concentration of 0.13 ppm will be exceeded 22% of the time, or about 80 days a year, and the concentration that is exceeded only 1 day per year rises to 0.26 ppm.

PROBLEMS 26.1A Show how to construct the axes of an extreme-value probability graph on which a Weibull distribution will plot as a straight line. 26.2A Figure 26P.1 shows the frequency distributions of SO2 at two locations in the eastern United States from August 1977 to July 1978. Using the data points on the figure, fit lognormal distributions to the SO2 concentrations at Duncan Falls, Ohio and Montague, Massachusetts. If the data points do not fall exactly on the best-fit lognormal lines, comment on possible reasons for deviations of the measured concentrations from lognormality.

FIGURE 26P.1

26.3A Figure 26P.2 shows the frequency distributions of CO concentrations measured inside and outside an automobile traveling a Los Angeles commuter route. Fit lognormal distributions to the exterior 1-min and 30-min averaging time data. If the data points do not fall exactly on the best-fit lognormal lines, comment on possible reasons for deviations of the measured concentra­ tions from lognormality. What can be said about the relationship of the best-fit parameters for the lognormal distributions at 1-min and 30-min averaging times. 26.4B Table 26P.1 gives CO concentrations at Pasadena, California, for 1982—in particular, the weekly maximum 1-h average concentrations and the monthly mean of the daily 1-h average maximum concentrations.

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STATISTICAL MODELS

FIGURE 26P.2

TABLE 26P.1 Carbon Monoxide Concentrations at Pasadena, California, for 1982: Some Statistics of 1-Hour Average Data (ppm) Month January February March April May June July August September Octobera November December

Weekly 1-h average maxima

Highest 1-h average value

Mean of daily 1-h average maxima

17, 12, 8, 10 10, 8, 9, 7 5, 7, 6, 6 3, 5, 4, 5, 2 3, 4, 3, 3 5, 4, 3, 4 2, 3, 3, 4, 3 6, 3, 5, 4 7, 6, 4, 8 8, 12, 11, —, 11 13, 10, 10, 11 15, 14, 20, 8, 13

17 10 7 5 4 5 4 6 8 (12) 13 20

7.2 5.3 4.0 2.9 3.2 2.5 2.4 2.8 3.8 (7.2) 6.9 8.9

a

No data were available for the fourth week of October.

a. Plot the weekly 1-h average maximum values on log–probability paper and extreme-value probability paper. b. Determine the two parameters of both the lognormal and Weibull distributions by the method of moments and the method of quantiles. On the basis of your results, select one of the two distributions as representing the best fit to the data. c. What was the expected number of exceedances of the weekly 1-h average maximum CO concentration at Pasadena of the National Ambient Air Quality Standard for CO of 35 ppm? d. What is the expected number of weekly 1-h average maximum observations between successive exceedances of 35 ppm? e. What is the variance of the number of weekly 1-h average maximum observations between successive exceedances of 35 ppm? f. For the expected exceedance of 35 ppm to be once in 52 weeks, how must the parameters of the distribution change?

REFERENCES

26.5B Consider a single elevated continuous point source at a height h of strength q. Assume that the wind blows with a speed that is lognormally distributed with parameters μug and σug and with a direction that is uniformly distributed over 360°. No inversion layer exists. a. Determine the form of the statistical distribution of hc(x, y, 0)i in terms of the known quantities of the problem. b. Assuming that the source strength changes from q to κq, determine the form of the new distribution of hc(x, y, 0)i. c. Assuming that the air quality standard is related to the value that is exceeded only once a year, derive an expression for the change in this value as a function of location resulting from the source strength change. 26.6B Given that the CO emission level from the entire motor vehicle population is reduced to one-half its value at the time the data in Figure 26P.2 were obtained, calculate the expected changes in the three CO frequency distributions. (Note that in order to calculate this change, one needs the results of Problem 26.3) How does the frequency at which a level of 50 ppm occurs change because of a halving of the emission rate? How do the expected return period and its variance change for the 50 ppm level from the old to the new emission level? 26.7D Calculate the contributions of sources (vegetative burning, primary crude oil, motor vehicles, limestone, ammonium sulfate, ammonium nitrate, and secondary OC) to the PM2.5 concentra­ tions observed in Fresno, CA during 1988–1989 (Table 26.2). Use the CMB for nitrate, sulfate, ammonium, EC, OC, Al, Si, S, K, and V, with the source profiles given in Table 26.1. Use only the measurement uncertainties and compare your results with the estimates of Chow et al. (1992) in Table 26.3. 26.8D Using the results of Problem 26.7 as an initial guess, apply the effective variance method to the same problem. Discuss your results. 26.9D Verify the calculations of Section 26.3.1 applying the principal-component analysis method to the correlation matrix for elements measured in Whiteface Mountain, New York (Table 26.4).

REFERENCES Abdi, H., and Williams, L. J. (2010), Principal component analysis, Wiley Interdisc. Rev. Comput. Stat. 2, 433–459. Ashbaugh, L. L., et al. (1984), A principal component analysis of sulfur concentrations in the western United States, Atmos. Environ. 18, 783–791. Ashbaugh, L. L., et al. (1985), A residence time probability analysis of sulfur concentrations at Grand Canyon National Park, Atmos. Environ. 19, 1263–1270. Begum, B. A., et al. (2005), Evaluation of the potential source contribution function using the 2002 Quebec forest fire episode, Atmos. Environ. 39, 3719–3724. Bencala, K., and Seinfeld, J. H. (1976), On frequency distributions of air pollutant concentrations, Atmos. Environ. 10, 941–950. Cheng, M. D., et al. (1988), Resolving PM10 data collected in New Jersey by various multivariate analysis techniques, in PM10: Implementation of Standards, C. V. Mathai and D. H. Stonefield (ed.), Air Pollution Control Association, Pittsburgh, PA, pp. 472–483. Chow, J. C., et al. (1992), PM10 source apportionment in California’s San Joaquin Valley, Atmos. Environ. 26A, 3335–3354. Chow, J. C., et al. (1993), PM10 and PM2.5 compositions in California’s San Joaquin Valley, Aerosol Sci. Technol. 18, 105–128. Coulter C. T., et al. (2004), EPA-CMB8.2 Users Manual, Office of Air Quality Planning & Standards, Research Triangle Park, NC.

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STATISTICAL MODELS Curran, T. C., and Hunt, W. F., Jr. (1975), Interpretation of air quality data with respect to the national ambient air quality standards, J. Air Pollut. Control Assoc. 25, 711–714. David, H. A. (1981), Order Statistics, 2nd ed., Wiley, New York. Draxler, R. R. (1999), HYSPLIT4 User’s Guide, NOAA Technical Memo ERL ARL-230, NOAA Air Resources Laboratory, Silver Spring, MD. Gebhart, K. A., et al. (1990), Empirical orthogonal function analysis of the particulate sulfate concentrations measured during WHITEX, in Visibility and Fine Particles, C. V. Mathai (ed.), AWMA TR-17, Air & Waste Management Association, Pittsburgh, PA, pp. 860–871. Georgopoulos, P. G., and Seinfeld, J. H. (1982), Statistical distributions of air pollutant concentrations, Environ. Sci. Technol. 16, 401A–416A. Gerlach, R. W., et al. (1983), Review of the Quail Roost II receptor model simulation exercise, in Receptor Models Applied to Contemporary Pollution Problems, S. L. Dattner and P. K. Hopke (eds.), Air Pollution Control Association, Pittsburgh, PA, pp. 96–109. Glover, D. M., et al. (1991), Source apportionment with site specific source profiles, J. Air Waste Manage. Assoc. 41, 294–305. Gumbel, E. J. (1958), Statistics of Extremes, Columbia Univ. Press, New York. Henry, R. C. (1983), Stability analysis of receptor models that use least-squares fitting, in Receptor Models Applied to Contemporary Pollution Problems, S. L. Dattner and P. K. Hopke (eds.), Air Pollution Control Association, Pittsburgh, PA, pp. 141–157. Henry R. C. (1987), Current factor analysis receptor models are ill-posed, Atmos. Environ. 21, 1815–1820. Henry, R. C. (1997), History and fundamentals of multivariate air quality receptor models, Chemom. Intell. Lab. Syst. 37, 37–42. Henry, R. C., and Hidy, G. M. (1979), Multivariate analysis of particulate sulfate and other air quality variables by principal components, I. Annual data from Los Angeles and New York, Atmos. Environ. 13, 1581–1596. Henry, R. C., and Hidy, G. M. (1981), Multivariate analysis of particulate sulfate and other air quality variables by principal components, II. Salt Lake City, Utah and St. Louis, Missouri, Atmos. Environ. 16, 929–943. Henry, R. C., and Kim, B. M. (1989), A factor analysis receptor model with explicit physical constraints, in Receptor Models in Air Resources Management, J. G. Watson (ed.), APCA Transactions 14, Air Pollution Control Association, Pittsburgh, PA, pp. 214–225. Henry, R. C., et al. (1991), The relationship between empirical orthogonal functions and sources of air pollution, Atmos. Environ. 24A, 503–509. Holland, D. M., and Fitz-Simons, T. (1982), Fitting statistical distributions to air quality data by the maximum likelihood method, Atmos. Environ. 16, 1071–1076. Hopke, P. K. (1985), Receptor Modeling in Environmental Chemistry, Wiley, New York. Hopke, P. K. (1991), Receptor Modeling for Air Quality Management, Elsevier, New York. Huber, P. J. (1981), Robust Statistics, Wiley, New York. Kaiser, H. F. (1959), Computer program for varimax rotation in factor analysis, Educ. Psychol. Meas. 33, 99–102. Koutrakis, P., and Spengler, J. D. (1987), Source apportionment of ambient particles in Steubenville, OH using specific rotation factor analysis, Atmos. Environ. 21, 1511–1519. Kowalczyk, G. S., et al. (1982), Identification of atmospheric particulate sources in Washington, D.C. using chemical element balances, Environ. Sci. Technol. 16, 79–90. Larsen, R. I. (1971), Mathematical Model for Relating Air Quality Measurements to Air Quality Standards, EPA Publication A-89, US Environmental Protection Agency, Research Triangle Park, NC. Mage, D. T. (1980), The statistical form for the ambient particulate standard annual arithmetic mean versus geometric mean, J. Air Pollut. Control Assoc. 30, 796–798. Miller, M. S., et al. (1972), A chemical element balance for the Pasadena aerosol, J. Colloid Interface Sci. 39, 165–176. Norris, G., et al. (2007), EPA Unmix 6.0 Fundamentals & User Guide, US Environmental Protection Agency, Office of Research and Development, Washington, DC. Paatero, P. (1997), Least squares formulation of robust non-negative factor analysis. Chemom. Intell. Lab. Syst. 37, 23–35. Paatero, P. (1999), The multilinear engine—a table-driven, least squares program for solving multilinear problems, including the n-way parallel factor analysis model. J. Comp. Graph. Stat. 8, 854–888.

REFERENCES Paatero, P., and Tapper, U. (1994), Positive matrix factorization: A non-negative factor model with optimal utilization of error estimates of data values, Environmetrics 5, 111–126. Paatero, P. (2000), User’s Guide for the Multilinear Engine Program ME-2 for Fitting Multilinear and Quasi-multilinear Models, Univ. Helsinki, Helsinki, Finland. Paatero, P., et al. (2002), Understanding and controlling rotations in factor analytic models, Chemom. Intell. Lab. Syst. 60, 253–264. Paatero, P., and Hopke, P. K. (2009), Rotational tools for factor analytic models, J. Chemometrics 23, 91–100. Parekh, P. P., and Husain, L. (1981), Trace element concentrations in summer aerosols at rural sites in New York State and their possible sources, Atmos. Environ. 15, 1717–1725. Pearson, K. (1934), Tables of the Incomplete Beta Function, Biometrika Office, London. Polissar, A. V., et al. (1998), Atmospheric aerosol over Alaska: 2. Elemental composition and sources. J. Geophys. Res. 103, 19045–19057. Sarhan, A. E., and Greenberg, B. G. (1962), Contributions to Order Statistics, Wiley, New York. Shlens, J. (2014), A Tutorial on Principal Component Analysis, arXiv Preprint arXiv:1404.1100. Sun, Y. L., et al. (2011), Characterization of the sources and processes of organic and inorganic aerosols in New York City with a high resolution time-of-flight aerosol mass spectrometer, Atmos. Chem. Phys. 11, 1581–1602. Tsukatami, T., and Shigemitsu, K. (1980), Simplified Pearson distribution applied to an air pollutant concentration, Atmos. Environ. 14, 245–253. Watson, J. G. (1979), Chemical Element Balance Receptor Model Methodology for Assessing the Source of Fine and Total Suspended Particulate Matter in Portland, Oregon, PhD thesis, Oregon Graduate Center, Beaverton. Winchester, J. W., and Nifong, G. D. (1971), Water pollution in Lake Michigan by trace elements from pollution aerosol fallout, Water Air Soil Pollut. 1, 50–64. Wolff, G. T., and Korsog, P. E. (1985), Estimates of the contributions of sources to inhalable particulate concentrations in Detroit, Atmos. Environ. 19, 1399–1409. Wolff, G. T., et al. (1985), The influence of local and regional sources on the concentration of inhalable particulate matter in south-eastern Michigan, Atmos. Environ. 19, 305–313. Zeng, Y., and Hopke, P. K. (1989), Three-mode factor analysis: A new multivariate method for analyzing spatial and temporal composition variation, in Receptor Models in Air Resources Management, J. G. Watson (ed.), APCA Transactions 14, Air Pollution Control Association, Pittsburgh, PA, pp. 173–189. Zhang, Q., et al. (2011), Understanding atmospheric organic aerosols via factor analysis of aerosol mass spectrome­ try: A review, Anal. Bional. Chem. 401, 3045–3067.

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APPENDIX

A

Units and Physical Constants

The units used in this text are more or less those of the International System of Units (SI). We have chosen not to adhere strictly to the SI system because in several areas of the book the use of SI units would lead to cumbersome and unfamiliar magnitudes of quantities. We have attempted to use, as much as possible, a consistent set of units throughout the book while attempting not to deviate markedly from the units commonly used in the particular area. For an excellent discussion of units in atmospheric chemistry, we refer the reader to Schwartz and Warneck (1995).

A.1 SI BASE UNITS

TABLE A.1 SI Base Units SI base unit Base quantity Length Mass Time Electric current Thermodynamic temperature Amount of substance Luminous intensity

Name

Symbol

meter kilogram second ampere kelvin mole candela

m kg s A K mol cd

Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, Third Edition. John H. Seinfeld and Spyros N. Pandis.  2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

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APPENDIX A: UNITS AND PHYSICAL CONSTANTS

A.2 SI DERIVED UNITS Derived units are expressed algebraically in terms of base units or other derived units (including the radian and steradian, which are the two supplementary units). For example, the derived unit for the derived quantity molar mass (mass divided by amount of substance) is the kilogram per mole, symbol kg mol 1. Additional examples of derived units expressed in terms of SI base units are given in Table A.2. Certain SI derived units have special names and symbols; these are given in Table A.3. Table A.4 presents examples of SI derived units expressed with the aid of SI derived units having special names and symbols. Table A.5 presents standard prefixes. Concentration units are used in connection with chemical reaction rates and optical extinction. Traditional concentration units are mol L 1 (the SI unit of the liter is the cubic decimeter, dm3), mol cm 3, or mol m 3. Concentrations are also expressed as number concentration, molecules cm 3. (In expressing number concentration, centimeter-based units are more widely used than meter-based units.)

TABLE A.2 Examples of SI Derived Units Expressed in Terms of SI Base Units SI derived unit Derived quantity Area Volume Speed, velocity Acceleration Wavenumber Mass density (density) Specific volume Current density Magnetic field strength Amount-of-substance concentration (concentration) Luminance

Name

Symbol

square meter cubic meter meter per second meter per second squared reciprocal meter kilogram per cubic meter cubic meter per kilogram ampere per square meter ampere per meter mole per cubic meter candela per square meter

m2 m3 ms 1 ms 2 m 1 kg m 3 m3 kg 1 Am 2 Am 1 mol m 3 cd m 2

TABLE A.3 SI Derived Units with Special Names and Symbols, Including the Radian and the Steradian SI derived unit Derived quantity Plane angle Solid angle Frequency Force Pressure, stress Energy, work, quantity of heat Power, radiant flux

Special name radian steradian hertz newton pascal joule watt

Special symbol rad sr Hz N Pa J W

Expression in terms of other SI units

Nm Nm Js 1

2

Expression in terms of SI base units m m 1=1 m2m 2 = 1 s 1 m kg s 2 m 1 kg s 2 m2 kg s 2 m2 kg s 3

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SI DERIVED UNITS

TABLE A.3 (Continued ) SI derived unit Derived quantity Electric charge, quantity of electricity Electric potential, potential difference, electromotive force Capacitance Electric resistance Electric conductance Magnetic flux Magnetic flux density Inductance Celsius temperature Luminous flux Illuminance

Special symbol

Special name coulomb volt

C V

farad ohm siemens weber tesla henry degree Celsius lumen lux

F Ω S Wb T H °C lm lx

Expression in terms of other SI units WA

sA m2 kg s

1

CV 1 VA 1 AV 1 Vs Wb m Wb A cd sr lm m

Expression in terms of SI base units

2 1

2

3

A

1

m 2 kg 1 s4 A2 m2 kg s 3 A 2 m 2 kg 1 s3 A2 m2 kg s 2 A 1 kg s 2 A 1 m2 kg s 2 A 2 K cd sr m 2 cd sr

TABLE A.4 Examples of SI Derived Units Expressed with the Aid of SI Derived Units Having Special Names and Symbols SI derived unit Derived quantity

Name

Symbol

Expression in terms of SI base units

Angular velocity Angular acceleration Dynamic viscosity Moment of force Surface tension Heat flux density, irradiance Radiant intensity Radiance Heat capacity, entropy Specific-heat capacity, specific entropy Specific energy Thermal conductivity Energy density Electric field strength Electric charge density Electric flux density Permittivity Permeability Molar energy Molar entropy, molar heat capacity

radian per second radian per second squared pascal second newton meter newton per meter watt per square meter watt per steradian watt per square meter steradian joule per kelvin joule per kilogram kelvin joule per kilogram watt per meter kelvin joule per cubic meter volt per meter coulomb per cubic meter coulomb per square meter farad per meter henry per meter joule per mole joule per mole kelvin

rad s 1 rad s 2 Pa s Nm Nm 1 Wm 2 W sr 1 W (m2 sr) 1 JK 1 J (kg K) 1 J kg 1 W (m K) 1 Jm 3 Vm 1 Cm 3 Cm 2 Fm 1 Hm 1 J mol 1 J (mol K) 1

m m 1 s 1=s 1 m m 1 s 2=s 2 m 1 kg s 1 m2 kg s 2 kg s 2 kg s 3 m2 kg s 3 sr 1 kg s 3 sr 1 m2 kg s 2 K 1 m2 s 2 K 1 m2 s 2 m kg s 3 K 1 m 1 kg s 2 m kg s 3 A 1 m 3sA m 2sA m 3 kg 1 s4 A2 m kg s 2 A 2 m2 kg s 2 mol 1 m2 kg s 2 K 1 mol

1

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APPENDIX A: UNITS AND PHYSICAL CONSTANTS

TABLE A.5 Standard Prefixes Factor 18

10 10 15 10 12 10 9 10 6 10 3 10 2 10 1 101 102 103 106 109 1012 1015 1018

Prefix

Symbol

atto femto pico nano micro milli centi deci deca hecto kilo mega giga tera peta exa

a f p n μ m c d da h k M G T P E

A.3 FUNDAMENTAL PHYSICAL CONSTANTS

TABLE A.6 Fundamental Physical Constants Speed of light in vacuum Planck constant Elementary charge Electron mass Avogadro number Faraday constant Molar gas constant Boltzmann constant Molar volume (ideal gas), RT/p (at 273.15 K, 101.325 × 103 Pa) Stefan–Boltzmann constant Standard acceleration of gravity

c h e me NA F R k vm σ g

2.9979 × 108 6.626 × 10 34 1.602 × 10 19 9.109 × 10 31 6.022 × 1023 96485 8.314 1.381 × 10 23 22.414 × 10 3 5.67 × 10 8 9.807

Source: Cohen and Taylor (1995).

A.4 PROPERTIES OF THE ATMOSPHERE AND WATER

TABLE A.7 Properties of the Atmospherea Standard temperature Standard pressure Standard gravity Air density Molecular weight Mean molecular mass Molecular root-mean-square velocity (3RT0/M0)1/2

T0 = 0°C = 273.15 K p0 = 760 mm Hg = 1013.25 millibar (mbar) g0 = 9.807 m s 2 ρ0 = 1.29 kg m 3 M0 = 28.97 g mol 1 = 4.81 × 10 23 g = 4.85 × 104 cm s 1

ms 1 Js C kg mol 1 C mol 1 J mol 1 K JK 1 m3 mol 1 Wm 2K ms 2

1

4

1095

PROPERTIES OF THE ATMOSPHERE AND WATER TABLE A.7 (Continued ) = 3.31 × 104 cm s 1 c p̂ = 1005 J K 1 kg 1 c ̂v = 717 J K 1 kg 1 c p̂ /c ̂v = γ = 1.401 N = 2.688 × 1019 (at 273 K) = 2.463 × 1019 (at 298 K) σ = 3.46 × 10 8 cm λa = 6.98 × 10 6 cm μ = 1.72 × 10 4 g cm 1 s 1 k = 2.40 × 10 2 J m 1 s 1 K (n 1) × 108 = 6.43 × 103 6 4 ‡ 2:95  102 ‡ 2:55  102 146 λ 41 λ

Speed of sound (γRT0/M0)1/2 Specific heats Ratio Air molecules per cm3 Air molecular diameter p Air mean free path 2π Nσ2 Viscosity Thermal conductivity Refractive index (real part)

1

1

…λ in μm†

Dry air at T = 273 K and 1 atm.

a

TABLE A.8 US Standard Atmosphere, 1976 Water vapor Height (km) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 30 35 40 45 50 70 100

Pressure (hPa)

Temperature (K)

Density (g m 3)

1.013 × 103 8.98 × 102 7.950 × 102 7.012 × 102 6.166 × 102 5.405 × 102 4.722 × 102 4.111 × 102 3.565 × 102 3.080 × 102 2.650 × 102 2.270 × 102 1.940 × 102 1.658 × 102 1.417 × 102 1.211 × 102 1.035 × 102 8.850 × 101 7.565 × 101 6.467 × 101 5.529 × 101 4.729 × 101 4.048 × 101 3.467 × 101 2.972 × 101 2.549 × 101 1.197 × 101 5.746 2.871 1.491 7.978 × 10 1 5.220 × 10 2 3.008 × 10 4

288 282 275 269 262 256 249 243 236 230 223 217 217 217 217 217 217 217 217 217 217 218 219 220 221 222 227 237 253 264 271 220 210

1.225 × 103 1.112 × 103 1.007 × 103 9.093 × 102 8.194 × 102 7.364 × 102 6.601 × 102 5.900 × 102 5.258 × 102 4.671 × 102 4.135 × 102 3.648 × 102 3.119 × 102 2.666 × 102 2.279 × 102 1.948 × 102 1.665 × 102 1.423 × 102 1.217 × 102 1.040 × 102 8.891 × 101 7.572 × 101 6.450 × 101 5.501 × 101 4.694 × 101 4.008 × 101 1.841 × 101 8.463 3.996 1.966 1.027 8.283 × 10 2 4.990 × 10 4

Moist stratosphere (g m 3) 5.9 4.2 2.9 1.8 1.1 6.4 × 10 3.8 × 10 2.1 × 10 1.2 × 10 4.6 × 10 1.8 × 10 8.2 × 10 3.7 × 10 1.8 × 10 8.4 × 10 7.2 × 10 5.5 × 10 4.7 × 10 4.0 × 10 4.1 × 10 4.0 × 10 4.4 × 10 4.6 × 10 5.2 × 10 5.4 × 10 6.1 × 10 3.2 × 10 1.3 × 10 4.8 × 10 2.2 × 10 7.8 × 10 1.2 × 10 3.0 × 10

1 1 1 1 2 2 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 6 7 4

Dry stratosphere (g m 3) 5.9 4.2 2.9 1.8 1.1 6.4 × 10 3.8 × 10 2.1 × 10 1.2 × 10 4.6 × 10 1.8 × 10 8.2 × 10 3.7 × 10 1.8 × 10 8.4 × 10 7.2 × 10 3.3 × 10 2.8 × 10 2.4 × 10 2.1 × 10 1.8 × 10 1.5 × 10 1.3 × 10 1.1 × 10 9.4 × 10 8.0 × 10 3.7 × 10 1.7 × 10 7.9 × 10 3.9 × 10 2.1 × 10 1.8 × 10 1.0 × 10

1 1 1 1 2 2 3 3 3 4 4 4 4 4 4 4 4 4 4 5 5 5 5 6 6 6 7 9

1096

APPENDIX A: UNITS AND PHYSICAL CONSTANTS

TABLE A.9 Properties of Water Mass basis

Molar basis

Specific heat of water vapor at constant pressure c p̂ w Specific heat of water vapor at constant volume c v̂ w Specific heat of liquid H2O at 273 K

1952 J kg 1463 J kg 4218 J kg

Latent heat of vaporization At 273 K At 373 K Latent heat of fusion, 273 K Surface tension (water vs. air)

2.5 × 106 J kg 1 2.25 × 106 J kg 1 3.34 × 105 J kg 1 0.073 J m 2

1

1

1

1

K K 1 K

1

35.14 J mol 26.33 J mol 75.92 J mol

1

1

1

1

K K 1 K

1

4.5 × 104 J mol 1 4.05 × 104 J mol 1 5.94 × 103 J mol 1

A.5 UNITS FOR REPRESENTING CHEMICAL REACTIONS The rate of a bimolecular reaction between substances A and B may be written as dcA ˆ kcA cB dt where cA and cB are the concentrations of A and B, respectively. The second-order rate constant k has units concentration 1 time 1. Possible units for k are cm3 molecule 3

m mol

1

s

1

s

1

1

Table A.10 compares ranges of concentrations and bimolecular and termolecular rate constants pertinent to atmospheric chemistry for these two units. For example, the hydroxyl (OH) radical has an average tropospheric concentration of about 8 × 105 molecule cm 3. This is equivalent to 1.3 × 10 12 mol m 3 (1.3 pmol m 3).

TABLE A.10 Typical Ranges of Values in Atmospheric Chemistry System Concentration unit Bimolecular rate constant unit Termolecular rate constant unit

Molecule (cm3 units) 104–1019 molecule cm 3 10 18–10 10 cm3 molecule 10 36–10 29 cm6 molecule

Mole (m3 units) 1

s 2 s

1 1

10 14–10 mol m 3 1–108 m3 mol 1 s 1 1–107 m6 mol 2 s 1

Source: Schwartz and Warneck (1995).

A.6 CONCENTRATIONS IN THE AQUEOUS PHASE Concentrations of substances dissolved in water droplets and present in particulate matter are of great importance in atmospheric chemistry. A commonly used unit of concentration in the chemical thermo­ dynamics of solutions is molality, mole of solute per kilogram of solvent (mol kg 1). One advantage of the use of molality is that the value is unaffected by changes in the density of solution as temperature changes. For aqueous-phase chemical reactions the commonly used concentration unit is mol L 1. Aqueous solutions in cloud and raindrops are characterized by concentrations in the range of μmol L 1. For a

1097

REFERENCES

dilute aqueous solution molality (mol kg 1) is approximately equal to molarity (mol L 1). The conversion between molar concentration c and molality m is mˆ

ρ

c Mc

where ρ is the density of solution (kg m 3); M is the molecular weight of solute (kg mol 1), and c is concentration (mol m 3).

A.7 SYMBOLS DENOTING CONCENTRATION The following symbols for concentration are used in this book: nA cA or [A] [A] mA

Gas-phase number concentration of species A (molecule cm 3) Gas-phase molar concentration of species A (mol m 3) Aqueous-phase concentration of solute A (mol L 1) Molality of solute A (mol kg 1)

REFERENCES Cohen, E. R., and Taylor, B. N. (1995), The fundamental physical constants, Phys. Today (Aug.) BG9–BG16. Schwartz, S. E., and Warneck, P. (1995), Units for use in atmospheric chemistry, Pure Appl. Chem. 67, 1377–1406.

APPENDIX

B

Rate Constants of Atmospheric Chemical Reactions

See Tables B.1–B.10. TABLE B.1 Second-Order Reactions: k(T) = A exp( − E/RT) A (cm3 molecule

Reaction Ox and O…1 D† reactions O + O3 → O2 + O2 O(1 D) + O2 → O + O2 O(1 D) + H2O → OH + OH O(1D) + N2 → O + N2 O(1D) + N2O → N2 + O2 → NO + NO O(1D) + CH4 → OH + CH3 HOx Ox reactions O + OH → O2 + H O + HO2 → OH + O2 OH + O3 → HO2 + O2 OH + HO2 → H2O + O2 HO2 + O3 → OH + 2 O2 HO2 + HO2 → H2O2 + O2 M

→ H2 O 2 ‡ O 2 NOx reactions O + NO2 → NO + O2 O + NO3 → O2 + NO2 OH + NH3 → H2O + NH2 HO2 + NO → NO2 + OH NO + O3 → NO2 + O2 NO + NO3 → 2 NO2 NO2 + O3 → NO3 + O2 OH + HNO3 → H2O + NO3

8.0 × 10 12 3.3 × 10 11 1.63 × 10 10 2.15 × 10 11 4.63 × 10 11 7.25 × 10 11 1.75 × 10 10 1.8 × 10 3.0 × 10 1.7 × 10 4.8 × 10 1.0 × 10 3.0 × 10

11

2.1 × 10

33

11 12 11 14 13

[M]

5.1 × 10 12 1.0 × 10 11 1.7 × 10 12 3.3 × 10 12 3.0 × 10 12 1.5 × 10 11 1.2 × 10 13 See footnotea

1

s 1)

E/R (K) 2060 55 60 110 20 0 0

k(298 K) (cm3 molecule

1

s 1)

8.0 × 10 15 4.0 × 10 11 2.0 × 10 10 3.1 × 10 11 4.95 × 10 11 7.75 × 10 11 1.75 × 10 10

180 200 940 250 490 460

3.3 × 10 5.9 × 10 7.3 × 10 1.1 × 10 1.9 × 10 1.4 × 10

11

920

4.6 × 10

32

210 0 710 270 1500 170 2450

1.0 × 10 1.0 × 10 1.6 × 10 8.0 × 10 1.9 × 10 2.6 × 10 3.2 × 10

11

11 14 10 15 12

[M]

11 13 12 14 11 17

Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, Third Edition. John H. Seinfeld and Spyros N. Pandis.  2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

1098

1099

Appendix B: Rate Constants of Atmospheric Chemical Reactions

TABLE B.1 (Continued ) Reaction HOx reactions OH + CO → CO2 + H OH + CH4 → CH3 + H2O OH + HCN → products OH + CH3CN → products HO2 + CH3O2 → CH3OOH + O2 HCO + O2 → CO + HO2 CH2OH + O2 → HCHO + HO2 CH3O + O2 → HCHO + HO2 CH3O2 + CH3O2 → products CH3O2 + NO → CH3O + NO2 C2H5O + O2 → CH3CHO + HO2 C2H5O2 + NO → products CH3C(O)O2 + NO → CH3C(O)O + NO2 FOx reactions OH + CH3F → CH2F + H2O OH + CH2F2 → CHF2 + H2O OH + CHF3 → CF3 + H2O OH + CH3CH2F → products OH + CH3CHF2 → products OH + CH2FCH2F → CHFCH2F + H2O OH + CH3CF3 → CH2CF3 + H2O OH + CH2FCHF2 → products OH + CH2FCF3 → CHFCF3 + H2O OH + CHF2CHF2 → CF2CHF2 + H2O OH + CHF2CF3 → CF2CF3 + H2O OH + CH2FCF2CHF2 → products OH + CF3CF2CH2F → CF3CF2CHF + H2O OH + CF3CHFCHF2 → products OH + CF3CH2CF3 → CF3CHCF3 + H2O OH + CF3CHFCF3 → CF3CFCF3 + H2O ClOx reactions O + ClO → Cl + O2 O + OClO → ClO + O2 OH + HCl → H2O + Cl OH + HOCl → H2O + ClO OH + ClNO2 → HOCl + NO2 OH + CH3Cl → CH2Cl + H2O OH + CH2Cl2 → CHCl2 + H2O OH + CHCl3 → CCl3 + H2O OH + CH2ClF → CHClF + H2O OH + CHFCl2 → CFCl2 + H2O OH + CHF2Cl → CF2Cl + H2O OH + CH3CCl3 → CH2CCl3 + H2O OH + C2HCl3 → products OH + CH3CFCl2 → CH2CFCl2 + H2O OH + CH3CF2Cl → CH2CF2Cl + H2O OH + CH2ClCF2Cl → CHClCF2Cl + H2O OH + CH2ClCF3 → CHClCF3 + H2O OH + CHCl2CF3 → CCl2CF3 + H2O OH + CHFClCF3 → CFClCF3 + H2O OH + CH3CF2CFCl2 → products OH + CF3CF2CHCl2 → products OH + CF2ClCF2CHFCl → products

A (cm3 molecule

1

s 1)

E/R (K)

k(298 K) (cm3 molecule

1.5 × 10 13 (1 + 0.6 patm) 2.45 × 10 12 1.2 × 10 13 7.8 × 10 13 4.1 × 10 13 5.2 × 10 12 9.1 × 10 12 3.9 × 10 14 9.5 × 10 14 2.8 × 10 12 6.3 × 10 14 2.6 × 10 12 8.1 × 10 12

0 1775 400 1050 750 0 0 900 390 300 550 365 270

1.5 × 10 6.3 × 10 3.1 × 10 2.3 × 10 5.2 × 10 5.2 × 10 9.1 × 10 1.9 × 10 3.5 × 10 7.7 × 10 1.0 × 10 8.7 × 10 2.0 × 10

13

2.5 × 10 12 1.7 × 10 12 5.2 × 10 13 2.5 × 10 12 8.7 × 10 13 1.05 × 10 12 1.1 × 10 12 3.9 × 10 12 1.05 × 10 12 1.6 × 10 12 6.0 × 10 13 2.1 × 10 12 1.3 × 10 12 9.4 × 10 13 1.45 × 10 12 6.3 × 10 13

1430 1500 2210 730 975 710 2010 1620 1630 1660 1700 1620 1700 1550 2500 1800

2.1 × 10 1.1 × 10 3.1 × 10 2.2 × 10 3.3 × 10 9.7 × 10 1.3 × 10 1.7 × 10 4.4 × 10 6.1 × 10 2.0 × 10 9.2 × 10 4.4 × 10 5.2 × 10 3.3 × 10 1.5 × 10

14

2.8 × 10 11 2.4 × 10 12 1.8 × 10 12 3.0 × 10 12 2.4 × 10 12 2.4 × 10 12 1.9 × 10 12 2.2 × 10 12 2.4 × 10 12 1.2 × 10 12 1.05 × 10 12 1.64 × 10 12 8.0 × 10 13 1.25 × 10 12 1.3 × 10 12 3.6 × 10 12 5.6 × 10 13 6.3 × 10 13 7.1 × 10 13 7.7 × 10 13 6.3 × 10 13 5.5 × 10 13

85 960 250 500 1250 1250 870 920 1210 1100 1600 1520 300 1600 1770 1600 1100 850 1300 1720 960 1230

3.7 × 10 1.0 × 10 7.8 × 10 5.0 × 10 3.6 × 10 3.6 × 10 1.0 × 10 1.0 × 10 4.1 × 10 3.0 × 10 4.8 × 10 1.0 × 10 2.2 × 10 5.8 × 10 3.4 × 10 1.7 × 10 1.4 × 10 3.6 × 10 9.0 × 10 2.4 × 10 2.5 × 10 8.9 × 10

11

15

1

s 1)

(1 + 0.6 patm)

14 14 12 12 12 15 13 12 14 12 11

14 16 13 14 14 15 14 15 15 15 15 15 15 16 15

13 13 13 14 14 13 13 14 14 15 14 12 15 15 14 14 14 15 15 14 15

(continued )

1100

APPENDIX B: RATE CONSTANTS OF ATMOSPHERIC CHEMICAL REACTIONS

TABLE B.1 (Continued ) Reaction

A (cm3 molecule

HO2 + ClO → HOCl + O2 Cl + O3 → ClO + O2 Cl + CH4 → HCl + CH3 Cl + OClO → ClO + ClO Cl + ClOO → Cl2 + O2 → ClO + ClO ClO + NO → NO2 + Cl ClO + ClO → Cl2 + O2 → ClOO + Cl → OClO + Cl BrOx reactions O + BrO → Br + O2 Br + O3 → BrO + O2 BrO + ClO → Br + OClO → Br + ClOO → BrCl + O2 BrO + BrO → 2 Br + O2 S reactions OH + H2S → SH + H2O O + OCS → CO + SO OH + OCS → CO2 + HS OH + CH3SH → products OH + CH3SCH3 → CH2SCH3 + H2Ob NO3 + CH3SCH3 → CH3SCH2 + HNO3 HOSO2 + O2 → HO2 + SO3

2.7 × 10 2.3 × 10 7.3 × 10 3.4 × 10 2.3 × 10 1.2 × 10 6.4 × 10 1.0 × 10 3.0 × 10 3.5 × 10

12

1.9 × 10 1.6 × 10 9.5 × 10 2.3 × 10 4.1 × 10 1.5 × 10

11

6.1 × 10 2.1 × 10 1.1 × 10 9.9 × 10 1.2 × 10 1.9 × 10 1.3 × 10

12

1

s 1)

E/R (K)

11 12 11 10 11 12 12 11 13

11 13 12 13 12

11 13 12 11 13 12

a

The following apply:

k3 ‰MŠ 1 ‡ k3 ‰MŠ=k2 k0 ˆ 7:2  10 15 exp…785=T† k2 ˆ 4:1  10 16 exp…1440=T† k3 ˆ 1:9  10 33 exp…725=T†

k…T† ˆ k0

b

The rate constant given is for the DMS–OH abstraction path. Source: Sander et al. (2011).

k(298 K) (cm3 molecule

220 200 1280 160 0 0 290 1590 2450 1370

5.6 × 10 1.2 × 10 1.0 × 10 5.8 × 10 2.3 × 10 1.2 × 10 1.7 × 10 4.8 × 10 8.0 × 10 3.5 × 10

12

230 780 550 260 290 230

4.1 × 10 1.2 × 10 6.0 × 10 5.5 × 10 1.1 × 10 3.2 × 10

11

75 2200 1200 360 280 530 330

4.7 × 10 1.3 × 10 1.9 × 10 3.3 × 10 4.7 × 10 1.1 × 10 4.4 × 10

12

11 13 11 10 11 11 15 15 15

12 12 12 12 12

14 15 11 12 12 13

1

s 1)

1101

Appendix B: Rate Constants of Atmospheric Chemical Reactions

TABLE B.2 Termolecular Reactions Low-pressure limita

High-pressure limita

n k0 …T† ˆ k300 0 …T=300† 6 2 1 (cm molecule s )

m k1 …T† ˆ k300 1 …T=300† 3 1 1 (cm molecule s )

k300 0

Reaction

6.0 × 10 9.0 × 10 2.5 × 10 7.0 × 10 1.8 × 10 2.0 × 10 4.5 × 10 2.3 × 10 5.3 × 10 1.5 × 10 9.7 × 10 1.8 × 10 1.6 × 10 5.2 × 10 3.3 × 10

O + O2 + M → O3 + M O + NO + M → NO2 + M O + NO2 + M → NO3 + M OH + NO + M → HONO + M OH + NO2 + M → HNO3 + M NO2 + NO3 + M → N2O5 + M CH3 + O2 + M → CH3O2 + M CH3O + NO + M → CH3ONO + M CH3O + NO2 + M → CH3ONO2 + M CH3O2 + NO2 + M → CH3O2NO2 + M CH3C(O)O2 + NO2 + M → CH3C(O)O2NO2 + M ClO + NO2 + M → ClONO2 + M ClO + ClO + M → Cl2O2 + M BrO + NO2 + M → BrONO2 + M OH + SO2 + M → HOSO2 + M

k300 1

n 34

— 3.0 × 10 2.2 × 10 3.6 × 10 2.8 × 10 1.4 × 10 1.8 × 10 3.8 × 10 1.9 × 10 6.5 × 10 9.3 × 10 1.5 × 10 3.0 × 10 6.9 × 10 1.6 × 10

2.4 1.5 1.8 2.6 3.0 4.4 3.0 2.8 4.4 4.0 5.6 3.4 4.5 3.2 4.3

32 31 31 30 30 31 29 29 30 29 31 32 31 31

m 11 11 11 11 12 12 11 11 12 12 11 12 12 12

— 0 0.7 0.1 0 0.7 1.7 0.6 1.8 2.0 1.5 1.9 2.0 2.9 0

a

Rate constants for association reactions of the type

A ‡ B ƒ ‰ABŠ∗ → AB M

are given in the form

k0 …T† ˆ k0300

T 300

n

cm6 molecule

2

s

1

where k300 accounts for air as the third body. Where pressure falloff corrections are necessary, the limiting high-pressure rate 0 constant is given by

k1 …T† ˆ k300 1

T 300

m

cm3 molecule

1

s

1

To obtain the effective second-order rate constant at a given temperature and pressure (altitude z), the following formula is used:

k…T; z† ˆ

Source: Sander et al. (2011).

2 k0 …T†‰MŠ 0:6f1‡‰log10 …k0 …T†‰MŠ=k1 …T††Š g 1 ‡ …k0 …T†‰MŠ=k1 …T††

1

1102

APPENDIX B: RATE CONSTANTS OF ATMOSPHERIC CHEMICAL REACTIONS

TABLE B.3 Alkane–OH Rate Constants: k(T) = CT2 exp( − E/RT) Alkane Ethane Propane n-Butane n-Pentane 2-Methylbutane n-Hexane 2-Methylpentane 2,3-Dimethylbutane n-Heptane 2,2-Dimethylpentane 2,4-Dimethylpentane 2,2,3-Trimethylbutane Methylcyclohexane

k(298 K) × 1012 (cm3 molecule

1

s 1)

0.254 1.12 2.44 4.00 3.7 5.45 5.3 5.78 7.02 3.4 5.0 4.24 10.0

C × 1018 (cm3 molecule

1

s 1)

E/R (K)

15.2 15.5 16.9 24.4

498 61 145 183

15.3

414

12.4 15.9

494 478

8.46

516

Source: Atkinson (1997).

TABLE B.4 Alkene–OH Rate Constants at 760 torr Total Pressure of Air:a k(T) = A exp( − E/RT) Alkene b

Ethene Propenec 1-Butene cis-2-Butene trans-2-Butene 2-Methylpropene 1-Pentene 3-Methyl-1-butene 2-Methyl-1-butene 2-Methyl-2-butene 1-Hexene 2,3-Dimethyl-2-butene 1,3-Butadiene Cyclohexene a

k(298 K) × 1012 (cm3 molecule

1

s 1)

8.52 26.3 31.4 56.4 64.0 51.4 31.4 31.8 61 86.9 37 110 66.6 67.7

Except for ethene and propene, these are the high-pressure rate constants k1. k1 = 9.0 × 10 12 (T/298) 1.1 c k1 = 2.8 × 10 11 (T/298) 1.3 b

Source: Atkinson (1997).

A × 1012 (cm3 molecule

1

s 1)

E/R (K)

1.96 4.85 6.55 11.0 10.1 9.47

438 504 467 487 550 504

5.32

533

14.8

448

1103

Appendix B: Rate Constants of Atmospheric Chemical Reactions

TABLE B.5 Alkene–O3 Rate Constants: k(T) = A exp( − E/RT) k(298 K) × 1018 (cm3 molecule

Alkene Ethene Propene 1-Butene cis-2-Butene trans-2-Butene 1-Pentene 2-Methyl-2-butene 1-Hexene 2,3-Dimethyl-2-butene 1,3-Butadiene Cyclohexene

1

s 1)

A × 1015 (cm3 molecule

1.59 10.1 9.64 125 190 10.0 403 11.0 1130 6.3 81.4

1

s 1)

E/R (K)

9.14 5.51 3.36 3.22 6.64

2580 1878 1744 968 1059

6.51

829

3.03 13.4 2.88

294 2283 1063

Source: Atkinson (1997).

TABLE B.6 Alkene–NO3 Rate Constants: k(T) = A exp( − E/RT) k(298 K) (cm3 molecule

Alkene Ethenea Propene 1-Butene cis-2-Butene trans-2-Buteneb 2-Methyl-2-butene 2,3-Dimethyl-2-butene 1,3-Butadiene Cyclohexene k = 4.88 × 10 k = 1.22 × 10

a

18 2

b

18 2

2.05 × 10 16 9.49 × 10 15 1.35 × 10 14 3.50 × 10 13 3.90 × 10 13 9.37 × 10 12 5.72 × 10 11 1.0 × 10 13 5.9 × 10 13

T exp( 2282/T) over the range 290–523 K. T exp (382/T) over the range 204–378 K.

Source: Atkinson (1997).

1

s 1)

A (cm

3

molecule

4.59 × 10 3.14 × 10

13

1.05 × 10

12

13

1

s 1)

E/R (K) 1156 938

174

1104

APPENDIX B: RATE CONSTANTS OF ATMOSPHERIC CHEMICAL REACTIONS

TABLE B.7 Aromatic–OH Rate Constants at 298 K: kab = Abstraction Path and kad = Addition Path k(298 K) × 1012 (cm3 molecule

Aromatic

k(T) = 2.33 × 10 k(T) = 1.81 × 10

12

b

12

kab =kab ‡ kad

s 1)

1.22a 5.63b 7.0 13.6 23.1 14.3 11.9 18.6 11.8 32.7 32.5 56.7 58 12.9 27 41 68 50

Benzene Toluene Ethylbenzene o-Xylene m-Xylene p-Xylene o-Ethyltoluene m-Ethyltoluene p-Ethyltoluene 1,2,3-Trimethylbenzene 1,2,4-Trimethylbenzene 1,3,5-Trimethylbenzene Styrene Benzaldehyde Phenol o-Cresol m-Cresol p-Cresol a

1

0.05 0.12 0.10 0.04 0.08

0.06 0.06 0.03

exp(193/T). exp(338/T).

Source: Calvert et al. (2002).

TABLE B.8 Oxygenated Organic–OH Rate Constants: k(T) = CTn exp( − E/RT) Organic Aldehydes HCHO CH3CHO CH3CH2CHO HOCH2CHO Ketones CH3C(O)CH3 CH3C(O)CH2CH3 α-Dicarbonyls (CHO)2 CH3C(O)CHO CH3C(O)C(O)CH3 Alcohols CH3OH CH3CH2OH Ethers CH3OCH3 C2H5OC2H5 Carboxylic acids HCOOH CH3COOH Hydroperoxides CH3OOH Unsaturated carbonyls CH2 = CHCHO CH2 = C(CH3)CHO CH3CH = CHCHO CH2 = CHC(O)CH3 Source: Atkinson (1997).

k(298 K) × 1012 (cm3 molecule

1

s 1)

C (cm3 molecule 1.20 × 10 5.55 × 10

14

0.219 1.15

5.34 × 10 3.24 × 10

18

11.4 17.2 0.238

1.40 × 10

0.944 3.27

1

s 1)

n

E/R (K)

1 0

287 311

2 2

230 414

18

2

194

6.01 × 10 6.18 × 10

18

2 2

170 532

2.98 13.1

1.04 × 10 8.91 × 10

11

0 2

372 837

0.45 0.8

4.5 × 10

0

0

5.54

2.93 × 10

12

0

190

1.86 × 10

11

0

175

4.13 × 10

12

0

452

9.37 15.8 19.6 9.9

19.9 33.5 36 18.8

12

18

18

18

13

1105

Appendix B: Rate Constants of Atmospheric Chemical Reactions

TABLE B.9 Biogenic Hydrocarbon–OH Rate Constants: k(T) = A exp ( − E/RT) k(298 K) × 1012 (cm3 molecule

Biogenic Isoprene (2-methyl-1,3-butadiene) α-Pinene β-Pinene Myrcene Ocimene (cis- and trans-) Camphene 2-Carene 3-Carene Limonene α-Phellandrene β-Phellandrene Sabinene α-Terpinene γ-Terpinene Terpinolene

1

s 1)

A × 1012 (cm3 molecule

101 53.7 78.9 215 252 53 80 88 171 313 168 117 363 177 225

1

s 1)

E/R (K)

31.0 12.1 23.8

350 444 357

Sources: Sander et al. (2011) and Atkinson (1997).

TABLE B.10 Biogenic Hydrocarbon–O3 and −NO3 Rate Constants: k(T) = A exp ( − E/RT) kO3 …298 K†  1018 (cm3 molecule

Biogenic Isoprene (2-methyl-1,3-butadiene) α-Pinene β-Pinene Myrcene Ocimene (cis- and trans-) Camphene 2-Carene 3-Carene Limonene α-Phellandrene β-Phellandrene Sabinene α-Terpinene γ-Terpinene Terpinolene k(T) = 7.86 × 10 k(T) = 1.01 × 10 c k(T) = 3.03 × 10 d k(T) = 1.19 × 10 a

b

15

exp( 1913/T). exp( 732/T). 12 exp( 446/T). 12 exp(490/T). 15

Source: Atkinson (1997).

12.8a 86.6b 15 470 540 0.90 230 37 200 2980 47 86 21100 140 1880

1

s 1)

kNO3 …298 K† (cm3 molecule 6.78 × 10 13c 6.16 × 10 12d 2.51 × 10 12 1.1 × 10 11 2.2 × 10 11 6.6 × 10 13 1.9 × 10 11 9.1 × 10 12 1.22 × 10 11 8.5 × 10 11 8.0 × 10 12 1.0 × 10 11 1.4 × 10 10 2.9 × 10 11 9.7 × 10 11

1

s 1)

1106

APPENDIX B: RATE CONSTANTS OF ATMOSPHERIC CHEMICAL REACTIONS

REFERENCES Atkinson, R. (1994), Gas-phase tropospheric chemistry of organic compounds, J. Phys. Chem. Ref. Data (monograph 2), 1–216. Atkinson, R. (1997), Gas-phase tropospheric chemistry of volatile organic compounds: 1. Alkanes and alkenes, J. Phys. Chem. Ref. Data 26, 215–290. Calvert, J. G., et al. (2002), The Mechanisms of Atmospheric Oxidation of Aromatic Hydrocarbons, Oxford Univ. Press, Oxford, UK. Sander, S. P., et al. (2011), Chemical Kinetics and Photochemical Data for Use in Atmospheric Studies, Evaluation 17, Jet Propulsion Laboratory, Pasadena, CA (available at http://jpldataeval.jpl.nasa.gov/).

APPENDIX

C

Abbreviations

AAO Ac ACE ADOM AGAGE AGWP AIM AIOMFAC ALE-GAGE AMF AMS AO AOD APi-TOF aq ARB as As ASME AVHRR AWMA bc BC BGK BrC BVOC CAM CAPE CAPRAM Cb Cc CCN CCR

Antarctic oscillation altocumulus (cloud) Aerosol Characterization Experiment Acid Deposition and Oxidant Model Advanced Global Atmospheric Gases Experiment absolute global warming potential Aerosol Inorganics Model Aerosol Inorganic–Organic Mixture Functional-Group Activity Coefficient Atmospheric Lifetime Experiment–Global Atmospheric Gases Experiment aerosol mass fraction aerosol mass spectrometer (Aerodyne Research, Inc.) Arctic oscillation aerosol optical depth atmospheric-pressure interface time-of-flight (mass spectrometer) aqueous-phase (in subscripts and preceding chemical-compound terms, e.g., aqSOA) Air Resources Board (California) aerosol–surface (system) (in subscripts) altostratus (cloud) American Society of Mechanical Engineers Advanced Very High Resolution Radiometer (NASA) Air & Waste Management Association below-cloud (in subscripts) black carbon Bhatnagar–Gross–Krook (model) brown carbon biogenic volatile organic compound Canadian Aerosol Module convective available potential energy Chemical Aqueous Phase RAdical Mechanism (database) cumulonimbus (cloud) cirrocumulus (cloud) cloud condensation nuclei carbon–climate response

Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, Third Edition. John H. Seinfeld and Spyros N. Pandis.  2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

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1108

APPENDIX C: ABBREVIATIONS CDNC CERES CERN CFC Ci CIRPAS CMB CN CNPG Cs CSTRIPE Cu cut DBP DEAD DMA DMS DOP DRH DU EC EDGAR EESC ELVOC ENSO EOF eq ERBE erf ERF erfc ERH ESRL FCT FGOP FTP GAW GCM GCR GEOS GFED GHG GICC GISS GMD gs GWP HCFC HFC HMS HMSA HOA HYSPLIT IARC IEPOX IN INDOEX

cloud droplet number concentration Clouds and Earth’s Radiant Energy System European Organization for Nuclear Research (Conseil Européen pour la Recherche Nucléaire) chlorofluorocarbon cirrus (cloud) Center for Interdisciplinary Remotely-Piloted Aircraft Studies chemical mass balance condensation nuclei carbon number–polarity grid cirrostratus (cloud) Coastal STRatocumulus Imposed Perturbation Experiment (in 2003) cumulus (cloud) cuticle (in subscripts) dibutyl phthalate Dust Entrainment and Deposition model differential mobility analyzer; dimethylamine dimethyl sulfide dioctyl phthalate deliquescence, relative humidity of Dobson unit elemental carbon Emission Database for Global Atmospheric Research equivalent effective stratospheric chlorine extremely low-volatility organic compound El Niño–Southern Oscillation empirical orthogonal function equilibrium (in subscripts) Earth Radiation Budget Experiment error function effective radiative forcing error function, complementary efflorescence relative humidity Earth System Research Laboratory flux-corrected transport functional group oxidation potential Federal Test Procedure Global Atmosphere Watch General Circulation Model galactic cosmic rays Goddard Earth Observing System Global Fire Emissions Database greenhouse gas Global Inventory for Chemistry–Climate (studies) Goddard Institute for Space Studies Global Monitoring Division (appears in “NOAA/ESRL/GMD datasets”) ground surface (in subscripts) global warming potential hydrochlorofluorocarbon hydrofluorocarbon hydroxymethanesulfonate hydroxymethanesulfonate ion hydrocarbon-like organic aerosol HYbrid Single-Particle Lagrangian Integrated Trajectory International Agency for Research on Cancer isoprene epoxydiol ice nuclei Indian Ocean Experiment

APPENDIX C: ABBREVIATIONS IPCC IPY ITCZ IUPAC IVOC JPL KEMS KM-GAP Kn LAI LCL LES LHS lu LUC LV LVOC LV-OOA LW LWC LWP MACR MANGE MASE MB MBL MBO MCM ME MEGAN MEK MESA μeq MISR MNB MLOPEX MP MPAA MPAN MSA MV MVK NAAQS NAM NAPAP NAT NCAR NEI NH NMHC NMVOC NOAA Ns OA OC ODE ODP

Intergovernmental Panel on Climate Change International Polar Year intertropical convergence zone International Union of Pure and Applied Chemistry intermediate-volatility organic compound Jet Propulsion Laboratory Knudsen effusion mass spectrometry Kinetic Multilayer model of GAs–Particle interactions in aerosols and clouds Knudsen number leaf area index lifting condensation level large-eddy simulation left-hand side (of equation) land use (in subscripts) land-use category less volatile (in subscripts) low-volatility organic compound low-volatility oxygenated organic compound longwave liquid water content liquid water path methacrolein mean absolute normalized gross error Marine Stratus/Stratocumulus Experiment mean bias marine boundary layer methyl butenol Master Chemical Mechanism mean error; multilinear engine Model of Emissions of Gases and Aerosols from Nature methyl ethyl ketone Multicomponent Equilibrium Solver for Aerosols microequivalent multiangle imaging spectroradiometer mean normalized bias Mauna Loa Photochemistry Experiment Marshall–Palmer methacrylic peroxy acid methacryloylperoxy nitrate methane sulfonic acid more volatile (in subscripts) methyl vinyl ketone National Ambient Air Quality Standards northern annular mode National Acid Precipitation Assessment Program nitric acid trihydrate National Center for Atmospheric Research National Emissions Inventory Northern Hemisphere nonmethane hydrocarbons nonmethane volatile organic compounds National Oceanic and Atmospheric Administration nimbostratus (cloud) organic aerosol organic carbon ordinary differential equation ozone depletion potential

1109

1110

APPENDIX C: ABBREVIATIONS OOA OPE ORVOC PAH PALMS PAN PCA pdf PETM PG PM PMF POA POLARCAT POZ ppb ppbv pphm ppm ppt PSC PSCF PSSA Re Rf RF RGM RH RHC RHS SAGE SAM SAPRC SBUV Sc SC SCAQS Sh SH SI SLCP SOA SOM SONEX SPADE SSA SST st St SVOC SV-OOA SW TA TC TGM TKE

oxygenated organic aerosol ozone production efficiency other reactive volatile organic compounds polycyclic aromatic hydrocarbon particle analysis by laser mass spectrometry peroxyacetyl nitrate principal-component(s) analysis probability density function Paleocene–Eocene Thermal Maximum Pasquill–Gifford particulate matter positive matrix factorization primary organic aerosol POLar study using Aircraft, Remote measurements and models, Climate, chemistry, Aerosols, and Transport primary ozonide parts per billion parts per billion by volume parts per hundred million parts per million parts per trillion polar stratospheric cloud potential source contribution function pseudo-steady-state approximation Reynolds number flux Richardson number radiative forcing reactive gaseous mercury relative humidity critical relative humidity, right-hand side (of equation) Stratospheric Aerosol and Gas Experiment southern annular mode Statewide Air Pollution Research Center solar backscattered ultraviolet Schmidt number; stratocumulus (cloud) seasonal category Southern California Air Quality Study Sherwood number Southern Hemisphere International System of Units (Système International d’Unités) short-lived climate pollutant secondary organic aerosol statistical oxidation model SASS Ozone and Nitrogen oxide Experiment Stratospheric Photochemistry, Aerosols, and Dynamics Expedition single scattering albedo sea surface temperature stomata (in subscripts) stratus (cloud); Stokes number semivolatile organic compound semivolatile oxygenated organic compound shortwave total ammonia total (particulate) carbon total gaseous mercury turbulent kinetic energy

APPENDIX C: ABBREVIATIONS TN TOA TOMS TS TSI TSP TTL UCAR UCM UNIFAC USAEC USEPA UT/LS VBS VOC VTDMA WCRP WMO WSOC ZSR

total nitrate top of the atmosphere total ozone mapping spectrometer total sulfate total solar irradiance total suspended particulate matter tropical tropopause layer University Corporation for Atmospheric Research unresolved complex mixture UNIversal Functional group Activity Coefficient (model) US Atomic Energy Commission (also AEC) US Environmental Protection Agency (also EPA) upper troposphere/lower stratosphere volatility basis set volatile organic compound volatility tandem differential mobility analyzer World Climate Research Program World Meteorological Organization water-soluble organic carbon Zdanovskii–Stokes–Robinson

Notes: Abbreviated names of chemical compounds (OCS, NO, etc.), SI unit abbreviations (kPa, Tg, etc.), and extremely common terms (2D, 3D, UV, etc.) have not been included in this list.

1111

Index

absorption: cross section of NO2, 114 cross section of O2, 106 cross section of O3, 107 efficiency, mass, 648 spectrum: atmospheric, 101 molecular oxygen, 101, 105–108 ozone, 101, 105–108 water vapor, 101 accommodation coefficient, 500 accretion reactions, 612–613 accumulation mode, 50 ACE-Asia, 988 acetaldehyde, 34, 205, 616 acetone, 34, 207–208 acetylene, 33 acid deposition, 874–881 acid rain, see Acid deposition actinic flux, 95, 108 activated complex, 74 activity, 407 coefficient, 407 adiabatic cooling, 720 adiabatic lapse rate, see Lapse rate advection equation, 764 aerodynamic resistance, 834–835 aeronomy, 5 aerosol: accumulation mode, 50 Aitken mode, 50 classification of sizes, 50 coarse mode, 50, 343 composition, 352–354 condensation mode, 342 definition, 55 desert, 349 droplet mode, 342 fine mode, 50, 343 free troposphere, 348 gravitational settling, 372 marine, 344–346 mass distribution, 330 mean free path, 384

mean particle diameter, 334 mobility, 381 modes, 50 non-spherical, 388–393 nuclei mode, 50 number distribution, 328 organic, 440 polar, 349 remote continental, 348 rural continental, 347 sources, 54 stratospheric, 155 surface area distribution, 330 terminal settling velocity, 373 urban, 343–344 aerosol mass spectrometer (AMS), 580–582 agglomeration, see coagulation AIM, 441 AIOMFAC, 441 air parcel, 662 air pollutants: hazardous, 57–59 air pollution: air quality standards, 56, 1080–1083 definition, 18 Aitken, John, 480 albedo, 941 cloud, 993–995 single-scattering, 973 susceptibility, 997 alcohols, 231–232, 254 aldehydes, 34 tropospheric chemistry, 230 aldol condensation, 611 alkalinity, carbonate, 876 alkanes, 32 alkenes, 33, 222–225 alkoxy radicals, 217–219 alkyl peroxy radicals, 216 alkyl radicals, 33 alkynes, 33 alpha-pinene, 38 atmospheric chemistry, 241–244 altocumulus clouds, 266

Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, Third Edition. John H. Seinfeld and Spyros N. Pandis.  2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

1112

1113

Index amines, 32, 245–246, 255 ammonia, 31 aqueous-phase equilibrium, 278–279 atmospheric concentrations, 31 emissions, global, 31 ammonium bisulfate, 410 ammonium nitrate, 430–438 ammonium sulfate, 427–430 hygroscopic growth, 410 Köhler curve, 714 Ångstrom exponent, 650–651 anthropocene, 4 anticyclone, 899 aromatics, 33–34 tropospheric chemistry, 229–230 Arrhenius form, 75 Arrhenius, Svante, 933 aryl group, 254 Asian Pacific Regional Aerosol Characterization Experiment (ACE-Asia), 988 asymmetry parameter, 635 atmosphere: general circulation, 891 preindustrial, 948 total mass, moles, molecules, 9 US standard, 12–13 atmospheric diffusion equation, 768, 1026–1030 autoxidation, 216, 579–580 backscatter ratio, hemispheric, 637 backward differentiation methods, 1040 band saturation effect, 104 baroclinic instability, 893–894 Beer-Lambert law, 93 benzene, 33, 254 best raindrop distribution, 750 beta-pinene, 38 bicarbonate ion, 272 bilinear model, 1053 biogenic hydrocarbons, 36–39 atmospheric chemistry, 233–244 biogeochemical cycles, 18 biomass burning, 53–54 bisulfate ion, 23 bisulfite ion, 23 black carbon, 52 blackbody radiation, 89 Boltzmann distribution, 459 Boussinesq approximation, 686 Brewer-Dobson circulation, 165, 905 bromine: cycles, 143–144 Brownian motion, 377–379 Bry chemical family, 144 Brunt-Vaisala frequency, 667 Buckingham π theorem, 693 n-butane:

atmospheric oxidation, 221 Buys Ballot’s law, 899 canopy resistance, 830 carbon: black, 52 brown, 987 cycle, 912–920 carbonate ion, 272 carbon dioxide: absorption spectrum, 104–105 aqueous-phase equilibrium, 272–274 atmospheric concentration, 912–913 lifetime, 921–923 carbon disulfide, 23 carbon monoxide, 39–40 atmospheric chemistry, 181–184 global budget, 40 carbon number polarity grid, 605 carbonyl sulfide, 23, 26–27 global budget, 27 carbon tetrachloride, 41 carbonyls, 34 carboxylic acid, 255 carene, 38 CCN, see cloud condensation nuclei CFCs, see chlorofluorocarbons Chapman mechanism, 122–127 Chapman-Enskog theory, 366 Chapman function, 111 Chapman, Sydney, 119 Chappuis band, 108 chemical family, 81–83, 127 chemical mass balance method, 1053–1059 chemical potential: definition, 397 electrolytes, 408 ideal gas mixture, 403–404 ideal gas, 403 ideal solution, 404 nonideal solution, 407 pure solids, 407–408 water, 411 chemical transport models, 1012–1020 chlorine: chemical family, 142 cycles, 140–143 chlorine monoxide radical (ClO), 140 chlorine nitrate, 141–142 chlorofluorocarbons, 40–42 global warming potential, 962 lifetime, 962 naming, 40–42 cirrus clouds, 266 Clausius-Clapeyron equation, 412 climate, 5 feedbacks, 935, 955–958

1114

INDEX climate (Continued ) sensitivity, 934–935, 950 CLOUD, 482–487 cloud: condensation nuclei (CCN), 49, 730–733 ice, 743–746 liquid water content, 270 optical depth, 993 types, 266 ClOx chemical family, 140 Cly chemical family, 142 coagulation, 544–547 Brownian motion, 544–547 effect of force fields, 562–567 equation, 551–556 gravitational settling, 561 laminar shear flow, 560 timescale, 554 turbulent flow, 560 coarse mode, 50 collision: efficiency, 870 frequency factor, 73 of raindrops and aerosols, 868–870 theory, 71 collision rate: gas-kinetic, 73 condensation: aerosol, 538–543 coefficient, 724 equation, 589 continuum regime, 493 convective available potential energy (CAPE), 680–681 Coriolis force, 892, 895–896 cosmic rays, 949–950 Coulomb forces, 564 Courant limit, 1045 Crank-Nicholson algorithm, 1042 Criegee intermediate, 227–228 critical saturation, 771 Crutzen, Paul, 119 cumulative size distribution, 325 cumulonimbus clouds, 266 cumulus clouds, 285 cuticular resistance, 840 cyclone, 899 Dahneke theory, 499 deliquescence relative humidity, 410–418 denitrification, 28 deposition: dry, 19 measurement, 849–851 velocity, 829–830 wet, 19 dew temperature, 673–675 diameter:

aerodynamic, 391 electrical mobility equivalent, 393 median, 335 mode, 335 number mean, 335 Stokes, 390 surface area mean, 335 surface area median, 335 vacuum aerodynamic, 392 volume equivalent, 388 volume mean, 335 volume median, 335 dielectric constant, 564 diene, 33, 255 diffusivity: Brownian, 380 eddy, 767 gases in air, 367 dihydrofuran, 218 dimethyl disulfide, 23 dimethyl sulfide, 23, 26, 247–249 dimethyl sulfoxide, 23 dinitrogen pentoxide, 155–160 dispersion parameters: cloud droplet, 1001 Gaussian, 791–796 distribution factor, gas/aqueous phase, 290 Dobson unit, 45 doldrums, 893 drag coefficient, 371 drift velocity, 381 drizzle, 998 droplet mode, 373, 802 dust, 47 dystrophic lakes, 877 Earth Radiation Budget Experiment, 1084 eddy, 722 diffusivity, 787 viscosity, 741 effective radius, 999 effective saturation mass concentration, 590 efflorescence relative humidity, 419 Einstein relation, 381 Ekman layer, 900 electrical migration velocity, 376 electrical mobility, 376 electroneutrality equation, 273 emission factors, 55 emission inventories, 55–56 empirical orthogonal function models, 1070–1071 energy, bond dissociation, 98 energy balance, global, 931–932 enol, 610 enone, 612 entrainment, 721 epoxide, 255

1115

Index epoxydiol (IEPOX), 236 EQUIL, 440 equivalent particle diameters: aerodynamic, 392 classical aerodynamic, 391 electrical mobility, 393 Stokes, 390 vacuum aerodynamic 392 volume, 388 error function, 337 ethene, 33, 223 ethers, 231, 255 ethylene, see ethene Euler method, 1037 eutonic point, 417 evaporation coefficient, 450, 453 excited species, 77–78 exosphere, 6 extinction, 633 coefficient, 633 extreme values, 1079–1080 factor analysis, 1059–1061 Farman, Joseph, 146 feedback factor, 954 Fenton reaction, 688 Ferrel cell, 893 Fick’s law, 493 finite difference methods, 1032–1034 finite element methods, 1034 flux: matching, 497 Maxwellian, 495 fog, 964–965 force, electrostatic, 411 formaldehyde, 34, 189–190, 320 aqueous-phase chemistry, 312 aqueous-phase equilibrium, 282–283 formic acid: aqueous-phase equilibrium, 283–284 free molecule regime, 497 free troposphere, 7 friction velocity, 693 Fuchs: theory of transition regime, 497–498 Fuchs–Sutugin theory, 498 fume, 47 functional group oxidation model, 605 furan, 255 galactic cosmic rays, 949–950 gas constant, moist air, 671 gas-particle partitioning, 590–591, 594–596 Gaussian (normal) distribution, 336 Gaussian plume equation, 784–792 gem-diol, 611 general dynamic equation for aerosols, 559

geometric standard deviation, 338 geostrophic layer, 897 windspeed, 897–898 Gibbs-Duhem equation, 399 Gibbs free energy, 398 Gibbs-Thompson equation, 500 global warming potential, 962 glycolaldehyde, 616 glyoxal, 616, 619 Henry’s Law constant, 619 glyoxylic acid, 620 Gorham, Eville, 955 gradient method, 925 Greenfield gap, 870 greenhouse: effect, 936–941 gases, 945–946 Haagen-Smit, Arie, 175 Hadley cell, 893 halflife, 76 halocarbons, 40 halogen, 40–44 cycles, 139–144 tropospheric chemistry, 249–251 halons, 40 Hamaker constant, 563 Hartley band, 108 hazardous air pollutants, 57–59 haze, 47 hemiacetal, 218, 612 Henry’s law, 268–269 coefficients for atmospheric gases, 269 effective coefficient, 273 thermodynamic data for coefficients, 272 Herzberg continuum, 108 heterocyclic compounds, 256 Hoppel gap, 740 horse latitudes, 893 HOx: chemical family, 137, 183 cycles, 134–139 Huggins bands, 108 humidity, relative: definition, 12 specific, 11 hydrocarbons: classification, 32–34 hydrocarbon-like organic aerosol, 580–581 hydrochlorofluorocarbons, 40 numbering system, 42 tropospheric chemistry, 251–252 hydrofluorocarbons, 40–42 tropospheric chemistry, 251–252 hydrogen chloride: role in stratospheric chemistry, 144–146 hydrogen cyanide, 246

1116

INDEX hydrogen peroxide, 203 aqueous-phase equilibrium, 281–282 hydrogen sulfide, 23 hydrologic cycle, 905–906 hydroperoxyl radical, 181 hydroxyl radical, 176 formation by O(1D)+H2O, 176–178 global budget, 197–198 production from alkene-O3 reaction, 228 source in aqueous phase, 618–619 ice: clouds, 743–745 ice nucleating particles, 746–747 Indian Ocean Experiment (INDOEX), 988 indirect forcing, 990 Intergovernmental Panel on Climate Change (IPCC), 947 intertropical convergence zone (ITCZ), 893 inversion, temperature, 668–669 ionic strength, 287 ionosphere, 7 iron: aqueous-phase equilibria, 292 catalyzed oxidation of S(IV), 291–293 irradiance, 91 solar, 90, 101 isobaric cooling, 720 isomerization, 244–245 isoprene, 38 chemistry, 233–241 hydroxyhydroperoxides (ISOPOOH), 236 IEPOX, 236 ISORROPIA, 440 Jacob, Daniel, 200 jetstreams, 894 Johnston, Harold, 119 kappa factor, 733–735 Keeling, Charles, 912 Kelvin effect, 419–421 ketones, 34 tropospheric chemistry, 230–231 kinetic regime, see free molecule regime Knudsen number, 362 Köhler theory, 712–715 extended, 751–759 Koschmeider equation, 645 K theory, 767 Lagrangian: approach to atmospheric diffusion, 772–774 correlation function, 849 models, 1024–1026 Laminar sublayer, 831–832

Langevin equation, 377 lapse rate: dry adiabatic, 663–664 moist adiabatic, 675–678 law of mass action, 487–488 letovicite, 428 Levy, Hiram, 176 lifetime, 20 chemical reaction, 70 organic compounds, 232 lifting condensation level, 671–673 limonene, 38 liquid water content, 266, 409, 615 liquid water mixing ratio, 266 liquid water path, 997 Liss and Merlivat formula, 843 logarithmic law for wind velocity, 693–694 lognormal distribution, 335–340 geometric standard deviation, 338 median diameter, 337 Loyalka theory, 499 manganese: catalyzed oxidation of S(IV), 293 MARS, 440 Marshall-Palmer distribution, 750 mass absorption efficiency, 648 mass scattering efficiency, 648, 981 mass transfer coefficient, 523 Maunder minimum, 943 Maxwellian flux, 495 mean free path, 362 air, 365 gas in binary mixture, 365–367 mercury, 55 atmospheric chemistry, 253 mesophyllic resistance, 839–840 mesoscale, 15 mesosphere, 6 meterology, 5 methacrolein, 239–241 methane, 34–36 atmospheric chemistry, 188–192, 200–203 concentrations ambient, 34 global budget, 36 global warming potential, 962 lifetime, 963–964 methane sulfonic acid, 23 methanol, 231 methoxy radical, 189 methyl bromide, 42–43 methyl chloride, 42–43 methyl chloroform, 41, 927 methyl ethyl ketone (MEK), 230 methylglyoxal, 616 methyl hydroperoxide, 189, 203 methyl mercaptan, 23

1117

Index methyl nitrite, 246 methyl peroxy radical, 188–189 methyl radical, 188 methyl vinyl ketone, 239–241 microscale, 15, 720 Mie theory, 634, 647 Milankovitch cycles, 942 mineral dust, 53 mist, 47 mixing: atmospheric timescales, 14–15 external, 985–986 internal, 985–986 mixing length: model, 690–692 mixing ratio, 11 mobility, 381 electrical, 376 mode diameter, 335 molality, 407 molarity, 407 Molina, Mario, 119 Monin-Obukhov length, 695–696, 698 monoterpenes, 38 atmospheric chemistry, 241–244 moments, 360 method of, 1076–1078 Montreal Protocol, 121 Mt. Pinatubo, 48, 161–162, 984 multilinear engine, 1067 myrcene, 38

nitrous acid, 204 nitrous oxide, 28, 30 global budget, 28 role in stratospheric chemistry, 129–131 stratospheric concentration, 130 normal distribution, see Gaussian distribution NOx, 28 aqueous-phase oxidation, 311 concentrations ambient, 31 cycles, 131–134 family, 192 NOy, 30 NOz, 210 nucleation: acid-base chemical model, 486–487 atmospheric, 480–487 binary, 468–473 classical theory, 449–464 critical cluster size, 455 density functional theory, 468 experimental measurement of, 465–467 growth rates, 482 H2SO4–H2O system, 473–475 ion-induced, 478–480 on an insoluble foreign surface, 475–478 role of amines, 484 role of cosmic rays, 482–484 role of NH3, 481, 485 role of organics, 487 nuclei mode, 590 numerical diffusion, 1043

Navier-Stokes equations, 369 nimbus cloud, 266 nitrate: aqueous chemistry, 303 radical, 133–134, 225, 238–239 nitric acid: aqueous-phase equilibrium, 280–281 nitrification, 29 nitriles, 246 nitrites, 246 nitrogen: containing compounds, 27–28 cycle, 29 fixation, 28 odd, 37 reactive odd, 30–31 nitrogen dioxide: absorption cross section, 115 photolysis, 114–116 quantum yield for production of NO and O, 114–115 nitrogen oxides, 29 aqueous equilibria, 311–312 chemical family, 133, 183 emissions, 30 stratospheric chemistry, 131–134

Odin, Svante, 875 olefins, see Alkenes oligomers, 612 operator splitting, 1032, 1035–1037 optical: depth, 93, 643, 981 thickness (see Optical depth) organic aerosol: global budget, 622–623 oxalic acid, 620 oxidation state, 24 organic compounds, 576–578 oxygen: absorption cross section, 106 odd, 124 photolysis, 98 oxygen atom: singlet, 112–113 oxygenated organic aerosol, 580–581 ozone, 44–47 absorption cross section, 107 air quality standard, 56 budget, 195 depletion potential, 167 hole, 146–154

1118

INDEX ozone (Continued ) isopleth plot, 209–210 layer, 119 photolysis, 112–114 production efficiency, 184–186 weekend effect, 211–212 PAHs, see Polycyclic aromatic hydrocarbons PAN chemical family, 207 paraffins, 32 particulate matter, 19, 47–55 Pasquill-Gifford curves, 795–796, 799 Pasquill stability classes, 698 pentane, 218–219 perhalocarbons, 40 peroxides, 35 peroxyacetyl nitrate (PAN), 204–208 peroxyacetyl radical, 205 peroxyhemiacetal, 612 pH: definition, 272 ocean, 918, 923 phase function, scattering, 638 phellandrene, 38 photodissociation, 69 photon, 88 photosphere, 90 photostationary state, 180 Pinatubo, Mt., 48, 161–162 planetary boundary layer, 692 plume: models, multiple source, 805–807 rise, 796–798 polar: cell, 893 stratospheric clouds, 149 vortex, 153–154 polycyclic aromatic hydrocarbons, 52 positive matrix factorization (PMF), 1064–1067 potential source contribution function (PSCF), 1068–1069 potential temperature, 663 predictor-corrector methods, 1042 pressure, units, 7 primary organic aerosol, 606–607 principal component analysis (PCA), 1061–1064 propane, atmospheric oxidation, 220 propene, atmospheric oxidation, 226–227 propylene, see propene pseudo-steady state approximation, 76–77 quantiles, method of, 1075–1076 quantum yield, 96–97, 99 quasi-laminar resistance, 835 radiance, 91 radiant flux density, 91

radiation, 88–91 radiative forcing, 933 indices, 960–961 radiative forcing efficiency, 984 radius, effective, 999 raindrop size distribution, 750 rainout, 854 Raoult’s law, 405 Rayleigh: atmosphere, 639, 645 scattering regime, 638–640 reaction: order, 69 photochemical, 69 termolecular, 78–80 receptor models, 1051 reduced mass, 72 refractive index, 635 relative humidity, 12 deliquescence, 412–419 efflorescence, 419 reservoir species, 144–146, 203–208 residence time, 20 Reynolds number, 369 Reynolds stresses, 689 Richardson number, flux, 695 roughness length, 694 Rossby waves, 895 Rowland, F. Sherwood, 119 sabinene, 38 scale height: pressure, 8–9 scattering, light: efficiency, 633 elastic, 634 geometric, 638 inelastic, 634 Mie, 634 quasi-elastic, 634 Rayleigh scattering, 638–639 scavenging: coefficient, 858–859 Schmidt number, 871 Schumann-Runge bands, 108 Schumann-Runge continuum, 108 secondary organic aerosol, 574–576 models, 596–605 yield, 576 seasalt, 354 SEQUILIB, 440 Sherwood number, 527, 860 ship tracks, 992–993 single scattering albedo, 634 Sitarski–Nowakowski theory, 499 S(IV), 286 oxidation by dissolved O3, 286

1119

Index oxidation by H2O2, 289–290 oxidation by OH radical, 304, 307 oxidation by organic peroxides, 290 oxidation by oxides of nitrogen, 308–309 oxidation catalyzed by Fe and Mn, 291–293 uncatalyzed oxidation by O2, 291 slender plume approximation, 789 slip correction factor, 371 Smith, Robert Angus, 874 smog, 47 smoke, 47 sodium chloride: crystallization RH, 419 deliquescence RH, 410 Köhler curve, 714 sodium nitrate: deliquescence RH, 414 solar: constant, 90, 936 spectral irradiance, 90, 942 solubility, of salts in water, 413 solute effect, 712–714 solution: ideal, 404–406 non-ideal, 407 soot, 47, 51 Southern California Air Quality Study (SCAQS), 37 spectrum: electromagnetic, 88–89 solar, 90 speed, molecular mean, 72 splitting methods, 1032–1034 stability, atmospheric, 665–668 standards, air quality, 56 statistical oxidation model, 604–605 Stefan flow, 496 stiff ordinary differential equations, 1037–1039 Stokes–Einstein relation, 380 Stokes’ law, 368–370 Stokes number, 388, 871 stomatal resistance, 841 stop distance, aerosols, 387 stratocumulus clouds, 266, 991 stratosphere, 6–7 aerosol layer, 155 transport, 165–167 stratus clouds, 266 sunspot cycle, 944 sulfate: ion, 23 sulfur: compounds, 23–27 global cycle, 908–911 global emissions, 25 oxidation states, 24 residence times, 908–911

sulfur dioxide, 23 aqueous-phase equilibrium, 274–276 global emissions, 25 reaction with OH, 246 sulfuric acid, 28 aqueous solution properties, 423–426 hydrates, 473 saturation vapor pressure, 475 summation convention, 764–765 sun, 89 sunspots, 943–944 surf zone, 165 surface layer, 693 surface layer resistance, 839–842 surface tension: of selected compounds, 459 water, 701–702 synoptic scale, 16 temperature: dew-point, 673–674 global, record, 944 potential, 663 terminal settling velocity, 372–375 terpinene, 38 terpinolene, 38 thermal wind relation, 902–904 thermodynamic diagrams, 681–683 thermosphere, 6 thiol, 254 thiophene, 255 toluene, 228–230 transition probability density, 766 transition regime, 497 transition state theory, 74–75 transmittance, 94 triangle plot, 581 Troe theory, 80 tropical pipe, 165 tropical tropopause layer, 166 tropopause, 6 troposphere, 5–6 -stratosphere transition, 199 turbulent flow, 687–688 two-film model, 841–842 UNIFAC, 441 upper troposphere/lower stratosphere, 166 updraft velocity, 721–722 upscatter fraction, 973, 979 uptake coefficient, 83, 156 upwind algorithm, 1043 van der Waals forces, 562–563 van Krevelen plot, 578–579 van’t Hoff equation, 268

1120

INDEX visibility, 644–647 volatile organic compounds (VOCs), 36 volatility: organic compounds, 582–583 volatility basis set (VBS), 597–601, 603–604 volatility tandem differential mobility analyzer (VTDMA), 593–594 volcanoes, 984 effect on stratospheric chemistry, 160–162 von Karman constant, 693

refractive index, 636 saturation vapor pressure, 709 wavenumber, 88 weekend ozone effect, 211 Weibull distribution, 1074 Weinstock, Bernard, 175 wet deposition velocity, 860 windspeed: empirical formula, 804 geostrophic, 897–898

Wanninkhof formula, 844 washout, 854 ratio, 859 water: homogeneous nucleation rate, 465

xylene, o-, m-, and p-, 34 Zeldovich nonequilibrium factor, 472 zenith angle, solar, 94 ZSR relation, 433

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