The Fluid Environment of the Earth: Atmosphere and Ocean 3031315383, 9783031315381

Table of contents :
Preface
Acknowledgements
Contents
Nomenclature
1 The Environment in the Solar System
1.1 Radiative Energy Balance of a Planet
1.1.1 Properties of a Black Body
1.1.2 The Effective Radiating Temperature
1.1.3 The Greenhouse Effect
1.2 The Point of View of Entropy
1.2.1 The Entropy Budget of a Planet
Bibliography
2 The Fluid Environment of the Earth
2.1 The Atmosphere
2.1.1 Chemical Composition
2.1.2 Atmospheric Pressure
2.1.3 Vertical Changes of Pressure and Temperature
2.1.4 The Point of View of Entropy
2.2 The Ocean
2.2.1 The Equation of State of Seawater
2.2.2 Incompressibility: Change of Pressure with Depth
Bibliography
3 Thermodynamics of the Atmosphere and the Ocean
3.1 Effects of Water Vapor
3.2 Static Stability of the Atmosphere
3.3 Static Stability of a Moist Atmosphere
3.4 Stability of Ocean Stratification
3.4.1 Potential Temperature and Density
Bibliography
4 Chemical Kinetics
4.1 Gas Phase Reactions
4.2 Photochemistry
4.3 Aqueous Solutions
Bibliography
5 Fluid Dynamics
5.1 Some Tool of the Trade
5.1.1 Total Derivative
5.1.2 Continuity Equation
5.1.3 The Diffusion Equation
5.2 The Equation of Motion in a Rotating Earth
5.3 Geostrophic Motion
5.3.1 The Effect of Friction
5.4 Thermal Wind Equation
5.5 Thermodynamic Equation
5.6 Equation of Vorticity
5.6.1 Some Implications: Rossby Waves and Zonal Flow
Bibliography
6 General Circulation of the Atmosphere
6.1 The Requirements for the General Circulation
6.2 The General Circulation of the Atmosphere
6.3 The Margules Formula
6.4 The Tropical Circulation
6.5 The Mid-latitude Circulation: The Ferrel Cell
6.5.1 Baroclinic Instability
6.6 The Circulation at Mid-latitude
Bibliography
7 General Circulation of the Ocean
7.1 The Wind Driven Circulation
7.1.1 The Ekman Pumping
7.1.2 The Sverdrup Balance
7.1.3 The Western Boundary Current
7.2 The Deep Circulation
7.2.1 The Mass Overturning Circulation (MOC)
7.2.2 A Few Numbers
7.2.3 The Horizontal Component
Bibliography
8 Biogeochemical Cycles
8.1 The Earth System
8.2 The Carbon Cycle
8.2.1 Carbonate Chemistry in the Ocean
8.2.2 How Much normal upper C normal upper O Subscript 2CO2 Takes up the Ocean
8.2.3 There Is a Way to Measure F?
8.3 Box Model of the Carbon Cycle
8.4 The Nitrogen Cycle
8.5 The Sulfur Cycle
8.5.1 The Sulfur Atmospheric Cycle
8.6 The Oxygen Cycle
Bibliography
9 Greenhouse Effect-Chemistry Climate Connection
9.1 The Height of the Troposphere
9.2 The Radiative Forcing
9.2.1 The Evaluation of the Radiative Forcing
9.3 Climate Sensitivity and Water Vapor Feedback
9.4 Cloud Feedback
9.5 Theory of Feedback
9.6 The Climate-Ocean Chemistry Connection
9.7 The Future of the Earth
Bibliography
10 The Perturbed Atmosphere
10.1 The Hydroxyl Radical
10.2 The Oxidation of Methane normal upper C normal upper H Subscript 4CH4 and CO
10.2.1 The Oxidation of Carbon Monoxide
10.2.2 The Methane Oxidation
10.3 The Volatile Organic Compounds (VOC)
10.4 The Polluted Atmosphere
10.5 The Stratospheric Ozone
10.6 The Ozone Hole
10.6.1 Other Threats to the Ozone Layer
Bibliography
11 Some Chemistry of the Sea
11.1 Some Data on the Hydrological Cycle
11.2 The Sea Water Composition
11.2.1 Salinity and the Age of the Ocean
11.3 Distribution of Elements in the Ocean
11.3.1 Respiration and Photosynthesis
11.4 The Air Sea Exchange
Bibliography
12 Aerosols, Clouds and Rain
12.1 What a Cloud Weighs
12.2 Sources of Atmospheric Aerosols
12.3 Particle Size Distribution
12.4 The Interaction of Aerosols and Radiation
12.4.1 Color for Non-absorbing Sphere
12.4.2 Effects on the Albedo
12.5 Inside a Cloud with the Help of a Radar
12.6 Generation and Growth of Particles Within a Cloud
12.6.1 Growth by Condensation
12.6.2 Droplet Growth by Collision and Coalescence
12.7 Acid Deposition
12.7.1 Effect of Acid Rain
Bibliography
13 Atmosphere–Sea Interactions
13.1 The Sea Breeze
13.2 The Monsoon
13.3 The Hurricanes
13.3.1 Generalities About Hurricanes
13.3.2 The Hurricane as a Carnot's Engine
13.3.3 Dissipation as Additional Fuel
13.4 El Nino Southern Oscillation (ENSO)
13.4.1 The Delayed Oscillator
13.4.2 The Charge–Discharge Oscillator
Bibliography
14 The Fluid Environment in the Solar System
14.1 The Rotating Annulus
14.1.1 Dimensionless Parameters
14.2 The Atmosphere Dynamics
14.2.1 The Atmospheric Circulations of Venus and Mars
14.3 Dynamics of Jovian Planets
14.4 Atmospheric Chemistry of Terrestrial Planets
14.4.1 Venus Atmospheric Chemistry
14.4.2 Mars Atmospheric Chemistry
14.5 The Atmosphere of Jovian Planets
14.5.1 Chemistry on the Giant Planets
14.5.2 Clouds on the Jovian Planets
14.6 Oceans and Seas in the Solar System
14.6.1 Internal Processes
Bibliography
Index

Citation preview

Guido Visconti

The Fluid Environment of the Earth Atmosphere and Ocean

The Fluid Environment of the Earth

Guido Visconti

The Fluid Environment of the Earth Atmosphere and Ocean

Guido Visconti Physical and Chemical Sciences Università dell’Aquila Coppito, L’Aquila, Italy

ISBN 978-3-031-31538-1 ISBN 978-3-031-31539-8 (eBook) https://doi.org/10.1007/978-3-031-31539-8 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

This book is dedicated to the memory of Prof. Luciano Onori. He was an outstanding medical doctor and a friend of mine who healed the body and soul.

Preface

The idea of this book had haunted me for years so that, as soon as I got some time, I seriously thought about gather all the material I had accumulated in years of teaching in and out of this subject. In particular, I have taught for years a Physics of the Atmosphere and Ocean course. The inspiration was to imitate the Daniel Jacob book and to write something for undergraduate without too much math. Unfortunately, I soon found out that this target was impossible and so I resolved to the approach of writing simple chapters complemented with much heavier appendices. Of course, the novelty with respect to the previous Fundamentals of Physics and Chemistry of the Atmosphere is that the book includes the ocean so that cover title is justified although everyone knows that also the interior of the planet is fluid while they still call it solid Earth. Half of the book is dedicated to introduction and it is heavily based on our previous efforts while completely new is the part dedicated to the ocean which includes beside thermodynamics, general circulation, chemistry and the chapter on the interaction between the atmosphere and the sea. Also, completely new is the chapter on biogeochemical cycles, while the treatment of the perturbed atmosphere is updated. Each paragraph of the book could be a subject of a new book so that we have to make a choice and neglect some fascinating subject. In particular, we make some mention on the use of the entropy concept and to some applications of the MEP (Maximum Entropy Production). There are many circles around the geophysics realm that because of some inferiority complex have the claim to find the theory of everything and this temptation (dealing with urban pollution is too trivial when compared to black holes!) is very strong when you talk about the atmosphere and the ocean. We think this claim is quite foolish because there are many respectable scientific endeavors that do not use math at all and fall in the same paradox that Richard Lewontin used to mention and that is about an elephant and a mouse that fall in the same way from the Pisa tower but nobody asks why they came to such different sizes. It is like using the cargo culture of Feynman legacy: the use of math does not guarantee that you are doing good science. So we skipped the reductionist temptation from many subjects like general circulation and climate theory. On the other hand, it was inevitable to mention other environments in our Solar System which constitute a fashionable subject anyway. vii

viii

Preface

Beside these topics, there are other new interesting chapters the most relevant being the biogeochemical cycles, the one on the interaction of the atmosphere and the ocean and the chemistry–climate connection. The sea–atmosphere interactions include topics hardly treated in textbook like the monsoon circulation and the hurricanes debate. The latter includes the discussion of the so-called dissipation engine where the hurricane is assimilated to a Carnot machine that could use as energy source the heat generated by dissipation. Some though is also given to the far future of the Earth. Also interesting is the discussion on the salinity of the sea. The book is directed to students at the undergraduate level can study the introductory part of each chapter, while the most experienced ones could use the appendices and the vast bibliography at the end of each chapter. When necessary the book also provides MATLAB scripts as useful exercises. Coppito, Italy

Guido Visconti

Acknowledgements

This is a good point to extend my acknowledgement to Antonella Di Nisio of the scientific library at my university that was kind enough to track down many old and buried papers. The same acknowledgement goes to Paolo Ruggeri who was too busy for his new job and for his role of young father to collaborate to this book but helped with MATLAB and Latex and read the manuscript. The same manuscript was sent to Prof. Vincenzo Rizi and Francesco Mulargia that provided useful comments. I have to mention my family again for bearing the weight of my absence from family life. The work on this book was carried out during the COVID-19 pandemic and my friend Luciano Onori survived it but had to succumb to another obscure disease. It was a great loss to me and gladly I dedicated this book to him. Shine your light on Luciano!

ix

Contents

1

The Environment in the Solar System . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Radiative Energy Balance of a Planet . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Properties of a Black Body . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 The Effective Radiating Temperature . . . . . . . . . . . . . . . . 1.1.3 The Greenhouse Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Point of View of Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 The Entropy Budget of a Planet . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3 6 7 8 16

2

The Fluid Environment of the Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Chemical Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Atmospheric Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Vertical Changes of Pressure and Temperature . . . . . . . . 2.1.4 The Point of View of Entropy . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Ocean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 The Equation of State of Seawater . . . . . . . . . . . . . . . . . . . 2.2.2 Incompressibility: Change of Pressure with Depth . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 17 19 21 23 25 25 28 37

3

Thermodynamics of the Atmosphere and the Ocean . . . . . . . . . . . . . . 3.1 Effects of Water Vapor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Static Stability of the Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Static Stability of a Moist Atmosphere . . . . . . . . . . . . . . . . . . . . . . 3.4 Stability of Ocean Stratification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Potential Temperature and Density . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 39 42 43 45 48 56

4

Chemical Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Gas Phase Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Photochemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 57 59

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4.3 Aqueous Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 68

5

Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Some Tool of the Trade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Total Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 The Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Equation of Motion in a Rotating Earth . . . . . . . . . . . . . . . . . . 5.3 Geostrophic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 The Effect of Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Thermal Wind Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Thermodynamic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Equation of Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Some Implications: Rossby Waves and Zonal Flow . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69 69 69 71 72 73 74 76 77 80 82 85 99

6

General Circulation of the Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The Requirements for the General Circulation . . . . . . . . . . . . . . . . 6.2 The General Circulation of the Atmosphere . . . . . . . . . . . . . . . . . . 6.3 The Margules Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 The Tropical Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 The Mid-latitude Circulation: The Ferrel Cell . . . . . . . . . . . . . . . . 6.5.1 Baroclinic Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 The Circulation at Mid-latitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101 101 104 107 109 112 113 116 135

7

General Circulation of the Ocean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 The Wind Driven Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 The Ekman Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 The Sverdrup Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 The Western Boundary Current . . . . . . . . . . . . . . . . . . . . . 7.2 The Deep Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 The Mass Overturning Circulation (MOC) . . . . . . . . . . . . 7.2.2 A Few Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 The Horizontal Component . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

137 137 138 140 141 144 144 146 148 157

8

Biogeochemical Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 The Earth System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 The Carbon Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Carbonate Chemistry in the Ocean . . . . . . . . . . . . . . . . . . 8.2.2 How Much CO2 Takes up the Ocean . . . . . . . . . . . . . . . . . 8.2.3 There Is a Way to Measure F? . . . . . . . . . . . . . . . . . . . . . .

159 159 162 163 166 168

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8.3 8.4 8.5

Box Model of the Carbon Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Nitrogen Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Sulfur Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 The Sulfur Atmospheric Cycle . . . . . . . . . . . . . . . . . . . . . . 8.6 The Oxygen Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

169 170 172 175 176 183

Greenhouse Effect-Chemistry Climate Connection . . . . . . . . . . . . . . . 9.1 The Height of the Troposphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 The Radiative Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 The Evaluation of the Radiative Forcing . . . . . . . . . . . . . . 9.3 Climate Sensitivity and Water Vapor Feedback . . . . . . . . . . . . . . . 9.4 Cloud Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Theory of Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 The Climate-Ocean Chemistry Connection . . . . . . . . . . . . . . . . . . . 9.7 The Future of the Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

185 185 190 193 195 198 199 202 204 220

10 The Perturbed Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 The Hydroxyl Radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 The Oxidation of Methane CH4 and CO . . . . . . . . . . . . . . . . . . . . . 10.2.1 The Oxidation of Carbon Monoxide . . . . . . . . . . . . . . . . . 10.2.2 The Methane Oxidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 The Volatile Organic Compounds (VOC) . . . . . . . . . . . . . . . . . . . . 10.4 The Polluted Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 The Stratospheric Ozone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 The Ozone Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.1 Other Threats to the Ozone Layer . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

223 223 225 226 227 228 230 233 235 237 249

11 Some Chemistry of the Sea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Some Data on the Hydrological Cycle . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Sea Water Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Salinity and the Age of the Ocean . . . . . . . . . . . . . . . . . . . 11.3 Distribution of Elements in the Ocean . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Respiration and Photosynthesis . . . . . . . . . . . . . . . . . . . . . 11.4 The Air Sea Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

251 251 252 254 256 257 259 273

12 Aerosols, Clouds and Rain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 What a Cloud Weighs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Sources of Atmospheric Aerosols . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Particle Size Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 The Interaction of Aerosols and Radiation . . . . . . . . . . . . . . . . . . . 12.4.1 Color for Non-absorbing Sphere . . . . . . . . . . . . . . . . . . . . 12.4.2 Effects on the Albedo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Inside a Cloud with the Help of a Radar . . . . . . . . . . . . . . . . . . . . .

275 275 277 279 281 284 285 286

9

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12.6 Generation and Growth of Particles Within a Cloud . . . . . . . . . . . 12.6.1 Growth by Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6.2 Droplet Growth by Collision and Coalescence . . . . . . . . 12.7 Acid Deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.1 Effect of Acid Rain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

289 291 292 293 295 309

13 Atmosphere–Sea Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 The Sea Breeze . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 The Monsoon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 The Hurricanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.1 Generalities About Hurricanes . . . . . . . . . . . . . . . . . . . . . . 13.3.2 The Hurricane as a Carnot’s Engine . . . . . . . . . . . . . . . . . 13.3.3 Dissipation as Additional Fuel . . . . . . . . . . . . . . . . . . . . . . 13.4 El Nino Southern Oscillation (ENSO) . . . . . . . . . . . . . . . . . . . . . . . 13.4.1 The Delayed Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.2 The Charge–Discharge Oscillator . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

311 311 313 316 316 318 321 323 324 327 338

14 The Fluid Environment in the Solar System . . . . . . . . . . . . . . . . . . . . . . 14.1 The Rotating Annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.1 Dimensionless Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 The Atmosphere Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1 The Atmospheric Circulations of Venus and Mars . . . . . 14.3 Dynamics of Jovian Planets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Atmospheric Chemistry of Terrestrial Planets . . . . . . . . . . . . . . . . . 14.4.1 Venus Atmospheric Chemistry . . . . . . . . . . . . . . . . . . . . . . 14.4.2 Mars Atmospheric Chemistry . . . . . . . . . . . . . . . . . . . . . . . 14.5 The Atmosphere of Jovian Planets . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.1 Chemistry on the Giant Planets . . . . . . . . . . . . . . . . . . . . . 14.5.2 Clouds on the Jovian Planets . . . . . . . . . . . . . . . . . . . . . . . 14.6 Oceans and Seas in the Solar System . . . . . . . . . . . . . . . . . . . . . . . . 14.6.1 Internal Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

339 339 340 343 344 351 354 355 356 358 359 361 364 366 377

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

Nomenclature

αT β χ η ⎡, ⎡d , ⎡s λ μ ν ω, Ω φ ψ ρ σ τ τ θ ξ, ζ Bu BΔ cp cν LR Lρ R0 Rθ S0 Ta A a APE

Thermal expansion Coriolis parameter gradient Size parameter Deviation from reference altitude, entropy Lapse rate Wavelength Coefficient of viscosity Kinematic viscosity Angular velocity, vorticity, wave frequency, solid angle, single scattering albedo Geopotential, velocity potential Stream function Density Growth rate, density anomaly, Stefan–Boltzmann constant Stress Optical thickness, viscous stress, time constant Potential temperature Relative vorticity Burger number Planck function Specific heat at constant pressure Specific heat at constant volume Rhines scale Rossby radius of deformation Rossby number Thermal Rossby number Solar flux Taylor number Albedo Radius of the Earth Available potential energy xv

xvi

D e, es f G g H h I J k M N p q R Ra Re S s T U z

Nomenclature

Diffusion coefficient, downward flux Vapor pressure, saturation Coriolis parameter Gravitational constant Acceleration of gravity, asymmetry factor Scale height Planck constant, enthalpy Intensity of radiation Entropy flux Wave vector, Boltzmann constant Molecular mass Buoyancy frequency Pressure Specific humidity Gas constant Rayleigh number Reynolds number Salinity, entropy Entropy Temperature Potential energy, internal energy, upward flux Altitude

Chapter 1

The Environment in the Solar System

The climate of the Earth is mainly determined as we will see by the balance between the energy that the Earth absorbs from the Sun and the one re-emitted as heat (infrared radiation). At local level (regional) the situation is much more complex because of the important role of the atmospheric and oceanic circulations, that is, the currents flowing in the atmosphere and the ocean. The main role of the currents is to compensate for the excess heat at low latitude resulting from the radiation imbalance (more energy absorbed than re-emitted). On the other hand, the circulation is determined by the temperature distribution in the atmosphere and temperature and salinity in the ocean. The role of the atmospheric chemical composition becomes clear as well as the interactions of the oceans with the solid Earth and the cryosphere (polar and mountain ice). The chemical composition of both the atmosphere and the ocean depends very much from the existence of life on our planet that probably arose because of the very peculiar environmental conditions occupied by the Earth in the solar system. We have indicated a quite long and complicated trail that will try to follow in the book.

1.1 Radiative Energy Balance of a Planet The simplest case for calculating the energy balance of a planet is to consider a planetary body devoid of the atmosphere like the Moon or Mercury. Also we need to consider that the planet behaves like a black body which corresponds to an ideal body which absorbs all the radiation. A good exemplification of a black body is old, monumental church sunbathed in a square. If you look from a distance to the entrance you see the inside perfectly black and this corresponds to the fact that once the sunlight has penetrated into the church it cannot longer escape.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 G. Visconti, The Fluid Environment of the Earth, https://doi.org/10.1007/978-3-031-31539-8_1

1

2

1 The Environment in the Solar System

1.1.1 Properties of a Black Body We will have occasions to talk about the theory of radiative transfer and here we will start to give just the basis. We can divide the entire electromagnetic spectrum of radiation in two bands, the visible comprised between wavelength of 0.4 and 0.7 µm and the infrared between 0.7 and 1 µm. The visible region refers essentially to the radiation coming from the Sun. On the other hand, planets are warmed by the Sun and can emit radiation that we call “heat”. Such radiation is comprised mainly in the infrared range. The different behavior of the Sun and planets can be explained by the theory of black body radiation. The intensity of the black body radiation is described by Planck’s function which gives the power emitted for unit of surface, unit of wavelength, and unit of solid angle Bλ (T ) =

2hc2

(1.1)

λ5 (ehc/kT λ − 1)

where h is Planck’s constant (6.626 × 10−34 J s), k is the Boltzmann constant (1.381 × 10−23 JK−1 ), and c is the speed of light. Bλ is measured in watt m−2 m−1 ster−1 . It seems we have a new quantity ster that corresponds to the unit of solid angle defined by a cone which contains the intensity of radiation. In Fig. 1.1, two Planck functions are reported corresponding to 6000 K (roughly the temperature of the Sun) and 300 K which is the temperature at which the Earth emits its radiation. We notice that the maxima of the two spectra are around 0.5 µm for the Sun while for the Earth is around 10 µm. The maximum values for the intensity differ of almost seven order of magnitude. The wavelength for which we have the maximum of emission can be found by imposing 300 102

SUN

SUN 200 10-2

100

EARTH

10-6

EARTH x 2.5 105 0 4

8

12

16

20

10-1

1

101

102

WAVELENGTH (μm)

Fig. 1.1 The spectral irradiance in kw m−2 sr−1 µm−1 for two black bodies at 6000 K (the Sun) and 300 K (the Earth). On the left linear scales are used and the Earth irradiance is multiplied by a factor 2.5 × 105 . On the right logarithm scales are used

1.1 Radiative Energy Balance of a Planet

3

d Bλ =0 dλ

(1.2)

That once solved would give λmax =

α , T

α=

hc = 2897 µm K 5k

(1.3)

This relation is known as Wien displacement law and for a black body at 6000 K, λmax is around 0.482 µm that is yellow-orange. On the other hand for the Earth λmax ≈ 9.65 µm and the two spectral region are well separated. As we have repeated many times Bλ is the intensity of radiation while in order to calculate the temperature of a planet we need the radiative flux, Fλ that is the power integrated all over the solid angle which is measured in watt m2 . We show in the appendix that the relation between flux and intensity is given by Fλ (T ) = π Bλ (T )

(1.4)

So that the total flux integrated over the wavelength (

∞

FB (T ) =

Fλ (T )dλ = σT 4

(1.5)

0

which represents the Stefan–Boltzmann law while σ is known as Stefan–Boltzmann constant. 2π 5 k 4 W σ= = 5.67 × 10−8 2 4 (1.6) 2 3 15c h m K

1.1.2 The Effective Radiating Temperature Following the Stefan–Boltzmann law we can write the total power emitted from the Sun 2 4 T⊙ (1.7) P⊙ = 4πσ R⊙ which corresponds to 3.95 × 1020 W m−2 and where R⊙ and T⊙ are radius and temperature of the Sun. If D S E is the distance between Earth and Sun (1.5 × 108 km) the power per unit area received at the orbit of the Earth is given by ( S=

σT⊙4

R⊙ DS E

)2 (1.8)

At the Earth’s orbit S ≈ 1.37 kW m−2 . The equilibrium is established when the emitted power from the Earth is equal to the absorbed power from the sun

4

1 The Environment in the Solar System

Table 1.1 The relevant data to determine the radiative equilibrium temperature of the terrestrial planets Venus Earth Mars Distance from the Sun (106 km) Albedo Radiative equilibrium temperature (K) Surface temperature (K)

108 0.76 239 740

150 0.3 255 290

228 0.15 226 235

Table 1.2 The data for the outer (Jovian) planets. The table also lists the presumed internal heat source Jupiter Saturn Uranus Neptune Solar flux (Wm−2 ) Geometric albedo Absorbed flux (Wm−2 ) Emitted flux (Wm−2 ) Internal source (Wm−2 ) Radiative equilibrium T (K) Observed bolometric T (K)

53.3 0.503 6.613 14.098 7.485 113 125.72

15.2 0.242 2.88 4.89 2.01 84 96

2 2 4π R⊕ σTe4 = (1 − A)πS R⊕

3.76 0.215 0.738 0.696 −0.042 60 59

1.52 0.215 0.297 0.73 0.433 48 60

(1.9)

where R⊕ is the radius of the Earth and A is the albedo that is the fraction of the incident radiation reflected by the Earth. The temperature Te results then [

S (1 − A) Te = σ 4

]1/4 (1.10)

Although there are uncertainties on the value of the albedo, for the Earth, it is around 0.3 and the temperature Te ≈ 255 K that is −18 ◦ C. As we know the average temperature of the surface of the Earth is around 15 ◦ C and the difference is due to what is called greenhouse effect that must be attributed to the atmosphere which is between the emission level and the surface. The same calculation we have made for the Earth can be repeated for the so-called terrestrial planets of the solar system, that is, Venus and Mars and the results are given in Table 1.1. The same exercise made for the outer planets (Jupiter, Saturn, Uranus, and Neptune) gives a lot of surprises. For these planets, it is possible to measure the absorbed and emitted radiation and except for Uranus all planets emit more radiation than what they absorb. This indicates that there must be an internal source of energy for those planets. Table 1.2 reports the details for each. Notice that the table shows the solar flux to have more direct comparison.

1.1 Radiative Energy Balance of a Planet

5

There are several interesting things in this table that give insight in the concept of equilibrium temperature. First of all Eq. (1.9) can be read in a coincidence in different way if written as σTe4 =

S (1 − A) 4

(1.9a)

This means that the intercepted solar radiation (π R 2 S) is redistributed over the surface of the planet (4π R 2 ) if the planet is in rapid rotation. Then each square unit of surface emits at equilibrium the radiation σTe4 . Rapid rotation means that the rotation period is much shorter than the period of revolution. In the case of Jupiter, the cross section is an ellipse with polar radius of 66,854 km and equatorial radius of 71,492 km. The ratio between the cross section and the global surface is near 1/4 that is 0.2488 and with an albedo measured from the Cassini mission of 0.503, the absorbed radiation is 53.3 × 0.2488 × (1 − 0.503) = 6.613 W m−2 . It is rather evident that the planet emits more radiation than it receives (14 W m−2 , against 6.614 W m−2 ) and since its discovery the imbalance has been justified with an energy source in the interior of the planet. The same problem is present for Saturn and is much less evident for Uranus. A possible explanation of the internal source is the contraction of the planet. The outer planets are made up mostly of a mixture of gases (mainly helium and hydrogen) and in very loose sense they are “missing stars” where the equilibrium is between the gravitational pull that tends to squeeze them which is resisted by the internal pressure. For such a body we can easily evaluate the potential energy dU = −

Gm(r )dm r

where dU correspond to the change in potential energy when a layer of mass dm is added to a preexisting mass m(r ). Considering that dm = 4πρr 2 dr,

m(r ) =

4 πρr 3 3

Assuming for simplicity a constant density ρ the potential energy for a mass M with radius R is given by 3 G M2 (1.11) U =− 5 R For a constant mass, the energy becomes more negative if the radius decreases (i.e. there is contraction of the planet). In the case of continuous contraction gravitational energy is released. This was the mechanism envisaged to explain the energy for the Sun before nuclear fusion was discovered. However if that does not work for the sun it could well work for a planet. We get for the power emitted dU d R 3 G M2 d R dU = =− d R dt 5 R 2 dt dt

(1.12)

6

1 The Environment in the Solar System

So that the required contraction is dR 20π ΔW R 4 = dt 3 G M2

(1.13)

where ΔW is the internal source of the planet given by ΔW =

1 dU 4π R 2 dt

(1.14)

Using the appropriate data M = 1.9 × 1027 kg, R = 7 × 7 m, G = 6.67 × 10−11 N m2 kg−2 , ΔW = 7.485 W m−2 we obtain d R/dt ≈ 0.5 mm/year. This number is so small that in practice cannot be measured.

1.1.3 The Greenhouse Effect In Table 1.1, it is shown also the surface temperature of the planet that is quite different from the radiative equilibrium temperature. Usually the surface temperature is higher than the radiative equilibrium and the difference is attributed to the so-called greenhouse effect that we will describe in detail later (we do not have the necessary tools now). The same cannot be said for the outer planets because we do not know where the surface (if exist!) is located. The name of the effect is rather misleading and makes you think about the warm environment of a greenhouse. However, this is determined by the fact that the greenhouse is closed and so the air motions inside are blocked. The same effect you have if you leave your car with the windows shut in the Sun. On the other hand what is meant by greenhouse effect is the following. The greenhouse is warmed by the solar radiation because the walls are transparent to that. On the other hand, the greenhouse does not lose heat because the walls are opaque to the infrared radiation. This imbalance makes the inside of the greenhouse to warm. The dominant effect in an greenhouse is the absence of convection, that is, the air movements. However on a planet with the atmosphere it could happen that the solar radiation reaching the ground is absorbed. For example for the Earth about 70% of the incoming solar radiation. The warmed surface of the planet emits infrared radiation that is absorbed by the atmosphere. The efficiency of heating depends on the chemical composition of the atmosphere and on the amount of the absorbing gases. The atmosphere of Venus contains large amount of carbon dioxide CO2 having a surface pressure of about 90 atm while the Earth contains much less CO2 but considerable amount of water vapor H2 O. Mars has a very tiny atmosphere with surface pressure of about 0.006 atm. A very simple model of the greenhouse effect can be built with the few things we know. We now assume as in Fig. 1.2 that our system is made up of planet surface at temperature Ts which emits as a black body σTs4 . The absorbing atmosphere is a single layer at temperature Ta which absorbs a fraction ∊ of the

1.2 The Point of View of Entropy

7

Fig. 1.2 The simple scheme to illustrate the greenhouse effect

4

(1-ε)σTa

ATMOSPHERE

Ta

4

σTs

SURFACE

Ts

radiation coming from the surface, ∊σTs4 so that after traversing the atmosphere the radiation lost to space would be (1 − ∊)σTs4 . The atmosphere being at some temperature Ta will emit radiation ∊σTa4 in both the upward and downward directions. However, the atmosphere is transparent to the solar radiation that will be deposited at the ground in the amount (1 − A)S/4. We can write the energy balance of the surface S (1 − A) = (1 − ∊)σTs4 + ∊σTa4 (1.15) 4 While the balance for the atmosphere reads ∊σTs4 = 2∊σTa4

1

⇒ Ts = 2 4 Ta

(1.16)

After substituting into (1.15) and with some laboring, we get [

S (1 − A) Ts = σ 4

]1 ( 4

1−

( ∊ )−1 ∊ )−1 = Te 1 − 2 2

(1.17)

We see that the radiative equilibrium temperature is increased by a factor (1 − 0.5∊)−1 . To obtain the observed mean temperature Ts = 288 K we need ∊ = 0.77 and the corresponding atmospheric temperature is Ts = 241 K. This is a very simple model and can be improved by introducing several layers of the atmosphere in what it is known as gray model. Hopefully we will deal later with that.

1.2 The Point of View of Entropy There is another interesting way to look at the equilibrium temperature and that involves the concept of entropy. We know from thermodynamics that if we provide an amount of heat δ Q to a system which is a temperature T then its entropy S increases by an amount

8

1 The Environment in the Solar System

dS =

δQ T

(1.18)

We can make a practical example by considering two large reservoirs of heat at temperatures T1 and T2 with T2 > T1 . The two reservoirs exchange heat by conduction so that heat will flow from the hot reservoir T2 to the cold T1 at a rate F. Then the cold reservoir will gain energy and its entropy will increase by the amount F/T1 while the hot reservoir will lose energy and its entropy will decrease by the amount −F/T2 . The conduction process does not affect the entropy of the system so that the total change will be d S1 d S2 dS = + =F dt dt dt

(

1 1 − T1 T2

)

( =F

T2 − T1 T1 T2

) ≥0

(1.19)

As we can imagine the transport of heat from the hot to the cold reservoir will result after a long time in a negligible difference on temperature between the two reservoirs. This is what is called thermodynamic equilibrium. Imagine now that instead of simple conduction part the heat from the hot reservoir (T2 ) is converted into work W using a reversible cycle so that the cold reservoir receives a flux F − W . The total change of entropy is then F−W F dS = − =F T1 T2 dt which implies that W ≤

(

1 1 − T1 T2

) −

W ≥0 T1

T2 − T1 F = ηF T2

(1.20)

(1.21)

where η = 1 − (T1 /T2 ) is the efficiency of the Carnot cycle.

1.2.1 The Entropy Budget of a Planet The few things we have summarized about thermodynamics can be applied to the radiative equilibrium for a planet. In our case energy is being transferred from the Sun (Tsun = 5800 K) to the Earth (Te = 255 K) so that the entropy production amounts to ( ) S 1 1 dS = (1 − A) − (1.22) ≈ 0.89 W K−1 m−2 4 Te Tsun dt Actually this production is the result of a number of processes that can be detailed in a rough energy balance of the Earth’s atmosphere system as shown in Fig. 1.3. In this figure a simplified energy budget of the Earth is shown. The 30% of incident solar radiation (340 W m−2 ) is reflected so that 340 × 0.3 = 102 W m−2 is reflected and 340 − 102 = 238 W m−2 enters the atmosphere. Of this radiation about 40%, that is,

1.2 The Point of View of Entropy

9

Reflected radiation 102 Wm-2

Ta = 255 K Incoming solar radiation 340 Wm-2

Surface net emission 40 Wm-2

Absorbed by the ground 142 Wm-2

Latent and sensible heat 102 Wm-2

Ts = 288 K

Fig. 1.3 The simplified energy budget of the Earth

95 W m−2 are absorbed by the atmosphere while the rest is absorbed by the ground (142 W m−2 ). We will discuss in detail the surface balance while for the time being we just assume that the power absorbed by the ground partly is dissipated by the latent and sensible heat flux (102 W m−2 ) while the remaining (40 W m−2 ) are emitted as infrared radiation. Latent and sensible heat fluxes are related to the turbulent transport in the layer in contact with the ground. On the other hand the infrared flux is the “net” flux that takes into account the difference between the upward and downward flux. We just need to fix a few temperatures, Tsun , the temperature of the Sun; Ta , the atmospheric temperature (255 K; and Ts , the surface temperature (288 K). Then the entropy fluxes are ( ) ˙Stur b = 102 1 − 1 , Ta Ts ( ) 1 1 ˙Ssw,a = 98 − , Tsun Ta

( ) ˙Ssw,s = 142 1 − 1 Ts Tsun ( ) 1 1 ˙Slw,a = 40 − Ta Ts

(1.23)

Using the appropriate values it is easy to find that (in Wm−2 K−1 ) S˙tur b = 0.046,

S˙sw,s = 0.469 S˙sw,a = 0.367,

S˙lw,a = 0.018

(1.24)

The total entropy flux is just the sum that amounts to the value found in (1.22). Actually the incident flux from the Sun is made up of very low entropy photons because the temperature of the Sun is 5800 K and a detailed calculation (see, for example, Wu and Liu (2010) gives a value of 0.079 W m−2 K−1 for the incident radiation and 0.110 W m−2 K−1 for the reflected radiation. These values are much smaller than the entropy flux of the outgoing longwave radiation. A more correct

10 Table 1.3 Production of entropy within the Earth System

1 The Environment in the Solar System Process

%

Absorption SW radiation Absorption LW radiation Water cycling Atmospheric circulation Life Humans

92.8 2.6 2.6 1.1 0.7 0.04

calculation (see appendix) gives a value around 1.25 W m−2 K−1 . This means that the low entropy of the incident energy is degraded within the Earth system by a number of processes which are summarized in Table 1.3. We can see that the major source of entropy within the Earth is the absorption of solar radiation while life and human activity are apparently quite negligible. On the other hand, the absorption of infrared radiation (LW) and the transfer of heat by evaporation/condensation processes at the surface (water cycling) are order of magnitude less than absorption of SW radiation. The conclusion is that entropy production is mainly affected by the albedo and much less by the partitioning at the surface between radiative and turbulent fluxes. It is interesting to see how the radiative temperature Te is affected by something we have neglected so far which is the poleward transport of heat. We will discuss in detail some of these processes later in the book. For the time being, we can consider the system made up by two boxes representing the tropical region and the polar region. The tropical region absorbs ST radiation and emits Q T while the same quantities for the polar regions are S P and Q P . The exchange of energy between the two boxes is assumed to be Q H T . We can write the balance for the two boxes St − Q t − Q ht = 0 (1.25) S p − Q p + Q ht = 0 where the flux of energy Q H T is simply proportional to the difference of temperature between tropic (Tt ) and the pole (T p ) Q ht = k(Tt − T p )

(1.26)

where k is some kind of conductivity whose dimensions are W m−2 K−1 . The emitted infrared radiation can be written as linear with temperature expressed in ◦ C, Q t, p = A + BTt, p . This approximation was used for the first time by Mikhail Budyko and will be discussed later in the book. Up to now the temperatures are expressed in degree Celsius. With the approximations just mentioned we can rewrite (1.25) as St − A − BTt − k(Tt − T p ) = 0 S p − A − BT p + k(Tt − T p ) = 0

(1.27)

1.2 The Point of View of Entropy

11

This is a simple linear system that can be solved for Tt and T p giving Tp =

St − A ΔS(k + B) − , B B(2k + B)

Tt = T p +

ΔS 2k + B

(1.28)

where ΔS = St − S p . Once the temperatures of the two boxes are determined it is possible to evaluate the entropy production that can be separated in the radiative entropy ) ( (1.29) S˙rad = (Q t /TT − St /Tsun ) + Q p /TP − S p /Tsun And the transport entropy S˙ht = Q ht (1/TT − 1/TP )

(1.30)

where TT = Tt + 273.15 and TP = T p + 273.15 are the absolute temperatures. Figure 1.4 shows the results of the above calculations as a function of the conductivity coefficient k. The calculations were made for St = 300 W m−2 and S p = 170 W m−2 . The entropy production is dominated by the radiative component being of the order of 1.5 W m−2 K−1 while the transport component account only for 1% of the total in accordance with the value listed in Table 1.3. Also notice there is maximum in the entropy production to which corresponds a temperature difference between the two boxes of roughly 30 ◦ C. According to the hypothesis of the Maximum Entropy Production (MEP), the atmospheric circulation adjusts in such a way to maximize the entropy production. This can be seen in Fig. 1.4 where an increase in the coefficient

0.09 40

TEMPERATURE (°C)

0.08 30

0.07 TROPICS 0.06

20

0.05 10

0.04 POLAR REGIONS

0

0.03 0.02 ENTROPY PRODUCTION

-10

ENTROPY PRODUCTION (W m-2 K-1)

0.1

50

0.01 0

-20 10 -2

10-1

10 0

10 1

10 2

EXCHANGE COEFFICIENT (W m -2 K-1)

Fig. 1.4 The temperatures of the tropics (upper curve) and polar regions (lower curve) as a function of the exchange coefficient k. The dashed line shows the entropy production as a function of k

12

1 The Environment in the Solar System

k corresponds to a decrease of the temperature difference between the tropics and the pole. We can see that this corresponds to a mechanism of negative feedback. If for some reason there is an increase in the coefficient k the gradient of temperature is reduced and so it is the intensity of the circulation that leads to a restoration of the initial gradient. We will return to the MEP principle several times in the book starting from the appendix.

Appendix Stefan–Boltzmann Law We have given Planck’s law (1.1) without proof and it would be easy that integrating Eq. (1.1) with respect to wavelength we would get the total power emitted per unit surface (the Stefan–Boltzmann law (1.5)). Actually this was found much earlier than Planck’s law and just for curiosity we give here its derivation based only on thermodynamics. We can start from the relation between energy density and radiation pressure. The Poynting vector S which represents the power per unit area so that S = power/ar ea = (ΔF/Δt)Δx/ar ea = pr essur e Δx/Δt. In the case of electromagnetic radiation Δx/Δt it is just the speed of light so that we came to the conclusion that pressure of radiation p is p = S/c. Considering now the intensity of radiation I which has been defined in (1.1) impinging on a reflecting surface with an incidence angle θ (Fig. 1.5), the relation between pressure and intensity is now 2I cos2 θ dp = c dΩ We get the total pressure by integrating on the total solid angle Ω = 4π. We have, considering that dΩ = 2π sin θdθ, p=

4π I c

( 0

π/2

cos2 θ sin θdθ =

4π I u = 3c 3

(1.31)

where we have used the relation between energy density u and intensity of radiation u = 4π I . We can now refer to a volume V so that the total internal energy U is given by U = 3 pV and then the first principle can be written as d Q = dU + pd V = d(3 pV ) + pd V = 4 pd V + 3V dp

(1.32)

Appendix

13 2πr2sinθ dθ θ

A

I cos θ dθ r θ

A/cos θ

Fig. 1.5 The calculation of radiation pressure for a perfectly reflecting surface. On the left the radiation is incident with angle θ. On the right the calculation of the infinitesimal solid angle is shown

And from the definition of entropy dS = which implies

So that

dQ p V = 4 d V + 3 dp T T T

∂S p =4 , ∂V T

(1.33)

∂S V =3 ∂p T

4 p dT ∂2 S = −4 2 , ∂V ∂ p T T dp

∂2 S 3 = ∂V ∂ p T

(1.34)

From the quality of (1.34) we have dT dp =4 T p

(1.35)

That once integrated gives p = CT 4

⇒

u = 3 p = 3C T 4 = aT 4

(1.36)

with a as constant. Notice that the relation between energy density u and power flux is that S = uc so the product ac corresponds to the Stefan–Boltzmann constant. At the time of this derivation, Planck’s law was not known yet so that a was determined experimentally. Knowing the relation for Planck’s function (1.1) σ = ac can be determined as (1.6).

14

1 The Environment in the Solar System

Black Body Radiation Entropy Flux We define L the entropy per unit solid angle then the entropy spatial density s can be found by integrating over the solid angle s=

1 c

( LdΩ =

2πL c

(

π

sin θdθ =

0

4πL c

(1.37)

Given L as a vector with assigned angle θ with respect to a surface the flux of entropy J in the normal to the same surface will be (

( J=

π/2

L cos θdΩ = 2πL

sin θ cos θdθ = πL

(1.38)

0

As in the case of (1.37) we assume L to be isotropic that is uniform in all the directions. Now using (1.32) we write (

( S=

ds =

dQ = T

(

dU + pd V = T

(

ud V + pd V T

(1.39)

where from (1.31) u = p/3 and u = aT 4 so that the substitution would give S=

4 3 aT V 3

(1.40)

That represents black body radiation entropy. Eliminating L between (1.37) and (1.38) we have c S c4 3 4 c = aT = σT 3 (1.41) J= s= 4V 43 3 4 where we have used ac = σ.

Is MEP Working on the Planets? Richard Goody has been a long time friend of entropy and atmospheric circulation. In a 2007 paper, he went to test in a very simple way if the MEP hypothesis could be applied also to a few other planets in the Solar System. As we have already done he divided each planet in a tropical region (1) and northern region (2). The solar radiations received by the two regions are S1 and S2 while the emitted infrared radiations are E 1 and E 2 . According to (1.25) we have F = S1 − E 1 = E 2 − S2

(1.42)

Appendix

15

where F is the energy flux between the two zones. The production of entropy is ( s˙ = F

1 1 − T1 T2

) (1.43)

Equation (1.42) implies S1 + S2 = E 1 + E 2 so that (1.42) becomes s˙ =

E 2 − S2 S1 − E 1 − T1 T2

Now we know that E 1 and E 2 are dependent on T (see (1.27)) so that the minimization with respect to T1 is the same of minimization with respect to E 1 and (1.43) is equivalent to E 2 − S2 S1 − E 1 S1 S2 s˙ = − =2− − (1.44) E2 E1 E1 E2 so that

S1 S2 d E 2 d s˙ = 2+ 2 =0 d E1 E1 E2 d E1

(1.45)

where from (1.42) d E 2 /d E 1 = −1 so that we have maximum production if S2 = S1

(

E2 E1

)2 (1.46)

Table 1.4 Solar and thermal emissions for Venus, Earth, and Mars. Adapted from Goody (2007) Fluxes (W m−2 ) E equ E mep E obs Venus

Earth

Mars

Mean S1 S2 S1 /S2 Mean S1 S2 S1 /S2 Mean S1 S2 S1 /S2

157.3 192 123 1.56 237 300 173 1.73 116.7 140 93 1.51

157.3 175 140 1.25 237 270 204 1.32 116.7 129 105 1.23

157.3 161 153 1.06 237 258 215 1.20 116.7 139 94 1.48

16

1 The Environment in the Solar System

This result can be tested with the data for the three planets of our interest (Venus, Earth, and Mars) as listed in Table 1.4. In this table E equ corresponds to a situation of radiative equilibrium that is when S1 = E 1 and S2 = E 2 compared to the observed data E obs and the data calculated from (1.46) (E mep ). The calculation starts assuming the ratio S1 /S2 is equal to the radiative equilibrium and that the average value 0.5(E 1 + E 2 ) coincides with the observed mean value. Notice that the S1 /S2 value in the E mep column is just the square root value of the E equ column. The MEP calculations show significant discrepancies with the observed values with difference on 0.19 for Venus, 0.12 for Earth, and 0.25 for Mars. The uncertainties of the observed date are however large even in the case of the Earth.

Bibliography Textbooks Jacob DJ (1999) Introduction to Atmospheric chemistry. Princeton University Press Kleidon A (2016) Thermodynamic foundations of the earth system. Cambridge University Press Kleidon A, Lorenz RD (2004) Non-equilibrium thermodynamics and the production of entropy: life, earth, and beyond (Understanding Complex Systems). Springer Marshall J, Plumb RA (2007) Atmosphere, ocean and climate dynamics: an introductory text. Academic

Articles Goody R (2000) Sources and sinks of climate entropy 62. J Roy Met Soc 126:1953 Goody R (2007) Maximum entropy production in climate theory. J Atmos Sci 64:2735 Lorenz RD, Lunine JI, Withers PG, McKay CP (2001) Titan, mars and earth : entropy production by latitudinal heat transport. Geophys Res Lett 28:418 Wu W, Liu Y (2010) Radiation entropy flux and entropy production of the Earth system. Rev Geophys 48:RG2003

Chapter 2

The Fluid Environment of the Earth

In the previous chapter, we gave a very general introduction to the Earth’s environment which is mainly determined by the interaction of our planet with the Sun. We have mentioned however the greenhouse effect produced by the presence of tiny quantities of radiative active gases in the atmosphere. We have also mentioned that energy exchanges between the surface and the atmosphere contribute to determine the surface temperature. It is quite evident from the role of the fluid Earth in this context. For fluid Earth we intend the atmosphere and the ocean.

2.1 The Atmosphere 2.1.1 Chemical Composition The atmosphere of the Earth is made up by a mixture of gases with the most important being nitrogen (N2 ) and oxygen (O2 ). Their abundance is specified by the mixing ratio. The volume mixing ratio is the ratio between the mole of a gas i and those of the atmosphere and is just a number. Just consider the gas law pV =

m RT = n RT M

(2.1)

where p is the pressure, V is the volume, m is the mass of the gas, M its molecular mass, n is the number of moles, R is the gas constant, and T is the temperature. From (2.1) we see that the volume mixing ratio is the same as the ratio between the partial pressure of the gas pi and the atmospheric pressure p. Equation (2.1) can be written for the gas i and the atmosphere pi V = n i RT ,

pV = n RT

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 G. Visconti, The Fluid Environment of the Earth, https://doi.org/10.1007/978-3-031-31539-8_2

17

18

2 The Fluid Environment of the Earth

So that the mixing ratio by volume Ci =

pi ni = p n

(2.2)

In the same fashion we can define the mass mixing ratio as the ratio between densities which are related to pressure by ρ=

pM , RT

ρi =

pi Mi RT

And the mass mixing ratio χi χi =

Mi pi Mi = Ci M p M

(2.3)

Another important quantity is the number density, n i which is the number of molecules per unit volume and is measured in cm−3 . The number density is related to the volume mixing ratio because the mass density is the product of the number density and the mass of a single molecule which is the molecular mass (M) divided by the Avogadro number Av . ρi = n i

pa Mi pi Mi pi = , ⇒ ni = = Ci RT kT kT Av

(2.4)

where k is the Boltzmann constant defined as R/ Av . Equation (2.4) can be used to find the number density of the atmosphere at the surface where the pressure is 1.013 × 105 Pa (Pascal) na =

1.013 × 105 = 2.69 × 1025 molecules cm−3 1.38 × 10−15 × 288

where we have used the Boltzmann constant k = 1.38 × 10−18 JK−1 and a temperature of 15◦ . The number density has great importance in the chemistry of the atmosphere and the radiative calculations. An important parameter is the columnar density, N defined as the number of molecules contained in an air column of height z t . We have ( zt

Ni =

n i dz

(2.5)

0

At this point we can summarize in Table 2.1 the data for the chemical composition of the terrestrial atmosphere. The gases with larger concentration (N2 , O2 e Ar) are chemically inert. On the other hand, other gases with much lower concentrations have major impact on the climate of the Earth. In the table, it seems that the presence of water vapor (H2 O) to be neglected. Actually its mixing ratio is quite variable ranging from 10−6 to 10−2 mole/mole for a maximum mixing ratio of 10 g/kg. This gas in

2.1 The Atmosphere

19

Table 2.1 The chemical composition of the atmosphere of the Earth. r m v indicates the volume mixing ratio and r m m is the mass mixing ratio in in grams/kg r m v (mole/mole) r m m (g/kg) Gas Nitrogen (N2 ) Oxygen (O2 ) Argon (Ar) Carbon dioxide (CO2 ) Neon (Ne) Ozone (O3 ) Helium (He) Methane (CH4 ) Krypton (Kr) Hydrogen (H2 ) Nitrous oxide (N2 O)

0.78 0.21 0.0093 400 ×10−6 18 ×10−6 (0.01 – 10) ×10−6 5.2 ×10−6 1.9 ×10−6 1.1 ×10−6 500 ×10−9 360 ×10−9

750 230 12 0.6 0.0093 1.65 ×10−5 –1.65 ×10−2 7.1 ×10−4 9.3 ×10−4 4.91 ×10−3 3.4 ×10−5 5.43 ×10−4

the environmental conditions of the Earth can exist in its three different phases (ice, liquid, and gas) and we will deal with its thermodynamics at length. Beyond argon the concentrations are so low that we talk about trace gases and their concentrations are expressed in ratio of 10−6 and 10−9 , respectively. Consequently neon has a mixing ratio of 18 ppm while nitrous oxide has a mixing ratio of 360 ppb. ppm, parts per million or ppb, parts per billion corresponding to volume mixing Although the atmosphere is a mixture of gases from a thermodynamic point of view, in most cases it behaves as a perfect gas. We can evaluate the molecular mass Ma using a weight average of the main gases Ma =CN2 · MN2 + CO2 · MO2 + CAr · MAr = 28 · 0.78 + 32 · 0.21 + 39.4 · 0.01 = 28.96 g/mole

(2.6)

So that air is a perfect gas with molecular mass 29.

2.1.2 Atmospheric Pressure The atmospheric pressure is defined as the weight of the atmosphere above the unit surface so that it is measured in units of newton/m2 that corresponds to 1 pascal, Pa. At the surface of the Earth the pressure amounts to 1.013 × 105 kg m−1 s−2 that is 1.013 × 105 Pa. The atmosphere, atm corresponds to 1.013 × 105 Pa. In the cgs system the basic unit is the bar with 1bar = 105 Pa. From this we derive the millibar, mb which corresponds to 1 100 Pa. Today the mb is substituted with the hectopascal, hPa, 1 mb = 1 hPa.

20

2 The Fluid Environment of the Earth

The knowledge of the atmospheric pressure pa allows to evaluate the atmospheric mass m a so that the weight is simply m a g, with g the acceleration of gravity. The surface of the Earth is 4π a 2 with a radius of the Earth. We have ma =

4πa 2 pa g

(2.7)

Using a = 6400 km we find m a = 5.2 × 1018 kg and with the molecular mass of 29 in the atmosphere there are 5.2 × 1021 /29 = 1.8 × 1020 mole of air. As an application of the mass mixing ratio we can evaluate the total quantity of carbon dioxide emitted in the atmosphere starting from the industrial revolution. Using (2.4) we can write the mass difference as ( ) 44 ΔMCO2 = 5.2 × 1018 400 × 10−6 − 280 × 10−6 = 9.45 × 1014 kg 29

(2.8)

where we have assumed an initial volume mixing ratio of 280 ppm and a final mixing ratio of 400 ppm so that the total mass is about 945 billions of tons. Expressed in mass of carbon this corresponds to (12/44) × 9.45 × 1014 = 2.58 × 1014 kg that is about 260 billion of tons. The atmospheric pressure is a good measure to establish the masses of the different atmospheric regions. Based only on the temperature behavior (see Fig. 2.1) the troposphere is the region where the temperature decreases constantly with height that extends from the surface up to 8 − 18 km (depending on latitude and season). The stratosphere shows a temperature increasing with altitude and extends from the top of the troposphere to about 50 km. Finally the mesosphere which extends up to 80 km with decreasing temperature.

Fig. 2.1 The behavior of temperature and pressure in the Earth’s atmosphere

80

80

ALTITUDE, km

MESOPSPHERE 60

60

40

40

20

20

STRATOSPHERE TROPOSPHERE

0 0.01 0.1

1

10

100 1000

PRESSURE, hPa

0

200

240

280

TEMPERTURE, K

2.1 The Atmosphere

21

The region which separates the troposphere from the stratosphere is known as tropopause and is found at a pressure of roughly 100 hPa. We can easily compare the atmospheric mass of the troposphere with the total mass using just the pressure of the tropopause so that the mass fraction f tr op is f tr op = 1 −

p(tr opopause) = 0.90 p(0)

The troposphere contains about 90% of the total atmospheric mass. With the same criteria we can establish that the stratosphere contains about 99% of the remaining 10%. The troposphere is the region where we live and where almost all the human activities develop (including the airline flights).

2.1.3 Vertical Changes of Pressure and Temperature It is quite easy to find the pressure variation of pressure with altitude once the temperature profile is known. On the other hand, the latter is a quite complicated matter that will be discussed in due time. To find the variation of pressure with height just consider the pressure difference between two horizontal surfaces at heights z and z + dz. Because the pressure decreases with altitude the difference [ p(z) − p(z + dz)]A (where A is the horizontal surface) determines a force directed upward which is balanced by the weight of the air contained in the elementary volume Adz so we have ρg Adz = [ p(z) − p(z + dz)]A (2.9) That becomes

dp = −ρg dz

(2.10)

Substituting ρ = pM/RT we obtain Mg dp =− dz RT p

(2.11)

We see how the change in pressure depends on the temperature. In the case of an isothermal atmosphere (T = cost) the integration of (2.11) gives ln p(z) − ln p(0) = − So that

Mg z RT

( ) Mg p(z) = p(0) exp − z RT

(2.12)

22

2 The Fluid Environment of the Earth

The quantity RT /Mg has the dimensions of a length and it is called scale height H H=

RT Mg

(2.13)

The same behavior is observed by the density which is proportional to ρ ρ(z) = ρ(0)e−z/H

(2.14)

With an average value T = 250 K we obtain H = 7.4 km. This means that the pressure decreases by a factor e every 7.4 km. Another simple solution can be obtained for a linear change of temperature T = T0 − Γz. In this case the equation to be integrated is dp Mg dz Mg dT =− = (2.15) p R T0 − Γz RΓ T With the result that p = p0

(

T T0

) Mg RΓ (2.16)

Using Γ = 6.5 K/km we have Mg/RΓ = 5.26. The differences in the pressure ratios p/ p0 between the isothermal case and linear case are 0.76, 0.78 at 2 km up to 0.25, 0.26 for 10 km. The calculation of the vertical profile of temperature is not an easy task. The simplest approach is to recur to the approximation of an adiabatic state of the atmosphere. We can consider an air parcel as a mass of air which is in its interior is thermally isolated from the outside so that it exchanges no significant amount of heat with its surrounding. In their very interesting book, Bohren and Albrecht compare the air parcel to the atom of the atmosphere. Such an air parcel in contact with the ground may heat up and expand and so acquire a positive buoyancy. This can be explained in a simple way when you consider that the pressure inside the parcel must always equalize the pressure outside. This means that with the same pressure the density will go as the inverse of the temperature. The parcel which has a volume V is subject to two opposite forces. If its density is ρ , the weight of the parcel will be ρ , V g. The other force is Archimedes’ ρV g where ρ is the density of the surrounding air. The net force will be (ρ − ρ , )gV so that if ρ , < ρ there will be positive buoyancy with acceleration directed upward while if ρ , > ρ there will be a negative buoyancy. Heating the parcel results in a decrease in density and so positive buoyancy. When the parcel cools the opposite will happen. Heating and cooling will be in any case adiabatic that is the exchange with the surrounding will be zero and according to the first principle cv dT + pd V = 0 It is convenient to write this equation for unit mass

2.1 The Atmosphere

23

Cv dt + pdα = 0

(2.17)

where Cv is the specific heat at constant pressure per unit mass and α is the specific volume. The equation for the perfect gas is then written as pV =

m RT M

⇒ pα = RT

(2.18)

where α = V /m and R is defined as R/M = 8.31 × 103 /29 = 287 JK−1 kg−1 . Equation (2.17) can be written as (Cv + R)dT = −αdp = αρgdz = gdz

(2.19)

where we have used the hydrostatic equation dp = −ρgdz and used α = 1/ρ. Deriving Eq. (2.19) with respect to z we obtain Cp

dT = g, dz

⇒

dT g = = Γd dz Cp

(2.20)

where Γd is called the dry adiabatic lapse rate and gives the vertical rate of change of temperature for an adiabatic atmosphere. It only depends on the gravity acceleration and specific heat at constant pressure. In the atmosphere of the Earth C p ≈ 1000 J/K−1 kg−1 so that Γd ≈ 9.8 K km−1 .

2.1.4 The Point of View of Entropy The adiabatic lapse rate is connected strictly with the concept of stability and again we can use the entropy to discuss the problem. Suppose we shift adiabatically an air parcel from the height z to z + dz. The parcel will conserve the entropy s(z) and its pressure will be p , (z) = p(z + dz) and its density ρ(s, p , ). If the parcel should be stable this density must be higher than the surroundings at z + dz which has the same pressure but different entropy s , = s(z + dz). For stability we must have ,

,

,

ρ( p , s) > ρ( p , s ),

( ⇒

∂ρ ∂s

) p

ds 0 Cp T T T dz T dz

Substituting dp/dz = −ρg we have

(2.21)

24

2 The Fluid Environment of the Earth

T

dT ds = Cp +g>0 dz dz

Which implies −

dT g < dz Cp

(2.22)

That is vertical movements are stable and the temperature gradient in the atmosphere is less than the adiabatic lapse rate. In the opposite case convection sets in starting mixing processes that bring the temperature gradient back to its equilibrium value. Actually the real gradient in the atmosphere is much less than Γd being around 6.5 K km−1 . The reasons we will try to illustrate later in the book. There is another application of the entropy (nowadays as Bohren and Albrecht says it is used even in the fashion trade) and refers to some conservation property. If we use its definition and assume adiabatic process in which ds = 0 we have from (2.21) observing that V /T = R/ p R dp C p dT − =0 p dz T dz Once integrated this gives

( θ=T

p0 p

(2.23)

) R/C p (2.24)

where T is the temperature at pressure p and θ is a conserved quantity for adiabatic motion that we call potential temperature. We calculate the logarithm of (2.24) R R ln p + ln p0 Cp Cp

ln θ = ln T − And its differential d ln θ =

R dp dT − Cp p T

If this is compared with the definition of entropy ds = C p

dT dp −R T p (

we obtain ds = C p d ln θ,

θ ⇒ s − s0 = C p ln T0

) (2.25)

It is evident from (2.25) that in the adiabatic approximation ds = 0 the potential temperature is conserved θ = T0 . Besides differentiating the ln θ with respect to z we have

2.2 The Ocean

25

1 g 1 dθ 1 dT R 1 dp 1 dT + = − = T Cp θ dz T dz C p p dz T dz So that

dθ θ = dz T

(

g dT + Cp dz

) (2.26)

And from the stability condition (2.22) we have stability for dθ/dz > 0 or instability for dθ/dz < 0.

2.2 The Ocean The ocean covers about 71% of the Earth’s surface and has a total volume of 1.33 × 1018 m3 and a mass of 1.35 × 1021 kg. The average density is about 1.015 kg m−3 which is slightly higher than the freshwater because it contains salt. As curiosity the ocean water represents 96.5% of all the water existing on the planet with the remaining 3.5% made by freshwater of which 69% is present as ice. Should this melt the average level of the ocean would rise approx. of 100 m. It is quite difficult to calculate this number because some of the ice is floating in the sea. It is to notice that surface freshwater (rivers, lakes, etc.) represents only the 0.02% of the total mass.

2.2.1 The Equation of State of Seawater The seawater contains ions of several elements as summarized in Table 2.2. The density of water not only depends on the pressure temperature but also on the salt content. The equation of state can be written formally as ρ = ρ(T , S, p)

(2.27)

where T is the temperature, S the salinity, and p the pressure. The salinity is a non-dimensional parameter because it is measured in grams of salt per kg of water, “per thousands” (o /oo ). Based on Table 2.2 water has an average salinity of 34.5 psu, practical salinity units. Actually rather than the density is the density anomaly σ which is defined as (2.28) σ = ρ − ρr e f where ρr e f = 1000 kg m−3 . In Fig. 2.2 the behavior of the density anomaly is shown as a function of temperature and salinity. We notice that the density increases with salinity and decreases with the temperature. Freshwater (S = 0) reaches a maximum in density for a temperature of 4 ◦ C. Freshwater colder than this value is less dense. This is the reason why in the lakes ice is formed in surface water. In winter the low

26

2 The Fluid Environment of the Earth

Table 2.2 The average composition of seawater. The values shown are quite independent from the geographical region

Salt

g/kg

Chlorine Sodium Sulfur Magnesium Calcium Potassium Bicarbonate Others Total salinity

18.98 10.56 2.65 1.27 0.40 0.38 0.14 0.11 34.48

40

SALINITY (psu)

SALINITY (psu)

15

10

10

10

B

36.0

15

15

35.5 35.0

5

34.5

0

34.0

30

33.5

10

A

5

5

10

15

20

25

35

TEMPERATURE(°C)

0

26.5 26.0 25.5 25.024.524.0

5

10

.

5

15

.

0

.

-5

27.0

0

.

.

0

.

20

28.5 28.0 27.5 .

29.0

20

25

20

20

.

25

30

36.5

25

30

20

TEMPERATURE (°C)

Fig. 2.2 The density anomaly of ocean water as a function of salinity and temperature. The gray regions represent the normal environmental conditions in the ocean

temperatures form ice at the surface rather than sink in the water. This is a very important property because if ice had a higher density than water all the ocean would solidify. In the range of normal oceanic conditions, we see that the influence of temperature prevails with respect to that of salinity. If we refer to the figure we notice that the same density (27.5) is found for a temperature of 0 ◦ C and a salinity of 34.2 psu (point A) or for a temperature of 11.4 ◦ C and a salinity of 36 psu (point B). The thermal expansion coefficient for water is defined as αT = − While the same relation for salinity

1 ∂ρ ρr e f ∂ T

(2.29)

2.2 The Ocean

27 DENSITY (g cm-3) 1.023 1.025 1.027

0

Pycnocline

TEMPERATURE (C) 5 10 15

SALINITY’ 33 34

32

Thermocline

Halocline

1

2

3

4

pycnoclino

DEPTH (km)

DEPTH (km)

Fig. 2.3 The top of the figure shows the profiles of temperature, salinity, and density of the ocean. The gray regions indicate the regions where the changes are more evident and correspond to the pycnocline for the density, to the thermocline for the temperature and to the halocline for the salinity. At the bottom the figure shows the qualitative behavior with latitude and depth

surface layer

1 2 3 acqua fredda e densa

4 5 60

βS = −

30

0 LATITUDE (°)

30

60

1 ∂ρ ρr e f ∂ S

(2.30)

These two quantities are necessary to write down the equation of state σ = σ0 + ρr e f [−αT (T − T0 ) + β S (S − S0 )]

(2.31)

This equation is actually a linearization around the point S0 , T0 for the variables a S and T . Typical values at the surface are αT = 10−4 ◦ C−1 and β S = 7.6 × 10−4 psu−1 . The value of σ0 is calculated for T = T0 and S = S0 . The pressure dependence does not appear explicitly in (2.31) but could be taken into account changing the expansion coefficients which have a weak dependence on pressure. At this point, we would like to give an idea of the general behavior of temperature and salinity in the ocean. It is quite evident that these variables have a rather complex climatology but just to give an idea we show in Fig. 2.3 the general behavior of the temperature and salinity as a function of the depth. We notice as in the first few hundred meters such quantities are constant. The reason is that this layer of the ocean is in contact with the atmosphere and the surface water is mixed constantly by the surface winds. As a matter of fact this region is called mixed layer Below the mixed layer the temperature has a tendency to decrease while the salinity increases. This region has a thickness of about 1 km and corresponds to the so-called thermocline which somehow represents the transition between the mixed

28

2 The Fluid Environment of the Earth

layer and the deep waters. The values shown in the figure are purely indicative in the sense that the mixed layer thickness may change between 30 and 200 m while the thermocline between 800 and 1000 m.

2.2.2 Incompressibility: Change of Pressure with Depth A rather interesting problem is the calculation of pressure dependence with depth. In the elementary textbooks, they assume the water incompressibility and then after the integration of the hydrostatic equilibrium, assuming a constant density the following relation is obtained: (2.32) p(z) = p0 − ρgz where p0 is the pressure at the surface z = 0 while the depth z is negative. The linear dependence of pressure with depth is a consequence of the assumption of constant density. On the other hand, the constancy of the density is a consequence of the assumption of the water incompressibility. This conclusion however implies a paradox because if the equation of state is p = f (ρ, T ). The constancy of ρ requires the constancy of p. The truth actually is that the water has a finite compressibility and the equation of state must be substituted by ρ(z) = F(S, T , p(z))

(2.33)

1 dρ ∂T ∂S ∂p =α +β +γ ρ dz ∂z ∂z ∂z

(2.34)

That once differentiated gives

where α=

1 ∂ρ , ρ ∂T

β=

1 ∂ρ , ρ ∂S

γ =

1 ∂ρ ρ ∂p

(2.35)

Observations show that the last term in (2.35) dominates so that we can write 1 ∂ρ ∂z g 1 ∂ρ = =− 2 ρ ∂z ∂ p cs ρ ∂p

(2.36)

where we have used the hydrostatic equilibrium ∂ p/∂ z = −ρg and noticed that the sound velocity cs2 = ∂ p/∂ρ. As we have made for the atmosphere we can introduce a scale depth D(z) defined as c2 (2.37) D(z) = s g So we need to integrate the equation

Appendix

29

dρ dz =− ρ D(z)

(2.38)

Then sound velocity in the seawater is roughly 1500 ms−1 and this implies a scale depth of more than 200 km that is between 210 km at the surface and 240 km at depth so that we can make the constancy assumption without making a great error taking into account the fact that the ratio z/D is comprised between 2% and 4%. The density results to be (2.39) ρ(z) = ρ(0) e−z/D ≈ ρ(0)(1 − z/D) And increases only of some percent. Using this relation in the hydrostatic equilibrium we arrive at ( z ) p(z) = p(0) + g Dρ(0)[ex p(−z/D) − 1] ≈ p(0) − gzρ(0) 1 − (2.40) 2D Neglecting the factor z/2D we obtain the (2.32). As we said the scale depth is analogous to the scale height for the atmosphere where for the isothermal case ∂ p/∂ρ = RT and H = RT /g

Appendix Flat Earth and Atmospheric Mass The calculation of the atmospheric mass that gives the (2.7) as result is quite simple because is based on the assumption of a flat Earth. Actually for an isothermal atmosphere where the density changes according to (2.14) one can show that the total mass m s per unit area is given by (

∞

ms =

ρ(z)dz

0

So that substituting (2.14) we obtain (

∞

m s = ρ(0)

e−z/H dz = ρ(0)H

(2.41)

0

So the atmospheric mass per unit area correspond to that of a layer of thickness H with constant density equal to the surface value ρ(0). This relation is the same as (2.14) because pa = ρ(0)RT so that m a = 4πa 2 ρ(0)

RT = 4πa 2 m s g

(2.42)

30

2 The Fluid Environment of the Earth

Fig. 2.4 Geometry for the calculation of atmospheric mass

Ar

A0 z a

Considering that H = RT /g. The problem is that the Earth is spherical so that the geometry to consider is shown in Fig. 2.4 so that the element to consider is not a parallelepiped as in the case of flat Earth but rather a truncated cone. We need to consider two different effects one related to the change of the gravity acceleration with altitude and the other related to the volume of the truncated cone. The first effect can be treated considering the definition of acceleration of gravity g g=

( GM ( z )−2 z )−2 GM ≈ = g 1 + 1 + 0 a2 a a (a + z)2

(2.43)

where g0 = G M/a 2 with a, radius of the Earth. The ratio between the area Ar at the base of the truncated cone at distance r and the area at the surface of the Earth (A0 ) is given by ( (a + z)2 z )2 Ar = ⇒ A = A (2.44) 1 + r 0 A0 a2 a The equation for the hydrostatic equilibrium gives us Ar Δp = gΔm spher e

(2.45)

where Δm spher e is the mass contained in the truncated cone. Substituting the (2.45) in (2.44) we have z )4 A0 Δp ( z )4 ( = 1+ Δm piatta = Δm s f era 1+ g0 a a

(2.46)

We see that the mass of the spherical atmosphere is larger than the one of the flat Earth of the amount

Appendix

31

(

1+

( ) ( ) 4z z )4 4H ≈ 1+ ≈ 1+ a a a

(2.47)

The solution (2.47) can be justified the left-hand side of (2.47) (

4z A0 ρ 1 + a

) dz = A0 ρ(0)e

−z/H

(

4z 1+ a

) dz

That can be integrated by parts with the result (

4H A0 ρ(0) 1 + a equivalent to

)

( ) 4H m spher e = 1 + m f lat a

It is a very small because (1 + 4H/a) ≈ 1.00125 so that the mass of the spherical atmosphere is about half of the 0.25%.

Average Temperature and Geopotential The equation for hydrostatic equilibrium (2.10) can be integrable between the heights z 1 and z 2 to obtain ( R p2 T dp (2.48) z2 − z1 = − g p1 p where p1 is the pressure at z 1 and p2 at height z 2 . The average temperature ⟨T ⟩ between the two heights is defined as ( ⟨T ⟩ =

p2 p1

T dp p

/(

p2 p1

dp p

(2.49)

So that the (2.48) becomes z2 − z1 =

R ⟨T ⟩ ln g

(

p1 p2

) (2.50)

So that the thickness between two surfaces at constant pressure is proportional to the average temperature of the layer between the two pressures. In meteorology geopotential, φ is defined as

32

2 The Fluid Environment of the Earth

(

(

z2

φ 2 − φ1 =

p2

gdz = −

z1

p1

dp ρ

(2.51)

So that for (2.50) we have ( φ2 − φ1 = −R⟨T ⟩ ln

p1 p2

) (2.52)

The geopotential gives an immediate measure of the average temperature of a layer contained between two isobars. It is measured in [m2 s−2 ], that is, energy per unit mass but most commonly is measured in m following (2.43).

Energy of a Hydrostatic Atmosphere The internal energy U per unit area of the atmosphere is given by (

∞

U=

ρcv T dz

(2.53)

0

While the potential energy P

(

∞

P=

ρgzdz

(2.54)

0

From the equation of hydrostatic equilibrium ρgdz = −dp so that the (2.54) becomes (

∞

P =−

zdp 0

Integrating by parts ( P=−

∞

pdz −

0

pz|∞ 0

(

∞

=

pdz 0

Because the pressure goes to zero more rapidly than z goes to infinity. And from the gas equation ( ∞

P=

ρ RT dz

(2.55)

0

The sum of the potential and internal energies gives (

∞

U+P= 0

(

∞

ρ(cv + R)T dz = 0

ρc p T dz = H

(2.56)

Appendix

33

where H is called enthalpy per unit area. If the atmosphere is heated up its enthalpy increases for two reasons, on one side its internal energy increases and on the other the altitude of its center of gravity ⟨z⟩ increases. (∞

ρgzdz P = ⟨z⟩ = (0 ∞ Ma g 0 ρgdz

(2.57)

where Ma is the total mass of the atmosphere.

Density of Seawater The most important variables which determine the density of seawater are the temperature and salinity. The salinity is strongly influenced by the geographical region because the dissolved salt depends on the river transport, rocks erosion, etc. The density of seawater varies between 1020 kg m−3 and 1030 kg m−3 while the freezing point is mostly influenced by salinity. The density as a function of salinity can be written as (2.58) ρ = ρ0 + AS + B S 1.5 + C S 2 where both ρ0 and the coefficients A, B, C are functions of temperature. In particular ρ0 is the density of freshwater given by ρ0 = 999.842594 + 6.793952 × 10−2 T − 9.095290 × 10−3 T 2 + 1.001685 × 10−4 T 3 − 1.120083 × 10−4 T 6 + 6.536332 × 10−9 T 5

(2.59)

While the coefficients A, B, C A = 0.823997 − 0.40644 × 10−3 T + 7.6455 × 10−5 T 2 + 8.332 × 10−10 T 3 + 5.4961 × 10−12 T 4 B = −5.5078 × 10−3 + 9.7598 × 10−5 T − 1.6218 × 10−6 T 2 C = 4.6106 × 10−4 (2.60) Figure 2.2 is based on these relations.

Steric Level of the Ocean An interesting application of the (2.31) is the evaluation of the ocean steric level. “Steric” refers to global changes in sea level due to thermal expansion and salinity variations. Especially in the last few decades the changes in sea level have been

34

2 The Fluid Environment of the Earth

Fig. 2.5 The calculation of the steric level of the ocean

η z=0

H δA z=-H

studied in connection with the consequences of global warming. Consider a water column as shown in tale Fig. 2.5 with base at level −H and the top at level η with respect to the reference altitude z = 0. The mass of the fluid column will be ( δm(t) = δ A

η(t) −H

ρ(z, t)dz

(2.61)

where δ A is the horizontal section. Deriving with respect to time we have [ ] ( η(t) ∂ρ ∂η ∂(δm) = δ A ρ(η, t) + dz ∂t ∂t −H ∂t The quantity ρ(η, t) is the surface density ρs so that we obtain 1 ∂η =− ρs ∂t

(

η(t)

−H

∂ρ 1 ∂ dz + ∂t ρs ∂t

(

δm δA

) (2.62)

This gives the changes in the ocean level as a function of time which results from a steric contribution due to the changes of density and a contribution due to the changes of mass (called mass component). The latter may be produced by the melting of alpine ice masses or the Antarctic pole. We can consider theta the water column is comprised between the atmospheric pressure patm and the pressure at the bottom pb so that pb = patm + g(δm/δ A). The (2.62) can be written as ∂η 1 =− ∂t ρs

(

η(t)

−H

[ ] ∂ρ 1 ∂ pb ∂ patm dz + − ∂t ρs g ∂t ∂t

(2.63)

Appendix

35

The changes in density can be reduced to temperature changes (thermosteric component) or change in salinity (halosteric component) so that referring to the (2.31) we write Δρ = αT ΔT − β S ΔS (2.64) − ρ At global level we neglect the changes due to the salinity and following the work of Williams et al. (2012) Eq. (2.64) can be written as Δη = αT ΔT D

(2.65)

where D indicates the global average depth of the ocean. We can now relate the temperature changes ΔT to the anomalies in the heat flux H. The heat capacity per unit area of the column of depth D is given by ρC p DΔT and so we have

That substituted in (2.65) gives

ρC p DΔT = HΔt

(2.66)

αT H Δη ≈ ρC p Δt

(2.67)

We can use the average global values for αT = 1.57 × 10−4 K−1 , C p = 3995 J kg−1 K −1 e ρ = 1026 kg m−3 to find that αT /ρC p ≈ 1.26 mm/year. In the period given by the mentioned paper (1961–2008) the flux change is 0.53 w m−2 so that for the same period we can get a change of the order of 0.66 mm/year.

Box Models The study of the ocean is a good excuse to introduce the use of the box models that we will use at length in the future. The example we make deals with the exchange between surface and deep ocean. We reduce the ocean to two boxes as shown in Fig. 2.6, the upper one that corresponds to the surface ocean and the lower one that corresponds to the deep ocean. Our purpose is to find the concentration of a substance in the two regions once the exchange fluxes are assigned between the two boxes. We call R the volume of water that the rivers pour annually in the surface ocean (in m3 /year) while Criv is the concentration of the substance transported by the rivers (in mole/m3 ). In the same fashion, we indicate with Vd and Cs the volume of water subtracted from the surface ocean (downwelling flux) and the concentration of our substance in the same region. Finally, the quantities that characterize the deep water are Vu (upwelling flux) and Cd . We also assume that the evaporation rate E be equal to the flux from the rivers and define Vd = Vu = Vmi x . The balance for the surface waters will be

36

2 The Fluid Environment of the Earth

Fig. 2.6 Two box model for the ocean

E Surface water

R

Cs

Vu

Vd

P

Cd Deep water

B

Vmi x

dCs = RCriv − Vmi x (Cs − Cd ) − P dt

(2.68)

where P is the particulate flux coming from the surface waters. In the same way we can write for the deep ocean Vmi x

dCd = Vmi x (Cs − Cd ) + P − B dt

(2.69)

where B indicates the burial flux in the sediments. At the steady state must be dCd dCs = = 0, dt dt

RCriv = B

(2.70)

At this point we calculate the efficiency of bioremoval from the surface as particle g defined as P bior emoval = (2.71) g= RCri v + Vmi x Cd input While the fraction f of buried particles is given by f = B/P f =

RCri v P

(2.72)

That can be written as f =

RCri v

[ ] RCri v Vmi x (Cd − Cs ) −1 = 1+ RCriv + Vmi x (Cd − Cs )

(2.73)

The product f g is an interesting parameter. As a matter of fact g is the fraction of the substance that reach the surface water and is eliminated as particulate, while f

Appendix

37

is the fraction of such particulate that survive the destruction. The product of these two quantities gives the fraction that is removed during a mixing cycle of the ocean. That is the transfer from the deep ocean to the surface waters. The product of (2.71) and (2.72) gives ( ) Vmi x Cd −1 fg = 1 + (2.74) R Criv For the phosphorus Vmi x ≈ 20R ≈ 1.9 × 1014 m3 /anno. The duration of an entire ocean cycle in which all the surface waters are substituted is about Tmi x = 1600 years so that the time to remove a specific element will be τ=

1600 T mi x = years fg fg

(2.75)

For phosphorus we have Cd /C f iu ≈ 5 and Cs /C f iu ≈ 0.25 and we obtain g ≈ 0.95. From (2.73) we obtain f ≈ 0.01 and from (2.75) we have a residence time of 160.000 years in the ocean. This means that substances that contain this element make 100 cycles of 1600 year between the deep ocean and the surface waters before going to the rest in the sediments where it remains for hundreds millions of year before being raised to the surface of the Earth by the tectonic movements.

Bibliography Textbooks Bohren CFBA (1998) Albrecht, atmosphere thermodynamics. Oxford University Press Broecker WS (1978) Chemical oceanography. Harcourt Brace Jacob DJ (1999) Introduction to atmospheric chemistry. Princeton University Press Marshall J, Plumb RA (2007) Atmosphere, ocean and climate dynamics: an introductory text. Academic Millero F (2013) Chemical oceanography. CRC Press Olbers D, Willebrand J, Eden C (2012) Ocean dynamics. Springer

Articles Prather MJ, Hsu JC (2019) A round Earth for climate models. PNAS 116:19330 Williams RG, Goodwin P, Ridgwell A, Woodworth PL (2012) How warming and steric sea level rise relate to cumulative carbon emissions. Geophys Res Lett 39:L19715

Chapter 3

Thermodynamics of the Atmosphere and the Ocean

In the previous chapters, we have used some elementary thermodynamics. Now we have the basis to go some step further and above all to apply some of our knowledge to the ocean. One of the conclusions of the previous chapter was that the atmosphere can be treated as a perfect gas. One of the gases present in the atmosphere is water vapor which can change phases in the environmental conditions of the atmosphere. That is, it can be in the gaseous form or liquid form or solidify as ice and this complicates the simple thermodynamics we used so far. These characteristics have some important consequences, for example, on the temperature profile of the atmosphere. As for the ocean the only thermodynamics we know so far is the equation of state and we will be able to obtain some interesting development concerning the stratification and stability.

3.1 Effects of Water Vapor Water vapor is a gas that can change phase (liquid, solid, or gas) in the physical conditions of the Earth’s atmosphere. The main quantity which defines the presence of water vapor is the vapor pressure defined as the partial pressure of the gas in equilibrium with its liquid. Think about some liquid water in a pan with a cover. The water molecules are constantly exchanged between liquid phase and the gas phase in the space under the cover. The equilibrium is reached when the number of molecules that evaporates from the liquid equals the molecules that return from the gas phase to the liquid. It is clear that the higher the temperature of water the larger is the number of molecules that evaporates and so is the equilibrium pressure. The same pressure corresponds to the saturation pressure. When the vapor pressure is higher than the saturation pressure we have condensation. The phase rule helps to determine how many variables are necessary to determine the saturation. For a number of c chemical components with p phase the number of variables n is given by

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 G. Visconti, The Fluid Environment of the Earth, https://doi.org/10.1007/978-3-031-31539-8_3

39

40

3 Thermodynamics of the Atmosphere and the Ocean

n =c+2− p

(3.1)

and in our case c = 1, p = 2 and so there is only one variable to determine the pressure which is temperature. The vapor pressure is usually indicated with e and so the saturation pressure with es and the equation which gives es as a function of temperature (that we will demonstrate in the appendix) is known as Clausius– Clapeyron that is ( ) ( ) 1 L 1 es − = (3.2) ln T e0 Rv T0 where e0 = 611 Pa for T0 = 273.15 K while L is the latent heat of vaporization (that is the energy needed to evaporate 1 kg of water) which amounts to 2.5 × 106 Jkg−1 while Rv è is the gas constant for water vapor 461 Jkg−1 K−1 . Rv is the ratio between the constant and the water molecular weight 8.31 × 103 /18. Up to this point we have considered the equilibrium between liquid and vapor phases while the same thing can be done for the interface vapor ice. The only thing we have to change is the heat of sublimation which substitutes the constant L in the (3.2). For the ice we have (Fig. 3.1) ( ) ( ) 1 Ld 1 es − = (3.3) ln T e0 Rv T0 where L d is the heat of sublimation 2.83 × 106 Jkg−1 . In Fig. 2.1 the so-called phase diagram for water is shown. We notice that the saturation pressure of vapor–ice is slightly higher than the liquid–vapor below zero. This means that in a clouds the formation of ice particles will prevail with respect to the liquid droplets. As a matter of fact this is a quite complex problem we will discuss in the chapter on aerosols. Equations (3.2) and (3.3) can be manipulated a little and put in a more simple form. So that the (3.2) becomes (3.4) es = AebT

100 Vapor Pressure , hPa

Fig. 3.1 The phase diagram of water. The dashed line represents the vapor pressure with respect to ice. The liquid below zero can exist only super-cooled

10

1

0.1 -40

-20

0 20 Temperature (°C)

40

3.1 Effects of Water Vapor

41

where A = 6.11 hPa, b = 0.067 ◦ C e T is in degrees centigrade for the vapor–liquid interface. The saturation pressure at −10 ◦ C will be 2.87 Pa. For the vapor–ice interface only b change to the value 0.078. Directly connected to the saturation pressure is the definition of relative humidity, r h that is the ratio between the partial pressure and the saturation pressure at the same temperature e r h(%) = 100 (3.5) es (T ) This is what you read on your home barometer. On the other hand, the dew point temperature Td is the temperature at which the air parcel would be saturated with respect to liquid water (3.6) e = es (Td ) And the frost point temperature T f is the same thing but with respect to ice. The amount of water vapor can also be indicated as mass mixing ratio q (also called specific humidity) that following the definition is given by q=

ρv ρ

(3.7)

where ρv is the water vapor density and ρ the density of air. q can also be expressed as a function of pressures q=

e e e M H2 O = ε = 0.622 p p p M

(3.8)

where ε is the ratio between the molecular weight of water and that of air. We consider now the mixture of air and vapor so that the total pressure is given by (3.9) p = pd + e where pd is the pressure of dry air. In the same way for the densities

The ratio between e and p

ρd = ρ − ρv

(3.10)

e q = p [ε + (1 − ε)q]

(3.11)

And the density ρ expressed as a function of q, p, and T ρ=

p p = RTv [RT (1 − q + q/ε)]

42

3 Thermodynamics of the Atmosphere and the Ocean

where Tv is the virtual temperature Tv = T (1 − q + q/ε) = T (1 + 0.6078q)

(3.12)

Virtual temperature is indication of the humidity content of an air mass. If we use a representative value of q = 0.01 ≈ 10 grams/kg the virtual temperature is about 0.5% higher than ambient temperature that means a few degrees difference.

3.2 Static Stability of the Atmosphere We have seen in Paragraph 2.1.4 that the temperature gradient has a direct influence on the stability of the atmosphere. We like to explore a little more this aspect. Consider that air parcels move only in the vertical and heir motion to be adiabatic. If the parcel moves from the height z 0 and temperature T0 to a height z 0 + Δz where its temperature will be T0 − Γd Δz (see Fig. 3.2). The ambient temperature will have a value T0 − ΓΔz. The parcel pressure will be in equilibrium with the external pressure so that its density ρ p will be different from the environment ρ = p/RT . The net force on the parcel will be the Archimedes force decreased by its weight, that is, (Fig. 3.2) F = V (ρ − ρ p )g With V volume of the parcel and the acceleration ρ − ρp F = g m ρp

(3.13)

Densities go as the inverse of the temperatures so that because T p = T0 − Γd Δz and T = T0 − ΓΔz, (3.13) becomes Tp − T Γ − Γd d2 Δz = g= gΔz T dt 2 T

T0-ΓdΔz z0+Δz HEIGHT

Fig. 3.2 Stability of a parcel for a vertical displacement Δz. Γd indicates the profile for a dry adiabat while Γ is the environment gradient

(3.14)

T0-ΓΔz z0

T0 TEMPERATURE

3.3 Static Stability of a Moist Atmosphere

The quantity N2 = −

43

Γ − Γd g T

(3.15)

has dimension of frequency s −2 and it is called Brunt–Väisälä frequency so that (3.14) can be written as d2 Δz + N 2 Δz = 0 (3.16) dt 2 This is the equation for the harmonic oscillator so that if N 2 > 0 (that is Γd > Γ) the parcel will oscillate around an equilibrium position and we have a stable situation. On the other hand if Γd < Γ, N 2 < 0 and we have an imaginary frequency and the parcel will move indefinitely from the equilibrium position. For a typical gradient of 6.5 ◦ C km−1 the oscillation period is about 2π/N ≈ 10 min. The relation of N with the potential temperature discussed in Chap. 2 can be found easily if in (3.15) we substitute to Γd = g/c p and Γ = −dT /dz we have g N = T 2

(

dT g + dz cp

) =

g ∂θ . θ ∂z

(3.17)

Of course (3.17) relates the stability to N 2 and the potential temperature gradient. For a typical gradient dT /dz we have dθ/dz ≈ −2.5 ◦ C km−1 .

3.3 Static Stability of a Moist Atmosphere The presence of water vapor changes our previous conclusions about static stability of the atmosphere. Now to the temperature changes consequence of the vertical motions we should associate the processes of condensation or evaporation. The process will no longer be strictly adiabatic. If the water vapor condenses it will release the latent heat of condensation and warm the parcel. On the other hand if the water evaporates it will cool the parcel. If dq is the amount of heat released or absorbed we will write according to the first principle −Ldqs = c p dT − αdp = c p dT + gdz

(3.18)

where L is the latent heat of condensation. Notice that if (dq < 0) that is vapor condenses the parcel will be heated while the evaporation corresponds to dq > 0 and the parcel will cool. If we derive (3.18) with respect to z we have −L

dT L dqs dT dT dqs dT = cp +g⇒− = + Γd dz c p dT dz dz dT dz

(3.19)

44

3 Thermodynamics of the Atmosphere and the Ocean

From which we obtain the moist adiabatic lapse rate Γs Γs = −

Γd dT = dz 1 + (L/c p )/(dqs /dT )

(3.20)

It is interesting to note that Γs < Γd because the quantity dqs /dT > 0 and from the Clausius–Clapeyron equation we have Lqs dq = Rv T 2 dT And write (3.19) in slightly different way −

1 L dqs = T cp T

) ( g dT + dz cp

And from the potential temperature definition −

L dqs dθ = cp T θ

That can be integrated in the approximation that dqs /T is temperature independent while the integration is carried out up to the point qs = 0 and the temperature θe . We have (q ) s (3.21) θe = θ exp cT where θe is indicated as equivalent potential temperature that can be interpreted as follows. We consider a moist air parcel which expands while ascending. The water vapor in the parcel will condense until the parcel will be completely dry. During this process the temperature of the parcel will change according to a moist adiabat while at the end of condensation it will behave as a dry parcel. If from this point we bring the air mass back to the reference pressure its temperature will be equivalent potential temperature θe . These considerations lead to reformulate the stability criteria for a moist atmosphere with the help of Fig. 3.3. We consider the air parcel initially at point O that moves toward lower pressure. Until the condensation starts the particle will follow the dry adiabat (indicated with Γd ). From the condensation point (A) the temperature change will follow the moist adiabat (indicated with Γs ) until the temperature of the parcel is above the ambient temperature (point B). From this point on the parcel will move freely. This corresponds to n instability known as conditionally instability because it represents a situation of stability with respect to the dry atmosphere that may become unstable if the air masses are forced beyond the free convection level.

3.4 Stability of Ocean Stratification Fig. 3.3 The conditional instability in a moist atmosphere. The parcel follows initially the dry adiabat until the condensation starts (point A). From this point the temperature change following the moist adiabat up to the point of free convection (point B)

45

Free convection level

PRESSURE

B A Forced convection level

O

Γs

Γd Γ

TEMPERATURE

3.4 Stability of Ocean Stratification To illustrate the static stability of the ocean we will follow the approach we already used for the atmosphere. We start from Eq. (2.34) that gives the change of background density ρ¯ as a function of depth 1 ∂ρ¯ ∂ T¯ ∂ S¯ = −γT ρ¯ g − αT +β ρ ∂z ∂z ∂z

(3.22)

We made some modification to (2.34), namely, ∂ p/∂ z was substituted with −ρ¯ g and we add the index T to the thermal expansion coefficient α and to the compressibility coefficient γ to distinguish the former from the specific volume. At this point to derive the static stability conditions we consider a parcel of water at depth z with density ρ0 (z) and salinity S0 and temperature T0 . This parcel is raised to the depth z + δz and let the parcel to expand at the lower pressure. The density will be ( )( ) ∂ρ ∂ρ0 ∂p δz = ρ0 (z) + δz ∂p ∂z ∂z ( )( ) 1 ∂ρ ∂p δz = ρ0 (z) − γa ρ02 δz = ρ0 (z) + ρ0 ∂ p ∂z

ρ0 (z + δz) = ρ0 (z) +

(3.23)

where γa = (1/ρ0 )(∂ρ/∂ p) is the adiabatic coefficient because we assume the parcel displacement is adiabatic. We can calculate now the acceleration of the parcel as gΔρ/ρo using (3.23)

46

3 Thermodynamics of the Atmosphere and the Ocean

g

Δρ g g = [ρ¯ (z + δz) − ρ0 (z + δz)] = ρ0 ρ0 ρ0 ( ) 1 ∂ρ¯ =g + γa ρ0 g δz = −N 2 δz ρ0 ∂z

) ( ∂ρ¯ δz − ρ0 (z) + γa ρ02 δz ρ¯ (z) + ∂z

(3.24) That is the acceleration is proportional to the displacement and we have an equation similar to (3.14) where the buoyancy frequency N 2 is now ( N 2 = −g

) { ] ∂ T¯ ∂ S¯ 1 ∂ρ¯ + γa ρ0 g = g (γt − γa )ρ0 g + αT −β ∂z ∂z ρ0 ∂z

(3.25)

where we have substituted (3.22) for the derivative of density. Notice now that c12 =

1 ρ0 (γT − γa )

(3.26)

Is the speed of sound in a liquid so that the buoyancy frequency (

) ∂ T¯ ∂ S¯ g N = −g αT −β + 2 ∂z ∂z c1 2

(3.27)

When we plug in the data we obtain oscillation periods ranging from a few minutes up to a few hours in the deep ocean. It is of interest the case when the salinity is constant with depth ∂ S/∂z = 0. Based on (3.24) the acceleration may vanish for N 2 = 0 that is αT

∂T g + 2 = 0, ∂z c1

⇒

∂T g =− ∂z αT c12

(3.28)

This implies that the temperature of the displayed particle (T0 (z + δz) must be equal to temperature of the surrounding T (z + δz) because for N = 0 also the densities and pressures are equal. Then the adiabatic temperature change for the particle will be ( ¯) ∂T g δz = − 2 δz δT0 = ∂z αt c1 which implies ( δT0 = −δ The quantity

gz αt c12

)

(

= 0,

gz ⇒ δ T0 + αT c12

) ( gz θ = T0 + αT c12

) =0

(3.29)

(3.30)

3.4 Stability of Ocean Stratification

47

is conserved in a stratified fluid and has the same role of the potential temperature. ◦ Typical values for αT ≈ 10−4 , c1 ≈ 1500 ms−1 give dT /dz ≈ 4.3 × 10−2 C m−1 which is a reasonable value in the thermocline. However stability in the real ocean is quite different and we show in the appendix that the adiabatic gradient Γd is given by Γd =

gαT T cp

(3.31)

With c p specific heat at constant pressure and we see that now the gradient depends on the temperature. The condition for stability reads now ) ( dT ∂s >0 −β α T Γd + dz ∂z

(3.32)

Again we can think of a particle displaced of a depth δz that experience an acceleration a. We define stability E the quantity 1 E =− δz

( ) a g

(3.33)

If we move the particle downward δz < 0 and the acceleration is upward the situation is stable and E > 0. We show in the appendix that gE = N2

(3.34)

And so we have the classical situation for which E > 0, E = 0, E < 0,

stable neutral unstable

(3.35)

The standard values for the stability parameter in the ocean are 1 − 10−6 m−1 ,

depths < 1000 m

10−6 m−1 ,

depths > 1000 m

10

−8

−1

m ,

deep tr enches

(3.36)

48

3 Thermodynamics of the Atmosphere and the Ocean

3.4.1 Potential Temperature and Density Following (3.30) we can now redefine the potential temperature {

z

θ=T− 0

T αT gdz , cp

(3.37)

It is evident that for a layer of thickness δz the potential temperature can be obtained by subtracting a temperature δT {

z

δT = 0

αT T αT gdz , ≈ gδz cp cp

(3.38)

This relation can be applied to obtain the temperature change for a water that moves, for example, from the surface to some depth. The adiabatic gradient can be written as a function of pressure ( δT =

∂T ∂p

) δp =

Γ T αT ∂ T ∂z δp = δp = δp ρg ρc p ∂z ∂ p

(3.39)

Using T = 288 K, c p = 4200 Jkg−1 K−1 and αT = 2.2 × 10−4 K−1 for δp = 500 bar (that is around 5000 m) we obtain a temperature difference of 0.74 K. The temperature change is much smaller than the actual difference which is of the order of 18 K. ρθ = ρ( pr , θ, S)

(3.40)

The reason of such large difference has to be found that in the ocean the large vertical movements are inhibited while the quasi-horizontal motions develop along isopycnical lines as we will see later. The potential density is defined as the density of seawater parcel would attain if it were lifted adiabatically from a pressure level p to a reference pressure pr at the surface z = 0. So the potential density id defined as with the potential temperature evaluated following (3.37). Some of these properties can be visualized in a diagram as shown in Fig. 3.4 where the observations are summarized in T-S diagram. In particular, this refers to data from 150 to 5000 m depth in the Atlantic Ocean at 9S latitude. At the data are superimposed the curves of density anomalies σt . The numbers on the data are the depth in hectometers.

Appendix t= 16 t

1.5

25

*

.5 = 25

-

TEMPERATURE (°C)

Fig. 3.4 T-S diagram for a water sample of the Atlantic Ocean described in the text. SAAI: Sub-Antarctic Intermediate water; NAD: North Atlantic Deep water; AAB: Antarctic Bottom water

49

t

12

2

6 = 2

*

2.5

.5 = 26 t

*

3

*

8

4

*

t

4

5

7 = 2

*

6

*

7* 8* 10 9* *

SAAI

.5

=

27

t

AAB

0 34.0

NAD * 16 ** 20 * **30 NAB **35 *40 28 = t * *45 14

*

12

50

35.0

36.0

SALINITY (psu)

Appendix Clausius–Clapeyron Equation In Fig. 3.5, the isothermal for water vapor is shown when condensation takes place. The gas is compressed up to the volume VD where condensation starts. This point on the temperature remains constant until all the vapor has condensed (point V A ). Fluid can be considered incompressible. Consider now two isotherms at temperature T and T − δT and connect them with two adiabats A − B and C − D so that we have an elementary Carnot’s cycle. The efficiency of this cycle is dT /T while the work done corresponds to the area of the cycle des (VL − VG ) where VG indicates the volume of the gas and VL the same for the liquid. We have then des (Vg − VL ) = L

dT ≈ des VG T

Because VG >> VL . Now VG = Rv T /es so that Les des = Rv T 2 dT The once integrated gives Eq. (3.2). The same equation can now be written in the form { ( )] ( ) ( ) L 1 es 1 L L = exp − = exp exp − Rv T0 T e0 Rv T0 Rv T

3 Thermodynamics of the Atmosphere and the Ocean

PRESSIONE

50

T+ΔT T

VA

Δes

B; Α:

ΔT

VD

VL

C D

VG

VOLUME

Fig. 3.5 On the left a couple of isotherms are represented in presence of the two phases (vapor and liquid). On the right is represented the elementary Carnot cycle

where we can also write L L L = = Rv (T0 + C) Rv T0 Rv T

(

1 1 + C/T0

)

L = Rv T0

(

C 1− T0

)

where C is the temperature in degrees centigrade and we use the approximation for C/T0 0 dz dz

(3.58)

which is our (3.32).

Static Stability of the Ocean The previous conclusions can be used to relate the static stability to buoyancy frequency N . Consider a water parcel at pressure p and depth −z with salinity S that we move vertically without exchanging salt or heat with the environment up to the level −z + δz and pressure p − δp. At this level the properties of the parcel are ρ , , s, T + δT . The change in temperature δT corresponds to the change in pressure δT = (dT /dp)ad δp. Substituting for the hydrostatic equilibrium we have

Appendix

55

δT = −(dT /dp)ad ρgδz = −Γd δz where Γd is the adiabatic lapse rate. The force acting on the parcel at level −z + δz is F = δV g(ρ2 − ρ , )

(3.59)

Dove δV is the volume of the parcel and ρ2 is the density of the environment. The acceleration is then F δV g(ρ2 − ρ , ) a= = (3.60) m δVρ , The densities appearing in the (3.60) can be found in such way that [ ) ( ) ][ ) ]−1 ( ( ∂ρ g ∂ρ 1 ∂ρ δz − ρ − δz δz a= ρ+ 1+ ∂z ρ ∂z ρ ∂z e p p

(3.61)

where e indicates the environment and p the parcel. The environment change can be written as ] { { ] ∂ρ ∂ S ∂ρ ∂ T ∂ρ ∂ p ∂ρ δz = + + δz ∂ S ∂z ∂ T ∂z ∂p ∂z e ∂z e While for the particle {

∂ρ δz ∂z

] e

{ ] ∂ρ ∂ρ ∂ p = − Γd + δz ∂T ∂p ∂z p

At this point we can make a few simplifications. In (3.61) for δz → 0 the denominator becomes 1. Also (∂ p/∂ z)w = (∂ p/∂ z)e and also (∂ρ/∂ p)w = (∂ρ/∂ p)e so that the (3.61) becomes { ( )] 1 ∂ρ ∂ S ∂ρ ∂ T a = + + Γd δz (3.62) ρ ∂ S ∂z ∂ T ∂z g It is clear that if δz is negative, that is, the particles moves downward and the acceleration a is positive we have a stable situation and this means that the term in square parenthesis is negative. As in the case of the atmosphere in this case the parcel will oscillate around its initial position. We can define as stability the ratio 1 E =− δz

( ) a g

(3.63)

So that we will have stability for E > 0 and instability for E < 0. To find the Brunt– Väisällä frequency we observe that in (3.62) the term involving salinity is an order of magnitude less than the other terms and it can be neglected. So that we remain with 1 a = ρ g

{

∂ρ + ∂z

(

∂ρ ∂T

) ad

(

∂T ∂z

) ] ad

1 δz = ρ

{

∂ρ + ∂z

(

∂ρ ∂z

) ] δz ad

56

where

3 Thermodynamics of the Atmosphere and the Ocean

(

∂ρ ∂z

(

) = ad

∂ρ ∂ p ∂p ∂z

) =− ad

g c2

(3.64)

where we have used the hydrostatic equilibrium and c is the velocity of sound (a little bit different from (3.26)). The buoyancy frequency is then {

g 1 ∂ρ − 2 N = gE = − c ρ ∂z 2

] (3.65)

Data show that the oscillation periods change from a few minutes to a few hours in the deep ocean.

Bibliography Textbooks Bohren CFBA (1998) Albrecht, atmospheric thermodynamics. Oxford University Press Gill AE (1982) Atmosphere-ocean dynamics. Academic, San Diego Jacob DJ (1999) Introduction to atmospheric chemistry. Princeton University Press Marshall J, Plumb RA (2007) Atmosphere, ocean and climate dynamics: an introductory text. Academic Olbers D, Willebrand J, Eden C (2012) Ocean dynamics. Springer Pond S, Pickard GL (1983) Introductory dynamical oceanography. Elsevier

Chapter 4

Chemical Kinetics

In the next few chapters we will use heavily some chemistry. We will start by reviewing some of the most important features of the atmospheric chemistry including what happens in the stratosphere, troposphere and the precipitation. The chemistry of the ocean will be another important topic together with the geochemical cycles which are strictly intertwined with the atmosphere and the ocean. Also the study of the particulate matter in the atmosphere will use relevant chemical concepts. Each one of these topics is reflected in an immense literature so we do not pretend to invent anything but simply give the basis to understand most of them. It is then important to give an introduction to the particular aspect of the chemical kinetics which is one of the taboos of the average science student.

4.1 Gas Phase Reactions We start by the bimolecular reaction when two compounds A and B collide to yield two other products C and D. We write the reaction A+B →C+D

(4.1)

A and B will disappear and its rate will be −

d d d d [ A] = − [B] = [C] = [D] = k[A][B] dt dt dt dt

(4.2)

where k is the rate constant for the reaction. We can make an example to introduce some caveat to the (4.2) rule. The simplest way to write the photosynthesis is the reaction (4.3) 6CO2 (g) + 6H2 O(l) → C6 H12 O6 (s) + 6O2 (g)

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 G. Visconti, The Fluid Environment of the Earth, https://doi.org/10.1007/978-3-031-31539-8_4

57

58

4 Chemical Kinetics

where 6 molecules of carbon dioxide in the gas phase react with 6 molecules of water (in the liquid phase to give one molecule of organic material in the solid phase plus 6 molecules of oxygen. We notice that the number of atoms present in the reactants is conserved in the products to the right. Also (4.2) must change in following way −

1 d d 1 d 1 d [CO2 ] = − [H2 O] = [C6 H12 O6 (s) + 6O2 ] = [O2 ] 6 dt dt 6 dt 6 dt

(4.4)

In general if we have a reaction like a A + bB + · · · → gG + h H

(4.5)

The rate equation should be written as −

1 d 1 d 1 d 1 d [ A] = − [B] + · · · = [G] = [H ] a dt b dt g dt h dt

(4.6)

Beside the bimolecular reaction (4.1) we may have the three body reaction A+B+M →C+M

(4.7)

Example of such reaction is the formation of ozone in the stratosphere O + O2 + M → O3 + M In this case M is the nitrogen molecule which absorbs the excess energy of the reaction. The rate of formation for ozone is then d [O3 ] = k[O][O2 ][M] dt The concentration of [M] and [O2 ] are constant so that the rate coefficient is a complicated function of atmospheric pressure. Situation We consider now a possible situation of chemical equilibrium when we have a reversible process like A+B →C+D

(4.8)

C+D → A+B

(4.9)

And its inverse

with rates k f and kr . Equilibrium is reached when the forward rate equals the reverse k f [A][B] = kr [C][D] From which we can define an equilibrium constant

(4.10)

4.2 Photochemistry

59

Ke =

kf [C][D] = [ A][B] kr

(4.11)

This is the case of the formation of nitrogen pentoxide N2 O5 from nitrogen nitrate NO3 and and nitrogen dioxide NO2 NO2 + O3 → NO3 + O2

(k1 )

NO3 + NO2 + M → N2 O5 + M

(k2 )

(4.12)

The condition for equilibrium is that every single step being in equilibrium that is k1 f [NO2 ][O3 ] = k1r [NO3 ][O2 ] and k2 f [NO3 ][NO2 ][O3 ] = k2r [N2 O5 ] Solving for [NO3 ] from the first and substituting in the second we have Ke =

k1 f k2 f [N2 O5 ][O2 ] = [NO2 ]2 [O3 ] k1r k2r

(4.13)

The rate constant that we have indicated with kr may depends on temperature T and pressure. The most common temperature dependence takes the form of the Arrhenius relation (4.14) kr = A exp(−E/kT ) where A is related to the collision rate of the molecules, E is the activation energy and k in the Boltzmann constant. For those interested we further discuss this point in the appendix. The dimension of k depend on the number of reactants. For two reactants and the concentration measured in molecules cm−3 the rate for a reaction like (4.1) is cm3 s−1 . Because the Boltzmann constant ha the dimension of energy/temperature the ratio E/k has dimension of temperature that characterize the speed of the reaction. For example the reaction CH4 + OH → CH3 + H2 O

(4.15)

which as we will see initiates the methane (CH4 ) oxidation has a ratio E/k ≈ 1815 K and the rate coefficient decrease with altitude in the troposphere.

4.2 Photochemistry There are chemical compounds that under the action of sunlight may experience dissociation. Formally this reaction is written as

60

4 Chemical Kinetics

X + hν → Y + Z

(4.16)

where hν indicate a single photon whose energy is the product of the Planck constant h and the frequency of radiation ν. The rates of this reaction can be written as −

d d d [X ] = [Y ] = − [Z ] = J [X ] dt dt dt

(4.17)

where J is known as a photodissociation coefficient. In order to calculate J we consider a slab of air of vertical of thickness dz and unit horizontal air. The probability that a photon incident on [X] is absorbed by the molecule is given by σ/A where A is the cross-sectional area of the molecule and σ its absorption cross section measured in cm2 /molecule. The fraction on the incoming photons absorbed in the elemental slab is given by σ [X ]dz. The flux of photons (photons cm−2 s−1 ) is indicated with I and the number of molecules photolyzed in the slab will be then qσ [X ]I dz. q indicates the so-called quantum yield that is the probability that the absorbed photon will cause photolysis. The photolysis rate is obtained by dividing this number by the total molecules [X ]dz, that is qσ I dz. Now it is obvious that q, I and σ are function of the wavelength λ (that is frequency) so we could write for J { J=

λ

q(λ)σ (λ)I (λ)dλ

(4.18)

And it measured in s−1 . The flux I (λ) also known as actinic flux is a function of the zenith angle and the absorbing gases present in the atmosphere. We will discuss in detail in these aspects. For the moment it enough to say that if we have a beam of radiation incident upon a layer of thickness dz at the altitude z and zenith angle θ (Fig. 4.1) the intensity will be attenuated as dI = −I dτ

(4.19)

where τ is the optical thickness along the beam that is

Fig. 4.1 The attenuation of the intensity of solar radiation, I in an absorbing layer of thickness dz at some altitude z

I θ dz / cos θ dz z I-ΔI

4.3 Aqueous Solutions

61

τ (λ, z) =

1 Σ σi (λ)Ni (z) cosθ i

(4.20)

where Ni (z) is the columnar density of that particular constituent that has the absorption cross section σi { ∞

Ni (z) =

n i (z)dz

(4.21)

z

An example of this calculation is given in the appendix. At this point it is clear that a specie subject to photodissociation will disappear according to d [X ] = −J [X ] dt

(4.22)

So that its concentration will change in time as [X (t)] = X 0 exp(−J t)

(4.23)

Usually the integral (4.18) is reduced to a summation over the relevant spectral interval and usually the actinic flux is given in watt m−2 so that it must be converted in photons m−2 s−1 . For each spectral interval at wavelength lambda and power Δw the number of photons is given by Δwλ/ hc with c velocity of light. The solution (4.23) is a good occasion to introduce the half-life of a reactant. In practice the half-life t1/2 is the time it takes to half its concentration. So we have ln

X 1 0.693 = −J t = ln → t1/2 = X0 2 J

(4.24)

So any process of the kind A → pr oducts to which corresponds a rate equation like d [ A] = −k[ A] dt will have a half life t1/2 = 0.693/k.

4.3 Aqueous Solutions As we know water plays a very important role in the Earth’s environment. One of the reasons for such importance is the unique property of water to dissolve many substances. Solids may dissolve in water maintaining intact the molecule (like sugars) or breaking up into ions. The most common example is the common salt (Na Cl) that dissolve according to the reaction

62

4 Chemical Kinetics

Na Cl(s) ↔ Na+ (aq) + Cl− (aq)

(4.25)

where the (+) and (−) signs mean that the ions carries a unit positive and negative charges. Aqueous solutions containing ions are electrical conducting and are called electrolytes. In the solutions ions are surrounded by water molecules with the positive side of the water dipole (hydrogen atoms) in front of the negative ions and the negative side (oxygen atoms) toward the negative ions. Electrolytes are distinguished in strong when all the molecules are dissolved and weak when only part of the molecules becomes ions. An example of strong electrolyte is the hydrochloric acid (HCl) which dissolves as (4.26) HCl(g) → H+ (aq) + Cl− (aq) In this case the arrow is only in one direction and that means that the revere reaction is quite unlikely. An example of weak electrolyte is water itself that can dissociates as (4.27) H2 O(l) ↔ H+ (aq) + OH− (aq) At the equilibrium the rate of dissolution kd [H2 O] must be equal to the rate of recombination kr [H+ ][OH− ] so that kd [H2 O(l)] = kr [H+ (aq)][OH− (aq)] [H+ (aq)][OH− (aq)] kd =K = [H2 O(l)] kr

(4.28)

Now the concentration of H2 O molecules in water in mole/liter is given by 1000 g/18 = 55.55 and the number of ions is too small that the concentration of water may be considered constant. We may well consider a new constant given by K w = K [H2 O(l)] = [H+ (aq)][OH− (aq)] = 55.55 K

(4.29)

where we have expressed the concentration in mole/liter. At 25 ◦ C K w = 10−14 . At the same temperature the concentrations of the ions are easily calculated because at the equilibrium [H+ (aq)] = [OH− (aq)] so that (4.29) becomes K w = [OH− (aq)]2 and [OH− (aq)] = 10−7 M where M indicates the concentration in mole/liter. Some of these concepts will be applied when studying the solution of carbon dioxide CO2 in the ocean water. In that case the initial reaction will be between water and carbon dioxide forming carbonic acid (H2 CO3 ) CO2 + H2 O ↔ H2 CO3

(4.30)

The weak carbonic acid will form bicarbonate (HCO3 ) and carbonate (CO2− 3 ) with the sequence

Appendix

63

H2 CO3 ↔ H+ + HCO− 3 2− + HCO− 3 ↔ CO3 + H

(4.31)

2− Now we have at the least five unknowns (H2 CO3 , H+ , OH− , HCO− 3 , CO3 9) and an insufficient number of equations. We need to involve the dissolution of calcium carbonate (Ca CO3 ) to solve the problem. It is something we will do dealing with the geochemical cycle and the acidity of the ocean.

Appendix Collision Theory of Gaseous Reactions The simplest (bur wrong) approach to calculate the reaction rate between two gases with density n A and n B is to assume one of the gas (B) stationary while the molecules are assimilated to “balls” with radius a. In this case the A molecules will sweep a volume πa 2 v¯ with v¯ the average velocity. This volume will contain a number of B molecules π a 2 v¯ n B that also represents the number of collisions per unit time. In its nice little book Peter Hobbs ask how long will take the gas to disappear if each collision corresponds to a reaction. We assume to have a gas at a temperature of 273 K and 1 atm pressure. Using the relation P = nkT with k Boltzmann constant and n number of molecules per unit volume we obtain n=

p 1.013 × 105 = = 2.68 × 1025 mol m−3 ˙ kT 1.38 × 10−27 273

(4.32)

And then the number of collisions will be π a 2 v¯ n B n a = π(3 × 10−10 )2 (5 × 102 )(2.68 × 1025 ) = 1035 m−3 s−1

(4.33)

where we have taken a = 3 × 10−10 m and v = 500 ms−1 . The time required to consume both gases will be 2.68 × 1025 ≈ 3 × 10−10 1035

(4.34)

which is very short time indeed. It is then necessary to consider other factors beside the collisions. One possibility is the fact that not all collisions result in a reaction because the kinetic energy E of the molecules in a gas obeys to the Boltzmann distribution that is / ( )3/2 E 1 f (E) = 2 exp{−E/kT } (4.35) π kT

64 2.0

Fraction j / mole x 10-4

Fig. 4.2 The distribution of molecules as a function of kinetic energy in two different gases with temperature of 25 and 250 ◦ C

4 Chemical Kinetics

25°C

1.6

1.2

0.8 250°C 0.4

0 0

0.4

0.8 1. 2 1.6 Kinetic Energy (j /mole ) x 104

2.0

In this case f (E) represents the number of molecules normalized to unity with energy between E and E + d E. The energy is referred as joule/mole in order to deal with reasonable numbers. This is obtained simply by substituting k, the Boltzmann constant, with R the gas constant. The distribution is very sensitive to the temperature as it is shown in Fig. 4.2 where the distribution for T = 298 K and T = 523 K are illustrated. We see that the total number of molecules having energy greater than a fixed value E a increase considerably with higher temperatures. We may then expect that the reaction rate increase with increasing temperature as it was found experimentally by Svante Arrhenius. He noticed that for many reactions a plot of ln K (K is the reaction rate) versus 1/T would result in straight line that is ln K = ln A −

Ea RT

So that the rate could be expressed as K = A exp{−E a /RT }

(4.36)

The quantity A known also as pre-exponential factor is the reaction rate when T → ∞ while E a is also known as activation energy. The interpretation of the activation energy is explained qualitatively by a diagram like that shown in Fig. 4.3 where the internal energy of two possible reactants ( AB + C) is plotted as a function of the reaction progress. When energy E a is added to the system it may be enough to overcome the energy barrier for the reaction and a transition state will be formed. When the reaction comes to completion, forming A + BC, some energy E p may be released. If E p > E a the reaction will be exothermic. The activation energy can be interpreted as the minimum kinetic energy that reactants must have in order to form products. This means that only those collisions energetic enough may result in the final product. On the other hand the exponential factor represents the fraction of

Appendix

65

ENERGY

Ea

[AB]+[C] Ep [A]+[BC]

PROGRESS OF REACTION

Fig. 4.3 Qualitative energy profile for the reaction [AB] + [C] → [A] + [BC]. E a is the activation energy and E p the energy released at the completion Table 4.1 Photodissociation data for Hydrogen Peroxide Wavelength (1) Flux (2) Flux (3) 295–305 305–315 315–325 325–335 335–345 345–355

0.018 0.027 0.64 1.1 1.1 1.2

0.026 0.42 1.04 1.77 1.89 2.09

Cross section (4) 0.71 0.42 0.25 0.14 0.08 0.05

(1) nanometer, (2) mw cm−2 , (3) 1015 photons cm−2 s−1 , (4) 10−20 cm2

collisions having energy larger than E a . On the other hand the pre-exponential factor A is a measure of the number of collisions that occur irrespective of their energy.

Photodissociation Coefficient We can make an example for the calculation of the photodissociation coefficient for hydrogen peroxide (H2 O2 ). Table 4.1 reports the values for the relevant quantities. At this point we just make the product of the third and fourth column according to (4.18) and sum over the wavelength assuming a quantum yield 1. The resulting photodissociation coefficient is J = 9.588 × 10−6 s−1 . Beside photodissociation H2 O2 → 2OH hydrogen peroxide may react with the hydroxyl radical OH to give H2 O2 + OH → HO2 + H2 O

(4.37)

66

4 Chemical Kinetics

producing another hyperoxyradical HO2 with a reaction rate of K = 1.7 × 10−12 mol cm−3 s−1 . The concentrations of H2 O2 is 5 ppbv while the number density of OH is 2 × 106 cm−3 . We can evaluate the lifetime of hydrogen peroxide for the two processes. The photodissociation is simply 1/J and that means 1/9.588 × 10−6 ≈ 1. × 105 s that is 1.2 days. For the other process the loss of H2 O2 is K [OH][H2 O2 ] so that we have d 1 [H2 O2 ] = K [OH] (4.38) [H2 O2 ] dt And the time constant is simply 1/K [OH] ≈ 2.9 × 105 s that is 3.5 days. And then the loss of H2 O2 by the second process is much slower.

The Wrong Theory of the Ozone Layer (Chapman) In the 1930’s Sidney Chapman one of the fathers of modern geophysics knowing that ozone absorbed the UV radiation and so was an obstacle to astronomy observations advocated the possibility to make “holes” in the ozone layer. This is one of the proof that science can contradict itself. Anyway Chapman in the same years formulated a theory on the ozone layer that could not be verified because there were not enough data and when they became available (after the II World War) proved the theory to be wrong. We will treat this problem in some detail but it is nice to illustrate now the Chapman model. The ozone formation starts with the dissociation of molecular oxygen in the UV J2 (4.39) O2 + hν → O + O, and the recombination of atomic oxygen and the molecular oxygen produce ozone O2 + O + M → O3 + M,

k2

(4.40)

where [M] is the third component absorbing the excess energy (it could be nitrogen). Then ozone is destroyed by two processes, the photodissociation O3 + hν → O2 + O,

J3

(4.41)

And the recombination with atomic oxygen O3 + O → 2O2 ,

k3

(4.42)

where k, s are the reaction rates. Lets call X and Y the concentrations of O and O3 , respectively, and the rate equations become

Appendix

67

dX = 2J2 [O2 ] + J3 Y − (R2 + k3 Y )X dt dY = R2 X − ( J3 + k3 X )Y dt

(4.43)

where with R2 we have included [M][O2 ] which is constant. Summation gives d (X + Y ) = 2J2 [O2 ] − 2k3 X Y dt

(4.44)

At the steady state d/dt = 0 and X = J2 [O2 ]/k3 Y

(4.45)

We could then substitute X into the first of (4.4) and solve for Y . At this point we can make some simplification and introduce a new reactant that we call odd oxygen given by [X ] + [Y ] and from (4.44) its lifetime is simply τ Ox =

1 [X ] + [Y ] ≈ 2k3 [X ][Y ] 2k3 [X ]

(4.46)

because below 30 km [O] 0 and so cold air is moved from cold region (T ) to warmer region (T + ΔT ) and the local temperature change will be a cooling. This is also called negative advection. On the contrary if u is directed along the negative x-axis the product u∂ T /∂ x < 0 and the local temperature change will be a heating and in this case we talk about positive advection. We will have several occasion to use the Lagrangian derivative.

T

T+ΔT

u

T+2 ΔT

T

u 6T < 0 6x

T+ΔT

u

T+2 ΔT

u 6T > 0 6x

y

6T/ 6x

Fig. 5.1 Temperature advection when the isotherms are parallel to the y-axis and the velocity field is along x

5.1 Some Tool of the Trade

71

5.1.2 Continuity Equation We now claim the conservation of mass and refer for this to Fig. 5.2 where we consider an elementary volume of dimension δx, δy, δz. The box is immersed in fluid that moves and we start to consider the mass budget in the x-direction with an entering flux Fx (x, y, z) = ρu on the left face δyδz. The flux is measured in kg m−2 s−1 so that the mass entering the left face is ρuδyδz in kg s−1 . On the opposite face we have an exiting flux Fx (x + δx, y, z) = F(x, y, z) + ∂ F/∂ yxδx and the net flux will be [Fx (x, y, z) − Fx (x + δx, y, z)]δyδz = −

∂ (ρu)δxδyδz ∂x

(5.4)

The net mass contribution in kg s−1 to the elementary volume δxδyδz would be ∂ρ ∂ ∂ ∂ δxδyδz = − (ρu)δxδyδz − (ρv)δxδyδz − (ρw)δxδyδz ∂t ∂x ∂x ∂x And the result is the continuity equation ∂ρ = −∇ · (ρV) ∂t

(5.5)

where ∇· is the divergence of the flux vector ρV. Now we know the vector rule such that ∇ · (ρV) = ρ∇ · V + V · ∇ρ so that (5.5) becomes ∂ρ + V · ∇ρ = −ρ∇ · V ∂t The left-hand member is just the total derivative so that our conservation law reads

Fig. 5.2 Elementary volume used to obtain the continuity equation

Fz(x,y,z+δz) Fy(x,y+δy,z)

δz Fx(x+δx,y,z)

Fx(x,y,z)

z y

δy

Fy(x,y,z,) δx x

Fz(x,y,z,)

72

5 Fluid Dynamics

dρ + ρ∇ · V = 0 dt

(5.6)

The first important consequence of (5.6) is that for a fluid with ρ = const the divergence of velocity is zero. Equation (5.6) is known as the continuity equation and has been obtained in the hypothesis that there are not other contributions to the mass changes beside the fluid flux. If within the volume there is a production P and loss L, the equation is slightly changed dρ + ρ∇ · V = P − L dt

(5.7)

It is interesting to give an interpretation to the form (5.5) and (5.6). The former says that the local change of density ∂ρ/∂t is proportional to the so-called threedimensional mass divergence while (5.6) requires that the individual rate of change of density 1 dρ ∂u ∂v ∂w − = + + ρ dt ∂x ∂y ∂z be proportional to the three-dimensional velocity divergence. This implies that negative divergence will fill the volume with fluid while positive divergence will remove fluid from the volume.

5.1.3 The Diffusion Equation One purpose of this book is to illustrate how to model the transport of chemical in the atmosphere. To this end we need to modify a little continuity equation that we write in the most general form of (5.5) in that case the flux F is simply ρV . When both chemical production and loss are present we write ∂n +∇ · F = P −L ∂t

(5.8)

where n is the concentration of our chemical reactant. The form of F depends really on the process which determines the transport. In the case of simple advection then we have F = nV . The process of diffusion takes place when we have concentration gradient that could refer to any kind of scalar quantity like temperature of heat. In the old days, when smokers were allowed everywhere you could have a smoke gradient in a room with the smoke migrating slowly filling slowly any space. This process is called diffusion and the flux is assumed to be proportional to the gradient of the concentration in the direction of the gradient that is Fx = −n a K

dC dx

5.2 The Equation of Motion in a Rotating Earth

73

where n a is the number density of the background atmosphere, C is the concentration of the chemical, and K is the diffusion coefficient. That can be generalized for the three-dimensional case as (5.9) F = −n a K ∇C So that the continuity equation reads ∂n = ∇ · (n a K ∇C) + P − L ∂t

(5.10)

The same equation could be used for temperature and in its simplest form reads ∂T ∂2T =K 2 ∂t ∂x

(5.11)

which is most simple form of the diffusion equation. However the diffusion coefficient appearing in (5.11) (whose dimensions are m2 s−1 ) is not the same as the molecular diffusion coefficient we know. As a matter of fact we can make a “scale analysis” of Eq. (5.11) and calculate the order of magnitude it would take to warm up a room with molecular diffusion. Indicating with L the typical dimension of the room we have dimensionally ΔT ΔT Δx 2 =K −→ ΔT ≈ (5.12) Δt Δx 2 K If K is the molecular diffusion coefficient its value would be ≈5 × 10−5 m2 s−1 and with L ≈ 5 m it would take ΔT ≈ 5 days to warm up the room. This means that K is not the molecular value but is determined by the turbulent motions which develop in the room due to the convection determined by the heat pipe. If we call ΔT the temperature difference determined by the heat pipe the acceleration of the heated air would be a ≈ gΔT /T ≈ 9.8 × 5/300 ≈ 0.1/m s−2 . And the characteristic time would be / L 2L Δt ≈ ≈ ≈ 10 s v a That could be too optimistic but gives an idea of the efficiency of the process. Actually the turbulent diffusion coefficient could be of the order of 1 m2 s−1 so that using (5.12) the time is however much shorter than the molecular diffusion time. In the appendix we give some application of the diffusion equation.

5.2 The Equation of Motion in a Rotating Earth Before going to examine some basic of the fluid motion we should specify the reference system. It is evident we should consider a spherical Earth where the reference axes are those illustrated in Fig. 5.3. For a generic point P, the x-axis is oriented

74

5 Fluid Dynamics

Fig. 5.3 Geometric elements of the rotating Earth. In a the reference system is shown. In b is the polar view of the rotating Earth and in c the local vertical component of the angular velocity is illustrated

along the parallel at the appropriate latitude φ and the y-axis tangent to the local meridian at longitude λ. The z-axis coincides with the local vertical and is along the radius passing through P. The components of the velocities along the x, y, x-axes are u, v, w, respectively. The problem now is to write the equation of motion in a system which is not “inertial” and rotates with an angular velocity Ω around the polar axis. Simple vector calculus allows us to relate the motion in the inertial system Σ ' to rotating frame Σ and we show in the appendix that the motion can be studied essentially in the x − y-plane where the acceleration d V /dt reduces to dV 1 = f V × k − ∇p + F dt ρ

(5.13)

where f is the Coriolis parameter f = 2Ω sin φ, k is the vertical unit vector, and ∇ p is the pressure gradient. The first term on the right is the Coriolis acceleration and F is any other force (for example, friction). In the following sections, we will explain with very simple arguments some features of Eq. (5.13).

5.3 Geostrophic Motion The title of this section should not impress the reader because we will keep things simple following the approach given by Lindzen. To this end we consider as in Fig. 5.3 a rotating fluid around the z-axis coincides with the direction of the gravity acceleration g. We know that in the vertical direction pressure changes according to the hydrostatic equilibrium

5.3 Geostrophic Motion

75

1 ∂p = −g ρ ∂z That is, the acceleration of gravity is proportional to the vertical component of the pressure gradient. In the radial direction equilibrium is established between the centrifugal force ρ(Ω + ω(r ))2 r and the pressure gradient ρ(Ω + ω(r ))2 r = ρ(Ω2 + 2Ωω + ω2 )r =

∂p ∂r

(5.14)

In Eq. (5.13) Ω is the angular velocity of the Earth and ω(r ) is the angular velocity of the fluid. We now define a pressure p0 such that ρΩ2 r =

∂ p0 ∂r

(5.15)

And the pressure p = p0 + p ' so that ρ(2Ωω + ω2 )r =

∂ p' ∂r

(5.16)

Notice that (5.15) could be interpreted as the “radial” component of the hydrostatic equilibrium because the acceleration of gravity is slightly modified by the centrifugal force. In general however we have ω 0. This means that in this sector the advection of planetary vorticity moves the waves westward while the advection of relative vorticity moves the waves eastward. The relative contribution of the two effects can be evaluated from the ratio |−v·∇ f| U ( f /L) fL ≈ ≈ 2 | − v · ∇ζ | U (U/L ) U

(5.119)

We see that the contribution of the long waves to the planetary vorticity advection increases with the wavelength and so long waves move slower than short waves. It is interesting to discuss stationary waves (c = 0). The number of waves n and wavelength L contained in a latitude circle at latitude φ is given by n=

2π cos φ L

(5.120)

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5 Fluid Dynamics

That substituted in (5.117) with c = 0 gives / n s = a cos φ

β U

(5.121)

At mid-latitude for a 10 ms−1 we have between five and six waves. Then we would have progressive waves c > 0 for n > 6 and retrograde waves for n < 5.

An Application of Potential Vorticity We consider the conservation of potential vorticity ζ+ f = const Δz

(5.122)

And assume that Δz is the distance between two surfaces at potential temperature Δθ. We also have Δθ (ζ + f ) = const (5.123) Δz which is a particular form of Ertel potential vorticity. We now consider the potential temperature as given by θ = θ¯ + θ ' and indicate Δθ/Δz ≈ θz with the subscript indicating the derivative. Then we can write ( ) θ' (ζ + f )θz ≈ θ¯ z (ζ + f ) 1 + z θ¯ z We then observe that θz' =

N 2 θ¯ , g

(5.124)

θz' ρ' ≈− z ρ¯ θ¯ z

And substitution (5.124) gives ( ( ) ) g ρz' 1 ∂ 2 p' ¯ ¯θ z (ζ + f ) 1 − = θ z (ζ + f ) 1 + 2 N 2 ρ¯ N ρ¯ ∂ z 2

(5.125)

We now use the definition of streamfunction ψ = p/ρ f and the fact that at middle latitude ζ