Dynamics of the tropical atmosphere and oceans [First edition] 9780470662564, 0470662565

Table of contents :
Climatology of the tropical atmosphere and oceans --
Hydrological and heat exchange processes --
Fundamental dynamical processes --
Kinematics and tropical waves --
Simple models of the tropics --
Equatorial waves in simple basic states --
Waves in a longitudinally and vertically varying basic state --
Moist processes and large-scale tropical dynamics --
Extratropical influence on the tropics --
Tropical influence on extratropics : a zonally averaged perspective --
A tropical-extratropical synergy --
Dynamics of arid and desert climates --
Near-equatorial precipitation --
Large-scale, low-frequency coupled ocean-atmosphere systems --
Intraseasonal oscillations in the tropical atmosphere --
Dynamics of the large-scale monsoons --
The coupled monsoon system --
The changing tropics --
Some concluding remarks

Citation preview

Dynamics of the Tropical Atmosphere and Oceans

Advancing Weather and Climate Science Series Scientific advances in weather and climate science over the past decade are now making their way into the teaching literature, and climate change has become a major issue that pervades all aspects of weather and climate science. The Royal Meteorological Society (RMetS) is a world-leading professional and learned society in the field of meteorology. Based in the UK, it encourages and facilitates collaboration with organisations that are active in earth science. It serves its professional and amateur members and the wider community by undertaking activities that support the advancement of meteorological science, its applications and understanding. The RMetS Advancing Weather and Climate Science Book Series, in conjunction with Wiley-Blackwell, brings together both the underpinning principles and new developments in meteorology in a unified set of books suitable for undergraduate and postgraduate study, as well as being a useful resource for the professional meteorologist or Earth system scientist. Topics covered include; atmospheric dynamics and physics, earth observation, weather forecasting, climate variability and climate change science, impacts and adaptation. A full list of titles currently available in the series can be seen below. Other titles in the series: Radar Meteorology: A First Course Robert M. Rauber and Stephen L. Nesbitt Published: March 2018 ISBN: 978-1-118-43262-4 Hydrometeorology Christopher G. Collier Published: August 2016 ISBN: 978-1-118-41497-2 Meteorological Measurements and Instrumentation Giles Harrison Published: December 2014 ISBN: 978-1-118-74580-9 Fluid Dynamics of the Mid-Latitude Atmosphere Brian J. Hoskins, Ian N. James Published: October 2014 ISBN: 978-0-470-79519-4 OperationalWeather Forecasting Peter Inness, University of Reading, UK and Steve Dorling, University of East Anglia, UK Published: December 2012 ISBN: 978-0-470-71159-0 Time-Series Analysis in Meteorology and Climatology: An Introduction Claude Duchon, University of Oklahoma, USA and Robert Hale, Colorado State University, USA Published: January 2012 ISBN: 978-0-470-97199-4 The Atmosphere and Ocean: A Physical Introduction, 3rd Edition Neil C. Wells, Southampton University, UK Published: November 2011 ISBN: 978-0-470-69469-5 Thermal Physics of the Atmosphere Maarten H.P. Ambaum, University of Reading, UK Published: April 2010 ISBN: 978-0-470-74515-1 Mesoscale Meteorology in Midlatitudes Paul Markowski and Yvette Richardson, Pennsylvania State University, USA Published: February 2010 ISBN: 978-0-470-74213-6

Dynamics of the Tropical Atmosphere and Oceans

Peter J. Webster Georgia Institute of Technology Atlanta Georgia, US

This edition first published 2020 © 2020 John Wiley & Sons Ltd. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions. The right of Peter J. Webster to be identified as the author of this work has been asserted in accordance with law. Registered Offices John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK Editorial Office 9600 Garsington Road, Oxford, OX4 2DQ, UK For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com. Wiley also publishes its books in a variety of electronic formats and by print-on-demand. Some content that appears in standard print versions of this book may not be available in other formats. Limit of Liability/Disclaimer of Warranty While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. Library of Congress Cataloging-in-Publication Data Names: Webster, Peter J., author. Title: Dynamics of the tropical atmosphere and oceans / Peter J. Webster (Georgia Institute of Technology). Description: First edition. | Hoboken, NJ : Wiley-Blackwell, 2020. | Series: Advancing weather and climate science | Includes bibliographical references and index. Identifiers: LCCN 2019041158 (print) | LCCN 2019041159 (ebook) | ISBN 9780470662564 (hardback) | ISBN 9781118648438 (adobe pdf) | ISBN 9781118648452 (epub) Subjects: LCSH: Tropical meteorology. | Tropics–Climate. | Climatology. Classification: LCC QC993.5 .W43 2020 (print) | LCC QC993.5 (ebook) | DDC 551.6913–dc23 LC record available at https://lccn.loc.gov/2019041158 LC ebook record available at https://lccn.loc.gov/2019041159 Cover Design: Wiley Cover Image: Courtesy of Peter J. Webster Set in 10/12pt TimesNewRomanMTStd by SPi Global, Chennai, India Printed in the UK by Bell & Bain Ltd, Glasgow. 10 9 8 7 6 5 4 3 2 1

For: Benjamin David, Chloe, Caitlyn and Jack Webster and Clara Whelan, and, especially Judith Curry, and my students, postdoctoral fellows, colleagues and members of my research group through the years.

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Contents Preface xvii Acknowledgments xix Abbreviations xxiii 1 Climatology of the Tropical Atmosphere and Upper Ocean 1

1.1 1.2

1.3

1.4

The Growth of Tropical Meteorology 1 Seasonal Characteristics 4 1.2.1 Zonal Variability 5 1.2.1.1 Sea–Surface Temperature 5 1.2.1.2 Temperature and Humidity 5 1.2.1.3 Precipitation 6 1.2.1.4 Wind Fields 7 1.2.2 Spatial Variability in the Tropics 8 1.2.2.1 Surface Temperature 8 1.2.2.2 Precipitation 9 1.2.2.3 Surface Pressure 11 1.2.2.4 Wind Fields 12 1.2.2.5 Moisture Flux 13 1.2.3 Variability Along the Equator 14 1.2.3.1 Temperature and Moisture 14 1.2.3.2 Wind Fields 15 Macro-Scale Circulations 16 1.3.1 Hadley’s Circulation 16 1.3.2 Walker’s Circulation 17 1.3.3 Monsoon Circulations 19 1.3.3.1 Asian–Australasian Monsoons 20 1.3.3.2 Monsoons of the Americas 22 1.3.3.3 African Monsoon 23 1.3.4 Large-Scale Characteristics of Tropical Oceans 23 A Myriad of Variability 24 1.4.1 High-Frequency Variability 24 1.4.1.1 Waves in the Easterlies 25 1.4.1.2 Tropical Cyclones and Monsoon Depressions 26 1.4.1.3 The “Great Cloud Bands” 29 1.4.1.4 Mesoscale Convective Systems 29 1.4.2 Subseasonal Variability 29 1.4.3 Interannual Variability 31 1.4.3.1 El Niño and the Southern Oscillation 31 1.4.3.2 Atlantic Oscillations 32 1.4.3.3 Stratospheric Oscillations 33

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1.4.4 Notes 34

Overlapping of Variance Bands: Waves Within Waves

33

2 Hydrological and Heat Exchange Processes 37

2.1

Water on Earth 38 2.1.1 An Inventory 38 2.1.2 Global Disposition of Rainfall 39 2.2 Thermodynamics of Water and Earth’s Climate 39 2.2.1 Implications of Clausius–Clapeyron 40 2.2.2 Role of Water in the Evolution of Earth’s Climate 41 2.2.3 Estimate of the Planetary Radiative Surface Temperature 42 2.3 Water and the Tropical System 43 2.3.1 Atmosphere 43 2.3.1.1 Clausius–Clapeyron and the Vertical Profiles of Temperature and Humidity 43 2.3.1.2 Distribution of Water Vapor and Liquid/Ice Water 44 2.3.1.3 Moisture, Lapse Rates and Gradients of Atmospheric Buoyancy 45 2.3.2 Ocean 46 2.3.2.1 Ocean Surface Layer: Warm and Fresh 49 2.3.2.2 Abyssal Water: Cold and Saline 49 2.3.2.3 The Thermocline 50 2.4 Buoyancy, Differential Buoyancy, and the Generation of Horizontal Body Forces 50 2.4.1 Concept of Buoyancy 50 2.4.2 Zonal Variability of Buoyancy Induced by Radiative Forcing 51 2.4.3 Poleward Heat Transport 51 2.5 Integrated Column Heating 53 2.5.1 Components of Total Heating 53 2.5.2 Latitudinal Distribution of Latent Heat Flux and Condensational Heating 54 2.5.3 Latitudinal Distributions of Total Columnar Heating 55 2.5.4 Longitudinal Disposition of Total Columnar Heating 56 2.5.5 Annual Cycle of Total Columnar Heating 56 2.6 Buoyancy in the Tropical Ocean 57 2.6.1 Net Heating of the Upper Ocean 58 2.6.2 Fresh Water Flux into the Tropical Ocean 59 2.6.3 Distribution of Ocean F B Buoyancy Flux 59 2.6.4 Observations of Ocean–Atmosphere Fluxes in the Tropics 61 2.6.4.1 Western Pacific Ocean Circulation Experiment (WEPOCS) 61 2.6.4.2 Surface Fluxes in the Bay of Bengal During JASMINE 63 2.7 Translations to the Broader Scale 66 2.7.1 Large-Scale Columnar Heating Gradients 66 2.7.2 Upper-Ocean Heating 68 2.8 Convection–SST Relationships and the Vertical Scale of Tropical Motions 68 2.9 Coupled Global Ocean–Atmosphere Synergies 70 2.9.1 The Notion of Interactive Zones 70 2.9.2 A Stable Global Interactive System 71 2.9.2.1 The Tropical Circulation 71 2.9.2.2 State of the Stratosphere 72 2.9.2.3 The Return Atmospheric Flow Between the Tropics and the Poles 72 2.9.2.4 The Polar Ocean–Atmosphere Interface and the Formation of Deep Water 72 2.9.2.5 The Return Ocean Flow Between the Poles and the Equator 73 2.9.2.6 Maintenance of the Warm Pool 73 2.10 Synthesis 73 Notes 73

Contents

3 Fundamental Processes 77

3.1

Some Fundamentals of Low-Latitude Atmospheric Dynamics 79 3.1.1 Basic Equations 79 3.1.2 Scaling Atmospheric Motions in the Tropics 80 3.1.2.1 Is the Tropical Atmosphere Hydrostatic? 81 3.1.2.2 A Consequence of Making the Hydrostatic Assumption: Total Kinetic Energy and the “Traditional Approximation” 82 3.1.2.3 Scaling Thermodynamic Variability in the Tropics 83 3.1.3 Early Interpretations 84 3.1.4 Conundrums 86 3.1.4.1 Hypothesis I: Tropical Convection Is Driven by Extratropical Forcing 86 3.1.4.2 Hypothesis II: Convection Occurs Because of Reduced Static Stability in Regions of Convection 86 3.1.5 Geostrophic Adjustment in the Low-Latitude Atmosphere 87 3.1.5.1 Rossby Radius of Deformation (R) 87 3.1.5.2 Inertial Motion 88 3.1.5.3 Rotational or Buoyancy Waves? 89 3.1.5.4 Heating and Tropical Circulations 90 3.1.6 Overview 90 3.2 Dynamics of the Low-Latitude Upper Ocean 91 3.2.1 Scales of Motion 91 3.2.2 Geostrophic Adjustment in the Low-Latitude Ocean 93 3.2.3 Sverdrup Wind-Driven Transport 95 3.2.4 Ekman Transports 96 3.2.4.1 Formulation 96 3.2.4.2 Why Is the Total Integrated Ekman Transport Orthogonal to the Surface Wind? 97 3.2.5 Induced Geostrophic Currents 98 3.2.6 Low-Latitude Wind-Driven Currents 99 3.2.6.1 Global Wind-Stress Fields and Surface Current Climatology 99 3.2.6.2 Geostrophic Currents 100 3.2.6.3 Currents and Counter-Currents 101 3.2.6.4 Equatorial Undercurrents 101 3.2.7 Overview 103 Notes 104 107 Phase and Group Velocities, and Energy Propagation 107 4.1.1 Wave Characteristics in a Quiescent Basic State 107 4.1.1.1 Golf Ball in a Pond 107 4.1.1.2 Analysis of the Perturbation in the Pond 108 4.1.2 Kinematic Relationships Between Waves and Their Background Basic State 109 4.1.2.1 General Wave Kinematics in a Variable Basic Flow 110 4.1.2.2 Dispersion of Energy Away from a Source 111 4.1.2.3 Dispersion in a Quiescent or Constant Basic State 111 4.1.2.4 Constant Basic State 111 4.2 Dispersive and Non-dispersive Waves 111 4.3 Overview 112 Notes 113

4 Kinematics of Equatorial Waves

4.1

5 Fundamental Prototypes of Tropical Systems 115

5.1

The Laplace Shallow Fluid System 115 5.1.1 Governing Equations 115 5.1.1.1 Use the Equatorial 𝛽-Plane Approximation

115

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5.1.1.2 Define Total Depth of the Fluid and the Background Basic State 116 5.1.1.3 Integrate the System in the Vertical 116 5.1.1.4 Linearization of the System 116 5.1.2 Doppler and Non-Doppler Effects 117 5.1.3 Equatorial Wave Equation 117 5.2 Upper Ocean 118 5.2.1 Governing Equations 118 5.2.2 Ocean Wave Equation 119 5.3 A Stratified Atmospheric Model 119 5.3.1 Separation of Variables 120 5.3.2 Basic Equations 120 5.3.3 Coupled Horizontal and Vertical Structure Equations 120 5.4 Forced and Free Solutions and the Choice of H 121 5.5 Some Remarks 123 Notes 123 125 Atmospheric Modes in a Constant Basic State: Constant U 125 6.1.1 Governing Equations for a Motionless Basic State (U = 0) 125 6.1.2 Governing Equations for a Constant Basic State (U = Constant) 125 6.1.3 General Dispersion Relationship for a Constant Basic State (U = Constant) 126 6.1.4 Eigenfrequencies for a Constant Basic State 127 6.1.4.1 Dispersion Diagrams 127 6.1.4.2 The Ubiquitous Nature of MRG and K Waves 128 6.1.4.3 Eigenfrequency Dependence on H 129 6.1.5 Eigensolutions 129 6.1.5.1 Equatorial Rossby Waves (ER) 129 6.1.5.2 Inertia-Gravity Waves (WIG, IG) 137 6.1.5.3 Mixed Rossby-Gravity Wave (MRG) 139 6.1.5.4 The Kelvin Wave 141 6.2 Atmospheric Waves in Latitudinal Shear Flow: U = U(y) 144 6.2.1 Regions of Shear in the Tropics 144 6.2.2 Impacts of Latitudinal Shear 145 6.2.2.1 Rossby Waves in Shear Flow 146 6.2.2.2 Mixed Rossby-Gravity Wave in Shear Flow 146 6.3 Physics of Equatorial Trapping 146 6.3.1 Simple Potential Vorticity Arguments 147 6.3.2 Induced Relative Vorticity in Simple Basic States 148 6.3.2.1 Motionless Basic State U = 0 148 6.3.2.2 Constant Non-Zero Basic State (U = constant) 149 6.3.2.3 Shear Flow: U = U(y) 150 6.4 Large-Scale Low-Latitude Ocean Modes 151 6.4.1 Simple Model of the Upper Ocean – Geopotential Surfaces 151 6.4.2 Rotational Ocean Waves 152 6.4.3 Impact of Boundaries on Near-Equatorial Ocean Modes 153 6.4.4 The Longwave Approximation 156 6.5 Overview 157 Notes 158

6 Equatorial Waves in Simple Flows

6.1

7 Waves in Longitudinally and Vertically Varying Flows 159

7.1

Horizontal and Vertical Coupling of Equatorial Modes 160 7.1.1 Coupled Group Speeds 160 7.1.2 Coupled Dispersion Relationships and Group Speeds 160

Contents

7.1.2.1 Equatorial Rossby (ER) Waves 160 7.1.2.2 Mixed Rossby-Gravity (MRG) Wave 162 7.1.2.3 Kelvin Wave 163 7.2 Coupled Free and Forced Solutions of the Vertical Structure Equation 163 7.2.1 Free Solutions 164 7.2.1.1 Isothermal Atmosphere 164 7.2.1.2 Constant Lapse Rate Atmosphere 166 7.2.1.3 Construction of Realistic Temperature Profiles 166 7.2.2 Forced Motions in an Isothermal Atmosphere 166 7.2.2.1 Case 1: External Solutions 168 7.2.2.2 Case 2: Oscillatory Solutions 168 7.3 Wave Characteristics in a Zonally Varying Basic State U = U(x) 169 7.3.1 Rays in the Longitudinal Plane 170 7.3.1.1 Rossby Wave Characteristics 170 7.3.1.2 MRG Wave Characteristics 172 7.3.2 Impact of Longitudinal Displacement of Wave Sources in a Zonally Varying Flow 7.3.2.1 Accumulating Modes 174 7.3.2.2 Anomalous Non-accumulating Propagating Mode 174 7.4 Numerical Substantiation of the Analytic Ray-Tracing Results 176 7.4.1 Equatorial Accumulation 176 7.4.2 Equatorial Emanation Regions to Higher Latitudes 177 7.5 Zonally Varying Basic State and the “Longwave Approximation” 181 7.6 Vertical Trapping, Accumulation, and Lateral Emanation 182 7.7 Quasi-Biennial Oscillation (QBO) 183 Notes 184 8 Moist Processes and Large-Scale Tropical Dynamics 185

8.1 8.2 8.3

Convection and Large-Scale Budgets 186 Emerging Perspective on Tropical Convection 188 Comparison of Observed Waves and Waves from Theory 190 8.3.1 Stratospheric Modes 190 8.3.2 Transient Tropospheric Modes 190 8.3.3 Stationary Modes in the Tropics 191 8.4 Dry and Moist Modes in the Tropics 191 8.5 Processes 193 8.5.1 Convective Dissipation 194 8.5.2 Stability and Convection 196 8.5.3 Surface Flux Feedbacks 197 8.5.4 Convective Instability of the Second Kind: CISK 198 8.5.5 Spatial Variation of the Basic State 199 8.6 Synthesis 201 Notes 203 205 Lateral Wave Propagation in a Zonally Symmetric Basic State 205 Equatorial Wave Propagation in a Zonally Varying Basic State 208 9.2.1 Numerical Experiments in a Zonally Varying Basis State 210 9.2.1.1 Weak Equatorial Easterlies (Basic State A) 210 9.2.1.2 Weak Westerly Zone (Basic State B) 210 9.2.1.3 Strong Westerly Zone (Basic State C) 212 9.2.1.4 Latitudinal Distributions of PKE 212 9.2.2 Synthesis 212 Equatorward Wave Propagation in a Three-Dimensional Basic State 214 9.3.1 Structure of the Mean Fields in the Upper and Lower Troposphere 214

9 Extratropical Influence on the Tropics

9.1 9.2

9.3

173

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9.3.2 Transient Behavior 216 9.3.3 One-Point Correlation Fields in the Horizontal Plane 217 9.3.4 One-Point Correlation Fields in the Vertical Plane 218 9.3.5 Impacts of Extratropical Wave Incursion into the Tropics 219 9.4 Overview 221 Notes 221 10 Tropical Influence on the Extratropics: A Zonally Averaged Perspective 223

10.1 Axisymmetric Meridional Circulation Models 223 10.2 Zonally Averaged Perspective of Meridional Circulations 225 10.2.1 An Atmospheric Zonally Averaged Model 225 10.2.2 Observed Eddy Momentum and Heat Fluxes 226 10.2.3 Eddies and the Mean Circulation 227 10.3 Perspective 230 Notes 230 11 A Tropical–Extratropical Synergy 231

11.1 Mean and Transient Potential Vorticity on the 370 K Isentrope 231 11.2 Impermeability 234 11.3 Shallow Fluid Experiments 237 11.3.1 Potential Vorticity Substance in a Shallow Fluid 237 11.3.2 Simulations with “Equatorial” and “Monsoon” Heating 238 11.3.3 Role of PVS Fluxes in Determining the Zonally Averaged Tropical Circulation 239 11.4 Recursively Breaking Rossby Waves 240 11.5 Conclusions 241 Notes 243 12 Arid and Desert Climates 245

12.1 12.2 12.3 12.4 12.5

Dynamics of Deserts 245 Radiative and Surface Fluxes 248 Diurnal Cycle of Divergence 250 Tropospheric Energy Balance 251 Nocturnal Stabilization of the Boundary Layer 251 12.5.1 Development of a Nocturnal Desert Jet Stream 251 12.5.2 Dynamics of the Low-Level Nocturnal Jet 251 12.5.3 A Subsiding Lateral Exhaust 254 12.6 Desert–Monsoon Relationships 255 Notes 256

13 Near-Equatorial Precipitation 257

13.1 Near-Equatorial Distributions of Precipitation 258 13.1.1 Relationships Between Convection, MSLP, and SST 258 13.1.2 Theories of the Location of Near-Equatorial Convection 260 13.1.2.1 Collocation of Convection and Maximum SST 260 13.1.2.2 Locus of Near-Equatorial Disturbances 261 13.1.2.3 Zonally Symmetric Arguments 261 13.1.2.4 Dynamic–Thermodynamic Optimization and Feedbacks 261 13.1.2.5 Coupled Ocean–Atmosphere Explanations 262 13.2 Dynamic Instabilities Associated with a Cross-Equatorial Pressure Gradient 262 13.2.1 Distributions of Absolute Vorticity 263 13.2.2 Geophysical Context for Inertial Instability 265 13.2.3 Concept of “Perpetual” Instability 266 13.2.4 Near-Equatorial Inertial Instability 268 13.2.5 Processes Determining the Distribution of Absolute Vorticity 270

Contents

13.2.6 Vertical and Latitudinal Structures 272 13.2.7 Is the Existence of a CEPG a Sufficient Condition for Inertial Instability? 273 13.2.8 Dynamic Estimate of the Latitude of the Mean ITCZ in Regions of Strong CEPG 275 13.2.9 Low-Level Near-Equatorial Westerlies in Regions of Strong CEPG 276 13.2.10 A Potential Vorticity View of Ameliorating Secondary Circulations 278 13.3 Transient States of the Intertropical Convergence Zone 280 13.3.1 Character of the Transients 281 13.3.2 Transient Composites 284 13.3.3 Diagnostics of ITCZ Transients 288 13.3.4 Origin of “Easterly Waves” 289 13.4 The Great Cloud Bands 290 13.4.1 Climatology of the GCBs 291 13.4.2 Variability within the GCBs 291 13.4.3 Theories of the Formation, Location, and Orientation of the GCBs 292 13.4.3.1 Anchoring 292 13.4.3.2 Orientation of the GCBs 292 13.4.3.3 Continental and Orographic Forcing 295 13.4.4 High-Frequency Variance in the GCBs 295 13.5 Some Conclusions 298 Notes 299 14 Large-Scale, Low-Frequency Coupled Ocean–Atmosphere Systems 301

14.1 The Walker Circulation 302 14.1.1 Early Depictions 302 14.1.2 Nature of Zonal Circulations 303 14.1.3 Simple Model Simulations of Zonal Circulations 303 14.1.4 Role of an Interactive Ocean 304 14.2 The Southern Oscillation, El Niño and La Niña 305 14.2.1 Evolution 309 14.2.2 Annual Cycle of the Upper Pacific Ocean 309 14.2.3 Interannual Variability of the Annual Cycle 313 14.2.4 Large-Scale Signals of El Niño and La Niña 313 14.2.5 ENSO Theories 318 14.2.5.1 The “Bjerknes Hypothesis”: Positive Feedback Between the Ocean and the Atmosphere 319 14.2.5.2 ENSO Theories with Negative Feedbacks 320 14.2.6 Predictability of ENSO 326 14.2.6.1 Annual Cycle of Persistence and the Boreal Springtime “Persistence Barrier” 14.2.6.2 The Boreal Springtime “Predictability Barrier” 327 14.2.6.3 Real-Time Forecasts of ENSO Variability 328 14.2.6.4 Can Forecasts of ENSO Extrema and Their Impacts Be Improved? 331 14.3 Indian Ocean Interannual Oscillations 332 14.3.1 The 1961 Event 332 14.3.2 The 1997–1998 Event 333 14.3.3 Association of the IOD with ENSO 334 14.3.4 What Produced the 1997–1998 IOD Episode? 336 14.3.5 Composite Structure of the IOD 337 Notes 342 15 Intraseasonal Variability in the Tropical Atmosphere 345

15.1 Introduction 345 15.2 Structure of the Austral Summer ISV 15.2.1 Early Constructions 345 15.2.2 More Recent Analyses 346

345

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15.3 Variability of Austral Summer ISVs 348 15.4 Mechanisms 351 15.4.1 A Local Instability Mechanism for the Initiation of an ISV Event 351 15.4.1.1 Destabilization Phase 353 15.4.1.2 A Convective Phase 355 15.4.1.3 Restoration Phase 355 15.4.2 The Indian Ocean as an ISV Generation Region 356 15.4.2.1 Feedbacks from Wave Dynamics 356 15.4.2.2 Extratropical Influence 357 15.4.2.3 Impact of Climatological State 358 15.5 Conclusions 358 Notes 358 361 16.1 Overview 361 16.1.1 Slow Component (Months to Years) 362 16.1.2 Intermediate Component (Weeks to Months) 362 16.1.3 Faster Components (2–15 Days) 362 16.1.4 Connective Components (All Timescales) 364 16.2 Theories of the Monsoon and Its Variability 364 16.2.1 Early Descriptions 364 16.2.1.1 Sir Edmund Halley’s Tropical Wind Climatology 364 16.2.1.2 Halley’s Differential Buoyancy Hypothesis 365 16.2.1.3 Determining the Origin of Monsoon Flow 365 16.2.2 Attempts to Determine Remote Influences on the Monsoon 366 16.2.2.1 Walker’s Surmise 366 16.2.2.2 The Demise of the Walker Relationships: A Mid-century Conundrum 367 16.2.2.3 Relationships Revisited Using a Longer Data Series 367 16.2.3 Circulations Associated with Strong and Weak South Asian Monsoons 370 16.2.3.1 Identification of an Anomalous Monsoon 370 16.2.3.2 Impact of an Anomalous Monsoon on the Indian Ocean SST 372 16.2.3.3 Indian Ocean SST Anomalies and Monsoon Precipitation 372 16.2.3.4 Annual Cycle of Anomalous 850 and 200 hPa Winds 373 16.3 Macroscale Structure of the Summer Monsoon 374 16.3.1 Mean Seasonal PV Distributions 374 16.3.2 Annual Cycle of PV Fields 375 16.3.3 A Physical Basis for the Character of the Macroscale Monsoon 376 16.3.3.1 Anomalous Location of South Asian Monsoon Precipitation 376 16.3.3.2 Seasonal Distribution of Mean Upper Tropospheric Temperature and Specific Humidity 378 16.3.3.3 Mean Monthly Geopotential Sections 379 16.3.3.4 Comparison of Surface Heat Fluxes 380 16.3.3.5 Comparison of Vertical Temperature Profiles over the Gangetic Plains and the HTP 381 16.3.3.6 Precipitation over the HTP and the Evolution of the Elevated Surface Cyclonic Vortex 382 16.3.3.7 A Heating Threshold for a Subtropical Meridional Circulation 382 16.3.4 West African Summer Monsoon 386 16.4 Macroscale Structure of the Winter Monsoon 388 16.4.1 Siberian Cold Anticyclone 389 16.4.2 Limitations on Central Pressure 389 16.5 Subseasonal Summer Monsoon Variability 391 16.5.1 Identification of Propagating Intraseasonal Signals 391 16.5.2 Impacts of Monsoon Intraseasonal Oscillations (MISOs) 392

16 Dynamics of the Large-Scale Monsoon

Contents

16.5.3 Inter-event Variability 393 16.5.4 Composite Structure of the MISO 394 16.5.5 Theories Regarding the Genesis and Maintenance of the MISO 396 16.5.5.1 External Forcing of the MISO 396 16.5.5.2 MISO as an Internal Mode: Self-Induction and Self-Regulation 397 16.5.5.3 Extensions of the Internal Mode Theory 398 16.5.5.4 Northward Propagation of the Boreal Summer MISO 398 16.6 Higher-Frequency Monsoon Variability 400 16.6.1 Summer Monsoon 400 16.6.2 Winter Monsoon 401 16.7 Some Comments 405 Notes 405 17 The Coupled Monsoon System 407

17.1 Coupled Characteristics of the Indian Ocean Region 407 17.1.1 Monsoonal Moisture Fluxes 408 17.1.2 Annual Cycle in the Indian Ocean Region 408 17.1.3 The Surface Flux–SST Tendency Paradox in the Indian Ocean 408 17.2 Processes Determining the Indian Ocean SST 411 17.2.1 Ocean Heat Transport 411 17.2.2 Changes in Ocean Heat Storage 413 17.2.3 Spatial and Temporal Variability of Ocean Heat Flux 413 17.3 Do Ocean Heat Fluxes Regulate the Annual Cycle of the Monsoon? 415 17.3.1 Balanced Interhemispheric Heat Fluxes 415 17.3.2 Cross-Equatorial Ocean Ekman Heat Transport 415 17.4 Variability Within the Coupled Monsoon System 416 17.4.1 Biennial Variability 416 17.4.1.1 Ocean-Atmosphere Feedbacks 417 17.4.1.2 Pacific Warm Pool Seasonal Cycle Instability 418 17.4.1.3 Indian Ocean Feedbacks I, The Meehl Theory 418 17.4.1.4 Indian Ocean Feedbacks II: A Dynamic Ocean 419 17.4.1.5 ENSO and Internal Dynamics 419 17.4.2 Intraseasonal Variability in the Indian Ocean 421 17.5 An Holistic View of the Monsoon System 421 17.5.1 Indian Ocean Sector 422 17.5.2 Speculations on the Interaction of the Indian and Pacific Ocean Sectors 424 Notes 428 429 18.1 Tropical Warm Pool 429 18.1.1 Changes in the Ocean Warm Pool During Last Century 429 18.1.2 The Mid-Twentieth Century SST Plateau 430 18.1.3 Longer-Term Changes in the Tropical SST 431 18.1.4 Relationship Between SST and Convection in the OWP 432 18.1.4.1 Surface Energy Balance Regulation 433 18.1.4.2 Cloud-Radiation Feedbacks 433 18.1.4.3 Ocean Feedbacks 433 18.1.5 SST and Column Integrated Heating (CIH) 433 18.1.6 Why Is the Area of Organized Convection Relatively Constant? 435 18.2 Circulation Changes 438 18.2.1 Definition of a Broad-Scale Monsoon System 438 18.2.2 Variability and Trends of the Northern Hemisphere Monsoon System 439

18 The Changing Tropics

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18.2.3 Why Has the Northern Hemisphere Monsoon System Intensified? 441 18.3 Summary and Conclusions 442 Notes 444 19 Some Concluding Remarks 445

Notes 447 Appendix A Thermal Wind Relationship 449 Appendix B Stokes’ Theorem

451

Appendix C Dry and Moist Thermodynamical Stability 453 Appendix D Derivation of the Wave Equation (5.11) 455 Appendix E Conservation of Potential Vorticity of Shallow Water System 457 Appendix F Solutions to the Vertical Structure Equation for a Constant Lapse Rate Atmosphere Appendix G Nonlinear Numerical Model

461

Appendix H Derivation of the Potential Vorticity Equation on an Extratropical 𝜷-Plane 463 Appendix I Derivation of the Barotropic Potential Vorticity Equation (13.25) with Friction and Heating 465 Appendix J Steady State Model of the Tropics Appendix K Intermediate Ocean Model References 471 Index 493

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Preface My interest in tropical meteorology evolved in a number of ways. After finishing my undergraduate work and completing my meteorologist training with the Bureau of Meteorology in Australia, I was assigned as a very junior forecaster to the Darwin Airport forecast office in the far north of Australia. There appeared to be two seasons in Darwin: the “dry” and the “wet” or, alternatively, the “boring” and the “unknown.” During the dry, except for occasional early morning fog, a day of fine weather and southeasterly trade winds tended to follow a day of fine weather and etc., rather repetitively. The wet was completely different and each day was a challenge. Satellite meteorology was in its infancy. Beyond the very strong diurnal cycle there appeared to be no overriding physical explanation or schema that would help anticipate the development and migration of convective weather events and the occasional tropical cyclone. All forecasts were made using hand drawn synoptic charts and there was little upper air data, especially to the north of Australia. However, I had obtained a copy of Riehl’s (Riehl 1954) and later Riehl 1979 masterpiece on tropical meteorology and it seemed to me that perhaps there might be some order within the seemingly chaotic wet season if one could only find it. Further, Riehl and Malkus (1958) had suggested organization on the grand scale, posing a theory that heat and momentum transports within deep penetrating convection in the equatorial trough were integral parts of the general circulation of the planet. I found both works rather exciting! So motivated, and armed with a lack of humility, I moved to the United States for graduate work at Florida State University with Professor Michael Garstang who was a fine empiricist. I was lucky enough to be chosen to attend the National Center for Atmospheric Research graduate student summer workshop on thermal convection and where I was intrigued with the developing field of low-latitude dynamics. It was there I met Professors Jule Charney and Norman Phillips, who were to become my advisors at MIT. Charney was attempting to find a balanced dynamic system for the tropics that was equivalent to the quasi-geostrophic system he had formulated for the extratropics. At that

time he was also trying to understand the rapid spin-up of tropical cyclones. Phillips was particularly concerned with modeling the global general circulation, handling the tropics adequately and accounting for convection in a realistic manner. Both were aware of the very recent pioneering work of Professor Taroh Matsuno (1966) on the existence of wave-forms having maximum variance close to the equator. But no one had much idea of how these tropical modes were excited. The development of an adequate in situ instability process proved difficult and the propagation of extratropical waves through the tropical easterlies had its own theoretical problems. To a large degree, the interdependence of the tropics and the tropics was not understood at all. It was within this environment that I commenced my PhD work. There was an immediacy in the solution of some of these issues. During the early years of numerical weather prediction with its concentration on the extratropics, it was apparent that what occurred in the tropics influenced higher latitudes very rapidly. Hence, a forecast with a time horizon beyond a few days needed to be generated by a global model that was fueled by a global initial data. This meant that predicting mid-latitude weather required understanding tropical processes and the convective elements of which they are comprised. Perhaps because of this immediacy, our knowledge of tropical atmospheric, oceanographic, and land-surface processes would progress substantially during the next few decades. These advances have come from increased observations and a surge of theoretical insights. Much of progress has been built on the results of a large number of dedicated field campaigns specifically designed to increase understanding of components of the tropical system. Understanding the dynamics of tropical circulations is important, not only for predicting mid-latitude weather, but because weather and climate variations in the tropics impact about half of the global population directly. Within the monsoon regions, for example, vagaries in annual rainfall create periods of either agricultural scarcity or abundance, hardship or plenty. “Active” and “break” periods of the monsoon impart periods of

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short-lived drought and flood to the region and they have been notoriously difficult to predict and remain so to a large degree today. My interest in tropical meteorology and participation in a number of field experiments in tropical regions led me toward applying some of the growing knowledge of tropical dynamics for prediction in monsoon regions. I was fortunate to spend a sabbatical (1991–1992) at the European Centre for Medium Range Weather Forecasts working with Tim Palmer. This was a time when numerical weather prediction was undergoing a major transformation toward ensemble prediction that would enable probabilistic forecasts at longer lead times. Such techniques allowed Tim Palmer, Tom Hopson, Jun Jian and me, among others, to produce probability flood forecasts for the monsoon regions of Bangladesh that gave users the knowledge to make decisions based on the likelihood of an event occurring and the occurrent loss if action were not taken. Rural communities, anticipating the occurrence of a flood based on these forecasts were able to save the equivalent of an annual income by moving livestock to higher ground, early harvesting, and evacuation. This text rests on both a “reductionist” and “holistic” perspective. The book describes the basic physics of individual elements of the large-scale circulations like equatorially trapped waves, the El Niño, and the monsoon. Such is the “reductionist” perspective. The “holistic” approach acknowledges that new or emergent features of tropical circulation cannot be deduced from the properties of the individual elements alone. Without a knowledge of the basic physics of the components of a complex system, it is difficult to assess whether the results of a complex coupled climate model, for example, are realistic. However, without a complex model that encompasses the individual phenomena as well as the broader context and interactions with the larger-scale environment, it is difficult to determine a range of possible outcomes.

The text is aimed at the advanced undergraduate or an early career graduate student. A basic level of fluid dynamics and thermodynamics would be an advantage, but I have attempted to place complicated concepts in a broader and simpler context. The book is also intended to serve as a general reference book for scientists interested in tropical phenomena and their relationship with the broader climate system. The focus of this text is on the fundamental aspects of the large-scale coupled dynamics of the tropical system. Tropical cyclones are considered principally in terms of genesis processes related to the large-scale environment. There are a number of texts that deal with important details of tropical cyclones. To do this here in an adequate manner would double the size of the text. I started this project many years ago and it has progressed with many stops and starts. Part of the problem is accounting for changes in a rapidly evolving field. What is recorded in the text has been developed for and presented in many graduate classes. From one class to the next, new knowledge is relatively easy to incorporate by updating and changing class notes. However, in writing a text eventually one needs to draw a line. I think Sir Winston S. Churchill may have said it best: Writing a book is an adventure. To begin with it is a toy and an amusement. Then it becomes a mistress, then it becomes a master, then it becomes a tyrant. The last phase is that just as you are about to be reconciled to your servitude, you kill the monster and fling him to the public. So, for better or for worse . . . . ! Reno, Nevada, USA September, 2019

PJW

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Acknowledgments Throughout my career, I have had the privilege to work with many gifted students and colleagues. In fact, there are 31 young scientists whom I mentored on their way to their doctorates, in addition to many postdoctoral fellows who were part of our research group through the years. They constituted a diverse and international cadre, hailing from Romania, China, Australia, the United Kingdom, Brazil, Colombia, the United States, Japan, Russia, India, Taiwan, and Korea. Many have risen to positions of prominence, making significant contributions in their chosen careers in academia, research laboratories, and the private sector. Most of all I have to thank my students and post-docs. I think one of the great privileges of being a professor is to be associated with bright young minds, watching them develop through hard work to become shining skeptical scientists. I know that the faculty–student interaction is often thought to be “top-down” but I feel that it is at least an equal interaction or perhaps even weighted “bottom-up.” Over the years, I think all of us have enjoyed the to and fro of what were often exciting group meetings. I would like to acknowledge two members of my research group in particular, Drs. Hai-Ru Chang and Violeta Toma, both of whom contributed substantially to earlier versions of the manuscript. Hai-Ru has been an integral component of my research group since the mid-1980s. His rigor and theoretical knowledge of fluid dynamics have been a great benefit to our collective efforts. Violeta is a gifted diagnostician and was instrumental in producing analyses of near-equatorial phenomena, designing models and conducting many simulations. She has also made fundamental contributions to the concept of global synchronicity described in Chapter 11. I would also like to thank both Hai-Ru and Violeta for their critical assessment of various parts of the text. Thanks are also due to Dr. Ferdinand Hirata who carefully produced many of the figures. The work of our research group is embodied in much of the text. I would like to mention some contributions that are particularly relevant to its major themes: Hye-Mi Kim for her diagnostics of

intraseasonal variability and its relationship to tropical cyclone development; Song Yang who pointed out the non-stationarity of the interaction of the El Niño–Southern Oscillation (ENSO) and the Asian monsoon; Chidong Zhang who did initial work on the interaction of equatorially trapped modes and complex basic states and his more recent work on intraseasonal variability; Bill Lau for his insightful coupled ocean–atmosphere modeling over 40 years ago; Hai-Ru Chang who developed the concept of “wave energy” accumulation and emanation; Robert Tomas for determining the role that ocean heat transport played in the evolution and stability of the Asian monsoon and also, with Violeta Toma, on deciphering structure of near-equatorial convection; Carolyn Reynolds who introduced the group into uncertainties brought into forecasts and the regional growth of errors; Johannes Loschnigg and Chris Torrence who have made great strides towards understanding linkages between ENSO and monsoon variability, adding clarity to Normand’s (1951) surmise regarding the lag and or lead of variability between ENSO and the Asian monsoon; Victor Magana and Sanjay Dixit who contributed to our understanding of monsoon and ENSO prediction; John Fasullo for perceptions relating westerly wind bursts and the heat balance of the tropical warm pool; Gill Compo for determining the fundamental structures of Asian cold surges and, with John Fasullo, how they alter the meteorology of the equatorial Pacific; Initial work on the Indian Ocean Zonal mode and its utility in prediction was undertaken by Christie Oefke Clarke and Daria Halkidies; Carlos Hoyos for his work on the evolution of the tropical warm pool and the development of a Bayesian prediction scheme for monsoon rainfall; Paula Agudelo for deciphering the physics of the transition of the suppressed phase of intraseasonal variability to its convective phase; Kam Sahami and Galina Chirokova for insightful modeling of Indian Ocean variability; Fernando Hirata and Dave Lawrence for deducing the intricacies of intraseasonal behavior in both the summer and winter monsoon; Matt Widlansky and Fernando Hirata for making great strides towards solving the

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mystery of the Great Cloud Bands; Manuel Zuluaga who advanced understanding of the role of aerosols in tropical climates; and Sebastian Ortega for developing fundamental insights into how the tropics and the extratropics interact. Matt Wheeler, of course, made fundamental contributions to the manner in which equatorial modes and convection intertwine. I appreciate the hard and inspirational work Tom Hopson, jun Jian and Kris Shrestha who developed long-horizon prediction schemes for Bangladesh and Pakistan. Also, thanks to Tom, Jun, jun, Bob Grossman, Subbiah-Ji of RIMES and Tom Brennan (of USAID), with whom I shared adventures in South Asia, Bangladesh and Pakistan attempting to implement these forecasting schemes. We may even have actually done something useful! Over the years, I have had many colleagues to whom I have moaned and groaned about writing a book. They have remained cheerful, more so than me I must admit, and convinced me that it is a worthwhile endeavor. They have made both substantive and philosophical contributions that I have welcomed very much. George Kiladis and Matt Wheeler persisted (patiently) in convincing me in how convection and equatorial modes interact. George has been very generous with his time in discussing large sections of pertinent text. Robert Houze of the University of Washington has reminded me on many occasions how convective elements comprise an integral part of the dynamics of the tropics. Thanks are due to my Australian colleagues, Greg Holland, John McBride, and also “fellow (field expedition) traveller” Bob Grossman, who all nudged me on many occasions towards the realization that there is reality beyond theory or model results. I have come to realize that the empiricism embodied in their work and that offered by scientists such as Richard Johnson and Bob Houze is a vital component of the tropical puzzle. With respect to empiricism, I appreciate the inspiration of Prof. Michael Garstang, with whose research group I was briefly associated when I first arrived in the US. He was rather forceful in reminding many of us that the tropics was a complicated system and that not everything can be encapsulated in a neat set of equations. I also appreciate my colleague Professor Tim Palmer at Oxford for introducing me to the wonderful concept of uncertainty, chaos, and probabilistic forecasting during a sabbatical at ECMWF that has had a profound influence on subsequent work. Professor Sharon Nicholson of Florida State University provided data and ideas about the meteorology of near-equatorial Africa and how meteorology and climate differs markedly from one tropical region to another. I appreciate Graeme Stephens of NASA, who at an early stage of my career explained painstakingly the role of clouds in radiative forcing of the tropical atmosphere. We started our discussions in

our cramped shared office in Australia. His work has gone onwards to influence climate research globally. He surely has influenced my way of thinking about the role of clouds in global climate. Professor Bin Wang’s support, scientific collaboration, generosity and friendship through the years have remained very important to me. Much of our joint research appears throughout the text. Roger Lukas, my hard-working and dedicated “partner in TOGA COARE crime,” insisted I realize that the tropics must be thought of as a closely coupled ocean–atmosphere system across all time scales. I think his influence permeates much of the text. A large part of my career has been involved in field expeditions. I have come to believe that these times in the field account for much of the progress we have made in tropical meteorology. The data collected is invaluable but they are catalysts for thought. Although expeditions arise from curiosity, implementation is often beyond the capabilities of just the curious. The transformation of ideas to results requires expert logistical support. Such support has come from the University Corporation of Atmospheric Research’s (UCAR) Joint Office for Science Support led for many years by Karyn Sawyer. I first met Karyn in 1978 in Kolkata, India, during the Summer Monsoon Experiment and in the many subsequent field adventures that have followed. On numerous occasions her office has made it possible to transform a myriad of ideas to an organized reality, with the result of providing essential data to the larger scientific community. It is a pleasure to acknowledge two colleagues who, early in my career, changed the way I think about science. The first was A. J. “Sandy” Troup of Australia, who showed that you could do quiet, personal research with dignity and without the expectation acknowledgement and be successful and influential. He explained the coupled nature of the Southern Oscillation 5–10 years before it became an accepted theory, but received little credit for his publications, externally or internally. Just doing the science was sufficient for him. I met Shri Dev Sikka-ji during my first visit to India in 1976. He remained an influential colleague for the next 40 years. He was a monsoon polymath with extraordinary physical insight and his contributions were rooted in his profound urge to understand the monsoon system fully and make this knowledge useful to society. His enthusiasm and dedication were infectious! I wish to thank my doctoral advisor Professor N. A. Phillips of Massachusetts Institute of Technology, who patiently introduced me to careful science. In addition, I am indebted for the support I have received from Federal and International organizations, especially from the US National Science Foundation and the late Dr. Jay S. Fein.

Acknowledgments

I am deeply indebted to Beth Tully of the University of Washington and Patricia Bateson of Bateson Publishing (UK) who edited much of the text, both making many useful suggestions and adding clarity. Their collective rigor has been an important counterpoint to my sometimes wobbly English and logic. Finally, I would like to thank Judith Curry, my close companion and a supporter of what I have tried to achieve in this text and in so many other things. Her

continual support and encouragement is appreciated at a most fundamental level. I would like to thank my family for their love and for putting up with the long absences due to conferences, seemingly endless committee work, and field expeditions. Reno, NV September, 2019

Peter J. Webster

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Abbreviations AIRI AMEX AMG AMO ANN APE AV BP C-C CAPE CEPG CIH CIH CISK CMIPi CMORPH dalr DJF DOF DWP DYNAMO ECC ECMWF ED EE EI EIG EMEX

All-India Rainfall Index The Australian Meteorological Experiment Asian Monsoon Gyre Atlantic Multidecadal Oscillation annual Available potential energy Absolute Vorticity Before present Clausius-Clapeyron Convective Available Potential Energy Cross-Equatorial Pressure Gradient column integrated heating Atmospheric Column Integrated Heat Convective Instability of the Second Kind Version “i” of the IPCC Coupled Climate Model Simulations NOAA Climate Prediction Center Morphing Technique Dry adiabatic lapse rate December, January, February degrees of freedom Dynamic Warm Pool Dynamics of the Madden-Julian Oscillation field experiment Equatorial counter current European Centre for Medium Range Weather Forecasts Eastward Decaying intraseasonal event Eastward equatorial basis state Eastward Intensifying intraseasonal event Eastward propagating Inertial gravity wave The Equatorial Mesoscale Experiment

ENSO EOF EPIC

ER EW FFT GARP GATE GCB GPCP HTP IBTrACS ICSU IFA IGY IIOE IMD IMET IOD IOZM IPCC ISV ITCZ JAA JASMINE JJA JJAS K LGM LGM

El Niño-Southern Oscillation Empirical Orthogonal Function East Pacific Investigation of Climate Processes in the Coupled Ocean-Atmosphere System Equatorial Rossby wave Westward equatorial basis state Fast Fourier Transform Global Atmospheric Research Programme The GARP Atlantic Tropical Experiment Great Cloud Bands Global Precipitation Climatology Project (GPCP) Himalayan-Tibetan Plateau International Best Track Archive for Climate Stewardship International Council of Scientific Unions Intensive Flux Array of TOGA COARE International Geophysical Year International Indian Ocean Expedition Indian Meteorological Department Improved METeorological buoy Indian Ocean Dipole Indian Ocean Zonal Mode International Programme on Climate Change Intraseasonal variability Intertropical Convergence Zone Japanese Aerospace Agency Joint Air-Sea Monsoon Interaction Experiment June, July, August June, July, August, September Equatorial Kelvin wave Last Glacial Maximum Last Glacial Maximum

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Abbreviations

MALR MAM MCZ MCS MISO MJO MONEX MRG MSLP MYFZ NAO NASA NCAR NCEP NEC NH NIMBUS III Niño-1+2 Niño-3 Niño-3.4 Niño-4 OLR OWP PE PIRATA PKE PMIP PV PVS PVU QBO QBW RAMA

SACZ SAM SD SEC SH

moist adiabatic lapse rate March, April, May season maximum cloudiness zone mesoscale convective system Monsoon Intraseasonal Oscillation The Madden-Julian Oscillation The GARP Monsoon Experiment Mixed Rossby-gravity wave Mean sea level pressure Mei-Yu-Baiu Frontal zone North Atlantic Oscillation National Aeronautic and Space Administration National Center for Atmospheric Research National Center for Environmental Prediction North Equatorial current Northern Hemisphere NASA satellite launched 1969 East Pacific region (5∘ N-5∘ S, 90∘ W-80∘ W) Central-East Pacific region (5∘ N-5∘ S, 150∘ W-80∘ W) Central Pacific region (5∘ N-5∘ S, 170∘ W-120∘ W) West-Central Pacific region (5∘ N-5∘ S, 90∘ -80∘ W) Outgoing Longwave radiation at top of atmosphere (TOA) Ocean Warm Pool (SST> 28∘ C) Potential energy Prediction and Research Array in the Atlantic perturbation or eddy potential energy Paleoclimate Model Intercomparison Study potential vorticity potential vorticity substance: PVS = qS = 𝜎q potential vorticity units (10−6 m2 s−1 K kg−1 ) Quasi-Biennial Oscillation Quasi-Biweekly Wave Research Moored Array for African-Asian-Australia monsoon Analysis and Prediction South Atlantic Convergence Zone South Asian Monsoon standard deviation South Equatorial Current Southern Hemisphere

SICZ SMONEX SO SOI SON SPCZ SRES SSM/I SST TAO TOA TOGA TOGA COARE

TP TRMM TUTT UARS UCAR WEPOCS WIG WISHE WMO WMONEX yr

South Indian Convergence Zone Summer component of GARP MONEX Southern Oscillation Southern Oscillation Index September, October, November season South Pacific Convergence Zone Special Report on Emission Scenarios Special Sensor microwave imager sea-surface temperature Tropical Atmosphere Ocean Array Top of atmosphere The Tropical Ocean Global Atmosphere Experiment The TOGA Coupled Ocean-Atmosphere Response Experiment Triple point of water Tropical Rainfall Measurement Mission Tropical upper-tropospheric trough Upper-air Research Satellite University Corporation for Atmospheric Research West Pacific Ocean Climate Study Westward propagating Inertial gravity wave The Wind-Induced Surface Heat Exchange mechanism World Meteorological Organization Winter component of GARP MONEX Year

Symbols a A B 𝛼p 𝛼c 𝛼 th 𝛼v BE B BO BF

Earth planetary radius first Airy function second Airy function planetary albedo cloud albedo thermal expansion coefficient for sea water specific volume (1/𝜌) vertically averaged atmospheric moisture transport buoyancy force per unit mass Ocean buoyancy force per unit mass buoyancy flux

Abbreviations

𝛽 𝛽S C C c′ CD cg ̃cg cgx cgy Cp cp cpx cpy cgz cpz ̃c D De Do Dv 𝛿 - function 𝛿 E EK ev esv 𝜀m 𝜀 𝜂 𝜂𝜃 𝜃 𝜃e f fr F ̃ F FB FO FR FW g ̃ g G

latitudinal gradient of Coriolis parameter df/dy salinity contraction coefficient of sea water Symbolic Coriolis force integrated circulation vortex tube circulation drag coefficient wave group speed group velocity vector (cgx̃i + cgỹj + cgz ̃ k) zonal wave group speed meridional wave group speed specific heat of air at constant pressure wave phase speed (gH)1/2 zonal wave phase speed meridional wave phase speed vertical wave group speed vertical wave phase speed reduced gravity phase speed (̃ gH)1∕2 dissipation of kinetic energy depth of the ocean Ekman layer depth of no ocean motion Divergence (𝜕u/𝜕x + 𝜕v/𝜕y ) Dirac delta function increment evaporation rate Ekman number water vapor partial pressure saturated vapor pressure long-wave radiation emissivity energy density of a wave absolute vorticity (𝜁 + f ) absolute vorticity on an isentropic surface (𝜁 𝜃 + f ) potential temperature equivalent potential temperature vertical component of Coriolis parameter (2Ω sin 𝜑) representative frequency used in scaling radiative flux dissipation or frictional force vector buoyancy flux ocean buoyancy Froude number surface fresh water flux into the ocean (E-P) gravitational acceleration reduced gravitational acceleration atmospheric scale height (RT/g)

G 𝛾s Γd Γs Γ2 H HS (y) hB (x,y) h (x,y) hs Hn i, j and k ̃ ̃

IE IS I S↓ I S↑ JA J𝜃̇ JF JT J 0 (z) k KE ki 𝜉 𝜅 ln(p) L l L, D L0 LH LP

̃

scale height (RT/g) dissipation rate dry adiabatic lapse rate dry adiabatic lapse rate wave refractive index mean depth of a shallow fluid slope of the shallow fluid needed to maintain a geostrophic background flow topography of the lower boundary in a shallow fluid perturbation surface displacement in a shallow fluid: equivalent depth ocean steric height nth order Hermite polynomial Cartesian unit vectors in the x, y and z directions emitted terrestrial radiation to space at top of atmosphere net surface longwave radiation downward longwave radiation at surface upward longwave radiation at surface advective flux of potential vorticity substance on an isentropic surface potential vorticity substance flux on an isentropic surface due to non-adiabatic heating potential vorticity substance flux on an isentropic surface due to dissipative processes total potential vorticity substance flux on an isentropic surface (=JA + J𝜃̇ + Jf ) vertical heating function longitudinal or zonal wave number kinetic energy general wavenumber in direction “i” wave action density ratio of gas constant and specific heat at constant pressure R/C p log pressure lateral scale of a gravity wave latitudinal or meridional wave number representative length and height scales used in scaling diameter of pond surface latent heat flux into the atmosphere latent heat release due to precipitation

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Abbreviations

Lz LE LF LS 𝜆 M ie Mg 𝛿M g Mi M MA m mo m μ ̃ n N2 𝜐 Φ 𝜑 𝜙 P p Pm p(y), q(y) ps or p0 (P-E)

𝜓i 𝜓 Q̇ QS QT QSP qv qvs qs qg

vertical scale latent heat of evaporation-condensation latent heat of fusion latent heat of sublimation-deposition longitude vertically integrated Ekman ocean mass flux in direction i. geostrophic mass transport in the ocean incremental geostrophic mass transport in the ocean total vertically integrated ocean mass flux in direction i. mass of a reservoir of a water substance angular momentum vertical wave number map scaling factor magnitude of mass source/sink latitudinal structure function of mass source/sink μ normal unit vector Brunt-Väisälä frequency specific volume anomaly potential latitude geopotential Precipitation rate (e.g., mm/ay) pressure measure of the persistence of SST functions defined in equ. (5.11) reference or standard pressure often 1000 hPa Precipitation rate minus evaporation rate or net fresh water flux at ocean or land surface radiation penetration depth in ocean Stokes streamfunction total diabatic heating total heat flux at the surface total columnar heating of an atmospheric or ocean column sensible heat surface cooling of precipitation specific humidity saturation specific humidity potential vorticity substance of an isentropic surface;qS = 𝜎q geostrophic potential vorticity

q

qh qp qs R r R R RC RE Rd Rs RE RE Ri RNET Ro RS Rv 𝜌 𝜌0 𝜌w 𝜌a SR S0 S Sm

SH SP SS S s s0 𝜎 SB 𝜎 T TA TR TS

potential vorticity on an isentropic surface or Ertel’s potential vorticity potential vorticity in a shallow fluid potential vorticity on an isobaric surface potential vorticity substance real part of an expansion correlation coefficient specific gas constant Rossby Radius of Deformation Radius of curvature equatorial Rossby Radius of Deformation specific gas constant for dry air specific gas constant for moist air net radiation at top of atmosphere Reynold’s Number Richardson number net radiative flux into an atmospheric or ocean column Rossby number net surface radiation gas constant for water vapor density standard density density liquid water air density at ocean surface flux of water substance into a reservoir solar radiation arriving at Earth: Solar Constant net incoming solar radiation top of atmosphere mass source/sink in shallow fluid or time rate of change of mass between isentropes surface sensible turbulent heat flux sensible heat flux from precipitation net surface shortwave radiation static stability salinity standard value of salinity Stefan-Boltzman constant isentropic mass density temperature mean temperature of an atmospheric column residence time of a water substance in a reservoir standard temperature

Abbreviations

T0 TA TB To TS T SST t 𝜏i U U WD U WS U, V, W

u u* ue ug 𝜐 V ̃ V V𝜓 V𝜒 ̃g V v ve

primitive, equivalent or radiating planetary temperature average temperature of planetary atmosphere infrared cloud brightness temperature surface temperature of a planet surface temperature surface temperature of ocean surface time wind stress component in direction i background zonal wind magnitude of mean zonal wind in westerly duct basic wind shear between 850 hPa and 250 hPa representative scales of the zonal, meridional and vertical velocity components used in scaling zonal wind component deviation of the zonal velocity component from zonal average zonal Ekman current component geostrophic zonal wind component solar radiation absorption coefficient background meridional wind velocity vector rotational component of velocity divergent component of velocity vector geostrophic velocity meridional wind component meridional Ekman current component

vg vT v*

Wp w wZ wp we Ω 𝜛 𝜔 𝜔d ̃ V 𝜒 𝜁 𝜁T yT z z0 zm Z

geostrophic meridional wind component measure of the convergence of the trade winds in Pacific Ocean deviation of the meridional velocity component from zonal average precipitable water in an atmospheric column vertical wind component (dz/dt) dZ/dt: vertical velocity in Z (= −G ln(p∕ps )) coordinates. vertical wind component in pressure coordinates (dp/dt) Ekman vertical velocity component rotation rate of the planet angular velocity modal frequency Doppler-shifted modal frequency Vector velocity with components u i, v j and w k ̃

̃

̃

Velocity potential horizontal component of relative vorticity (𝜕v/𝜕x − 𝜕u/𝜕y ) total horizontal component of relative vorticity(mean plus perturbation) turning latitude height reference height in the atmosphere height of the Himalayan-Tibetan Plateau vertical coordinate = −G ln(p∕ps )

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1 Climatology of the Tropical Atmosphere and Upper Ocean This brief introduction identifies a number of phenomena that characterize the spatial and temporal variability of the tropical climate. Planetary scale circulations include Hadley-like circulations, the more zonal Walker systems, westerly ducts (WDs), tropical upper tropospheric troughs (TUTTs), quasi-biennial lower stratospheric oscillations, and the monsoon systems. Each of these features is “planetary” in scale and oscillates on interannual time scales. Within these slowly varying features of the tropical atmosphere and ocean, there is a myriad of circulation features with higher-frequency variability. These include the Madden–Julian Oscillation (MJO), quasi-biweekly variability, easterly waves, tropical cyclones and depressions, and a high-amplitude diurnal cycle. These higher-frequency scales of variability are not independent and are modulated by lower-frequency events such as the MJO, which, in turn, is modulated by the annual cycle and interannual variability. In subsequent chapters we will clarify the underlying mechanisms of these circulation features and their roles in weather and climate.

1.1 The Growth of Tropical Meteorology Prior to the 1970s, the abiding concern in meteorology was the development of numerical models for the extratropical weather prediction. In the early twentieth century V. Bjerknes (1904) had identified the necessary physical laws to describe atmospheric motion and established the concept of prediction as an initial value problem. He also realized that the equations could not be solved analytically. Richardson (1922),I though, showed that solutions could be attained numerically using a finite difference approach for solving the nonlinear differential equations that, eventually, would form the basis of weather and climate modeling. However, Richardson’s first attempts at weather prediction, calculated for a small region by hand, possessed extremely large errors. It would be 30 years later when numerical

prediction proved fruitful with the development of the electronic computer and the identification of approximations that would reduce Richardson’s numerical errors. In the decades that followed, a deeper understanding of fundamental modes of the midlatitudes and their instabilities developed. Rossby (1940), for example, had explained that waves in the extratropical westerlies resulted from the conservation of potential vorticity. Charney (1947) and Eady (1949) revealed that these waves arose from instabilities of the westerly wind regime growing at the expense of the energy of the background flow and formed efficient agents for energy and momentum transfer. With the emergence of the electronic computer, and an increasing network of atmospheric data, extratropical numerical weather prediction became a reality. A very useful history of numerical weather prediction is given by Harper et al. (2007), marking the 50th anniversary of the first US operational numerical weather predictions made by the Joint Numerical Weather Prediction Unit in July 1954. Whereas the focus of early numerical weather prediction was the northern hemisphere (NH) middle latitudes, the need to monitor and understand tropical phenomena grew rapidly. It was soon realized that to forecast events in an extratropical region for even short forecast horizons, a global model would be required. For such a model to work, global initial data were necessary. It became clear that errors arising from the neglect of the tropics or the southern hemisphere (SH) led to a rapid degradation of forecast skill in the NH. Thus, for very practical reasons, both an improved tropical database and a keener understanding of tropical phenomenology were required. Prior to the 1960s, atmospheric data were especially scarce in the tropics. Nevertheless, some key climatological features were well known. Fields of sea-surface temperature (SST), obtained mostly along shipping routes, varied slowly in space and time compared to higher latitudes. Furthermore, temperature and pressure in both the atmosphere and the ocean possessed smaller horizontal gradients than at higher latitudes. The trade wind

Dynamics of the Tropical Atmosphere and Oceans, First Edition. Peter J. Webster. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

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1 Climatology of the Tropical Atmosphere and Upper Ocean

regimes of both hemispheres were known to be steadier than any other wind system on the planet. Monsoons, and their distinct seasonality, had been described, at least grossly, for over 300 years and were thought to be explained largely in terms of differential heating of the ocean and the land. Near the equator, the equatorial atmosphere was characterized by a state of relative inactivity and stagnation called, initially, the doldrums.II Within these regions, there existed considerable “unsteadiness” with squalls and propagating convective disturbances. Also, it was realized that the tropics were the source region of the most violent storms on the planet, tropical cyclones, that impacted the tropics, subtropics, and extratropics alike. Interestingly, Halley (1686) had identified many of these large-scale characteristics of the tropics and produced an atlas of surface winds based on ships’ logs that had been painstakingly collected over the years. These observations were of such widespread interest and economic value that the compilations were published in 1690 by the Royal Society. These climatologies were being used by mariners setting out on commercial ventures around the world. Scientists attempted to explain the physics behind the patterns of observed wind and weather.1 The numerical weather models being developed in the 1960s required initial data that were far more detailed than a mere climatological description of the tropics. Driven by this growing need for data, the World Meteorological Organization (WMO) and the International Council of Scientific Unions (ICSU) in 1967 launched a 15-year project called the Global Atmospheric Research Program (GARP). Besides improving the atmospheric data stream, it organized a system for global data collection and spawned several important field experiments including GARP Atlantic Tropical Experiment (GATE, boreal summer 1974: Houze and Betts (1981)). GATE is still the largest and most complex international scientific experiment ever undertaken, with 10 nations – Brazil, Canada, France, Federal Republic of Germany, German Democratic Republic, Mexico, Netherlands, USA, UK, and USSR – working in close collaboration contributed 39 specially equipped ships, 13 large research aircraft, several meteorological satellites, and some 5000 personnel to an intensive three-month study of weather systems in the tropical eastern Atlantic Ocean. A further 50 countries in Africa and South America participated by making special land surface and upper-air observations. A subsequent major international field program, under the auspices of GARP, gathered data on the Asian monsoon. This was the Monsoon Experiment 1 These early efforts to explain the trade winds and monsoons are discussed extensively in Chapters 10 and 14.

(MONEX), with a field phase occurring in 1978–1979. There were a number of subprograms contained within MONEX: the summer and winter monsoon experiments (Summer MONEX and Winter MONEX). There were three major emphases: an Arabian Sea component (June–July 1979), the Arabian desert campaign and the Bay of Bengal component (BOB: August 1979), and Winter MONEX (December–February (DJF) 1978–1979). Each phase of MONEX also had an oceanographic component. By the 1970s, a number of influential studies had highlighted the importance of the tropics in global weather and climate. At the same time, research began to unravel some of the basic physics of low-latitude meteorology and oceanography. For example: (i) Matsuno (1966)III showed that there were classes or families of atmospheric planetary scale waves that were trapped about the equator and propagated both with and against the prevailing winds of the tropics. Here “trapped” has a special connotation. Trapped waves have a peculiar structure, essentially decreasing in amplitude away from the equator. Matsuno’s waves turned out to be subsets of global modes that, depending on the parameter range chosen, become increasingly trapped about the equator, depending on the equivalent depth (see Chapter 6) of the fluid and the Doppler-shifted frequency of the mode. These waves were quickly connected to the propagating convection across the tropics. Further, it was shown that similar families of wind-driven planetary scale waves existed in the ocean, also with maximum variance at low latitudes. We will explore the nature of these waves in some detail, including the impact of a varying basic state on their structure, in Chapters 6 and 7. (ii) In the 1920s, Sir Gilbert WalkerIV had shown that there was a planetary scale 2–5-year oscillation in surface pressure across the tropics between the eastern and western hemispheres.2 He referred to this mode as the “Southern Oscillation” or SO. However, its physical nature remained unexplained, and largely unexplored, for the next 40 years. Troup (1965) and Bjerknes (1969)V had the inspiration to suggest that Walker’s oscillation had roots in both the atmosphere and the ocean. In essence, these studies enabled the conjoining of the low-frequency variability of the atmosphere and the ocean by explaining the basic physics of a phenomenon that has been shown to alter the weather and climate of vast swathes of the planet. The oceanic part of the phenomenon was El Niño and the coupled phenomena became known as El 2 Walker (1924a,b, 1928).

1.1 The Growth of Tropical Meteorology

Niño–Southern Oscillation (ENSO). Thus emerged the concept of coupled ocean–atmospheric modes where the scale, amplitude, and period of the phenomena are determined jointly by both the dynamics of the atmosphere and the ocean through a series of feedbacks. The need for data to clarify joint connections between the atmosphere and ocean led to the multi-year Tropical Ocean–Global Atmosphere (TOGA) and the TOGA Coupled Ocean–Atmosphere Response Experiment (TOGA COARE). (iii) A third major development came with the observation that the tropics contain distinctly unique subseasonal phenomena not seen in the extratropics. Madden and Julian (1971), against the challenges of scare data, identified a large-scale eastward propagating entity, evident in both pressure fields and convection, appearing to form in the Indian Ocean and to propagate eastward across the equatorial ocean. Now named the MJO,3 this phenomenon possessed strong variance in the 30–60 day period band. Like ENSO, it too seemed to have strong teleconnections influencing remote parts of the tropics and subtropics. Recently, Li et al. (2018) reported on an earlier systematic study by Xie et al. (1963), where radiosonde stations in the warm pool region of the Pacific Ocean were analyzed. A robust signal was uncovered with a 40–50 day oscillation. The oscillation commenced with an acceleration of low-level westerlies over South India and South Asia, followed by an eastern propagation of low-frequency wave-like phenomena. Xie et al. noted a strong association of typhoon genesis and that this newly discovered phenomena may have predictive utility. This early work complements for a smaller region and over a more limited time span the more global results of Madden and Julian (1971). We also believe that it is important to acknowledge what Li et al. (2018) refer to as “hidden gems” that exist in journals that, hitherto, were not globally available. Many field campaigns were designed to clarify these new ideas. The TOGA program gathered simultaneous data from the atmosphere and the ocean to test Bjerknes’s hypothesis with the hope that it would lead to the prediction of the ENSO phenomena. The initial field experiment has grown into the Global Tropical Monitoring Array,VI of which there are three components: Tropical Atmosphere Ocean (TAO) array in the equatorial Pacific Ocean, Research Moored Array for African–Asian–Australia Monsoon Analysis and Prediction (RAMA) in the Indian Ocean, and Prediction 3 Madden and Julian (1971, 1972).

and Research Array in the Atlantic (PIRATA) in the Atlantic. Further field campaigns were designed specifically to understand the role of moist atmospheric processes in the tropics. Prior to GATE, early thinking centered around the vision that tropical convection was made up almost exclusively of convective towers with an incidental stratiform component. Following GATE, it was realized that the stratiform layer extended some hundreds of kilometers out from the convective clouds and accounted for about 50% of oceanic precipitation.4 However, more data of the coupling of dynamics and moist convection were required to understand the nature of the connections. This need led to additional field campaigns such as the Equatorial Mesoscale Experiment (EMEX),5 which was conducted in Northern Australia during January–February 1987 together with the Australian Monsoon Experiment (AMEX).6 The South China Sea Monsoon Experiment (SCSMEX)7 was carried out in the May–August period in 1988. The TOGA Coupled Ocean–Atmosphere Response Experiment (TOGA COARE), perhaps the largest international field program since GATE, was conducted in the western equatorial Pacific. The Intensive Observing Period (IOP) occurred from 1 November through 28 February 1993. During these four months, nearly 1200 people from more than 20 nations conducted more than 700 days of ship operations, released nearly 12 000 rawinsondes, completed 125 aircraft flights, and maintained continuous operation of 30 moored instrument systems. Details of the experimental plans can be found in Webster and Lukas (1992). A central scientific objective was the detailed determination ocean–atmosphere fluxes during disturbed and undisturbed conditions over the warm pool. A preliminary summary of the results of the experiment appears in Godfrey et al. (1998). A follow-up program (Joint Air–Sea Monsoon Experiment: JASMINE8 ) took place in the BoB in the summer of 1998 to seek similarities and differences between the Pacific and Indian Ocean warm pools. Later, the “Dynamics of the MJO” experiment (DYNAMO) was designed and implemented in the equatorial Indian Ocean and Indonesian archipelago during late 2011–early 2012.9 During the last few decades, the tropical database has improved immensely. First, a multitude of satellites, carrying diverse instrumentation, provide observations at an unprecedented frequency and resolution. Now, 4 5 6 7 8 9

E.g. Schumacher and Houze (2003). Webster and Houze (1991). Holland et al. (1986). Lau et al. (2000). Webster et al. (2002a, b, c). Gottschalck et al. (2013).

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1 Climatology of the Tropical Atmosphere and Upper Ocean

the total tropical data set consists of an amalgam of satellite data, together with observations from more traditional sources (e.g. surface meteorological stations, upper air soundings, commercial aircraft observations, and moored and floating and profiling ocean buoys). However, the resultant body of data is inhomogeneous in space and time and is, thus, difficult to ingest and analyze. The mix of data led to the creation of the “reanalysis system” aimed at rendering the vast fields of data into systematic gridded data. This is accomplished by using a weather prediction or a climate model as a data assimilator. Every 6–12 hours spatially and temporally non-homogeneous data is ingested into a model that performs spatial and temporal averaging. Out of this system comes a gridded data stream advancing through the years that is dynamically and thermodynamically consistent. Such data sets are routinely produced by a number of international agencies. The initial concept of reanalysis was created by Bengtsson and Shukla (1988). A comprehensive overview of reanalysis products can be found in the University Corporation of Atmospheric Research (UCAR)/ National Center for Atmospheric Research (NCAR) Climate Data Guide located at https://climatedataguide .ucar.edu/climate-data/atmospheric-reanalysis-overviewcomparison-tables. The original NCAR/National Center for Environmental Prediction (NCEP) data set was described in Kalnay et al. (1996). The ERA-40 reanalysis data set is described in Uppala et al. (2005). In the early days of numerical weather prediction, there was an expectation that if the initial state of the atmosphere and the ocean was better described, and if the forecast model was increasingly representative of the physical system, that the quality of the forecast would improve. This is true in one respect as an egregiously poor initial data or model will produce an egregiously bad forecast. Yet, the discovery of nonlinear error growth (Lorenz 1969) changed the philosophy of numerical weather and climate prediction irrevocably. Rather than seeking a deterministic outcome (i.e. the weather will be x at day y) one seeks a probabilistic forecast from an ensemble of forecasts (the weather at a particular location will x at day y but at a certain probability: e.g. Leutbecher and Palmer 2008). The European Centre for Medium Range Weather Forecasts (ECMWF) conducts forecasts with time scales of weeks some 100 times each day with slightly perturbed initial conditions. In essence, this form of forecasting places an even greater premium on initial data. The poor documentation of a remote system in the tropics, for example, can introduce errors and very rapidly influence the probability distribution of weather and climate globally. Probabilistic forecasts allow reasoned decision making (e.g. Palmer 2002VII ) through the simple relationship

that risk of an event occurring (R) is given by the product of the probability of the occurrence (P) of the event times the potential cost of the event (C). That is, R = P × C. Such forecasts allow a user to decide whether or not to take action, whether it is to determine the risk of a cholera outbreak (Thompson et al. 2006) or the possibility of a major broad-scale flooding (Webster et al. 2010; Hopson and Webster 2010). All of these factors place additional demand on describing and understanding tropical phenomena. This chapter commences the study of tropical phenomena with a description of low-latitude climatology and the associated variability using the reanalysis database. In subsequent chapters we will attempt to understand these phenomena from a fundamental dynamical and thermodynamical perspective.

1.2 Seasonal Characteristics To analyze the zonally axisymmetric nature of the tropical climate it is convenient to define zonal and time averages of an arbitrary variable 𝛼 as 2𝜋

[𝛼(𝜑, z, t)] = 𝛼(𝜆, 𝜑, z) =

𝛼(𝜆, 𝜑, z, t) d𝜆 and

∫0 t2

∫t1

𝛼(𝜆, 𝜑, z, t) dt

(1.1a)

where 𝜆, 𝜑, z, and t are longitude, latitude, height, and time, respectively. Collectively, we can write: t2

[𝛼(𝜑, z)] =

∫t1 ∫0

2𝜋

𝛼(𝜆, 𝜑, z, t) d𝜆 dt

(1.1b)

to define a quantity averaged both in longitude and time. 30 TSST (φ) 25 20 °C

4

15

DJF JJA Annual

10 5

40°S 30°S 20°S 10°S 0 10°N 20°N 30°N 40°N latitude

Figure 1.1 Latitudinal distribution of the zonally averaged sea-surface temperature (SST: [T SST (𝜑)]) for the two solstitial seasons (DJF and JJA) and the annual average (Annual). In all seasons and in the annual mean, maximum SST occurs in the summer hemisphere and not at the equator. Units: ∘ C. Source: Data from NCEP reanalysis.

1.2 Seasonal Characteristics

Temperature and specific humidity 100 200

p (hPa)

Figure 1.2 Zonally averaged vertical atmospheric temperature [T(𝜑, z)] (∘ C: color bar) and specific humidity [qv (𝜑, z)] (10−3 kg kg−1 contours) for (a) JJA and (b) DJF. Isotherms 0). Also, maximum winds in the winter hemisphere occur near where 𝜕[T]∕𝜕𝜑 is a maximum. The magnitude of the zonal mean meridional wind is far weaker than the zonal wind, especially outside the tropics. During JJA, the low-latitude surface meridional wind is northward crossing the equator, with a southward return flow in the upper troposphere. During DJF, the circulation is reversed. The meridional flow is also confined vertically with maximum values in the lower and upper troposphere. All these features are part of the zonal mean Hadley Circulation, a direct circulation raising warmer air and subsiding cooler air in the subtropics. The zonally averaged vertical velocity is also plotted in the bottom panels of Figures 1.4d. The strongest fields occur in the two solsticial seasons11 between the lower level convergence of the meridional wind and the upper level divergence. 1.2.2

Spatial Variability in the Tropics

Whereas the latitudinal structure offers only weak latitudinal gradients at low latitudes, much richer fields occur in longitude. 1.2.2.1

Surface Temperature

The distribution of surface temperature, T s (𝜑, 𝜆), between 45∘ N and 45∘ S for all four seasons is displayed in Figure 1.5 over both the land and ocean. The heavy black contour over the oceans encloses SST ≥28 ∘ C an area often called the ocean warm pool (OWP).12 The black contour over the land encloses a mean temperature ≥36 ∘ C. The longitudinal gradients of SST in the tropical ocean basins are much weaker than at higher latitudes. At all times of the year, the warmest SSTs reside in the western parts of the Pacific and Atlantic basins. The gradients are reversed in the Indian Ocean. The combination of the warm water of the Pacific and Indian 11 The word solstice comes from the Latin for sun sol and stil or stare or to stand reflecting the most poleward apparent migration of the sun. Equinox is also from the Latin indicating equal night nox and day. 12 The 28 ∘ C delimiter of the warm pool is somewhat arbitrary.

Oceans form the conglomerate “Indo-Pacific warm pool” that straddles the Indonesian Archipelago. There is a seasonal shift of the warm pool moving slightly about the equator into the summer hemisphere roughly following the insolation maximum. During the boreal spring (MAM), the Indian Ocean SST possesses the warmest ocean temperatures on the planet. At this time, the near-equatorial zonal surface pressure difference across the Indian Ocean is near zero compared to about 2–3 hPa during the other seasons. Changes have occurred in the equatorial Pacific Ocean as well. The longitudinal pressure difference along the equator is 8 mm day− 1, corresponding to an annual rate of just under 3 m. There are some persistent features that appear throughout the year. For example, there is a broad maximum oriented from northwest to southeast across the South Pacific (the South Pacific Convergence Zone: SPCZ) and, although

30° 28°

40°

50°

36°

of lesser intensity, a similarly oriented band extending across the South Atlantic from Brazil. The latter band is referred to as the South Atlantic Convergence Zone (SACZ). Satellite data available in the early 1970s allowed these bands to be identified for the first time as coherent climatological features that also migrated eastward during an El Niño.14 Another prominent feature is the narrow band of rainfall extending across the entire Pacific just north of the equator, often referred to as the Intertropical Convergence Zone (ITCZ). A similar feature, which also remains within the NH throughout the year, occurs in the tropical Atlantic. 14 E.g. Streten (1973).

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Figure 1.6 Spatial distributions of the precipitation rate, P(𝜆, 𝜑), for (a) JJA, (b) SON, (c) DJF, and (d) MAM fields (mm/day). Blue contour encloses precipitation rate >8 mm day−1 . Dashed line represents the equator. Source: Data sets from the NOAA Climate Prediction Center MORPHing technique (CMORPH, Joyce et al. 2004).

Seasonal Precipitation Rate (mm/day)

(a) JJA 40°N 20°N EQU 20°S 40°S

0° (b) SON

60°E

120°E

180°E

120°W

60°W

0°

0° (c) DJF

60°E

120°E

180°E

120°W

60°W

0°

0° (d) MAM

60°E

120°E

180°E

120°W

60°W

0°

0°

60°E

120°E

180°E

120°W

60°W

0°

40°N 20°N EQU 20°S 40°S 40°N 20°N EQU 20°S 40°S

40°N 20°N EQU 20°S 40°S

2

4

6

8

10

12 14 mm/day

16

18

There are several interesting seasonal rainfall distributions: 1. Boreal summer JJA (Figure 1.6a): The NH rainfall rates are at their most intense. There are high rainfall rates in the warm pool region of the western Pacific but the largest occurs in the Northern BoB (16–18 mm day−1 ), totaling nearly 2 m in JJA. Whereas this center of precipitation is associated with the South Asian Monsoon, the maximum rainfall is not over land; in fact, it decreases in intensity northwest along the Ganges Valley. Heavy rainfall

20

30

40

can be found in Southeast Asia coastal regions but with decreasing intensity through Northern China. A secondary maximum occurs to the west of Sumatra in the near-equatorial SH. There is also substantial rainfall across Central and West Africa, the latter associated with the West African Monsoon when the Pacific and the Atlantic ITCZ are at their maximum intensity. In the eastern Pacific the ITCZ is located at its most poleward position. 2. Boreal autumn SON (Figure 1.6b): Rainfall rate maxima can be found in equatorial Africa and South America associated with the first of the two

1.2 Seasonal Characteristics

equinoctial rainfall periods. The Pacific Ocean ITCZ remains strong in the NH, similar to the location of the Atlantic ITCZ. These NH features are sufficient to explain the location of the zonally averaged maximum apparent in Figure 1.3. During SON, especially at the latter end of the period, the Indian Ocean coast of Africa receives substantial rainfall. These are referred to as the “short rains.” 3. Austral summer DJF (Figure 1.6c): The locus of maximum rainfall has moved south of the equator with extensive rainfall over southern Africa, North Australia, and Brazil. The SPCZ remains a prominent precipitation feature throughout the year but during DJF it attains its most intense state. The SACZ also reaches its annual intensity maximum. The eastern Pacific ITCZ, and its weaker Atlantic counterpart, still remain in the NH. 4. Austral autumn MAM (Figure 1.6d): The second equinoctial maximum occurs over equatorial Africa, Indonesia, and equatorial South America. Rainfall also occurs along the Indian Ocean coast of equatorial East Africa. These are the “long rains.” The Pacific ITCZ still remains in the NH, although at its weakest state of the year but just strong enough to help explain the NH relative maximum in the MAM zonally averaged rainfall rate (Figure 1.3). There is an indication of a very weak ITCZ just south of the equator in the Eastern Pacific Ocean, suggesting a double ITCZ. For future reference, this weak band occurs at the time of not only the warmest eastern Pacific SSTs but also when the cross-equatorial SST gradient is smallest. Finally, it is worth reiterating the observation of near interhemispheric symmetry of the total rainfall both annually and seasonally appearing in Figure 1.3. Considering the very different geography of the two hemispheres and the range of types of rainfall that occur, it is a most surprising result. The rainfall results are reminiscent of the findings of Stephens et al. (2015), which showed that the difference between hemispheres of the annual integrated net radiation at the top of the atmosphere (TOA) is near zero. 1.2.2.3

Surface Pressure

The distributions of mean sea-level pressure (MSLP hPa: colored background) for each of the four seasons are plotted in Figure 1.7a–d. In the simplest sense, the tropics are a zone of low pressure between the subtropical high-pressure systems. The minimum pressure meanders through the summer tropics. This is often called the equatorial or near-equatorial trough, or, over South Asia, the monsoon trough. This low-pressure

band, though, is not always linked to precipitation. The surface trough passes through the great subtropical desert regions where at the time of minimum MSLP and maximum temperature there is little, if any, precipitation. The desert troughs are made up of thermally induced low-pressure systems. In summary, the near-equatorial oceanic trough is collocated with either a maximum in SST or surface land temperatures but not necessarily with maximum wind convergence or precipitation. The statistics of the relative locations of MSLP, SST, precipitation, and wind convergence are explored below. The bands of subtropical anticyclones, located at roughly 30∘ of latitude on either side of the equator, are quasi-permanent features. These high-pressure zones appear near the region of descent of the Hadley cells (Figure 1.4), altered somewhat by the distribution of land and ocean and time of year. During the boreal summer (Figure 1.7a), the location of the high-pressure centers is determined by the relatively colder oceanic temperatures and by the relative warmth of the landmasses. In fact, the entire Eurasian landmass is marked by anomalously low pressure, promoting onshore monsoon flow. This is referred to as the Asiatic Low. During the boreal autumn (Figure 1.7b), pressures tend to rise over Eurasia as insolation decreases. At the same time, surface pressures start to fall over equatorial Africa and South America, creating convergence and rainfall (Figure 1.7b). The surface pressure over North Australia has started to fall and reaches a minimum in the boreal winter (Figure 1.7c), promoting the north Australian monsoon. During the SON and DJF (Figure 1.7b and c), NH SSTs are generally warmer than the land surface temperature of the land. The strength of the ocean anticyclones decreases and shifts closer to or over land and may be replaced, as in the North Pacific, by a low-pressure system. We note, too, that the high-pressure system in the North Atlantic (often called the Bermuda High) shifts eastward to the vicinity of the Azores Islands during the boreal winter (Figure 1.7c), where it assumes a new seasonal name, the Azores High. The winter high-pressure system extending over the northeastern part of Eurasia (the Siberian High) becomes the strongest of all continental high-pressure systems and is the result of intense radiational cooling to space. The winds associated with the eastern branch of the Siberian High flow southward across the South China Sea to form the winter monsoon. In general, the locations of the SH subtropical high-pressure systems remain much the same throughout the year, varying little in longitude and latitude. The southeast trade winds on their northeast flanks of the

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Figure 1.7 (a) Mean sea level pressure (MSLP: shading relative to color bar: hPa) between 60∘ N and 40∘ S for (a) JJA, (b) SON, (c) DJF, and (d) MAM. Warm colors denote relative high pressure while cool colors denote relative low pressure. The vectors represent the 925 hPa wind field relative to the vector scale. Source: Data from NCEP reanalysis.

Mean sea-level pressure and surface winds 60°N

(a) JJA

45°N 30°N 15°N Equ 15°S 30°S 60°N

(b) SON

45°N 30°N 15°N Equ 15°S 30°S 60°N

(c) DJF

45°N 30°N 15°N Equ 15°S 30°S 60°N

(d) MAM

45°N 30°N 15°N Equ 15°S 30°S 0° 990

60°E 994

998

120°E

180°

120°W

60°W

0°

1002 1006 1010 1014 1020 1024 1028 1032 1034 1038 hPa

anticyclones continue throughout the year. The difference in the steadiness of the pressure regions between hemispheres is perhaps determined by the smaller percentage area of landmasses in the SH such that the positions of the high-pressure systems are determined mostly by the SST distribution.

10 m s–1

1.2.2.4

Wind Fields

The vectors in Figure 1.7 represent the climatological seasonal near-surface wind fields. Equatorward of the NH anticyclones are the northeast trades that converge toward the oceanic ITCZ. Similarly, equatorward of the SH high-pressure systems are the southeast trade winds.

1.2 Seasonal Characteristics

The trade winds15 are the steadiest surface winds on the planet and during their trek toward lower latitudes gather water vapor through evaporation, which will be eventually condensed through ascent nearer to the equator. In effect, the trade winds act as vast “solar collectors.” As the SST warms toward the equator, the saturated vapor pressure rises, allowing increasing water vapor transport as the trade winds converge. Equatorward of the trades are relatively calm regions known as the doldrums. The trades merge together into the equatorial trough region – a region marked by storminess and strong convective activity, but also by heat, humidity, and uncertain wind. Winds also converge into the tropical landmasses of the summer hemispheres, producing the monsoon circulations of South and East Asia, West Africa, North Australia, and South America. This convergence is clearly evident in JJA in the Indian Ocean, where a strong cross-equatorial gyre “feeds” the Asian monsoon. During DJF, similar cross-equatorial flows, although not as strong, occur in the Atlantic Ocean into South America and across Indonesia into North Australia. A more detailed view of the regional monsoons is presented below. The trade winds and monsoon flow are important components of the coupled ocean–atmosphere system. These tropical wind systems drive ocean currents that are also significant transporters of heat. Substantial spatial variability in winds also occurs in the upper troposphere. The 200 hPa horizontal winds are displayed in Figure 1.8a to d. The blue contour separates easterlies from westerlies, thus marking the U(𝜆, 𝜑) = 0 contour. Compared to the zonally averaged wind plotted in Figure 1.4 these fields are much more complex. There are a number of notable features. Westerly jetstreams (marked as “W”) dominate the subtropics and extratropics. During SON and DJF the jetstreams tend to move equatorward and poleward during MAM and JJA. Downstream of the jets are regions of westerlies that extend between the extratropics and the equatorial regions. These are referred to as “westerly” ducts (WDs). During JJA, a band of easterlies extends spasmodically around the planet. The strongest easterlies occur to the south of Asia. This is referred to as the monsoonal easterly jet (E), which is part of the monsoon gyre forced by the monsoonal heating over South Asia. The easterly jet discovered by 15 It is generally thought that the name “trade winds” had a commercial origin because of their use by early traders crossing the Atlantic Ocean for the Americas. However, Philander (1996) suggests that the name has a nautical origin reflecting the steadiness of the winds coming from the word “tread,” which refers to the steady path of a ship’s progress, probably emerging from the Middle English word “trade” meaning “path” or “track.”

Koteswaram (1958) is a quasi-geostrophic easterly wind maximum jet stream overlying southern Asia in the high troposphere (∼100 hPa) with a core near 15∘ N. Looking downstream along the jet, temperatures decrease from right to left across the current. During SON, the easterly jet decreases in magnitude and distinct zones of westerlies appear over the equatorial Pacific and Atlantic oceans. These regions of westerlies continue through DJF and MAM. The upper tropospheric equatorial westerlies are particularly strong over the central and eastern Pacific during the boreal winter. This is a region referred to as the WD, a zone where high-amplitude extratropical disturbances appear to propagate into the tropics.16 In addition, principally in the boreal summer, the westerly duct is replaced by the Tropical Upper-Tropospheric Trough (or TUTT), marked by “TT” in Figure 1.9. The term TUTT was introduced in Sadler (1976), a dynamic feature he hypothesized to be associated with the genesis of typhoons. Hanley et al. (2001) provide an extensive study of the general association of tropical cyclones and TUTTs. 1.2.2.5

Moisture Flux

The horizontal moisture transport in an atmospheric column across the tropics and subtropics is characterized by the vertically integrated moisture transport Bq , defined as: TOA

Bq (x, y) =

∫0

̃ (𝜑, 𝜆, z)dz qv (𝜑, 𝜆, z)V

(1.2)

̃ (𝜑, 𝜆, z) represent the distriwhere qv (𝜑, 𝜆, z) and V butions of the time-averaged specific humidity and horizontal velocity within a column, respectively. “TOA” refers to the “top of the atmosphere” or where p → 0. Bq is strongly weighted toward the lower troposphere, as specific humidity decreases rapidly with height, as clearly shown in Figure 1.2. Both the boreal summer and winter distributions of the moisture flux vector are depicted with colored shading denoting magnitude. A comparison with Figure 1.6 shows that the divergence of water vapor flux is associated with regions of (P–E) > 0 and convergence with (P–E) < 0, where P is precipitation and E is surface evaporation. The largest values of water vapor flux appear in the northern Indian Ocean near South Asia during the boreal summer. This arises from the accumulation of water vapor, collected by the southeast trades south of the equator and across the Arabian Sea and then condensing over India and the BoB. A divergent water vapor flux (P > E) exists to the east of Indonesia, feeding the warm pool precipitation maximum. In the Atlantic Ocean, the Be maximum, far 16 Webster and Holton (1982), Tomas and Webster (1994), and Tomas et al. (1999).

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Figure 1.8 Same as Figure 1.8 but for 200 hPa horizontal wind fields for (a) JJA, (b) SON, (c) DJF, and (d) MAM. Bold blue line separates easterlies from westerlies. “W”, “WD”, and “TT” denote westerly maxima, westerly ducts, and TUTTs, respectively. “E” refers to the location of the boreal summer easterly jet stream to the south of India. Magnitude of the wind field is color coded relative to the bar. Source: Data from NCEP reanalysis (Kalnay et al. 1996).

200 hPa wind field 60°N

(a) JJA

45°N

W

30°N 15°N Equ

TT

E

15°S W

30°S 60°N (b) SON 45°N W

30°N 15°N Equ

WD

15°S W

30°S 60°N

(c) DJF

45°N W

30°N

W

15°N Equ

WD

15°S 30°S 60°N (d) MAM 45°N W

30°N 15°N Equ

WD

15°S W

30°S 0° 0

60°E 10

120°E

180° 20

120°W 30

60°W 40

0° 50

m s–1

weaker in both the boreal summer and winter, appears to be associated with the precipitation maxima in the Carribean and nothern South America, respectively. 1.2.3

Variability Along the Equator

Here, we now examine the seasonal structure in the longitudinal-height plane between 5∘ N and 5∘ S.

1.2.3.1

Temperature and Moisture

Figure 1.10a and b plots the equatorial sections of T(𝜆, z) and qv (𝜆, z) for the boreal summer and winter, respectively. Both fields are remarkably flat. The “flatness” of the T(𝜆, z) field, and other thermodynamic fields, created considerable discussion in the 1960s and early 1970s.

1.2 Seasonal Characteristics

Vertically Integrated Moisture Flux Be (λ,φ)

103 kg m s–1

100

40°N

(a) JJA

0

10

100

20°N

0 30

100

300

100

10 0 10 0

EQU

100

100

20°S

100

100 100

10

0

40°S

100°W 40°N

50°W

0°

50°E

100°E

150°E 0 10

0

10 (b) DJF

100

20°N

100

300

10 0

EQU

100

100

100

100

20°S

160°W

10

100

100

0

100

40°S

100

100°W

50°W

0°

50°E

100°E

150°E

160°W

kg m s–1 0

100

200

300

400

500

600

Figure 1.9 Distribution of mean vertically integrated moisture transport, Bq (𝜆, 𝜑), from Eq. (1.2) for the period (a) June–September (JJA) and (b) December–February (DJF) relative to the vector key. Viewed in the context of moisture transport, the Asian–Australian monsoon system appears in both the boreal summer and the boreal winter as strong interhemispheric systems with moisture sources clearly defined in the trade wind regimes of the winter hemisphere. The African summer and winter monsoons are less clearly defined in terms of moisture transport and are similar in magnitude to the North Australian summer monsoon. Weak moisture fluxes into northwest Africa may be seen, for example, but the region is dominated by strong westward moisture fluxes associated with the trade wind across the Atlantic and into the Americas. Source: Adapted from Webster and Fasullo (2002).

Figure 1.10 does offer a surprise, though. The specific humidity fields have a much more longitudinal structure than the temperature fields. Gradients of specific humidity can only be explained if there are dynamic circulations in the longitude–height plane. In fact, the maxima in qv (𝜆, z) correspond to the ascending regions of the Walker Circulations and the minima to regions of descent. 1.2.3.2

Wind Fields

The structure of the zonal and meridional wind fields along the equator (Figure 1.11a and b) is far more complex than the corresponding temperature fields of Figure 1.4. In particular, they do not resemble the zonally averaged wind fields appearing in Figure 1.4 at all. For example, Figure 1.4a(i) shows that the zonally averaged JJA upper-tropospheric zonal wind field is easterly with magnitudes of 5–10 m s−1 . The corresponding fields in DJF have magnitudes 10 m s−1 . The zonal average of these winds results in weak upper tropospheric easterlies (Figure 1.4c). During both seasons the lower tropospheric zonal winds are out of phase with those in the upper troposphere. During both JJA and DJF westerlies extend through the lower and middle troposphere of the eastern hemisphere, reversing in sign in the western hemisphere. Figure 1.11b indicates that, generally, the longitudinal structure of meridional winds (V (𝜆, z)) follows the mean distribution portrayed in Figure 1.4a and c. During the boreal summer, the meridional wind component is toward the south in the upper troposphere. In the

15

1 Climatology of the Tropical Atmosphere and Upper Ocean

(a) JJA

Figure 1.10 Longitudinal-height section of temperature T(𝜆, p) (∘ C, color bar) and specific humidity qv (𝜆, p) (10−3 kg kg–1 contours) averaged between 5∘ S and 5∘ N for (a) JJA and (b) DJF. Near the surface, the isotherms tend to follow the SST closely. Higher in the troposphere, the longitudinal temperature gradient is almost flat whereas the specific humidity fields show greater longitudinal variation. Source: Data from NCEP reanalysis but we note that there is no moisture data available above 300 hPa.

Temperature and specific humidity (5°N-5°S)

100 200

p(hPa)

300 400 500 600 700 800 900 1000

0

60E

120E

180 longitude

120W

90W

0

60E

120E

180

120W

90W

0

(b) DJF 100 200 300 p(hPa)

16

400 500 600 700 800 900 1000 0

T(oC) –80 –70 –60 –50 –40 –30 –20 –10

0

10

boreal winter there is a distinct reversal in sign. During both seasons, the upper tropospheric meridional winds are out-of-phase with the lower troposphere. Yet there is still considerable variability in longitude to suggest that the mean zonally averaged meridional cells are a composite of a number of smaller scale meridional cells. These cells may be associated with equatorial orographic features. For example, maxima (northward in JJA and southward in DJF) are collocated with the East African Highlands 40∘ E–60∘ E, the islands of the Indonesian Archipelago 90∘ E–130∘ E, and the Andes 120∘ W.

1.3 Macro-Scale Circulations The circulations of the tropical atmosphere and ocean are characterized by a wide range of spatial scales possessing different periods of variability. Here we describe

20

30

seasonal characteristics of the largest of these phenomena: the Hadley, the Walker, and the monsoon circulations, as well as major climatological features of the tropical oceans. 1.3.1

Hadley’s Circulation

A meteorological cause celebre during the seventeenth and eighteenth centuries was how to explain why the trade winds moved from east to west, opposite to the rotation of the planet. This problem occupied some of the great minds of the time including Johannes Kepler and Galileo Galilei. Halley (1686) argued that the apparent westward movement of the solar heating, as Earth rotated about its axis, created a westward moving heat source that induced buoyancy and rising motion at the location of the greatest solar heating. The apparent westward movement of the differential buoyancy was thought to produce westward winds as the sun moved to

1.3 Macro-Scale Circulations

(b) Meridional wind component: V(λ, z), 5°N-5°S (i) JJA

(a) Zonal wind component: U(λ, z), 5°N-5°S (i) JJA 200 pressure (hPa)

pressure (hPa)

200 400 600 800 1000

0° 60°E (ii) DJF

120°E

180°

120°W

60°W

0°

400 600 800 1000

120°E

0°

120°E

180°

120°W

60°W

180° 120°W longitude

60°W

0°

200 pressure (hPa)

pressure (hPa)

200

0° 60°E (ii) DJF

400 600 800

400 600 800

1000

1000 0°

60°E

120°E

180° 120°W longitude

60°W

0°

60°E

m s–1 –12 –10 –8 –6 –4 –2 0

2

4

6

0° m s–1

–2

8 10

–1

0

1

2

3

Figure 1.11 (a) Cross-sections of the zonal wind component, U(𝜆, p), and (b) the meridional wind component, V (𝜆, p), along the equator for (i) JJA and (ii) DJF relative to the color bar (m s–1 ). The black contour denotes the location of the zero magnitude. Note, especially, the distinct changes in the sign of the zonal wind component throughout the troposphere, suggesting vastly more character than the weak easterlies of the zonal averages (Figure 1.4). Source: Data from NCEP reanalysis (Kalnay et al. 1996).

the west. Yet Halley was apparently unaware of angular momentum conservation and it was left to Hadley (1735) to explain its role in producing the trade wind regimes. Although his theory went almost unnoticed for over 100 years, Hadley had produced the first viable explanation of the westward nature of the trade winds. Hadley argued that the observed surface trade winds were part of the lower branch of an axially symmetric convective cell driven by the latitudinal heating gradients. This idea was quite revolutionary at the time, especially since there were no upper atmospheric observations to support the hypothesis. In acknowledgment of this seminal work, the thermally driven large-scale direct circulation that extends roughly between the Tropics of Cancer and Capricorn, with rising air in the vicinity of the equator and subsidence in the subtropics, is referred to as the “Hadley Circulation.” These distributions of vertical velocity appear in Figure 1.4. Furthermore, Figure 1.3a and c indicate that the ascending regions of the Hadley Circulations are collocated throughout the year, in a general sense, with a maximum zonally averaged precipitation rate. Likewise, the dry subtropics are collocated with the subsiding regions

of the cell. The extratropical rainfall winter maxima occurring further poleward, on the other hand, are related to the position of the extratropical jet streams and their inherent instabilities. Broadly, the Hadley Circulation has determined where human civilization has developed and where it has flourished through its influence on the general locations of precipitation. Temporal variations in the location and strength of the Hadley Circulation are thought to be responsible for the demise of early civilizations in North Africa, Mesopotamia, and North America.17 Further, on much shorter time scales, the trade winds associated with the lower tropospheric leg of the Hadley Circulation are intimately linked to the interannual ENSO phenomenon that imparts drought or flood to different parts of the globe. 1.3.2

Walker’s Circulation

While the Hadley Circulation is driven by meridional heating gradients, longitudinal large-scale zonal 17 E.g. Weiss and Bradley (2001).

17

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1 Climatology of the Tropical Atmosphere and Upper Ocean

overturning circulations in the equatorial regions are perhaps driven by their zonal equivalents. Bjerknes (1969) introduced the term “Walker Circulation” to describe the Pacific equatorial longitudinal circulation. In the early 1970s, T. N. Krishnamurti18,IX compiled a climatology of what he termed “tropical east–west circulations.” Within his climatology was the Bjerknes Walker Circulation, in addition to similar circulations in the Atlantic and Indian Oceans. The strongest of all circulations was centered over the South Asian region. Noting that the Asian Monsoon contributed a large amount of the power to the zonally symmetric Hadley cell, he referred to this prominent regional meridional circulation in South Asia as the “Hadley Circulation.” Bjerknes (1966) argued that equatorial SST gradients (warm in the western Pacific, cooler in the east) drive the Walker Circulation through ocean–atmosphere coupling. To a large degree he was correct. However, it is the differential heating between columns in the western and eastern oceans that is the most important driver. Deeper convection produces the primary gradient of condensational heating between the warm and colder columns and subsiding air around the large-scale convection. In turn, the cloudiness associated with deep convection reduces the loss of radiation to space whereas in the cooler clear regions of the tropics, long-wave radiation is lost more freely to space. The radiative heating gradient is about a factor of three smaller than the condensational heating gradient but the gradients are of the same sign. Similar total columnar heating arguments also apply to the meridional Hadley Circulation. Figure 1.12a describes the vertical sections along the equator as vectors constructed from the divergent part of the zonal velocity component, U(x), and the vertical velocity in pressure coordinates wp (i.e., dp/dt) for the June–September (JJAS) and December–February (DJF) of 1984/1985. This was a moderately weak La Niña year and, as such, is fairly representative of non-El Niño periods. The scale of the zonal velocity is 50 times the scale of the vertical velocity. The distributions of SST (∘ C) and out-going long wave radiation (OLR W m2 ) are shown at the base of the panels, reflecting the broad collocations discussed in the previous paragraph. Figure 1.12b shows the same section but for the DJF El Niño of 1986–1987. OLR (W m−2 ), measured from a satellite, is a quantity often used to infer precipitation rate. Minima in OLR indicate that OLR is radiated from the tops of deep convective clouds and thus, at least in the tropics, is associated with precipitation. Larger values indicate that the OLR comes from closer to the surface and hence there is a lack of deep convective clouds. The 18 Krishnamurti (1971).

basic physics comes from the Stefan–Boltzman law, F(W m−2 ) = 𝜀m 𝜎 SB T 4 , where T is the temperature of the emitting body and 𝜀m is its emissivity and 𝜎 SB is the Stefan–Boltzman constant (= 5.67 × 10−8 W m–2 K–4 ) . For a black body emitter, emissivity is set at unity. More exacting and direct satellite estimates of precipitation have been developed but OLR is especially useful as the data record goes back to the start of the satellite era. In both solstice seasons, DJF and JJA, maximum ascent occurs over the Indo-Pacific warm pool with a strong descent in the Eastern Pacific. However, there is also a reverse Indian Ocean component with a common ascending region over Indonesia but with descent over East Africa and the Middle East. Smaller-scale circulations do exist (e.g. over the Atlantic–South American sector) but these lesser circulations are overshadowed by the Pacific–Indian Ocean circulation. From the SST diagrams at the base of Figure 1.12a(i) and b(i) and the MSLP distributions shown in Figure 1.7a, it is seen that the rising branch of the Walker circulation is associated with a high SST and a low MSLP. Relatively cooler SSTs and higher MSLPs are collocated with descent. Hence, similar to the Hadley cell, the Walker circulation is thermally direct, with rising motion over the warm regions and subsidence over the colder regions. These regions are connected by easterly surface winds across the equatorial central Pacific with ascent over the western Pacific warm pool, a westerly return flow in the upper troposphere, and subsidence over the eastern Pacific region. Over the Indian Ocean, the circulations are opposite in sign to those over the Pacific but they are still direct. The solstitial east–west circulations along 25∘ N appear in the middle panels of Figure 1.12a and b. The two seasons are very different. In many respects, the summer profile is similar to the cross sections along the equator, except for enhanced ascent during JJA over South Asia, associated with the Asian Summer Monsoon. Note, too, the strong descent over North Africa and Southwest Asia. During the boreal winter, most of Asia is under the influence of strong subsidence. Ascent does occur along the East Asian coast. The lowest set of panels of Figure 1.12 a and b, show longitudinal cross sections along 90∘ E for JJA and DJF. Both the summer and winter monsoon circulations can be readily identified. Very strong rising motion occurs over South Asia and India during JJA coupled with strong descent over the Indian Ocean to the south of the equator. During DJF, on the other hand, ascent is located just to south of the equator. Descent occurs over South Asia although the strength of the overall winter monsoon circulation is much weaker than its summer component.

1.3 Macro-Scale Circulations

A

(i) Equ

C

100

300

300

500

500

700 850 1000

700 850 1000

60°E

D

120°E

E

200

180°E 120°W 60°W 0°

350

F

100

100

300

300

500

500

700 850 1000

700 850 1000

0°

60°E

G

100

(iii) 90oE

(a) JJA 250

100

0°

(ii) 25oN

B

220

120°E 180°E 120°W 60°W 0° H 190 I 360

A

(b) DJF 260 B

220

0°

60°E

D

60

0°

60°E

C

120°E 188°E 120°W 60°W

E

160

0°

F

120°E 180°E 120°W 60°W 200 I

0°

G 200 H 100

300

300

500

500

700 850 1000

700 850 1000

90°S 60°S

30°S

Equ

30°N

60°N 90°N

90°S 60°S

30°S

Equ

30°N

60°N 90°N

vector scale 4×10–4 hPa s–1

2 m s–1

Figure 1.12 Variation of flow fields made up of the zonal wind U d (𝜆, p) and the vertical velocity wp (𝜆, p) as a function of longitude and pressure for (a) June–August, 1985 and (b) January 1984–March, 1985; 1985 was a weak La Niña year. Panels (i) and (ii) show the 19–20 vectors in the equator and 25∘ N vertical planes, respectively. Panels (iii) show sections of the meridional wind Vd (𝜆, p) and the vertical velocity wp (𝜆, p) as a function of latitude along 90∘ E, providing a cross-section of the South Asian Monsoon. The scale at the upper right refers to the magnitude of the vector with the vertical scale of the vector about 1/50th the scale of the horizontal vector. The distribution of SST (red contours ∘ C) and outgoing longwave radiation (OLR: shaded contours) enclosing areas 40 mm day−1 . This is a region of moisture flux divergence where Bq (defined in Eq. (1.2)) drops to a minimum. During the austral summer, the surface

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1 Climatology of the Tropical Atmosphere and Upper Ocean

(b) Monsoon of the Americas (i) JJA

40°N 30°N 20°N 10°N 0° 10°S 20°N 30°N

icas. During JJA, the East Pacific ITCZ has intensified with maximum precipitation to the south of the Central American coast. Rainfall occurs along the coastal Sierra Madres Mountains and into the southwest of the United States, forming the North American Summer Monsoon. During the DJF, the ITCZ weakens but still remains north of the equator. Major precipitation is now located over Brazil and extends southward over South America. Precipitation is restrained to the east of the Andes. From Figure 1.9b, it appears that the South American rainfall maximum appears to be fed by a cross-equatorial flow similar to that occurring in the South Asian Monsoon.

(c) African Monsoon System

40°N 130°W

40°N

110°W

90°W

70°W

50°W

(ii) DJF

(i) JJA 30°N 20°N

30°N

10°N

20°N

0°

10°N

10°S 0°

20°S

10°S

30°S

20°N

30°W

30°N 130°W

1

2

4

110°W

90°W

70°W

50°W

6 8 10 12 14 16 18 20 30 40 mm/day 925 hPa wind (m s–1) 15

Figure 1.13 (Continued)

wind convergence (and moisture flux divergence) moves south of the equator. Figure 1.12a and b (panel iii) show cross-sections of the Asian–Australian Monsoon system along 90∘ E for both DJF and JJA. During the boreal summer, broad ascent occurs in the 15∘ N–30∘ N band, with strong subsidence over the SH subtropics. In the boreal winter, DJF, the ascending column moves to just south of the equator with subsidence over the South Asian sector. Monsoons of the Americas

Figure 1.13b describes the two solstitial precipitation patterns and low-level wind distributions over the Amer-

10°E

30°E

50°E

10°E

30°E

50°E

40°N (ii) DJF 30°N 20°N SL

10°N

GC

0° 10°S 20°S 30°S 30°W 1

1.3.3.2

10°W

2

10°W

4 6 8 10 12 14 16 18 20 30 40 mm/day 925 hPa wind (m s–1) 15

Figure 1.13 (Continued)

1.3 Macro-Scale Circulations

1.3.3.3

flow driven by the anticyclonic winds of the subtropical high-pressure systems. Two subtropical anticyclonic gyres are found in the NH and three in the SH. The flow in these three gyres is poleward on the western side of an ocean basin and includes currents such as the Gulf Stream and the Kuroshio in the NH and the Brazil, Agulhas, and East Australian Currents in the SH. These currents, flowing away from the equatorial regions toward the pole, are “warm” currents. Warm and cold currents are defined as being anomalously warm or cold relative to the latitudinal average SST in the particular ocean basin. Such currents advect warm or cool equatorial waters to higher latitudes, thus maintaining warmer climates in the higher latitude sectors of continents such as western Europe. An example of the moderation of a climate by a warm climate is London, U.K. The latitude of London, U.K. is 51.5∘ N or about the same latitude as Canada’s southern end of Hudson’s Bay. The mean winter temperature of London is about 5 ∘ C, some 30 ∘ C warmer than at Hudson’s Bay. The more temperate climate of the United Kingdom is attributed to the moderating impact of the Gulf Stream.

African Monsoon

The boreal summer West African monsoon precipitation extends from the eastern Atlantic Ocean, just to the north of the Gulf of Guinea, and into central North Africa. It may be noticed that the distribution of rainfall over West Africa is almost a mirror image of the North Australian Monsoon (Figure 1.13a(ii)); that is, rainfall decreases almost linearly poleward from the coast. The boreal winter rainfall is generally confined to Central Africa and to areas south of the equator. The Atlantic ITCZ, still located in the NH, also weakens in a fashion similar to its East Pacific counterpart. 1.3.4 Large-Scale Characteristics of Tropical Oceans Figure 1.14a shows the major annually averaged surface ocean currents. There are two major types of current: currents that flow generally in the direction of the wind and those that flow against the prevailing surface wind. The latter are termed “counter currents” and exist along the equator of the three tropical oceans in the opposite direction to the converging trade winds. The first class of currents, referred to as “subtropical gyres,” contain

(a) Ocean Currents and Continental Orography

(b) Indian Ocean Seasonal Currents (i) Surface currents NE (winter) monsoon

90°N

ash io

rre nt Cu ia al m So

Ku ro n

36°E

54°E

72°E

90°E

108°E

(ii) Surface currents SW (summer) monsoon 30°N

Circum-Antarctic

20°N

Orography (m)

0°

re

nt

10°N

0°E 180°W 12

Cu r

1 °W 20°W 60° W

180

MC

ia

90°S

An

n India

South Equatorial Current

20°S

al

b

tic Su

Antarc

South

gyre

South Equatorial

Current

Equatorial Counter Current

Equatorial Counter

Equatorial Counter South Equatorial

Circum-Antarctic r bpola c Su tarcti

0° 60 °E

60°S

Circum-Antarctic polar

North Equatorial

10°S

m

Braz

tlantic

South A

South Pacific

gyre North Equatorial

0°

SW mon soon cu rrent

So

gyre

il

Peru

gyre

30°S

Benguela

A

South Equatorial

EAH

th Sou atorial Equ

North Pacific

alia

Equatorial Counter

10°N

Oy

North Equatorial

o

North Equa toria Equat orial C l ounter

shi

HTP

Canary

a

North Equatorial

0°

North Atlantic Drift

gyre

rni

lifo

gyre

am

tre

lf S

Gu

20°N

Agulh as Mo zam biq ue

RSM

Ca

North Pacific

r No

ustr

r

Alaska

30°N

d

n st nla Ea ree G

Wes tA

Labrado

60°N

30°N

an gi we

10°S

0

1000

2000

3000

4000

5000

6000

7000

8000

South Equatorial Current

20°S 36°E

54°E

72°E

90°E

108°E

Figure 1.14 (a) Annually averaged major upper ocean currents. Cold and warm currents (see text for definition) are depicted as red and blue arrows, respectively. Major current systems are identified by name. Over the continent contours, the surface elevation is displayed relative to the color code. Note, especially, the major orographic features of the Himalayan–Tibetan Plateau (marked HTP), the East African Highlands (EAH), the Rockies–Sierra Madras complex (RSM), and the Andes (A) as well as the relative flatness of Australia and North Africa. (b) Currents in the Indian Ocean for the boreal winter during the winter northeast monsoon (panel i) and summer southwest monsoon (panel ii). Major currents are identified. Surface current schematics in the northern basin are adapted from Figure 11.3 of Tomczac and Godfrey (1994). Currents to the south of 20∘ S show little seasonal variability.

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1 Climatology of the Tropical Atmosphere and Upper Ocean

The eastern side of all subtropical gyres is characterized by equatorward flow. Most notable in the NH are the California and Canary currents, while in the SH there are the Humboldt Current (or Peru Current) off the western coast of South America and the Benguela Current to the west of South Africa. Besides advecting cold water from higher latitudes, these currents are cooled by wind-driven Ekman transports (see Section 3.2.4) that upwells nutrient-rich colder subsurface water to the surface. This process is responsible for the creation of productive fishing areas. In the Eastern Indian Ocean there is also a northward current; off the Australian coast there is a counter-current called the Leeuwin Current. It is strongest in the austral summer and strongly influences the climate of Western Australia. Evaporation from this warm current is responsible for a large proportion of the precipitation over southwest Australia. In the southern oceans, where there is no continental barrier and very strong westerly winds, the ocean currents are to the east. Collectively, these currents are referred to as the “West Wind Drift” or the “Circum-Antarctic Current.” Between the subtropical gyres there are currents that flow from east to west in all three tropical ocean basins. These are termed the North or South Equatorial currents depending on their location relative to the equator. These are directly wind-driven currents going in the direction of the prevailing easterly winds. However, between these two currents are the equatorial countercurrents flowing from west to east. Townsend Cromwell discovered the Pacific Equatorial Counter-current (Cromwell 1953).X The Indian Ocean Equatorial Counter-current flows only during the boreal winter and is a little south of the equator. Figure 1.14a refers to annual mean currents. Because of the extremely strong annual cycle of monsoonal winds over the Indian Ocean, the ocean currents also have a distinct seasonal variability. These are plotted separately for the boreal summer and winter in Figure 1.14b. In both seasons, there is a strong ocean gyre in the South Indian Ocean driven by the persistent southeast trade winds. In the western equatorial Indian Ocean, the Madagascar Current flows southward, merging with the Agulhas Current. However, the North Indian Ocean is strongly affected by the monsoon winds. During the boreal summer a second gyre extends from the African coast to Sumatra. Close to the coast is the shallow (150 m) and narrow (100–150 km) Somalia Current, forced by the atmospheric Somali Jet to reach wind speeds >20 m s–1 . These winds drive the ocean boundary Somalia Current along the coast of Somalia with speeds of 2–3.5 m s–1 when the boreal summer monsoon is fully established.

The upper tropical oceans are dominated by a warm and fresh layer that comes about by a combination of positive net radiational heating of the surface layer and a positive fresh water flux: the sum of fresh water input into the upper ocean from precipitation (P) less the amount of evaporation (E). In the subtropics, (P–E) is usually negative while in the deep tropics it is positive. A positive fresh water flux ((P–E) > 0), in addition to net radiational heating, tends to stabilize the upper ocean by decreasing the density of surface layers. The stability is important in maintaining the warm temperature of the upper tropical ocean by reducing the upward mixing of cold subthermocline water. In the subtropics, even though downwelling solar radiation is intense (less cloudiness in the subsident region of the large-scale meridional circulations), evaporation exceeds precipitation and the upper ocean is less stable, leading to a general subsidence or the subduction of subtropical water (Section 2.9.2). There we will argue that the warm fresh layer that dominates the tropics is a major factor in the stability of planetary climate.

1.4 A Myriad of Variability In preceding sections, we have found large spatial variability in the mean seasonal fields, far greater than was anticipated 50 years ago. It also turns out that the tropics possess transient modes at almost every time scale ranging from slow oscillations with periods of years to convective activity on the time scale of a day. Figure 1.15 provides an overview of the temporal variability of tropical OLR variance (units in (W m−2 )2 ) in four spectral bands: 2–10 days, 10–20 days, 20–60 days, and 1–10 years for the two solstitial seasons, DJF and JJA, and the equinoctial seasons, SON and MAM. OLR is a proxy measurement for intensity of convection, with deep convective clouds radiating to space with cold temperatures and thus low OLR. These bands represent “high-frequency weather,” “quasi-biweekly waves (QBWs),” “intraseasonal,” and “interannual” variance. An immediate observation from Figure 1.15 is that regions of maximum variance in each of the bands are often collocated. Given that the spatial distribution of the three higher frequency bands are quite similar, it would seem that the rainfall maxima of Figures 1.3, 1.6, and 1.13 receive considerable contributions from a wide range of frequencies; that is, there is “spectral overlap.” 1.4.1

High-Frequency Variability

The magnitude of the 2–10 day variance is about a factor of two larger than either the QBW or the intraseasonal variance, the latter two having a similar magnitude. In both DJF and JJA, synoptic variance extends

1.4 A Myriad of Variability

(a) OLR Variance JJA (W m–2)2 (i) 2 -10 day

(b) OLR Variance SON (W m–2)2 (i) 2 -10 day 40°N

20°N 0° 20°S 40°S

20°N 0° 20°S 40°S

(ii) 10 - 20 day

1000 800 600 400 200 0

(W m–2)2

40°N

(ii) 10 - 20 day

20°N 0° 20°S

20°N 0° 20°S

40°S

40°S

(iii) 20 - 60 day 40°N

20°N 0° 20°S 40°S

20°N 0° 20°S 40°S

(iv) 1 - 10 year

(iv) 1 - 10 year 40°N

20°N 0° 20°S

20°N 0° 20°S

40°S

300 250 200 150 100 50 0

40°S 120°E

180°

120°W

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Figure 1.15 OLR variance ((W m−2 )2 ) for (a) JJA, (b) SON, (c) DJF, and (d) MAM in four spectral bands: (i) “synoptic” (2–10 days), (ii) quasi-biweekly (QBW) (10–20 days), (iii) intraseasonal (20–60 days), and (iv) interannual (1–10 years). Variance magnitude is shown in bars to the right. Note the change of scale between frequency bands. Source: Following Ortega et al. (2017).

across large areas of the tropics although the variances most prominent in the monsoon regions and the ITCZs are over the Pacific and the Atlantic Oceans.

Prior to the satellite era, considerable evidence had emerged for the existence of propagating tropical disturbances. A number of early studies,19 using the special upper air data set created for the nuclear tests in the Pacific, found that near-equatorial waves propagated toward the west with roughly a 3000–4000 km horizontal wavelength and periods of about 3–5 days in both the Atlantic and Pacific. Figure 1.16 replicates the early analysis of satellite data by Chang (1970) in the form of a Hovmöller diagram,XI which is particularly useful for determining the progression of waves. This figure was constructed across the Pacific Ocean domain using successive daily shortwave reflectance measured from the TIROS V satellite for the 10∘ N–15∘ N latitude band.

The period of these waves appears to be 3–5 days and the propagation speed is about 10 m s–1 relative to the surface, or about 5 m s–1 to the west relative to the background basic wind. Figure 1.16b shows an enlargement of the period July 11–13, 1967, detailing the westward propagation of an organized convective structure. Westward propagating near-equatorial waves in the Atlantic develop in equatorial West Africa, prior to propagating into the central Atlantic and before decaying further to the west. One of the motivations for the GATE program was to gain understanding of the physics of the waves in the Atlantic easterlies. Not only did they represent a major form of near-equatorial variance but they would also occasionally develop into tropical cyclones. It was once thought that the waves in the easterlies circumnavigated the tropics westward from their genesis region over Africa.20 However, there appears to be only occasional evidence of waves propagating from the Atlantic basin into the Pacific.

19 Palmer (1952) and Riehl (1954).

20 Frank (1970).

1.4.1.1

Waves in the Easterlies

25

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1 Climatology of the Tropical Atmosphere and Upper Ocean

(c) OLR Variance DJF (Wm–2)2 (i) 2 - 10 day

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Figure 1.15 (Continued)

Figure 1.16a(i) suggests that a second Pacific maximum occurs near the date line and a third in the western Pacific near Indonesia. Subsequent studies21 have suggested that these waves in the easterlies develop in situ in both the Pacific and the Atlantic. 1.4.1.2

Tropical Cyclones and Monsoon Depressions

We use the generic term “tropical cyclone” to describe discrete and strong tropical vortices rather than the regional nomenclature such as “typhoons” in the West Pacific or “hurricanes” in the western hemisphere. Tropical cyclones are low-pressure anomalies on the scale of about 500–1000 km with cyclonically rotating winds and cloud bands. The system contains an intense inner vortex of extremely high winds and low pressure, and extends throughout the troposphere with anticyclonic outflow aloft. Tropical cyclones derive their energy from the condensation of water vapor gathered up in the subtropics and tropics rises, releasing this gathered latent heat through condensation. The heating, increasing toward the center of the disturbance, creates a “warm core” system; that is, the central part 21 E.g. Serra et al. (2008) and Toma and Webster (2010a, 2010b).

of the tropical storm is warmer than its surroundings. A consequence of this structure is that the strongest winds are near the surface22 and are, thus, particularly destructive to society. Figure 1.17a presents a plot of the global distribution of tropical cyclones between 1945 and 2007. The tracks are color coded to denote the intensity of the tropical cyclone ranging from weakest (cold colors) to most intense (warm colors). There are four major tropical cyclone “centers of action” in the NH: the North Atlantic, East Pacific, West Pacific, and the North Indian Oceans. In the first three regions, maximum activity occurs in June to October. In the North Indian Ocean there are two principal tropical cyclone seasons: prior to the South Asian Monsoon (May and June) and following the summer monsoon (October and November). The SH possesses two major regions: the 22 To first order, the increase or decrease of wind with height is determined by the “thermal wind equation.” This states that a geostrophic flow existing within an environment with a horizontal temperature gradient must change with height; that is, it will have vertical shear. The thermal wind equation is derived formally in Appendix A and also in Section 10.2.3.

1.4 A Myriad of Variability

150°E

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Figure 1.16 Examples of easterly waves: (a) Depiction of easterly waves in the Pacific Ocean in the 5∘ N–10∘ N band for the period July and August 14, 1967, using TIROS V visible imagery. Chang’s Hovmöller diagram depicts a westward migration of disturbances with wavelengths of 2500–4000 km propagating at speeds of order 8–10 m s−1 (yellow lines) with a period of about four days (vertical dashed yellow line). (b) Section between 0∘ N and 20∘ N for the three-day period July 10–12, 1967 in the red rectangle and yellow line in panel (a) depicting location of westward trajectory along 6∘ N–9∘ N. Source: Following Chang (1970).

27

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1 Climatology of the Tropical Atmosphere and Upper Ocean

July 10, 1967 20°N

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Figure 1.16 (Continued)

Southwest Pacific and the Southern Indian Ocean, with tropical cyclones forming generally in the austral summer. A secondary maximum occurs off the northwest Australian coast. The most frequent location of tropical cyclones is in the western Pacific Ocean followed by the eastern Pacific Ocean (Figure 1.17b). In all of the tropical cyclone regions, cyclogenesis takes place in tropical and subtropical latitudes, drifting first to the west and then curving to higher latitudes following an anticylonic trajectory. On reaching higher latitudes the warm core system undergoes an “extratropical transition,”23 developing a cold core structure and often intense winds and precipitation. Tropical cyclones thus add to the high-frequency variance appearing in Figure 1.16a, especially in the western regions of the ocean basins in the summer hemispheres as they recurve anticyclonically to higher latitudes. 23 Jones et al. (2003) provides an excellent review of the transition of tropical cyclones into extratropical storms.

During the wet phase of the monsoon cycle there is another major rain-bearing phenomenon. These are the monsoon depressions that form in the June–September period in the North Indian Ocean and over the South Equatorial Indian Ocean and Australia during December to February. There are about three monsoon depressions forming in the BoB each year that then migrate in a northwesterly direction up the Ganges Valley, often bringing copious periods of rainfall in northern India. The Indian Meteorological Department’s criterion for a monsoon depression is that the anomalous surface wind lies between 8.5 and 16 m s−1 . These disturbances occur during the established monsoon when very strong negative shear (westerly winds below easterly winds) exists over the North Indian Ocean. Such shear is not conducive to tropical cyclone development.24 Also, monsoon depressions are cold-cored and thus have weaker winds in 24 Gray (1968).

1.4 A Myriad of Variability

(a) IBTrACS tropical storm tracks 1979-2007 by intensity 80°N

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Figure 1.17 Global distribution of tropical cyclones for the period 1979–2007 based on “best-track” data, a subjectively smoothed representation of a tropical cyclone’s location and intensity over its lifetime. (a) All IBTrACS storm tracks colored by their category on the Saffir-Simpson Hurricane Scale shown in the inset top-left. IBTrACS (International Best Track Archive for Climate Stewardship) is a compilation of best track archives from a number of international organizations. There are six tropical cyclone regions: NI (North Indian Ocean), WP and EP (western and eastern Pacific), NA (North Atlantic), SI (South Indian Ocean), SP (South Pacific), and SA (South Atlantic) (b) Tracks and intensities of tropical cyclones 1851–2006. Frequency of the IBTrACS storm tropical cyclonic storms contoured at 2, 5, 10, 20, and 30 storms per decade. Source: From Knapp et al. (2010).

80°S 60°E

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the lower troposphere. Nonetheless, they are important rain-bearing phenomena accounting for a substantial fraction of the summer rainfall. Estimates of the contribution that monsoon depressions make to the total rainfall over Central India range from 50 to 70%.25 Monsoon depressions occur over North Australia as well, although their frequency is less than those over India (Figure 1.15). 1.4.1.3

The “Great Cloud Bands”

In both solsticial seasons, DJF and JJA, OLR minima extend from the western tropical oceans toward the southeast as cloud bands extend out of the warm pools toward higher latitudes. These cloud bands are especially evident in the Pacific and Atlantic oceans and are most intense in the austral summer. Respectively, these are known as the SPCZ and SACZ. Each has power in the 2–10, 10–20, and 20–60 day period bands indicating spectral overlap.

120°E

Mesoscale Convective Systems

Prior to GATE, it was thought that the majority of tropical rainfall came from deep cumulus that extended 25 Ramage (1971)

120°W

60°W

0°

throughout the troposphere. The anvil cloud was thought to be relatively non-active and just a plume of condensed moist air extending outwards from the convective turret. In fact, the area of cumulus cloud to stratiform cloud coverage was estimated to be about 5 : 1. Since GATE, with improved satellite products, estimates of the relative coverage has been reversed. It has been established that the majority of tropical rainfall is the result of the so-called mesoscale convective systems (MCSs). These are organized cloud systems with horizontal scales of 100–1000 km with a temporal scale of hours to one day. In contrast to the pre-GATE view of tropical convection MCSs contribute between 50 and 70% of total oceanic rainfall associated with a tropical disturbance. The percentage rainfall attributed to MCSs may increase to 80% over continental areas. 1.4.2

1.4.1.4

180° longitude

Subseasonal Variability

There is also strong variance displayed in Figure 1.15 in the 10–20 day period band. This is often termed the QBW band. Strong QBW variance may be seen in South and East Asia and the eastern Pacific/Rockies

29

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1 Climatology of the Tropical Atmosphere and Upper Ocean

(a) Original depiction of MJO

(b) Composite structure of MJO OLR (W m–2) and 850 hPa wind speed (m s–1)

EAST LONGITUDE WEST LONGITUDE 20° 60° 100° 140° 180° 140° 100° 60° 20° 15N

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Figure 1.18 Intraseasonal variability in the tropics. (a) Madden and Julian’s original depiction of variability along the equator. The convective element appears to form in the Indian Ocean and progresses eastward. Maximum convection, as well as the lowest surface pressure, occur in panel A. The amplitude of both convection and surface pressure anomaly decrease as the circulation moves eastward. Blue and orange shading have been added to emphasize low and high MSLP variations, respectively. Adapted from Madden and Julian (1972). (b) Composite of the MJO in the 25∘ N–15∘ S band as a function of longitude. Background colors indicate OLR anomalies relative to the lower scale. Contours indicate anomalous 850 hPa zonal velocity. Source: Courtesy of Dr. H.-M. Kim.

area during the boreal summer and over South America and North Australia in the boreal winter. Krishnamurti and Bhalme (1976) were among the first to identify South Asian monsoon QBW variability associated with the South Asian monsoon. The variance band appears to occur across the tropics, suggesting that it was a manifestation of a global instability, although the specific mechanism was not identified. During the latter period, similar power extends from southern Africa, across the Indian Ocean, and to the eastern South Pacific. Also, during DJF both the QBW and synoptic bands are prominent over South America and strong variance exists in the SPCZ throughout the year. We will return to the QBW during our discussion of higher frequency monsoon variability in Section 14.5.

Variance of a similar magnitude to the QBW occurs on the longer intraseasonal time scales. This band contains a variety of subseasonal phenomena referred to generally as the MJO in honor of Madden’s and Julian’s fundamental research. The phenomenon was discovered through a careful examination of the spectral decomposition of serial radiosonde data at Canton Island (3∘ S, 172∘ W). The results were catalogued in Madden and Julian (1971). A subsequent study (Madden and Julian 1994) determined that the oscillation propagated eastwards along the equator, extended throughout the troposphere as a divergent circulation (convergence in the lower troposphere and divergence in the upper troposphere), and was associated with strong convective activity. The phase speed of the propagation was about +5 m s–1 . Figure 1.18a and b describes the eastward

1.4 A Myriad of Variability

(a) Annual correlations of surface pressure with Darwin 40°N

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Figure 1.19 Interannual variability in the tropics: (a) Distribution of the correlation of Darwin surface pressure (location D) with surface pressure elsewhere. Both Darwin (D) and Tahiti (T) are very close to the nodes of the correlation pattern that define Walker’s (1923) Southern Oscillation Index. The four rectangles represent the regions where SST indices are calculated: Niño-1 + 2, 0∘ S–10∘ S, 90∘ W–80∘ W (black), Niño-3, 5∘ N–5∘ S, 170∘ W–120∘ W (black), Niño-3.4, 5∘ N–5∘ S, 170∘ W–120∘ W (blue), and Niño-4, 5∘ N–5∘ S, 160∘ E–120∘ W (red). (b) Mean monthly time series of the Walker’s Southern Oscillation Index (SOI: pressure difference between Tahiti and Darwin) from the late 1800s to the present. Negative values are associated with El Niño and positive with La Niña. (c) Spectra of the SOI showing maximum variance of the SOI between three and five years. Red dashed line indicates 95% statistical significance.

0 −1 −2

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propagation of the phenomena, both as originally deduced from in situ data and using more modern data. A number of studies have related MJO activity in other regions of the tropics and subtropics, such as an association with a clustering of tropical cyclones in the North Atlantic.26 1.4.3

Interannual Variability

The equatorial regions of the planet contain many modes of interannual variability that either appear to originate in the tropics or impact tropical circulations. 1.4.3.1

El Niño and the Southern Oscillation

Figure 1.15 shows that the OLR variance associated with interannual phenomena is about a factor of three smaller than synoptic variance and a further factor of two smaller than the QBW and MJO variance. Furthermore, variance during the boreal summer is a factor of two smaller than during DJF. In Section 1.1.2 we discussed the discovery of the SO a century ago by Sir Gilbert Walker. The Southern Oscillation Index (SOI) is defined as the anomalous pressure gradient between Tahiti and Darwin (see locations on Figure 1.19a). A correlation map of the SO (actually, the correlation of the time series of the Darwin surface pressure with time series of surface pressure elsewhere) is displayed in Figure 1.19a. Walker’s original choice of location was 26 E.g. Belanger et al. (2010).

not Darwin but Batavia, now Djakarta,27 Indonesia. Darwin was finally chosen because of the continuous nature of its observations. A time series of the SOI for the period 1875–2013 is given in Figure 1.19b, representing Walker’s observation of: … a swaying of pressure on a big scale backwards and forwards between the Pacific Ocean and the Indian Ocean … Walker (1924a, b) Walker had hoped to use the changing phase of the SO to “foreshadow” the state of the forthcoming Indian monsoon. Spectra of the SOI (Figure 1.19) show a broad and statistically significant peak in the 3–5 period band. The SOI correlates well with the anomalies of the central-eastern Pacific SST such that persisting values of the SOI > +2 are generally associated with La Niña conditions; that is, the anomalous Tahiti surface pressure is lower compared to the surface pressure of Darwin. In turn, a positive SOI corresponds to positive SST anomalies in the western Pacific Ocean and negative in the east, characterizing La Niña. Sustained values of the SOI > −2 indicates El Niño with a reverse anomalous pressure gradient and a reverse SST anomaly gradient. Even though the initial goal of forecasting monsoon rainfall was not entirely fulfilled, Walker’s discoveries 27 See Trenberth (1984) and Trenberth and Shea (1987) for discussion.

31

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1 Climatology of the Tropical Atmosphere and Upper Ocean

(a) Atlantic Multidecadal Oscillation (AMO)

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Figure 1.20 Long-term oscillations in the Atlantic Ocean. (a) The Atlantic Multidecadal Oscillation (AMO) and (b) The North Atlantic Oscillation (NAO). Upper panels (i) show the principle SST patterns associated with the AMO and the NAO (∘ C). Lower panels (ii) show the time evolution of the AMO and NAO indices described in the text. Source: Diagrams from The Climate Data Guide: Atlantic Multi-decadal Oscillation (AMO). Retrieved from https://climatedataguide.ucar.edu/climate-data/atlantic-multi-decadal-oscillation-amo.

are nevertheless considered monumental. His ideas of large-scale atmospheric oscillations were embraced by Troup (1965) and Bjerknes (1969), the latter study noting that the atmospheric SO was coupled to an evolving oceanic oscillation, El Niño/La Niña, to provide the coupled ocean–atmosphere ENSO phenomenon that exhibited long-term predictability (Zebiak and Cane 1987), although with limited predictability across the boreal spring. In Chapter 18, we will consider an ENSO-like Oscillation we refer to as the mega-ENSO. Similar in pattern to ENSO, it oscillates on a longer time scale, is broader in latitude, and appears to be associated with variability of the large-scale monsoon. 1.4.3.2

Atlantic Oscillations

There are a number of independent oscillations within the Atlantic Ocean. On an interdecadal time scale, North Atlantic Ocean SSTs vary over a broad area on time scales of 30–40 years with an amplitude of ±0.2 ∘ C. This oscillation is referred to as the Atlantic

Multidecadal Oscillation (AMO).28 It appears to have a strong influence on mean global and NH temperatures and also on NE Brazilian and African Sahel rainfall and the number of North Atlantic tropical cyclones.29 The pattern is shown in panel (i) of Figure 1.20a. The AMO Index, defined as the difference between the global SST between 60∘ N and 60∘ S and the average SST of the North Atlantic is plotted in panel (ii). In calculating the AMO index the ENSO signal has been removed. The second Atlantic oscillation is the North Atlantic Oscillation (NAO: e.g., Folland et al. 2009). The anomalous SST pattern associated with the NAO is shown in Figure 1.20b(i). The NAO index, essentially the MSLP difference between the Icelandic Low in the northern Atlantic Ocean pattern and the Azores High in the NH subtropics, appears in Figure 1.20b(ii). Depending on the sign of the index, the low-level 28 E.g. Lamb (1978a, b), Latif and Barnett (1994), and Hastenrath (1990). 29 Knight et al. (2006).

1.4 A Myriad of Variability

Quasi-Biennial Oscillation (QBO): Zonally-averaged U(z) 30 26 30 22

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Figure 1.21 Zonally averaged zonal mean monthly velocity [U(z)] between 1 and 30 km over the 18-year period 1993–2011. Yellow shading denotes westerly winds, blue shading easterlies. Source: Diagram constructed using data from The Climate Data Guide: QBO: Quasi-Biennial Oscillation. Retrieved from https://climatedataguide.ucar.edu/climate-data/qbo-quasi-biennial-oscillation.

westerlies will be enhanced or decreased consistent with the anomalous cooler or warmer SSTs. It appears that much of the influence of the NAO on climate is restricted to the NH. However, there seem to be moderate links of the boreal winter NAO and austral summer precipitation intensity in southeastern Africa.30 1.4.3.3

Stratospheric Oscillations

Following the eruption of the Krakatoa volcano in 1883 on the southern tip of Sumatra (6∘ S, 105∘ E), ejecta in the high reaches of the atmosphere was observed to move eastward encircling the globe at about 30 m s–1 . These became known as the Krakatoa easterlies and substantiated the belief at the time that the high altitudes winds above the equator were easterly. Some 25 years later upper-air balloon ascents over Lake Victoria (1∘ S, 33∘ E) found westerlies.31 These became known as the Berson westerlies and raised questions regarding the observations of the Krakatoa easterlies. Rather than a mono-signed phenomenon, an oscillating system was found,32 as illustrated in Figure 1.21. A zonally symmetric circulation reversing sign every 24–26 months emerged with a downward propagating phase with a rate of about 1 km month–1 . The oscillation is symmetric about the equator with a very small or negligible meridional wind component. Initially it was suggested that the phenomenon arose from photochemical processes as the relaxation time for ozone was about 26 months. However, more plausible theories emerged resulting from vertically propagating tropospheric equatorial waves.33 30 31 32 33

McHugh and Rogers (2000). Berson (1910). Veryard and Ebdon (1961). See Lindzen (1987) for an historical account.

Associations between the tropospheric weather and the Quasi-Biennial Oscillation (QBO) have been difficult to substantiate. For example, monsoon precipitation possesses a biennial component as described in Sections 16.1 and 17.4.1, but there are plausible hypotheses that the internal dynamics of the monsoon determine this variability. However, as we will develop in Chapter 7, equatorial modes do propagate vertically and the vertical propagation is influenced by the vertical structure of the background wind so viable influences on tropospheric phenomena can be envisaged. In fact, it has been suggested that the MJO described above in Section 1.4.2, changes its character depending on the phase of the QBO. Zhang and Zhang (2018), for example, have investigated a QBO–MJO connection finding stronger MJO activity in the easterly phase of the QBO than in its westerly phase. As the MJO is strongly connected to equatorial wave dynamics (see Chapter 15) this may not be a surprise. The influence of the QBO further afield from the tropics is fairly established. … Through modulation of extratropical wave propagation, the QBO has an effect on the breakdown of the wintertime stratospheric polar vortices and the severity of high-latitude ozone depletion. The polar vortex in the stratosphere affects surface weather patterns, providing a mechanism for the QBO to have an effect at the Earth’s surface. Baldwin et al. (2001, p. 179) 1.4.4 Overlapping of Variance Bands: Waves Within Waves We have noted that the spatial characteristics of variance (Figure 1.15) in the synoptic, biweekly, and intraseasonal bands appear to be similarly spatially located.

33

1 Climatology of the Tropical Atmosphere and Upper Ocean

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Figure 1.22 Co-variability of modes in the South Asian Monsoon over the central northern Bay of Bengal (85∘ E–95∘ E, 12∘ N–22∘ N): (a) Average spectrum of the three-hourly TRMM rainfall over the Bay of Bengal from 1998 to 2004. The shaded region encloses the average spectrum ±1 standard deviation. (b) The three-hourly TRMM rainfall variability in different time scales (bands) during one full intraseasonal event over summer 2001. The figure shows rainfall in the 25–80 day band (dashed line), 5–20 day band (thick solid line), and the diurnal cycle (thin solid line). Source: From Hoyos and Webster (2007).

However, there appears to be a phase locking in time as well such that the variance of convection, between temporal scales, appears to be modulated by a longer period variability. An example of this coherency appears in Figure 1.22. Figure 1.22a presents the spectra of a three-hourly Tropical Rainfall Measurement Mission (TRMM)XII rainfall over the BoB from 1998 to 2004. The shaded area encloses the variability of the spectra from year to year within ±1 standard deviation. Figure 1.22b plots the TRMM rainfall over the BoB in the different period bands during one full intraseasonal cycle (dashed line) during the period April 20, 2016 (Julian day 110) through June 8 (Julian day 160). Within the cycle, the maximum amplitudes of both the quasi-biweekly, synoptic, and diurnal variations occur at the same time. It would seem that the intraseasonal variability is modulating the higher frequency oscillations such as the biweekly variability, which in turn modulates even higher frequencies.

Such a spectral overlap has been found in many parts of the tropics. The first systematic study was undertaken by McBride (1983), who conducted a careful and systematic examination of satellite data over the Northern Australian Monsoon region during the Winter MONEX period of December 1, 1978 to March 5, 1979. The study uncovered: … synoptic scale organizations of convection …. McBride (1983, p. 189) that propagated systematically both to the west and the east. The careful analysis was important as it was the first to suggest that the complexity of observed convection can be diagnosed in terms of eastward propagating Kelvin waves and westward propagating Rossby waves. Some years later Nakazawa (1988) using very high-resolution satellite data was able to discern similar structures in MJO propagating through the West Pacific.

Notes I Lewis Fry Richardson (1981–1953) was an English

physicist and mathematician with a deep interest in meteorology. He was also a pacifist and spent WWI as part of a Quaker ambulance unit attached to a French Infantry Unit. He was a man of great integrity who ceased work in fields where his work could be used to produce weapons or aid in warfare. In 1923, he proposed a prototype numerical prediction system. Without the availability of fast computing, he devised a “Weather Factory” where a large number of “operators” strategically placed in a great hall, and as such acting as grid points, made numerical calculations and passed these on to the

surrounding “grid points.” The Richardson Factory is described in https://en.wikipedia.org/wiki/Lewis_ Fry_Richardson. II 2 The Merriam Webster Dictionary defines the word “doldrum” as “a state of inactivity or stagnation” or “a dull, listless, depressed mood; low spirits.” The word appears to have emerged from “dold” for dull and stupid. A vivid description of the doldrums appears in Verse 8 of Coleridge’s (1798) Rime of the Ancient Mariner: “Day after day, day after day;/ We stuck. No breath no motion./ As idle as a painted ship,/ Upon a painted ocean.” The doldrums were often the region of demise of sailing ships and their crews. A

1.4 A Myriad of Variability

III

IV

V

VI

VII

VIII

detailed description of deprivations associated with becoming becalmed in the doldrums can be found in The Circumnavigators (D. Wilson 1989). Professor Taroh Matsuno (1934-) is a Japanese fluid dynamicist who showed that there were special sets of waves trapped close to the equator. His work laid the foundation for equatorial atmospheric and ocean wave dynamics. He has received many awards, most recently the International Meteorological Prize of the World Meteorological Organization in 2010 and the prestigious Blue Planet Award in 2013. Sir Gilbert Walker (1868–1958) was a British mathematician, statistician, and physicist. In 1889, while at Cambridge University, he earned the singular honor of being the Senior Wrangler, a title bestowed on the top student of the Mathematical Tripos. He was Director-General of the Indian Meteorological Department (1904–1924) and developed objective statistical forecasting techniques, some of which are described in the main text. His papers were revolutionary, suggesting that slowly varying remote climate events can influence local weather. Jacob Bjerknes (1897–1975) was a Norwegian-born meteorologist and climatologist. He was the son of the Wilhelm Bjerknes with whom he worked developing the midlatitude cyclone model that was to provide a physical foundation for weather forecasting. He founded the Department of Meteorology at UCLA (now Atmospheric and Oceanic Sciences) in 1940, where he hypothesized about the coupled nature of El Niño. See https://www.pmel.noaa.gov/gtmba: with components: PIRATA (Prediction and Research Moored Array in the Atlantic); RAMA (Research Moored Array for African–Asian–Australian Monsoon Analysis and Prediction), TAO (Tropical Atmosphere Ocean) array in the Pacific Ocean. Timothy N. Palmer CBE, FRS (1952–) is the Royal Society Professor of Climate Physics at the University of Oxford. Palmer is a British mathematical physicist who graduated from the University of Bristol in Mathematics and Physics and received his PhD in General Relativity from the University of Oxford. Palmer turned his attention to weather and climate and joined the European Centre for Medium Range Weather Forecasts (ECMWF), where he led efforts to introduce ensemble techniques and the concept of uncertainty into weather and climate prediction. Palmer has received numerous awards, including using the Carl-Gustav Rossby Research Medal from the American Meteorological Society and the Dirac Gold Medal from the Institute of Physics. The zonally averaged precipitation rate (mm day−1 ) and total rainfall by volume (m3 day−1 ) were calculated in the following manner. The CMORPH precipitation data set provides a 2.5∘ × 2.5∘

IX

X

XI

XII

latitude–longitude (Δ𝜑 × Δ𝜆) data set. The average rainfall rate (mm day−1 ) is calculated as the arithmetic mean of rainfall rate of all boxes in a set Δ𝜑 band. The rainfall by volume is the integral of the rainfall in each box around a latitude circle, noting that Δ𝜆 = Δ𝜆(𝜑) decreases with increasing latitude. T. N. Krishnamurti (1971–2018) was the Lawton Distinguished Professor Emeritus: Department of Earth, Ocean, and Atmospheric Science, Florida State University). He was a prominent Indian meteorologist who has worked on fundamental problems in tropical meteorology. He was awarded the Rossby Research Medal from the American Meteorological Society among many other honors. The current was initially named the Cromwell Current after its discoverer. Townsend Cromwell (1922–1958) died in a plane crash on the way to an oceanographic field experiment. He was a Research Associate at Scripps Institution of Oceanography, La Jolla, California and a Senior Scientist with the Inter-American Tropical Tuna Commission. Ernest Hovmöller (1912–2008), a Danish meteorologist, invented a useful technique for following ridges and troughs in space and time (Hovmöller 1949). On such a diagram, time may be the ordinate and longitude the abscissa. The longitudinal variability of some quantity (say pressure) would be plotted at set times (e.g. daily). The completed diagram would show the progression of pressure patterns in time. Also, the technique is useful in distinguishing between phase and group velocity of a wave (see Chapter 4). The US NASA and the Japan Aerospace Agency (JAA) partnered in a joint mission to monitor and study tropical rainfall. The satellite contained an active Precipitation Radar (PR), a Microwave Imager (the TRMM MI, TMI), a Visible and Infrared Scanner (VIS), a Lightning Imaging Sensor (LIS), and a Clouds and Earth Radiation Energy Sensor (CERES). An imager (TMI), which measures a variety of cloud properties (e.g. Wentz and Schabel 2000), is central to these composites, providing reliable measurements of SST, precipitation, and cloud liquid water. Surface winds and convergence were derived from a QuikSCAT scatterometer (Freilich and Dunbar 1999), providing collectively with TRMM a comprehensive view of the tropics and the subtropics. The satellite was launched in 1997 with an expected life of 2–3 years. The satellite operated for 17 years, providing an unprecedented precipitation data set. A description of the instrumentation and the data, as well as access to the archives, can be found at https://pmm.nasa.gov/ trmm.

35

37

2 Hydrological and Heat Exchange Processes Webster The Earth’s climate system is determined by a conjunction of influences from the ocean, the atmosphere, the lithosphere, the cryosphere, and the biosphere. Permeating among each of these components is water through a plethora of hydrological processes. Earth appears in stark contrast to its inner solar system neighbors, both in its abundance of water and in the multiple phases in which water is present. In addition to large land masses, the surface of the planet exhibits vast reservoirs of water and large expanses of ice. Figure 2.1 plots the relative distribution of land and ocean as a function of latitude. Overall, more than 70% of the planet is covered by ocean. In the northern hemisphere the ratio of land to ocean is about 2 : 3. In the southern hemisphere (SH), the ratio is 1 : 4. In the tropics (defined here as between 30∘ N–30∘ S) the ratio is very similar to the SH. In terms of absolute area, the majority of ocean surface lies in the tropics. In the atmosphere, large evolving areas of clouds appear throughout the depth of the troposphere. Moreover, the movement of the clouds reflects dynamic and thermodynamic time scales, ranging from diurnal to interannual and fluid motions that link geographically remote regions. The most rapid time scale of hydrological processes in Earth’s climate system occurs in clouds. Clouds and the ambient water vapor mass, within which clouds reside, dominate the radiative budget of the planet. At any one time, 50% of the planet is covered by clouds. The three forms of water that constitute a cloud have decidedly different radiative properties. All forms of water are strong absorbers of longwave radiation. Water ice is a strong reflector of shortwave radiation but is a weak absorber. Liquid water is a weak reflector although in abundance, such as in a cloud, multiple reflections of shortwave radiation can produce substantial aggregated reflection and high albedos. Liquid water is a weak absorber of solar radiation, which, together with liquid water in the column, accounts for a reduction of about 15% of the incoming solar radiation. Successful weather

forecasting and climate simulation require a very careful simulation of these cloud processes. In assessing the impact of the anthropogenic increase of CO2 on the global climate, it is the role of water that remains the greatest mystery. A warming atmosphere leads to an increase of water vapor (from increased evaporation) and an enhancement of the greenhouse effect. By increasing the emission of infrared radiation, water vapor leads to further warming and increased thermal emission. On the other hand, a warming planet may lead to an increase or decrease of cloudiness that, in turn, would modulate the global temperature by an increased or decreased reflection of solar radiation. Thus, the net radiative effects of an increase in CO2 concentration remain uncertain. What is certain is that hydrological processes stand at the center of the climate of the Earth system. However, the simulation of clouds in models remain an elusive problem and the largest sources of errors in climate simulations. This chapter discusses the influence of hydrological processes on the interaction of the components of the climate system, concentrating on processes that occur in the tropical regions. The exponential variation of saturation water vapor pressure with temperature indicates that the signal of hydrological processes is largest where the temperature is warmest. Overall, the tropical regions of Earth possess the warmest SSTs, the deepest atmospheric convection, the most copious precipitation, and, some 17 km above the surface, the coldest air temperatures on the planet (see Figure 1.2). Within the tropics, strong radiative and latent heating gradients exist that drive vigorous circulations. Between the equator and the poles, the ocean and the atmosphere are connected over wide spatial and temporal scales by fluxes of heat, water, and momentum. In particular, the equator-to-pole heating gradients are magnified by the exponential variation of saturation vapor pressure. In its myriad forms, water permeates the thermodynamic and dynamic structure of the ocean and atmosphere and imparts an indelible and dynamic signature on the Earth’s climatic structure.

Dynamics of the Tropical Atmosphere and Oceans, First Edition. Peter J. Webster. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

2 Hydrological and Heat Exchange Processes

Relative Area of Land and Ocean 250

200 AREA (106 km2)

38

land area

150

total surface area

ocean area 30

100

20

JJA 50 0 90°S

10

Sea Surface Temperature 60°S

30°S

equ

30°N

60°N

SST (oC)

DJF

0 90°N

Latitude Figure 2.1 The relative areas of land and ocean as a function of latitude. In the tropics (defined here as from 30∘ N to 30∘ S) the ratio of ocean to land is roughly 3 : 1 and reduced to 5 : 1 south of 30∘ S. North of the tropics the ratio reduces to near parity. The red curves show the sea-surface temperature (SST∘ C) distribution, indicating that the majority of the tropical oceans and hence the global oceans have SSTs greater than 20 ∘ C.

2.1 Water on Earth 2.1.1

An Inventory

Figure 2.2 defines the main water reservoirs of the planet (the atmosphere, the ocean, the cryosphere, and the lithosphere) and their subreservoirs (biosphere, mid-ocean, abyssal1 ocean) together with the fluxes of water from one reservoir to another. The atmosphere is divided into two parts: the volume of air over land and ocean. The reservoirs are connected to one another by advection or flux of water and through precipitation, evaporation, transpiration, and river runoff.I Our inventory comes from Chahine (1992). The atmosphere contains only about 0.001% (15.5 GT/1 458 000 GT) of the total water in the Earth system, mainly in vapor form. Liquid and ice content of the atmosphere is only 0.05% of the water vapor by mass. However, this extremely small proportion of the total water produces clouds and, thus, is responsible for the reflection of a large proportion of the solar radiation away from the planet.

1 “Abyssal” comes from the Latin “abyssus” for “deep, hell, abyss” and is used to refer to the very deep reaches of the ocean. In Section 3.2 we refer to this zone as below the “level of no motion” where recycling processes are very slow.

Oceanic SB Ocean water mass exceeds land water (surface, land ice and snow, and the water contained in the land biosphere) by a factor of nearly 25. The ocean water mass is also 100 000 times greater by mass than atmospheric water. About a third of the land-based water is subterranean. Assuming that the land areas constitute 30% of the surface area of the planet, the overall rain rate over the oceans is about 25% greater than over land. Furthermore, over the ocean, evaporation exceeds precipitation by about 8.3%, with the difference being advected by atmospheric motion over the land. Over land, on the other hand, precipitation exceeds evaporation and transpiration by 50%. The net gain of water over the continents is balanced by river runoff into the ocean. Using data from Figure 2.2 it is possible to calculate residence times of water in the various reservoirs. If the mass of a reservoir is M kg and the flux into (out of) the reservoir is SR kg/year, then the residence time is T R years, i.e.: TR = M∕SR

(2.1)

Thus, the residence time of an individual molecule of atmospheric water vapor over the oceans is 10 days (i.e. T R = 11 Pkg/434 Pkg year−1 ). In the atmosphere over land, the residence time of water is a little longer, at 15 days. The residence time of all atmospheric water is very small, just 11 days. The total evaporation from the ocean is estimated to be 434 Pkg year−1 , which is equivalent to an annual reduction of sea level of 1.2 m. This loss of ocean water is balanced by a river runoff of 36 Pkg year−1 , which matches the flux of water from the marine atmosphere into the terrestrial atmosphere. The overall residence time of water in the oceans is over 3000 years (i.e. 1 400 000/(398 + 36) kg year–1 ). p calculations become more difficult as the ocean contains three main layers: the deep abyssal layer (depths > 1 − 2 × 103 m), the thermocline layer (200 to 103 m), and the upper surface mixed layer (0–200 m). The residence time for the upper ocean, assuming that the precipitation and river runoff are the sources of water, is greater than 100 years. The residence time of deeper water is much longer and is defined by the time scales of the planetary scale thermohaline overturnings of the ocean, referred to as the “conveyor belt” (Broecker 1991). To calculate this long time scale we would need to determine the rate of generation of cold deep water in the Arctic and the Antarctic. However, modeling and radiocarbon dating have provided residence time scales of 1000–2000 years for abyssal water (Gebbie and Huybers 2011, 2012). For these isotopic time scales to be correct deep water would have to be created at a rate of 250–450 × 1015 kg year−1 .

2.2 Thermodynamics of Water and Earth’s Climate

Figure 2.2 The global water cycle and water reservoirs (units: Pkg: 1015 kg). Arrows indicate fluxes of water from one reservoir to another (Pkg year−1 ). Water in the atmosphere accounts for 0.001% of the total water mass in the climate system. Residence times (𝜏) of water in each reservoir are noted in the text. These time scales range from 10 to 15 days in the atmosphere to thousands of years in the deep ocean. Source: Data from Chahine (1992). Figure from Webster (1994).

Terrestrial Atmosphere 4.5 (15 days) evaporation and transpiration 71

11 (10 days)

precipitation 107

evaporation 434

precipitation 398

Land

Oceans

ice and snow 43,400 surface water 360 (5 yrs) sub-sfc water 15,300 (1000 −10,000yrs)

mixed layer 50,000 (100 yrs) thermocline 460,000 abyssal 89,000 (1–3000yrs)

river runoff 36

59,000

1,400,000

reservoirs 1015 kg

The transitions between reservoirs described above involve phase changes of water (e.g. condensation/ evaporation, melting/freezing, and sublimation/deposition) that involve energy exchanges, but, ultimately, the only energy available to change the phase of water is from net radiative heating/cooling of the planet. Given the geographic distribution of land and ocean (Figure 2.1), we may expect large spatial and temporal differences in the magnitude and location of these phase changes. 2.1.2

Marine Atmosphere

advection 36

Global Disposition of Rainfall

It is worthwhile considering the zonal averages of seasonal and annual rainfall displayed earlier in Figure 1.3 in more detail. In particular we are interested in Figure 1.3c, which shows the annual volume of precipitation as a function of latitude for the total rainfall over (i) land and the ocean, (ii) the ocean, and (iii) land. Table 2.1a summarizes the global rainfall distributions shown in Figure 1.3. The table lists the global rainfall rate (1011 m3 day–1 ) over the entire year (ANN) and for each season.2 The rates appear almost constant with seasonal rates differing from the annual mean by about 1%. However, while the contribution from precipitation over land and ocean do differ, they add up to provide the same total rainfall rate. Calculations using the more

2 The Global Precipitation Climatology Project (GPCP): Adler et al.(2003) data set.

fluxes 1015 kg/year

latitudinally constrained CMORPH3 data set yielded very similar results. The numbers on the figures denote the total rainfall by mass in each hemisphere. Tables 2.1b and c list the rainfall rates for each hemisphere. We note that the total rate of precipitation in both hemispheres is nearly identical! Given the vast differences between the geography of the two hemispheres (the ratio of land to ocean is 1 : 1.5 in the NH and 1 : 4 in the SH (see Figure 2.1)), the magnitude and distribution of the SST, and the very different cloud populations, these are surprising results. Clearly, it would seem that there are perhaps overall constraints on climate that enforce this symmetry. The rainfall results seem to concur with the findings of Stephens et al. (2015) showing that the difference between hemispheres of the integrated annual net radiation at the top of the atmosphere is near zero. We will revisit these issues in Chapter 11.

2.2 Thermodynamics of Water and Earth’s Climate We now discuss a relationship that is extremely important to the climate of Earth. The essential feature of Earth’s climate is the proximity of the environment to the triple point of water: the thermodynamical state where the liquid, vapor, and solid phases may coexist in equilibrium. 3 NOAA CPC Morphing Technique (CMORPH): Joyce et al. (2004).

39

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2 Hydrological and Heat Exchange Processes

Table 2.1 Rainfall rates for (a) the globe, (b) the SH, and (c) the NH. Rates are listed for an annual basis (ANN) and by season for total rainfall (ocean plus land) and ocean and land separately. (a) Global rainfall rate (1011 m3 day−1) Total

Ocean

Land

ANN

13.56

10.47

3.10

JJA

13.74

10.35

3.39

SON

13.53

10.53

3.01

DJF

13.56

10.55

3.04

MAM

13.40

10.44

2.95

(b) Southern hemisphere rainfall rate (1011 m3 day−1 )

ANN

6.875

5.640

1.235

JJA

5.700

5.083

0.616

SON

5.855

4.780

1.076

DJF

7.999

6.145

1.854

MAM

7.950

6.554

1.393

(c) Northern hemisphere rainfall rate (1011 m3 day−1 )

ANN

6.683

4.823

1.856

JJA

8.045

5.267

1.776

SON

7.676

5.747

1.929

DJF

5.563

4.404

1.159

MAM

5.448

3.889

1.589

2.2.1

Implications of Clausius–Clapeyron

Figure 2.3a shows a plot of the phase transitions of water as a function of temperature and the partial pressure of water vapor (ev ) or vapor pressure.II The solid lines show where two water phases (e.g. vapor and liquid) can coexist in equilibrium. The triple point of water is located at point TP where the three phase lines of water meet and where all phases of water can coexist in equilibrium. This occurs at a vapor pressure of 6.11 hPa and a temperature of 0.01 ∘ C (272.15 K), values commonly encountered in the Earth’s environment. The diagram also shows the melting/freezing point of water. As pressure decreases, the melting point (denoted by the phase transition line between liquid and solid) decreases, but only slightly. On the other hand, the boiling point temperature (transition between liquid and vapor) decreases with decreasing pressure. The complexity of hydrological processes in the Earth system is underlined by the nonlinearity of the phase equilibrium lines, especially at the vapor–liquid

interface where the saturation vapor pressure phase transition increases exponentially with temperature. If condensation occurs in a column warmer than its neighbor, there is the potential for a greater release of latent heat in the warm column compared to the cooler column for the same vertical displacement. In the tropics, we would expect the impact of hydrological phase conversions to be amplified compared to higher latitudes. The Clausius–ClapeyronIII equation describes the nonlinear dependency of the saturated vapor pressure es on temperature T and may be written as: es LE,S des (2.2) = dT Rv T 2 where LE and LS are the latent heats of evaporation– condensation and sublimation–deposition, respectively, and es (T) is the saturation vapor pressure at some temperature T. Rv is the gas constant for water vapor. LE and LS vary very slightly with temperature, with an 8% decrease between ±40 ∘ C, roughly the temperature range at which condensation can occur in the tropics. However, if LE and LS are assumed to be independent of T, Eq. (2.2) can be integrated between two temperatures T 1 and T 2 to provide a simpler expression for the saturation vapor pressure over liquid water or an ice surface as a function of temperature: ( ( )) LE,S 1 1 − (2.3) es (T2 ) = es (T1 ) exp Rv T2 T1 where es (T 1 ) and es (T 2 ) are the saturated vapor pressures at temperatures T 1 and T 2 . The expression shows that saturation vapor pressure is, to a good approximation, a sole function of temperature and increases exponentially with temperature.4 This is broadly referred to as the “Clausius–Clapeyron effect.” It represents processes that are fundamental to the climate of Earth. The three latent heating coefficients for the transitions from vapor to liquid, from vapor to solid, and from liquid to solid (i.e. higher to lower phase) are: LE = 2.5 × 106 J∕kg, LS = 2.84 × 106 J∕kg, and LF = 3.3 × 105 J∕kg

(2.4)

evaluated at T = 273. Note that the sign of the coefficients refers to transitions from a higher phase state of water (e.g. vapor) to a lower phase (e.g. liquid) and refers to the gain of heat by the environment (e.g. condensation) per 1 kg of water involved in the transition. To determine the heat required to change water from a lower to a higher thermodynamic state (e.g. liquid to vapor), the signs of the latent heat coefficients in Eq. (2.4) are reversed. 4 Curry and Webster (1999), Chapter 4, provides the full derivation of the Clausius–Clapeyron equation.

2.2 Thermodynamics of Water and Earth’s Climate

0 0.01

100

374oC

103

LE = 2.5×106 J kg–1 LS = 2.8×106 J kg–1 LF = 3.3×106 J kg–1 M

Standard melting point

SOLID

Standard boiling point

B

LIQUID TP

6.11 0.13

triple point

194

VAPOR

273 273.1

373

Partial Pressure Water Vapor (hPa)

Partial Pressure Water Vapor (hPa)

–79

–75

Temperature (log K) (a)

–25

0

101 100 10

25

50

75

100oC

LIQUID

102

triple point

Venus

SOLID Earth

–1

VAPOR 10–2 10–3 200

647K

–50

Mars 225

250

275

300

325

350

375K

Temperature (K) (b)

Figure 2.3 (a) The phases of water as a function of the partial pressure of water and temperature. The solid curves are the phase transition curves between liquid water, vapor, and ice defined by Eqs. (2.2, 2.3). The triple point of water (solid circle, labeled TP) occurs at the location where the three phases of water can coexist in equilibrium. There are six possible phase transformations: sublimation ↔ deposition, condensation ↔ evaporation, and freezing ↔ melting. Transition either liberates heat (transition to right: i.e. high phase state to lower) or requires heat (transition to left: lower phase state to higher). The transitional latent heats are also indicated. Points M and B denote melting and boiling points. (b) The background gray arrows are hypothetical climate trajectories of Venus, Earth, and Mars as they may have evolved through time (after Rasool and de Bergh, 1970). The trajectories emanate from the primitive temperatures of the planets defined in Eq. (2.5). As a planetary atmosphere develops and the optical thickness increases, the development of climate is constrained by interception of a trajectory with a phase transition line. Source: Adapted from Figure 1, Rasool and De Bergh (1970).

2.2.2 Role of Water in the Evolution of Earth’s Climate Goody and Walker (1972) argued that the nonlinearity of water phase transitions may have played a crucial role in setting the basic climatic regimes of the inner planets. Prior to the evolution of an atmosphere, the initial surface temperature of a planet would be the same as the equivalent black body temperature of the planet radiating away the heat gained from the intercepted solar stream. This they referred to as the primitive temperature of the planet; that is, for a given albedo 𝛼 and solar irradiance S0 (W m–2 ), the energy available to the planet would be S0 (1 − 𝛼 p )𝜋a2 , where a is the radius of the planet, 𝛼 p is the average planetary albedo and S0 is the annual average solar irradiance (1380 W m–2 ) at Earth’s distance from the sun. We assume that the planet would radiate this energy away to space as a black body at some temperature T 0 , so that, at equilibrium, the energy exiting the planet would be 4𝜋a2 𝜎SB T04 , where 𝜎 SB is the Stefan–Boltzman constant. This “primitive” temperature or “equivalent” temperature is obtained by equating the incoming and outgoing energy, giving: T0 = (S0 (1 − 𝛼p )∕4𝜎SB )1∕4

(2.5)

where, taking into account the spherical geometry, the average solar radiation is S0 /4 W m−2 spread across the

planetary disk. Calculated in this manner, the primitive temperatures for atmosphere-free Venus, Earth, and Mars are 300, 255, and 205 K, respectively. These temperatures are marked on the abscissa of Figure 2.3b. If the Goody–Walker hypothesis is correct, the question then becomes how do the three planets end up with the current observed averaged surface temperatures of 700, 290, and 220 K? Goody and Walker assumed that volcanic activity was continual on all planets and that the concentration of water vapor and other greenhouse gases increased with time. Mars, starting with a cold primitive temperature, intercepted the sublimation–deposition phase transition early in its evolution. According to the hypothesis, further increases in Martian atmospheric water vapor would then be unlikely and subsequent increases in water vapor would be subject to deposition. Thus, Mars would have a “truncated” greenhouse evolution. Venus, on the other hand, commenced with the warmest primitive temperature (abscissa in Figure 2.3b) and the highest saturation vapor pressure of the three planets. Goody and Walker hypothesized that no phase transition curves would be encountered and a “runaway greenhouse” with an ever-increasing surface temperature and a massive atmosphere would ensue. Here we define the greenhouse effect in its simplest form. Surface outgoing longwave radiation is absorbed

41

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2 Hydrological and Heat Exchange Processes

within the atmospheric column and re-emitted upward and downward, thus increasing the net radiation at the surface to levels greater than that received directly from the Sun. Furthermore, the hypothesis does not take into account an absence of a magnetic field and consequently the loss of an atmosphere through the impact of the solar wind. Nor does it take into account every long-term orbital change of Mars or any planet for that matter Clearly, the equilibrium climate state that a planet finds itself in depends on many factors. Earth, starting with a primitive temperature between that of Mars and Venus, followed a “Goldilocks”IV climate trajectory and intercepted phase lines near the triple point of water. There, water vapor concentration can be modulated by transitions between the three phases of water. Vast oceans are possible and act as a continual source of water vapor, increasing the greenhouse effect and warming. However, cloud formation from water evaporated at the ocean surface increases reflection, hence leading to increased cooling. Thus, proximity to the critical point at the end of the Goldilocks trajectory appears as a climate stabilizer or a climate attractor. In essence, the proximity of the triple point to Earth’s planetary trajectory substantially increases the number of its climate degrees of freedom (dof). This dof describes the number of independent ways by which a dynamic system can move without violating any constraint imposed on it.5 This proximity may point to why Earth has possessed a climate within a rather narrow range over the eons despite large changes in solar radiation due to the aging of the Sun and orbital changes. Clearly, this discussion is very simplistic and is introduced only to illustrate the importance of Clausius– Clapeyron thermodynamics. For example, there seems to be growing evidence that the Martian climate went through a distinct pluvial phase sometime in the past. Clearly, many other factors are at work in the climate trajectory of planets. 2.2.3 Estimate of the Planetary Radiative Surface Temperature The primitive temperature of a planet derived in Eq. (2.5) is useful in determining the broad-scale characteristics of the Earth’s climate. However, we need to extend the model to estimate the current radiative surface temperature of the planet. Consider a two-layer atmosphere (Figure 2.4a) with a mean surface temperature T 0 and a mean atmospheric 5 The number of degrees of freedom also refers to the number of independent pieces of information in a data set. This is important in determining the statistical significance of a periodicity (say) found in a sequence of measurements.

temperature of T A . We let the total reflectance of the atmospheric column (cloud and surface) to be 𝛼 and assume that there is some fractional absorption of incoming solar radiation 𝜐. In addition, we propose that the atmosphere is “gray” in the infrared with an emissivity 𝜀m, so that there is partial absorption within the column but also a transmission of longwave radiation to space at a fractional rate of (1 − 𝜀m ). The steady state balance equations for the top of the atmosphere and the surface can be written as: 𝜀m 𝜎SB TA4 + (1 − e)𝜎SB T04 − S(1 − 𝛼p (1 − 𝜐)2 ) = 0, z = zTOP

(2.6a)

𝜎SB T04 − 𝜀m 𝜎T04 − S(1 − 𝛼p )(1 − 𝜐) = 0, z=0

(2.6b) 𝜀m 𝜎TA4

In both equations the term is the re-radiation of absorbed surface longwave radiation by the atmosphere. In the second equation 𝜀m 𝜎TA4 represents the greenhouse effect at the surface. The second term of (2.6a) expresses the proportional transmission to space of the surface longwave radiation. Eliminating T A between the two equations provides an expression for T 0: ( )1∕4 (1 − 𝛼p )(1 − 𝜐) + (1 − 𝛼p (1 − 𝜐)2 ) T0 = S 𝜎(2 − 𝜀m ) (2.7) T 0 is plotted in Figure 2.4b as a function of emissivity and albedo. In this example, the solar absorptivity is set at 10% (i.e. 𝜐 = 0.1). Without an atmosphere, 𝜀 and 𝜐 are both zero and the surface temperature reduces to the primitive temperature T 0 depending just on the albedo of the planet and the magnitude of solar radiation. T 0 is marked on the figure as the large red dot corresponding to 𝜀m = 0 with 𝛼 p = 0.2. The observed mean surface temperature of the planet is denoted by a heavy black dot and corresponds to the solution of Eq. (2.7) for 𝛼 = 0.3, 𝜀m = 0.7, and 𝜐 = 0.1. It is interesting to note that in this simple model, the emissivity has to be greater than about 0.5 for the planet to have a mean temperature greater than 273 K. It is not difficult to understand why the cloud radiation parameters, albedo and emissivity , would be related. Both cloud albedo and cloud emissivity increase with increasing cloud thickness (Stephens 1978). As these two radiative properties have the opposite effect on surface temperature, they tend to partially compensate for each other. However, as a cloud thickens, the rates of change of the albedo and emissivity are different. Emissivity tends to increase faster than albedo as the water mass of a cloud increases; that is, a cloud

2.3 Water and the Tropical System

(b) T0 as a function of α and ε 1.0

(a) Energy balance in simple climate model (1

4

0.8

) T0

z = zTOP S (1

TA

(1

)2 )

Emissivity (ε)

T A4

x y sz 300

z’ x’ s’ y’

0.6

280

0.4 273

z=0

T0 T A4

T04 S (1

(1

))

0.2

260

Te 0

0

.2

240

220

.4 .6 Albedo (α)

200 180

.8

1.0

Figure 2.4 (a) Radiative balance of a simple gray atmosphere of emissivity 𝜀m , albedo 𝛼 p , and solar absorptivity 𝜐. The arrows indicate the net shortwave and longwave radiative fluxes at the surface and the top of the atmosphere. T 0 and T A represent ground and atmospheric temperatures, respectively. These balances are expressed in Eqs. (2.6a) and (2.6b). (b) Surface radiative temperature T 0 plotted as a function of albedo 𝛼 and emissivity 𝜀 for a set value of 𝜐 = 0.1. The yellow colored region to the right of the diagonal solid line (the 273 K isotherm) denotes the values of albedo and emissivity for which the mean radiative surface temperature would be less than the freezing point of water. The observed mean temperature of the planet (solid black circle) matches albedo 0.3 and emissivity 0.7. The primitive surface temperature T e is plotted as a solid red circle matching an albedo of 0.3 and is an extremely small emissivity. Source: After Webster (1994).

becomes darker (in an infrared sense) more quickly with increasing thickness than it becomes whiter (in terms of shortwave radiation). An interesting example is that of upper tropospheric cirrus clouds, which may create a net surface heating, as the back-longwave radiation, for thin clouds, may be greater than the shortwave reflectance. Both the magnitude and sign of the radiative compensation are very important for a number of issues. For example, if changes in cloudiness were to accompany increased greenhouse gas concentrations, the impact on the net radiative flux at the ocean–atmosphere interface could be very different depending on whether changes occurred preferentially in deep clouds or shallow clouds. If the albedo-emissivity compensation is exact, clouds will have a neutral impact on the surface and will follow the trajectory s–s’ indicated on Figure 2.4b. However, if 𝛼 increases more rapidly than 𝜀 with increasing cloud coverage, the impact on the surface will be cooling (trajectories y–y′ or trajectory z–z′ ) if Δ𝛼 p < Δ𝜀m . However, if 𝛿𝛼 p > 𝛿𝜀m then a net cooling would occur. In the present climate it is generally assumed that clouds have a small net cooling effect at the surface.

2.3 Water and the Tropical System Simple models and basic theory suggest water is instrumental in fashioning the structure of the climate system. Here, we examine observational evidence to confirm this

linkage and show how the impact of hydrological effects is amplified in the tropics. 2.3.1

Atmosphere

2.3.1.1 Clausius–Clapeyron and the Vertical Profiles of Temperature and Humidity

The zonally averaged distribution of specific humidity (Figure 1.2) showed a strong variability in both height and latitude. Figure 2.5 provides a different depiction of the water vapor distribution relative to the saturation vapor pressure curve (labeled C–C for Clausius– Clapeyron obtained from Figure 2.3a). The climatological zonally averaged vertical profiles of temperature and vapor pressure are plotted for the equator, 40∘ N and 75∘ N for DJF and JJA. Values of water vapor pressure at 1000, 900, 700, and 500 hPa appear as solid black circles. A number of interesting thermodynamical characteristics emerge from Figure 2.5: (i) Each profile is more or less parallel to the non-linear Clausius–Clapeyron saturation vapor pressure curve. Thus, even though the parcels in the mean are below saturation, their water vapor content follows the shape of Clausius–Clapeyron. (ii) Overall, surface vapor pressure in the tropics is a factor of almost two greater than values in the middle latitude columns, and an order of magnitude greater than in the polar regions.

43

2 Hydrological and Heat Exchange Processes

Temperature-water vapor profiles

36

CC

10

24

45oN

12

9

0 –48

70oN7 –32

10 9 4 4

Equ 9

10 7

Vapor Pressure (hPa)

48

(b) JJA

36

Equ

24

9 10 9

12

70oN

5 5

–16 0 Temperature (oC)

16

32

10

CC

(a) DJF

48 Vapor Pressure (hPa)

44

0 –48

4

–32

10 7 9

7 5 5

–16 0 Temperature (oC)

45oN

7

16

32

Figure 2.5 Atmospheric vapor pressure (hPa) as a function of temperature for mean atmospheres located at the equator (black), 45∘ N (blue), and 70∘ N (gold). Pressure levels are marked in hundreds of hPa for (a) the boreal winter December–January–February (DJF) and (b) the boreal summer June–July–August (JJA). The red curve (marked C–C) denotes the Clausius–Clapeyron saturation vapor pressure as a function of temperature. Note how the environmental curves are consistently parallel to C–C at all temperatures. The ratio of qv (T 1 ) and qvs (T 1 ) defines the relative humidity. Note that the scaling is logarithmic in Figure 2.3a but linear in Figure 2.5. Source: From Webster (1994).

(iii) The warmer the column, the greater the vapor pressure with the equatorial JJA column being the moistest. The relative humidity, the ratio of the specific humidity at a particular level and saturated specific humidity for a given temperature, gives values from 75% in the tropical boundary layer to about 20–40% in the upper tropical troposphere. 2.3.1.2 Water

Distribution of Water Vapor and Liquid/Ice

There is also much to be found in looking at the latitudinal distribution of total columnar water in all of its forms. We can also define a useful quantity called precipitable water (Wp mm) as z

Wp =

1 e(z) dz 𝜌w ∫z=0

(2.8)

Wp is the amount of water in an atmospheric column per unit area (m−2 ) either in total or in one of water’s forms, converted to a depth of liquid water; that is, it is the measure of the total water in a column. Here 𝜌w is the density of liquid water. Figure 2.6 shows the distributions of Wp (in mm) for water vapor (left ordinate) and liquid water/ice (right ordinate) in an atmospheric column as a function of latitude. The insert in the figure plots the relationship of Wp with underlying SST. Some interesting features emerge. (i) The distribution of Wp shows that the majority of water vapor is contained within the tropics, decreasing linearly with increasing latitude. (ii) Wp is strongly tied to the underlying SST (inset in Figure 2.6). For the most part the integrated

water vapor distribution almost follows the C–C curve but with slight deviations in two regions. Wp is depressed in the 10–20 ∘ C SST range in the subtropics, which is a region of pronounced subsidence. The second region of suppression occurs in the polar regions, where there is also strong subsidence. Subsidence implies stability and a reduction in the vertical mixing of moisture. Thus, the distribution of W is also influenced by circulation dynamics as well as the underlying thermodynamics associated with SST. (iii) Figure 2.6 presents further evidence of the joint influence of thermodynamic and dynamic processes controlling the distribution of water in an atmospheric column. The dashed contour represents the integrated total cloud liquid/ice water in non-precipitating clouds extracted from the Special Sensor Microwave Imager (SSM/I) satellite (Greenwald et al. 1993). This quantity is almost independent of SST and appears to be determined by the atmospheric circulation. The liquid/ice content of a column is about 2 orders of magnitude by mass less than the total water vapor in a column. The peaks in the liquid water distribution are associated with net rising motion in the ITCZ (as observed in Figure 1.6) and the belt of extratropical cyclonic storms, both of which are characterized by strong vertical motions that induce saturation and increase the liquid water concentration. The implications of the strong temperature dependency of atmospheric water vapor pressure are important from both radiative and latent heating perspectives. Atmospheric water vapor, liquid water,

2.3 Water and the Tropical System

60

0.25

60 50 es (kg m–2)

50

40

Wp (mm)

40

0.2

water vapor

C-C DJF

30 20

JJA

0.15

10 0

30

0

10

20

30

SST (oC)

0.1

(mm)

Figure 2.6 Satellite estimates of the total liquid water content (Wp ) defined in Eq. (2.8) in vapor form (solid curve; in millimeters: left-hand ordinate) and liquid and ice form (dashed curve; in millimeters: right-hand ordinate) as a function of latitude (Greenwald et al. 1993). All forms of water derived from the satellite data are transformed into an equivalent liquid state for easy comparison. Inset: Wp (mm) as a function of SST (Stephens 1990). Except in certain regions (discussed in the text) the distribution follows SST. The liquid and ice forms are correlated with atmospheric circulation features whereas Wp follows the SST. Source: Adapted from Greenwald et al. (1993).

20 liquid water and ice

10 0 90°S

60°S

30°S

0°

30°N

60°N

0.05

0 90°N

Latitude ΔT (°C): Moist versus dry ascent

8 6 ΔT (°C)

and ice are strong longwave absorbers. The single scattering properties of cloud particles (i.e. optical depth and single scattering albedo) are a function of the amount of condensed water phase (i.e. solid or liquid) and particle size distribution. Collectively, these factors determine the albedo and emissivity of the atmosphere. Together with latent heat release, the radiative properties of clouds alter atmospheric buoyancy. Thus, from the moisture distributions shown in Figure 2.6, the radiative effects of water may be expected to possess a strong geographical variation.

JJA 4

DJF

2 0 0

20

40

60

80

Latitude 2.3.1.3 Moisture, Lapse Rates and Gradients of Atmospheric Buoyancy

Moisture content and the temperature of a parcel determine to a large degree the potential buoyancy of the parcel. Consider the lifting of saturated parcels at different latitudes assuming that the initial temperature is inversely proportional to latitude. Let each parcel start at 900 hPa and be lifted 1 km. The difference in temperature between moist adiabatic ascent and dry adiabatic ascent as a function of latitude is portrayed in Figure 2.7. For the same vertical displacement, assuming saturation has been reached, the relative temperature changes induced by the latent heat release are a factor of at least two larger in the tropics than in the middle latitudes during summer and a factor of three in winter. By extension of these thermodynamic arguments, it can be expected that convection in the warm pool regions of the tropical oceans will be deeper and more intense and associated with greater rainfall than convection at higher latitudes. The latter is confirmed by the difference between precipitable water, Wp (Figure 2.6), between the tropics and higher latitudes.

Figure 2.7 Temperature change difference as a function of latitude for DJF (solid curve) and JJA (dashed) between moist and dry ascent when a parcel is lifted 1 km above the 900 hPa level. Climatological temperature values are used as initial data. Near the equator, dry ascent of a parcel will cool by 5–6 ∘ C more than a saturated parcel. At higher latitudes, the difference is less. The latitudinal difference in ΔT is a result of the exponential increase of saturated vapor pressure as a function of temperature (Figure 1.2). The nonlinear properties of water allow 3–5 times the temperature gain for the same amount of work in lifting a tropical air parcel compared to the same action at higher latitudes.

The greater potential release of latent heat through ascent in the tropics compared to the higher latitudes for the same amount of work against gravity has many important implications for the tropical lapse rate dT/dz. Consider a dry parcel undergoing a slow vertical displacement. As the environmental pressure decreases with ascent the parcel expands and thus does work on the environment. The rate of change of temperature with height in a dry atmosphere is given by g dT = Γd =− dz Cp

(2.9)

45

2 Hydrological and Heat Exchange Processes

where g is the acceleration due to gravity and C p is the heat capacity of dry air. Γd is referred to as the dry adiabatic lapse rate.V Physically, Eq. (2.9) is easy to understand for a system in a state of hydrostatic balance where the gravitational force (directed downwards) is balanced by the pressure gradient force (directed upwards). Consider a small element of thickness Δz with pressure p1 above and p2 below where p1 < p2 . If the downward force of gravity is balanced by the upward force of the pressure gradient across the slab, then: 𝜕p0 = −𝜌0 g|Δz→0 (2.10) (p1 − p2 )∕Δz = −g𝜌 or 𝜕z where 𝜌0 and p0 represent the background environment. Such a balance is referred to as hydrostatic equilibrium. Equation (2.10) states that the vertical pressure gradient is determined by gravity so that the stronger g, the greater is 𝜕p0 /𝜕z. Thus, for a unit vertical displacement, a parcel will be forced to expand more depending on the magnitude of g. Finally, the degree of cooling will be inversely proportional to the specific heat of the gas (C p : J kg−1 K−1 ). As long as there is no mixing, the displacement is thermodynamically reversible. Consider, now, a moist column of air in hydrostatic equilibrium. Assume, once again, that a parcel is incrementally raised vertically in the atmosphere. The parcel will cool as it expands, working against the environment but, depending on the vapor pressure of the parcel, condensation may occur, liberating latent energy and warming the column. As the parcel is raised further, condensation will continue until all vapor is converted to liquid water. The column now has within it a cloud. If there is no precipitation, the cloud will eventually evaporate and the net heating of the column will be naught, although there would have been a redistribution of heat vertically. When there is precipitation, a fraction of the water mass leaves the atmospheric column and there will be a net warming from condensation. The lapse rate in the moist tropics is less than at higher and drier latitudes. Formally, Γs , the moist adiabatic lapse rate, can be written as ) ) ( ( rLE 2 es LE es / 1+ (2.11) Γs = Γ d 1 + Rd T cpd Rd T 2

Moist Adiabatic Lapse Rate (K/km) 3

200

Pressure (hPa)

46

400

600

800

1000 200

7

6 5

240 260 Temperature (K)

280

9

220

8

4

300

Figure 2.8 Moist adiabatic profiles defined by Eq. (2.11) as a function of temperature and pressure. For “cool” temperatures and hence low specific humidities, the lapse rates are close to the dry adiabatic of −9.8 K km−1 (left-hand section of the figure), but as temperatures and saturation vapor pressures increase (right-hand section of graph), the vertical lapse rates decrease.

profiles defined by Eq. (2.11) as a function of temperature and pressure. As the temperature decreases and the saturated specific humidity also decreases, the lapse rates are close to the dry adiabatic lapse rate of −9.8 K km–1 . However, as temperatures and saturation vapor pressures increase, the moist adiabatic lapse rates decrease. 2.3.2

Ocean

where Rd is the gas constant for dry air, Rs is the gas constant for moist air, and r = RS /RD . The saturated vapor pressure is given by es and Γd is defined in Eq. (2.9).6 Note that the denominator is larger than the numerator so that Γs < Γd with Γs decreasing with increasing temperature. These dependencies are displayed in Figure 2.8, which shows the moist adiabatic

In the previous section we have considered the vertical structure of a moist and compressible atmosphere in which phase changes occur as motion takes place from one part of the atmosphere to another. We now consider the thermodynamics of an almost incompressible fluid: the ocean. It is conventional to plot profiles of ocean density as a function of temperature and salinity on a so-called “temperature–salinity or (T–s)” diagram, such as shown in Figure 2.9. There is no analytic equation of state of seawater such as we have found for an ideal gas, and the relationship between T, s, and density has to be determined empirically.7 The isopleths represent density (kg m–3 ) less the density of pure water (1000 kg m–3 ). Thus, an isopleth labeled 20 refers to a density of 1020 kg m−3 . In all circumstances saline water is denser than pure water. The relationship is highly nonlinear and suggests that the density will be differentially sensitive to both temperature and salinity.

6 A complete derivation of Eq. (2.11) is given in Chapter 4 of Curry and Webster (1999).

7 Curry and Webster (1999) provides an extensive description of ocean thermodynamics.

2.3 Water and the Tropical System

(a) T-s characteristics

(b) T-s profiles

35

35 17.7 19.1 20.6

Temperature (oC)

25

15

B

25

Y 26.5 27.9

S

25.0

29.4

V

C

A

0

E

ΔTC

X

5

30

D

23.5

10

–5

ΔsW

22.1

20

W

ΔTW Temperature (oC)

30

32

22.1

15

23.5

0 100

30°N 100

26.5 27.9

200 200

29.4

10

60°N 0

25

100

600 1000 1000 2000 5000

30.9

0

ΔsC 30

EQU 0

20.6

20

5

30.9

19.1

34

36

38

40

–5 30

Salinity (psu)

32

34

36

38

40

Salinity (psu)

Figure 2.9 (a) Sensitivity of ocean density as a function of temperature and salinity. Isopleths denote density (less 1000 kg m−3 ). Two water masses are chosen representing typical tropical surface water (marked “W”) and high-latitude water (“C”). The nonlinear aspects of ocean density are illustrated by the mixing of two water parcels (A and B) of equal mass to produce a denser parcel X. Mixing of two tropical water parcels D and E of the same density will produce a parcel (Y) that is almost the same as D and E. The two solid circles marked “S – surface” and “V – volume” show the mean surface and volume temperatures and salinities of the world’s oceans from Figure 2.10. (b) The temperature and salinity of three typical northern hemisphere water columns at the equator (red), 30∘ N (green), and 60∘ N (blue), plotted as a function of temperature and salinity. Points at 100 m, 200 m, …, 1 km, 2 km, and 5 km are indicated on the profiles. Most of the differences between ocean columns occur in the first few hundred meters. The tropical profile is capped by a warm and fresh surface layer. In contrast, the subtropics are cooler and more saline. At high latitudes the surface water is still cooler but fresher. At depths greater than 600 m, the characteristics of all profiles become very similar, denoting cool very saline abyssal water. Source: Data are from Levitus (1982).

Consider two water masses in Figure 2.9a marked “W” and “C” representing warm and cold water parcels, respectively,. Density increases with salinity in both masses. Density decreases with increasing temperature. Now consider the required changes in temperature to change the density by 2 units (25 to 23 for W and 27 to 25 for C). The temperature change required for W is far less than for C; that is, ΔT W < ΔT C . On the other hand, the change in salinity is similar for both parcels so that ΔsW ≈ ΔsC . These relative sensitivities are quantified in terms of two coefficients: thermal expansion (𝛼 th ) and salinity or haline contraction (𝛽 S ), defined formally as 𝛼th = −

1 𝜕𝜌 || 𝜌0 𝜕T ||s

and

𝛽s =

1 𝜕𝜌 || 𝜌0 𝜕s ||T

(2.12)

respectively. Overall, 𝛼 th changes most drastically as a function of temperature, with a factor of four difference in magnitude between 2.5 and 30 ∘ C (from 781 to 3413 × 10−7 K−1 ). The salinity contraction coefficient (𝛽 s ), on the other hand, varies far less percentage-wise between the same water masses (8010 × 10−7 to 7490 × 10−7 psu−1 ). In the tropics the processes that affect temperature will thus have a greater effect on density than processes that affect salinity. This differential sensitivity will turn out to be an important factor in determining the buoyancy of the upper ocean.

There is another nonlinear property of ocean density, although one that is not particularly important in the tropical upper ocean. Consider mixing two equal volumes of water masses A and B, each with the same density but different temperatures and salinities. The density of the mix is (A + B)/2 (point X) that is greater than either of the initial parcels and this denser parcel will be negatively buoyant. The increase in density is due to a slight decrease in volume of the mixture. This is referred to as “cabbeling”.8 Given the “shape” of the isopycnals, it would seem that the density differential of the mixture would be greatest when the ocean is very cool. If a similar mixing of water masses were made between D and E for the warm ocean column, the resultant water mass, Y, would have almost the same density as either D or E. Figure 2.9b shows three climatological vertical ocean sections at the equator, 30∘ N and 60∘ N. Below 1 km, each sounding coalesces to common values that represent a dense, cool regime with temperatures in the 3–4 ∘ C range and high salinity values near 34.5–35 density units. Above this level the T–s structure is very different for the tropics compared to the higher latitudes, with the tropics being both warmer and fresher and the subtropics cooler and more saline. Overall, there 8 McDougall (1987).

47

2 Hydrological and Heat Exchange Processes

are three main sectors in the ocean: a cold and high salinity deep ocean, a warmer upper ocean relatively fresh layer, and an intermediate transition zone, referred to as the thermocline.

These regimes can be examined further from a volumetric and areal perspective of the entire ocean. Figure 2.10a plots the ocean temperature and salinity distribution by per thousand (‰) volume as a function

(a) Volume (o/oo) Temperature (oC) 0

10

20

SUM

30

0

40

0 38

0

Salinity (o/oo)

1

2

2

2

36 46 422 287 81 4

1

1

38 24

5

5

9 12

6

4

3

2

1

1

1

1

1

1

7 11 5

3

2

2

2

1

1

1

1

2

1

1

1

1

1

34

1

926 55 8 0

32

0 0

30

0

SUM 47 424 293 88

44

31 19 17

(b) Surface (o/oo)

9

7

4

4

4

3

3

2

1

Temperature (oC)

0

10

20

SUM

30

0

40

38

Salinity (o/oo)

48

36

1

1

1

6

6

1 4

2

1

11

35 24 16 10

7

4

9

12

22 23

5

8

2

9

15

17

1

3

3

5

5

4

34 1

7

6

2

39

21

6

4

1

1

1

7

156

23

18

17

14

12 30 71 69

366

6

14

25 28

46 56 81 30

312

2

5

9 8

3 1

32

2

3

3

18 37 15 4

2

1

95

4

17

2 2

3 1

1

4

30 SUM 21

54 34

35

32

30 33 34

41

46 52

64 82 131 200 109

0

0

Figure 2.10 Inventory of ocean water by temperature (∘ C) and salinity (‰) by volume (a) and surface (b). The summed temperature and salinity are shown on the abscissas and ordinates. The boxed areas enclose temperature and salinity pairs occupied by more than 2% volume or area. Over 70% of water by volume is less than 3 ∘ C. However, by surface area, nearly 60% of the water has a temperature greater than 18 ∘ C. Such temperatures occupy less than 2% by volume. The numbers along the abscissas and ordinates (marked as “SUM”) denote the per mil distribution of temperature and salinity. The mean temperature and salinity values by volume and surface area are plotted on Figure 2.9a as the two red circles “V” and “S,” respectively.

2.3 Water and the Tropical System

of temperature and salinity. Overall, the temperature of the ocean is very cool with a mean temperature near 4 ∘ C. The mean salinity of the ocean is about 35.5 ‰. This mean value is plotted in Figure 2.9a as the full red circle labeled V. Water with temperatures >300 K (27 ∘ C) occupies only about 0.08% of the total water mass! Thus, the warm SSTs in the tropics associated with deep convection and precipitation, the drivers of the large-scale circulations such as the Hadley and Walker cells, constitute only an extremely small part of the total ocean mass. The vast majority of the ocean is comprised of cool and saline water. If the oceans were pure water, the 277 K (4 ∘ C) water would be close to maximum density. However, depending on the salinity of an ocean parcel, maximum density can be closer to the freezing point of fresh water. Figure 2.10b shows the temperature and salinity distribution as a function of surface area (km2 ). A very different distribution emerges compared to that determined by volume. The mean surface SST is 292 K (19 ∘ C) while the mean salinity is 35.2 density units (i.e. 1035.2 kg m−3 ), far fresher than the deep ocean. About 11% of the surface is covered by water >28 ∘ C, 31% >26 ∘ C, and 57% >20 ∘ C. Thus, the surface water, so critical for the thermodynamical forcing of atmospheric circulations, is relatively warm and fresh, and essentially a skin layer overlaying a cold and saline deep ocean with a connecting layer, the thermocline. A simple calculation shows that if water >20 ∘ C were spread around the globe it would only be about 2 m thick compared to the average ocean depth of 4 km. 2.3.2.1

Ocean Surface Layer: Warm and Fresh

The physical processes that maintain the warm and fresh tropical surface layer remained something of a mystery until careful observations were made during the joint U.S.–Australian West Pacific Ocean Climate Study (WEPOCS).9 This was the first time that serial vertical profiles of temperature and salinity of the warm pool region were made across a range of environmental conditions. Before WEPOCS, the consensus of opinion was that the upper reaches of the tropical ocean were well mixed through the first 100 m and homogeneous, such as appears in Figure 2.11a. There, the difference between the vertical profile of actual density of sea-water in density units (𝜌 less 1000 kg m−3 , 𝜌′ ) is plotted in addition to profiles of temperature and salinity. However, the WEPOCS observations showed that this homogeneity was relatively rare and profiles like in Figure 2.11b were 9 Lindstrom et al. (1987) and Lukas and Lindstrom (1991). There were two phases of WEPOCS, June–August 1985 (WEPOCS I) and January–February 1986 (WEPOCS II).

far more common. In fact, the homogeneous profile only occurred following a severe wind event such as a westerly wind burst event or the passage of a tropical cyclone. Lukas and Lindstrom (1991) noted that the stratification of the most common tropical upper ocean profile was very stable and relatively buoyant. They coined the term “barrier layer,” labeled in Figure 2.11b, as the stable region reducing the entrainment of colder thermocline water that would otherwise reduce the surface temperature. In the development of the stable upper ocean, they noted the importance of the freshening of the warm pool surface by precipitation. For example, the tropical warm pool of the western Pacific receives about 5 m year–1 of precipitation and 3 m year−1 of evaporation, resulting in a net 2 m year–l of fresh water flux. The stable surface layers of the tropical warm pool are then a direct result of the precipitation excess in addition to strong radiational heating. Thus, the warm and fresh ocean surface layer is a distinct signature of coupled ocean–atmospheric hydrological processes.VI 2.3.2.2

Abyssal Water: Cold and Saline

The mean temperature of the deep ocean is warmer than the primitive planetary surface temperature but considerably cooler than the planetary radiative surface temperature when absorbing gases are taken into account. This radiative temperature is about 284 K (11 ∘ C), assuming a global mean emissivity of 0.7 and an albedo of 0.3, or 10 K warmer than the deep oceans. Therefore, the temperature of the deep water is not determined globally by radiative loss to outer space. In fact, over most of the planet the deep ocean does not interact directly with the atmosphere. Once laid down, the deep ocean is well insulated by a stable warm and fresh surface layer. One thought was that the temperature of the deep ocean is a remnant of past cooler climates. However, the residence times of the deep ocean appear to be too short and are estimated at 1000–2000 years (see Section 2.1). Therefore, a more rapid source of the cold deep water is needed. Over 200 years ago, Count von RumfordVII speculated that: … if the water of the ocean, which, on being deprived of a great part of its heat by cold winds, descends to the bottom of the sea, cannot be warmed where it descends, as its specific gravity is greater than that of the water at the same depth in warmer latitude, it will immediately begin to spread out on the bottom of the sea, and flow towards the equator, and this must produce a current at the surface in an opposite direction … von Rumford (reproduced in 1969)

49

2 Hydrological and Heat Exchange Processes

(a) Well-mixed upper ocean 0 ρʹ

s mixed layer

T

s

50

100

barrier layer

100

150

150

24

26

34

35

T(oC)

28

s(psu)

22

30

36

24

26

23

In essence, von Rumford noted that the deep ocean temperature could only be created by the loss of heat caused by the cold continental winds blowing across and cooling the high latitude oceans. In this he was slightly in error as the concept of radiational cooling to space was still 75 years in the future. Support for the von Rumford view can be found in Figure 2.10b. Only 5–6% of the globe has SSTs that match the mean temperature of the deep oceans. Thus, these areas in the northern Atlantic and along the Antarctic coast are likely to be the source regions of the abyssal water; that is, high latitude surface water, cooled radiationally or by loss of heat through encountering the cold dry continental winds, achieves close to maximum density and sinks to the deep ocean. The Thermocline

The thermocline is the zone of transition between the surface layer and the abyssal water with a sharp vertical temperature gradient evident in Figure 2.9b. There are various sources of this intermediate water, including subducted subtropical water and water masses advected from higher latitudes. From a dynamical perspective in the tropics, the thermocline is extremely important.

2.4 Buoyancy, Differential Buoyancy, and the Generation of Horizontal Body Forces Here we consider the concept of buoyancy in both the ocean and the atmosphere and how radiative forcing, modified by hydrological processes, drives tropical circulations. First, we need to determine the factors that

T(oC)

28

30

35 s(psu)

34

ρʹ (kg m–3)

2.3.2.3

mixed layer

ρʹ

T

50

Figure 2.11 Profiles of upper ocean potential temperature (T ∘ C), salinity (s ‰), and density (𝜌 ’ = 𝜌 − 1000 kg m−3 ) for (a) a very well mixed upper ocean layer and (b) a density-stratified upper ocean. Note the distinctive fresh warm lens near the surface in the stratified profile produced by strong insolation and heavy rainfall over the warm pool. Source: After Lukas and Lindstrom (1991).

(b) Density-stratifield upper ocean

0

depth (m: dbar)

50

22

36 23

ρʹ (kg m–3)

produce buoyancy. Second, we need to understand how buoyancy, differing between neighboring columns, can produce horizontal body forces that drive atmospheric and oceanic motion.

2.4.1

Concept of Buoyancy

Vertical motion in a fluid parcel occurs if there is differential heating per unit mass of the fluid between itself and its neighbor. Essentially, if a parcel has a lower or greater density than its neighbor, it will be more or less buoyant. Consider an environment that is in hydrostatic balance where the force of gravity is balanced by the vertical pressure gradient force as expressed in Eq. (2.10). Consider now a small vertical displacement of a parcel within this environment that has an initial pressure p′ . Following Newton’s second law of motion, the vertical acceleration of the parcel is equal to the sum of the pressure gradient and gravitational forces. That is: dw 1 𝜕p′ = −g − dz 𝜌 + 𝜌′ 𝜕z

(2.13)

where 𝜌′ 0), producing the jet streams evident in Figure 1.4. Jet streams are finite in latitude extent whilst, as can be seen from Figure 1.1, the temperature gradient continues toward the poles. Therefore, on the poleward side of the jet there must be an efficient mechanism to transfer momentum (and heat) poleward. The jet stream is unstable to perturbations, as found by Charney and Eady in the late 1940s. The process is baroclinic instability that occurs in the form of large-scale eddies.

T the temperature of the gas, we obtain an expression for the change of pressure with height:

2.5 Integrated Column Heating

p dp =− g (2.16) dz RT This is referred to as the “barometric equation”.10 If we assume, for simplicity, that the columns are isothermal,11 we find after integration between z = 0 and z = z that: ( g ) z (2.17) p(z) = p(0) exp − RT Consider, now, two adjacent atmospheric columns with temperatures of T 1 and T 2 , where T 2 > T 1 is consistent with Figure 2.12c. From Eq. (2.17) the pressure in the equatorward (warmer) column will decrease more slowly with height compared to the cooler, more polar, column. Thus, at any height z = z0 , the pressure will reduce toward the poles promoting a pressure gradient force toward higher latitudes. That is: ( ) p2 (z0 ) − p1 (z0 ) p(0) g = exp − z Δy Δy R(T2 − T1 ) (2.18)

While the latitudinal gradient of net radiative forcing of an atmospheric column is the ultimate forcing factor of motion, the radiative stream reaching the surface of the planet is transformed, at least partially, into different forms of heating.

The warmer column does work on the cooler column promoting the poleward advection of mass and heat.X The generation of this latitudinal force is, in essence, the result of differential buoyancy between the two columns. Values of the vertical variation of pressure and the induced latitudinal pressure gradient force are listed in Table 2.2. This simple argument will produce a vertically varying body force between two columns with different temperatures. However, the atmosphere is rotating and horizontal gradients of temperature translate into 10 See Berberan-Santos et al. (1997) for an extremely interesting historical discussion of the “barometric formula.” 11 Calculations with T(z) with full variability are more complicated but the results are in agreement with the broad properties found here using isothermal approximation.

2.5.1

Components of Total Heating

Incident radiation heats the planetary surface, which is transferred to the atmospheric column through conduction in the skin layer. The heated surface, warmer than the atmosphere, radiates back to the atmosphere. The third form of heat transfer is the flux of latent heat, LH , where radiative heating evaporates water either from the moist land surface or from the ocean. The resultant increase of water vapor in the atmospheric column, being at a higher phase form than the liquid water that was evaporated, has the potential of becoming available as a diabatic heat source if condensation occurs in the column. Condensation may occur in locations far removed from where the radiational energy was initially used to evaporate water. The total heating of an atmospheric column, QT (W m−2 ), may be written as QT = RNET + SH + LP

(2.19)

where RNET is the net radiation in and out of the column, SH is the sensible heating of the column, and LP is the latent heating of the column from the release of latent heat of condensation occurring during the precipitation process. Differences in the net heating between adjacent columns develop differential buoyancy, producing forces that drive motion and drive atmospheric and oceanic advection. Water vapor that is evaporated at the surface of the planet comes from the surface latent heat flux LH .

53

54

2 Hydrological and Heat Exchange Processes

LP =

w

approximation as

L E P (W m 2 )

n–1

n

SH ≈ Cp 𝜌w′ T ′

n+1

(2.23)

Similarly, we can write an expression for the latent heat flux: LH (in)

Lp

LH ≈ LE 𝜌w′ q′v

LH (out) P

LH (y) Figure 2.13 Schematic representation of the relationship of latent heat flux (LH ∶W m−2 ) and the release of latent heat in an atmospheric column (LP , W m−2 ). Precipitation P is the result of the net dynamic convergence of moisture into a column.

LP comes from the proportion of water vapor that converges into the column that precipitates, i.e.: LP = 𝜌w LE P

(2.20)

where P (m3 s–1 ) is the total precipitation occurring within the column. Figure 2.13 presents a schematic diagram relating LH and LP . The vertical transfers of sensible and latent heat are the result of turbulent motions in the planetary boundary layer. Surface heating creates an unstable lower boundary layer that is ameliorated by turbulent eddies that transfer the heat upwards, replacing it with cooler air from aboveXI . Friction at the surface of the atmosphere can also create turbulence. This transfer renders the boundary layer stable. Transfers of heat, momentum, and moisture are accomplished if the upward moving eddies have properties different from the downward moving eddies. Thus, turbulence will move water vapor upward and into the atmosphere if the surface air has a higher specific humidity than the air above. The upward transfer of sensible heat is given by the product of the vertical velocity and the temperature of a parcel. That is: SH = Cp 𝜌wT

(2.21)

where C p is the specific heat of air and 𝜌 is its density. Here, w represents vertical velocity. The overbar represents a time average. It is simpler to expand quantities into time mean and deviations from that mean such that wT = (w + w′ )(T + T ′ ) = wT + w′ T ′

(2.22)

where we have taken into account that time means of deviations (i.e. w′ and T ′ ) are zero by definition. Also, the amplitude of vertical velocity of the turbulent eddies is far greater than the mean vertical velocity, so that we can rewrite Eq. (2.21), the flux of sensible heat, to a good

(2.24)

where LE is the latent heat of vaporization or condensation. Together, Eqs. (2.23, 2.24) state that if the vertical velocity deviation is correlated with the temperature or moisture deviation, then there will be a positive or negative vertical transport of enthalpy or moisture, respectively. For example, if the product of w′ and q′v is positive, on average, then there will be an upward transport of latent heat into the column. If the time variations of temperature, specific humidity, and vertical velocity can be measured, then these important fluxes can be calculated directly. However, such measurements require sophisticated instrumentation. To circumvent this problem and to allow calculation of surface fluxes from more easily observed quantities, empirical relationships have been developed.12 Turbulence is assumed to be proportional in a linear sense to the mean boundary layer wind speed U times a drag coefficient, C D . The other quantities needed are vertical gradients of temperature and moisture measured between the surface and a standard height above the surface. These so-called “bulk aerodynamic” formulations of fluxes of heat into the column are expressed as: SH = Cp 𝜌CD |V a |(T0 − Ta ) and LH = LE 𝜌CD |V a |(q0 − qa )

(2.25)

Here T 0 is the surface temperature and q represents the specific humidity. The subscript “a” and “0” refers to measurements made at an elevation of 10 m above the surface and at the surface. |V a | represents the magnitude of the lower boundary layer (10 m) wind speed. Over the ocean q0 is assumed to be saturated vapor pressure for the SST. 2.5.2 Latitudinal Distribution of Latent Heat Flux and Condensational Heating In the paragraphs above, we anticipated that the source of the water vapor in a column may come from evaporation in different locations, perhaps greatly removed. The zonally averaged distributions of both quantities during the DJF and JJA seasons appear in Figures 2.14a and b, respectively. During both seasons there is a distinct difference in the distributions [LH ] and [LP ], especially in the subtropics. The subtropics are the regions of 12 Based on the pioneering work of Priestley and Taylor (1972).

2.5 Integrated Column Heating

10

10 (a) JJA

8

[LH]

6

6

[LH]

1014 W

1014 W

(b) DJF

8

4 [Lp]

2

4 2

[Lp]

0

0 −80 −60 −40 −20

0

20

40

60

−80 −60 −40 −20

80

0

20

40

60

80

Latitude Figure 2.14 Comparison of the latitudinal distribution of mean zonal latent heat flux (LH ) and the condensational heating (LP ) for (a) DJF and (b) JJA (units: 1014 W).

10

10 (b) DJF

(a) JJA

8

[LP]

6

6

4

4

2

[SH]

0 –2

[QT]

1014 W

1014 W

8

Total column heating (1014 W)

[RNET]

2

[SH]

0 –2

–4

–4

–6

–6

–8

[LP] [QT]

[RNET]

–8 −80° −60° −40° −20° 0° 20° 40° 60° 80° Latitude

−80° −60° −40° −20° 0° 20° 40° 60° 80° Latitude

Figure 2.15 Zonally averaged net heating of an atmospheric column ([QT ], red curve) and its components, condensational heating ([LP ], green), sensible heating ([SH ], blue), and net radiation ([RNET ], black) according to Eq. (2.19) for (a) DJF and (b) JJA. Units are 1014 W, thus taking into account the spherical nature of the planet.

large evaporation that converge toward the low latitudes where maximum precipitation occurs. The profiles suggest that the majority of the tropical precipitation in the summer hemisphere originates in the winter hemisphere subtropics.

2.5.3 Latitudinal Distributions of Total Columnar Heating The zonal and annual averages of columnar heating components of Eq. (2.19), as well as the net heating, are shown in Figure 2.15a and b for DJF and JJA. At the surface, net radiation heats the surface and evaporated surface water. In this manner, a portion of the radiative heat is transferred into both sensible and

latent heating. Due principally to the condensational heating, the total heating, QT , is generally positive in the summer hemisphere whereas the total heating in the winter hemisphere is negative. In the development of the simple poleward body force expression in Eq. (2.18), we had considered only radiative forcing to determine differential buoyancy. Now we see that the buoyancy of a column depends on QT , which includes both latent and sensible heating in addition to the net radiational heating of the column. The difference in net heating from one column to the next (i.e. ΔQT (y)) constitutes the net zonally averaged latitudinal forcing. Within the uncertainty of the estimates, an approximate balance between planetary heating and cooling

55

56

2 Hydrological and Heat Exchange Processes

occurs. There are a number of interesting characteristics of the latitudinal distribution of the heat budget: (i) The maximum in total columnar heating lies in the summer hemisphere. (ii) Greater radiational cooling occurs in the subtropics of the winter hemisphere than in the summer hemisphere because of the incidence of deep convection in the summer low latitudes and the subsequent reduction in radiational loss to space. (iii) The maximum release of latent heat (LP ) does not correspond to the location of maximum evaporation. Maximum evaporation occurs in the subtropics and through dynamical processes a convergence of LH occurs, usually in the vicinity of the warmest SST (e.g. western tropical Pacific) or, as we shall see later, in the summer hemisphere poleward of strong cross-equatorial pressure gradients. (iv) The strongest sensible heating of the atmospheric column occurs in the subtropics associated with the great continental deserts.

The subtropical distributions of QT (Figure 12.16b) are very different from their equatorial counterparts. The JJA heating is dominated by LP associated with the monsoon rainfall over southern and eastern Asia and the reduced radiative loss to space due to the associated deep convection. The boreal summer total columnar heating has a maximum over Asia, but QT rapidly decreases to the east in accord with the decrease of LP . Over the more arid regions of western Asia and Africa, there is a substantial increase in SH . The DJF profiles show little contribution from LP , as expected. Overall, QT is universally negative across the boreal winter (DJF) subtropics. Figures 2.15 and 2.16 illustrate that the meridional and latitudinal gradients of the total heating are much the same. This similarity is testimony to the importance of moist processes in determining the magnitudes of both radiative and latent heating contributions. 2.5.5

2.5.4 Longitudinal Disposition of Total Columnar Heating We now discuss its longitudinal variability. Figure 2.16a and b shows plots of QT and its components LP , SH , and RNET along the equator and 25∘ N for both solstitial seasons. Along the equator, LP and RNET dominate QT , the total columnar heating. The two solstitial distributions are very similar. There are three maxima in the LP distribution, corresponding to the Indonesian western Pacific Warm pool, Africa, and South America, which are collocated with the precipitation maxima identified in Figure 1.6. The distribution of RNET (middle panel of Figure 2.16a) reflects the decrease in longwave loss to space in regions of deep convection. In the less cloudy regions of the equatorial eastern Pacific Ocean, the net radiative loss to space is about 130 W m−2 , but in the western Pacific Ocean the radiative loss is a factor of two smaller. The resulting distribution of QT (panel iii) has a broad maximum extending across the eastern hemisphere with a smaller maximum over equatorial South America. Net cooling, attributed to the minimum SST and generally subsiding air, exists across the eastern Pacific and, to a lesser extent, across the equatorial Atlantic. Clearly, two cooperative heating gradients determine the total heating along the equator. Condensational heating decreases to the east across the Pacific, corresponding in accordance with the radiative loss increasing to the east. Both contribute to the same gradient of total heating with the total dominated by LP by a factor of 5 : 2.

Annual Cycle of Total Columnar Heating

Figure 2.17 displays the spatial distribution of total columnar intergrated heating (CIH) for all four seasons. Warm colors represent net columnar heating while cool colors represent net cooling. Over the oceans, regions of net heating coincide with regions of maximum SST. Net heating also occurs over the summer subtropical continents at the time of the solstices, or the equatorial land masses at the time of the equinoxes. Maximum heating also occurs in the regions of convergence along the equator, especially in the eastern Pacific Ocean north of the equator and south of the equator in DJF. Strong heating may be found in the South Pacific Convergence Zone (SPCZ) that stretches from the West Pacific warm pool to the southeast across the South Pacific. Columns are connected by the advection of moisture (Bq (𝜆, 𝜑) in Figure 1.9a) and its subsequent convergence or divergence. Strong columnar heating also occurs in the vicinity of both the Kuroshio CurrentXII to the east of Asia and the Gulf Stream off North America. Atmospheric heating exists in these locations in all seasons but maximizes in the boreal winter and spring. These maxima are the result of cold and dry continental air advection across the warm ocean, resulting in large positive values of LH and SH . The strongest columnar heating in the tropics occurs in the region of strongest convergence of Bq , which coincides with maximum LP and minimum RNET , such as South Asia during JJA and the Brazilian region in DJF. Columnar net cooling, on the other hand, is largest in the vicinity of the large-scale subtropical anticyclones in all three ocean basins. From Figure 2.16 these are

2.6 Buoyancy in the Tropical Ocean

(a) Along equator

(b) Along 25oN (i) Lp and SH

(i) Lp and SH 350

DJF JJA

300 W m–2

200

150

100

100 SH

50

0 0°

100°E

160°W

60°W 0°

(ii) RNET DJF JJA

−70 W m–2

SH

50

−50

0°

−110

−110

−130

−130 100°E

160°W

60°W 0°

(iii) QT

400

100

100

0

0

−100

−100 100°E

100°E

160°W Longitude

60°W 0°

−200

160°W

60°W 0°

(iii) QT DJF JJA

300 200

0°

DJF JJA

0°

200

60°W 0°

(ii) RNET

400 DJF JJA

300

160°W

−70 −90

0°

100°E

−50

−90

−200

Lp

200

Lp

150

0

DJF JJA

300 250

250

W m–2

350

0°

100°E

160°W Longitude

60°W 0°

Figure 2.16 Distribution of total columnar heating, QT , and its components along (a) the equator and (b) 25∘ N for DJF (blue) and JJA (red) as a function of longitude (units: W m−2 ).

the regions of minimum precipitation, minimal deep convection, and, consequently, large radiative cooling to space. Similar net cooling to space occurs over the great deserts of the planet. There is a clear association between SST and columnar heating. Figure 2.18 presents the total annual columnar heating occurring in 1 ∘ C SST bins (plotted as bars) between 30∘ N and 30∘ S. Units are PW (1015 watts). Also plotted is the average columnar integrated heating occurring in each of the bins (bold black line). In the current climate, 25%, 48%, and 27% of the

total heating occurs within the 27–28 ∘ C, 28–29 ∘ C, and 29–30 ∘ C bins, respectively. The transitional SST between heating and cooling occurred at about 27 ∘ C for the period 1979–2001, with convection occurring at warmer SSTs.

2.6 Buoyancy in the Tropical Ocean Analogous to the definition of atmospheric buoyancy, Eq. (2.14), the ocean buoyancy, BO , is a function of the relative density of an ocean parcel compared to average

57

58

2 Hydrological and Heat Exchange Processes

Figure 2.17 Distribution of seasonally averaged total atmospheric columnar integrated heating (CIH) in the tropics for (a) DJF, (b) MAM, (c) JJA, and (d) SON. Warm colors denote net heating and cold colors net cooling relative to the scales on the right. Units are W m−2 . Source: Data from the ECMWF ERA-40 Atlas available at: http:// nwmstest.ecmwf.int/research/era/ERA-40_ Atlas/docs.

Total Columnar Heating (W m–2) (a) DJF 40°N 20°N 0° 20°S 40°S 0°

60°E

120°E

180°W

120°W

60°W

0°

(b) MAM 40°N 20°N 0° 20°S 40°S 60°E

120°E

180°W

120°W

60°W

0°

60°E

120°E

180°W 120°W

60°W

0°

120°E

180°W 120°W

60°W

0°

(c) JJA 40°N 20°N 0°

W m–2

0°

300 200 100 0 –100 –200 –300

20°S 40°S 0°

(d) SON 40°N 20°N 0° 20°S 40°S 0°

60°E

conditions (e.g. Gill 1982). That is: 𝜌 −𝜌 Bo = g 0 ≈ g(𝛼th (T − T0 ) − 𝛽s (s − s0 )) (2.26) 𝜌0 where 𝛼 th and 𝛽 s are the thermal expansion and salinity contraction coefficients defined in Eq. (2.12). T 0 and s0 represent background environmental temperature and salinity, respectively. Thus, buoyancy changes as a function of the heat and fresh water flux at the ocean surface. 2.6.1

Net Heating of the Upper Ocean

The total heat flux at the ocean surface is given by: QS = RNET + SH + SP + LH

(2.27)

where RNET is the net radiation: RNET = SS − I S

(2.28)

where SH is the surface turbulent sensible heating of the atmosphere and LH is the latent heat flux into the atmosphere, the heat loss occurring through evaporation. In (2.28) the subscript S refers to the ocean surface. We have introduced a second sensible heating SP to account for the ocean cooling by rainfall. The usual assumption is that the temperature of raindrops is equivalent to the near-surface wet bulb temperature. This is about 0.2 ∘ C less than the SST and thus rainfall constitutes a negative sensible heat flux into the ocean. Gosnell et al. (1995) showed, using a theoretical model, that the wet-bulb temperature approximation is valid. The value of SP is generally small and within the error bars of the heat flux into the ocean. However, in periods of heavy rainfall it can be substantial, as illustrated in Section 2.7. LH and SH are defined in Eq. (2.27). For the ocean column, SH , SP , and LH are usually negative quantities. RNET is

2.6 Buoyancy in the Tropical Ocean

(a) Mean SST (°C) and CIH (1979–2001)

(b) CIH and average heating 300

4

200

2

100

0

0

40°N

CIH (PW)

20°N Equ 20°S 40°S

–100

–2 45°E

90°E 20

21

135°E 22

23

180° 24

135°E

25 26

27

90°E 28

Average CIH (W m–2)

6

20–21 22–23 24–25 26–27 28–29 22–22 23–24 25–26 27–28 29–30

45°E SST (°C)

1°C SST Bins

29

Figure 2.18 (a) Long-term annual mean SST distribution (1979–2001). Contours represent columnar integrated heating (CIH). Continuous black contours denote positive heating, dashed negative. Contours are plotted for ±20, 50, 100, 200, and 300 W m−2 . (b) Total annually averaged tropical atmosphere column integrated heating in the 30∘ S–30∘ N band plotted in 1 ∘ C SST bins (bars: units PW). The black line shows the average column integrated heating in each SST interval; 25, 48, and 27% of the total heating occurs between, respectively, the 27–28, 28–29, and 29–30 ∘ C SST contours for the 1971–2001 period. Column integrated heating distribution from the ECMWF ERA-40 Atlas available at: http://nwmstest.ecmwf.int/research/era/ERA-40_Atlas/docs. SST from NOAA Extended reconstructed SST v2 data. Source: From Hoyos and Webster (2011).

positive over the tropics, attaining only negative values in the wintertime higher latitudes. The annual mean heat flux into the ocean is plotted in Figure 2.19a. The major regions of heat gain occur in the eastern ocean basins corresponding, principally, to the solar heating maxima. The maxima along the equator in both the Atlantic and Pacific oceans arise through a combination of a solar flux maximum and a reduction in long wave loss at the surface by the deep convection. The major regions of heat loss correspond to the regions of strong evaporation by the trade winds and also in the vicinity of the Kuroshio Current and Gulf Stream, where there is a strong sensible and latent heat loss. Over the continents, there is no net annual heating. This is because, as distinct from the oceans, there is little or no heat storage in the surface layers of the land areas, and the surface heating must then be in balance over the year. In the desert regions where there is sparse moisture and cloud, the large solar flux is balanced by a sensible heat flux and upward longwave radiation. In more forested regions of the tropics (e.g. central Africa) the balance contains a latent heat component.

2.6.2

2.6.3

Distribution of Ocean F B Buoyancy Flux

To determine distributions of ocean buoyancy, it is necessary to understand buoyancy in terms of heat flux and fresh water flux. Following Gill (1982),XIII the total buoyancy flux, F B , is given by FB =

g𝛼th QS g𝛼 + g𝛽s s(P − E) = th 𝜌cw 𝜌cw

(RS + SH + SP + LH ) + g𝛽s s(P − E)

Fresh Water Flux into the Tropical Ocean

The surface fresh water flux (mm/day) is given by Fw = E − P

To the north of the equator, near the location of the Pacific ITCZ, P exceeds E by +6 mm day–1 (+2.1 m year−1 ). Even larger differences are found in the warm pool region of the western Pacific, with rates of +8 mm day−1 (3 m year−1 ). In the northeast Bay of Bengal and the eastern Pacific in the south near 120 ∘ W during JJA (not shown), the mean fresh water plus exceeds 20 mm day−1 or, totaled over a season, nearly 2 m/90 days, creating extremely strong ocean barrier layers. Areas where E > P are smaller in magnitude but are spread over much larger areas. For example, in the subtropical high-pressure zones, F w exceeds +5 mm day−1 (−1.8 m year−1 ).

(2.29)

Figure 2.19b displays the annual average distribution of F w . Blue and green hues indicate that E < P. Yellow hues indicate that E > P.

(2.30)

where cw is the specific heat of water. The terms on the right side of Eq. (2.30) denote the impact of net heating and fresh water flux on the buoyancy flux. If the net heating is positive (heat added to the ocean column), then a parcel will become more buoyant than its neighbors. If (P − LH ) > 0, when precipitation exceeds evaporation, buoyancy will also be increased. On the other hand, if

59

60

2 Hydrological and Heat Exchange Processes

(a) Annual net ocean-atmosphere surface heat flux (W m–2) 90°N 60°N

30°N Equ

30°S 60°S 90°S 60°E

120°E

180°

120°W

60°W

0° W

–500 –300

–200

–100

–50 –30 –10 10

30

50 70

net heating of atmosphere

100

200

60°E

m–2

300 500

net heating of ocean

(b) Annual net fresh water flux: E-P (mm/day) 90°N 60°N 30°N Equ

30°S 60°S 90°S 60°E

120°E –30 –17

–13

180° –10

–8

–6

P>E

120°W –4

–2

60°W

–1 –0.2 0.2

1

0° 2

4

6

60°E mm/day 8

10

P P, an increase of density and a loss of buoyancy. Source: Data from ECMWF ERA-40 Atlas, available at: http://nwmstest.ecmwf.int/research/era/ERA-40_ Atlas/docs.

2.6 Buoyancy in the Tropical Ocean

Figure 2.20 Components of the annual average ocean buoyancy flux F B from Eq. (2.31). (a) Total buoyancy flux, (b) buoyancy flux from heating, and (c) buoyancy flux from fresh water flux. Warm colors show an increase in buoyancy while cold colors denote loss of buoyancy. Note the relatively small contribution of fresh water flux compared to surface heating. Units: 10−8 m2 s−3 relative to the color bar on the right.

(a) Total Buoyancy Flux (10−8 m2 s−3) 50°N

8

25°N

4

0°

0

20°S

−4

50°S

−8 0°

69°E

120°E

180°E

120°W

60°W

0°

(b) Buoyancy Flux from Heating (10−8 m2 s−3) 50°N

8

25°N

4

0°

0

20°S

−4

50°S

−8 0°

69°E

120°E

180°E

120°W

60°W

0°

(c) Buoyancy Flux from Fresh Water (10−8 m2 s−3) 50°N

8

25°N

4

0°

0

20°S

−4

50°S

−8 0°

(P − LH ) < 0, the ocean will lose buoyancy. Similarly, buoyancy is lost if the net heating is negative. Figure 2.20a to c displays the total net buoyancy flux and the heating and fresh water components, respectively. Overall, the ocean buoyancy flux in the tropics is dominated by heating effects rather than the freshwater flux with a ratio of about 4 : 1. In fact, we had anticipated the ascendency of thermal over salinity effects on density in the examination of the (T–s) diagram in Figure 2.9. In the tropics, then, we can simplify Eq. (2.30) to F B ≈ 𝛼gQS ∕cw

(2.31)

This approximation has been used in early modeling efforts in tropical ocean–atmosphere prediction models of the EI Nino–Southern Oscillation phenomena.13 All of these patterns have a strong annual cycle. Figure 2.21a and b displays the total upper ocean buoyancy for DJF and JJA, respectively. Given the domination of the net heating on buoyancy, the large 13 E.g. Cane and Zebiak (1985) and Anderson and McCreary (1985).

69°E

120°E

180°E

120°W

60°W

0°

hemispheric scale cooling translates to a loss of buoyancy in the upper ocean in the winter hemisphere but a substantial buoyancy gain in the summer hemisphere. 2.6.4 Observations of Ocean–Atmosphere Fluxes in the Tropics On a day-by-day basis there is great variability of the surface energy balance with some components varying by orders of magnitude. Here we examine observational data showing how the fluxes vary between disturbed and calm periods within the tropics. 2.6.4.1 Western Pacific Ocean Circulation Experiment (WEPOCS)

A vivid depiction of the variability of SST and sea-surface salinity (SSS) is found in the scatterplot in Figure 2.22 (Lukas 1990). The data were collected in the western Pacific Ocean warm pool during a 24-day period of the WEPOCS I experiment,14 mentioned earlier. The labeled axes denote a number of regimes. 14 Described in detail by Lukas and Lindstrom (1987, 1991).

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2 Hydrological and Heat Exchange Processes

(a) DJF Total Buoyancy Flux (10−8 m2 s−3) 50°N

8

25°N

4

0°

0

20°S

−4

50°S

−8 0°

60°E

120°E

180°W 120°W

60°W

Figure 2.21 Same as Figure 2.20 except for means of F B averaged over (a) JJA and (b) DJF. Units: 10−8 m2 s−3 relative to the color bar on right.

0°

(b) JJA Total Buoyancy Flux (10−8 m2 s−3) 50°N

8

25°N

4

0°

0

20°S

−4

50°S

−8 0°

60°E

120°E

180°W 120°W

60°W

0°

A: Diurnal heating WEPOCS III, LEG 2

Days 187-211

30.4 30.2 30.0 Temperature (oC)

62

29.8

Figure 2.22 Scatterplot of SST and sea surface salinity (SSS) pairs obtained during a 25-day period of the Western Equatorial Pacific Ocean Circulation Study (WEPOCS) experiment in the equatorial western Pacific Ocean. The labeled straight lines, identified in the text, denote the impacts of diurnal radiative heating (A: red), precipitation (B: green), and changes in the upper ocean associated with strong winds (C: blue). Source: From Lukas (1990).

29.6 29.4 29.2 29.0

B: Rain 28.8 28.6 28.4 32.8

33.0

33.2

33.4

33.6

33.8

Salinity (psu)

34.0

34.2 34.4 34.6 C: entrainment/ evaporation

(i) Axis A (red) shows the impact of diurnal radiational heating and cooling. During these diurnal changes the salinity remains constant, irrespective of its initial value, and may be seen as a series of vertical data points. The strongest diurnal variations occur when the winds are lightest, being associated with minimal evaporation as expressed in the empirical relationship of Eq. (2.25). (ii) Axis B (green) illustrates the dual effect of rainfall on surface conditions, causing both SST and SSS to decrease. The lowering of SST results from two effects: the physical cooling of the surface layers by the raindrops, which are colder

than the SST (sensible heat of rainfall Sp ), and also from the reduced net radiation flux at the surface with increasing cloudiness during rain events. Surface salinity decreases due to the freshening effect of the surface layers by rainfall. (iii) Along Axis C (blue) the impact of entrainment and evaporative cooling. Strong winds cause deep-ocean mixing and the entrainment of cooler, saltier water into the surface layer. Evaporation, on the other hand, is also a strong function of wind strength and cools the surface while increasing surface salinity.

2.6 Buoyancy in the Tropical Ocean

IS

IS

S0

0 cm

z

1

10 1 10 20 A m 30 B 40 C 50 60

2

D cool

heat

cool

heat

Figure 2.23 Attenuation of radiation in the upper ocean: (left) infrared radiation and (right) solar radiation. Upward and downward longwave fluxes are denoted by IS↑ and IS↓ ; 𝜓 1 and 𝜓 2 are the attenuation depths of red and blue-green parts of the solar spectrum (S0 ). The near-infrared and longwave components are attenuated within a few micrometers of the surface of the ocean. Lettering (A–D) refers to mean thermocline depths, indicative of various regions in the tropics. Source: Adapted from Simpson and Dickey (1981).

Consider the processes occurring along Axis A. The ocean is heated by direct absorption of solar radiation, S0 . The red end of the solar spectrum is absorbed very close to the ocean surface, whereas the blue-green end of the spectrum e-folds at about 7–8 m.15 This differential solar absorption is illustrated in the right-hand section of Figure 2.23. This shallow attenuation depth occurs because of the impurities in the ocean column. If the ocean consisted of pure fresh water, the attenuation depth would be much greater, allowing heat to penetrate to deeper reaches of the upper ocean. The surface longwave radiation I S has two components, one upward and the other downward, i.e. IS = IS↑ − IS↓

(2.32)

As the surface of the ocean is generally warmer than atmospheric downward radiating bodies (e.g. cloud or a moist boundary layer) then I S > 0 as I S↑ > I S↓ . Both the upwelling and downwelling longwave streams are confined to the ocean skin layer. During the day solar heating is spread through the ocean column according to Figure 2.23, together with the downwelling atmospheric radiation. At nighttime, the heating of the ocean depends only on the net infrared radiation, which is confined to the surface layer. 15 Simpson and Dickey (1981) and Chang and Dickey (2004). Divers are well aware of the differential attenuation. With increasing depth, greens predominate and to see the full color of corals, for example, they carry a flashlight that produces white light.

Dynamic processes above the thermocline occur on much more rapid timescales than lower in the ocean. If the attenuation depth (e.g. 𝜓 2 in Figure 2.23) is greater than the mixed layer depth, then slower dynamic processes will have to adjust to the added heating. The lettering on Figure 2.23 shows the average thermocline depths for the summer extratropics (A), the average tropics (C), and the eastern (B) and western (D) equatorial Pacific Ocean, respectively. During warm events in the Pacific Ocean (i.e. during EI Niño episodes), the average thermocline depth across the entire Pacific Ocean is somewhere between B and D. The importance of the magnitudes of the attenuation depths is now apparent. If 𝜓 2 < z(A, …, D) then the radiative heating of the ocean column will be mixed rapidly by near-surface mixing processes. However, if 𝜓 2 > z(A, …, D), then the slower dynamics of the deeper ocean must compensate for the diurnal heating at depths greater than the thermocline depth. It has been suggested that during a period of rapid transition in the Pacific Ocean (e.g. during the onset of an EI Niño event), a fraction of the solar heating may be absorbed below the thermocline and would not be redistributed rapidly by mixed layer dynamics.16 2.6.4.2 Surface Fluxes in the Bay of Bengal During JASMINE

During the boreal summer of 1999, the JASMINE experiment17 was conducted in the southern and central sectors of the Bay of Bengal. Here we concentrate on Phase II of JASMINE as it encompasses both a quiescent and active period of the early southwestern Asian monsoon. This experiment was the second of a series, the first being the TOGA COARE (the TOGA Coupled Ocean–Atmosphere Response Experiment) conducted in 1993 in the western Pacific warm pool.18 Figure 2.24 shows the track of the NOAA Research Ship “Ronald H. Brown” and details of the track are listed in Table 2.3, but essentially it consisted of four north–south legs along 89∘ E between the equator and 11∘ N and two “star” patterns (top right-hand inset) at the northern limit. The star patterns were repeated continually over a five-day period. Figure 2.25 shows the trajectories plotted against time (Julian day) and latitude against a background of (a) OLR (W m–2 ) and (b) near-surface wind (vectors: m s−1 ). Here the two star patterns appear as the near-horizontal lines at 11.9∘ N and 11.2∘ N. The OLR during the Star 1 period indicated near-clear skies near calm winds. In contrast, 16 Woods (1984). 17 Webster et al. (2002b). 18 See Webster and Lukas (1993) for the motivation and design of TOGA COARE and Godfrey et al. (1998) for a summary of the results.

63

64

2 Hydrological and Heat Exchange Processes

(a) Phase II: Jasmine 1999 20°N

150 (May30)

11.5°N

(b) TOGA COARE Array

20°N Wake

11.4°N o

15°N

star 1

145 (May 25)

11.2°N

star 2

Guam

10°N

11.1°N 11°N

10°N

Vaisala VIZ TOGA Radar MIT Radar

11.3°N

Chuuk

Yap

Pohnpei

Koror

LSA

Kwajalcin Majuro

OSA

140 (May20)

10.9°N 88°E 88.1°E 88.2°E 88.3°E 88.4°E 88.5°E

Kapingamarangi

Equ

5°N

Bjak

135 (May 15) Madang

0° 130 (May 10)

5°S

10°S

Govc Darwin

Tarawa

Moana Wave Nauro Shiyan 3 IFA Kexue 1

Manus Kaviong

Kanton

Funafuti

Thursday Islang

Misima

Honiara

Santa Cruz Pago Pago

Willis islang

10°S 80°E 85°E 90°E

125 (may 5) 95°E 100°E 105°E 110°E

20°S

Towvnsvillc

140°E

Efale

160°E

Nandi

180°E

Figure 2.24 Track of the NOAA research ship “Ronald H. Brown” in the southern part of the Bay of Bengal during Phase II of the Joint Air–Sea Monsoon Interaction Experiment (JASMINE) held during the boreal summer of 1999. During this phase of JASMINE, meteorological and oceanographic measurements were made along 89∘ E. Two star patterns, described in the text, were deployed at the northern end of these transects. Color bar on right shows the timing of the tracks. Source: From Webster et al. (2002b). Table 2.3 Track of the NOAA Research Ship “Ronald H. Brown” in the south-central Bay of Bengal during May and June, 1999. Right-hand column refers to points on Figure 2.17. Phase II:

Section

Dates 1999

Figure 2.17

Singapore–

Transect 1

5–10 May

1–2

Darwin:

Star 1

10–15 May

3

Transect 2

15–18 May

3–4

Transect 3

18–21 May

4–5

1 May–8 June 1999 (UTC: 122–159)

Star 2

21–26 May

5

Transect 4

26–29 May

5–6

Return to Darwin

30 May–June 9

Star 2 was very convective with heavy precipitation and strong winds. An extremely strong diurnal variation persisted through the disturbed period with convective lines propagating from the north at roughly 50 km h−1 (14 m s–1 ). During each of the intraseasonal events occurring throughout the summer of 1999, propagations of diurnal signals appear along trajectories 2000–3000 km in length with propagation speeds in the range of 50–60 km h−1 . The convective disturbances usually form in the early afternoon over the land areas in the northern reaches of the Bay of Bengal. The disturbances adopt a southwest–northeast orientation roughly parallel to the surface isobars and extend for 300–500 km. In the direction normal to propagation (perpendicular to the wind), the disturbances are much narrower with scales of about 50 km. These features have characteristics of propagating gravity waves.

In a multiyear study of intraseasonal monsoon variability, Hoyos and Webster (2007) found the Phase II-Star 2 to be typical of disturbed and periods of the monsoon, indicating similar ocean–atmosphere interactive processes operated over wide ranges of the tropics. Using a multiyear global cloud archive, Yang and Slingo (2001) also found persistent diurnal propagations in the Bay of Bengal with convection commencing over the land at the head of the Bay. A major objective of the JASMINE pilot study was to document surface fluxes during an intraseasonal oscillation of the South Asian monsoon. A further objective was to determine the degree of similarity of surface fluxes between Indian Ocean and Pacific Ocean intraseasonal variability, the latter being described by Godfrey et al. (1998) for the TOGA COARE experiment. The conditions encountered in JASMINE appear similar to “active” (i.e. pluvial) and “break” (dry) periods of the established monsoon (e.g. Webster et al. 1998). For example, the buoy data from the Indian surface moorings in the Bay of Bengal19 show SST oscillations of about 1.5 ∘ C. These intraseasonal time scales turn out to match the variations observed during JASMINE. Moderate winds occurring during the first northward transect in Phase II lessened to –250 W m−2 . The sensible heat loss by the ocean due to turbulent transfer and rain cooling made only a small contribution

to the net heating during the undisturbed period. However, SH became a larger contributor during Star 2 when increased winds enhanced low-level turbulence. Also, instantaneous values of the sensible heat loss caused by rain cooling of the ocean surface (QSP ) were occasionally greater than –200 W m−2 , but for relatively short periods of time. The average value of SP during Star 2 was −7 W m−2 . Overall, the differences in the net surface flux between the two star periods were caused by a severe reduction in net surface radiation and increases in the turbulent fluxes, although offset slightly by a decrease in net longwave radiation loss. The changes in turbulent fluxes can be accounted for, to a large degree, by the large increases in surface wind strength. Figure 2.27 shows the daily averaged net surface flux during JASMINE. The average during Phase II was +27 W m−2 . QS changed from positive in Star 1 (+92 W m−2 ) to negative during the course of Phase II (−89 W m−2 ). The reason for the change was that the incident surface solar radiation dimmed considerably

65

2 Hydrological and Heat Exchange Processes

(a) Surface Heat Fluxes during STAR 1 (W m–2) 1000

RAIN FLUX

0

SENSIBLE

50

NET LONGWAVE

100 LATENT

150

Net Heat Flux into Ocean: Phase II (W m–2) transect 1

Star 1

transect 2 Star 2 transect 3

100 Flux (W m–2)

Heat Flux (W m–2)

150

NET SOLAR

500

50

mean +27 W m–2

0 –50

200 –100

250

Net flux into ocean: +92 W m–2

300 142

143

144 Julian day

145

–150

146

125

130

135 140 Julian day

145

150

1000 Figure 2.27 Daily averaged net heat flux (units W m−2 ) into the ocean during Phase II of JASMINE. The shaded areas indicate the two star patterns. During the relatively undisturbed period, the total heat flux exceeded 100 W m−2 . During the disturbed period negative fluxes > −150 W m−2 were observed. Net flux into the ocean was calculated to be +27 W m−2 .

NET SOLAR

500 Heat Flux (W m–2)

66

0 50

RAIN FLUX

SENSIBLE NET LONGWAVE

100

similar. This result needs to be tempered by noting the different observational periods and the different times of the year.

150 200

LATENT

250 300

Net flux into ocean: -89 W m–2 142

143

144 Julian day

145

2.7 Translations to the Broader Scale

146

−2

Figure 2.26 Components of the surface energy balance (W m ) for (a) Star 1 and (b) Star 2. Five surface fluxes are shown: the solar radiative flux, sensible turbulent heat flux, latent turbulent heat flux, net longwave flux (outgoing surface minus incoming atmospheric), and the sensible heat flux of rainfall (units: W m−2 ). Note the change of scale for net solar radiation. Net flux into the ocean was +92 W m−2 during Star 1 and − 89 W m−2 during Star 2. Source: From Webster et al. (2002b).

during the convective period whilst latent heat loss increased. The SST of the North Indian Ocean in late spring is very similar to the SST in the Pacific warm pool (see Figure 1.5). The meteorology is similar such that both regions have prolonged stable periods (e.g. Star 1 of Phase II in JASMINE and the first 30 days of Figure 2.25). Both are influenced by propagating disturbances and both possess strong intraseasonal variability. It is instructive then to compare the components of the heat balance of the two regimes. This comparison appears in Table 2.4. Overall, the components of the surface heat balance are remarkably

In previous sections we have examined the distributions of columnar heating and net fluxes into the upper ocean and these assessments have allowed us to estimate buoyancy gradients in the coupled ocean–atmosphere system. We have also calculated the distribution of total heating in the atmospheric column in a zonally averaged sense (Figure 2.16a), as a function of longitude along the equator and 25∘ N (Figure 2.16b), and the columnar heating distribution throughout the tropics (Figure 2.18). Further, we have calculated the net flux of heat, in addition to the fresh water flux, into the upper ocean. We consider how these processes are associated with the major circulation systems of the tropics. 2.7.1

Large-Scale Columnar Heating Gradients

In considering Figure 2.16 we noted that a cloud-free column radiates more efficiently to space, causing a column to cool. At the same time, the surface below this column will heat because solar radiation is less impeded. However, in the presence of convection, the radiation balance within the column and at the surface changes considerably. Clouds will deplete the incident

2.7 Translations to the Broader Scale

Table 2.4 Comparison of the energy balance of the western equatorial Pacific Ocean and the northern Indian Ocean during TOGA COARE (November 1992–February 1993) and during JASMINE (April-September, 1999). Experiment

TOGA COARE

JASMINE

Undisturbed

QS

QL

QSEN

QLH

QSP

QT

+247

−57

−5

−84

−1

+99

Disturbed

+158

−43

−11

−150

−5

−51

Net

+198

−49

−8

−105

−3

+34

Undisturbed

+260

−49

−5

−115

0

+92

Disturbed

+128

−31

−17

−162

−7

−89

Net

+217

−42

−6

−109

−2

+62

solar radiation through reflection, but the column, now much more moist, will absorb surface-emitted longwave radiation and efficiently reradiate it back to the surface. However, the largest change occurs in the loss of radiation to space that is now largely from the cloud top and hence significantly reduced compared that which would occur in a clear column. In this manner, a radiative heating gradient develops between the convective regions and the suppressed, subsiding regions. If convection occurs at some location, the local release of latent heat provides further heating compared to the non-convective, non-precipitating column. The spatial latent heating gradient and the radiational heating are of the same sign, as indicated in Figure 2.16. For example, between the western and eastern Pacific Ocean along the equator, the magnitude of the radiational cooling difference is about 70 W m−2 (−60 W m−2 in the west compared to −130 W m–2 in the east). The latent heating gradient is roughly 240 W m–2 (300 W m−2 in the west, 60 W m−2 in the east). Although the latent heating gradient is a factor of about 3 or 4 greater than the radiational heating gradient, they are of the same sign. It is important to note that the radiational heating gradient does not exist until convection develops! Once there is convection, however, the overall longitudinal heating gradient is enhanced. We can make a first-order of magnitude assessment of the total atmospheric heating gradients associated with major circulation systems using data from Figures 2.16 and 2.17. We choose two major tropical atmospheric circulation features: the South Asian monsoon and the near-equatorial Walker Circulation. The differences in total columnar heating are marked on the divergent wind circulations shown in Figure 1.12a and b for JJA and DJF, respectively. Along the equator during JJA (Figure 1.12a(i)), there is a difference in total columnar heating on either side of the Indonesian-western Pacific region of 220 W m−2

to the west (between A and B) and 250 W m−2 to the east (between B and C). Associated with this distribution of heating are two east–west overturnings or Walker Cells, discussed in Section 1.3.2. Figure 1.12a(ii) indicates that the heating gradients along 25∘ N are of the same sign but 50% larger in magnitude compared to the equator. The east–west gradient between the arid regions of North Africa (D) and the Asian monsoon (E) is >200 W m−2 and between the monsoon of South Asia (E) and the eastern Pacific Ocean (F) is 350 W m−2 . This gradient is the result of columnar radiative cooling over the arid regions. The lower values of radiative cooling over the convective regions of Asia and the condensational heating are associated with monsoon rainfall. Two resulting zonal circulations are evident with maximum ascent over South and East Asia. Gradients of total columnar heating were also calculated similarly along 90∘ E for the June–September period. The strong monsoon circulation may be seen in Figure 1.12a(iii) with strong ascent over South Asia and subsidence over the South Indian Ocean. Between 30∘ S and 30∘ N there is a difference of 360 W m−2 in total columnar heating between (G) and (H.) A reverse, although weaker, gradient spans North Asia (H) and (I). Similar estimates of the heating gradients, but for the DJF winter monsoon, are shown in Figure 1.12b. The heating differences along the equator (Figure 1.12b(i)) are quite similar to those found in the boreal summer, also driving a strong Walker Circulation. The greatest difference occurs along 25∘ N where the magnitudes of the heating gradients have diminished by a factor of 4 (e.g. between (D) and (E) and between (E) and (F)). There is upward motion off the coast of East Asia but this is driven more by the heating associated with cold and dry continental air flowing across the warm Kuroshio Current. The meridional section along 90∘ E shows a weaker thermal monsoon circulation with decreased heating differences compared to JJA.

67

68

2 Hydrological and Heat Exchange Processes

2.7.2

Upper-Ocean Heating

Here we estimate the circulations induced from upper ocean heating between the cloudy and the clear regions. The overall net flux into the upper ocean in the tropical warm pool (Table 2.3), as measured from JASMINE and TOGA COARE, is about +45 W m−2 . This is larger than the ERA-40 climatological values for similar periods shown in Figure 2.17, which are closer to +20 W m−2 . In the eastern Pacific, the total heat flux at the surface is roughly +110 W m−2 . The change of temperature of a layer is given by the net flux into the layer, i.e.: 1 dQT dT =− dt 𝜌Cp dz

(2.33)

Assuming that the downward heat flux at the base of the mixed layer is zero, the gradient of heating of the upper ocean ranges from −0.1∘ C month−1 in the western Pacific to +0.5 to +1.0 ∘ C month−1 in the eastern Pacific, depending on the choice depth of the mixed layer. In the western ocean, the anomalous radiative flux convergence in the atmosphere (which produces a relative atmospheric heating compared to non-convective regions) is accompanied by a reduced solar flux at the surface of the ocean (which results in a slight ocean cooling). However, the relative cloudless eastern ocean allows significant surface heating. Alternatively, increases (decreases) in atmospheric buoyancy are matched by decreases (increases) in ocean buoyancy. In summary, the heating gradient of the ocean is in the opposite sense to the radiational and latent heating gradient in the atmosphere. Given a positive correlation between warm SST and cloudiness, it would seem that the ocean–atmosphere feedback might be negative and imply a natural governor on the coupled system. With an increase in SST, cloudiness increases which, in turn, reduces the radiative heating of the ocean and modulates the SST increase. A decrease in cloud amount, though, will increase the heating of the ocean. It has been suggested that this feedback keeps the ocean temperature within rather narrow bounds.XIV

2.8 Convection–SST Relationships and the Vertical Scale of Tropical Motions There are three important features of the major tropical atmospheric circulation systems. First, the vertical scales of the circulations are extremely deep. Near the equator convection often penetrates to 16–17 km above the surface. The Walker Circulation along the equator, the ITCZ, and the interhemispheric monsoons

extend throughout the entire troposphere, as apparent in Figure 1.12. Second, the principal circulation systems are driven and defined by heating gradients associated with hydrological processes, as discussed in the last section. Third, the convection associated with these circulations is either near the continental-ocean margins, as in the case of the monsoon, or in the warm pools of the tropical ocean. Here we examine why there is such a great difference in the height of convection from one point in the tropics to another, and also why this height varies with latitude. Classical fluid dynamics does not predict the observed vertical scale of convective motion in the tropics. Prior to the existence of upper-atmospheric soundings, JeffriesXV (1923) used tidal theory to estimate the vertical scale of the monsoon. This height was thought to be 3–4 km, based on the incorrect assumption that the monsoon was driven by surface heating and not by the distribution of middle tropospheric convective heating. Clearly, the form of heating and its distribution in the vertical are important in determining the vertical scale of tropical motions. Later, based on much longer periods of observations, Riehl (1954) made a key association, noting that tropical storms and disturbances, and deep convection in general, tend to be confined to regions of the warmest SST. We now argue that the reconciliation of Jeffries’s estimate of the depth of tropical circulations with observations and the juxtaposition of convection with the warmest SST have the same root. Consider two columns residing above SSTs of T 1 and T 2 , where T 2 > T 1 , representing the western and eastern equatorial Pacific Ocean or the tropical warm pool and the subtropics. Figure 2.28a shows the Clausius–Clapeyron relationship (discussed in Section 2.3.1) with the saturation vapor pressure es plotted as a function of SST. If we take representative SSTs of the western and eastern Pacific to be 29 and 22 ∘ C, we find that esv (29∘ C)/esv (20∘ C) ≈ 1.7, reaffirming that moist processes are considerably more important in the warm pools than in the colder regions of the tropics or elsewhere on the globe. This point was made earlier relative to Figure 2.7 where it was illustrated that vertical displacement of a saturated parcel in a warm column invokes a far greater change in temperature for the same amount of work than in an Colder column. To determine the importance of the differences in vapor pressures between the warm and cold columns, we now calculate the equivalent heat cntent of surface parcels in each column, assuming that they completely condense their water vapor and use that heat to raise an air parcel upward. The convective penetration height of a moist parcel is defined here as the level at which a saturated parcel, lifted through the atmosphere, will

2.8 Convection–SST Relationships and the Vertical Scale of Tropical Motions (a) Clausius-Claperyron

(b) Moist-dry asymptotes

Height (z)

esv(T2)

T2

T1

T1 Temperature

(c) Penetration (SST)

(d) Heating profile

Height (z)

Penetrative Height

SST

T2

T2 T1

T2

T1 SST

finally expend its latent energy. Equivalently, it is the level at which the moist adiabatic lapse rate asymptotes to a dry adiabatic lapse rate and where the parcel loses its buoyancy, becoming neutrally buoyant. This process is shown schematically in Figure 2.28b. Assuming that the parcels are initially saturated near the surface, the convective penetration height is a strongly nonlinear function of the SST. Figure 2.28c plots the convective penetration height as a function of SST. From this simple construct, the magnitude of the latent heat release in each column must also be a function of the SST and the vertical distribution of the heating will be determined by the convective penetration in each column. In summary, the warmer and moister column (T 2 ) has a higher and larger maximum latent heating rate than its cooler, dryer neighbor (T 1 ). Figure 2.28d shows a schematic representation of the heating profiles in the warm and cold columns. To first approximation at low latitudes, the heating is almost balanced by adiabatic cooling. Such cooling can only be accomplished by rising motion. The scaling that supports this conclusion will be discussed in Chapter 3. Thus, the vertical heating profiles in Figure 2.28d may

at iab t ad at iab ad

C

C-

esv(T1)

is mo

Saturated vapor pressure

height of asymptote

y dr

Figure 2.28 Schematic diagram relating the distribution of organized convection over the oceans with the SST. Two columns with temperatures T1 and T2 (T2 > T1 ) represent, for example, the eastern and western Pacific Ocean, respectively, or the tropics and the subtropics. (a) shows the Clausius–Clapeyron relationship (i.e. es = es (T)) and its relation to the convective penetration height of a saturated parcel (b and c) over the ocean. Assuming the same vertical distribution of relative humidity in each column and that convection has occurred, the magnitude of the heating in the warm column will be greater and will occur exponentially higher (d) than in the cold column.

Heating rate

also be thought of as vertical motion profiles such that where the heating is positive, the vertical motion will be upward, and vice versa. In summary, over the warm pools of the tropics (and over the major convective regions of the tropical continents during the summer), the magnitude of the heating and the associated vertical velocity will be stronger and at a higher elevation than over other regions of the globe. How do the convective penetration heights compare with the observed scales of atmospheric circulations? Assuming that atmospheric parcels have surface relative humidities of 85%, the convective penetration heights for SSTs of 30, 25, and 20 ∘ C are 17, 15, and 12 km, respectively. These compare well with observed tropopause heights as a function of SST20 . In essence, one may argue that the height of the troposphere is a function of penetrative convection. Noting that the convective penetrative height is a function of SST, it is logical to assume that the coolest air temperatures should lie over the warmest SST. Indeed, the coldest part of the troposphere–stratosphere occurs over the equator (see Figure 1.2). 20 Peixoto and Oort (1992).

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2.9 Coupled Global Ocean–Atmosphere Synergies Heating in the tropics and cooling in the polar regions drive and govern the circulation patterns of the Earth’s climate systems, systems that are required to maintain a global heat balance. Poleward heat transports are a combined effort between the atmosphere and the ocean. The question we now raise is how the ocean and the atmosphere work together to perform the balancing of the global heat budget and how, through this effort, the stability of the climate system is maintained. Furthermore, are hydrological processes involved? We propose a hypothesis describing the entire Earth’s climate system as a robust interactive entity. Robustness refers to a resilient configuration of the climate state. Interaction refers to an interdependency of the various spheres of the climate system that maintain its robustness. Within this scheme, the hydrology cycle, so important on fairly short time scales, is hypothesized to impart a powerful constraint on the overall climate on very long time scales as well. To develop this hypothesis, a very simple prototype of Earth is used where the planet is assumed to be ocean covered. Clearly, this is an oversimplification as the division of the ocean into ocean basins is critically important for the explanation

of the thermohaline circulation. However, the simplification is made to elucidate some very basic interactions and controls. This simple prototype, first introduced by Webster (1994), suggests linkages over long time scales and between remotely located geographical regions of the coupled system.

2.9.1

The Notion of Interactive Zones

The ocean and the atmosphere have very different time scales. The atmosphere possesses a rapidly evolving troposphere but a more slowly evolving stratosphere. The ocean possesses an upper ocean and a deep ocean characterized, respectively, by relatively fast and slow time scales. Thus, as a working hypothesis, suppose that the regions of most rapid variability of the ocean and the atmosphere are those that interact – namely, the upper ocean and the troposphere. This interactive zone, so defined, is demarked by the yellow area in Figure 2.29 and shows a marked latitudinal structure. The deepest part of the atmospheric interactive zone exists at low latitudes, while the deepest region of interaction in the ocean lies at high latitudes. Latitudinally, the interactive zones of the ocean and the atmosphere are completely reversed. We can suggest why structural reversal in latitude exists. The tropopause separates the slow dynamics

Ocean-Atmosphere Interactive Zones 40

height (km)

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Figure 2.29 Schematic diagram of the interactive zones of the ocean and the atmosphere. Interactive zones are defined as regions in the ocean and the atmosphere that have comparable time scales and thus directly influence each other. Vertical temperature sections through the atmosphere and the ocean appear: (left) the tropics and (right) the polar regions. The interactive zones encompass the most rapidly evolving regions of the two spheres such as the troposphere, the upper ocean, and the entire ocean at higher latitudes, where deep water is formed. The deep ocean and the stratosphere evolve on much longer time scales; note the “reversed” mirror image of the ocean and the atmosphere. Convective instability occurs in the tropical atmosphere and in the high-latitude oceans. Both regions are responsible for the formation of deep penetrative circulations or currents. Source: After Webster (1994).

2.9 Coupled Global Ocean–Atmosphere Synergies

of the stratosphere from the rapid dynamics of the troposphere. Similarly, the thermocline separates the slow deep ocean from the more rapid upper ocean. In that sense the tropopause and the thermocline fulfill similar roles. The tropical tropopause can be found at about 15–17 km but at less than 10 km in the high latitudes. The depth of the troposphere, at least in the tropics, is determined by the degree of convective penetration, with the tropopause showing the limit of the mean free moist convective height, which, in turn, is determined by the SST. In the tropics, the thermocline is the clear demarcation of the rapid and slow regions of the tropical ocean. At higher latitudes, the demarcation between time scales extends much deeper. The wind-driven circulation, less constrained by the thermocline, which is not as strong as in the tropics, extends to greater depths. Furthermore, the higher latitude oceans are regions of penetrative convection, or subsidence, caused by radiative cooling of the upper surface and salt rejection during the freezing process. Figure 2.29 suggests a “reversed mirror image” between the atmosphere and the ocean, where the tropical troposphere and the high-latitude oceans play very similar roles. Both accomplish deep mixing into their respective interiors. Both regions are dominated by the same physical process: gravitational instability. The vertical temperature profiles of both the ocean and the atmosphere are plotted on the sides of Figure 2.29. In the tropics, the warm surface layer is surmounted by the deep tropical atmosphere but capped by the coldest part of the atmosphere and the lowest saturated vapor pressure (see Figure 1.2). This zone is referred to as the “cold trap,” the physics of which will be discussed in Section 2.9.2.2. In the stratosphere, the temperature lapse rate reverses because of ozone absorption of ultraviolet radiation. Below the thermocline, the temperature rapidly decreases to the temperature of the abyssal zone. At higher latitudes, the tropopause is lower and warmer than in the tropics. The upper ocean temperature is cool and nearly isothermal into the deep ocean, as is clearly evident in Figure 2.9b. This conceptualization of the coupled system now allows the processes of ocean-atmosphere interaction to be placed on a global perspective. In the next section, the simple model will be extended to show how hydrological processes bind the interactive zones and the non-interactive zones together. 2.9.2

A Stable Global Interactive System

Figure 2.30 provides a schematic synthesis of the global-scale interactions within the ocean–atmosphere couplet. The broad arrows represent Lagrangian circulations in the ocean and the atmosphere between the

equator and the poles. These circulations illustrate a response to the imposed latitudinal radiational heating gradient. In this simple system, major convection takes place in two locations: in the tropical atmosphere (upward) and the polar ocean (downward). The convective (gravitational) instability in the atmosphere is a result of very moist air rising over the warmest SST. The convective instability in the polar oceans is caused by the negative buoyancy flux related to the intense radiational cooling at the surface of the ocean during winter, with ice formation affecting negative buoyancy fluxes as salinity of the upper ocean increases during the freezing process. Similar effects occur in the subtropics, where strong evaporation causes ocean descent and the subduction of cooler saline water under the equatorial thermocline, thus increasing the static stability of the upper ocean. The polar atmospheric circulation is also stable with descending and warming air producing a low-level inversion. The warm SST and the tropical atmosphere determine the state of the upper atmosphere, and the polar ocean mixed layer determines the abyssal characteristics of the ocean. Can this simple model be used to tie together the many seemingly disparate components of the coupled ocean and atmosphere system? The tropical warm pool is chosen as a starting point. 2.9.2.1

The Tropical Circulation

Driven principally by gradients of SST, warm and moist air converges into the equatorial regions, especially into the warm pool regions of the tropical oceans (e.g. Lindzen and Nigam 1987). Wind trajectories along an SST gradient increase the moisture content of the column through evaporation. Over the warm pools, in the region of maximum surface convergence, the air is forced to rise, producing deep penetrative moist convection, precipitation, and large releases of latent heat in the middle troposphere. The large-scale circulations that feed the tropical convection act as a form of solar heat collector, solar energy being stored in the evaporated water vapor picked up as the winds converge toward the equator across the ocean basins. The effect is emphasized by the exponential increase in the saturated vapor pressure toward the equator as temperatures increase. The collected solar energy is released in a relatively small volume over the warm pool, driving vigorous upward motion. The vertical limit of convection, demarked by the tropopause, is determined principally by the magnitude of the SST (Section 2.8). Furthermore, the warmer the SST, the stronger the convection and vertical penetration are and, at great heights, the greater the adiabatic cooling. Thus, there is a strong inverse relationship between the magnitude of

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o Temperature of the cold trap determined by SST

20km cold-trap

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o Stratospheric water vapor concentration determined by tropical cold-trap temperature

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o Very dry atmospheric column allows efficient surface radiative cooling to space

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UNSTABLE

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5km

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o Radiative cooling and salt rejection (sea-ice formation) creates unstable ocean and subsidence of cold salty water.

POLE

Figure 2.30 A schematic sequence of the role of hydrological processes in maintaining the general circulation of the coupled ocean–atmosphere system. In the tropics, complicated ocean–atmosphere fluxes together with high insolation and the very stable ocean configuration hold the SST in rather narrow bounds. The deep penetrative atmospheric convection produces a very cold tropopause, which provides a “cold trap” for moist air passing through to the stratosphere, thus ensuring that the stratosphere remains dry at least to the saturated vapor pressure at the temperature of the cold trap. A cold, dry stratosphere assists the rapid surface cooling in the polar regions, the formation of sea ice, and the formation of cold saline deep water. The slow equatorial encroachment of the polar water and the subduction of saline water from the subtropics below the equatorial warm pools enhance the stability of the upper ocean structure at low latitudes and thus help maintain the high temperature of the tropical water. In this manner the ocean–atmosphere system, with a strong involvement of the hydrology cycle, maintains a long-term balance and stability. Source: After Webster (1994).

the SST maximum and the coldness of the tropopause. Over the tropical warm pools (e.g. 28–30 ∘ C) the air temperature at the tropopause (17–18 km) is less than 190 K. 2.9.2.2

State of the Stratosphere

The atmospheric Lagrangian circulation (i.e. the path a parcel of air would actually follow), driven by the equator-to-pole heating gradients and the breaking of vertically propagating waves, is similar to a large Hadley-type cell with equatorial motion in the troposphere and poleward motion in the stratosphere. This is the Brewer–Dobson cell used to explain the ozone and moisture distribution in the stratosphere.XVI Thus, all of the moist air ascending through convection (the shaded area in Figure 2.30) must pass through the cold tropical troposphere.21 Except for the oxidation of methane (CH4 ), this injection through the troposphere is the only source of stratospheric water vapor. The temperature of the tropopause T T must then define the maximum vapor pressure in the stratosphere, methane aside. Indeed, observations indicate that the warmer stratosphere is unsaturated with a vapor pressure of 21 Danielson (1968, 1993) and Holton and Gettelman (2001).

es (T T ) or, roughly, a constant vapor pressure of 4 ppm In that sense, the cold tropopause acts as a moisture “cold trap” where tropospheric water vapor of a higher vapor pressure than es (T T ) would be “freeze dried.” The excess water, in the form of ice crystals, is hypothesized to precipitate back into the tropical troposphere.XVII 2.9.2.3 The Return Atmospheric Flow Between the Tropics and the Poles

The stratospheric air, dried efficiently by the equatorial cold trap, moves poleward down the pressure gradient. Nearer the pole, the dry air, cooling slowly by radiation to space, subsides and warms adiabatically, creating a stable inversion over the poles. As the air is always unsaturated while in the stratosphere, and as it warms during subsidence, the formation of clouds is retarded. The dry, clear stratosphere allows efficient radiational cooling of the surface and the atmosphere to space and a minimization of the greenhouse effect at high latitudes. 2.9.2.4 The Polar Ocean–Atmosphere Interface and the Formation of Deep Water

Intense radiational cooling of the polar ocean surface during the winter causes the formation or extension of sea ice. Both surface cooling and ice formation (through

2.10 Synthesis

salt ejection) produce an unstable density gradient in the upper ocean, allowing the dense water (cool and saline) to mix downward convectively. Just as the tropical ascent is the rising part of the atmospheric direct circulation (i.e. warm air rising), the sinking motion of the ocean is the thermally direct part of the ocean circulation (i.e. cold water sinking). A thermodynamically direct circulation converts potential energy into kinetic energy. These processes are thought to constitute the formation of the deep ocean waters. 2.9.2.5 The Return Ocean Flow Between the Poles and the Equator

An oceanic meridional circulation (the thermohaline circulation) spreads out away from the poles until it slowly ascends toward the south, as suggested by the slight upward tilt of the isentropes. The cold, saline ascending water mass has characteristics that are in sharp contrast to the warm surface layers of the tropics and the warmer, though very saline, water sinking from the subtropics. The intersection of the water masses produces an extremely stable stratification below the surface layer and the tropical thermocline. This part of the ocean circulation is analogous to the descending branch of the atmospheric Lagrangian circulation over the poles. 2.9.2.6

Maintenance of the Warm Pool

The warm SST supports the generation of deep convection. The reduced density of the surface layer formed by heating and fresh water flux creates a stable upper ocean resistant to vertical mixing and cooling. Very strong

stratification exists in the ocean and the strong evaporation in the subtropics also plays a role. Cooler, very saline water subducts below the equatorial mixed layer maintaining its vertical stability. Of all of the locations on the planet, the equatorial warm pool possesses the steadiest character.

2.10 Synthesis The emphasis of this chapter has been on the role that water plays, in all of its forms, on establishing the stable climate of the planet. We have also shown that the gradients of heating are a combination of radiative forcing and latent heating. The former gradient depends on the existence of cloud in a column while the latter depends on the convergence of moisture in an atmospheric column. The gradients introduced by these two factors are of the same sign both in longitude and latitude. In disturbed conditions, turbulent sensible heating and the negative sensible heating from rainfall may become important. Moist processes, in impacting the heating or cooling of the ocean surface and the fresh water flux, are important in determining the buoyancy of the upper ocean. The joint atmospheric–oceanic interaction leads to a stable fresh and warm upper ocean that is dynamically stable and sets, to a large degree, the climate of the planet. Finally, we have argued that this stability is a global phenomenon where processes in the tropic atmosphere and the ocean are coupled with polar phenomena.

Notes I Moustafa Chahine (1935–2011) was a research

scientist at the Jet Propulsion Laboratory in Pasadena, California, for 51 years. He was responsible for the formation of the Division of Earth and Space Sciences. He was appointed the JPL Chief Scientist in 1984. Among many scientific achievements, he developed an exact mathematical method for the inverse solution of the full radiative transfer equation, called the Relaxation Method. The method is used today for deriving atmospheric temperature and composition profiles from satellite observations of Earth, Venus, Mars, and Jupiter. He then developed a multispectral method using infrared and microwave observations of remote sensing of clouds. He was a Fellow of the National Academy of Engineers and won the prestigious George W. Goddard Award from the International Society for Optical Engineering.

II The partial pressure of a gas was discovered by John

Dalton (1766–1844), an English chemist, physicist, and meteorologist, and perhaps the father of atomic theory. The “Law of Partial Pressures of Gases” stipulated that the total pressure of a gas was the sum of the pressures of individual component gases. If a gas (e.g. water vapor) reached saturation at some temperature T, it would do so even in the presence of other gases. This is why we can discuss transitions in terms of just water vapor pressure. See Curry and Webster (1999). III Rudolf Clausius (1812–1888), a German physicist and mathematician, and Benoit Paul Emile Clapeyron (1799–1864), a French physicist, worked on fundamental aspects of thermodynamics. The discovery of “latent heat” (the heat released or required for a phase change of a substance to occur at a particular temperature and vapor pressure)

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IV

V

VI

VII

preceded their work by over 50 years. Joseph Black (1728–1799), the Scottish chemist and physician, developed the concept of latent heat. He also discovered carbon dioxide and the element magnesium and the property of the “specific heat” of a substance. http://en.wikipedia.org/wiki/The_Story_of_the_ Three_Bears. In astrobiology, a “Goldilocks zone” refers to the habitable orbit around a star being neither too far away from, nor too close to, a star to support life. Either of these two extremes would result in a planet incapable of supporting life, at least as we know it! A planet within this zone, with an environment close to the triple point of water, and thus able to maintain the three phases of water, is often called a “Goldilocks Planet.” A simple way of thinking of the negative sign of the dry adiabatic lapse rate is to consider raising a parcel upwards to a level of lower pressure. The parcel must expand and in so doing does work on its environment. There is only one source for the energy to do the work of expansion, which is the internal energy of the parcel that is proportional to its temperature. Thus, with upward motion there must be cooling. Note from Eq. (2.9) that the lapse rate is proportional to g. A larger g means a greater lapse rate. However, from the hydrostatic equation it also means a greater vertical pressure gradient. Therefore, for the same vertical displacement in a world with a higher g, the work required for the parcel to expand and match its new environmental pressure must be greater and the lapse rate larger. In our view, the Lukas–Lindstrom study makes a seminal contribution on the understanding of the couple ocean–atmosphere physics in the tropics. It shifted paradigms and guided climate scientists toward understanding how coupling between the ocean and the atmosphere occurs. Most importantly, it provided critical clues to why the tropical warm pool is a stable entity in climate. The paper was instrumental in the design of the TOGA COARE experiment of 1992–1993 in the equatorial western Pacific Ocean (Webster and Lukas 1993). Count Von Rumford (1753–1814): (Benjamin Thompson), an American born physicist, soldier, and inventor. We came across the quote by von Rumford in Warren and Wunch (1981). This work led us to the von Rumford Volume 2 (see references) that describes many of the careful, albeit intuitive, experiments that Von Rumford conducted on heat exchange. It is well worth the time spent on reading this compilation, especially the chapters on “The Propagation of Heat in Fluids” (referred to here) and “An Experimental Inquiry Concerning the Source of

the Heat which is excited by Friction.” Von Rumford was a Royalist and moved to Britain ahead of the American Civil War. He was knighted by the British and, in Bavaria, was raised to the level of a Count for his contributions to the military. There was a very practical side to von Rumford. He modified the design of household chimneys that increased the updraught, made combustion more efficient, and reduced internal house pollution. His chimney, known as the Rumford Chimney, formed the basis of the design of modern household chimneys. VIII Stefan–Boltzman’s Law states that radiation emitted from a body, integrated across all wavelengths is proportional to the fourth power of the body’s temperature. The law was developed separately by the Slovenia physicist Jozef Stefan (1835–1893) in 1879, based on experimental results, and the Austrian physicist Ludwig Boltzman (1844–1906) in 1884, using thermodynamic arguments. IX As the equatorial regions are warmer than the polar regions, there will a net positive radiative transfer toward the poles. Consider two columns with mean temperatures T EQ and T P representing the equatorial and polar latitudes, respectively, where T EQ > T P . We start with the Stefan–Boltzman radiative transfer equation: F = 𝜎SB T 4 where F (W m−2 ) is the radiant flux, T (K) is temperature and 𝜎 SB is the Stefan–Boltzman constant 5.67 × 10−-8 W m−2 K−4 . Consider now two columns located in the tropics (temperature T EQ ) and and higher latitudes (T P ). The change in temperature in each column by radiative transfer between the two columns will be 1 dF SB 1 𝜎 dT =− =− (T 4 − TP4 ) dt 𝜌Cp dy 𝜌Cp Δy EQ Choosing T EQ = 300 K, T P = 240 K, and Δy = 5000 km (5 × 106 m), we find the heating and cooling rate to be roughly 0.3 K/3 months, far too slow to accomplish the necessary heat transport. Even if the distance between columns were reduced by a factor of 5, the resultant heating rates would still be an order of magnitude less than observed. Thus, lateral radiative transfer cannot balance the difference in radiative gain at low latitudes and the net loss at higher latitudes over the period of a season. X A similar derivation follows for an atmosphere with a constant lapse rate with T(z) = T 0 − Γz, where Γ is a positive constant, and can be found relative to Eq. (1.48) (see Curry and Webster 1999). The variation of latitudinal pressure gradient with height is very

2.10 Synthesis

XI

XII

XIII

XIV

XV

similar to the isothermal case. Here we have made the simplifying (although unnecessary) assumption that the surface pressure of the two columns is the same. On a hot day, one may notice shimmering of images close to the ground above a roadway. The shimmering is the result of diffraction of light by the high-frequency turbulent eddies. The Kuroshio Current (“black tide” or “black stream”) is a strong warm boundary current flowing eastward in the central North Pacific and is part of the North Pacific gyre. In essence, it is the Pacific counterpart to the Atlantic Gulf Stream. Adrian E. Gill (1937–1986) was an Australian meteorologist and oceanographer who was a Senior Research Fellow in the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge. He was elected to Fellow of the Royal Society (FRS) in 1986. His work spanned a wide range of topics in geophysical fluid dynamics, especially low-latitude meteorology and oceanography. He is perhaps best known for his textbook Atmospheric–Ocean Dynamics and for his leadership in the early stages of the Tropical Ocean–Global Atmosphere Programme (TOGA). This is the basis of the so-called “thermostat hypothesis” suggested by Ramanathan and Collins (1991), which will be discussed in Section 18.1.4(b). Sir Harold Jeffries (1891–1989) was a British geophysicist, fluid dynamicist, statistician, and seismologist located at St. John’s College

Cambridge, UK. He was instrumental in the development of the WKB approximation (named after founders Wentzel, Kramers, and Brillouin), whereby approximate solutions to partial differential equations can be found. We will use this methodology in Chapter 7. More correctly, the method should be called the WKBJ method in honor of Jeffrie’s contribution. His statistical work produced a revival in probabilistic Bayesian statistics that was to make an important contribution to the study of predictability. He was a strong opponent of what appeared to be the controversial theories of continental drift and plate tectonics proposed by the German scientist Alfred Wegener (1880–1930), who was also known for his theory (the Wegener–Bergeron–Findeisen theory) of ice-crystal growth. XVI The Brewer–Dobson circulation was proposed by Alan Brewer in 1949 and Gordon Dobson in 1956 to explain the stratospheric ozone distribution. See also Plumb and Eluszkiewisz (1999). XVII Potter and Holton (1995) offer an alternative explanation of how the lower stratosphere remains so dry. They envisaged buoyancy waves, generated by tropospheric convection, producing ice clouds through local lifting in the stratosphere. The ice crystals formed in the clouds precipitate downward, as suggested by Danielson (1968). Irrespective, the cold trap is instrumental in keeping the stratosphere relatively dry.

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1 An isopycnal surface is a surface of constant density in the atmosphere or ocean. It is derived from the Greek iso, meaning equal, and puknos, meaning dense.

200

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0.4 0.5 400 pressure (hPa)

As articulated in Chapter 1, inroads into tropical dynamics were made by the discovery of families of atmospheric and oceanic dynamic modes that are trapped close to the equator. Distinct tropical phenomena with intraseasonal time scales were discovered and coupled ocean–atmosphere interactions between the trade winds and the low-latitude ocean were found to be critical ingredients in producing the low-frequency El Niño. The kinematics of weather extratropical systems had been described with some eloquence earlier in the 20th century (see, for example, Henry 1922). Rossby (1939)I had described the basic structure and governing physics of extratropical waves in the westerlies that would become known as “Rossby waves” and provided an explanation for their kinematics using conservation of absolute vorticity arguments. Research by CharneyII (1947) and EadyIII (1949) provided physical explanations of the structure and development of extratropical high-frequency phenomena based on the instability of the highly baroclinic extratropical atmosphere. A baroclinic atmosphere is one for which density is a function of both temperature and pressure. Baroclinicity is defined by the magnitude of ∇𝜌 × ∇ p (where 𝜌 and p represent atmospheric density and pressure, respectively). The cross-product is non-zero when pressure and density surfaces intersect. In a barotropic atmosphere, in contrast, density depends only on pressure. Theoretical developments described how large-scale eddies would grow in regions of the planet where baroclinicity is most pronounced, such as in the winter extratropics. Figure 3.1 provides an average cross-section of isopycnals1 plotted as a function of pressure. A simple measure of baroclinicity is given by the angle between the isobaric and isopycnal gradients, that is most pronounced in the winter hemispheres of the extratropics in the vicinity of jet streams. In the

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Figure 3.1 Cross-section of the zonally averaged isopycnals plotted against pressure for the NH winter (1998–2010: December–February). In the middle latitudes, in the vicinity of the westerly maximum, the atmosphere is highly baroclinic, as indicated by the large angle between the 𝜌 and p gradients (vectors ∇p and ∇𝜌). In the tropics, however, the gradients are almost parallel, indicating a lack of baroclinicity.

tropics, however, the gradients appear near parallel, indicating a far more barotropic atmosphere. Clearly, the fundamental dynamical structure of the tropics and the extratropics are vastly different. Further differences are evident in the zonal-height distributions of the potential temperature [𝜃] and equivalent potential temperature [𝜃 e ] plotted in Figure 3.2a and b for the JJA and DJF seasons, respectively. The two temperatures are defined as 𝜃 = T(ps ∕p)R∕Cp

(3.1a)

𝜃e = 𝜃(LE es ∕Cp T)

(3.1b)

and

where p is pressure, ps is a standard pressure, often assumed to be 1000 hPa, C p is the specific heat of dry air at a constant pressure, es is the saturated vapor pressure at temperature T, and LE is the latent heat of condensation or evaporation (2.5 × 106 J kg−1 ). Formally, 𝜃 is

Dynamics of the Tropical Atmosphere and Oceans, First Edition. Peter J. Webster. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

3 Fundamental Processes

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Figure 3.2 Zonally averaged distributions of the (a) JJA and (b) DJF potential temperatures ([𝜃]: upper row) and the equivalent potential temperatures ([𝜃 e ]: lower row) as a function of pressure. The two temperatures are defined in Eqs. (3.1a) and (3.1b). The 370, 330, 290, and 250 ∘ K isentropes are emphasized in bold red.

the temperature that a parcel of air would achieve if it were compressed or expanded in an adiabatic reversible manner from a given state, p and T, to the standard pressure ps . A surface of constant 𝜃 is referred to as an isentropic surface upon which entropy is constant. The equivalent potential 𝜃 e (z) includes the impact of atmospheric water vapor that is potentially condensed during ascent. For example, consider a moist parcel that is raised to saturation and that is continually lifted until all moisture has condensed. The latent heat of condensation will heat the parcel during ascent. The parcel is then lowered adiabatically to p = ps to produce the equivalent potential temperature 𝜃 e (z). As is clear from Eq. (3.1b), if es is non-zero then 𝜃 e > 𝜃 for a given parcel with the largest differences occurring in the tropics where es is largest. Figure 3.2a indicates that the potential temperature [𝜃] increases with height at all latitudes reflecting a strong static stability in a vertical column for dry processes (see Appendix C). Isentropes slope upward toward higher latitudes from a region of small latitudinal gradient that spans the equator to a much larger

gradient at higher latitudes. The distributions of [𝜃 e ] (Figure 3.2b) are quite different; [𝜃 e ] still increases toward the poles along an isobaric surface but in the tropics from the surface to the lower- mid-troposphere, [𝜃 e ] > > [𝜃], reflecting the greater tropical moisture. Within about 20∘ around the equator 𝜕[𝜃 e ]∕𝜕z is weakly negative, indicating potential convective instability. Higher in the column above a relative minimum of [𝜃 e ], 𝜕[𝜃 e ]∕𝜕z becomes positive, indicating convective stability. It had been known for a long time that convection in the tropics extends through the entire troposphere and the minimum in [𝜃 e ]. One of the persistent questions in tropical meteorology is how this very deep convection is maintained in a stable middle and upper troposphere. Figure 3.3 presents a typical horizontal section of potential temperature 𝜃 at the 500 hPa level between 30∘ S and 60∘ N The colored shading represents anomalies in the outgoing longwave radiation (OLR) relative to the lower bar. There are regions of large-scale deep convection (blue shading) in both the tropics and the extratropics, although, as noted relative to Figure 3.2,

3.1 Some Fundamentals of Low-Latitude Atmospheric Dynamics

Figure 3.3 Contours of potential temperature, 𝜃(𝜆, 𝜙), at 500 hPa between 30∘ S and 60∘ N on January 4, 2011. Background colored shading shows the anomalous outgoing longwave radiation (OLR) with blue shades indicating deep convection and red shades its absence. At higher latitudes the isentropes are tight, consistent with Figure 3.2b and wavelike, associated with extratropical Rossby waves. In the tropics the isentropes are relatively flat with minimal gradient. However, large areas of deep convection (blue shading) are evident.

60°N

OLR anomaly and 500 hPa θ:

Jan. 4, 2011

30°N

0

30°S 90°E

120°E

150°E

–100

there is little similarity of the distribution of 𝜃. In the extratropics there are large variations in the value of 𝜃. In the tropics, the isentropic surfaces are rather flat. The convoluted 𝜃 fields poleward of the subtropics represent baroclinic extratropical waves drawing upon the potential energy (PE) of the background flow. Both Charney and Eady were able to show that the most rapidly growing, baroclinically unstable modes occurred with a longitudinal wavenumber between k = 5 and 8, where k is defined as 2𝜋a/L. Here L represents the wavelength of the mode and a is the radius of the planet. The understanding of extratropical phenomena resulted from a very systematical scaling of the equations of motion using observed spatial and temporal characteristics. Such analyses indicated that the extratropical atmosphere was almost in geostrophic balance. The remaining small deviations from geostrophy formed a second set of equations, called the “quasi-geostrophic” set, which described unstable and growing modes in the westerlies. The quasi-geostrophic description of the extratropical atmosphere pointed toward the plausibility of numerical weather prediction that would eventually replace qualitative prediction, at least in the middle and higher latitudes. However, quasi-geostrophic theory offered little insight into the generation or maintenance of tropical phenomena such as those appearing in Figures 1.15 or 3.3. By contrast, the absence of tropical baroclinicity leads to the suggestion that the coherent patches of convection in the tropics appearing in Figure 3.3 must be formed by processes other than baroclinic instability. The existence of organized tropical phenomena raised the following questions: What forces these motions in the atmosphere in the absence of baroclinicity? Are convective tropical disturbances forced by influences from higher latitudes? Is the kinetic energy (KE) of tropical systems generated by barotropic processes? Are there some hitherto undiscovered instabilities lurking in the tropics?

–60

180° –20

150°W 20

120°W 60

90°W 100

60°W

W m–2

3.1 Some Fundamentals of Low-Latitude Atmospheric Dynamics 3.1.1

Basic Equations

Based on the success of the scaling approach used to identify the fundamental balances of the extratropics, Charney (1963) proposed a similar procedure for the tropics. Consider a set of governing equations: ̃ dV ̃ +F ̃ 𝜌 = −∇p + 𝜌̃ g − 𝜌2Ω × V (3.2a) dt d𝜌 ̃ = 𝜌∇ ⋅ V (3.2b) dt Q̇ d ln 𝜃 Cp = (3.2c) dt T p = 𝜌RT (3.2d) representing the conservation of momentum, the conservation of mass, the conservation of energy, and an atmõ , p, 𝜌, T, and 𝜃 spheric equation of state, respectively. V represent fluid velocity, pressure, density, temperature, and potential temperature and ̃ g is a vertical gravitational vector. Noting that Q̇ represents a heating rate (J s−1 ) from external processes (e.g. radiative heating or cooling, water phase changes or dissipation), Eq. (3.2c) states that following a parcel 𝜃 can only change with the ̃ addition or subtraction of heat external to the parcel. F represents dissipative processes. The Lagrangian or total derivative following a parcel is given byIV 𝜕 | d ̃ ⋅ ∇ |t = 𝜕 ||𝜆,𝜑,z +V = | | dt 𝜕t |𝜆,𝜑,r 𝜕t | u v 𝜕 | 𝜕 | 𝜕 | + |t,𝜑,z + | t,𝜆,z + w | | a cos 𝜙 𝜕𝜆 a 𝜕𝜑 𝜕z |t,𝜆,𝜑 (3.3a) ̂ where the vector V is given by ̃ = r cos 𝜙 d𝜆 i + r d𝜑 j + dr k = ui + vj + wk V ∼ ∼ ∼ dt ∼ dt ∼ dt ∼ (3.3b)

79

80

3 Fundamental Processes

Ω z P(λ, ϕ, r) z r λ

ϕ

y

~ V

x

kw ~

~iu jv ~ P(λ, ϕ, r)

a

Figure 3.4 The spherical (𝜆, 𝜙, r) coordinate system used in Figure 3.4. Radius of Earth, a, is assumed constant so that r, the distance of point P relative from the center of the planet, can be set as r = a + z, where z is the height of P above the surface. Thus, ̃ (𝜆, 𝜙, z) is the 𝜕∕𝜕r = 𝜕∕𝜕z and 1∕r ≈ 1∕a as a >> z. V three-dimensional velocity vector relative to the rotating sphere with u, v, and w its zonal, meridional, and vertical velocity components, respectively.

where ∼i, j and k∼ are Cartesian unit vectors in the ∼ longitudinal, latitudinal, and vertical directions with corresponding velocity components u = r cos 𝜑d𝜆/dt, v = rd𝜑/dt, and w = dr/dt, as depicted in Figure 3.4. The spatial dimensions 𝜆, 𝜙, and z represent longitude, latitude, and height, respectively. The distance between a parcel and the center of the planet is given by r. It is useful to expand the vector equations in scalar form. Also, we note that the vertical coordinate may be expanded as r = a + z, assuming a spherical planet, where z is the height above the surface. As a ≫ z, r ≈ a, and, as a is constant, dr = dz. The basic sets of Eq. (3.1) then become du −1 𝜕p = + 2Ωv sin 𝜑 − 2Ωw cos 𝜑 dt 𝜌a cos 𝜙 𝜕𝜆 uv tan 𝜑 uw + (3.4a) − − F𝜆 a a u2 tan 𝜑 wv dv −1 𝜕p = − 2Ωu sin 𝜑 − − − F𝜑 dt 𝜌a 𝜕𝜑 a a (3.4b) dw −1 𝜕p u2 + v2 = − g + 2Ωu cos 𝜑 + − Fz dt 𝜌 𝜕z a (3.4c) 𝜌 d𝜌 =− dt a cos 𝜑

(

) 𝜕u 𝜕 𝜕w w + (v cos 𝜑) − 𝜌 − 2𝜌 𝜕𝜆 𝜕𝜑 𝜕z a (3.4d)

Equation set (3.4) possesses both the horizontal (terms double underlined) and vertical (underlined) components of the Coriolis force. The geometric terms, those relating to the choice of the coordinate system, are identified by a single overbar. These terms change from one coordinate system to another in order to facilitate the identical expression of an equation in different frames of reference. For example, set (3.4) could be written relative to a Mercator projection so that different geometric terms would be necessary. However, as long as the new coordinate choice is orthogonal, they can be translated back and forth from one geometry to another (e.g. spherical to Mercator to cylindrical and so on) without losing their generality or physical consistency.2 The horizontal components of the Coriolis force are proportional to cos 𝜙, whereas the vertical components are shown as sin 𝜙. Thus, it might seem that the horizontal component of the Coriolis force is greater than the vertical component close to the equator. To determine whether or not these horizontal Coriolis terms should be retained or if they really achieve importance in the tropics, we first have to perform a careful scaling of the equations of motion. The important issue is to make sure that when certain terms are neglected in the scaling process, the fundamental angular momentum balances are not destroyed. In particular, it is critical that sets of corresponding terms are eliminated consistently. 3.1.2

Scaling Atmospheric Motions in the Tropics

The aim in scaling the governing equations is to deduce systematically the magnitude of individual terms for observed scales of motion. Through this method dominant physical processes may be isolated such as the approximate geostrophic balance that exists in the extratropics. The pertinent adjective in the last sentence is “approximate” as the deviations from an “almost-balance” are critical in describing the growth and decay of transient motions. Formally, scaling is accomplished by performing an asymptotic expansion in terms of some basis system-dependent parameter. The basis of scale analysis is that the behavior of the ocean and the atmosphere (or fluids in general) depends on some simple criteria that can be expressed as a number of dimensionless numbers or ratios. Dimensionless numbers, representing the ratio between physical forces (e.g. between inertial and rotational terms), are commonly used in expositions of atmospheric and oceanic dynamics. They allow a test of the dynamic similitude between simulations and reality. For instance, the flow in a laboratory tank is 2 See Haltiner (1971), especially Sections 1.9 to 1.11, pages 12–18.

3.1 Some Fundamentals of Low-Latitude Atmospheric Dynamics

Table 3.1 The definition of useful parameters describing the state of the atmosphere. The last four quantities are formed as ratios of terms in the governing equations. Quantity

Symbol

Definition

Explanation

Scale height

G

RT 0 /g

Vertical e-folding depth of thermodynamic variable Buoyancy frequency at which a parcel will oscillate in a stably stratified fluid. Named after David Brunt, Welsh meteorologist 1886–1965, and Vilho Väisälä, Finnish meteorologist and physicist 1889–1963

Brunt–Väisälä frequency

N

g 𝜕𝜃∕𝜕z 𝜃

Static stability

S

𝜕 ln 𝜃/𝜕z

2

S > 0 stable stratification S < 0 unstable stratification

Rossby number

RO

U∕f L

Ratio of inertial and rotational terms. Named after Carl Gustav Rossby, Swedish-American meteorologist 1898–1957.

Froude number

FR

U2 ∕gD

Ratio (squared) of the fluid speed to the gravity wave phase speed. After William Froude, English engineer 1810–1879.

Richardson number

RI

Rossby radius of deformation

R

NG∕f

Scale at which rotational effects become equal to buoyancy (or gravitational) effects

Reynold’s number

RE

GU𝜌∕𝜇

Ratio of inertial to viscous processes, where 𝜇 is the kinematic viscosity. Named after Osborne Reynolds, the Irish-English fluid dynamicist 1842–1912.

Ekman number

EK

Az ∕f d2

Ratio of the frictional and rotational forces. Az represents a vertical eddy viscosity and the depth of the surface friction layer. Named after the Swedish oceanographer Vagn Walfrid Ekman 1874–1954.

SgD∕U2 or 2

N2 G ∕U2

Ratio of potential and kinetic energy. After Lewis Fry Richardson, 1881–1953.

similar to that in the atmosphere if certain physical ratios are constrained to be the same. Computer models of weather and climate systems represent their real counterparts if there is a similitude between the dimensionless numbers within the model and those of the real system. The relative scales of the non-dimensional numbers allow a rapid determination of the physical state and the processes that need to be taken into account. To determine the relative scales of the terms in the governing equations we let U and W represent typical scales of horizontal and vertical velocities observed (or inferred) over particular spatial scales L (a typical horizontal scale) and D (a depth scale). Finally, P represents the magnitude of the local difference of pressure along a horizontal surface from the average pressure across that surface. fr is a measure the observed frequency of a perturbation in the atmosphere. Combinations of these typical scales can be formed to represent ratios of terms. For example, the Rossby number (RO ) gives the relative magnitude of the inertial and rotational forces. There is also the Brunt–Väisälä frequency (N2 ), the natural buoyancy frequency at which a parcel will oscillate in a stably stratified fluid, the Froude number (FR ) that compares the fluid speed to a gravity wave speed, and the Richardson number (RI ) that measures the relative scales of potential and kinetic energies. Each of these numbers acquires different magnitudes between higher and lower latitudes, between

different scales of motion, and also between observed and characteristic frequencies. These non-dimensional numbers are defined in Table 3.1, together with other important physical constants. Typical values of these numbers had been established for the extratropical atmosphere and used for scaling the equations of motion (Charney 1947). Charney (1963) used representative values of variables in the tropics to perform a similar scaling of the equations of motion. 3.1.2.1

Is the Tropical Atmosphere Hydrostatic?

A first (and critical) step is to determine which scales of tropical motion are hydrostatic and which are not. Specifically, does the vertical pressure gradient balance the gravitational force to a high degree of approximation for all scales of motion? We are interested in the relative scale of the third and fourth terms on the right-hand side of Eq. (3.4c) compared to the vertical pressure gradient and gravitational forces. Using typical values of quantities (i.e. U, W, L, D, P, and fr ) in Eq. (3.4c), we obtain the following relative scales: dw 1 𝜕p u2 + v2 = − −g + 2Ωu cos 𝜑 + dt 𝜌 𝜕z a (3.5) | P | | U2 | | |g| | | |Wf r | || |ΩU| | | | | 𝜌D | | a | −1 −2 With U = 10 m s , g = 10 m s , Ω ≈ 10−5 s−1 , and a ≈ 6400 km, the third and fourth terms on the

81

82

3 Fundamental Processes

right-hand side are orders of magnitude less than g. Thus, in Eq. (3.5) either dw/dt or − 𝜕p/𝜌𝜕z must balance the gravitational acceleration. To determine the balance we need to find scales for the pressure deviation P from the horizontal equations of motion. With frequency defined by U/L Eq. (3.4a) can be written as: uv tan 𝜑 1 du = − ∇p −2Ωv sin 𝜑 −2Ωw cos 𝜑 + dt 𝜌 a | U2 | | P | | | |f U| |ΩW| |Uf r | || || | a | | | | 𝜌L |

uw − a | UW | | | | a | | | (3.6)

Two P scales emerge depending on whether or not the flow is in geostrophic balance (i.e. fr > f ). If there is a balance between the first two terms on the right-hand side of Eq. (3.6), then there is a low-frequency or “geostrophic limit.” The second scale emerges if there is a balance between the pressure gradient force on the right-hand side of Eq. (3.6) with the inertial terms on the left. This occurs for high-frequency motions (i.e. fr >> f ). These two limits define the extremes of the perturbation pressure deviations. These are: (i) The inertial or high-frequency limit: fr >> f ∶ 𝜌LUf r >> 𝜌Lf U so that P = P1 ≈ 𝜌LUf r

(3.7a)

(ii) The geostrophic or low-frequency limit: fr 0, ζt < 0 ζ>0

ζ 0

ζ 0

ζ>0 w(z)

direction of propagation Figure 3.9 Schematic diagram of the response of the tropical atmosphere to deep convective heating with forcing on a scale >2𝜋R. The vertical velocity distribution associated with the heating produces vortex tube stretching in the lower troposphere, generating low-level cyclonic relative vorticity. Vortex tube shrinking in the upper troposphere produces anticyclonic vorticity and an anticyclonic outflow above the convective heating. The advection of planetary vorticity by resultant flow generates relative vorticity that produces a westward propagation of the disturbance with the low-level cyclone moving westward into the region of increasing cyclonic vorticity whilst the upper-level anticyclone also moves westward into the increasingly anticyclonic region.

divergent. This presented early investigators with some problems. Were large-scale convective events forced by modes propagating from higher latitudes or was there some unknown instability operating in the tropics? A discussion of the geostrophic adjustment added a further requirement for the forcing of convective events. Specifically, the forcing must occur on scales significantly longer than the Rossby Radius of deformation (R), the scale at which there is parity between rotational and buoyancy effects. Smaller scale forcing will produce phenomena that will radiate away as gravity waves. Finally, it was noted that R increased substantially toward the equator, insisting further on the need for very large-scale forcing. We explored two early theories regarding the initiation of tropical convection and the seeming absence of an in situ mechanism for the generation of tropical convection. The first was extratropical influences. However, this was puzzling because of the difficulty of waves propagating toward the equator through deep easterlies. A second theory suggested that if there is deep convection that the stability of the atmosphere would be sufficiently reduced that baroclinic modes could develop in situ in the tropics. However, the reduction of atmospheric stability required the pre-existence of convection. Thus, a local dynamic mechanism was still required.

3.2 Dynamics of the Low-Latitude Upper Ocean In the atmosphere, one is principally concerned with the pressure gradient, the advection of enthalpy and momentum, rotational effects (Coriolis), and frictional

forces that occur at the interface of the atmosphere and the ocean and at the bottom of the ocean. What occurs at the base of the atmosphere depends on whether the lower boundary is rigid (land) or moveable (liquid). From an angular momentum perspective, this difference is very important. Although there are interesting impacts on the atmosphere arising from mountain barriers, the atmosphere can, in general, be considered as continuous in the horizontal directions, responding to pressure gradient forces on a rotating platform. Oceans reside in specific largely contained basins so that “lateral sidewall” effects are very important. Reflections of modes occur at the sidewall boundaries and these will turn out to be very important in attempts to explain El Niño and La Niña in the Pacific Ocean. Pressure gradient forces are set up by two specific processes: (i) gradients of density caused by variations of temperature and salinity and (ii) wind-driven transfers of momentum at the ocean surface. Furthermore, for the modes that we will consider here, the intrinsic vertical scale will not be the complete depth of the ocean (∼3–4 103 m) but rather the depth of the thermocline (∼102 m). That depth scale also requires clarification. We will see that the vertical stratification of density, especially in the uppermost layer, reduces buoyancy and, thus, the effective vertical scale of the ocean. This is taken into account by the so-called “reduced-gravity” approximation. 3.2.1

Scales of Motion

In scalar form, the equations of momentum and continuity equations may be expressed as: du 1 𝜕p (3.46a) =− + fv + Fx dt 𝜌0 𝜕x

91

92

3 Fundamental Processes

1 𝜕p dv =− − fu + Fy dt 𝜌0 𝜕y 1 𝜕p dw =− − g + Fz dt 𝜌0 𝜕z ̃ =0 ∇⋅V

(3.46b) (3.46c) (3.46d)

Here we have omitted the horizontal Coriolis components (e.g. ∝ cos 𝜙) since they are much smaller than the vertical components. We have also assumed quasi-incompressibility, thus simplifying the continuity Eq. (3.46d). We have also employed the BoussinesqXII approximation, which states that if 𝜌 = 𝜌0 + Δ𝜌, where 𝜌0 is the mean density along a horizontal surface and Δ𝜌 is the spatial variability about the mean, then 1/𝜌 ≈ 1/𝜌0 as Δ𝜌 < < 𝜌0 . Frictional forces are important only in the boundary layers at the surface and bottom of the ocean. These may be expressed as F x = Ax

𝜕2u 𝜕2u 𝜕2u + Ay 2 + Az 2 2 𝜕x 𝜕y 𝜕z

(3.47)

with similar expressions for F y and F z . Here, the Ax,y,z represent eddy viscosity coefficients. In general, the horizontal structure of the tropics is relatively homogeneous compared to the vertical so that to good approximation: 𝜕2u 𝜕2v 𝜕2w , F y ≈ Az 2 , F z ≈ Az 2 (3.48) 2 𝜕z 𝜕z 𝜕z We now introduce typical scales of motion observed in the tropical upper ocean. Specifically: F x ≈ Az

U ∼ 0.01 − 0.1 m s−1 , L ∼ 106 m, H = 103 m, f0 = 10−5 − 10−4 s−1 , d ∼ 100 m, 𝜌 ∼ 103 kg m−3 , Δ𝜌 ∼ 0.003 × 103 kg m−3

(3.49)

Here, U, L, and H represent horizontal zonal current speeds and horizontal and vertical scales of motion, f 0 is the magnitude of the low-latitude Coriolis terms, while d is the thickness of the surface friction layer. In essence, d represents an e-folding depth below which the impacts of surface stress effects are not apparent. As we found convenient for the atmospheric, we define parameters that represent the ratio of the inertial, rotational parts and frictional parts of the flow. These are the Rossby number: RO = U/f L, the ratio of the inertial and rotational terms as defined earlier, and the Ekman number: EK = Az ∕f d2 , the relative scale of the frictional and rotational effects. Consider the vertical momentum Eq. (3.46c). The gravitational acceleration, g, is of order 10 m s−2 . For the system not to be hydrostatic, either the vertical acceleration (dw/dt) or the vertical frictional term must be of a similar scale. We introduce a basic frequency fr defined by U/L. From Eq. (3.49), fr is about 10−7 s−1 . Thus, for the vertical acceleration to be able to match

the gravitational acceleration, w would have to be greater than gL/U, which is not observed. The vertical dissipation term is also relatively small. Thus, to very good approximation, the ocean is hydrostatic. From Eq. (3.49), the Rossby number is O (10−2 –10−3 ), even quite close to the equator. Thus, away from the upper boundary of the ocean, essentially defined by d, the flow is strongly geostrophic as it is well above the boundary layer at the bottom of the ocean. In a steady state, we then have two major regimes. In the interior of the ocean, the flow is strongly geostrophic, but near to the surface (and at the bottom of the ocean) there is a three-way balance between frictional and rotational factors and the pressure gradient. Equation (3.46) can now be written as 1 𝜕p 1 𝜕𝜏x du − fv = − + (3.50a) dt 𝜌0 𝜕x 𝜌0 𝜕z 1 𝜕p 1 𝜕𝜏y dv + fu = − + (3.50b) dt 𝜌0 𝜕y 𝜌0 𝜕z 𝜕p (3.50c) = −𝜌0 g 𝜕z 𝜕u 𝜕v 𝜕w + + =0 (3.50d) 𝜕x 𝜕y 𝜕z Here, we have written the vertical mixing terms in terms of the surface wind stress such that 𝜕 2 u 𝜕𝜏 𝜕 2 v 𝜕𝜏y 𝜌0 Az 2 = x and 𝜌0 Az 2 = (3.51) 𝜕z 𝜕z 𝜕z 𝜕z where 𝜏 x,y represent the zonal and meridional wind stress, respectively, that transfer atmospheric momentum to the upper ocean. An empirical definition of the magnitude of the total wind stress as a function of the atmospheric wind speed is given as:7 − |→ 𝜏|=C 𝜌 u 2 (3.52) d a a

where C d is a drag coefficient, 𝜌a the atmospheric density at the ocean surface, and ua is the surface atmospheric wind speed measured a few meters above the surface. We also note that the inertial terms are O(Ro ) and that the stress terms are O(1) near the surface, thus balancing the pressure gradient and Coriolis terms. Figure 3.10 presents a simple schematic of the ocean between the surface z = 0 and the bottom z = −H. The layer of the upper ocean impacted by the atmospheric wind stress (𝜏 x, y ) has a depth z = −ze . At the base of the Ekman layer the vertical velocity is we . The advection associated with these vertical velocities may perturb the isobaric surface, inducing geostrophic currents. The total current field is the sum of the Ekman and geostrophic flow: i.e., u = ue + ug , v = ve + vg . At some greater depth, it is assumed that the horizontal pressure 7 E.g. Tomczak and Godfrey (1994, 2000).

3.2 Dynamics of the Low-Latitude Upper Ocean

ρ0 Az z=0

𝜕ue = 𝜏x 𝜕z

ρ0 Az

u = ue + ug v = ve + vg

𝜕ve = 𝜏y 𝜕z

24 5 km

w=0

23 p1 z = –zE

p2 p3 p4 p5 pi

z = –D0

pi+1

Ekman Layer w = we

𝜏x = 0 𝜏y = 0

22 1 𝜕p fug = – ρ0 𝜕y

1 𝜕p fvg = ρ0 𝜕x

21 𝜕p 𝜕p ug = vg = w = 0 = =0 𝜕x 𝜕y level of no motion

N

E

20

ocean bottom Ekman layer

z = –H

Figure 3.10 A schematic vertical section of the ocean indicating the basic forces at different levels. At the top of the Ekman layer, the ocean is acted upon by a wind stress 𝜏(x, y) that decreases its impact through the depth of the Ekman layer (z = 0 to z = −zE ). At the surface of the ocean, w = 0. At the bottom of the Ekman layer the vertical velocity is we , representing Ekman pumping or suction. These vertical motions induce gradients in the horizontal pressure fields (blue contours), which, in turn, induce geostrophic currents (ug and vg ). These gradients decrease with depth to the “level of no motion.”

gradients vanish. This is referred to as the “layer of no motion” located at z = −D0 . An Ekman boundary layer also exists at the bottom of the ocean just above the Earth’s surface but will not be discussed here. 3.2.2 Geostrophic Adjustment in the Low-Latitude Ocean Geostrophic adjustment to forcing also takes place in the ocean. The intrinsic time scale of rotation is given by inertial motion in the same manner as described for the atmosphere in Section 3.1.5. The pertinent issue is the same: what is the distance traveled by an oceanic gravity wave during an inertial period? In the absence of a pressure gradient force, we find that the acceleration of an ocean parcel is determined solely by rotational effects. That is: dv du = fv and = −fu (3.53) dt dt where, after eliminating u between the two equations, we obtain the equation of a simple harmonic oscillator: d2v + f 2v = 0 dt2 with solutions: u(t) = −V0 cos ft and v(t) = V0 sin ft

(3.54)

(3.55)

19 18

Figure 3.11 Example of the inertia motion in the Baltic Sea from Gustafson and Kullenberg (1936) transcribed from Gill (1982). The trajectory is a series of anticyclonic circulations slowly advected northward and with a period of about 0.6 day. Numbers represent days. Depth of the buoy was approximately 14 m.

√ where V0 = (u0 2 + v0 2 ) is the initial speed of the fluid parcel. The inertial motion is anticyclonic in both hemispheres. Thus, inertial circulations are opposite in sense to the planetary rotation. In essence, the trajectory of inertial motion is the projection of the fluid motion on the equatorial plane. The period of the inertial circle is given by 2𝜋/f or 𝜋/Ω sin 𝜙 and the radius of the inertial circle is V 0 /f orV 0 /2Ω sin 𝜙. Figure 3.11 shows (although not for the tropics!) the trajectory of a floating buoy released in the Baltic Sea.8 The period of the oscillation is about 0.6 days, matching the inertial period at 57∘ N. Table 3.2a lists the oceanic (and atmospheric) inertial periods at a number of latitudes. Inertial oscillations also exist in the atmosphere. In Figure 3.11 the buoy was released in a slowly northward moving current that progresses just 20 km in seven days or at an average rate of 0.23 m s−1 . Therefore, the inertial oscillations are easy to discern. However, in the atmosphere, where the background wind speed is two orders of magnitude greater than an ocean current, the oscillations are hard to depict. With a background wind of 10 m s–1 , the Baltic inertial circle one would be spread over a 500 km trajectory. However, careful 8 From the pioneering observations of Gustafson and Kullenberg (1936).

93

94

3 Fundamental Processes

observation in a quiescent situation does allow atmospheric inertial oscillations to be observed (e.g. Buajitta and Blackadar 1957). The basic question then, as we discussed for atmospheric motions, is how far will an oceanic gravity wave propagate during the period of an inertial oscillation? This distance will define the oceanic Rossby Radius of √ Deformation. Equation (3.34) states that R = gH∕f . The numerator is the phase speed of a gravity wave in a fluid of depth H. The gravity wave phase speed is then a critical parameter. However, what characteristic depth do we use for this calculation? If we were to assume H ∼ 103 m, the phase speed of a gravity wave would be >100 m s−1 . This gravity wave speed is far greater than is observed for planetary scale waves in the ocean. Only if the entire column of the ocean is perturbed is the total depth of the ocean the appropriate vertical scale. This may occur when a submarine earthquake lifts the entire column above, producing a tsunami.XIII Then gravity waves radiate away from the source, spreading rapidly across an entire basin. R, from this assumption, would be extremely large, on the order of >1000 km at 30∘ N, but wind-generated planetary scale waves that are associated with, for example, El Niño, are observed to have much slower phase speeds. The appropriate depth may be thought to be that of the friction layer (∼100 m). This assumption would produce phase speeds of about 30 m s−1 , still far larger than waves observed to propagate along the thermocline. This dilemma is solved by noting that the deep ocean (density 𝜌2 ) is surmounted by a slightly lower density surface layer of density 𝜌1 with a difference of Δ𝜌 = 𝜌2 − 𝜌1 < < 1, shown schematically in Figure 3.12. Stratification effectively reduces buoyancy such as would occur if gravity were reduced to a value of ̃ g = Δ𝜌g

(3.56)

where Δ𝜌/𝜌 ≈ 0.006. This is referred as the√“reduced gravity” approximation. The phase speed c = ̃ gH of a gravity wave at the interface of the layers is much smaller than the phase speed of an atmospheric gravity wave or an ocean wave using the other depths, discussed above. If H is 100 m √ then the √ phase speed of a gravity wave at the gH = 9.8 × 0.006 × 100 m s−1 or roughly interface is ̃ 2 m s−1 , which matches observed propagation speeds. The reduced gravity approximation takes into account the reduced buoyancy or restoring force between the two ocean columns in Figure 3.12. The buoyancy force is proportional to the horizontal pressure gradient along the horizontal A–B. In Figure 3.12a, the pressure differential along A–B is given by 𝜌2 gh, where h is the differential height between the two columns. In Figure 3.12b, the pressure differential along A–B is (𝜌2 − 𝜌1 )gh

ρ1 = 0 h

C B

A

Δ = ρ2

ρ2 D (a) ρ1 h

C B

A ρ2

Δ = ρ2 – ρ1

D (b) Figure 3.12 The concept of “reduced gravity.” (a) Two adjacent ocean columns of different depths but with no fluid above them. (b) Two adjacent ocean columns made up of two fluids of different densities, 𝜌1 and 𝜌2 , where Δ𝜌 = 𝜌2 − 𝜌1 and where 𝜌2 > 𝜌1 . A–B is a reference level between the two columns. The pressure differential along A–B is given by 𝜌2 gh in (a) where h is the differential height between the two columns and in (b) where the pressure differential along A–B is (𝜌2 − 𝜌1 )gh = Δ𝜌gh = ̃ gh.

= Δ𝜌gh = ̃ gh. The effect of the stratification on buoyancy is the same as reducing gravity by the factor Δ𝜌 and the phase speed of a gravity wave by the same factor. Figure 3.13 provides the first observational evidence of an eastward propagating wave across the equatorial Pacific Ocean (Knox and Halpern 1982). Previously, Knox (1976) had been the first to show evidence of the theoretical Kelvin wave in the Indian Ocean, using ocean buoy and tide gauge data along the equator. These observations utilized prototypes of the eventual Tropical Ocean-Global Atmosphere–Tropical Ocean Atmosphere (TOGA-TOA) array that would eventually span the entire tropical Pacific Ocean.XIV The figure shows three time sections of ocean transport at three equatorial locations in the Pacific Ocean (152∘ W, 110∘ W, and 91∘ W, the first two from moored buoy arrays along the equator and the last in the Galapagos Islands). Phase speeds of the wave, identified as an equatorial Kelvin wave, are between 2.4 and 2.6 m s−1 . The figure provides evidence of long-distance oceanic communication along the equator across the Pacific Ocean. Given the oceanic gravity wave speeds, the Rossby Radius of Deformation in the ocean is far smaller than in the atmosphere (see Table 3.2c). For example, with a depth of 100 m, R is 20 km at 30∘ N and 100 km at 5∘ N. As pointed out by Knox and Halpern (1982), relatively small-scale but sustained wind events (such as “westerly

3.2 Dynamics of the Low-Latitude Upper Ocean

(a) Location of stations 20°N

09 December 2014

10°N N8

EQU

NORPAX ARRAY

EPOCS ARRAY

10°S 15°S

N. Isabela

T9

GALAPAGAS

150°W 40°W 130°W 120°W 110°W 100°W 90°W 80°W 70°W SST (°C) 18.4 20.4 22.5 24.5 26.6 28.6 30.7

14.3 16.3

(b) Time series of mass transport 20

300

eastward transport per unit width about the equator (m2s–1)

N8 (152°W) 250

15

T9 (110°W)

10

N. Isabela (91°W) 200

5

150

0 –5

100 –10

2.52ms–1 50

2.43ms–1

–15

2.4ms–1

0 8

anomalous sea level height N. Isabela (hPa)

Figure 3.13 Propagation of an internal equatorial Kelvin wave in the Pacific Ocean. The existence of such a wave had been predicted theoretically, but Knox and Halpern (1982) showed that a remote meteorologically forced impulse in the western Pacific could influence the dynamical and thermodynamical structure of the eastern Pacific Ocean. (a) Location of the three stations used to determine the eastward mass transport per unit width between 0 and 250 m close to the equator. The array is set against 12/09/2014 NOAA-NESDIS Geo-Polar 5 km SST obtainable at http:// www.ospo.noaa.gov. (b) Calculations of the eastward transport surge indicating phase speeds 2.3–2.4 m s−1 , which match theoretical estimates discussed in the text. Blue and red curves denote measurements of the eastward mass transport per unit width (m2 s−1 : left-hand scale). The red curve is a proxy for mass transport and represents the anomalous sea level height at North Isabela Island of the Galapagos (hPa; right-hand scale).

–20 20 March

10 20 April

10 20 May

Year 1980

wind bursts” in the western equatorial Pacific Ocean) can produce a basin-scale oceanic response that may exist in the western-central Pacific and last from days to weeks. Their occurrence seems to be strongly tied to the phase of the MJO occurring more frequently during the periods when it is active in the western-central Pacific.9 Furthermore, westerly wind bursts often precede the formation of El Niño and may be integral components of the El Niño–Southern Oscillation (ENSO) cycle.10 3.2.3

Sverdrup Wind-Driven Transport

How do surface winds drive ocean currents? For largescale motions, where the inertial terms are very small compared to the Coriolis term (i.e. RO ≪ 1), we can assume that the flow is “steady” and does not 9 E.g. Fasullo and Webster (1999). 10 Eisenman et al. (2005).

accelerate. Moreover, the frictional effects of the wind are important only at the surface (where the Ekman number is order unity) and decrease rapidly with depth (Figure 3.10). Sverdrup (1947)XV developed a theoretical basis for computing the wind-driven mass transport in the ocean. An expression relating wind stress and transport can be derived in the following manner. First, Eq. (3.50a) is differentiated in the meridional direction and Eq. (3.50b) in the zonal, which, after subtraction, gives ( 2 ) ( ) 𝜕 𝜏y 𝜕 2 𝜏x du dv 1 f + + 𝛽v = − (3.57) dx dy 𝜌0 𝜕x𝜕z 𝜕y𝜕z Using the assumption that the vertical velocities at the surface (z = 0) and at z = − D0 are zero, the continuity Eq. (3.50d) becomes dM x dM y du dv + = 0 or + =0 (3.58) dx dy dx dy

95

96

3 Fundamental Processes 0

0

Here, u = ∫−D u dz and v = ∫−D v dz. The meridional 0

0

0

and zonal mass fluxes are defined as M y,x = ∫−D 𝜌0 v, udz. 0 Integrating Eq. (3.57) from z = −D0 to z = 0 gives ( 2 ) 0 𝜕 𝜏y 𝜕 2 𝜏x 1 𝛽v = − dz (3.59) 𝜌0 ∫−D0 𝜕x𝜕z 𝜕y𝜕z Since 𝜏 x and 𝜏 y are zero at depths below −ze (see Figure 3.10), Eq. (3.59) becomes ( ) 1 𝜕𝜏y 𝜕𝜏x 1 (3.60) 𝛽v = − = (∇ × 𝜏̃) 𝜌0 𝜕x 𝜕y 𝜌0 With the definitions of total vertically integrated meridional and zonal mass fluxes defined above, we can use Eq. (3.60) to give ( ) − 0 𝜏 ∇ ×→ 1 𝜕𝜏y 𝜕𝜏x My = 𝜌0 vdz = − = z ∫−D0 𝛽 𝜕x 𝜕y 𝛽 (3.61) Equation (3.61) represents a profound and fundamental relationship referred to as the Sverdrup balance, where the northward mass transport of wind-driven currents is equal to the curl of the wind stress. There are some latitudes where the curl of the wind stress will be zero. Such latitudes demark the extent of distinct ocean circulations or gyres. 3.2.4

Ekman Transports

In the next paragraphs we follow Ekman (1905) and develop expressions that define wind-driven Ekman transport. We then attempt to explain the curious result that the wind-driven mass transport is at 90∘ to the surface wind. 3.2.4.1

Formulation

Fridtjof NansenXVI was interested in ocean currents in polar seas. In 1893, he allowed his small wooden ship, the Fram, to freeze into the Arctic pack ice about 1100 km south of the North Pole. His goal was to determine how ocean currents affected the movement of pack ice, and, hopefully, drift across the North Pole. The Fram remained locked in pack ice for 35 months, drifting to within 400 km of the North Pole, the most northerly excursion at that time. However, during this progression, Nansen made a critical observation: the direction of ice and ship movement was consistently 20–40∘ to the right of the prevailing wind direction. Vagn Walfrid EkmanXVII was intrigued by Nansen’s observations and was able to explain the observations theoretically. He proposed a hypothetical situation where a wind stress acts on a quiescent ocean within which there are no horizontal pressure gradients and

thus a horizontally homogeneous ocean. In such a system, there is a balance between the vertical wind stress components and the Coriolis terms, as is evident from Eqs. (3.50a) and (3.50b), such that −fve = fue =

1 𝜕𝜏x 𝜌0 𝜕z 1 𝜕𝜏y

(3.62a) (3.62b)

𝜌0 𝜕z

where ue and ve represent the horizontal components of the wind-driven ocean circulation. This is referred to as “Ekman flow.” If we define the meridional and zonal mass fluxes as 0

M xe =

∫−ze

0

𝜌0 ue dz

and M ye =

∫−ze

𝜌0 ve dz

then with Eqs. (3.62), the latitudinal and longitudinal mass transports are M ye = −𝜏x ∕f

(3.63a)

and M xe = 𝜏y ∕f

(3.63b)

stating that the integrated mass transport within the Ekman layer is due only to the wind stress at the ocean surface. If the zonal component of the surface wind is zero (i.e. 𝜏 x = 0), the total mass transport will be in the zonal direction. On the other hand, if 𝜏 y = 0, the total mass transport will be in the meridional direction. If the Ekman mass transport takes place where there is a horizontal temperature gradient, then heat can also be transferred laterally. Equations (3.63) state that the integrated Ekman mass flux is orthogonal to the surface wind stress. Also, the mass flux is to the right of the wind in the northern hemisphere but to the left in the southern hemisphere. This change of direction of the transport relative to the surface wind between hemispheres provides a vital clue to explaining the orthogonality of the vertically integrated mass transport. We investigate the vertical structure of the Ekman transport and the direction of the surface current by rewriting Eqs. (3.62) using an eddy formulation of mixing in Eq. (3.51). That is, −fve = Az

𝜕 2 ue 𝜕z2

(3.64a)

and 𝜕 2 ve (3.64b) 𝜕z2 To simplify the algebra without losing generality, we consider a constant wind stress acting only in the zonal direction. Also, we assume that at some depth below the fue = Az

3.2 Dynamics of the Low-Latitude Upper Ocean

We can derive an equation for the meridional component of the Ekman velocity, ve , by first taking the second derivative of Eq. (3.64b) and substituting the result into Eq. (3.64a). Following the same procedure, an equivalent expression for ue can also be obtained. Two expressions emerge: ( )2 ( 4 ) 𝜕 ve Az ve = − (3.66a) f 𝜕z4 ( )2 ( 4 ) 𝜕 ue Az ue = − (3.66b) f 𝜕z4 These fourth-order differential equations have standard solutions given by ( ) ( ) (1 + i)𝜋 (1 + i)𝜋 ue = A1 exp z + A2 exp − z De De ( ) ( ) (1 + i)𝜋 (1 + i)𝜋 ve = A3 exp z + A4 exp − z De De (3.67) where the Ai are constants. In order for the wind-driven currents to reduce with depth as stipulated in Eq. (3.65), A2 and A4 must equal zero. Inserting Eq. (3.67) into Eqs. (3.66)√ and using the relationship (1 + i)4 = − 4, we find De = 2𝜋 2 Az ∕f , the e-folding depth of the Ekman layer. The value of A1 can be obtained from upper boundary conditions. Then we can use Eqs. (3.64) to obtain A3 . The results are √ ) ( 𝜏(1 − i) 𝜏 𝜋 1 (3.68a) De = exp −i A1 = 2𝜌0 𝜋Az 𝜌0 fAz 4 A3 = −iA1

(3.68b)

√ where the relationship 1 − i = 2 exp(−i𝜋∕4) has been used. Using trigonometric identities, the real parts of the general solutions are √ ( ) ( ) 𝜏 𝜋z 𝜋 𝜋z 1 ue = exp cos − 𝜌0 fAz De 4 De √ ( ) ( ) 𝜋 𝜋z 𝜏 𝜋z 1 ve = − sin (3.69) exp − 𝜌0 fAz De 4 De which is often referred to as the Ekman spiral. Figure 3.14 illustrates the balance of forces in the upper ocean Ekman layer and the variation of the wind-driven

Ocean Ekman Layer (NH) U0 (a) Vertical Section

V0 0 −10

z

−20 depth (m)

surface the Ekman wind-driven components become very small. This is a reasonable assumption given that the impact of surface winds will decrease with depth. Formally, we define boundary conditions on the ocean column such that 𝜕u 𝜏x = 𝜌0 Az e and 𝜏y = 0 at z = 0 𝜕z ue , ve → 0 at z = −∞ (3.65)

−30 −40 U0

−50 −60 (b) Horizontal Projection

45°

V0

y

Ekman Spiral x

Me Integrated Ekman Mass Transport

Figure 3.14 Current velocity vectors in the upper ocean relative to a surface wind of U0 . (a) Vertical section of the current vectors from the surface to the bottom of the Ekman layer showing an exponential decrease with depth and a clockwise rotation. Note that the surface current is at 45∘ to the surface wind following Eq. (3.70). (b) Horizontal projection of the current vectors illustrating the orthogonality of the vertically integrated mass transport (blue arrow) to the surface wind following Eq. (3.63).

current with depth. The observed Ekman spiral is rarely as well defined in observations as described by the theoretical profile. 3.2.4.2 Why Is the Total Integrated Ekman Transport Orthogonal to the Surface Wind?

Ekman’s theory explains the difference between surface current and surface wind and integrated transports. Later, we will see that if there is a horizontal temperature gradient in the upper ocean, Ekman processes can also transport heat. Setting z = 0 in Eq. (3.69), we can obtain an expression for the surface current: ue (z = 0) = −ve (z = 0)

(3.70) ∘ which states that the surface current is 45 to the right of the surface wind in the northern hemisphere, as shown in Figure 3.14. Moreover, as indicated in the figure and as expressed theoretically in Eq. (3.63), the integrated Ekman mass flux through the upper ocean is orthogonal to the surface wind.

97

98

3 Fundamental Processes

The explanation of this orthogonality is often just cast in terms of balance of forces. Near the surface, the wind stress balances the Coriolis force and as the depth increases, the impact of stress decreases. The Coriolis force, proportional to velocity, decreases as well and the fluid parcel moves toward geostrophic balance with the subsurface pressure gradient (Figure 3.14). This interpretation, as a changing balance of forces, is correct but does not explain the 90∘ angle specifically. To explain this orthogonality, we resort to the most basic of conservation laws: the conservation of angular momentum. The orthogonality of the surface wind and the overall ocean transport is a simple example of its conservation. Consider, first, a non-rotating planet with an arbitrary wind, caused perhaps by radiative heating differentials. The resulting stress will drive an ocean current in the direction of the wind. The loss of momentum by the atmosphere must be equal and opposite to the gain of angular momentum by the ocean. The direction of the wind in this non-rotating case is irrelevant and the angular momentum of the entire system will always be conserved. However, in a rotating system the wind direction is very important.

momentum by the atmosphere) as the atmosphere is in super-rotation relative to the ocean. An equatorward Ekman mass transport and an increase of low-latitude ocean depth constitutes a gain in angular momentum by the ocean and a conservation of angular momentum by the entire system. (ii) Consider now an easterly (westward) wind in subrotation relative to the rotating planet. The atmosphere gains angular momentum from the ocean (Figure 3.15b). The consequent poleward Ekman mass transport shallows the equatorial ocean depth, reducing the ocean angular momentum and thus maintaining the total angular momentum of the system. (iii) Consider next an arbitrarily directed wind. Through a combination of (i) and (ii), the mass flux will be orthogonal to the direction of the surface wind.

(i) Consider first a westerly (eastward) surface wind in the northern hemisphere (Figure 3.15a). Momentum is transferred to the ocean (loss of

Ekman transports can result in regional convergence or divergence of mass in the upper ocean, resulting in variations of the horizontal pressure gradient.

(a) Westerly surface winds (super-rotation) o Atmosphere provides angular momentum to ocean at its own expense o Equatorial Ekman mass flux increases ocean moment of inertia increasing its angular momentum Angular momentum of the oceanatmosphere system is conserved.

The determination of the role of the coupled ocean and atmosphere in the conservation of angular momentum provides a simple explanation of the relationship between wind stress and Ekman transport. 3.2.5

Induced Geostrophic Currents

(b) Easterly surface winds (sub-rotation) o Ocean provides angular momentum to atmosphere at its own expense o Polweard Ekman mass flux decreases ocean moment of inertia decreasing its angular momentum Angular momentum of the oceanatmosphere system is conserved.

Figure 3.15 Schematic of the interaction of the ocean and the atmosphere and the forcing of the integrated Ekman mass transports at right angles to the surface wind. (a) Surface westerly (eastward) winds (blue vectors). The atmosphere, in super-rotation relative to the ocean, loses momentum to the ocean. To compensate for the gain, equatorward wind driven mass transports (red vectors) increase the moment of inertia of the oceans, thus decreasing its momentum depicted by a change in sea level height (dashed line). Together, the total angular momentum of the ocean–atmosphere system is conserved. (b) The impact of easterlies (westward flow) on the ocean. In contrast to (a), Ekman transport compensates for the gain of momentum by the atmosphere.

3.2 Dynamics of the Low-Latitude Upper Ocean

The altered mass field will result in geostrophic currents. Returning to Eqs. (3.50), we note that below the surface layer the frictional effects become small compared to the Coriolis terms, so that −fvg = −

1 𝜕p 𝜌0 𝜕x

(3.71a)

and fug = −

1 𝜕p 𝜌0 𝜕y

(3.71b)

The terms ug and vg represent the horizontal geostrophic velocity components. Operating on (3.71a) with 𝜕/𝜕y and on (3.71b) with 𝜕/𝜕x and subtracting leads to ) ( 𝜕ug 𝜕vg (3.72) + + 𝛽vg = 0 f 𝜕x 𝜕y Then integrating from (the level of no motion) z = − D0 to the bottom of the Ekman layer, z = − ze , we obtain ) −ze ( 𝜕u 𝜕vg g + dz (3.73) 𝛽vg = −f ∫−D0 𝜕x 𝜕y −z

where vg = ∫−D E v dz. Using the continuity equation, and o noting from Figure 3.10 that w = wE at z = −zE and w = 0 at z = −Do , Eq. (3.73) becomes 𝛽 vg = −fwe

(3.74)

where we represents the vertical velocity at the bottom of the Ekman layer. We can obtain further interesting results by operating on Eq. (3.62a) with 𝜕/𝜕y and on Eq. (3.62b) with 𝜕/𝜕x and subtracting the two equations to give [ ( ) ( )] 𝜕ue 𝜕ve 1 𝜕 𝜏y 𝜕 𝜏x + = − (3.75) 𝜕x 𝜕x 𝜌0 𝜕x f 𝜕y f We again use the continuity equation together with Eq. (3.74) to obtain a solution for vg as a function of 𝜏 x and 𝜏 y . That is: [ ( ) ( )] f 𝜕 𝜏y 𝜕 𝜏x 𝛽vg = − (3.76) 𝜌0 𝜕x f 𝜕y f Moreover, using Eqs. (3.59a) in Eq. (3.72), we obtain [ ] 1 𝜕𝜏y 𝜕𝜏x 𝛽v = 𝛽(ve + vg ) = − (3.77) 𝜌0 𝜕x 𝜕y Equation (3.77) shows that total meridional mass M y transport is given by the sum of the vertically integrated geostrophic and Ekman transports (i.e. M y = M yg + M ye ). A solution for M yg as a function of surface wind stress can be obtained from Eq. (3.76): ( ) f 𝜏̃ M yg = ∇z × (3.78) 𝛽 f

Between Eqs. (3.74) and (3.76), and assuming that the vertical velocity at the top of the ocean is zero, the vertical velocity at the base of the Ekman layer, we , can also be expressed as a function of the wind stress at the ocean surface as ( ) 1 𝜏̃ (3.79) we = − ∇z × 𝜌0 f If we is 0, it is called “Ekman suction.” These vertical velocities have an extremely large influence on both the vertical and horizontal structures of the ocean. As the vertical velocity at the surface must be zero, the Ekman vertical velocity must be balanced by a geostrophic vertical velocity wg . That is: ( ) 1 𝜏̃ we = −wg = − ∇z × (3.80) 𝜌0 f From this equation it would seem that the Ekman pumping appears singular or undetermined at the equator. We will need to consider Ekman processes very carefully near the equator, especially in regions where there is a substantial cross-equatorial wind field such as in the Indian Ocean. 3.2.6

Low-Latitude Wind-Driven Currents

We now attempt to draw together some of the concepts developed earlier in Section 3.2. The treatment is not complete and a number of key principles central to physical oceanography have not been addressed. However, emphasis has been made on those processes that are central to the development of a physical understanding of coupled ocean–atmosphere phenomena. 3.2.6.1 Global Wind-Stress Fields and Surface Current Climatology

A global map of the annually averaged surface wind stress is presented in Figure 3.16. The colored background shows the Ekman upwelling and downwelling calculated from Eq. (3.79). The Ekman transports create regions of net mass convergence and divergence in the upper ocean. The background colored contours reflect this induced divergence field and show regions of we < 0 (Ekman pumping: warm colors) and we > 0 (Ekman suction: cold colors). From Eq. (3.79), this translates into regions of upwelling (regions “A” in Figure 3.16), providing a means of communication between the deeper ocean and the surface. These regions also correspond to major fishing grounds and are regions of coastal upwelling where directed flow on the eastern side of ocean basins (e.g. California coast, South American coast, South African coast denoted by “C”) drive off-shore Ekman mass transports, as shown

99

100

3 Fundamental Processes

Mean Annual Wind Stress (N m–2) & Ekman Vertical Velocity (m/yr) 60°N 40°N

B

20°N

C

0°

E

D

A

C

20°S

A

A

C

A C

40°S 60°S 0°

40°E

80°E –100

120°E –50

160°E

0 m/yr

50

160°W

120°W

100

80°W

40°W

0°

0.2 Nm–2

Figure 3.16 Average annual surface wind stress vectors (vectors: N m−2 ) for the 60∘ S–60∘ N band using ERA40 reanalyses data for the period 1990–2000. Background color shading shows Ekman pumping (red) and suction (blue) relative to the scale (m year−1 ). Location “A” marks the regions of the convergence of the subtropical trade winds in the Atlantic and Pacific associated with upwelling and the creation of the equatorial cold tongues (“E”). Location “B” marks the confluence of equatorial westerlies and the southeast trades during the boreal. Equatorial upwelling occurs at “E.” Locations “C” show examples of coastal upwelling. “D” is the unique location of equatorial downwelling where the background atmospheric basic state along the equator is westerly.

schematically in Figure 3.17. Region “D” refers to equatorial downwelling as equatorial westerlies create Ekman convergence in the equatorial Indian Ocean. Along the equator (Figure 3.16) the Ekman vertical velocity is undefined. This result comes directly from Eq. (3.79), which states that we ∝ f −1 . However, this singularity is removable and finite and well-behaved Ekman transports of mass and heat can cross the equator.11 3.2.6.2

Geostrophic Currents

The fields of wind stress are complicated and it is perhaps instructive to consider some specific examples. Figure 3.18 illustrates two cases of induced vertical Ekman velocity fields. The first example (Figure 3.18a) shows the confluence of the trade winds similar to what occurs near the equator in the eastern Pacific Ocean producing a divergence of Ekman transport (location “E,” Figure 3.17). The second example occurs between the easterly trade winds and the extratropical westerlies such as in the North Pacific, where there is a convergence of Ekman transport (location “B”). The first case (Figure 3.18a) leads to upwelling and a surface “valley” or indentation. The second case (Figure 3.18b) leads to a surface “dome” and downwelling. In both of these cases, the water near the surface is warmer than the deeper water. In the subtropical case (Figure 3.18b), the Ekman convergence and the downwelling produces a region of relatively warm water 11 Discussed in Section 17.3.2.

in the upper ocean. Following the Sverdrup theory deeper in the ocean, there is no geostrophic motion and consequently there cannot be horizontal pressure gradients. Thus, the weight of the columns across the whole domain must be equal so that to compensate for the warmer and less dense water, the surface must bulge upwards. However, the horizontal gradients of density produced by the bulge produce latitudinal pressure gradients that must be balanced by longitudinal geostrophic currents. To the north of the dome, the induced geostrophic currents are to the east and to the south of the dome the currents are to the west. For the equatorial case (Figure 3.18a), we assume that the winds are symmetric about the equator, similar to the situation found in the eastern Pacific Ocean where the trade winds converge. This is a region of divergence of Ekman transport, creating a region of Ekman upwelling along the equator. Thus, a cool pool is produced in the upper ocean. Using arguments identical to the subtropical case, an antisymmetric pressure gradient will be produced about the equator. The resulting geostrophic current is symmetric about the equator and to the west. Figure 3.19 provides a detailed description of the near equatorial current as a function of latitude and depth. These meridional overturning cells (MOCs) show the convergent Ekman flow near the surface and the convergent geostrophic flow to be consistent with Eqs. (3.74) and (3.80). Essentially, this figure is a detailed version of Figure 3.18a. Note that the orientation of the isotherms (cool at the equator, warmer to the north and south),

3.2 Dynamics of the Low-Latitude Upper Ocean

Coastal Upwelling (NH) (a) Horizontal section v0

τx < 0

τx > 0

wind field. The theory also explains why there are counter-currents. Sverdrup noticed that in the open ocean the wind is essentially zonal if 𝜕𝜏 y /𝜕x < < 𝜕𝜏 x /𝜕y so that to a good approximation Eq. (3.61) becomes 1 𝜕𝜏x My ≈ − (3.81) 𝛽 𝜕y Rewriting the continuity Eq. (3.58) in terms of vertically integrated mass flux, we obtain

y

𝜕M x 𝜕M y + =0 𝜕x 𝜕y

x

Substitution of Eq. (3.81) into Eq. (3.82) and using 𝛽 = 𝜕f /𝜕y = 2Ω cos 𝜑/a gives

(b) Vertical sections v0 X warm

warm cold z

we > 0 x

cold

we < 0

vg < 0

X vg > 0

Figure 3.17 Ekman upwelling along a coast on the eastern side of an NH ocean basin. (a) Horizontal sections showing the direction of the wind and the integrated Ekman transport (orthogonal to the wind vector) for a southward wind (left) and a northward wind (right). (b) Vertical sections: the westward-induced Ekman transport by the southward flow creates upwelling cool subsurface water and thus cooling along the coast and an upward tilt of the ocean surface toward the west. The induced geostrophic transport (v g ), produced by the changes in the horizontal pressure gradient, is in the same direction as the surface wind. For a northward surface wind the surface is tilted toward the coast producing a thicker warm thermocline and subsidence along the coast. The induced subsurface geostrophic current is toward the north in the same direction as the surface wind.

together with the meridional currents, lead to a net heat flux toward the poles. Figure 3.20 shows a plot of the long-term average surface currents in the Pacific Ocean. A comparison with Figure 3.16 shows currents that align themselves with the trade winds in both hemispheres, although at an angle as per Eq. (3.71). These are the North and South Equatorial Currents (NEC and SEC, respectively). In addition, there are currents that are in direct opposition to the winds. These are referred to as “counter currents.” For example, between the NEC and the SEC is the Equatorial Counter Current (ECC). Such counter-currents occur because of nuances in the wind profiles.

𝜕M y 𝜕M x 1 =− =− 𝜕x 𝜕y 2Ω cos 𝜑

Currents and Counter-Currents

Sverdrup transport theory allows the prediction of wind-induced surface currents for a given surface

(

) 𝜕𝜏x 𝜕 2 𝜏x a tan 𝜑 + 𝜕y 𝜕y2 (3.83)

where a is the radius of the planet. To find an expression for M x , Sverdrup integrated Eq. (3.83) in longitude from a longitudinal eastern boundary at x = 0, where he assumed a no-flux boundary condition so that at x = 0, M x = 0. Using the boundary condition described in Figure 3.21a, Eq. (3.83) then becomes ([ ] [ 2 ]) 𝜕𝜏x 𝜕 𝜏x Δx Mx = − tan 𝜙 + a 2Ω cos 𝜑 𝜕y 𝜕y2 [ 2 ] 𝜕 𝜏x Δxa ≈− (3.84a) 2Ω cos 𝜑 𝜕y2 ([ ]) 𝜕𝜏x a My = − (3.84b) 2Ω cos 𝜑 𝜕y Here Δx is the distance from the eastern boundary, with the square brackets denoting zonal averages. Noting that the first term on the right-hand side of Eq. (3.84a) is much smaller than the second term, we can calculate longitudinal mass flux, M x . Figure 3.21b illustrates Sverdrup’s calculations using the annual average wind stress. The wind speed is seen to diminish within the doldrums between the trade wind regimes of both hemispheres corresponding to a region of negative [𝜕 2 𝜏 x /𝜕y2 ] and leading to a positive longitudinal mass flux and the equatorial counter-current. Sverdrup’s calculation is limited. It does not contain dissipative effects at a western boundary and the magnitude of the computed flux grows toward the west. Later studies (e.g. Stommel 1948 and Munk 1950) would extend the analysis to entire ocean basins, providing excellent replications of the upper ocean currents. 3.2.6.4

3.2.6.3

(3.82)

Equatorial Undercurrents

One of the major consequences of the near equatorial Ekman transport is the development of a tilted sea surface height from east to west in the Pacific Ocean

101

3 Fundamental Processes

(a) Equator

(b) Extratropics dome

valley U0(y)

ρ1

depth

U0(y)

ρ6

ρ2 ρ3 ρ4

W

W

ρ5 ρ6

SH

trade winds

ρ5

ρ1

EQU

westerlies

ρ2 ρ3 ρ4

equatorial easterlies

Ekman layer

E

E S

NH

NH latitude

latitude

N

Figure 3.18 Examples of geostrophic induced currents by wind-driven near-surface Ekman transports. The sections are three-dimensional, showing isopycnal surface against depth, latitude, and longitude. Blue arrows denote surface wind. Black arrows illustrate Ekman vertical current velocities and lateral mass transports within the Ekman layer (yellow). The transports are orthogonal to the surface wind. Red and green vectors show induced westward and eastward geostrophic currents, respectively. (a) Section across the equator showing Ekman pumping and equatorial Ekman divergence and the development of a cold “valley” in surface height along the equator. The induced geostrophic currents are to the west in the same direction as the surface wind. (b) Section through the extratropical region where the trades and the midlatitude westerlies meet. In this region of Ekman convergence and Ekman “suction,” a “dome” forms in the surface height, reflecting the perturbed mass field, with geostrophic westerly currents to the south and easterly currents to the north. The solutions are essentially two-dimensional in the latitude–depth plane, as longitudinal boundaries are not considered here.

Section across eastern Pacific Ocean 0 depth (m)

102

50

27

28

28

26 27 26

100

1 𝜕p 1 𝜕𝜏x du =− + + ds u dt 𝜌0 𝜕x 𝜌0 𝜕z

20 150

8°S

4°S

equ latitude

4°N

divergence (Figure 3.18a) will cause water to accumulate near the western boundary of the basin. This may be seen from the following simple argument. Consider the zonal momentum (Eq. (3.50a)) where we have set f = 0. In a symbolic sense we have

8°N

Figure 3.19 Vertical cross-section of the equatorial Ekman layer current and temperature field. The two fields referred to as “meridional overturning circulations” (MOCs) are roughly symmetric about the equator. The figure is a detail of Figure 3.16a. Source: Adapted from Figure 1g of Perez and Kessler (2009).

along the equator. For the moment, let us assume that the planet is completely oceanic with no land areas.XVIII On this simplified planet, assuming the convergence of lower-tropospheric trade winds, upwelling would occur completely around the globe, producing a low-latitude sea-surface temperature (SST) relative minimum and continuous surface valley around the equator. Figure 3.18a would then be representative of the entire near-equatorial tropics. However, the equatorial oceans are bounded laterally, and the zonal geostrophic current induced by the Ekman

(3.85)

where d s is some frictional dissipation rate. If there is initially no variation in the zonal direction and no zonal pressure gradient, the zonal wind stress (𝜕𝜏 x /𝜕z < 0) will accelerate a near-surface current to the west (u < 0). The zonal mass transport to the west builds up a zonal pressure gradient toward the east. Equilibrium will be reached (i.e. du/dt = 0) when the frictional force equals the imbalance between the pressure gradient term and the wind stress. Naturally, it is not possible to continue to build up mass in the western part of the ocean. As the wind stress impacts decrease with depth (𝜕𝜏 x /𝜕z → 0), a new balance below the surface ensues, with ds u =

1 𝜕p 𝜌0 𝜕x

(3.86)

producing a down-the-gradient subsurface current toward the east. This pressure gradient drives an eastward current, referred to as the equatorial undercurrent (EUCXIX ), that transmits mass from the western ocean to the east. Similar undercurrents occur in the Atlantic

3.2 Dynamics of the Low-Latitude Upper Ocean

Mean Pacific Ocean Currents (1993-2013) 40°N 30°N

20°N NEC 10°N ECC 0°

SEC

10°S

20°S 30°S 40°S 120°E

140°E

160°E

180°

160°W

140°W

120°W

100°W

0.5 m s–1 0.05

0.1

0.15

0.2

0.25

0.3

0.35

Figure 3.20 Surface currents in the Pacific Ocean averaged for the 1993–2013 period from NOAA’s Ocean Surface Current Analyses (OSCAR) data set. Note the general westward drift along the equator and to the south of the equator. This flow is referred to as the South Equatorial Current (SEC). In the vicinity of 10∘ N, the current also flows to the west. This is the North Equatorial Current (NEC). However, in between the NEC and the SEC there is a narrow band of eastward currents. This feature is referred to as the Equatorial Counter Current (ECC). The term “counter” refers to the current being in the opposite direction to the prevailing westward trade winds.

Ocean. In the Indian Ocean there is a reversed undercurrent (east to west) but the physics are considerably more complicated. The magnitude of the EUC can be substantial, reaching >1 m s−1 . The longitude–depth temperature structure along the equator is shown in Figure 3.22. On average, and in a non-El Niño year, the slope of the surface is about 60–70 cm from west to east. When we were discussing induced geostrophic flow described in Figure 3.18a,b we drew upon the fact that below the mixed layer the pressure gradients vanish. In order for the pressure at depth to be constant across the entire Pacific basin, the thermocline has to deepen toward the west so that the weights of all the ocean columns remain the same across the basin. The EUC completes the mass balance circuit along the equator. 3.2.7

Overview

A very cursory description of fundamental physics of the low-latitude upper ocean was presented with

emphasis on wind-driven circulations. Ekman’s theory showed that the surface current drift is at 45∘ to the surface wind. Further, the integrated ocean mass transport was orthogonal to the surface wind, causing regions of upper ocean divergence and convergence with an induced vertical Ekman velocity. The tilting of the near-surface pressure fields, caused by the Ekman surface divergence and convergence, induce geostrophic currents in the upper ocean. It was also shown that Ekman and Sverdrup transport explain to a large degree the observed surface current structure in the tropical oceans. Finally, the longitudinal finiteness of the oceans was considered, noting that the westward surface wind produced a piling-up of mass in the western Pacific Ocean. Very simple arguments were presented to show the force balance that would produce a subsurface return easterly current from the warm pool of the western Pacific Ocean to the cooler East Pacific. One of the problems we need to consider is what happens to Ekman transports at the equator. The developments above suggest that there is a singularity at

103

3 Fundamental Processes

(a) Sea Level Height (10°N-10°S) 2013 m

the equator (e.g. Eq. (3.78)). This is especially important in the Indian Ocean where the monsoon wind gyre winds straddle the equator, reversing between summer and winter.

0.6 0.4 0.2 150°E

(a) Longitudinal boundary conditions y

m

200

–x

Mx

0 τx

coast

P

180° 150°E 120°W 90°W (b) Potential Temperature (10°N-10°S) 2013

100

My x

x=0

300 400 500 150°E

180°

150°E

120°W

90°W °C

(b) Wind-field, stress-terms and induced surface currents WIND FIELD

OCEAN CURRENTS 30°N

latitude

104

Northeast trades Doldrums Southeast trades

𝜕2τx 𝜕y2

20°N

North Equatorial Current

10°N

Equatorial Counter-current

equ 10°S

South Equatorial Current

Figure 3.21 (a) Configuration of the system for the derivation of the longitudinal boundary condition. (b) The zonal components of the wind-field (blue vectors), the stress term 𝜕 2 𝜏 x /𝜕y2 from Eq. (3.84a) and the induced zonal surface current (red vectors). Source: From Pond and Pickard (1983).

8

10 12 14 16 18 20 22 24 26 28

Figure 3.22 (a) Sea level height (m) as a function of longitude average between 10∘ N and 10∘ S for 2013 from NOAA’s Global Ocean Data Assimilation System (GODAS) data set. The rise in sea level in (a) reflects the piling-up of mass by the generally westward surface currents (see Figure 3.18). (b) Vertical cross-section of potential temperature as a function of depth (m) across the Pacific Ocean for the same time period. In general, the thermocline is much deeper in the west than the east. Mass balance is achieved by a return subsurface eastward current, the equatorial undercurrent (EUC), which rises toward the surface in the eastern Pacific. The EUC is shown schematically as white contours. The EUC is trapped latitudinally close to the equator and may achieve magnitudes in excess of 1 m s−1 .

Notes I Carl Gustav Rossby (1898–1957) was a Swedish

meteorologist who emigrated to the United States in the late 1930s. He created the Department of Meteorology at the University of Chicago. To a large degree, it could be claimed that he was the “father of modern dynamic meteorology,” being the doctoral advisor to H. R. Byers, R. Montgomery, M. Neiberger, V. Starr, D. Fultz, H.-L. Kuo, G. Platzman, T. C. Yeh, G. Cressman, Y.-P. Hsieh, J. Malkus Simpson, C. Newton, Bert Bolin, A. C. Wiin-Nielsen, B. Doos, J. Namias, and J. C. Charney, among many others. He first explained the fundamental kinematics of extratropical weather systems, in particular the fundamental propagation characteristics of extratropical waves, eventually referred to as Rossby waves.

II Jule Gregory Charney (1917–1981) was an American

scientist who made seminal contributions toward the understanding of basic dynamics, especially of the extratropics. While at the University of California at Los Angeles (UCLA), he developed the “quasi-geostrophic theory”, which allowed for a systematic methodology of investigation on how basic balances (e.g. geostrophy) change with time. From this emerged the quasi-geostrophic vorticity equation that enabled the development of operational meteorological forecasts and also the theory of baroclinic instability. Later, as a professor at the Massachusetts Institute of Technology, he continued the study of the dynamics of the atmosphere and ocean with increasing emphasis on the tropics. He was instrumental in the development of the Global

3.2 Dynamics of the Low-Latitude Upper Ocean

III

IV

V

VI

VII

Atmospheric Research Program (GARP) and its tropical component GATE. He was the doctoral advisor of a number of notable scientists, including J. Pedlosky, J. Holton, C. Leovy, I. Fung, K. Emanuel, J. R. Bates, B. N. Goswami, T. Shepherd, E. Kalnay, among many others. Professor Eric Eady (1915–1966), University of Reading, UK, was a British meteorologist who developed the theory of baroclinic instability independently at about the same time as Charney. Joseph-Louis Lagrange (1736–1813) was an Italian-born mathematician, astronomer, and physicist. His work is thought to be a bridge between the classical mechanics of Newton and the developing mathematical physics of the nineteenth century. His “Lagrangian” derivative described changes within a parcel (e.g. dT/dt) as it moved through a variable background field. Using the chain rule, we can expand the Lagrangian derivative assuming variations only in t and x so that dT/dt = 𝜕T/𝜕t|x + 𝜕x/𝜕t(𝜕T/𝜕t|t ) = 𝜕T/𝜕t|x + u𝜕T/𝜕x|t . Professor Norman Phillips (1923–2019) of the Department of Meteorology, Massachusetts Institute of Technology. He obtained his doctorate from the University of Chicago in 1951. After leaving MIT in the 1970s he was the Director of the National Meteorological Center of the U.S. National Meteorological Center. He created the first general circulation model of the atmosphere (Phillips (1956)). He became the foremost authority on numerical weather prediction modeling, concluding his career at the NOAA National Center for Environmental Prediction. He received numerous awards and is a Fellow of the U.S. Academy of Sciences. The first mathematical expression of the Coriolis force was introduced by French scientist and engineer Gaspard-Gustave de Coriolis (1792–1843) in 1835. His principle interests were the mechanics of rotating machinery and was unaware of the geophysical implications of his formulations. An understanding of its consequence in atmospheric and oceanic dynamics came much later. Edward N. Lorenz (1917–2008) was a professor of meteorology at MIT. He provided a fundamental understanding of the energetics of the global atmosphere (Lorenz 1967). In addition, he was the first to illustrate the nonlinear error growth in numerical models and the transition into chaos. He is known for the popular conceptualization of these effects through what is referred to as the “butterfly effect” whereby small perturbations in initial conditions may lead to drastically different solutions. These achievements are discussed by Palmer (2016).

VIII William M. Gray (1929–2016) was a professor in the

Department of Atmospheric Science at Colorado State University. He was a pioneer in hurricane research and described the environmental conditions necessary for the genesis of tropical storms. He used this knowledge to make the first seasonal forecasts of seasonal hurricane activity. He was a fierce foe of the concept of anthropogenic influence in climate change. He had a large number of students who rose to positions of prominence, especially in the field of tropical cyclone research and forecasting: J. McBride, G. Holland, C. Landsea, R. Maddox, J. Martin, J. Chan, and P. Klotzbach. IX The “Foucault pendulum” was invented by physicist Leon Foucault to demonstrate the rotation of Earth. The pendulum day is the time required for the plane of a freely suspended (Foucault) pendulum to complete a rotation about the local vertical. X We will note later that there are situations where the relative vorticity may exceed f especially close to the equator in the presence of a strong cross-equatorial pressure gradient. Such regions may be inertially unstable (see Section 13.2). XI This special gravity wave is the equatorial Kelvin wave that propagates eastward and is trapped by rotational effects to the equator. The issue of what happens to the initially westward propagating gravity wave will be addressed in Chapters 7 and 8. XII Joseph Valentin Boussinesq (1842–1929) was a French mathematician and physicist. The “Boussinesq approximation” allows density differences to be neglected except where they appear in terms multiplied by g, the acceleration due to gravity. XIII Japanese for “harbor wave.” A computer generated animation of the December 26, 2004 Indian Ocean tsunami can be seen at http://www.noaanews.noaa .gov/video/tsunami-indonesia2003.mov (animation by NOAA). A surface wave propagated from Sumatra to the east coast of Africa in roughly 24 hours. Speeds of signal propagation were about 2.8 km s−1 . XIV The Tropical Ocean-Global Atmosphere (TOGA) Tropical Ocean Atmosphere (TOA) buoy array. A complete archive can be found at http://www.pmel .noaa.gov/tao. XV Hans Ulrik Sverdrup (1888–1957) was a Norwegian oceanographer who held many leadership positions in oceanography including Director of the Scripps Oceanographic Institute from 1936 to 1948. Besides the theoretical work described in this section, he was also a sea-going oceanographer making many important ocean measurements. He was honored by having a volume flux transport unit named after him: 1 Sverdrup (Sv) is equivalent to 106 m3 s−1 . For example, the mass transport of the Gulf Stream is

105

106

3 Fundamental Processes

30 Sv off Florida and about 15 Sv near Newfoundland. The seasonal variation of cross-equatorial ocean mass transport in the Indian Ocean varies between between ±2 Sv as discussed in Section 14.6. XVI Fridtjof Nansen (1861–1930): truly, a modern day “renaissance man.” He was a polar explorer, zoologist, and neurologist, champion Olympic skier and ice skater, and diplomat, winning the Nobel Prize for Peace in 1922 for his work helping displaced people after WWI. He developed an instrument called the “Nansen Bottle,” which allowed the sampling of water at a specific depth, but it is his observations that relate the atmospheric and ocean boundary layers that are of most interest to us here. A lunar crater and a mountain in Antarctica have been named after him. XVII Vagn Walfrid Ekman (1874–1954): Swedish oceanographer. He is best known for the conversion of the observations of Fridjtof Nansen into a theory of the ocean boundary layer. XVIII It is sometimes convenient to consider the simplest prototype of the globe system where there are no

continents. This is referred to as an “aqua-planet” and is used to elucidate fundamental aspects of global motion. It is a useful artifact for atmospheric motions, but less so for the ocean, where lateral boundaries are of great importance where modal structures are modified. The term “aqua-planet” was introduced by Hayashi and Sumi (1986). XIX The Pacific Equatorial Undercurrent was initially named the “Cromwell Current” after Townsend Cromwell (1922–1958), an American oceanographer who died during an oceanographic expedition. He made key observations that allowed the identification of the Pacific Equatorial Undercurrent (Cromwell et al. 1954). Actually, the undercurrent was discovered toward the end of the nineteenth century in central Pacific soundings made in preparation for the laying of underwater cables. Buchanan (1886) noted a countercurrent so strong that measurements were difficult to obtain (transcribed from Gill 1982, pp. 462–463).

107

4 Kinematics of Equatorial Waves Before we examine the modal structure of the tropical atmosphere and ocean, it is instructive to consider the properties of waves: specifically wave kinematics.1 Kinematics refers to that branch of classical mechanics describing the motion of points or bodies or systems of bodies without consideration of the root cause of the motion. Kinematics concentrates the trajectories of points, lines, and other geometric objects and their differential properties such as velocity and acceleration. The mode of communication of information from one region of the planet to another may take the form of conduction, advection, or as wave propagation. Molecule to molecule transfer of information is extremely slow. Advection transfers information at the speed of the background flow. A highly efficient mode of information transfer is through wave transmission. Formally, a wave is a disturbance that transfers information in the form of energy from one point in a fluid to another. There are two major forms of waves: mechanical and electromagnetic. Both forms are important in geophysics but here we concentrate on mechanical waves. Such waves are associated with little or no transport. Consider standing on a long pier. Waves created by atmospheric forcing pass under you traveling toward the shore. Yet, if you watch a piece of flotsam, it does not move with the wave but drifts very slowly in the direction toward the shore but at a speed far slower than the crests of the wave.I The slower advection is referred to as “Stokes drift.” In essence, the movement of the flotsam signifies the transfer of mass. We are interested here in how the state of the background flow through which waves travel alter the characteristics of the waves. There are two results from Chapter 1 that are pertinent here. First, there are vast areas in the tropics where propagating waves exist, as exemplified by the OLR variance maps in Figure 1.15. Second, these waves exist within a very complicated background in a three-dimensional basic state that also varies seasonally, as shown in the zonal wind speed cross-sections along the equator in Figure 1.11. What 1 From the Greek kinêma for movement.

is the impact of these complicated basic states on the communication of information and transfer of energy by waves? We know that a Rossby wave has an inherent westward phase speed. We also know that when a westerly background state is added the wave may have an eastward Doppler shifted phase speed. However, what happens to the wave if the basic state varies spatially, for example if U = U(x)? Would the wave communication of a signal be altered? Here, we attempt to understand the impact of a varying background state on the characteristics of a wave.

4.1 Phase and Group Velocities, and Energy Propagation Given that we are particularly interested in how one region influences another, it is necessary to understand how this communication and transfer of energy takes place. Thus, we need to understand both the phase and group velocity of waves. 4.1.1 Wave Characteristics in a Quiescent Basic State We first concentrate on the propagation characteristics of waves in general. This is a subject that often leads to confusion as there are two major velocities associated with a wave: the phase velocity cp and the group velocity cg . In the simplest sense, the phase velocity of the mode is the speed in a particular direction of the crest of a wave. However, a particular forcing may produce a collection of waves. The group velocity refers to the integrated velocity of the envelope of waves and how the integrated energy is transmitted from a source region to other parts of the system. 4.1.1.1

Golf Ball in a Pond

Consider a large quiescent pond into which a golf ball lands, albeit errantly directed. The immediate impact is a narrow splash at the center of the pond. The impact may be thought of as a “Dirac delta function,”II

Dynamics of the Tropical Atmosphere and Oceans, First Edition. Peter J. Webster. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

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4 Kinematics of Equatorial Waves

which possesses only an amplitude at a point. Almost instantaneously, circular waves emanate from the center proceeding toward the edge of the pond. Eventually, the pond will become quiescent once again after the waves reach the shore and the initial energy introduced by the golf ball is frictionally dissipated. A careful observer will note that there are waves of different amplitudes and wavelengths and propagation speeds that appear to wax and wane as the disturbance envelope moves toward the shore. The multitude of waves of a different scale can be no surprise as a Fourier transform of a 𝛿-function projects on to a broad range of horizontal scales. Longer waves, possessing a greater phase velocity, appear to propagate more quickly toward the shore but then dissipate at the front of the disturbed envelope. Shorter waves, with a slower phase velocity, are overtaken and appear to dissipate at the rear end of the envelope.2 4.1.1.2

Analysis of the Perturbation in the Pond

Consider one of the propagating waves of wavenumber k, frequency 𝜔, and peak amplitude of A0 . The lateral scale of the waves, L, is given by L0 /k, where L0 is the diameter of the pond.III We can write an expression for the amplitude of the wave as A(x, t) = A0 cos(kx − 𝜔t)

(4.1)

where the sign convention adopted is arbitrary. Assuming that the wave does not change shape in time (i.e. A0 = constant) and if 𝜔 is constant, then the number of crests passing a particular point per unit time provides the speed of the crests. Thus, the phase speed, cp , so defined, is given by cp = 𝜔∕k

(4.2)

The propagation of this single wave-form is shown in Figure 4.1a. In our pond, it is clear that one wave member of the ensemble with one particular phase speed cannot define the transfer of energy from the center of the pond to the shore produced by the impulsive forcing of the golf ball, as the energy is spread across many wavelengths. Thus, the transfer must be accomplished by the ensemble of waves of different scales forced by the impulse. Therefore, to describe the manner in which energy is transferred, we require a different measure that takes into account all of the gravity wave family members. That is, we need a method of considering the entirety of the overall group of waves. Consider now two sinusoidal waves produced by the initial impulse with slightly different wavelengths 2 The golfer, who has to contend with a one-stroke penalty, loss of distance, and losing a ball, may take solace in observing the physical impacts of the perturbation so created.

and frequencies but (for convenience only) the same amplitude. These two waves are of the same form but differ in scale by 2Δk and frequency by 2Δ𝜔. Each wave has the same form as Eq. (4.1). The sum of the waves is given by A(x, t) = A0 [cos((k − Δk)x − (𝜔 − Δ𝜔)t) + cos((k + Δk)x − (𝜔 + Δ𝜔)t)]

(4.3)

where Δ is a small increase. Using a trigonometric identity,3 Eq. (4.3) becomes cos((kx − 𝜔t) ± ((Δkx − Δ𝜔t)) = cos(kx − 𝜔t) cos(Δkx − Δ𝜔t) ∓ sin(kx − 𝜔t) sin(Δkx − Δ𝜔t) which allows Eq. (4.3) to be transformed to A(x, t) = 2A0 cos(Δkx − Δ𝜔t) cos(kx − 𝜔t) (4.4) The expression represents a simple sinusoidal wave of frequency 𝜔 and wavenumber k, with a modulated amplitude of 2A0 cos(Δkx − Δ𝜔t). This wave is illustrated in Figure 4.1b. The transfer of information occurs with a change in the waveform requiring some modulation of the amplitude and/or the frequency. The modulation of the wave, or the speed of content transfer, is given by Δ𝜔/Δk. This is the phase velocity of the combined wave or the group of waves. It is the phase velocity of the group! As each amplitude wave contains a group of waves, it is referred to as the group velocity, cg . Generally: cg = 𝜕𝜔∕𝜕k

(4.5)

where the partial derivative is used, as 𝜔 can be a multidimensional function. In many circumstances in the atmosphere and the ocean, stationary (non-propagating) waves appear to be forced, perhaps by orographic features or stationary differential heating or wind stress patterns. This phenomenon is easy to understand from the construct described above. Consider two identical waves propagating in opposite directions at the same speed: A0 cos(kx ± 𝜔t) = A0 (cos kx cos 𝜔t ∓ sin kx sin 𝜔t) (4.6) These two modes, propagating in opposite directions, produce a standing wave given by A(x, t) = A0 cos(kx + 𝜔t) + A0 cos(kx − 𝜔t) = 2A0 cos(kx) cos(𝜔t)

(4.7)

Of course, the real world is far more complicated than the quiescent pond we used as an illustration. 3 Specifically, cos(a ± b) = cos a cos b ∓ sin a sin b.

4.1 Phase and Group Velocities, and Energy Propagation

Figure 4.1 (a) Amplitude distribution of a sinusoidal wave of amplitude A, and wavenumber k at times t and t + nΔt. The speed of the crest is the phase speed denoted by bold red line. (b) The combination of two waves of wavenumbers k and k + Δk and frequency (𝜔 + Δ𝜔) at times t and (t + Δt). The dashed lines denote the phase speed of the two modes. The bold line, joining the progression of the envelope, defines the group velocity of the combined mode.

t 1 0 −1 1

2

3

4

5

2

3

4

5

2

3 displacement (a)

4

5

2

5

t + Δt 1 0 −1 1

t + 2Δt 1 0 −1 1

t

2 1 0 −1 −2

2

4

6

2

4

6

2

4

6

2 1 0 −1 −2

2 1 0 −1 −2

8

t + Δt

10

8

t + 2Δt

8

10

12

14

12

14

12

14

displacement (b)

The real ponds are large so that the rotation of the planet becomes important, allowing rotational modes to be produced rather than the relatively simple surface gravity waves. In addition, there may be background currents through the planetary ponds that vary in space and time that alter the dispersion of the modes. 4.1.2 Kinematic Relationships Between Waves and Their Background Basic State We can now establish some simple rules, based on the kinematics of waves, that will help us understand the

progression of waves (and their modification) in both quiescent systems and more complicated basic states. At about the same time as the dynamics of equatorial waves were being explored, there were other important and relevant studies, particularly Bretherton and Garratt (1968) and LighthillIV (1978), that provided fundamental insights into the manner of how the frequency and the scale of waves are related and how phase speed and group speed each have their special roles. Here, we will consider a general system where the background flow varies slightly over the wavelength of a wave. In essence, we are using the WKBJ approximation.V

109

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4 Kinematics of Equatorial Waves

4.1.2.1 Flow

General Wave Kinematics in a Variable Basic

We commence by developing a general relationship between the absolute frequency, 𝜔, the frequency of a wave observed at a fixed point in space, and the relative frequency 𝜔d , the frequency of a wave observed from a moving point in space. This is the Dopplershifted frequency of the wave.VI Consider a general multidimensional wave of the form A exp[i𝛼] = A exp[i𝛼(x1 , x2 , x3 , … , t)]

(4.8)

where A is again the amplitude of the wave and 𝛼 is the phase function. The xi are the spatial dimensions of the wave and t is time. We now define the absolute frequency 𝜔 and wavenumber ki as 𝜔=−

𝜕𝛼 𝜕t

and

ki =

𝜕𝛼 𝜕xi

(4.9a)

and the Doppler-shifted frequency as 𝜔d = 𝜔 − V i (x1 , x2 , x3 )ki

(4.9b)

where the background flow is V i in direction i. Here the Einstein summation convention or summation notation has been used.VII A general dispersion relationship (relating the manner in which waves of different frequency and scale disperse) provides a relationship between frequency and scale of a mode so that generally 𝜔 = 𝜔(x1 , x2 , x3 , … , k1 , k2 , k3 , …) From the definitions of 𝜔 and k: 𝜕𝜔 𝜕𝜔 𝜕kj 𝜕2𝛼 − | =− 𝜕xi 𝜕t 𝜕kj 𝜕xi 𝜕xi k

We define the change of longitudinal wavenumber along a ray path as ( ) 𝜕𝜔d 𝜕k 𝜕k 𝜕k dk 𝜕k = + U(x) + = + cgxd dt 𝜕t 𝜕k 𝜕x 𝜕t 𝜕x (4.13) where cgxd is the longitudinal Doppler group velocity component. With Eq. (4.13), Eq. (4.12) becomes dU(x) dk = −k (4.14a) dt dx stating that the longitudinal wavelength, k, will vary along a ray path in a manner determined by the longitudinal variation of the background zonal basic state U. Here, d/dt is the total derivative following the local group velocity. If U = 0, k will remain constant. However, if the basic state changes in space along a ray then k must change. If dU/dx < 0 (i.e. U decreasing toward the east) then the wavenumber will increase and hence the mode will “shrink” longitudinally. If dU/dx > 0 (U increasing toward the east) then the mode will “extend” in longitudinal scale as k decreases.VIII Further: 𝜕𝜔 dx =U+ = U + cgx = cgxd dt 𝜕k

Following the same procedure, and assuming that U = U (x,y,z) and noting that dz 𝜕𝜔d = = cgdz dt 𝜕m

(4.10a)

(4.10b)

Consider now a more specific zonal background state, U(x), that varies in longitude. Then Eq. (4.9b) becomes 𝜔 = U(x)k + 𝜔d (k)

(4.16a)

and dy 𝜕𝜔d = = cgdy dt 𝜕l we arrive at the expressions:

Because 𝜕kj /𝜕xi = 𝜕ki /𝜕xj , we have 𝜕ki 𝜕𝜔 𝜕ki 𝜕𝜔 =− + 𝜕t 𝜕kj 𝜕xj 𝜕xi

(4.15)

(4.11)

where k is the longitudinal wavenumber. For later use, we define m and l as vertical and latitudinal wavenumbers. In subsequent sections we will obtain specific expressions for 𝜔 but here leave it in its general form. Then, between Eqs. (4.10a) and (4.10b), we arrive at ( ) 𝜕𝜔d 𝜕k dU(x) 𝜕k + U(x) + = −k (4.12) 𝜕t 𝜕k 𝜕x dx Consider “ray path,” the trajectory along which energy propagates through the atmosphere or ocean.

(4.16b)

dU(z) dm = −k dt dz

(4.14b)

dU(y) dl = −k dt dy

(4.14c)

and

Equations (4.14b) and (4.14c) show the change in the vertical and longitudinal wavenumbers in a flow where U = U(y, z). Equation (4.14b) states that within a positive vertical shear (i.e. dU∕dz > 0) the vertical wavenumber will increase (decrease) the vertical scale. A negative shear (dU∕dz < 0) will decrease m (increase the vertical scale). Equation (4.14c) states that within a negative lateral shear (dU/dy < 0) l will decrease the latitudinal wavenumber. On the other hand, if the lateral shear increases with latitude then l will increase. In Section 1.2, we noted that the equatorial basic state is a strong function of longitude, latitude, and height. With the three expressions contained within Eqs. (4.14a)

4.2 Dispersive and Non-dispersive Waves

to (4.14c) in mind, we may expect considerable modification of wave scales in realistic flows, depending on the magnitude of its variation. Based on the kinematic relationships (4.14a) to (4.14c), the wavenumber vector would appear to be a strong function of the latitudinal, vertical, and meridional variations of the basic state. Negative stretching deformation (dU∕dx < 0) will decrease the longitudinal scale. Enhanced latitudinal shear will increase the latitudinal wavenumber and negative vertical shear will increase the vertical wavenumber, and vice versa. An important point to note is that as the wavenumber changes, so will the group and phase velocities of the wave. These dependencies can be inferred from the development above, but they are addressed specifically in Chapter 7. What we can say at this stage is that in a variable basic state the scales of modes will change, thus altering group speeds and the rate of energy dispersion away from a source.

the group velocity as a function of the longitudinal wavenumber for all possible modes in the tropics. This will require a considerable calculation but the outcome will be pleasing as we will be able to explain the tropical equatorial centers of action from a relatively simple theory. These properties of equatorial modes will be explored in Chapters 6 and 7.

4.1.2.2

𝜕𝜀 𝜕𝜀 + cg =0 (4.22) 𝜕t 𝜕x and the wave energy density is advected along with the group velocity of the wave ensemble.

Dispersion of Energy Away from a Source

There are some interesting extensions of Eqs. (4.14a) to (4.14c). Fundamental wave theory defines wave action density along a ray as 𝜉 = 𝜀∕𝜔d

(4.17)

where the energy density of a wave, 𝜀, is defined as 𝜌gh2 /2, with h representing the amplitude of the wave. Bretherton and Garratt (1968) described wave action density as pure wave energy, excluding the energy associated with the background flow, at one Doppler-shifted frequency. Wave action density is also conserved along a ray. Thus, using the identical procedure employed in obtaining Eqs. (4.14), it follows that 𝜕𝜉 𝜕 (̃c 𝜉) = 0 + 𝜕t 𝜕xi gd

(4.18)

If 𝜔d and cgd are independent of xi , then, using the definition of 𝜉 and assuming that the flow varies only in the longitude direction, we can rewrite Eq. (4.18) as dU 𝜕𝜀 𝜕𝜀 + cgd = −𝜀 𝜕t 𝜕x dx so that dU d𝜀 = −𝜀 dt dx

(4.19)

4.1.2.3 State

Dispersion in a Quiescent or Constant Basic

Most studies4 consider equatorial waves in a quiescent basic state despite the large variation noted in Section 1.2. In a quiescent basic state where U = 0, the kinematic rules (4.14a) to (4.14c) reduce to dl dm dk = = =0 (4.21) dt dt dt so that along a ray path there is no change from the initial scale of motion. Also, in a quiescent basic state cgd = cg , so that

4.1.2.4

Constant Basic State

For a constant non-zero basic state the kinematic rules expressed in Eqs. (4.14) and (4.20) still hold. However, the wave energy density in Eq. (4.22) is modified by replacement of the group speed with the Dopplershifted group speed, becoming 𝜕𝜀 𝜕𝜀 + cgd =0 𝜕t 𝜕x

where U is again the zonal component of the background flow. Equation (4.20) states that the wave energy density 𝜀 will increase as the wave moves into a region of negative stretching deformation or decrease if dU/dx > 0. To interpret more fully the consequences of Eqs. (4.14a) to (4.14c) and (4.20) we need expressions of

d𝜀 =0 dt

4.2 Dispersive and Non-dispersive Waves Non-dispersive waves are waves that propagate through a medium without deformation. The phase speed of such a wave is not a function of the scale of the wave. Thus, if 𝜔 = c0 k then, from Eqs. (4.2) and (4.5), cg = cp

(4.20)

or

(4.23)

Thus, energy travels with the waveform and is not dispersed. Electromagnetic waves radiating through unbounded space are non-dispersive. Sound waves are almost non-dispersive.IX If they were more strongly dispersive a choir would sound less harmonious as the soprano’s voice would arrive at different times to our ears than that of the bass. 4 E.g. Matsuno (1966), Longuet-Higgins (1968), and Gill (1982).

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4 Kinematics of Equatorial Waves

t

Figure 4.2 Plot of the standing wave of amplitude A0 expressed in Eq. (4.7) at times (t + nΔt), where n = 0, 1, 2, . . . . Strictly speaking, the phase and group speeds of this stationary wave are zero.

2 1 0 −1 −2 1

2

1

2

3

4

5

3

4

5

4

5

t + Δt

2 1 0 −1 −2

t + 2Δt 2 1 0 −1 −2 1

2

3 displacement

Figure 4.1 can be interpreted as a non-dispersive wave. Rather than one wave imagine a series of waves of different spatial scales or k. If the waves are non-dispersive then all of the crests will move at the same phase speed. Further, the wave energy will move at the phase speed that is identical to the group speed. On the other hand, nearly all atmospheric and oceanic waves, with some very important exceptions, are dispersive, where the frequency is a more complicated function of spatial scale. i.e. symbolically 𝜔 = 𝜔(kn ) and so does not vary linearly with k, as in the case of non-dispersive waves. Then cg ≠ c p

(4.24)

thus signifying a dispersion of energy. Figure 4.2 presents an example of a dispersive wave. Similarly, the golf ball example represents a set of dispersive waves. It is possible to show simply the differences between dispersive and non-dispersive waves by noting that for a given depth of fluid all non-dispersive waves have the same phase speed irrespective of scale; i.e. 𝜔 + Δ𝜔 k + Δk 𝜔 + Δ𝜔 𝜔 − Δ𝜔 = so that = k + Δk k − Δk 𝜔 − Δ𝜔 k − Δk (4.25) Now, using the formula of componendo and dividento5 we can transform Eq. (4.25) to 𝜔 k 𝜔 Δ𝜔 = → = (4.26) Δ𝜔 Δk k Δk 5 Essentially the rule of ratios states: a∕b = c∕d → (a + b)∕(a − b) = (c + d)∕(c − d).

If Eq. (4.26) is used in Eq. (4.4), we find that the transformation speed of the modulated amplitude is equal to that of an individual wave. Finally, we need to discuss the implications of a spã =V ̃ (xi ). From Eqs. tially varying basic state where V (4.14a), (4.14b), and (4.14c), we expect the wavenumber of a wave to change. For a non-dispersive wave (such as a gravity wave or a Kelvin wave: see Chapters 6 and 7) entering such a region, the change in spatial scale is immaterial as neither cp nor cg are functions of scale. For a dispersive wave (e.g. Rossby and mixed-Rossby gravity waves: Chapter 7) a change in spatial scale will have a considerable influence on both cp and cg portending regional accumulations, or depletions, of wave energy density, as suggested by Eq. (4.20).

4.3 Overview We have used as a basic premise the fact that as a wave moves through a homogeneous or inhomogeneous medium or basic state the intrinsic frequency of the mode is conserved along its ray path. We were able to show from a kinematic perspective that this constraint has a number of important ramifications. (i) If the background basic state of the zonal flow varies in longitude then the scale of a wave may increase or decrease depending on the sign of dU /dx, as expressed in Eq. (4.14a). If the stretching deformation is negative, the longitudinal scale will decrease, but will increase if it is positive. Similar dependencies were found for the vertical and

4.3 Overview

meridional wavenumbers depending on vertical and lateral shear of U, respectively. (ii) We used the results that wave action flux is constant along a ray and developed an expression for the energy density showing that, depending on the structure of the basic flow, wave energy density can accumulate in certain regions and be depleted in others. Wave energy density is advected along a ray with the local group speed. Therefore, in a variable basic flow, wave energy density may increase or decrease along a ray depending on the sign of the stretching deformation dU/dx.

The changes in structure of dispersive modes (e.g. k increasing or decreasing and corresponding changes in group velocity) and the manner in which wave energy density is aggregated or diminished regionally depends very much on the relative magnitudes of a wave’s group speed and the variability of the basic state. To proceed, we need a more thorough exposition on equatorial modes in order to test whether the changes induced by the observed variable basics are of a sufficiently large magnitude to be important. We will make such a determination in Chapters 6 and 7.

Notes I This is referred to as “Stokes drift” named after

George Gabriel Stokes (1819–1903), an Anglo-Irish mathematician and physicist who spent his career at the University of Cambridge. In 1949 he became the Lucasian Professor of Mathematics. Previous holders of the chair included Isaac Newton, Charles Babbage, Paul Dirac, James Lighthill, and Stephen Hawking. Stokes (1847) postulated that in a pure wave motion, the Stokes drift velocity is the average parcel velocity when following a specific fluid parcel as it travels with the fluid flow. As a wave passes a point each parcel has an orbital trajectory in the direction of the wave and in depth. Flotsam, for example, is moved by the net movement of the parcel in the direction of the wave motion; that is, the difference in end-points after the orbit of a parcel is the “Stoke’s Drift.” II The δ-function was introduced by Professor Paul Dirac (1902–1982) for use in signal processing. The Dirac delta function (δ-function) is an extremely thin entity of finite amplitude at some point in space or time. Such functions have a long history in mathematics and were considered by such notables as Fourier and Heaviside. The Fourier transform of the δ-function is spread across all spatial scales. Dirac was an English theoretical physicist who made fundamental contributions to quantum electrodynamics and quantum mechanics. He shared the 1933 Nobel Prize in Physics with Erwin Schrödinger “for the discovery of new productive forms of atomic theory.” Dirac introduced the concept of the “unit impulse” in his book “Principles of Quantum Mechanics” (Dirac 1930). III It is an easy extension in terms of the definition of k to go from the scale of a pond to that of the global tropics. In the tropics, k = 1 if one mode extends around the entire tropics Its wavelength, then, is 2𝜋a/1, where a is the radius of the planet. In general, we can write k = 2𝜋/(2𝜋a/n) = n/a, where n = 1, 2, 3, …, so that ka is an integer = 1, 2, 3, . . . .

IV Sir Michael James Lighthill (1924–1998) was a

V

VI

VII

VIII

British fluid dynamicist and mathematician. He held many university posts including the Lucasian Chair of mathematics at Cambridge. Of particular interest here is work on wave kinematics and wave propagation. Late in his career, he became interested in the dynamics of monsoons, resulting in a book with R.P. Pearce entitled “Monsoon Dynamics.” In mathematical physics, the WKB or WKBJ approximation is a technique for finding approximate solutions to linear differential equations that possess spatially varying coefficients. WKB stands for the inventors Wentzl, Kramers, and Brillouin, who developed the technique separately in 1923. The “J” is for the mathematician Harold Jeffries who developed the same technique a year or so later. It is especially useful for some of the problems we will consider here, especially for waves propagating through a slowly varying background basic state whose spatial or temporal variability is greater than the scale of the wave. We use the technique here and in subsequent sections. Named after the Austrian physicist Christian Doppler (1803–1853). He postulated that the observed frequency of a wave depends on the relative speed of the source and the observer. The usual example is the change in frequency of a train siren from higher to lower frequencies as the train approaches and passes an observer (or listener). Einstein’s summation convention conveniently simplifies notation. It was first introduced in Einstein’s 1916 paper on general relativity “The Foundation of the General Theory of Relativity”: Annalen der Physik. An excellent introduction is given by Professor Steven Rawlings (1961–2012) of Oxford University. The lecture may be found at: http://www-astro.physics.ox.ac.uk/~sr/lectures/ vectors/lecture10final.pdfc. In Section 6.1.3 we will use the sign convention that the frequency, 𝜔, of a wave is always positive.

113

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4 Kinematics of Equatorial Waves

Furthermore, the longitudinal wavenumber k is positive for eastward propagating modes and negative for westward propagating modes. Other conventions let k be either negative or positive for westward and eastward propagation whilst 𝜔 ≥ 0 for all modes. The choice of convections is arbitrary. IX Acoustic waves are dissipated as they pass through a medium. The dissipation is a function of frequency.

Low-frequency waves are dissipated less than high-frequency sounds. This is why fog horns use low-frequency sounds. Per cycle, about the same amount of energy is dissipated irrespective of the frequency. However, as high frequencies have more cycles in a given propagation, dissipation increases with frequency.

115

5 Fundamental Prototypes of Tropical Systems In Chapter 4, where wave kinematics were explored, we noted that a variable basic state can change the basic characteristics of waves. That kinematic analysis was predicated by the observed complexity of the mean tropical wind fields discussed at length in Section 1.2. In order to examine the dynamics of wave motions in the tropics more closely, we need to build a number of model prototypes. We introduce three basic systems: a global shallow ocean to represent the basic structure of the atmosphere and the ocean, and a vertically stratified model of the atmosphere. The aim in this chapter is to provide a mathematical foundation for subsequent chapters, allowing a more unencumbered concentration on physical processes. A second aim is to form a conceptual basis for differentiating between “forced” and “free” equatorial modes, a subject that has received confusing treatment in the literature.

5.1 The Laplace Shallow Fluid System In 1775, Pierre-Simon LaplaceI set about trying to understand the physics of ocean tides. He developed a set of linear partial differential equations for a spherical shallow fluid of depth H that rotated at a rate of Ω s−1 or 2𝜋 c day−1 . Laplace’s equations provide a foundation for examination of the basic rotational waveforms on a sphere. 5.1.1

1 𝜕(gH − Φ) dv + fu + =0 dt a 𝜕𝜙 ( ) dw 1 1 𝜕u 𝜕v + + = M(𝜆, 𝜙, t) dt a cos 𝜙 𝜕𝜆 𝜕𝜙

(5.1c)

where u, v, and w denote the zonal, meridional, and vertical velocity components and d/dt represents the substantial or Lagrangian derivative. Equation (5.1c) is the continuity equation in an incompressible fluid. In set (5.1) geometric terms have been omitted. M(𝜆, 𝜙, T) represents a mass source or sink and Φ(𝜆, 𝜙, t) potential. The gradient of “potential forcing” changes the potential energy of a fluid parcel by external factors such as lunar gravitational forcing, solar heating, or by variations in slope of the lower boundary of the shallow fluid. M can be thought of as equivalent to heating or cooling induced by external factors. In general terms, M and Φ represent the forcing agents of motion in the shallow fluid. For now, we set these forcing functions to zero and make a number of modifications to Eqs. (5.1). 5.1.1.1

Use the Equatorial 𝜷-Plane Approximation

Longuet–Higgins (1968) solved Laplace’s equations and used them in full spherical form, producing sets of global modes. Here we are particularly interested in equatorial modes and introduce a simplifying geometry by expressing the equations on an equatorial 𝛽-plane. A Taylor expansion of f (=2Ω sin 𝜙) about the equator yields f (y) = f (y = 0) +

Governing Equations

The fluid Laplace considered was assumed to be incompressible so that the density of the fluid is constant in time and space. In Laplace’s original formulation, the fluid was assumed to be in solid rotation with the rotating underlying spherical planet. Figure 5.1 displays Laplace’s view of the planet. The system can be described by the following three equations: 1 𝜕(gH − Φ) du − fv + =0 (5.1a) 𝜕t a cos 𝜙 𝜕𝜆

(5.1b)

df y + · · · ≈ 0 + 𝛽y+ ≈ 𝛽y dy (5.2)

where 𝛽 = df /dy = 2Ω cos 𝜙/a, f (y = 0) = 0, and cos 𝜑 = 1. Thus, on this equatorial 𝛽-plane, f varies linearly about the equator. Note, too, that 𝛽 is a maximum at the equator with a value of 2.29 × 10−11 m−1 s−1 so that the maximum restoring force for a poleward displacement of a parcel occurs at the equator. Although the equatorial 𝛽-plane approximation simplifies subsequent analysis it also restricts the analysis to the tropics. The 𝛽-effect (and hence the restoring

Dynamics of the Tropical Atmosphere and Oceans, First Edition. Peter J. Webster. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

5 Fundamental Prototypes of Tropical Systems

5.1.1.2 Define Total Depth of the Fluid and the Background Basic State

Ω

B

B A

h(λ,φ,t)

HS (φ)

C

The total depth, H, has a number of components: H(𝜆, 𝜙, t) = H + HS (𝜙) + h(𝜆, 𝜙, t) + hB (𝜆, 𝜙) (5.3)

C D U (φ)

H

hB (λ,φ) A

D

Figure 5.1 Laplace’s spherical “shallow fluid” system (left panel). Section ABCD of spherical section (right panel). Mean depth of the fluid is H (m). The zonal flow U(𝜑) is in geostrophic balance with HS (𝜑). The perturbation displacement of the surface is h(x, y, t). H total depth of the fluid given in equation (5.3).

25

1.5

where H is the mean depth of the fluid and h(𝜆, 𝜑, t) represents the height field perturbation (see Figure 5.1), hB (𝜆, 𝜙) represents the topography of the lower boundary, and H S (𝜙) describes the latitudinal slope of the fluid necessary to support a geostrophic basic state U(y) such that, on a 𝛽-plane: −𝛽yU = g

dHs dy

(5.4)

where we have, for the moment, left the basic state only as a function of latitude. For a motionless basic state, Hs is constant with latitude. It is assumed that the lower boundary is flat so that hB = 0. H is often referred to as the equivalent depth of the fluid.

20 5.1.1.3

1.0

15

βy

10

f 0.5

error (%)

magnitude (10–4 s–1)

116

5 0

0

0

10

20 Latitude

30

40

Figure 5.2 Comparison of the magnitude of the Coriolis term assessed in full spherical coordinates (blue) and on a 𝛽 - plane (right ordinate) as a function of latitude. The error in using the 𝛽–approximation is shown on the right ordinate.

force) may be enhanced or decreased depending on the sign of the basic zonal wind field. In a westerly basic state (U > 0), for example, the 𝛽-effect is reduced and the mode extends further poleward than a mode in a motionless basic state or when there is an easterly basic flow. Figure 5.2 plots the ratio of f = 2Ω sin 𝜙 and 𝛽y as a function of latitude. The overestimation of the vertical component of the Coriolis force with the 𝛽–plane approximation increases slowly away from the equator with an error of less than 1% at 10∘ from the equator, 2% at 20∘ , and 4% at 30∘ . Clearly, for modes close to the equator, the equatorial 𝛽-plane is a good approximation. The errors listed here refer to a motionless basic state.

Integrate the System in the Vertical

We note that there are three equations in system (5.1) but four unknowns: the velocity components (u, v, and w) and the depth (H). To simplify the system, we make use of the boundary conditions at the top and bottom of the shallow fluid and eliminate the vertical velocity component w. From Eq. (5.1c), we know that the forcing is independent of z, so if the pressure gradient force is independent of z initially, it will always be independent of z. This initial z independence assumption implies that u and v are independent of z. We define a vertical average of a quantity as z=H

𝛼(𝜆, 𝜙, t) =

∫z=hB

𝛼(𝜆, 𝜙, z, t)dz

(5.5)

It is also noted that at the top and bottom of the fluid dh at z = H and w = 0 at z = 0 (5.6) dt Using Eqs. (5.5) and (5.6) in Eq. (5.1) will reduce the number of variables by one resulting in three equations in three unknowns: u, v, and h(𝜆, 𝜙, t). w=

5.1.1.4

Linearization of the System

The system can be linearized with u = U(y) + u′ (x, y, t), v = v′ (x, y, t), and h = H + Hs + h′ (x, y, t)

(5.7)

Here, primed terms represent small perturbations about mean quantities such that u′ , v′ u′ u′ , etc.

5.1 The Laplace Shallow Fluid System

We make the a priori assumption that the basic state is largely defined by an arbitrary zonal component U(y). Furthermore, we have assumed that H >> Hs . Note that we have set the basic state to be represented solely by U defined in (5.4) and that the other components are small by comparison.II Using Eqs. (5.2) to (5.7), the basic governing Eqs. (5.1) become { } [ ] 𝜕u′ 𝜕h′ dU 𝜕u′ + U + v′ − 𝛽yv′ + g =0 𝜕t 𝜕x dy 𝜕x (5.8a) { } 𝜕v′ 𝜕h′ 𝜕v′ + U + 𝛽yu′ + g =0 (5.8b) 𝜕t 𝜕x 𝜕y ] ( ′ { } [ ) 𝛽yUv′ 𝜕h′ 𝜕u 𝜕v′ 𝜕h′ +H + U − + =0 𝜕t 𝜕x g 𝜕x 𝜕y (5.8c) where we have assumed that H >> HS and M = 0. Returning to the scaling discussion in Chapter 3, it is interesting to note in Eq. (5.8c) that the greater the depth of the fluid, the more barotropic and non-divergent the flow regime becomes. That is, assuming a motionless basic state, as H becomes larger the system must become increasingly non-divergent for balance. Simply, if H is sufficiently large, the last term in Eq, (5.8c) dominates but to satisfy the equality the term (𝜕u′ /𝜕x + 𝜕v′ /𝜕y = ∇ ⋅ V h ) must become vanishingly small. Conversely, the shallower the fluid, the increasingly divergent the motions become. Here we choose to keep the equations in dimensional form and adopt this strategy throughout the text. Whereas the algebra is a little more cumbersome, the physical interpretations of the equations are easier to follow. For example, a standard practice is to non-dimensionalize the equations of motion using and (1/𝛽c)1/2 for (c/𝛽)1/2 for a characteristic length scale√ a characteristic time scale.1 Here c =

gH, the phase

speed of a pure gravity wave in a fluid of depth H. However, the length scale is R = (c/𝛽)1/2 , the Rossby radius of deformation, which non-dimensionally is unity so that in a non-dimensional sense, all rotational modes will appear to have the same R. If the system is non-dimensionalized it is necessary to redimensionalize the solutions to gain a measure of relative scales. We feel it is simpler to retain the dimensional nature of the system.

1 E.g. Lindzen (1967).

5.1.2

Doppler and Non-Doppler Effects

A non-zero basic state introduces both Doppler terms (curly brackets) and non-Doppler effects (square brackets) in Eqs. (5.8). Doppler effects are those that refer merely to the translation of a characteristic by the basic state. If the reference coordinate system were to move at the speed of the basic state, the Doppler terms would vanish and the system would be identical to that seen from a motionless basic state. This simple case is handled by a Galilean transformation of coordinates, where two coordinate systems move with different but constant relative velocities. On the other hand, non-Doppler effects result from intrinsic changes in fundamental balances needed to support the basic state. These impacts cannot be deciphered by noting relative constant differences between two reference systems. In fact, the effects are strictly non-Galilean and simple system-to-system transformations are not necessarily possible. Non-Doppler effects result from intrinsic changes in fundamental balances needed to support the basic state. Because the Coriolis force is a fictional force it exists only in a non-inertial coordinate system. 5.1.3

Equatorial Wave Equation

Because the coefficients of Eqs. (5.8) are independent of x and t, we can seek solutions that are sinusoidal in longitude and time. That is, we set u′ , v′ , h′ (x, y, t) = R[(u, v, h(y)) exp i(kx − 𝜔t)] (5.9) where R denotes the real part of the expansion. These represent the normal modes of a system.III The substitution of Eq. (5.9) into Eqs. (5.8) leads to ( ) dU −i𝜔d u + − 𝛽y v + ikgh = 0 (5.10a) dy dh 𝛽yu − i𝜔d v + g =0 (5.10b) dy dv ikc2 u − 𝛽yUv + c2 (5.10c) − i𝜔d gh = 0 dy √ where c = gH and 𝜔d = 𝜔 − kU is the Doppler-shifted frequency. We can form a single equation in v(y) by eliminating the zonal wind component u and height h. The algebra is rather tedious and is summarized in Appendix D. The final general form of the equation in v(y) for arbitrary U = U (y) from Eqs. (D.6a) and (D.6b) is dv d2v + p(y) + q(y)v = 0 2 dy dy

(5.11)

The full expansions of p(y) and q(y) can be found in Eq. (D.6).

117

118

5 Fundamental Prototypes of Tropical Systems

Clearly, the addition of a basic state with shear complicates the equations considerably. At first glance, the solution of Eq. (5.11) looks quite formidable and is not in the form of a standard wave equation. However, by making judicious assumptions, we can examine the impacts of the basic state (or its absence) incrementally and systematically. We will find in Chapter 6 that for a motionless basic state, for example p(y) and q(y), assume much simpler forms, with the former coefficient becoming zero. With U = constant, p(y) and q(y) are more complicated, but analytic solutions are still possible. For a basic state with latitudinal shear (U = U(y)) we will have to depend upon numerical solutions. Consider, for example, U = 0. The governing equations reduce to p(y) = 0 and q(y) =

𝛽2 2 𝜔2 𝛽k 2 − − y − k 𝜔 c2 c2 (5.12)

where the Doppler-shifted frequency has been replaced by the intrinsic frequency and Eq. (5.11) now has the form of a canonical wave equation: ) ( 2 𝛽k 𝛽2 2 𝜔 d2v 2 v=0 (5.13) + − − y − k 𝜔 c2 c2 dy2 For the special case of U = 0, the coefficients p(y) and q(y) assume their simplest form. This special case is considered in detail in Section 6.1.1. For general values of U, Eq. (5.11) does not have this simple form of Eq. (5.13) because p(y) is non-zero (see Eqs. D.5 and D.6). To modify the equation into a malleable form we use a standard transformation2 to eliminate the first derivative term (i.e. the second term in Eq. 5.12) by setting ) ( 1 p(y)dy (5.14) v(y) = V (y) exp − 2∫ so that Eq. (5.11) adopts the form of a wave equation: d2V + Γ2 (y)V = 0 2 dy

(5.15)

where Γ2 (y) is the refractive index squared of a wave given by 1 dp 1 Γ2 (y) = q(y) − − p(y)2 2 dy 4

(5.16)

for U = U (y) on an equatorial 𝛽-plane. Whereas Γ2 (y) is a complicated function, it possesses three major characteristics of importance. Specifically, is Γ2 (y) > 0, = 0 or < 0? The sign of the refractive index determines whether the solutions to the wave equation are wave-like or exponentially growing or decaying. 2 Hildebrand (1976, p. 46).

The latitude where Γ2 = 0 are called the turning latitude (i.e. y = yT ) and separates wave-like solutions at smaller values of y from exponential solutions at larger values of y. The overall boundary conditions on Eq. (5.15) are such that the solutions become vanishingly small at large y. Equation (5.15) is a form of Schrodinger’s equation.IV The region between y = ± yT is called the equatorial waveguide and may be thought of as a potential well. Clearly, yT depends on the basic state U, the Doppler shifted frequency 𝜔d , and the scale of the wave k. It also depends on the depth of the fluid that resides in the constant c2 , the square of the phase speed of a gravity wave. The components of the refractive index contain both Doppler and non-Doppler effects. In fact, if we set U= constant, there are still non-Doppler effects, which we will consider later. It should be noted, too, that many of these terms contain higher powers of k so they may be expected to become more important at smaller longitudinal scales. It should also be remembered that we have used the 𝛽-plane approximation, which contains slowly increasing errors as latitude increases away from the equator. Finally, Eqs. (5.10) and (5.15) are written for finite v(y) only. We have not yet looked into the possibility of solutions existing when v becomes vanishingly small or zero. As it turns out, seeking solutions as v → 0 will lead to the identification of one of the most important modes in the tropical atmosphere and ocean – the equatorially trapped Kelvin wave.

5.2 Upper Ocean We also need to develop an oceanic wave equation equivalent to Eq. (5.11) taking into account ocean structure. In its simplest construct, the ocean may be thought of as a shallow warm layer of density 𝜌 overlying an inert deep colder layer of density 𝜌 + Δ𝜌, with a rigid upper surface. 5.2.1

Governing Equations

The development of the ocean model follows almost identically that described for the Laplace model in Section 5.1.2. The vertical velocity of the system is eliminated by vertically integrating the equations and noting that w = 0 at the top surface and w = dh′ /dt at the interface of the two layers. Here, h′ is the deviation of the depth of the upper layer. The vertically integrated velocity components in the upper layer are u′ and v′ . The lower level is assumed to be motionless and the upper level has no background motion. The system is

5.3 A Stratified Atmospheric Model

w=0

τ(x,y,t) hʹ (x,y,t)

H

w = dh/dt

HD

ρ ρ+Δρ

hʹ (x,y,t) 𝜌 then 1∕𝜌 = 1∕(𝜌 + 𝜌′ ) ≈ 1∕𝜌. Furthermore, it is assumed that 𝜌u >> 𝜌′ u and so on. Here, 𝛾 = C p /C v is the ratio of specific heats. Finally, z has been chosen as the vertical coordinate in order to have the same dimensions as the horizontal coordinates. 5.3.3 Coupled Horizontal and Vertical Structure Equations We seek normal modes in longitude and time, assuming [u, v, w, 𝜌, p(x, y, z)]

5.3.2

Basic Equations

⁀ 𝜌 ⁀(y, z)] exp i(kx − 𝜔t) ⁀, p = R[u⁀, v⁀,w,

We can write a governing set of equations on an equatorial 𝛽-plane linearized about a motionless hydrostatic basic state given by3 dp = −g𝜌 dz

(5.28a)

p = 𝜌RT

(5.28b)

1 d𝜌 1 =− 𝜌 dz G G(z) =

( ) dG 1+ dz

RT(z) g

(5.28c) (5.28d)

Here, 𝜌(z), p(z), and T(z) are the density, pressure, and temperature of the basic state and G is the local vertical 3 Lindzen (1967), where solutions on two 𝛽-planes, one in the midlatitudes and the other in the equatorial regions are developed.

(5.30)

where R denotes the real part of the right-hand side. With Eqs. (5.30) and (5.28), we obtain −i𝜔u⁀ − 𝛽yv⁀ + ikp⁀ = 0 𝜕p⁀

−i𝜔v⁀ + 𝛽yu⁀ + =0 𝜕y ( ) 𝜕p⁀ dG ⁀ 1 1+ p + g𝜌⁀ = 0 − 𝜕z 2HS dz ( ) ⁀ 1 dG ⁀ 𝜕w w+ 1+ −i𝜔𝜌⁀ − + iku⁀ + 2HS dz 𝜕z

−i𝜔p⁀

−

i𝜔𝛾gGp⁀

(5.31a) (5.31b) (5.31c) 𝜕v⁀ =0 𝜕y (5.31d)

( ( ) ) dG ⁀ +g 𝛾 1+ −1 w dz

√ = (𝛾 − 1) 𝜌Q̇

(5.31e)

5.4 Forced and Free Solutions and the Choice of H

following the transformation

where

⁀ 𝜌 ⁀(y, z)] ⁀, p [u⁀, v⁀,w, √ √ √ √ √ = [ 𝜌u, 𝜌v, 𝜌w, p∕ 𝜌, 𝜌(y, z)∕ 𝜌 ] (5.32)

Strictly, each of the eigenfunctions (e.g. u⁀) should have the designation k, 𝜔 as there is a set of eigensolutions for each k and 𝜔. For simplicity, we assume these designations are understood. Substitution of Eq. (5.32) into Eqs. (5.31) and the ⁀ p ⁀, and 𝜌 ⁀ (in the same manner as in elimination of u⁀,w, Appendix D) provides, after (considerable) manipulation: ( 2 ) 𝜕 k k2 2 ⁀ − (𝛽y) v − Z(𝛽 2 y2 − 𝜔2 )v⁀ + g 𝜕y2 𝜔 𝜔2 (√ ) ( )( ) 𝜌Q̇ 𝜕 k 𝜕 1−𝜀 +𝜅 + 𝛽y − =0 𝜕y 𝜔 𝜕z 2H 𝜅−𝜀 (5.33) where Z is a vertical operator given by ( ) d d G 𝜕2 G Z≡ + 𝜅 − 𝜀 𝜕z2 dz 𝜅 − 𝜀 dz ( ( )) gk2 (1 − 𝜀)2 d (1 + 𝜀)∕2 − 𝜅 + − − 𝜅−𝜀 𝜔2 4(𝜅 − 𝜀)G dz (5.34) in which 𝜀 = −dH S ∕dz and 𝜅 = R/C p . Following the separation of variables procedure described above, and noting that the coefficients are strictly either a function of y or z, we can write ∑ v⁀(y, z) = Ψn (y) Zn (z) (5.35) where Ψn (y) and Zn (z) represent the vertical and latitudinal structure functions of v⁀(y, z), respectively. Following Lindzen (1967), it is assumed that the forcing function (last term in Eq. (5.33)) for a given vertical structure Sn (z) can be separated in a similar manner to the variables (i.e. Ψn (y)Sn (y)) so that the forcing can be written as ∑ Sn (z)Ψn (y) (5.36) (𝛽 2 y2 − 𝜔2 ) n

where, for a given pair of k and 𝜔, Ψn (y) is an eigenfunction of the equation ( ) k k2 𝛽 2 2 d2 − − 2 y Ψn 𝜔 dy2 𝜔 ( ) 1 k2 − − (5.37) (𝛽 2 y2 − 𝜔2 )Ψn = 0 ghn 𝜔2 Here, hn is the separation coefficient or the equivalent depth of a family of shallow water systems. Again following the separation of variables procedure (Section 5.3.1), we obtain a general vertical structure equation: d 2 Zn dz

2

+ a1 (z)

dZn S (z) (5.38) + a2 (z)Zn = 𝜅(𝜀 − 𝜅) n dz G

a1 (z) = a2 (z) =

𝜅−𝜀 d G dz 𝜅−𝜀 Ghn

−

(

G 𝜅−𝜀 (1 − 𝜀)2

) and

2

4G ( ) 𝜅 − 𝜀 d (1 + 𝜀)∕2 − 𝜅 − 𝜅−𝜀 G dz A horizontal structure equation emerges from Eq. (5.37): ( 2 ) d 2 Ψn 𝛽2 2 𝜔 k 2 + −k − 𝛽− y Ψn = 0 ghn 𝜔 ghn dy2 (5.39) This equation is similar to Eq. (5.13) but c2 = ghn has replaced c2 = gH, where H is the depth of the shallow fluid. However, here hn is a separation coefficient and will be determined by the solution’s vertical structure equation subject to some forcing function Sn (z). As we shall see in the forced case, the choice of a particular vertical forcing function determines the form of the horizontal response. Equation (5.39) has solutions of the form ) ( 1 (5.40) Ψn = An exp − 𝜉 2 Hn (𝜉) 2 1 where 𝜉 = (𝛽 2 ∕gh ) ∕4 y and H is an Hermite polynomial n

n

of order n. We note immediately the determination of the latitude structure by hn emerging from the vertical structure of Eq. (5.40), is a true solution of Eq, (5.39) when )√ ( ghn 𝛽k 𝜔2 2 −k + = 2n + 1, n = 0, 1, 2, … 𝜔 ghn 𝛽 (5.41) Thus, as hn is altered through the vertical structure, so will be the frequency. Therefore, for an 𝜔 and k pair, we can write a solution for v⁀(y, z) as ) ( ∑ 1 v⁀(y, z) = An Zn (z) exp − 𝜉 2 Hn (𝜉) (5.42) 2 n The important issue to be noted here is that, in general, the horizontal and vertical structure equations (i.e. (5.38) and (5.39)) are not independent but are coupled through the separation coefficient hn . One might think of the coupled set as an infinity of shallow water models for each set of hn , 𝜔, and k.

5.4 Forced and Free Solutions and the Choice of H In the preceding paragraphs, we have developed basic equations that describe the fundamental modes of

121

5 Fundamental Prototypes of Tropical Systems

oscillation of the tropical atmosphere and ocean. In subsequent chapters we will determine the eigenfunctions of the system. We will also discuss the choice of H and hn and whether they possess a predetermined value or whether it is a free parameter. Before proceeding, we need to determine what value of H should be chosen to represent atmospheric or oceanic motions. Clearly, the choice is important. For example, we have shown that in Eqs. (3.34) and (3.35) the Rossby radius of deformation (R is the scale at which rotational effects become as important as buoyancy) is a strong function in the choice of H. For example, in a quiescent basic state with H = 300 m and 𝛽= 2.3 × 10−11 s−1 m−1 , R ≈ 1500 km. For H = 25, 50, 100, 200, 400, and 1000 m, R possesses values of 650, 1000, 1157, 1400, 1655, and 2085 km, respectively. Clearly, the choice of H will have drastic impacts on the form of a mode and its spatial influence. Also, H is related to the vertical wavelength of a mode and hence to its vertical length scale.4 Formally, the depth of the fluid (i.e. the equivalent depth) appears as a principal component of the separation coefficient between the vertical and horizontal parts of the linear primitive equations describing an atmosphere at either rest or in solid rotation. For an isothermal atmosphere of temperature T, the vertical scale, Lz and H, are related by Lz = 2𝜋∕[𝜅∕(hs H) − 1∕(4G)]1∕2

(5.43)

where G = R T∕g represents a vertical scale height and T is the mean temperature of the layer and Lz is plotted as a function of H. Figure 5.4, provides a plot of the

40 vertical wavelength LZ (km)

122

30

20

10

0

5

200

400 600 equivalent depth H (m)

800

1000

Figure 5.4 Relationship between the vertical wavelength Lz (km) and equivalent depth H (m). Source: After Lindzen (1967). Extended by Webster and Chang (1988). 4 Lindzen (1967).

relationship between the vertical wavelength Lz (km) and equivalent depth H (m). We will see later that a wide range of values of H from 5 to 1200 m have been assumed in the literature. At the extremes, large values of H are non-divergent and extend throughout the troposphere in the vertical and laterally into the subtropics. The assumption of small values of H infers a strongly divergent mode that has a short vertical wavelength and is restricted in lateral extent close to the equator. Given the importance of H (or hn ) it is worth exploring how a particular value is chosen. We use the simplest version of the Laplace system (Section 5.1) with a zero basic state (U = 0) but forced with a mass source and sink. Retaining M from Eq. (5.1) and following the identical procedures, we obtain an equivalent version of set (5.10): −i𝜔u − 𝛽yv + ikgh = 0 𝛽yu − i𝜔v + g ikc2 u + c2

dh =0 dy

dv − i𝜔gh = 𝜇(y) dy

(5.44a) (5.44b) (5.44c)

where the 𝜇(y) the coefficients of Fourier expansion: M(x, y, t) = R[𝜇(y) exp i(kx − 𝜔t)]

(5.45)

as expanded in Eq. (5.9). Following the same procedure used in Section 5.1.3, we arrive at ( 2 ) (𝜔 − k2 c2 ) (𝛽y)2 k𝛽 d2v + − 2 − v 𝜔 c2 c dy2 1 d𝜇 k𝛽y 𝜇 (5.46) = 2 − c dy 𝜔c2 √ d = Making the transformation 𝜉 = 𝛽∕cy so that dy √ 2 2 𝛽 d d 𝛽 d and dy we arrive at an equation for forced 2 = c d𝜉 c dy2 motion in a shallow fluid. The transformation renders the equation into a classical form: (( 2 ) ) k𝛽 c d2v 𝜔 2 2 + −k − −𝜉 v 𝜔 𝛽 d𝜉 2 c2 √ 1 d𝜇 k 1 = √ + 𝜉𝜇 (5.47) c 𝛽c d𝜉 𝜔 𝛽c This equation represents those modes forced by the distribution of 𝜇(y). If 𝜇(y) = 0 we obtain solutions that represent all possible oscillations within the system relative to the boundary conditions v = 0 at large y. This free or unforced equation is (( 2 ) ) k𝛽 c 𝜔 d2v 2 2 + −k − − 𝜉 v = 0 (5.48) 𝜔 𝛽 d𝜉 2 c2

5.5 Some Remarks

This is Schrodinger’s harmonic oscillator equation. More generally: ) ( 2 d 2 − 𝜉 v + Ev = 0 (5.49) d𝜉 2 where the eigenvalue E can be expressed as ( 2 ) k𝛽 c 𝜔 2 E= −k − 𝜔 𝛽 c2

(5.50)

Suppose there is a family of eigenfunctions, V m , that satisfy Eq. (5.49) such that ∑ Bm Vm (y) (5.51) v(y) = m

so that (

) d2 2 − 𝜉 Vm + Em Vm = 0 d𝜉 2

In the forced problem, the response is determined by the scale and the frequency of the forcing function. Therefore, k and 𝝎 are set. Thus, the eigenvalue for the forced problem is H. Using the orthogonal property of eigenfunction V n gives for a particular n: ∑ ky (E − En )CBn = − A V V d𝜉 ∫ m 𝜔c m n m ∑ Am + √ V dV ∫ m c 𝛽c n m where C = ∫ V n V n d𝜉 so that summed over all m we arrive at the general forced solution

(5.52)

Also E = E(𝜔, k, H), where we have noted that c2 = gH. The 𝜔 in Eq. (5.48) represents a family of eigenvalues for a series of set values k and the depth of the fluid H. Thus: For the free, unforced situation, V m is the eigenfunction of the system and 𝝎 is the eigenvalue for sets of k and H. Consider now the forced problem. We can expand the forcing function (contained in the right-hand side of Eq. (5.47)) in terms of the eigenfunction V m ; that is, in parallel with Eq. (5.51), ∑ Am Vm (y) (5.53) 𝜇(y) = m

so that Eq. (5.47) may be written symbolically as ∑ (E − Em )Bm Vm ( √ ) ∑ 1 dV m 1 k = Am 𝜉V + √ (5.54) 𝜔 𝛽c m c 𝛽c d𝜉 Equation (5.54) points to large differences between free and forced solutions, particularly in the identification of the eigenvalues.

v(y) =

∑ n

Bn Vn =

(

×

∑ n

1 C(E − En )

) ∑ Am k𝛽y ∑ Am Vn Vm d𝜉 √ V dV − ∫ m c 𝛽c n m ∫ m 𝜔c (5.55)

5.5 Some Remarks We have developed a model basis for the exploration of atmospheric and ocean waves in a variety of basic states. The difference between the Laplace models developed for the atmosphere and the ocean is that the ocean basic state is assumed to be quiescent and, also, we have made use of the reduced gravity approximation. We also developed a fully stratified atmospheric model that will allow the study of waves in a basic state that varies in longitude and height. Finally, for later use, we have attempted to differentiate between “forced” and “free” solutions of Laplace’s tidal equation. We also discussed the choice of equivalent depth. This will be a continuing discussion as we proceed, especially in Chapter 8 where we will discuss the role of moist processes on the structure of fundamental tropical modes.

Notes I Pierre Simon Laplace (1749–1827) achieved great

prominence as a mathematician, astronomer, and statistician. He is best known for his mathematics although less so for his other contributions. For example, he was the first to suggest the existence of what has become known as a black hole, hypothesizing that some entities may be so massive and their gravitational attraction so strong that light cannot

escape. He was also the father of inductive probability and offered a theoretical basis for Bayesian statistics. Yet, at the same time, he proposed that “… the present state of the universe as the effect of its past and the cause of the future.” This is referred to as Laplace’s Demon and puts forward the concept of a deterministic world. This is pertinent to numerical weather prediction stating, essentially, that a perfect knowledge of the past

123

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5 Fundamental Prototypes of Tropical Systems

and present coupled ocean atmosphere system leads to a perfect prediction. This is contrary to Chaos Theory, where very minor variations render the system unpredictable. II Note that we have made the tacit assumption that the zonal variability of the basic flow is more important than the other components of the background flow, such as the mean meridional component. Yet, in certain regions of the tropics (e.g. the South Asian monsoon region), the mean meridional component may be of a similar scale to the zonal component. III Normal modes describe the oscillations of a system and represent an oscillatory pattern of motion where all parts of the system move sinusoidally relative to a fixed phase relationship. Each normal mode is independent and orthogonal and one normal mode does not produce another.

IV Erwin Schrödinger (1887–1961) was an Austrian

physicist noted for the development of “wave mechanics,” putting forth a basic tenet of quantum mechanics: the Schrodinger equation. This is a partial differential equation that describes a harmonic oscillator and how the quantum state of a physical system changes with time. For this work he received the Noble Prize in Physics in 1933. Of particular relevance here is the concept of the “potential well,” within which wave-forms are constrained to remain. Matsuno (1966) was the first to recognize the geophysical implications of the Schrödinger equation and its relevance to the trapping of equatorial modes in the ocean and the atmosphere.

125

6 Equatorial Waves in Simple Flows In Chapter 5, we derived wave equations for the tropical atmosphere and ocean on an equatorial 𝛽-plane. We will now seek solutions to these equations to isolate the fundamental modes in both spheres. Noting the variability of the background basic atmospheric state of the tropics (Chapter 1), we are particularly interested in the impact of this variability on characteristics of the modes we find. In this chapter, we will consider basic states that are quiescent (U = 0) or have constant speed (U = constant) or have latitudinal shear (U = U(y)). Our aim is to understand the relative degree of trapping in these various basic flows and determine their respective lateral extent toward higher latitudes. We will also use potential vorticity arguments to understand the physical basis of trapped waves and why the trapping varies in different basic flows. Finally, we will examine equatorially trapped waves in the upper ocean noting both similarities and differences to those found in the atmosphere. We will introduce the “longwave approximation” in order to simplify the system and to allow an investigation of the role of lateral boundaries.

6.1 Atmospheric Modes in a Constant Basic State: Constant U 6.1.1 Governing Equations for a Motionless Basic State (U = 0) In our examination of wave kinematics (Chapter 4), we found that along a ray there will be no change in wavenumber (4.14) nor any accumulation of energy (4.21) if the background basic state does not vary spatially. Thus, a system in which U = 0 in accord with earlier studies1 is ideal for examining the basic properties of equatorially trapped modes. In a quiescent

1 Specifically the seminal papers of Matsuno (1966), Longuet-Higgins (1968), and Gill (1980, 1982).

basic state, the coefficients of Eq. (5.11) adopt a simple form: p(y) = 0

(6.1a)

𝛽2 𝜔2 𝛽k − (6.1b) − k2 − 2 y2 2 𝜔 c c where the Doppler-shifted frequency has been replaced by the intrinsic frequency. As previously defined, c2 = gH. With Eq. (6.1), Eq. (5.11) then has the form of a wave equation: ( 2 ) 𝛽k 𝛽2 2 d2v 𝜔 2 + − (6.2) −k − 2y v=0 𝜔 c2 c dy2 q(y) =

describing the form of near-equatorial waves.2 The non-dimensional version of the wave equation can be formed using (c/𝛽)1/2 and (1/𝛽c)1/2 as characteristic length and time scales to give ( ) d2v k 2 2 2 + 𝜔 − k − v=0 (6.3) − y 𝜔 dy2 or d2v + (a − b2 y2 ) = 0 (6.4) dy2 where non-dimensionally k a = 𝜔2 − k 2 − and b = 1 (6.5a) 𝜔 Dimensionally, the coefficients for U = 0 are 𝛽 k 𝜔2 a = 2 − k2 − 𝛽 and b = (6.5b) 𝜔 c c Equation (6.3) is Matsuno’s (1966) classical equatorial wave equation. It is interesting to note that Matsuno correctly identified Eq. (6.3) as a geophysical form of Schrodinger’s Harmonic Oscillator equation. 6.1.2 Governing Equations for a Constant Basic State (U = Constant) Most early studies considered equatorial modes in a quiescent basic state. Eventually, studies included basic 2 Equation (6.2) is also listed as Eq. (5.13) with coefficients defined in Eq. (5.12).

Dynamics of the Tropical Atmosphere and Oceans, First Edition. Peter J. Webster. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

126

6 Equatorial Waves in Simple Flows

states closer to reality with non-zero basic flow both constant and with latitudinal shear.3 For zonal basic states, where U = constant, Eq. (5.15) becomes ( 𝜔d 2 k d2V + − k2 − 𝛽 2 2 𝜔 c dy d ( ) ) 2 𝛽U 𝛽 2 kU U + 2 y2 V = 0 − 2 − 2 1+ 𝜔d 2c c 4c

H2 (y) = 4y2 − 2,

H3 (y) = 8y3 − 12y,

… H4 (y) = 16y4 − 48y2 + 12, dHn = 2nHn−1 and Hn+1 = 2yHn − 2nHn−1 dy (6.13) Even polynomials (n = 0, 2, 4, …) are symmetric about the equator. Odd polynomials (n = 1, 3, …), on the other hand, are antisymmetric.

(6.6) which contains both Doppler and non-Doppler effects of the basic flow as defined in Section 5.1.3. The Doppler effects are contained within 𝜔d whilst the non-Doppler (or dynamic) aspects of the basic flow appear in other terms containing U. The Doppler-shifted frequency,𝜔d , is defined in Eq. (4.9b) in general form or specifically to the basic state we consider here in Eq. (4.11). We 2 can neglect terms −𝛽U∕2c2 and U ∕4c2 relative to the magnitude of the neighboring terms so that Eq. (6.6) is reduced to ( 2 𝜔d k d2V + − k2 − 𝛽 2 2 𝜔d c dy ( ) ) 𝛽2 kU y2 V = 0 (6.7) − 2 1+ 𝜔d c which has the general form: d2V + (a − b2 y2 ) = 0 dy2

(6.8)

where a=

𝜔d 2 k − k2 − 𝛽 𝜔d c2

(6.9)

b=

)1∕2 ( 𝛽 kU 1+ c 𝜔d

(6.10)

and

The standard solution for an equation of the form (6.7) is V (y) = Hn (b1∕2 y) exp(−by2 ∕2)

(6.11)

Hn (y) is the nth order Hermite polynomial : I

Hn (y) = (−1)n exp(y2 )

dn [exp(−y2 )] dyn

The form of the first few polynomials are H0 (y) = 1,

3 Zhang and Webster (1989).

H1 (y) = 2y,

(6.12)

6.1.3 General Dispersion Relationship for a Constant Basic State (U = Constant) The associated dispersion relationship,II the condition that Eq. (6.11) is a solution of Eq. (6.8), is a = (2n + 1)b

(6.14)

In full form, with Eqs. (6.9) and (6.10), Eq. (6.14) becomes )1∕2 ( 𝜔d 2 𝛽 k kU 2 −k − 𝛽 = (2n + 1) 1+ 𝜔d c 𝜔d c2 (6.15) The equation is a cubic in 𝜔d , suggesting that there are three major sets of solutions for various n and k. The order n refers to the number of latitudinal interceptions of the wave or the number of latitudinal “zeros.” For example, n = 0 indicates that there are no intercepts, n = 1, one intercept, etc., as can be seen easily from Eq. (6.13). The n = 0 mode has a maximum at the equator and exponentially decays toward the two poles. The n = odd modes possess a zero at the equator and are antisymmetric about the equator. Modes with n = even are symmetrical about the equator. The longitudinal wavenumber k in the assumed longitudinal structure solution (exp(ikx)) in Eq. (6.10) is related to the number of zeros around the equator. For example, a ka = 1 mode has two unique intercepts around a line of latitude, a ka = 2 mode has four, and so on. In general, there are 2 ka zeros in longitude. The functional form of the frequency in Eq. (6.15) is quite complex. The frequency depends on the longitudinal scale (k), latitudinal modal number (n), the magnitude and sign of the basic state (U), and the depth of the fluid (H, contained within c). Similar dependencies exist for the turning latitude and the so-called “equatorial Rossby radius,” which we will refer to as RE , to differentiate it from the general Rossby radius of deformation, R, defined in Eq. (3.35). The turning latitude yT is the latitude where wave-like solutions transform into exponentially decaying solutions. This occurs where a = b2 y2 in Eq. (6.8) so that

6.1 Atmospheric Modes in a Constant Basic State: Constant U

with Eq. (6.14), we find that [ ]1∕2 (2n + 1) yT = ± b [( )−1∕4 ] )1∕2 ( c k =± (2n + 1) U 1+ 𝛽 𝜔d (6.16) Noting the relationship between v(y) and V (y) in Eq. (5.14), we can rewrite Eq. (6.11) to give ) ) ( ( 1 kU vn (y) = Hn (b1∕2 y) exp − y2 b− 2 4𝜔d ) ( kU 2 = Fn (y) exp (6.17) y 4𝜔d where the meridional structure function, Fn (y), is defined as [ ( )2 ] ( ) y 1 y Hn Fn (y) = exp − (6.18) 2 RE RE

(ii) R, defined in Eq. (3.34), refers to the distance traveled by a gravity wave during an inertial period. For a forcing scale 2𝜋R), rotational effects overcome buoyancy effects and rotational modes will be generated. Thus, irrespective of the value of RE , it is R that determines whether or not a rotational low-frequency mode is generated. The full solutions for a constant basic state are ⎛ un ⎞ ⎛[u1 Fn+1 (y) + u2 Fn−1 (y)]∕i(𝜔d 2 − k2 c2 ) ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ vn ⎟ = ⎜ Fn (y) ⎟ ⎜𝜙 ⎟ ⎜[𝜙 F (y) + 𝜙 F (y)]∕i(𝜔 2 − k2 c2 )⎟ 2 n−1 d ⎝ n ⎠ ⎝ 1 n+1 ⎠ × exp[i(kx − 𝜔d t)] where

where −1∕2

RE = b

( )1∕2 ( )−1∕4 c kU = 1+ 𝛽 𝜔d

(6.19)

is the equatorial Rossby radius of deformation within a non-zero, but constant, basic flow. We note that the turning latitude and RE are related. From Eqs. (6.16) and (6.19) we find that yT = ±(2n + 1)1∕2 RE

(6.20)

with the two quantities having the same relationship with the basic flow. In general, we expect the equatorial radius of westward moving waves, and the turning latitude as well, to increase in a westerly basic flow. This is because, for k < 0 and U > 0, the denominator in Eq. (6.19) decreases and RE and yT both increase. Correspondingly, both the turning latitude and equatorial radius decrease in easterly flow. These changes are relative to scales in a quiescent basic state. We will explore the physical reasons for these changes in scale in Section 6.3. It is worth noting that the denominator can become imaginary if k < 0 and U ≫ 0. Before proceeding, it is important to differentiate between the two Rossby radii. The difference between RE and R can be best thought of in terms of free and forced solutions, discussed briefly in Section 5.4. (i) RE , defined in Eq. (6.19), represents a latitudinal e-folding distance of a particular mode defined by 𝜔d , k, n, U, and H. There is an infinity of free solutions relative to these parameters (see Section 6.1.4.1 below).

(6.21)

1 u1,n = − [𝛽(𝜔d + kU)RE + kc2 ∕RE ] 2

(6.22a)

u2,n = −n[𝛽(𝜔d + kU)RE − kc2 ∕RE ]

(6.22b)

1 𝜙1,n = − [𝛽(𝜔d U + kc2 )RE + 𝜔d c2 ∕RE ] 2

(6.22c)

𝜙2,n = −n[𝛽(𝜔d U + kc2 )RE − 𝜔d c2 ∕RE ]

(6.22d)

We now have access to the complete structure of equatorial waves noting that eigen-frequency,3 which is the eigenvalue of the free problem, can be determined from the dispersion relationship (6.15) for each set of modes. 6.1.4

Eigenfrequencies for a Constant Basic State

We now discuss the characteristics of the eigenfrequencies that emerge from the dispersion relationship (6.15). 6.1.4.1

Dispersion Diagrams

The dispersion relationship (6.15) contains the frequencies, or eigenfrequencies, of possible free solutions of Eq. (6.7). These solutions are each associated with a scale and possess specific phase and group velocities. The number and form of the solutions are subject only to the boundary conditions of the system and the characteristics of the system itself. Figure 6.1a shows the eigenfrequencies as a function of ka and 𝜔 for a range of n values. We have adopted the convention that modes with eastward phase velocities are designated by a positive longitudinal wavenumber k whilst modes with a westward phase possess a negative ka. Other studies (e.g. Matsuno 1966) choose different sign conventions. The choice of a sign convention is arbitrary and does not affect the physics of a system. In this alternative

127

6 Equatorial Waves in Simple Flows

2.0

Convention 1

1.75 1.5

0.5

EIG

0.7

1.0

1.0

n=1

1.0

MRG

0.25

n=1 K n = –1 n=4

0

0.75 0.5

0.7

n=0

1.25

n=0 n=1

n = –1

K ER

0.0 n = 4 –10 –5 0 5 Zonal wavenumber (ka)

(a)

2.0

0.5

n=4 WIG

n=4

WIG

Convention 2 2.0

–1.0

n=1 n=0 n=1

ER

10

2.0 4.0

MRG

4.0 2.0 1.0 0.7

4.0 8.0

1.0 Period (days)

Frequency (cycles/day)

128

n=4 –2.0

H = 400m

EIG 2 6 Zonal wavenumber (ka)

0.5 10

(b)

Figure 6.1 (a) Eigenfrequencies of families of equatorial modes: inertial-gravity (EIG: eastward-propagating and WIG: westwardpropagating), Rossby (ER), Kelvin (K), and the mixed Rossby-gravity mode (MRG). The modes are plotted as a function of frequency, 𝜔, and longitudinal wavenumber ka for various latitudinal modes n. The background basic state is at rest (U = 0) and the depth of the fluid (H) is 400 m. Negative ka represents waves with a westward phase velocity and positive k an eastward phase velocity. The dashed line joins 𝜕𝜔/𝜕k = 0, thus differentiating between eastward and westward group speeds. (b) An alternative but identical representation of the dispersion relationship but with all ka ≥ 0. In this configuration, positive frequency denotes eastward phase propagation and negative westward. Units: cycles day–1 left ordinate and days right ordinate.

convention, modes are assumed to have positive longitudinal wavenumbers. Modes with an eastward phase velocity are defined to have 𝜔 > 0 and 𝜔 < 0 for westward phase speeds. As the frequency is in this chosen convention universally positive, by definition, modes to the right of ka = 0 have eastward phase speeds and to the left westward ones. Modes with a positive slope (𝜕𝜔/𝜕k > 0) have group speeds that are from west to east, while negative slope modes (𝜕𝜔/𝜕k < 0) have reverse group speeds. The equivalent depth used in Figure 6.1 is set at 400 m. A second commonly used sign convention is illustrated in Figure 6.1b. Here ka is universally positive but the frequency may be negative or positive. Modes associated with negative frequency have negative phase speeds while those with positive frequency have positive phase speeds. The information conveyed in Figure 6.1b is identical to that in Figure 6.1a. In Figure 6.1a there are four clusters of solutions. The first is a set of equatorial Rossby waves (subsequently termed ER) with westward phase speeds at all scales. Group speeds, on the other hand, are westward for the longest waves but eastward for shorter waves. In addition there are sets of eastward-propagating and westward-propagating inertia-gravity waves (EIG and WIG waves, respectively) that possess eastward and westward phase speeds, respectively. In general, except for very large scales, the slope of the eigenfrequencies

is linear so that the group speeds of the EIG and WIG waves match for all k > 2 or so. This similarity of group and phase speeds illustrates the non-dispersive nature of these modes. These non-dispersive waves advect their wave energy with the phase speed of the wave (Section 4.2). A “hybrid” wave appears for n = 0 that seems to mimic westward-propagating Rossby wave characteristics at low frequencies but EIG ones at higher frequencies. This is the so-called mixed Rossby-gravity mode (MRG).IV The MRG has positive group speed (i.e. 𝜕𝜔/𝜕k > 0) at all scales. For positive k, the wave is close to being non-dispersive. The final mode is the Kelvin wave (K) appearing for n = −1. This mode is completely non-dispersive with identical eastward phase and group speeds. 6.1.4.2

The Ubiquitous Nature of MRG and K Waves

There is a particular and unique characteristic of both the MRG and K waves. Even though we are concentrating on free modes in this section, we can anticipate the impact of forcing. Returning to either Figure 6.1 we note that there is a wide frequency gap between the ER and the WIG and EIG modes. One may anticipate that for an inertia-gravity wave to be produced the forcing would have to be high frequency in order for there to be some form of resonant response.V Similarly, one may expect that to force an ER, the frequency of the forcing will have to be low. This low-frequency forcing is far removed

6.1 Atmospheric Modes in a Constant Basic State: Constant U

from the EIG and WIG eigenfrequencies so that there will be little or no high-frequency response to such forcing. Likewise, a high-frequency forcing will not force a low-frequency ER. Both the MRG and K waves tend to bridge this gap. Consider some forcing near 0.5–0.75 cycles per day somewhere between the frequency of the ER and WIG and EIG. However, the forcing intersects both the MRG and K waves and only those modes will be excited. In fact, noting the MRG and K eigenfrequency curves, both of these waves will respond to forcing occurring at any frequency. This ubiquitous property of these two waves will prove to be quite important in understanding the forced response within the tropical atmosphere and ocean. 6.1.4.3

Eigenfrequency Dependence on H

We had noted in the discussion of the dispersion curves that the eigenfrequencies depended on a number of factors, especially c and hence H. Figure 6.2 is identical to Figure 6.1a (in which H = 400 m) except showing eigenfrequencies for H ranging between 20, 50, 100, and 1000 m. These different choices of H impact the eigenfrequencies of all modes radically. Consider the n = 1 ER mode. For H = 20 m, for example, the period of the mode at ka = −5 is 20 days, at 50 m 14 days, and at 100 m 10 days. At H = 400 m (from Figure 6.1a), the ka = −5 ER mode has an 8-day period and for H = 1000 m, a 5-day period. The periods of modes where H < 200 m are not observed in nature. This statement refers to modes in a dry atmosphere. In H = 20 m

1

6.1.5.1

Equatorial Rossby Waves (ER)

Frequency Domain Inspection of the dispersion curves

(Figures 6.1 and 6.2) indicates that ER waves are low-frequency phenomena. Thus, we can seek asymptotics in the dispersion relationship for 𝜔2 < < 1. The dispersion relationship (6.15) can then be approximated as )1∕2 ( 𝛽 k k 𝛽 = (2n + 1) U (6.23) 1+ −k2 − 𝜔d c 𝜔d H = 100 m

H = 1000 m

n=2

n=2 n = –1

n=2

Eigensolutions

We now determine the latitudinal structure of the modes appearing in the dispersion diagrams.

H = 50 m

0.8 0.6

6.1.5

1.0 1.25

n = –1

1.67

n=1 0.4 n=0

n=0

0.2

n = –1

n=1 0 n=2 –10 –5

0 (a)

5

n=1

n=2 10 –10 –5

n=0

n=0

0 (b)

n = –1 5

2.5

n = –1

0 (c)

5.0

n=1

n=1

n=2 10 –10 –5

n = –1

Period (days)

Frequency (cycles/day)

Chapter 8 we will deal with modes in a moist convective atmosphere where strong vertical mixing and dissipation change the character of fundamental modes, in general decreasing their horizontal phase speeds. The phase speed of a Kelvin wave also varies over a wide range depending on the choice of H. With H = 20 m, the phase (and group speed) is 14 m s−1 . For H = 100 m, cp is 32 m s−1 . If H is assumed to be 1000 m, cp is 100 m s−1 . Finally, the inertia-gravity wave eigenfrequencies vary similarly with H as with other modes. For small H, the gravity wave eigenfrequencies are of order one to two days. For deeper equivalent depths, the characteristic periods decrease substantially to order 0.5 days. Clearly, many modal characteristics are extremely sensitive to the choice of H.

n=2 5

10 –10 –5

0 (d)

5

10

Zonal wavenumber (ka) Figure 6.2 Dispersion curves for different H = 50, 100, and 1000 m in the same format as Figure 6.1a. The eigenfrequencies are strong functions of the equivalent depth of the fluid. In general, for small H, frequencies of a particular mode are less than for a larger H. Units of frequency in cycles day–1 , left ordinate and period in days, right ordinate.

129

6 Equatorial Waves in Simple Flows

Solving for the Doppler-shifted eigenfrequency, 𝜔d , yields ⎞ ⎛ 𝛽 𝛽 2U ⎜ 2𝛽k2 − (2n + 1)2 2 − (2n + 1) r1∕2 ⎟ k c c ⎟ 𝜔d = − ⎜ 2 ⎟ 2⎜ 𝛽 2 4 k − 2 (2n + 1) ⎟ ⎜ c ⎠ ⎝ (6.24a) where

𝛽2 2 U − 4k2 𝛽U + 4𝛽 2 (6.24b) c2 Figure 6.3 shows the ER dispersion relationship as a function of ka for U = −10, 0, and 10 m s−1 . If U = 0 then r = 4𝛽 2 . Noting that a4 − b2 = (a2 + b)(a2 − b) we obtain r = (2n + 1)2

𝜔≈

−𝛽k k2 + (2n + 1)𝛽∕c

(6.25)

This is the classical Matsuno ER dispersion relationship in dimensional form. In general, the frequency increases as the magnitude of k increases up to where

(𝜕𝜔/𝜕k = 0), after which the frequency of the mode decreases. Also, from Eq. (6.24a) and in Figure 6.3 it is clear that the change of frequency is not a linear function of zonal flow. The frequency is greater for westerly flows compared with the stationary flow (U = 0), which, in turn, is greater than for easterly flow for all ka. Thus, the differences in frequency are in accord with Doppler effects, but non-Doppler effects become important especially in the westerly basic flow. Group and Phase Speeds Reflecting on our discussion

regarding forced and free modes (Section 5.4) it may seem strange to be considering the group speed of free modes. After all, unless there is some forcing agent that does work on the environment or an instability that converts potential energy to kinetic energy, there is no energy to transmit. The group speed of a free oscillation, however, will provide an indication of the forced response if there is a matching of the frequency of forcing and an eigenfrequency. Figure 6.3 Eigenfrequencies of the ER in background states from Eq. (6.23) with U = −10, 0, and +10 m s−1 from Eq. (6.23) for n = 1 (solid) and 2 (dashed) with (i) H = 50 m, (ii) 300 m, and (iii) 500 m. Units cycles day–1 .

ER Doppler frequency (cycles/day) (i) H = 50 m

Cycles/day

0.3 0.2

= –10 m s–1 U = 0 m s–1 = +10 m s–1 n = 1 (solid), n = 2 (dashed)

0.1 0.0 –8

–6

Cycles/day

–4

–2

0

–2

0

–2

0

(ii) H = 300 m

0.3 0.2 0.1 0.0

–8

–6

–4 (iii) H = 500 m

0.3 Cycles/day

130

0.2 0.1 0.0

–8

–6 –4 Zonal wave number (ka)

6.1 Atmospheric Modes in a Constant Basic State: Constant U

ER Doppler zonal group speed (m s–1) (i) H = 50 m

0 –10

–30

U = –10 m s–1 U = 0 m s–1 U = +10 m s–1

–8

–6

–4

cgdx (m s–1)

0 –10 –20 –30

–8

–6

–4

–2

0 –10 –20 –30

–8

–6 –4 Zonal wavenumber (ka)

–2

(a)

0

1000

4

(i) n = 1

–24 –20 –16

0

600

3

1

400

2 –8

200 1000

–28

–4

800

–12

0

(iii) H = 500 m

10

ER zonal group speed (m s–1)

0

(ii) H = 300 m

10

cgdx (m s–1)

–2

Equivalent depth (m)

–20

Equivalent depth (m)

cgdx (m s–1)

10

–10 (ii) n = 2

800

–8

–6

–12

–4

–2

–4

2

–16

1

600

0

0

–12

400 –8

200 –12

–10

–8

–6

–4

–2

0

Zonal wavenumber (ka)

(b)

Figure 6.4 (a) ER Doppler-shifted group speeds for n = 1 (solid) and 2 (dashed) as a function of ka for (i) H = 50 m, (ii) H = 300 m, and (iii) H = 500 m. Units m s−1 . (b) ER group velocity as a function of k and H for n = 1 and 2. The bold red line depicts the cgx = 0 contour and the bold blue contour outlines cg = −10 m s−1 , suggesting that in the region of upper tropospheric westerlies, the Doppler-shifted group speed could approach zero.

Dividing Eq. (6.24a) by k gives

cpdx |ER

⎞ ⎛ 𝛽 𝛽 2U ⎜ 2𝛽k2 − (2n + 1)2 2 − (2n + 1) r1∕2 ⎟ 1 c c ⎟ =− ⎜ 2 ⎟ 2⎜ 𝛽 2 4 k − 2 (2n + 1) ⎟ ⎜ c ⎠ ⎝ (6.26)

the Doppler-shifted phase speed of the ER. By differentiating (6.24a) with respect to 𝜔we obtain cgdx |ER = −

1 2s

𝛽 𝛽2 ⎡ ⎤ 6k2 𝛽 − (2n + 1)2 2 U − (2n + 1) r1∕2 ⎢ ⎥ c c ⎢ ⎥ 𝛽 ⎥ + 4(2n + 1) k2 Ur−1∕2 ×⎢ c ⎢ ⎥ ) ( 2 ⎢ 4k4 𝛽 −1∕2 ⎥ 2 2𝛽 2k 𝛽 − (2n + 1) 2 U − (2n + 1) r ⎢+ ⎥ c c ⎣ s ⎦ (6.27)

the Doppler-shifted group velocity where s = k4 − (2n + 1)2 𝛽 2 /c2 . Figure 6.4a shows the ER Doppler-shifted group speed for n = 1 (solid) and 2 (dashed) as a function of ka. For all values of constant U, large-scale waves possess rapid westward values. Only at large negative values of ka (≥−7) does the Doppler group speed become westerly. The impact of the basic state is twofold. First, for the same ka, the group speed is more easterly if U = +10 m s−1 and less so in an easterly zonal flow. The basic state makes only small changes to the transition from easterly to westerly group speeds. Second, the sign of the basic state makes very small changes to the transition between westerly and easterly group speeds. Figure 6.4b plots the distribution of the group speed with U = 0 as a function of H and zonal wavenumber ka. In general, large-scale ERs (small ka) possess westward group speeds and for small-scale ERs (large ka) the group velocity is to the east. In general, for H > 200 m, the cgdx = 0 contour remains in the vicinity of ka = −7 for

131

6 Equatorial Waves in Simple Flows

n = 1 and ka = −9 for n = 2. Thus, there is a small creep in the transition toward smaller scales with an increase of n. The largest changes occur with H. For H > 400 m, the group speeds are generally constant although larger for the n = 1 mode compared to n = 2. However, as H decreases to 100 m, the group speed decreases by a factor of two for all zonal scales. The determination of the zonal scale (kc ) at which the transition between easterly and westerly group speeds is straightforward, especially for U = 0. This is accomplished by setting cgdx to zero in Eq. (6.27). For U = 0, separation occurs where √ 𝛽 kC = − (2n + 1) (6.28) c where we have neglected the positive root. As n increases, the group speed transition moves to larger k. The transition from westward to eastward group speed can be seen in Figure 6.5a for different values of n. As n increases, kc moves to ever-increasing values of |ka|. For future reference, the eigenfrequency of the MRG is also plotted, even though there is no kc . Figure 6.5b shows that there is also a large dependence of kc on the equivalent depth, H. For small values of H, kc occurs at extremely small scales so that essentially all ERs have negative group speeds. As H increases beyond 200 m, kc tends to asymptote toward moderate longitudinal scales. The discussion in the last few paragraphs has some important implications. In a motionless basic state, large-scale forcing can be expected to transmit signals rapidly away from the source region. On the other hand, modes excited at smaller scales would remain locally near the forcing. In addition, the results portrayed in Figure 6.4b show that, once again, the choice of H is critical. For the same frequency of forcing, excited modes will radiate away from the source region much more rapidly for a large H than for a smaller H. Also, the group speeds of modes for small H are to the east but, as can be seen in Figure 6.4b, the magnitude is relatively small. Thus, in a small H environment the response to forcing (assuming there is a match between eigenfrequency and the frequency of the forcing) will remain relatively locally close to the forcing compared to modes where H is larger. There is one consideration that we will return to presently. By comparing the group speeds of the ER modes, it is possible that the Doppler-shifted group speeds become equal and opposite to the basic state as it propagates through the tropics. Are there implications? Equatorial Rossby Radius, Turning Latitudes, and Wave Structures The ER radius of deformation (RE ) is greatly

30 Zonal group speed (m s–1)

132

20

Zonal group speed for ER and MRG MRG n = 0 ER n = 1, 2, 3 0

10 1

0

3 2 1

–10 –20 –30 –19 –17 –15 –13 –11 –9 –7 –5 Zonal wavenumber (ka) (a)

–5

–3

–1

Critical wavenumber (kc) in terms of H and n 1

–10

2

3

4

kc –15

–20

–25 20

200

400 600 Equivalent depth (H) (b)

800

1000

Figure 6.5 (a) Zonal group speed (cgx ) as a function of ka for n = 1–3 for U = 0. For later reference, the group speed of the MRG is shown as the dashed line. Note that as n increases, the transfer for westward to eastward group speed occurs at smaller scales (greater k). (b) Critical scale (k0 ) where westward group speed turns to eastward as a function of equivalent depth. In general, as n increases for a given equivalent depth, the scale transition wavenumber increases.

influenced by the basic state, the scale of the mode and the value of H. Figure 6.6a shows that, for an n = 1, ka = 5 mode, with equivalent depths >500 m and U = 0, RE is roughly constant with a value of 1600 km with slow increases as H increases. For small values of H, the modes are trapped very close to the equator. For U = 0 and −10 m s−1 , there is a similar dependence on H, but in a westerly basic state all modes extend much further poleward, considerably poleward, and into the subtropics. Figure 6.6b indicates that RE is greater

6.1 Atmospheric Modes in a Constant Basic State: Constant U

ER Rossby radius and turning latitude ka = 5 ka = 9

5

90o

–10 0 +10

U

75o

4

60°

3

45°

2

30°

1

15°

0

0

200

600 400 800 Equivalent depth (m) (a)

latitude (°)

Rossby radius (106 m)

6

0° 1000

ER Rossby Radius dependence on ka

Rossby radius (106 km)

9

U

–10 0 +10

7

5

Doppler and Non-Doppler Effects The form of the eigen-

3

1

The dependence of the ER radius on the sign and magnitude of U has a strong influence on the meridional structure function. Figure 6.7a plots the first few Hermite polynomials defined in Eq. (6.13). The components of the function (6.21) are also plotted for different modal values in Figure 6.7b. The exponential component decays much more slowly with latitude in a westerly basic current than in an easterly flow. Similarly, the Hermite component of the function extends much farther poleward in a westerly flow. Thus, in combination, these two components conspire to provide a latitudinal extension of the wave regime (i.e. where Γ2 > 0) in a westerly flow. The issue of imaginary quantities aside, from a strictly phenomenological perspective there are interesting implications of the dependency of RE and yT (related through Eq. (6.20)) on k when U > 0. A mode of a particular frequency in the upper-tropospheric westerlies over the Pacific or the Atlantic (see Figure 1.8) would have a greater lateral extent than the same mode in upper-tropospheric easterlies. We will argue later that regions of upper-tropospheric westerlies are locations where tropical waves may emanate to higher latitudes. Also, these are regions where extratropical modes may propagate into the tropics.

–9

–8

–7 –6 –5 –4 –3 Zonal wavenumber (ka) (b)

–2

–1

Figure 6.6 The Rossby radius of deformation RE from Eq. (6.19) (a) Plotted as a function of the equivalent depth H (m) of depth for U = −10, 0, +10 m s−1 with ka = 5 and k = 9, respectively, and (b) dependence of RE on longitudinal scale (ka) for U = −10, 0, +10 m s−1 and H = 300 m. Source: Adapted from Figure 4, Zhang and Webster (1989).

in westerly flow than in easterly flow. As we found in Figure 6.6a, the smaller longitudinal scales have unique characteristics, especially for U = +10 m s−1 . Figure 6.6b displays the dependency of RE on a zonal scale. When U = 0 there is no longitudinal scale dependence, which follows from Eq. (6.19) where, when U = 0, RE = (c∕𝛽)1∕2 = constant. However, for U < 0 the factor (1 + kU∕𝜔d )−1∕4 must increase in magnitude as 𝜔d > 0 while ka < 0. Thus, RE (U = 0) > RE (U < 0). Further, if U > 0, then RE (U > 0) > RE (U = 0). However, this case isolates a problem. For larger negative value of ka, (1 + kU∕𝜔d ) < 0 and the solution becomes imaginary! We will deal with this issue shortly.

solutions reflects the impacts of the components of the meridional structure equation on the structure of the mode. The specific form of the eigensolutions may be computed from Eqs. (6.21) and (6.22) using the ER eigenfrequencies expressed in Eq. (6.24a). The zonal velocity, meridional wind, and geopotential solutions are plotted in Figure 6.8a to c, for n = 1 and 2 and ka = 5, respectively. The impact of an easterly basic state is to restrict the poleward extension of the mode in contrast to the impact of a westerly flow where modes extend more poleward. There is also a dependence on latitudinal and longitudinal modal numbers. As n and ka increase, the latitudinal extent of the wave regime increases as well. Finally, we note again, for later reference, that for the H = 50 m, the eigensolutions are trapped extremely close to the equator, much closer than for H = 500 m. This reiterates that the choice of a particular value of H is very important. How does the basic state impact fundamental modes? We note from the form of the solutions that the impact of the basic state is non-uniform. In all of the solutions, for example, the latitudinal extent of the modes is not equally displaced about the U = 0 solution, as would be expected from purely Doppler effects. Thus, the differential, non-symmetric effects of the basic state that arise are non-Doppler in origin. Both Eqs. (5.8a) and (5.8c) contain non-Doppler terms (square brackets)

133

6 Equatorial Waves in Simple Flows

Latitudinal structure of Hermite polynomials 40 30 20 n=2

10 n(y)

134

n=1

0

n=0 n=3

–10

n=4

–20 –30 –40 –30

–20

–10

0 Latitude (o)

10

20

30

(a) (i) n=1

(ii) n=2

1

1 –10

Exp(y)

0.5

U

0.5

0 +10

0 0°

20°

40°

60°

80°

20

0°

20°

40°

60°

80°

0°

20°

40°

60°

80°

0°

20°

40°

60°

80°

100

15 n(y)

10

50

5 0 0°

Fn(y)

20°

40°

60°

80°

2

3

1.5

2 1

1

0

0.5 0

−1 0°

20°

40°

60°

−2

80°

Latitude

Latitude (b)

Figure 6.7 (a) Latitudinal structure of the Hermite polynomial from Eq. (6.13) for the first few n with U = 0. For even n, the functions are symmetric about the equator. Odd functions are asymmetric. (b) The components of the meridional structure function (6.21) for (i) n = 1, ka = 5 (red) and (ii) n = 2, ka = 5 (blue) for U = -10 (red), 0 (black), and 10 m s−1 (blue). The three rows show the exponential factor (top row), the Hermite polynomial (middle row), and the complete meridional structure function of Eq. (6.23). Note the more extensive meridional structure if U = 10 m s−1 compared to the greater trapping in a U = −10 m s−1 basic state. Source: Adapted from Figure 5, Zhang and Webster (1989).

6.1 Atmospheric Modes in a Constant Basic State: Constant U

Zonal velocity (m s–1) 10

(i) H = 50m, n = 1

10

0

–5

–10

–10 (ii) H = 50m, n = 2

10

–10 0 +10

(iv) H = 500m, n = 2

5 ms–1

m s–1

5 0 –5 –10

U

0

–5

10

(iii) H = 500m, n = 1

5 ms–1

m s–1

5

0

–5

30°S

0° Latitude

30°N

–10

30°S

0° Latitude

30°N

(a) Meridional velocity (m s–1) (iii) H = 500m, n = 1

(i) H = 50m, n = 1

4

4

2

m s–1

m s–1

U 2

0

0

–2

–2

4

(ii) H = 50m, n = 2

4

–10 0 +10

(iv) H = 500m, n = 2

2 m s–1

2 m s–1

Figure 6.8 Eigensolutions for the ER wave for (a) zonal velocity component (m s−1 , (b) meridional velocity component (m s−1 ), and (c) geopotential (m2 s−2 ). Each figure displays the eigensolutions with H = 50 m for (i) n = 1 and (ii) n = 2, and H = 500 m panels (iii) and (iv). Blue, black, and red contours signify solutions for U = −10, 0, and 10 m s−1 . Source: Based on Figure 6, Zhang and Webster (1989).

0

0

–2

30°S

0° Latitude

30°N

–10

(b)

30°S

0° Latitude

30°N

135

6 Equatorial Waves in Simple Flows

Figure 6.8 (Continued)

Geopotential (m2 s–2) (i) H = 50m, n = 1

200

U 100 m2s–2

m2s–2

100 0

–100

–200

–200 (ii) H = 50m, n = 2 400

200

(iv) H = 500m, n = 2

m2s–2

200

0 –200 –400

–10 0 +10

0

–100

400

(iii) H = 500m, n = 1

m2/s2

200

m2s–2

136

0

–200

30°S

0° Latitude

30°N

–400

30°S

0° Latitude

30°N

(c)

although for the constant zonal state, the term vdU∕dy in Eq. (5.7a) has zero magnitude. If the remaining non-Doppler term is neglected in Eq. (5.7c) (i.e. −yUv), the wave characteristics would be identical to those found for U = 0 except that the frequency would be replaced by the Doppler-shifted frequency. Thus, the meridional structure of the flow would not be changed irrespective of the value of U. However, it is possible that a non-zero basic state may change the ambient potential vorticity (PV) gradient rather than merely advect its gradient. Figure 6.9 displays the associated eigensolutions for different values of equivalent depth. Consistent with arguments presented above and the form of the eigensolutions shown in Figure 6.8, the solutions for a very shallow fluid are tightly trapped about the equator. Eigensolutions for H = 400 m extend much further poleward. Behavior of Small Longitudinal Scale Modes in a Westerly Basic State (U > 0) We noted in Section 6.1.5.2 that

RE increased substantially as ka increased, as illustrated in Figure 6.6b. Is this a real phenomenon or a behavior resulting from some improper assumption or an extension of an assumption? Is it possible that

there is a shortwave cutoff suggesting a range of longitudinal scales where waves cannot exist? We can approach this question by considering the assumptions we have made during the derivations of the basic wave equations. There exists a range of parameters where r, defined in Eq. (6.24b), may be negative, leading to imaginary values of 𝜔d in Eq. (6.24a). Specifically, r < 0 when 4k2 𝛽U > (2n + 1)2

𝛽2 2 U − 4𝛽 2 c2

(6.29)

which can only occur if U > 0. The term r is positive definite if U < 0 or U = 0, so that 𝜔d remains a real positive number. If we follow the terms carefully through the derivation of Eq. (6.24a), we can find that the source of the term 4k2 𝛽U in Eq. (6.29) comes from the dynamic non-Doppler term in Eq. (5.8c), displayed in the square brackets. This term arises from the assumption of a geostrophic balance on a 𝛽 - plane. The other non-Doppler term (square brackets in Eq. (5.8a)) is zero unless U = U (y). Plots of the meridional distribution of the ER geopotential are displayed in Figure 6.10a. They are shown as a function of latitude for ka = −3, −5, and − 9 for U = −10, 0, and +10 m s−1 . All of the modes are well-behaved and trapped within the region where the equatorial 𝛽-plane

6.1 Atmospheric Modes in a Constant Basic State: Constant U

30°N

Eigensolution ER: ka = 5, n = 1 H = 400m

20°N Latitude

Figure 6.9 Impact of the choice of H on the eigensolutions of the ER for ka = 5, n = 2 mode for (a) H = 400m and (b) H = 50m. Arrows represent wind vectors and contours of the background geopotential.

10°N 0° 10°S 20°S 30°S 0°E 30°N

20°E

(a)

40°E

60°E

H = 50m

Latitude

20°N 10°N 0° 10°S 20°S 30°S 0°E

is applicable. The exception is with a westerly basic state and large ka. The reasons for the apparent shortwave cutoff comes more from a violation of the 𝛽 - plane approximation than from a physical instability. The basic state is defined by Eq. (5.4). For U > 0, the gradient of HS must be negative. Figure 6.10a illustrates the latitudinal variation of Hs necessary to support the U = −10, 0, and 10 m s−1 basic states. Plots of the total fluid depth as a function of latitude (Figure 6.10b) are revealing. The integration of Eq. (5.8) provides the distributions of HS necessary to support the geostrophic states.4 Consider a system with an equivalent depth of 500 m. In a quiescent basic state, HS in Eq. (6.4) is zero. In an easterly basic state, Hs must increase in latitude so that an increase in the total depth to 1000 m at 60∘ is required. This slope provides a gradient in HS sufficient to support a −10 m s−1 geostrophic basic state. However, for a positive basic state, Hs must have a negative slope with latitude and the total depth of the fluid becomes negative at latitudes poleward of 60∘ . One way of overcoming this problem is to use full spherical geometry and seek a distribution of Hs that 4 See Section 6.3.2.2, Eq. (6.56).

20°E

40°E Degrees longitude (b)

60°E

provides a basic state that is in solid rotation relative to the planet. However, this would mean that the background zonal wind speed would have to decrease to zero near the poles. Thus, for the study of near-equatorial phenomena, there is not much to gain from this more complicated geometry. 6.1.5.2

Inertia-Gravity Waves (WIG, EIGIG)

High-frequency equatorially trapped inertia-gravity modes are an important phenomenon in the tropical atmosphere taking part in the geostrophic adjustment to forcing when the scale of the forcing is less than RE . Consider the high-frequency limit (𝜔 >> f ) in the dispersion relationship (6.15). Then )1∕2 ( 𝜔d 2 𝛽 k 2 − k = (2n + 1) U (6.30) 1+ c 𝜔d c2 However, as kU∕𝜔d 0 (panel iii) basics states plotted between 90∘ N and S. Three longitudinal wavenumbers are shown: ka = −3 (red), −5 (black), and − 9 (green) with n = 1. Notice that for U > 0, the latitudinal scale of the modes increases substantially. (b) Variation of the total fluid depth for various basic states. In a quiescent basic state (black: U = 0), the depth is constant with latitude at 500 m. In an easterly basic state (blue: U = −10 m s−1 ) the fluid increases depth with latitude in order to support the geostrophic state. However, with a westerly basic state (red: U = +10 m s−1 ) the fluid possesses negative depth near 55∘ .

6.1 Atmospheric Modes in a Constant Basic State: Constant U

family (WIG) that are almost symmetric about 𝜔 = 0. Note, too, that for relatively small ka, the modes are slightly dispersive but as ka increases (both negatively and positively) then 𝜕𝜔/𝜕k ≈ 𝜔/k and the modes become increasingly non-dispersive. However, the impact of changes in the background basic state is so small that Figures 6.1 and 6.2 for U = 0 provide most of the modal information, even for U ≠ 0. 6.1.5.3

Mixed Rossby-Gravity Wave (MRG)

Noting that by definition we have chosen 𝜔d > 0, we accept the positive root. Except for Doppler effects, there is no influence of a basic state. A third asymptote occurs for large negative k and 𝜔d asymptotes toward zero, paralleling the n = 1 equatorially-trapped Rossby wave. The eigenfrequencies of the westward-propagating MRG appear in Figure 6.11a for various constant basic states. The eigenfrequency tends to be larger for the westerly basic state compared to the easterly state, in much the same manner as with the ER (Figure 6.5a).

Frequency Domain With n = 0 in Eq. (6.15), the disper-

sion relationship for the MRG mode is written as )1∕2 ( 𝜔2d 𝛽 k kU 2 −k − 𝛽= (6.31) 1+ 𝜔d c 𝜔d c2 Equation (6.31) is a sixth-order equation and difficult to solve. However, we can discern the character of the mode by looking at asymptotics for large and small 𝜔d . For large values of 𝜔d the last term in parentheses is «1 over the range of observed values of U. The term kU∕𝜔d also emerges from the dynamic term in Eq. (6.7c). At this limit, Eq. (6.31) reduces to 𝜔2d c2

− k2 −

𝛽 k 𝛽≈ 𝜔d c

(6.32)

To a good approximation. Equation (6.32) possesses three roots: (𝜔d + ck)(𝜔d 2 − ck𝜔d − c𝛽) = 0

(6.33)

The first solution, 𝜔 = − ck, is neglected since it leads to growing amplitudes at large y and this would violate boundary conditions. Using the quadratic formula we obtain the remaining two roots. These are: ck 1 √ 2 2 𝜔I = c k + 4c𝛽 (6.34a) + 2 2 and ck 1 √ 2 2 c k + 4c𝛽 (6.34b) − 𝜔II = 2 2 √ As ck∕2 < c2 k2 + 4c𝛽∕2 for all k, 𝜔I > 0 and 𝜔II < 0 for all k. However, by definition, 𝜔 must be positive and so the second root 𝜔II is neglected as it leads to exponentially growing solutions in latitude. Thus, the dispersion relationship for all of MRG waves is given by Eq. (6.34a). For large positive values of k, 𝜔I ≈ k, which approximates an eastward-propagating equatorial gravity wave. Overall, the impact of a basic state on the MRG for ka > 0 is minimal, as was found for the EIG. Another checkpoint occurs where ka = 0 in Eq. (6.32) gives 𝜔2d = 𝛽c

(6.35)

Phase and Group Speeds The phase speed of the MRG is

given by

√ 4c𝛽 c 1 cpdx |MRG = + c2 + 2 (6.36) 2 2 k At high frequencies the mode has a positive phase velocity while at lower frequencies (i.e. 𝜔I < 1) the zonal phase velocity is negative or westward. The group speed of the MRG is given by cgdx |MRG =

c c2 k + √ 2 2 c2 k2 + 4c𝛽

(6.37)

which is positive for all ka. For large negative values of k, the mode asymptotes to small positive values of cg . At ka = 0, cg = 0.5c increases to an asymptotic limit of cg for large positive values of ka. Dimensionally: √ (6.38) cgdx |MRG (k >> 0) → gH mimicking, for larger k, an n = 0 EGW. The group velocities of the MRG are shown in Figure 6.11b as a function of H and ka. Over the entire parameter range, the group velocity is positive. For small |ka| the group speed is large. For a given H the group speed decreases monotonically with increasing wave number. For a given ka, the group speed asymptotes to constant values as H increases. However, for H < 200 m, the longitudinal group speed decreases rapidly. For example, for ka = −4 and H = 400, the group speed is 17 m s−1 , but at H = 50 m, the speed has reduced to 8 m s−1 . Doppler and Non-Doppler Impacts of a Basic State The

impact of a non-zero basic state is minimal in the highfrequency limit and similar to that found for the inertia-gravity waves simply because, dimensionally, √ U 0 and large negative k. However, as argued in Section 6.1.5.1.4, this is an artifact of a breakdown of the 𝛽-plane approximation rather than a real physical phenomenon. 6.1.5.4

The Kelvin Wave

Edges, orographic barriers, and sloping terrain play an important role in both the ocean and the atmosphere. Here we will concentrate on the near-equatorial Kelvin wave, but note in passing that there are families of waves associated with slopes and barriers. Waves at a Boundary For over 100 years fluid dynam-

icists have studied surface wind-driven ocean gravity waves propagating toward a coast. Early investigators observed that a second family of waves with a maximum amplitude close to the coast was also present propagating parallel to the coast in both directions seemingly not directly driven by the wind field. These are transverse waves, an important part of beach building and erosion. They are of sufficiently small scale and high frequency not to be affected by the rotation of the planet. Sir William Thompson (later Lord Kelvin)VI became interested in larger-scale edge phenomena where planetary rotation was an important factor. In particular, Thompson was interested in coastal waves that propagated along the coast of the English Channel.5 Like the coastal transverse waves discussed above, they too appeared to be trapped against the coastline but, as distinct from the smaller-scale transverse waves, possessed unidirectional propagation. In the northern hemisphere they propagate eastward along the northern side of an east–west oriented coastline. They also propagate along north–south coastlines. For example, on the Pacific coast of North America such waves propagate northwards while on the Pacific coast of South America they move southwards. Common to both the small-scale transverse waves and the large-scale coastal waves is that there is no normal fluid motion at the boundary. That is, for the east–west oriented coastline, described above, the meridional component of flow must be zero (i.e. v = 0) at the coast line. For the north–south coastline, the zonal component vanishes (u = 0). In general, the boundary condition ̃ = 0, where ̃ n is a unit vector normal at a coast is ̃ n•V to the coast. Subsequently, trapped by boundaries or 5 Thompson (1879).

changes in the gradient of angular momentum, they have become known generically as “Kelvin waves.”6 Besides the large-scale oceanic transverse coastal waves studied by Lord Kelvin, there is another boundary wave of immense significance in both meteorology and oceanography. It would be almost a century later that the existence of the equatorial planetary scale edge wave, existing at the equator, was confirmed.7 That is, the equatorial Kelvin wave is trapped about the equator by the gradient of angular momentum on either side of the equator. Edge waves, in general, are non-dispersive gravity waves, as discussed in Section 4.3. The formation of equatorially trapped modes can be visualized using Laplace’s shallow fluid system.8 Consider a large-scale forcing function (mass source/sink or heat source) centered at the equator set in a quiescent background state (Figure 6.12a (i)). A mass sink (equivalent to heating) is located at the equator with an e-folding scale of about 60∘ longitude. The temporal dependence of the forcing is shown in panel (ii). Starting at t = 0, the forcing reaches a maximum at day 2.5 and then slowly decreases in intensity. Figure 6.12b describes the first seven days of the integration. At the end of day 1 (Figure 6.12b (i)), the forcing produces what appears as an almost circular geopotential response. At this stage flow is “down the pressure gradient.” At Day 2 (panel (ii)), striking changes have occurred. First, the response is elongated along the equator to the east, where the signal moves rapidly along the equator. Away from the equator the rotational effects create a pair of cyclonic circulations to the southwest and northwest of the initial forcing. These are the “attendant” or “shedded” Rossby waves. Throughout the period of integration, these vortices migrate slowly westward and by day 7 are centered near 150∘ E on either side of the equator. In the meantime, to the east of the forcing, a wave has propagated rapidly eastward at an average of 60∘ of longitude per day or at a rate of 69 m s−1 . This is very close to the speed of a theoretical gravity wave in a 500 m fluid. On day 4, the eastward wave completes its circumnavigation of the tropics and starts to interfere with the Rossby response. The two-mode structure of the equatorial response may be understood by considering separately the east and west sections relative to the position of the forcing. In both locations, the system adjusts to geostrophy. On the eastern side, the adjustment is down the pressure gradient in the form of a gravity (Kelvin) wave. Subsequent flow away from the equator will be channeled 6 Johnson (2007) provides an excellent historical and theoretical review of edge waves. 7 Matsuno (1966) and Longuet-Higgins (1968). 8 The details of the nonlinear free-surface model are given in Appendix G.

141

6 Equatorial Waves in Simple Flows

Figure 6.12 (a) Functions used to force modes along the equator using a nonlinear global shallow fluid model (H = 500 m). (i) Spatial distribution of a forcing function centered at 180∘ longitude and amplitude at day 5. (ii) The temporal variability of the forcing with units of m2 s−3 . (b): Sequence of the geopotential (m2 s−2 ) relative to the color bar below the diagram and the vector velocity fields. Distributions are shown for day 1, 2, 3, 4, 5, and 7 and show the rapid eastward progression of a Kelvin wave along the equator and a pair of Rossby modes that propagate very slowly westward.

(i) Spatial distribution of forcing at day 5

Latitude

60°N 30°N

0

equ

–1

0

30°S 60°S 0

60°E

120°E

180°

120°W 60°W

(ii) Temporal distribution of forcing at Forcing amplitude (m2s–3 × 102)

142

0.5 0 –0.5 –1.0 –1.5 –2.0 –2.5 –3.0 0

60°N

1

2

3

(i) Day 1

4 days (a)

30°N equ 30°S

60°S

60°S 0 60°E 120°E 180° 120°W 90°W 0 (ii) Day 2

7

8

(iv) Day 4

(v) Day 5

60°N 30°N equ 30°S

60°S

60°S 0 60°E 120°E 180° 120°W 90°W 0

0 60°E 120°E 180° 120°W 90°W 0 (iii) Day 3

6

0 60°E 120°E 180° 120°W 90°W 0

30°N equ 30°S

60°N

5

60°N

30°N equ 30°S

60°N

0°

180o

(vi) Day 7

60°N

30°N equ 30°S

30°N equ 30°S

60°S

60°S 0 60°E 120°E 180° 120°W 90°W 0

0 60°E 120°E 180° 120°W 90°W 0 10 m s–1

–270 –240 –210 –180 –150 –120 –90 –60 Geopotential (m2 s–2)

(b)

–30

0

+30

6.1 Atmospheric Modes in a Constant Basic State: Constant U

back toward the equator by the Coriolis force. In fact, in the lateral direction the flow is in geostrophic balance where 𝜌𝛽yU = −𝜕p∕𝜕y from Eq. (5.30b), but to the west of the forcing, the initial flow is deflected both north and south of the equator so that v is finite. Geostrophic adjustment is accomplished by the projection on to the Rossby waves. Slope Waves The waves considered above (transverse edge waves and Kelvin waves) depend on the normal velocity to an edge to be zero. Essentially, this is the ̃ = 0, where ̃ no-flux boundary condition ̃ n⋅V n is the normal unit vector to the edge. Kelvin waves have been observed along slopes together with a much slower unidirectional large-scale mode called “slope” waves. These slope modes are important in the ocean (Mysak 1980) and the atmosphere (Reason 1994). In essence, these are Rossby waves where the changes in the angular momentum caused by the slope permit a slow propagation along the boundary barrier. Energy remains in the lower troposphere trapped by low-level stable inversions.

where C is a constant. Now, (6.39a) and (6.39c) are algebraic equations in u and h for which solutions exist if 𝜔2d − c2 k2 = (𝜔d + ck)(𝜔d − ck) = 0 𝜔d = ±ck

or (6.41)

Applying these conditionals to Eq. (6.40) leads to two solutions, one of which grows exponentially with y, thus violating the boundary condition of finiteness of solution at large y, and the other decaying with increasing latitude. Therefore, the complete eigensolution and the dispersion relationship for the equatorial Kelvin wave is given by 𝜔d = ck

) ( k𝛽 2 h(y) = C exp − y ∕2 𝜔 (d ) kg k𝛽 2 u(y) = C exp − y ∕2 𝜔d 𝜔d v=0

(6.42)

Phase and Group Speeds The longitudinal phase and Frequency Domain of Equatorial Edge Waves The equatorial

Kelvin mode can be isolated by setting v = 0 in (5.45). It can also be derived elegantly using the full spherical equations, where it can be shown that for a Kelvin mode there is a vanishingly small meridional velocity component (Longuet-Higgins 1968). Starting with Eq. (5.45) for v = 0 and U = constant, we have −i𝜔d u + ikgh = 0

(6.39a)

𝛽yu + gdh∕dy = 0

(6.39b)

−i𝜔d gh + ikc2 u = 0

(6.39c)

The only impact of a constant basic state is included in 𝜔d , the Doppler-shifted frequency. We note in Eq. (6.39b) that the zonal wind is in geostrophic balance with the latitudinal pressure gradient. Very close to the equator, where f and v are small, the pressure gradient along the equator (𝜕h/𝜕x in Eq. (6.8a)) provides a zonal acceleration. In the absence of dissipation (or, as we will see later, reductions in static stability), it is expected that this √ near-equatorial mode will propagate eastward at gH. We note, too, that y changes sign at the equator and that 𝛽 decreases both to the north and south from its maximum value. Between (6.39a) and (6.39b) we obtain a first-order differential equation with solutions: ( ) 𝛽k 2 h(y) = C exp − y ∕2 (6.40) 𝜔d

group speeds are identical and both to the east such that in non-dimensional form: 𝜔 cpd |K = d = c − U and k √ 𝜕𝜔d (6.43) = c − U = gH − U cgd |K = 𝜕k Thus, the longitudinal phase and group speeds of a Kelvin wave are identical to the phase speed of a pure gravity wave in a fluid of depth H, plus advection by the basic state. The equal magnitude of the phase and group speeds indicates that the wave is non-dispersive. That is, all parts of the wave packet move at the same speed without deformation. For realistic values of U there is little impact on the Doppler-shifted group speeds, i.e. for H = 20, 50, 100, 400, and 1000 m, cpx and cgx have values of approximately 14, 22, 31, 63, and 100 m s−1 . For U = −10, 0, and 10 m s−1 , cpg and cgd are similar to the non-Doppler counterparts. Even for very small values of U the Doppler-shifted group speed remains positive for U = −10 m s−1 . Note also that for any reasonable value of the basic state, even for ka = 1, the solution 𝜔d = − kc still grows exponentially with latitude, thus violating the boundary condition at large y and justifying its neglect. Figure 6.13 shows the eigensolutions for the Kelvin wave deduced from Eq. (6.42). The u-field is clearly in geostrophic balance with the latitudinal pressure gradient and in quadrature9 with the zonal pressure gradient. 9 I.e., out of phase by 90∘ .

143

6 Equatorial Waves in Simple Flows

Figure 6.13 Eigensolutions for the Kelvin wave with H = 500 m relative; (ii) H = 50 m; and (iii) H = 20 m. The lateral extent of the Kelvin wave decreases rapidly with decreasing H.

Equatorial Kelvin Wave Eigensolutions 30°N (i) H = 500m

Latitude

20°N 10°N EQ 10°S 20°S 30°S 0

10°E

20°E

30°E

40°E

50°E

60°E

70°E

50°E

60°E

70°E

50°E

60°E

70°E

30°N (ii) H = 50m

Latitude

20°N 10°N EQ 10°S 20°S 30°S 0

10°E

20°E

30°E

40°E

30°N (iii) H = 20m

20°N Latitude

144

10°N EQ 10°S 20°S 30°S

0

10°E

20°E

30°E 40°E Longitude

What is particularly interesting is the increased trapping of the Kelvin wave about the equator as H decreases.

7

√ turning latitude goes as 2n + 1 so that higher modes would extend more poleward.

6.2.1

6.2 Atmospheric Waves in Latitudinal Shear Flow: U = U(y) We noted in Chapter 4 that if the zonal basic state varied as a function of latitude, the modal structure would change in order to maintain the constancy of its intrinsic frequency along a ray path. Equation (4.14c) suggested that the latitudinal wavenumber l would increase if 𝜕U∕𝜕y < 0. Furthermore, we noted in the last section that as n increases so does the latitudinal extent of the mode. In fact, Eq. (6.20) specified that the

Regions of Shear in the Tropics

Figure 6.14a shows the climatological DJF zonal wind field. Equatorial easterlies (EE) (blue shading) dominate the eastern hemisphere upper troposphere whilst moderate westerlies extend across the equatorial Pacific and Atlantic oceans. Two basic state profiles of U = U(y), referred to as EE and EW (Figure 6.14), appear representative of upper-tropospheric shear profiles, with easterlies near 140 ∘ E and westerlies near 120 ∘ W, respectively. EW possesses zonal winds of magnitude +10 m s−1 near the equator and + 20 m s−1 at higher latitudes. In EE, equatorial values of U are

6.2 Atmospheric Waves in Latitudinal Shear Flow: U = U(y)

Mean boreal winter 200hPa U-field and tuning latitude (yT) 20 30 40

30

20 10 0

−5 −1

−10

20

10

10

EE

30

0

0

−5

0

10

10

20

20

40°S

20

0

−5 10

20 30

30

2010 0

0

20°S

30

40

50 40

40

−5

60 70

50

50

20°N equ

30

40

20 30

0

40°N

Latitude

Figure 6.14 (a) The background boreal winter basic state U (x, y) at 200 hPa. Two basic states, representative of sections of the upper tropospheric zonal wind field through sections EE (along 140∘ E) and EW (120∘ W), are depicted as solid meridional lines. These longitudes are chosen to represent the profiles of strong equatorial easterlies and westerlies, respectively. For later reference, an estimate of the turning latitude Γ2 (y) = 0 is plotted as the bold red dashed line calculated from Eq. (6.18). (b) The two profiles EE and EW as a function of latitude. Source: Based on Figure 12, Zhang and Webster (1989).

30

60°E

20

20

30

20

EW

120°E 140°E 180° Longitude (a)

120°W

60°W

0

Meridional sections of zonal wind U

Zonal wind (U m s–1)

30

EE (140°E)

20

EW(120°W)

10 0 –10 90°S

60°S

near −10 m s−1 while the midlatitude westerlies reach +40 m s−1 . It should be mentioned that the discussion below is not a rigorous treatment of the impact of shear but an attempt to differentiate between relative impacts of the two major shearing domains of the tropics. We consider here simpler states. We suppose that EE and EW are symmetric functions so that the mean absolute vorticity (i.e. 𝜂 = 𝜁 + f ) is zero at the equator and positive to the north and negative to the south. In addition, the meridional component background basic flow is set to zero so that there is no advection of absolute vorticity across the equator, consistent with the absence of a cross-equatorial pressure gradient. With U = U (y), the equation set (5.8) takes on its full form and with it added complexity. As a result, the turning latitude, where the refractive index is zero, expressed in Eq. (6.16) becomes more complex through the addition of a further non-Doppler dynamic term in Eq. (5.8a). The bold red dashed contour in Figure 6.14a depicts yt . The turning latitude extends much further poleward where there are equatorial westerlies (e.g. in profile EW) compared to the region of EE.

30°S

equ Latitude (b)

30°N

60°N

90°N

Equation (5.11) cannot be solved analytically but the equation is amenable to solutions by numerical techniques.10 Below, we summarize the results of these numerical calculations.

6.2.2

Impacts of Latitudinal Shear

Figure 6.15 shows the eigenfrequencies for the equatorial modes in a sheared environment. For reference, the solid lines show the eigenfrequencies for the quiescent basic state. The short dashed contours denote the EE profile while the long dashed contours represent the EW state. In the discussion below, we will concentrate on the low-frequency modes: the ER and the westward-propagating MRG. We note here that the impact of shear on the high-frequency modes is not large, either increasing the eigenfrequency for 10 For example, Zhang and Webster (1989) rendered equation set (5.8) into a standard eigenvalue problem. This allowed solutions for arbitrary shear configurations. Details of the numerical procedure can be found in Section 5 of Zhang and Webster (1989).

145

6 Equatorial Waves in Simple Flows

2.5

Eigenfrequencies in Shear Flow (U(y)) U=0 U(y) = EE U(y) = EW

2.0 1.5 1.0

WIG

EIG E-MRG

W-MRG

K

0.5 0 0 EE

–0.5 EW

0.5 ER 0 –6 –5 –4 –3 –2 –1 0 1 2 3 Zonal wavenumber (ka)

–1

4

5

90°S

6

Figure 6.15 Eigenfrequencies in shear flow for all equatorial modes. The impact of the EE shear tends to increase the eigenfrequency slightly for eastward modes but reduce it for westward modes. Greatest changes occur in ER modes where the critical wavenumbers (k0 : denoted by red solid circles) tend to occur at larger scales.

eastward-propagating modes while decreasing it for westward modes. The impact of the sheared flow on the eigenfrequency and meridional structures of the Kelvin wave is rather small, similar to that found for the constant basic states. The reason for this is the absence of the two non-Doppler terms vdU∕dy and − yUv in Eqs. (5.8a) and (5.8c), respectively, as v is zero or vanishingly small. 6.2.2.1

ER Geopotential ka = 5, n = 1

1

Geopotential (m2 s–2)

3.0 Eigenfrequency (cycles/day)

146

Rossby Waves in Shear Flow

From Figure 6.15, it is clear that the n = 1 ER eigenfrequencies are modulated by shear flow very differently than by the constant zonal flow considered earlier. The modal frequency is larger for all ka in the EE basic flow (which possesses EE) than in the EW state that contains EW. The location of the k0 , where 𝜕𝜔/𝜕k = 0 (marked by a bold red circle indicating the ko separating eastward and westward group speeds), occurs at longer scales in EW than in EE although the transition occurs at longer scales for both shear flows compared to the motionless basic state. The transition occurs at about ka = −4 for the EE and ka = −3 for the EW. Thus, relatively large-scale waves (ka ≈ −3) in both shear flows have westerly group speeds. This compares with a transition near ka = −6 for U = 0. Earlier, we found that the Rossby wave exhibits larger meridional scales in a constant westerly basic flow than in a motionless basic state. Similar properties are found in shear flow as well. The meridional structure of geopotential appears in Figure 6.16 for n = 1, ka = 5.

60°S

30°S

0° Latitude

30°N

60°N

90°N

Figure 6.16 Meridional distributions of the ER n = I, ka = 5 R geopotential in sheared basic zonal flow; EE (red line), EW (blue), and in zero basic zonal flow (black). Source: From Zhang and Webster (1989), Figure 15.

The perturbation geopotential field extends significantly more poleward in the two shear cases compared to the zero basic state. 6.2.2.2

Mixed Rossby-Gravity Wave in Shear Flow

The impacts of shear flow on the westward-propagating MRG are similar to those found for the ER wave. The greater differences compared to the motionless state occur with the westward-propagating modes. The dispersion curves (Figure 6.15) show that the frequencies are increased for the eastward MRG, relative to the motionless basic state, but decreased for the westward MRG. The largest changes of this trend occur in the EW shear flow. The impact of the shear flow on the meridional structure of the eastward MRG is relatively small (Figure 6.17) but more considerable for the westward MRG. In both shear flows the modes extend farther poleward than in the motionless basic state.

6.3 Physics of Equatorial Trapping We now develop a physical basis for the existence of equatorially trapped modes. The basis is equally appropriate in explaining oceanic equatorially trapped modes, albeit with appropriate parameter choices such as the value of H and the use of reduced gravity. As described above, a non-zero basic state can impact equatorial modes through both Doppler and non-Doppler effects, the former merely advecting the solutions with the basic state while the latter alters the dynamical environment. The non-Doppler effects have been shown in the last section to alter significantly both

6.3 Physics of Equatorial Trapping

MRG Geopotential ka = 5, n = 0

Geopotential (m2 s–2)

1 EW

0.5

EE

0 0 –0.5 –1 90°S

60°S

30°S

0° 30°N Latitude

60°N

90°N

Figure 6.17 Meridional geopotential structures of westwardpropagating MRG in shear flow: EE (red line), EW (blue), and in zero basic zonal flow (black); ka = 5 and H = 400 m. Source: From Zhang and Webster (1989).

the eigenfrequencies and meridional structures of the Rossby mode and the westward-propagating MRG. The other equatorial modes (Kelvin and inertia-gravity waves) were far less affected by non-Doppler effects. For the ER and the westward-propagating MRG mode, the meridional structures appeared less trapped in a westerly than in an easterly basic flow. In addition, the transition scale between easterly and westerly group speeds occurs at larger scales than in a motionless basic state. Hence, waves propagating along the equator will find themselves in basic states with differing longitudinal structures, and thus become more trapped or less trapped. Such behavior may have important ramifications on where centers of action occur in the tropics and where equatorial wave energy will emerge toward higher latitudes. However, all of these issues raise the fundamental question: What are the actual physical processes that invoke changes to the latitudinal structure of the equatorially trapped modes? We examine this question by considering the basic generating mechanisms of the Rossby wave and the equatorially trapped Rossby wave. We use the conservation of potential vorticity as a useful starting point here applied to a shallow fluid.11 6.3.1

Simple Potential Vorticity Arguments

As an air parcel moves from one location to another, it encounters changes in the characteristics of the background flow, the impacts of rotation and stratification. 11 The following analysis is based on Zhang and Webster (1989).

In order to conserve the parcel’s potential vorticity, changes in the parcel’s relative vorticity (𝜁 = 𝜕v/𝜕x − 𝜕u/ 𝜕y) must be induced. The induced 𝜁 can be viewed as a restoring force promoting a Rossby wave oscillation.12 In the following development, we look for conditions that invoke a latitudinal variation in the restoring force. For a mode to be equatorially trapped, the restoring force to a lateral displacement must decrease with latitude on either side of the equator. In an environment that causes the restoring force to decrease rapidly with latitude, the mode will be more equatorially trapped than a mode in an environment of a less rapidly decreasing restoring force. In the latter case, the greater extent of the restoring force may allow the influence of the waveform to extend significantly poleward and perhaps even influence the extratropics. A general expression for potential vorticity in a shallow fluid, qh , is qh =

𝜁T + f 𝜁T + 𝛽y = H H + HS (y) + h(x, y, t)

(6.44)

where H, the total depth of the fluid, is defined in Eq. (5.3). The total relative vorticity is 𝜁T = 𝜁 + 𝜁 , the sum of the mean and perturbation vorticity. Assuming that h∕H 0)| greater trapping

less trapping (6.59)

Figure 6.18 also shows the normalized magnitude of the 𝜁 induced by a lateral displacement for non-divergent and divergent states where U = −10, 0 and 10 m s−1 . The e-folding distances are, respectively, infinity for a non-divergent system (no trapping) and approximately 15∘ , 20∘ , and 25∘ of latitude for the three basic states in a divergent background state. In essence, divergent modes in an easterly basic state are more trapped about the equator than divergent modes in a westerly basic state. On the other hand, non-divergent modes are not trapped at all. The physical impact of a basic zonal flow on midlatitude Rossby waves is well known. Here, we extend the midlatitude analysis to the tropics. Specifically, a basic zonal flow in a geostrophically balanced geopotential field modifies the latitudinal gradient of the background PV. In essence, this is equivalent to changing the magnitude of the 𝛽-effect. In a westerly basic flow, the 𝛽-effect is essentially enhanced, leading to a slower decrease in latitude of the induced relative vorticity. In an easterly basic flow, the effective 𝛽-effect is reduced, leading to a more rapid decrease of 𝜁 and a greater trapping of the mode about the equator. The potential vorticity interpretation of the nonDoppler effects is very helpful in illustrating the basic physics of equatorial trapping. The potential vorticity interpretation is summarized in Figure 6.19 for the following systems: (I) a non-divergent (rigid top) motionless basic state, (II) a divergent (free surface) motionless basic state, (III) a divergent westerly basic state, and (IV) a divergent easterly basic state. The second-column schematics describe the basic structure of the four systems as a function of latitude in the form of “equivalent” shallow fluid systems. The third column describes the normalized values of the induced relative vorticity obtained from Eq. (6.55) for the non-divergent case and from Eq. (6.57) for U = 0, >0, and 0; weak trapping

y |ζ1|

Δy

trapping about the equator as the induced vorticity is independent of latitude. System II represents a potentially divergent basic state with a shallow fluid height given by h = H + HS (y) + h′ , but as U = 0, Hs = 0. The ′ perturbation height field, h , is shown as the dashed line superimposed on H (denoted by the solid line). Moderate trapping occurs because the restoring force will change with latitude depending on the scale of the mode. With a geostrophic basic state U, Hs is no longer zero and possesses a negative slope for a westerly basic state and a positive slope for an easterly basic state. The divergent systems III and IV will have a total depth given by h = H + HS (y) + h′ . The perturbations field ′ h is the dashed line superimposed on H (solid line). For a westerly basic state (system III), the slope of the total fluid with latitude creates a substantial shallowing of the total of the fluid with increasing latitude (see Figure 6.10b). Depending on the sign and magnitude of the basic flow, the modes will be either trapped more weakly than in system II in a westerly basic state or more strongly in an easterly basic flow. In essence, the total height H + HS (y) becomes either shallower or deeper as a function of y, thus changing the degree of induced relative vorticity as the vortex tube is moved laterally. 6.3.2.3

no trapping

y h´

II - divergent - motionless basic state

DEGREE OF TRAPPING

Figure 6.19 Summary diagram of the degree of equatorial trapping of atmospheric waves. The left-hand column describes the physical nature in when a lateral poleward displacement is made of the response: (I), a non-divergent basic state where either H is very large or there exists a rigid surface, (II) divergent flow but in a quiescent basic state, (III) divergent in a westerly basic state, and (IV) easterly and westerly basic states. The sections shown refer to the Northern Hemisphere with the equator on the left and increasing latitude to the right. Source: After Zhang and Webster (1989), Figure 24.

U < 0; strong trapping

y

the shear originated from EE than when there were EW at the equator. Thus, equatorial modes extend more poleward in regions of reduced shear such as the EW region (central-western Pacific) than elsewhere in the tropics. Despite the complexity of the modification of modes by a latitudinally sheared flow, it is possible, at least qualitatively, to use potential vorticity arguments to understand why there is a general decrease in Rossby wave trapping in positive shear flow (i.e. dU/dy > 0). Even though the values of the sheared zonal flow (EE and EW) are of different signs at low latitudes, both possess strong positive values of dU/dy between the equator and the westerly maximum of the extratropics. In terms of the system used above, the free surface height (H + HS (y)) must reduce rapidly with increasing latitude to support the shear. That is: ΔHs (y)|(SHEARED) > ΔHs (y)|(NON-SHEARED) (6.60) where we have set H as a constant. The relative change in height has the impact of enhancing the 𝛽-effect even further than in a constant basic state. Consequently, the induced perturbation of relative vorticity reduces even more slowly with latitude, resulting in a reduction of the trapping. An inspection of the basic state (Figure 6.14) shows that the regions of strong EE occur at the same longitudes as the strongest latitudinal shear. On the other hand, upper-tropospheric EW correspond to regions of weaker latitudinal shear. For the Rossby wave, the combination of the sign of the basic state at low latitudes and the magnitude of the shear act in the same direction

6.4 Large-Scale Low-Latitude Ocean Modes

as in the case of the constant zonal flows. Thus, those waves whose zonal scales are smaller than the latitudinal variation of the zonal flow will be considerably more trapped in the eastern hemisphere than in the western hemisphere. Similarly, the eastward-propagating MRG that asymptotes to an n = 0 Rossby mode as ka increases will also be less trapped in the eastern hemisphere. The dispersion relationships of Figure 6.15 indicates that frequency varies with ka more rapidly for the EW basic zonal flow than for the EE flow. The dependence of the WIG modes upon the character of the basic zonal flow indicates that the variations of the atmospheric basic state must be included in order to explain the observed differences in wave characteristics. This difference implies that Doppler-shifted group velocity of the westward-propagating MRG should be faster in the EW than the EE. In fact, such variability has been observed in the lower tropical troposphere.14

6.4 Large-Scale Low-Latitude Ocean Modes We now examine some aspects of low-latitude ocean dynamics. Our first aim is to determine the pressure differences with depth and find a relationship with the slope of the ocean surface and the thermocline. We then examine the basic physics of ocean Rossby waves and determine the variability of their phase speeds as a function of latitude. We will then consider the differences between oceanic and atmospheric equatorially trapped waves, noting especially equatorial waves that have to deal with lateral boundaries. 6.4.1 Simple Model of the Upper Ocean – Geopotential Surfaces First, we determine the differences in geopotential (Φ, units m−2 s−2 ) between two isobaric surfaces in the ocean. Using the hydrostatic equation: 𝛼v dp = gdz = dΦ

(6.61)

where 𝛼 v (m3 kg−1 ) represents the specific volume (1/𝜌) and the geopotential difference between pressure surfaces p1 and p2 is p2

Φ(p1 ) − Φ(p2 ) =

∫p1

𝛼dp

(6.62)

We can express the specific volume as 𝛼 v (s, T, p) = 𝛼 v (35, 0, p) + 𝜐(s, T, p) such that, for any ocean salinity (s 14 Liebman and Hendon (1990).

PSU15 ), temperature (T ∘ C), 𝛼 v is the difference between the specific volume of a unit mass of seawater of salinity s and temperature T, and a standard unit mass of ocean water where s = 35 PSU and T = 0 ∘ C. Between Eqs. (6.61) and (6.62) the geopotential anomaly may be written as p2

Φ(p1 ) − Φ(p2 ) = ΔΦ = p2

−

∫p1

∫p1

𝛼v (35, 0, p)dp

𝛼v dp =

p2

∫p1

𝜐dp

(6.63)

Here 𝜐(s, T, p) is a specific volume anomaly and is the difference in volume between a standard unit of ocean water and a unit mass at temperature T and salinity s. The specific volume anomaly, 𝜐, is the change in volume per unit mass induced by a change in s and T relative to a standard specific volume. The geopotential anomaly in Eq. (6.63) is referred to as the dynamic height and for convenience it is scaled by 1/g to obtain the steric height (hs ), measured in meters. Specifically, dynamic height (units J kg−1 or m2 s−2 ) is the oceanic equivalent of geopotential in the atmosphere. Atmospheric geopotential is the amount of work needed to move a unit mass of gas vertically from the surface to some height above sea level. Dynamic height in the ocean is the amount of work required to move a unit mass of water vertically from sea level to a given depth. Steric height refers to the variation or change of the height of an ocean column due to density variations induced by temperature and salinity changes. Accordingly, the specific volume anomaly, 𝜐(s, T, p), expressed in Eq. (6.63) is often called the specific volume steric anomaly. Re-writing the specific volume anomaly as ) ( 𝜌 − 𝜌(s, T, p) 1 1 = 0 − 𝜐(s, T, p) = 𝜌(s, T, p) 𝜌0 𝜌0 𝜌(s, T, p) 𝜌0 − 𝜌(s, T, p) ≈ (6.64) 𝜌0 2 and using the hydrostatic approximation (6.61), the steric height can be expressed as hs (z1 , z2 ) =

z(p1 )

∫z(p2)

𝜌0 − 𝜌(S, T, p) dz 𝜌0

(6.65)

The steric height, hs , measures the change in height of a column of water between z1 and z2 in a column andwill change if T and/or S are changed. We use the 1 1/2 level model of Tomczak and Godfrey (1994, 2000) in which the ocean is divided into a deep ocean layer of density 𝜌2 and a much shallower layer 15 PSU (or Practical Salinity Unit) is a measure of salt concentration in units of 0/00. For a salinity value of 35 0/00 this means that there are 35 parts of dissolved salt per 1000 parts of water.

151

152

6 Equatorial Waves in Simple Flows

depth in the upper layer z1 gives

surface p = 0 ocean

VG

y

VG

hs (x, y, z1 ) =

isobars

As (znm − z1 )(𝜌0 − 𝜌2 )/𝜌0 and z1 Δ𝜌/𝜌0 do not vary horizontally they do not affect the horizontal distribution of hs (x, y, z1 ). Therefore, we can rewrite Eq. (6.66) as

ρ1 ρ2 isobars

(znm − z1 )(𝜌0 − 𝜌2 ) − (H(x, y) − z1 )Δ𝜌 𝜌0 (6.66)

interface thermocline H(x,y)

V2 = 0

V2 = 0 z Figure 6.20 The 11 /2 -layer ocean where the thermocline depth, H(x, y) is variable but the density difference Δ𝜌 = 𝜌2 − 𝜌1 between the layers is constant. Gradients of steric height (defined in Eq. (6.65)) are constant throughout the top layer. The geostrophic current VG are therefore independent of depth. Geostrophic currents are anticyclonic. The steric height measures the surface elevation and is proportional to the thermocline depth, as indicated in Eq. (6.66). Below the thermocline is the region of no motion, where steric height gradients are flat. Source: After Tomczak and Godfrey (1994).

of constant density 𝜌1 where 𝜌1 = 𝜌2 − Δ𝜌, as depicted in Figure 6.20. It is assumed that the lower layer is motionless (as assumed at “depth of no motion” in Figure 3.9). Here, the interface between the upper and lower layers appears as a bold dashed red line and the solid contours represent isobars. As density is assumed to be constant in the upper layer, the isobars are also isotherms. Below the interface separating the upper and lower layers, the horizontal pressure gradients are flat, consistent with no motion. In the upper layer there are horizontal gradients of pressure that support geostrophic currents. If we assume that the section in Figure 6.20 is oriented along a line of longitude with north to the right, the flow is to the west (into the page) on the equatorial side of the elevated isobars and to the east on the poleward side. Note that the pressure gradient (or temperature gradient) decreases with increasing depth. In an equivalent manner to the atmospheric thermal wind, the magnitude of the zonal current must increase toward the surface, as also must the meridional current. The depth of the interface between the two layers, H(x, y, t), varies in space and time, mimicking the behavior of the thermocline in the real ocean. The fluid in each layer is assumed to be in hydrostatic balance. The thickness of the lower layer is constant and only the upper layer thickness H(x, y) is allowed to vary. Integration of Eq. (6.65) between the level of no-motion (znm ) to some

hs (x, y) = −

Δ𝜌H(x, y) 𝜌0

(6.67)

Since H(x, y) » hs (x, y), the slope of the thermocline (lower bound of the upper layer) is very much larger than the slope of the surface, with a difference of at least two orders of magnitude. Note also that the negative sign in Eq. (6.67) indicates that if the thermocline slopes one way then the ocean surface will slope in the opposite direction. Across the equatorial Pacific Ocean the thermocline slope is about 140 m/10 000 km (1.4 × 10−5 m/m) east of 150∘ E and shoaling to the east. Similar slopes and orientation occur in the tropical Atlantic Ocean. In the equatorial Indian Ocean the slope is the mirror image of the Pacific Ocean with shoaling thermocline toward the west and a higher surface elevation in the east. 6.4.2

Rotational Ocean Waves

With the exception of the North Indian Ocean, large-scale high-pressure system circulations lie over the subtropical ocean basins, as apparent in Figure 1.7. In both hemispheres, Ekman transport in the upper ocean associated with anticyclonic surface winds will be toward the center of the high-pressure system, creating a positive anomaly in the depth of the thermocline. Anticyclonic geostrophic currents are associated with these depth anomalies (see Section 3.2.4). Figure 6.21 provides a schematic representation of the situation described above for the NH.16 It shows two surface height contours H 1 and H 2 , where H 1 < H 2 . Note that the interface thermocline depth (H(x, y)) slopes downward toward the center of the gyre. Consider four points A, B at latitude y1 and C and D at ′ ′ latitude y2 on the western side of the gyre, and A , B , ′ ′ C , and D similarly located on the eastern side. The height difference is given by ΔH = Δ𝜌

H1 − H2 𝜌0

(6.68)

If we assume that the system is represented by a 𝛽-plane, with the Coriolis force increasing linearly to the north,

16 Following Tomczak and Godfrey (1993), Chapter 3.

6.4 Large-Scale Low-Latitude Ocean Modes

C

D

Latitude

Y2

CONV

propagation

Y1

A

B

D´

C´

A´

and 𝛿y, and if these dimensions are small, we can seek the time rate of change of H over a small time interval 𝛿t such that gH ΔH𝛽𝛿y Δ𝜌 gH ΔH𝛽 Δ𝜌 H = 2 = 2 𝛿t f0 𝛿x𝛿y 𝜌0 f0 𝛿x 𝜌0 gΔ𝜌 ΔH𝛽 H = (6.72) 𝛿x f02 𝜌0

DIV

B´

The ratio of the first and fourth terms (specifically −(H/𝛿t)/(H/𝛿x)) leads to an expression for 𝛿x/𝛿t or the zonal speed of an isopleth of constant value H:

H2 H1 Longitude Figure 6.21 Horizontal view of the anticyclone constructed for the northern hemisphere where y2 > y1 . A and B are two points of the western side of the ocean eddy at latitude y1 separated by Δhs = Δ𝜌(hs1 − hs2 )∕𝜌. C and D are two similar points at latitude y2 = y + Δy. Total northward flow through AB must be greater than through CD low through Eq. (6.70) as f (y2 ) > f (y1 ). Due to the mass convergence in ABCD, the thermocline deepens. By analogous arguments, divergence of mass occurs in A′ B′ C′ D′ shallowing the thermocline. The eddy moves to the west (dashed arrow). Source: After Tomczac and Godfrey (1993).

the total mass transport through a layer of depth ΔH across latitudes yi by the geostrophic flow is Mg = H𝜌0 gΔH∕fi

(6.69)

We note that the flux across AB is greater than the flux across CD, resulting in mass convergence into the volume ABCD. On the other hand, the mass flux across ′ ′ ′ ′ A B is greater than the flux across C D , providing a ′ ′ ′ ′ net mass divergence within volume A B C D . Thus, the thermocline deepens in the west and shallows in the east, producing a net westward propagation of the gyre. We can use this convergence/divergence differential to determine the phase velocity of this large-scale oceanic mode and its dependence on latitude. This dependence will become important in helping to explain low-frequency variability observed in the Pacific and Indian oceans. From Eq. (6.69) we can calculate the net mass convergence between two latitudes y1 and y2 . The net mass geostrophic convergence (𝛿M g ) into ABCD is ( ) 1 1 − 𝛿Mg = gH𝜌0 ΔH (6.70) f2 f1 Introducing the 𝛽-plane approximation, where f = f i + 𝛽y, Eq. (6.70) becomes ) ( 𝛽𝛿y (6.71) 𝛿Mg = gH𝜌0 ΔH f02 where f 0 is given by (f 1 + f 2 )/2. Here we have also assumed that f » 𝛽. If the dimensions of ABCD are 𝛿x

cpx = 𝛽gH(Δ𝜌∕𝜌0 )∕f 2 (y)

(6.73)

This is the phase speed of an oceanic Rossby wave at some latitude y. As the denominator is a positive definite quantity, the phase speed of the large-scale ocean Rossby wave is to the west in both hemispheres and also decreases with increasing latitude. For a thermocline depth of 300 m and with Δ𝜌/𝜌0 = 3 × 10−3 , the phase speed of a long Rossby wave is 1.27 m s−1 at 5∘ from the equator such that propagation across the Pacific Ocean would take about six months. At 20∘ from the equator, the phase speed reduces to 0.08 m s−1 and at 40∘ to 0.002 m s−1 . Thus, the progression of a Rossby wave at higher latitudes across the Pacific extends to years or even decades.

6.4.3 Impact of Boundaries on Near-Equatorial Ocean Modes In Section 5.2, we developed a wave equation (5.22) for the equatorial ocean. The similarity of this expression to the atmospheric shallow-water equation suggests that we can expect a similar full range of solutions as was found for the atmosphere. Figure 6.22 plots the dispersion relationships for ocean modes using Eq. (6.15) with U = 0 and ̃c = 1.4 m s−1 . As ̃c2 = ̃ gH then H ≈ 35 m. The concept of reduced gravity, ̃ g, is discussed in Section 3.2.2 and defined in Eq. (3.56). The modal families, plotted as functions of longitudinal wavenumber ka (scaled as the inverse of the Rossby radius of deformation RE , from Eq. (6.19) or 1/250 km), and frequency 𝜔 (i.e., (𝛽̃c)1∕2 or about one cycle per two days), are of the same form as those found for the atmosphere (i.e. Kelvin (K), n = −1, MRG, n = 0, ER waves, and the inertial gravity (IG) waves for n = 1, 2, 3, …) but with different temporal scales. The bold dashed line (𝜕𝜔/𝜕k = 0) separates eastward and westward group speeds for n ≥ 1. The group speeds of the Kelvin wave and the MRG are universally to the east for all ka.

153

6 Equatorial Waves in Simple Flows

Dispersion curves for c = 1.4 m s–1 0.24 n=3

0.20 0.16

n=1 n=2

IG

6.25

∂ω/∂k = 0 n = 0

0.12

8.3 12.5

0.08 0.04 0 –5

K MRG n = –1 ∂ω/∂k = 0 n=1 ER n = 3 –4 –3 –2 –1 0 1 Scaled zonal wavenumber ka n=0

Period (days)

Scaled frequency (day–1)

154

25 50 2

Figure 6.22 Dispersion diagram for equatorially trapped oceanic modes using Eqs. (6.23), (6.33), and (6.43) for U = 0 and g = ̃ g as a function of frequency 𝜔 and longitudinal wavenumber ka. which , respectively. Periods of the modes are scaled by (𝛽̃c)1∕2 and R−1 E are plotted on the right-hand ordinate, frequencies on the left. As for the atmosphere, four families of modes appear: the Kelvin (K), n = −1, the mixed Rossby gravity (MRG), n = 0, the equatorial Rossby (ER), n = 1, 2, and 3, and the inertial gravity (IG) waves for n = 1, 2, and 3. The dashed line demarks 𝜕𝜔∕𝜕k = 0, calculated for n ≥ 1, separating the modes with eastward group speeds from westward group speeds. The shaded region denotes the period band within which westward group speeds do not exist. Like the Kelvin wave, the MRG has an eastward group velocity for all ka. Only long and low-frequency ER waves and the high-frequency IG waves possess group speeds to the west.

Despite the similarity of the atmospheric and oceanic modal families there are a number of important differences: (i) Because of the buoyancy in the upper ocean due to stratification (which is accounted for by the use of reduced gravity) there is a large difference between the period or frequency of atmospheric and oceanic modes. For example, the period of the n = 0, ka = −2 atmospheric MRG is roughly three days (Figure 6.1a) and for the n = 1, ka = −3 ER about seven days. In the ocean, the periods of these two modes have increased to about 30 and 50 days, respectively. Propagation speeds of atmospheric modes across the Pacific are about one to two weeks. In the ocean, propagation speeds are the order of months, much slower than its atmospheric counterpart. (ii) The differences in phase and group speeds between the ocean and the atmosphere define the fundamental time scales of the two systems. The atmosphere will respond to oceanic forcing (e.g. enhanced surface fluxes associated with regions of warm SST) very quickly, adjusting to the forcing in a few days or a week. On the other hand, the ocean will respond

much more slowly to impulsive atmospheric forcing (e.g. westerly wind bursts or anomalous trade winds). In this manner, an interesting simplification is that the atmosphere may be considered to be in continual equilibrium with the ocean! This approximation has proved useful in understanding low-frequency climate variability such as El Niño and, furthermore, it has simplified the basic couple ocean–atmosphere ENSO prediction models. For example, the Cane et al. (1986) prediction model used a time-evolving ocean model surmounted by a steady state atmosphere, thus taking advantage of the system simplification. (iii) Oceanic equatorially trapped modes, propagating either eastward (the Kelvin wave) or westward (the ER wave) along the equator will encounter lateral boundaries. Upon reaching a boundary, there are two possibilities. A wave may be reflected back along the equator or it may be transformed into an edge wave propagating poleward along the eastern boundary or equatorward along the western boundary. At this stage we can suspect that wave reflection may be an important component of interannual variability of climate in the Pacific Ocean. Consider the following thought experiment: Imagine an oceanic Kelvin wave propagating eastward and reaching the South American coast. Energy will have propagated eastward at the group speed of the Kelvin wave. We assume that the boundary is a rigid and vertical wall. There are two criteria for the reflected wave. First, it must have the same frequency as the incident Kelvin wave. Second, it must have a group velocity away from the boundary, in this case toward the west, in order to carry the reflected energy away from the boundary. In the reflection process, 𝜔 is the independent variable. However, from Figure 6.22, between periods of about a week to about 40 days (shaded region) there are no modes with a westward group speed. To establish what occurs in the 7–40 day period band, we need to return to fundamental wave theory. Written in terms of the longitudinal wave number, k, the dispersion relationships are 𝜔 (6.74a) kn=−1 = , for n = −1 ̃c 𝜔 𝛽 kn=0 = − , for n = 0 (6.74b) ̃c 𝜔 ( )]1∕2 [ 𝛽 𝛽 𝜔2 1 𝛽2 − − 4 (2n + 1) , ± kn≠0 = − 2𝜔 2 𝜔2 ̃c ̃c2 for n = 1, 2, 3, … (6.74c) We need to recall that, in forming the basic wave equations for the equatorial atmosphere and ocean, we

6.4 Large-Scale Low-Latitude Ocean Modes

The real and imaginary parts of k are, respectively: 𝛽 kr = − 2𝜔 √

(6.76a)

𝛽2 𝛽 𝜔2 − − 2 (6.76b) 2 ̃c 4𝜔 ̃c The condition for which k becomes a complex number is ki = ±

(√

n+1 − 2

(2n + 1)

(√ √ ) √ ) 𝜔 n n+1 n < √ < + , n≥1 2 2 2 ̃c𝛽

lower limit

upper limit

(6.77)

The upper limit of Eq. (6.77) corresponds to the minimum frequency of an IG for a given n. The lower limit corresponds to the maximum frequency of an ER, also for a particular n. The discussion above suggests that what happens at the eastern boundary of the incident Kelvin wave depends very much on frequency of the incident wave. A high-frequency Kelvin wave (periods < a week or so) may be reflected toward the west as IG waves of the same frequency as the incident wave. On the other hand, low-frequency Kelvin waves (periods > 30–40 days) may be reflected as large-scale Rossby waves of the same frequency. In these two cases, energy remains in the near-equatorial zone. However, in the frequency range between a week and 30–40 days, energy is ducted away from the equator by coastally trapped edge-waves. At a western boundary, the incident westward propagating Rossby waves can be reflected as Kelvin waves and short Rossby waves as each have a westward group speed. We can approach this process more formally. Figure 6.23 plots the real and imaginary parts of ka for n = 1 and 2 obtained from Eqs. (6.76a) and (6.76b), respectively. The black curve at the top of the figures represents the real values of the n = 1 and 2 high-frequency IG waves. The curve at the bottom of the figures represents low-frequency ER waves. Both of these waves can reflect at the eastern boundary as they possess westward group speeds, but only over a limited

Scaled frequency (ω)

3

n=1 IG

2

1 ER 0 –5

3 Scaled frequency (ω)

sought solutions of the form of Eq. (5.9) in Eq. (5.22) that possessed a longitudinal of the form exp(ikx). If k is real, then wave-like solutions would ensure the existence of a wave-form. If k is complex, then solutions would grow or decay in the longitudinal direction. The growing mode to the west is neglected as it violates the finiteness boundary condition. To determine the scales for which reflected modes are not possible, we seek the condition for when k becomes complex. This occurs when the radicand in Eq. (6.74c) becomes negative; i.e. when ) ( 𝛽 𝜔2 𝛽2 − ≤0 (6.75) − 4 (2n + 1) ̃c ̃c2 𝜔2

–4

–3 –2 –1 0 1 Scaled zonal wavenumber (ka) (a)

n=2

2

3

2

3

IG

2

1 ER 0 –5

–4

–3 –2 –1 0 1 Scaled zonal wavenumber (ka) (b)

Figure 6.23 Real and imaginary parts of the longitudinal wavenumber, ka, for (a) n = 1 and (b) n = 2. Real values of k represent the high-frequency IG and the low-frequency ER given by Eqs. (6.73, 6.73), respectively. Scaling of frequency and zonal scale following the same scaling procedure used in creating Figure 6.22 These modes possess westward group speed. The red contour denotes the zero limit of the radicand inequality of Eq. (6.75). However, within the red contour, where there are real values of wavenumber (black curve defined by Eq. (6.76a)), the wavenumber is complex, representing a westwardly exponentially decaying solution away from the boundary of the ocean. Within these limits, energy is ducted to higher latitudes as coastal Kelvin waves that transfer energy to higher latitudes.

frequency range. The third black line in each of the figures represents a real part of ka given by Eq. (6.76a). We note that it passes through the location of imaginary values of ka obtained from Eq. (6.76b). This means that in the intermediate zone between the IG and ER, complex solutions can exist, remembering that the solutions are proportional to exp(ikx) with an exponentially growing or decaying part and a waveform. Only the solution that decays to the west is permissible in order to satisfy finiteness of the solution toward the west. What happens to the energy from the incident wave at frequencies between the limits set out in Eq. (6.77)? In the simplest sense one can think of a build-up of mass at the eastern boundary which, in turn, produces an

155

6 Equatorial Waves in Simple Flows

along-coast pressure gradient and thus the generation of coastal Kelvin waves.VII

Dispersion curves for c = 1.4 m s–1 1

6.4.4

The Longwave Approximation

As in Eq. (6.25), we have assumed a motionless background state (U = 0). Emerging from this approximation are sets of westward-propagating non-dispersive waves, depicted in Figure 6.24 as straight solid lines, plus the Kelvin wave propagating to the east. Dashed lines indicate the omitted modes. Approximate group and phase

12.5

n=1

n=3

n=2 ER* K

0.5 MRG

25

n = –1

Period (days)

A much simpler view of the equatorial ocean tropics has been developed. Consider a spatially isolated and time-dependent forcing function similar to that appearing in Figure 6.12a. Fourier analysis in longitude and time would depict forcing across a wide spatial and temporal range. In a linear sense, each spatial–temporal mode would be expected to generate an array of normal modes according to Figure 6.22. Yet, the response illustrated in Figure 6.12b is relatively simple: a rapidly propagating non-dispersive eastward Kelvin wave and a slow westward-propagating pair of large-scale Rossby waves. The Kelvin wave has an eastward and rapid group speed to the east consistent with its steep slope, depicted in Figure 6.22. The sign of the group speed of the Rossby waves may be eastward or westward depending on scale. Large-scale waves (small k) have moderately fast westward group speeds. Smaller-scale waves (larger k) have eastward group speeds but their group speed (𝜕𝜔/𝜕k) diminishes with increasing k. Thus, for an imposed forcing, such as that shown in Figure 6.12a, the large-scale Rossby and Kelvin waves will transmit information to great distances whilst shorter-scale waves will remain in the region of the forcing. Given the smallness of the longitudinal group speed of an initially short Rossby wave (see Figure 6.4a) it has been presumed that such a mode cannot move far away from the region in which it was forced. The behavior of Rossby waves, discussed above, has allowed a substantial simplification of wave dynamics close to the equator. For both the atmosphere and the ocean a longwave approximation (ka small) has often been invoked. Specifically, it is assumed that 𝜕v/𝜕t = 0 in Eq. (5.8b) so that v becomes a diagnostic variable. This is similar to the development of the Kelvin wave dispersion relationship. Importantly, shorter Rossby waves, the inertial gravity waves, and the MRG do not satisfy these criteria. Only very long Rossby waves and the Kelvin wave do so. In this construct, the dispersion relationships of ER waves (6.25) take on a much simpler form: −𝛽k kc ≈− (6.78) 𝜔= 2 2n + 1 k + (2n + 1)𝛽∕c

Scaled frequency (day–1)

156

50 ER 0 –5

–4 –3 –2 –1 0 1 Scaled zonal wavenumber (ka)

2

Figure 6.24 Low-frequency detail of the dispersion diagram using the longwave approximation. Rossby waves are approximated as non-dispersive straight lines emanating from the origin as given by Eq. (6.75). Like the Kelvin wave (K), the Rossby modes (ER*) are rendered non-dispersive. The MRG disappears. These approximate Rossby waves have phase and group speeds that are −cpk ∕(2n + 1), where cpk is the phase or group speed of the Kelvin wave.

speeds of these non-dispersive Rossby waves are given by c cpL = cgL = − (6.79) 2n + 1 Alternatively, Eq. (6.79) can be derived directly by ′ assuming 𝜕v /𝜕t = 0 in Eq. (5.8b), which is tantamount to assuming that the zonal velocity is in geostrophic balance with the latitudinal variation of the height field. It is easy, then, to relate the phase speeds of these long Rossby modes with that of the Kelvin wave. If cpK is the phase speed of the Kelvin wave (n = −1), then cpLR (n = 1) = −cpK ∕3 cpLR (n = 2) = −cpK ∕5 cpLR (n = 3) = −cpK ∕7

(6.80)

and so on. That is, if cpK = 1.27 m s−1 , then the alongequator phase and group speeds of the n = 1, 2, and 3 equatorially trapped ocean Rossby waves will be −0.42, −0.25, and −0.19 m s−1 , respectively. The longwave approximation has simplified the dispersion relationships considerably. Non-dispersive waves, propagating both eastward and westward, have replaced the more complicated structures of Figure 6.22. Furthermore, they allow, at least to a first approximation, the behavior of equatorially trapped waves in a confined basin with lateral boundaries. The problem of the reflection of waves at boundaries is especially simplified. With the longwave approximation there are no frequency zones where reflection is not possible. For

6.5 Overview

a given frequency, there is always a long Rossby wave that can reflect at the eastern boundary and a Kelvin wave in the west. Also, there is no leakage out of the tropics by edge waves: in this approximate system they do not exist. The longwave approximation is used in many simple coupled ocean–atmosphere systems. The impact of the simplification on modeling interannual variability is not known. Similar approximations are commonly made for the atmosphere. However, there one needs to be even more careful. There are two issues that need to be considered. First, the longwave approximation eliminates the MRG family. Second, for the ocean we have made the assumption that U = 0. Yet, as we found out in Chapter 4, if U ≠ 0 then k can vary along a ray even as frequency is conserved. An interesting question is this: In a basic state where, for example, U = U(x), can the basic state induce an initially short Rossby wave to lengthen sufficiently, thus increasing the phase and group speeds and so promote an eventual remote response? In other words, are there circumstances where the longwave approximation is not appropriate in the low-latitude atmosphere?

long waves have group velocities that propagate westward whilst shorter waves propagate to the east. The higher-frequency MRG has both phase and group speeds to the east and, as the wavelength decreases, the mode becomes increasingly dispersive. At low frequencies it behaves like a Rossby wave with a group speed that is positive and to the east. • The form of all modes is dependent on the depth of the shallow fluid H. In general, the eigensolutions for small values of H are more closely trapped to the equator than larger values of H. Both the ER radius of deformation and the turning latitude go as 1∕4

•

6.5 Overview Using the Laplace shallow fluid system, modified for an equatorial 𝛽-plane, three basic families of waves were isolated, each with maxima at the equator or in its vicinity and decreasing amplitude poleward. These waves exist within a wave guide or potential well created by the rotation of the planet. These are low-frequency equatorial Rossby waves (termed ER waves), two sets of inertial gravity modes (IG) one with phase velocities to the east (EIG) and the other to the west (WIG), a Kelvin mode (K) propagating to the east, and a set of equatorial Rossby waves (ER) with westward phase velocities. In addition, there is a hybrid mode that spans the complete frequency range between the low-frequency Rossby waves and the high-frequency inertial-gravity modes. This mode is termed the MRG mode. Properties of these waves and the importance of parameters defining their structure (e.g. the depth of the shallow fluid, H) and the form, magnitude, and sign of the basic state are summarized below. • The Kelvin and the IG modes are non-dispersive and there is no differentiation between phase and group velocities, so that wave energy is advected identically by the group or phase velocities. The ER modes, on the other hand, are dispersive. All members of the ER family have westward phase velocities, but

•

•

•

H (see Eqs. (6.19) and (6.20)). Compared to R at H = 1000 m, the Rossby radius decreases by factors of 84, 56, 47, and 37% for H = 500, 100, 50, and 20 m, respectively. Thus, decreasing H increases the degree of trapping, essentially increasing the depth of the potential well. Clearly, the choice of H is important and not arbitrary. In general, waves are more latitudinally extensive in a westerly than an easterly basic state. Returning to the Rossby radius of deformation expression (6.19), the denominator (1 + kU∕𝜔d )1∕4 is smaller for westerlies than easterlies, i.e. for an ER, ka < 0, U> 0, and 𝜔d > 0, so the denominator decreases and R increases. In Section 6.4 very simple potential vorticity arguments were used to explain these varying degrees of equatorial trapping in different basic states. In essence, the background basic state changes the 𝛽-effect and essentially “reduces or increases the effective rotation rate of the planet,” thus widening or shrinking the latitudinal scale of the potential well. The impact of latitudinal shear on modal structure and latitudinal extent is also significant. We had anticipated from the kinematics of equatorial waves (Eq. (4.13c)) that this may be the case. In general, latitudinal shear tends to slightly increase the latitudinal extent of the modes. The group speeds of the low-frequency equatorial modes (ER and the westward MRG modes) are similar in magnitude to the background basic state. If this is the case, is it possible to have zero Doppler-shifted group speed that would create local maxima in wave energy density? Yet, at the same time, we note that there are variations in the background basic flow (e.g. U = U(x) . The question we need to approach is whether or not the changes in wavenumber occurring when a mode encounters a U(x) < 0 region (see Equation (6.13a)) make “accumulation” of wave energy (i.e. a convergence of group speed) more likely or less so. The normal modes of the equatorial upper ocean were also examined. The families of oceanic modes

157

158

6 Equatorial Waves in Simple Flows

were identical to those found in the atmosphere but with two major differences: the phase and group speeds of oceanic waves are vastly different, being an order of magnitude slower in the ocean than the atmosphere. This allowed the assumption that the atmosphere could be considered to be in continual equilibrium with the slowly evolving ocean. • As distinct from atmospheric waves, propagating ocean waves encounter coastal boundaries. At a western boundary, very high- or low-frequency eastwardly propagating Kelvin waves are reflected as inertial gravity waves or long Rossby waves, respectively. Modes with frequencies between about a week

and a month form coastally trapped waves that propagate poleward along the boundaries. Finally, it should be reiterated that the existence of a normal mode in the tropical atmosphere or ocean does not mean that it will occur. For example, the note middle-C is a normal mode of a musical instrument but unless one presses a key or draws a bow across the strings, the note will not be heard. Symphonies just do not start spontaneously! We have yet to determine processes that “ring” the normal modes of the tropical atmosphere. We have yet to determine the forcing functions or instabilities that produce these waves.

Notes I Hermite polynomials are a classical orthogonal

sequence named after the French mathematician Charles Hermite (1822–1901). In physics these functions give rise to the eigenstates of a quantum harmonic oscillator. They are formally defined in Gradshteyn et al. (2015). One of Hermite’s students was Henri Poincare, the philosopher, mathematician, and physicist described widely as a “polymath” or “renaissance man.” II Dispersion of the properties of a wave occurs when pure plane waves of different wavelengths have different propagation velocities, so that a wave packet of mixed wavelengths tends to spread out in space, as discussed in Section 4.2. A dispersion relationship relates the scale of a wave (e.g. wavelength or wavenumber) to its frequency, from which the characteristics of dispersion can be determined. III An eigenfrequency basically means a “characteristic” or “typical” frequency. “Eigen” comes from German meaning “own” or “typical” as in “da sist him eigen” or ... “that which is typical of it” …. Consider a vibrating rod fastened at both ends. Of an infinity of possible frequencies, there is only a select number of “eigenfrequencies” that are possible, given the size of the rod, its composition, and the boundary conditions, such as how the rod is held at each end or by the loading placed on the rod. IV The MRG, also referred to as the “Yanai” wave, was first observed in the stratosphere (Yanai and Maruyama 1966). Professor Michio Yanai (1934–2010) was a prominent Japanese meteorologist who spent the last 40 years of his

career at UCLA. Besides his fundamental work on tropical waves, he developed a systematic approach for estimating tropical heat sources and moisture sinks, and made fundamental contributions on the physical nature of the monsoons and the role of the Tibetan Plateau. V Resonant response. Forcing with a frequency close to an eigenfrequency of the system will tend to produce an oscillation of higher amplitude than a forcing with a frequency removed from an eigenfrequency. Such is a resonant response. An example of resonance known to oenophiles is the act of circling the rim of a wine glass with a damp finger. Friction will induce oscillations of the glass and if the frequency produced is close to an eigenfrequency of the glass, it will produce an ethereal ringing. Here is a pleasant experiment: drink a little wine and repeat the damp finger forcing. The frequency will change in accord with the new eigenfrequency of the wine glass as the wine diminishes. VI Sir William Thompson (Lord Kelvin, 1824–1907) was a British scientist born in Belfast, Ireland. His interests were wide-spanning thermodynamics, physics and engineering. He was knighted in 1866 and elevated to the peerage in 1892. His title comes from the River Kelvin in Glasgow where he held a faculty position for 50 years. VII The complete description of production of the poleward moving waves is quite complex and beyond the scope of this study. For further information we suggest the source references, e.g. Moore and Philander (1977).

159

7 Waves in Longitudinally and Vertically Varying Flows We have concluded that the sign and magnitude of the basic state and its latitudinal variations are important and will determine, for a given equivalent depth, the degree to which a mode will extend differentially to higher latitudes or to be more constrained about the equator. However, besides the existence of latitudinal shear, there are other large amplitude variations of the background state. Figures 1.11 and 1.12 indicate substantial vertical and longitudinal variations as well. The study of the kinematics of waves (Chapter 4) suggests that such variations of the basic state may also have an impact on the structure of equatorial modes. Equation (4.14a), for example, states that the longitudinal wavenumber k along a ray path is dependent on the longitudinal variability of zonal wind (i.e. U (x)). Furthermore, Eq. (4.14b) shows that the vertical wavenumber m is dependent on U (z). As group velocities are functions of wavenumber and the signal energy travels with the group velocity away from a source, it may be expected that transmission will change if U = U (x, z) varies along a ray. The aim of this chapter is to assess the impact of zonal and vertical variations of the background basic state on their modal structure. To assess whether the vertical shear (U = U (z)) and longitudinal variations (U = U (x)) of the background are important in modifying the characteristics of equatorial modes we will utilize the two basic Eqs. (5.38) and (5.39) derived for a fully stratified basic state. These two equations, representing the horizontal and the vertical structures, are coupled by a separation coefficient hn , the equivalent depth of a shallow fluid. It may be recalled that in deriving these two equations (i.e. Eqs. (5.38) and (5.39)) they are non-separable for a an arbitrary background state where U= U (x, z), To allow separation it was assumed that the background state in the stratified atmosphere was be motionless!

Although separation can now occur, the use of such a quiescent basic state may seem counterintuitive in to understand the impact of a background state where U= U (x, z).. But, as described in Section 4.1.2, we can utilize analytic solutions in a local frame of reference moving with the basic state. Then, with appropriate approximations we can determine the influence of the complex U (x, z) basic state on equatorial modes. However, we have already developed techniques that will allow a useful analysis by talking advantage of wave invariants that depend on the frequency and wavenumber of a mode. Invariance, as defined here, is a property of a wave that does not change during a transformation when observed relative to the motion of the basic flow. These were developed in Chapter 4 from kinematic wave theory. Using these invariants, solutions thus obtained can be related spatially to the local characteristics of the background flow. Subsequently, we can determine the modification of an equatorial wave as it moves through an inhomogeneous basic state. From these solutions, some rather exotic behavior of equatorial waves will emerge that help to explain the observed structures in the tropics and also some aspects of communication between the tropics and higher latitudes. The use of nonlinear and, thus, more complete models will be used to test the veracity of the simple kinematic theory in explaining near-equatorial dynamics. Alternatively, of course, we could choose to ignore analytic approaches, such as those developed in Chapter 5, and just use a four-dimensional and fully nonlinear general circulation model. We could then perform experiments setting forcing functions in different parts of the tropics and follow the response of the system. However, we choose, rather, to take the analytic route where possible as it explains more directly the physical nature of tropical modes and how they respond to forcing in different basic states.

Dynamics of the Tropical Atmosphere and Oceans, First Edition. Peter J. Webster. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

7 Waves in Longitudinally and Vertically Varying Flows

7.1 Horizontal and Vertical Coupling of Equatorial Modes

c~g (x,z) in longitude-heightplane ω1

7.1.1

Coupled Group Speeds cgx

We commence by determining relationships between horizontal and vertical group speeds. Away from an energy source region, where Sn = 0, and in an isothermal atmosphere, Eq. (5.38) becomes d 2 Zn dz where

2

c~g

cgz

c~g ω2

ka

m

160

+ m2 Zn = 0

(7.1) c~g

( m=±

𝜅 Ghn

)1∕2 −

1 2

c~g

(7.2)

4G from which we can compute a gravity wave speed: √ √ (7.3) cn = ghn = g𝜅∕(Gm2 + 1∕4G) where G(z) is defined in Eq. (5.28d). If the argument of the radical in Eq. (7.2) > 0, Eq. (7.1) will possess vertically propagating solutions. The positive root refers to upwards propagation and negative downwards. We note that m, the vertical wavenumber, is related inversely with 1∕2 hn , so that the smaller the equivalent depth, the larger m and, hence, the smaller the vertical scale. We now determine the full dispersion relationships in (k, m)-space for all equatorial modes. It is useful to relate the horizontal and vertical group speeds formally. The two-dimensional group velocity vector is 𝜕𝜔 𝜕𝜔 +k (7.4) 𝜕k ∼ 𝜕m where ∼i and k∼ (the latter not to be confused with k, the zonal wavenumber) are unit vectors pointing in the longitudinal and vertical directions, respectively. Thus, if we plot the dispersion relationships as functions of both ka and m, the group speed will be orthogonal to lines of constant 𝜔 and indicate its vertical and longitudinal signs. This relationship is illustrated in Figure 7.1 with the arrows denoting the sense of ̃cg (x, z). In the upper half, we present results for m > 0 with ̃cg (x, z) positive in longitude but negative in the vertical. However, from Eqs. (7.1) and (7.2) m may also be negative for wave-like solutions. For this case, the relationships are mirror images about the k-axis. So for m < 0, ̃cg (x, z) is then positive both in the vertical and longitude. ̃cg (x, z) = ∼icgx + k∼ cgz = ∼i

7.1.2 Coupled Dispersion Relationships and Group Speeds In Chapter 6, it was determined, for a given equivalent depth, the manner in which the eigenfrequency of a

ω2

ω1 Figure 7.1 The group velocity vector ̃ cg (x, z) as a function of k and m, the longitudinal and vertical wavenumbers from Eq. (7.3). The two scalar components, cgx and cgz , are functions of the longitudinal and vertical gradients of intrinsic frequency, 𝜔. The total group velocity vector, ̃cg (x, z) (red vectors) is the vector sum of these two components. The vertical wavenumber is based on m = 1, corresponding to about 30 km. Dispersion curves for m > 0 and m < 0 are shown, the latter being the mirror image about the ka-axis of the m > 0 case.

mode varied with zonal wavenumber ka. Figure 6.2 displayed dispersion relationships for a range of equivalent depths. Now, using Eq. (7.1) we can construct dispersion relationships as functions of k and m in terms of hn , thus encapsulating the full range of variability shown across Figure 6.2. 7.1.2.1

Equatorial Rossby (ER) Waves

Scaling Eq. (6.15) for the low-frequency Rossby wave (i.e. 𝜔 small) and using Eq. (7.2) leads to the dispersion relationship: 𝜔≈

(k2

−𝛽k + (2n + 1)M)

(7.5a)

where 2

M = 𝛽(m2 + 1∕(4G ))1∕2 ∕N

(7.5b)

Equation (7.5a) is very similar to the Rossby dispersion relationship (6.23) except, there, M was given by 𝛽/c. Here, hn is the separation coefficient (or equivalent depth) and n represents the latitudinal modal number. Further, N2 = g𝜅∕G. Equation (7.5b) can be approximated, to a good degree, to M ≈ 𝛽∕cn . Importantly, M now contains information about the vertical structure through the separation coefficient hn ,

7.1 Horizontal and Vertical Coupling of Equatorial Modes

cgx |ER =

𝜕𝜔 𝛽(k − (2n + 1)M) = 2 𝜕k (k + (2n + 1)M)2

0.05

0.100 200

5 10

2

15

1.5

20 m

hn(m)

0.125 400

6 3

600

25

0.150 1

30

5

6 3

5 10

200

2

15

1.5

20

800

m>0 1000 –14

Lz(km)

(a) ER dispersion (n = 1) 5

–12

0.175 –10

–8

–6 ka

–4

–2

0

600

0 2 3

800

4

2

1000 –14

(7.6a)

–12

and

–10

–8

–6 ka

m

400

–16 –12 –20 –24 –8 –28 –4 –2 –4

–2

25

1

30

6 3

5 10

2

15

1.5

20

Lz(km)

(b) ER cgx (m s–1) (n = 1)

hn(m)

which, through Eq. (7.2), is a function of m, the vertical wavenumber. The (m, k) dispersion relationship for the ER family is plotted in Figure 7.2a for n = 1. The relationship between equivalent depth hn , the vertical wavelength of a mode, Lz , and the vertical wavenumber, m (depicted on the ordinates), appears in Figure 5.4. For a given zonal wavenumber and vertical scale (or m), the group speeds change sign at some longitudinal wavenumber (ka). As the vertical scale becomes smaller (larger m) the transition takes place at increasingly smaller longitudinal scales. The transition appears as the bold dashed blue line. With the use of Eq. (7.5), two-dimensional group velocity vectors can also be deduced. The vectors point eastward or westward separated by the bold dashed blue line. The group velocity slopes downward for all ka signifying downward group velocities. However, the figure is drawn for m > 0. If m < 0, then the group velocities would be upward. The dispersion relationships would be a mirror image about the abscissa. Thus it would seem that group speed can be both upward and downward for all modes, depending on the location of the forcing. The longitudinal and vertical components of the group speed of the low-frequency Rossby waves can be derived quite easily from Eq. (7.4) to give

0

5

(7.6b)

Distributions of the ER zonal group speed were shown in Figure 6.4b. They are repeated here but in (k, m)-space to be consistent with the dispersion relationship (Figure 7.2a). These two components of the ER group speed are plotted in Figure 7.2b and c, respectively. The ER group speeds exhibit a strong dependence on zonal wavenumber. For the range of hn displayed, the group speeds of westward moving Rossby waves decrease monotonically as ka increases: that is, as the zonal scale decreases. For Rossby waves with a positive group speed, a maximum is reached between ka = 9 and 14. In the range of ka −6 to −3 and in the equivalent depth range from 300 to 1000 m, the value of cgx is almost independent of hn , but for hn < 200 m, cgx decreases rapidly. For very short waves (ka ≥ 11), cgx depends more strongly on hn than ka. An important observation is that for all ka > 2–3, the group speeds are similar (equal and opposite) to observed basic wind fields. This is especially so for small hn . The scale k0 at which the longitudinal group speed of the Rossby mode changes sign (i.e. cgx = 0) can be

–0.5

–1.0

200

–1.5 –2.0

400

–2.5 600

m

(2n + 1)𝛽 km∕(N M) 𝜕𝜔 = 𝜕m (k2 + (2n + 1)M)2

hn(m)

cgz |ER =

2

–3.0 –3.5

800

1

25

Lz(km)

(c) ER cgz (cm s–1) (n = 1) 3

30

–4.0 1000 –14

–12

–10

–8

–6

–4

–2

0

ka

Figure 7.2 (a) Dispersion curves (units day−1 ) for the n = 1 ER waves. Red arrows, which are orthogonal to the lines of constant frequency, denote the vertical-longitude group velocity vector from Eq. (7.3) and the bold blue dashed line separates the eastward and westward longitudinal group speeds. The vertical group speed is universally downward assuming m > 0. (b) ER longitudinal group speed (m s−1 ) and (c) ER vertical group speed (cm s−1 ). All quantities are plotted as functions of the longitudinal wavenumber ka and equivalent depth (hn , left ordinate) and vertical scale (Lz km) and vertical wavenumber (m) right ordinate.

161

(a) MRG dispersion: n = 0 (day–1) 0.2

5

(7.7) 0.3

200

and cgz |MRG = −

mc3n 2N2

( k+

k2 cn + 2𝛽 (c2n k2 + 4cn 𝛽)1∕2

) (7.9b)

1.5

20 25

0.6 800

m>0

1000 –5

–4

0.7 0.8 –3

–2

–1

0

1

1

30

6 3

5 10

2

15

1.5

20

2

ka 5

10 15 20

200

25 15

30 35

600

20

800 1000

m

400

40 45

25

–5

–4

25

Lz(km)

(b) MRG cgx (m s–1)

30

–3

50

–2

–1

0

1

30

6 3

5 10

2

15

1.5

20

55 60 1 2

ka (c) MRG cgz (cm s–1) 5 –1 200

–2 –4

m

Substituting cn from Eq. (7.3) for c in Eq. (6.31) leads to the more general dispersion relationship for the mixed Rossby-gravity (MRG) wave: [ ] 1 1 cn k + (c2n k2 + 4cn 𝛽) ∕2 (7.8) 𝜔= 2 Figure 7.3a presents eigenfrequencies as functions of ka and m. Frequency increases from ka = −5 to +2 and also with increasing vertical scale. All group vectors are in the same direction indicating eastward and downward group speed that increases with decreasing m and increasing ka. Using Eq. (7.8) we can write the two components of group velocity for the MRG as ) ( c cn k (7.9a) cgx |MRG = n 1 + 2 2 (cn k2 + 4cn 𝛽)1∕2

15

0.5

400

Mixed Rossby-Gravity (MRG) Wave

2

0.4 600

–3

7.1.2.2

5 10

m

hn(m)

400

hn(m)

and is plotted in Figu7.2a and b as the bold blue dashed line. The ER vertical group speeds (cm s−1 : Figure 7.2c) are universally negative or downward for m > 0. From Eq. (7.4), we note that the vertical phase speed is positive for m > 0 (noting ka < 0) and, thus, opposite to the group speed; i.e. for negative m, the phase speed would be downward and the group speed would be positive. In general, cgz is relatively independent of vertical scale except in the range ka = −2 to −6, where there is a rapid increase of group speed as hn becomes larger or as m decreases. We note, however, that in this zonal scale range, the vertical group velocity for hn between 50 and 100 m are 25 to 50% of the values for 400 m. From Figure 7.2b, we can also see that the zonal group speeds follow the same pattern. Thus, a signal will remain relatively closer to its source region, both horizontally and vertically, in a small equivalent depth fluid compared to a larger choice of hn . There is one other issue that bears mentioning. The development of the vertical structure above is for a quiescent atmosphere. In Section 7.1 we noted that as long as m2 > 0, we will have vertically propagating waves. However, for a non-resting atmosphere flow (U ≠ 0), U will enter an expression for m2 placing further restrictions on whether or not a solution will be propagating or exponentially growing or decaying in the vertical. For certain values of U, vertical propagation of a wave will not be possible.

6 3

–5 –6

600

25

Lz(km)

deduced from Eq. (7.6a) as ( )1∕2 𝛽(2n + 1) k0 = = ((2n + 1)M)1∕2 cn

Lz(km)

7 Waves in Longitudinally and Vertically Varying Flows

hn(m)

162

–8 –10

800

1

30

–15 –18 1000

–5

–4

–3

–2

–1

0

1

2

ka

Figure 7.3 Same as Figure 7.2 except for the mixed-Rossby gravity (MRG) wave (n = 0). (a) Dispersion curves and group velocity vectors (red arrows), (b) longitudinal group speed (m s−1 ) showing universally positive values, and (c) the vertical group speed (cm s−1 ).

The zonal group speed, cgx (Figure 7.3b), increases almost monotonically with hn for all ka. Figure 7.3c depicts the MRG vertical group speeds as a function of hn (or m) and ka. Universally, the vertical group speed (Figure 7.4c) is negative for m > 0 in the opposite sense to its phase velocity. For m < 0, the group speed will

7.2 Coupled Free and Forced Solutions of the Vertical Structure Equation

7.1.2.3

(a) Kelvin Wave dispersion (day–1) 5

m>0

5 10

3

15 20

1.5 0.4

600

25 0.6

800

0.8 1.0 1000 –5

–4

–3

–2

–1

0

1

30

3

5 10

Lz(km)

0.2

400

m

hn(m)

200

1.2 1

2

ka (b) cgx (m s–1): Kelvin wave 15 25 35 45 55

400

15 1.5

65

25

75

600

85

800 1000

20

m

hn(m)

200

1

30

3

5 10

Lz(km)

5

95 –5

–4

–3

–2

–1 ka

0

1

2

(c) cgz (cm s–1): Kelvin wave 5 –5

15

400

–15 –20 –25 –30 –35 –40 –45

600 800

1.5

20

m

hn(m)

–10

25 1

Lz(km)

200

30

1000 –5 –4

–3

–2

–1 ka

0

1

Kelvin Wave

Finally, the coupled dispersion relationship for the Kelvin wave is √ (7.10) 𝜔 = kcn = k g𝜅∕(Gm2 + 1∕4G)

2

Figure 7.4 Same as Figure 7.2 except for the Kelvin wave (n = −1). (a) Dispersion curves and group velocity vectors and (b) longitudinal group speed (m s−1 ). Note that for a given hn , the group speed is constant and equal to the Kelvin wave phase speed for that hn indicating its non-dispersive nature. (c) Vertical group speed (cm s−1 ) which varies as a function hn , indicating a dispersive nature in the vertical.

be universally upward. In general, cgz increases with decreasing ka and with increasing hn (small m). In a similar sense to the ER mode, the greatest change in the vertical group velocity occurs as hn becomes smaller and m increases.

Figure 7.4a shows the Kelvin wave dispersion relationship as a function of m and k. For a given vertical scale, the frequency increases almost linearly (approximately as 1/m) but with frequencies becoming extremely small for large m (small vertical wavelength, small hn ). The longitudinal phase and group speeds remain non-dispersive as 𝜔/k and 𝜕𝜔/𝜕k are not functions of k and, for a given m or hn , the longitudinal phase and group speeds are equal (Figure 7.4b) and appear as horizontal lines parallel to the ka-axis. We also note in Eq. (7.10) that 𝜔 = 𝜔(m) so that both cpz (𝜔/m) and cgz (𝜕𝜔/𝜕m) are functions of m and are, thus, vertically dispersive so that the group speed components are not equal. Figure 7.4c shows a plot of the Kelvin wave vertical group speed as a function of m and k in an isothermal atmosphere. For m > 0 (i.e. positive phase speed), the vertical group speed is negative in the same manner we have found for the ER and MRG modes. This may be shown simply by differentiating Eq. (7.10) with respect to m, leading to )1∕2 ( √ 1 (7.11) cgz = kG g𝜅∕ Gm2 + 4G Using Eq. (7.3), the definition of the vertical wavenumber for propagating solutions, and Eq. (7.2) leads to the following general relationship: 3∕2 √ √ g kGhn 1 𝜅 cgz = ∓ − (7.12) 2 𝜅 Ghn 4G which states that for m > 0 the vertical group speed will be negative, opposite to the sign of the phase speed. In contrast, when m < 0 (downward phase speed), the group speed is upward.

7.2 Coupled Free and Forced Solutions of the Vertical Structure Equation In Section 5.3 we laid the basis for exploring the interdependency of the horizontal and vertical structure of equatorial waves for both free and forced systems. There are two characteristics we need to be reminded of: (i) For the free system the frequency 𝜔 is the eigenvalue for sets of k and H (or hn in the latitudinal-vertical system) relative to the boundary conditions of the system. The frequencies (the eigenfrequencies) represent the possible modes of oscillation of the

163

7 Waves in Longitudinally and Vertically Varying Flows

system, which may or may not be excited by some forcing function. (ii) In the forced system, the frequency 𝜔 and longitudinal scale ka are given by the characteristics of the forcing function. Then the equivalent depth, hn , is the eigenvalue relative to the particular imposed forcing. First, we will consider the free solutions of the vertical structure equation. This will be followed by considering the response of the system to a specific forcing function.

211

233

253

273

293 °K

30 T(z) = constant G(z) = constant height (km)

164

20

10

7.2.1

Free Solutions

Consider solutions to the unforced vertical structure equation, Eq. (5.38) or, identically, Eq. (7.1), where in Eq. (5.33) Q̇ = 0 so that in Eq. (5.38) Sn (z) = 0. With these simplifications, Eq. (7.1) has non-trivial solutions for specific values of hn . The vertical structure equation is second-order and thus requires two boundary conditions to close the system. These solutions will provide families of possible of eigenfrequencies and eigensolutions that may or may not be excited by a given forcing. We consider an isothermal atmosphere, which is the simplest prototype of the tropical atmosphere. However, to show that the situation is not much more complicated for other vertical temperature structures, we also consider modes in a constant lapse rate atmosphere. These two simplifications of the vertical structure of the real tropical atmosphere are shown in Figure 7.5 relative to the climatological tropospheric atmosphere. 7.2.1.1

Isothermal Atmosphere

In an isothermal atmosphere, the scale height G is constant and 𝜀 (the vertical derivative of G) is zero. These approximations allow a simpler version of the vertical structure equation. Setting Sn = 0 in Eq. (5.38) leads to ( ) d 2 Zn 1 𝜅G 1 + − (7.13) Zn = 0 2 hn 4 dz2 G where G = RT∕g. Two boundary conditions are required: (i) a condition at z = 0 and (ii) a finiteness condition at z → ∞. The lower boundary condition requires that the normally directed velocity at the lower ̃ = 0. Generally, w = dhB /dt, boundary is zero, i.e. n∼ ⋅ V where hB is the height of the bottom surface, so that 𝜕h 𝜕h ̃ •∇hb (x, y) = 1 (U + u′ ) B ≈ U B wz=0 (x, y) = V a 𝜕x a 𝜕x (7.14) where it has been assumed that the background flow U is zonal and the system is linear as (U ≫ u′ ). The

T(z) = az G(z) = bz+c 0

–60

–40

–20 T (°C)

0

20

30 °C

Figure 7.5 Vertical structure of the tropical atmosphere for isothermal (T = constant, which corresponds to a constant G) and a constant lapse rate (T(z) = az + b corresponding to a linearly varying G). These are compared with a simplified climatological T(z).

V0 h(λ,ϕ)

n

n V0 = 0 z=0

Figure 7.6 Schematic of the lower boundary condition with a ̃ is the background finite lower boundary variation (hb (𝜆, 𝜙)). V 0 basic state and ̃ n is the outward-directed normal unit vector. The ̃ = 0. This condition is kinematic boundary states that n•V 0 expressed in Eq. (7.17).

assumption of only a zonal basic state is not necessary as Eq. (7.14) is easily expanded to allow, in addition, a meridional component in the basic state. Second-order terms have been neglected on the right-hand side of Eq. (7.14) so that if U = 0, then wz = 0 is also zero. Thus, a linear mode is unaware of undulations of the bottom surface in a quiescent atmosphere, to the assumed degree of approximation. The general z = 0 boundary condition is shown schematically in Figure 7.6. We can derive a formal lower boundary condition for w = 0 at z = 0 assuming that hB = 0 and Q(z = 0) = 0. Furthermore, an isothermal atmosphere implies dG∕dz = 0. Then, we can rewrite Eqs. (5.31c) and ⁀ 𝜕p (5.31e) as 𝜕z − 1 p⁀ + g𝜌⁀ = 0 and − i𝜔p⁀ + i𝜔𝛾gG𝜌⁀ = 0 2G which combined through elimination of 𝜌⁀ gives ( 𝜕p⁀ 𝛾)⁀ 1− p + 𝛾G =0 (7.15) 2 𝜕z

7.2 Coupled Free and Forced Solutions of the Vertical Structure Equation

vertical wavenumber m

Eliminating u⁀ from Eqs. (5.31a) and (5.31b) gives ( ) ⁀ 𝜕p 𝜕v⁀ 𝜕 k [𝜔2 − (𝛽y)2 ] = i𝜔 − 𝛽y (7.16) 𝜕z 𝜕y 𝜔 𝜕z Using Eqs. (7.15) and (7.16) we then obtain ( 𝛾)⁀ dv⁀ 1 1− v =0 + dz 𝛾G 2 which using the definitions 𝜅 = (𝛾 − 1)/𝛾 and with ∑ v⁀(y, z) = n (y) Zn (z) from Eq. (5.35) we finally obtain ) ( dZn 1 𝜅 Zn = 0| z=0 1− (7.17) − dz 2𝜅 G Equation (7.17) is a kinematic boundary condition specifying that the component of fluid velocity perpendicular to a solid boundary must vanish at the boundary itself, such as would occur if there is no flow through the boundary. Here it refers to the condition at a rigid, non-permeable surface. If the boundary is a fluid surface, the condition applies to the vector difference of velocities across the interface. A second boundary condition comes from the necessity of solutions remaining finite at great heights above the heat source. We will incorporate this particular condition as we examine the solutions. Equation (7.17) has two sets of solutions, each depending on the sign of the coefficient of the Zn (z) term. We will consider each in turn. First Solution Assume that hn is real and positive and

hn > 4𝜅G. Then, in Eq. (7.13), 𝜅G∕hn > 1∕4, providing exponential solutions of the type: √ √ ⎛ ⎞ ⎛ ⎞ 𝜅G 1 1 𝜅G ⎟ z⎟ + B exp ⎜− z − − Zn (z) = A exp ⎜ ⎜ 4 ⎜ hn ⎟ 4 hn ⎟ ⎝ ⎠ ⎝ ⎠ (7.18)

where A and B are constants. For Zn (z) to be finite where z → ∞, A must be zero. Using the lower boundary condition (7.17) for hB = 0, we find one allowable value of the equivalent depth, which we label as hL . This is hL =

𝛾RT G = 𝛾G = 1−𝜅 g

(7.19)

where 𝛾 = 1/(1 − 𝜅). The solution to Eq. (7.13) then becomes ) ( 2−𝛾 z (7.20) Z(z) ∝ exp − 2𝛾 This is the Lamb wave1 possessing an e-folding amplitude (the height at which the amplitude decreases by a factor of 1/e) of about 12–15 km. We note that 1 Lamb (1917). Also see Bretherton (1966) and Forbes et al. (1999).

Oscillatory

hm

Lamb

hL

m=0 40 γG γG 49

h

Figure 7.7 The equivalent depth, hn , for the Lamb wave and oscillatory waves in the vertical as a function of vertical wave number, m. Large values of hn are associated with long vertical wavenumbers whilst smaller values correspond to shorter wavelengths, consistent with Eq. (7.2a).

√ ghL = 𝛾RT corresponding to the speed of sound in an isothermal atmosphere. The Lamb wave is a large-scale compressional wave moving at the speed of sound. It has little physical significance in the troposphere and stratosphere and appears to be more important in the thermosphere than in the troposphere. In weather and climate modeling, the Lamb wave is normally removed by an “anelastic approximation” developed by Ogura and Phillips (1962), as it is the source of numerical instability. In essence, all sound waves are eliminated from the system. Figure 7.7 shows the equivalent depth of the Lamb wave, which is very large and of order 9000 m. √

Second Solution If hn is real and positive but hn < 4𝜅G,

so that 𝜅G∕hn < 1∕4, solutions of the vertical structure equation become oscillatory with the form: Z(z) = C exp(imz) + D exp(−imz) where m, the vertical wavenumber, is given by √ 𝜅G 1 − >0 m= hm 4

(7.21)

(7.22)

and C and D are constants yet to be determined. Both of these solutions are bounded as z → ∞. In the forced problem, one normally applies a “radiation condition” well away from a source region that permits only a wave with a particular sign of propagation. Only allowing outgoing waves, for example, away from the energy source, ensures that spurious energy sources will be excluded. For the free (unforced) case, we can determine, at best, the ratio between the coefficients of Eq. (7.21). To do this we rewrite Eq. (7.21) without loss

165

7 Waves in Longitudinally and Vertically Varying Flows

of generality using trigonometric functions. This transformation allows incorporation of the lower boundary condition (7.17). Equation (7.21) can be rewritten as Z(z) = E cos mz + F sin mz.

(7.23)

where E and F are constants. Between Eqs. (7.17) and (7.23) we find ( ) 1 G mE = F (7.24) − 2 hm Then, between Eqs. (7.19) and (7.22), we obtain 𝜅 10 hm = hL ( ) = hL ( ) 𝛾 m2 + 1∕4 49 m2 + 1∕4

Constant Lapse Rate Atmosphere

The second vertical profile shown in Figure 7.5 represents a constant lapse rate atmosphere. The solutions of the vertical structure equation for this atmosphere are developed in Appendix F. With T z = constant, G in Eq. (7.13) takes the form G = az + b, which transforms the simple wave equation to one with a more complicated structure. Equation (7.13) becomes ( ) 1 1 d2Z + (az + b) − Z=0 (7.26) gh 4 dz2 The solutions are still oscillatory but with a vertically dependent wavelength and, under certain conditions, exponential. The oscillatory solutions are not sinusoidal but Airy functions, which are wave-like for (𝜂 < 0) but exponential for (𝜂 > 0). Here, 𝜂 is a linear function of z defined in Eq. (F.5). The Airy functions are shown in Figure 7.8.I

(η)

1.0 0.5

(η)

0 –0.5 –10

(7.25)

noting we have an expression for hL already in Eq. (7.19). The values of hL and hm are compared in Figure 7.7. The Lamb wave equivalent depth is far larger than that of the oscillatory solutions. The vertical wavenumber m is any real number from 0 to ∞. When m = 0, hn possesses a maximum but as m increases (smaller vertical scales) hn asymptotes toward zero. This finding is consistent with Figure 7.2a. It is interesting to see if the modes described in Section 7.1 are oscillatory in the troposphere. Equation (7.22) remains positive and the solutions oscillatory for hn < 9000 m, after which the mode is so deep the atmosphere is essentially equivalent barotropic. An equivalent barotropic system is one in which temperature gradients exist, but are parallel to height gradients on a constant pressure surface. In such systems, height contours and isotherms are parallel everywhere and winds do not change direction with height. Here, for hn > 9000 m, the system is so deep that variations in the vertical through the troposphere are almost non-existent. 7.2.1.2

1.5 Airy Function amplitude

166

–8

–6

–4

η

–2

0

+2

Figure 7.8 Airy functions B(𝜂) and A(𝜂), which are the eigenfunctions of Eq. (7.23). Both functions are oscillatory for 𝜂 < 0 but exponentially growing or decaying (respectively) for 𝜂 > 0.

The same question can be asked about whether modes in the troposphere in a constant lapse rate atmosphere are oscillatory or external. Consider an atmosphere at Z = 0 (i.e. z = 0), the temperature is 300 K and at Z = 3 (z ≈ 30 km) T = 190 K, so that the coefficients of the stability (F.1) are a = −3.46 × 10−3 and b = 1.6× 10−2 so that 𝜂 in (F.2) is negative and the solutions are oscillatory over a wide range of hn . Thus, in a constant lapse-rate atmosphere modes will propagate through the troposphere. Note, though, that in a region of reverse temperature gradient, it is possible that the modes become external with reflection and absorption. 7.2.1.3

Construction of Realistic Temperature Profiles

One of the benefits of having solutions for both an isothermal and a constant lapse rate atmosphere is that one can build a reasonable vertical structure by using layers represented by one of the two lapse rates. After matching boundary conditions at the interfaces of the layers, analytic solutions are possible, although more conveniently obtained numerically.2 7.2.2 Forced Motions in an Isothermal Atmosphere In the linear forced problem, the forcing itself sets the spatial scales of the response (k, n) and its frequency, 𝜔. The eigenvalue is then hn , which is a function of k, n, and 𝜔 as expressed in Eq. (6.15). Solving for cn , we obtain, for U = 0: √ cn = ghn √ (2n + 1) ± (2n + 1)2 + 4k𝜔(1 + k𝜔∕𝛽)∕𝛽 = 2k(1 + k𝜔∕𝛽)∕𝜔 (7.27) 2 Geller (1968).

7.2 Coupled Free and Forced Solutions of the Vertical Structure Equation

(a) Heating function z

(b) External solutions z

(c) Oscillatory solutions

hn > 4κG

z

hn < 4κG Zn = An e–imz

z = z0

–κ2J0δ(z–z0)/G J0

z = z0

Zn = Cne–φz z = z0 Zn = Dne

Zn = Cn e–imz

+φz

Zn = Dne+imz

surface forcing z=0

Jn(z)

z=0

Zn(z)

z=0

Zn(z)

Figure 7.9 Characteristics of the forced problem in an isothermal atmosphere. In the forced problem, the spatial and vertical functions as well as the frequency are defined by the forcing function. The equivalent depth, hn , is the eigenvalue. (a) The heating distribution defined as a 𝛿-function of magnitude J0 at z = z0 . (b) The solutions for hn > 4𝜅G, which produces external (exponentially decaying) solutions about the z = z0 (red curves). The blue curve indicates the form of the external solution if the heating function was located at z = 0. (c) Solutions for hn < 4𝜅G, which are oscillatory. There are two oscillatory solutions below the heating function, satisfying the lower boundary (7.17) and one above that possesses the property of upward vertical energy transfer.

For each k, n, and 𝜔 set, there are two solutions for cn and thus for hn . We choose the positive root in order to ensure finiteness as z → ∞. Here, we will consider only the simplest problem where forcing of a specific frequency and horizontal scale is located in an infinitely thin layer at z = z0 . The governing vertical Eq. (5.39) written for an isothermal atmosphere (i.e. 𝜀 = 0, G = constant) is ( ) d 2 Zn S 1 𝜅G 1 + 2 − (7.28) Zn = −𝜅 2 n 2 hn 4 dz G G The heating is specified as a Dirac 𝛿-delta function (introduced in Section 4.1.1.1) of amplitude J 0 at level z = z0, The amplitude of the forcing is zero elsewhere. The heating distribution is plotted in Figure 7.9a. For such a system Eq. (7.28) becomes ( ) d 2 Zn 𝜅G 1 1 + 2 − Zn hn 4 dz2 G 𝜅2 = − J0 𝛿(z − z0 )z=z0 at z = z0 (7.29a) G and ( ) d 2 Zn 1 𝜅G 1 + 2 − Zn hn 4 dz2 G z>z = 0|z z0 and z < z0 (7.29b) Similar to the free problem, we note that Eq. (7.28) possesses two classes of solution depending on the sign of the coefficient of Zn . In Eq. (7.29a), the heating, of amplitude J 0 , forces the system at some prescribed k, n, and 𝜔. The heating distribution is not particularly physical but it is the simplest that will allow us to set up a forced wave problem in a stratified field.

The first two boundary conditions are the same as in the free problem: (I) Zn (z) is finite as z → ∞.

(7.30a)

(II) At z = 0, w = 0 as expressed in Eq. (7.14). (7.30b) We also have solutions above and below the heat source at z = z0 and need two more boundary conditions. Stipulating that Zn is continuous through z = z0 we obtain a third boundary condition: z +

(III) [Zn ]z00 − = 0 or

Zn (z0 +) = Zn (z0 −) (7.30c)

To obtain a fourth condition, we need a condition on the vertical derivative of Zn . This can be obtained by integrating the governing equation at z = z0 in the vertical between z0 − to z0 + immediately above and below z = z0 , respectively. That is: ( ) z0 + z0 + 2 d Zn 1 𝜅G 1 dz + 2 − Z dz ∫z0 − dz2 hn 4 ∫z0 − n G z0 + 𝜅2 = − J0 𝛿(z − z0 )dz G ∫z0 − Now making use of the condition (7.30c), the second term disappears. Making use of a property of the Dirac function that z0 +

∫z0 −

𝛿(z − z0 )dz = 1

leads to (IV)

dZn dZn 𝜅2 |z=z0 + − |z=z0 − = − J0 dz dz G

(7.30d)

167

168

7 Waves in Longitudinally and Vertically Varying Flows

providing the fourth boundary condition. This is a dynamic boundary condition between the atmosphere above the heat source and that below, which states that the pressure across an internal boundary (say at the interface between two fluids or at a free surface) must be continuous. We now have four boundary conditions, Eqs. (7.30a) to (7.30d). In addition we have an equation representing the fluid at z = z0 expressed in Eq. (7.29a) and an equation for the fluid above and below the level of the heat source, Eq. (7.29b). Within each of these equations there are two regimes depending on the sign of the coefficient of Zn . 7.2.2.1

Case 1: External Solutions

In Eq. (7.29b), (𝜅G∕hn − 1∕4) < 0 for hn > 4𝜅G or hn < 0. As the coefficient of Zn is negative for this case, solutions to the vertical structure equations above and below z0 are: z > z0 ∶

Zn (z) = An exp(−𝜗z) + Bn exp(+𝜗z) (7.31a)

z < z0 ∶

Zn (z) = Cn exp(−𝜗z) + Dn exp(+𝜗z) (7.31b)

where An , Bn , C n , and Dn are constants and √ 1 1 𝜅G 𝜗= − hn G 4 (7.32) For finiteness of Zn at large z, Eq. (7.30a) insists that in Eq. (7.31a) Bn = 0. Then, using the continuity of Zn through z0 , condition (7.30a), leads to An exp(−𝜗z) = Cn exp(−𝜗z) + Dn exp(+𝜗z) (7.33a) With the kinematic lower boundary condition (7.30b) we obtain ) ) ( ( 𝜅 𝜅 − 𝜗 + b Cn + 𝜗 − b Dn = 0 (7.33b) G G where b = 1 − 1/2𝜅. Finally, we apply the fourth boundary condition, the dynamic condition, expressed in Eq. (7.30d). Equation (7.33a) then becomes, at z = z0 : − An exp(−𝜗z0 ) + Cn exp(−𝜗z0 ) − Dn exp(+𝜗z0 ) 𝜅2

J0 (7.33c) 𝜗G Equations (7.33a) to (7.33c) constitute three algebraic equations in An , C n , and Dn . Solving for An in Eq. (7.33a) =−

and substituting the solution into Eq. (7.33c) we obtain 𝜅2

J0 exp(−𝜗z0 ) (7.34a) 2G𝜗 From Eq. (7.33b) we obtain C n as a function of Dn : ) ( G𝜗 − b Dn Cn = G𝜗 + b which using Eq. (7.31b) gives the following expression for C n : ( ) 2 G𝜗 − b 𝜅 J0 exp(−𝜗z0 ) Cn = − (7.34b) G𝜗 + b 2G𝜗 Then, using Eqs. (7.34a) and (7.34b): Dn = −

An = Cn + Dn exp(2G𝜗z0 ) ( ) 𝜅2 G𝜗 − b =− J0 exp(−𝜗z0 ) + exp(G𝜗z0 ) 2G𝜗 G𝜗 + b (7.34c) Substituting Eqs. (7.34a) to (7.34c) in Eqs. (7.31a) and (7.31b), and remembering that Bn = 0, we can reconstruct the vertical structure of the forced response. Figure 7.9b depicts these solutions. These are external solutions (exponential) where the response stays close to the source. The solution represents the vertical structure of the response to forcing (J 0 (z)) of the prescribed scale (k and n) and frequency 𝜔. These solutions correspond to very large values of hn (>9000 m). If the forcing were at the surface, there would be just one solution that decays with height, as shown by the blue curve in Figure 7.9b. 7.2.2.2

Case 2: Oscillatory Solutions

If (𝜅G∕hn − 1∕4) > 0 in Eq. (7.29b) there are oscillatory solutions. By defining a vertical wavenumber m as ( ) 1 𝜅G 1 m2 = 2 − (7.35) hn 4 G we have the following solutions: z > z0 ∶ Zn (z) = An exp(−imz) + Bn exp(imz) (7.36a) z < z0 ∶ Zn (z) = Cn exp(−imz) + Dn exp(imz) (7.36b) Here, we choose the positive root of m. As in Case 1, we use the boundary conditions to determine the four constants. The lower boundary condition (7.30b) leads to ( ) ) ( 𝜅 𝜅 − im + b Cn + im − b Dn = 0 (7.37a) G G

7.3 Wave Characteristics in a Zonally Varying Basic State U = U(x)

The condition of a continuous Zn (z) across z = z0 (7.30c) gives

Fn (𝜉) =

An exp(−imz0 ) + Bn exp(imz0 ) = Cn exp(−imz0 ) + Dn exp(imz0 )

(7.37b)

The condition regarding derivatives across z = z0 (7.30d) provides the third relationship: − imAn exp(−imz0 ) + imBn exp(imz0 ) = −imC n exp(−imz0 ) + imDn exp(imz0 ) −

𝜅

2

J0 G (7.37c)

The fourth boundary condition stipulates that, at great heights, energy will flow away from the energy source located at z = z0 . That is, there must be an upward energy flux away from the energy source as z → ∞. This means that we need to consider the two branches of the solution for z > z0 as expressed in Eq. (7.36a), noting that one must correspond to the boundary condition whereas the second will not. We simplify the solution as exp(imz) with m < 0 and allow this to denote the An branch. We let m > 0 denote the Bn branch. The simplest way to find those solutions with the correct sign of energy propagation is to calculate the direction of the pressure work, which is the work done by one volume of a gas at an anomalously higher pressure on a volume of gas at an anomalously lower pressure. Pressure work can be expressed as p′n w′n and will determine the direction of energy flux. If p′n w′n > 0 and z > z0 , this means that work is being done by the fluid below on the fluid above, thus defining the direction of energy flux. Therefore, if we find an expression for p′n w′n we will have created an appropriate radiation condition. Simply, those modes that do not support the condition that p′n w′n > 0 will be ignored. To obtain expressions for pressure and vertical velocity, we need to return to the basic dynamics of equatorially trapped modes discussed in Chapter 6. We write the full internal wave solution as ( 2) 𝜉 Hn (𝜉) (7.38) v′n = En exp i(kx + mz + 𝜔t) exp − 2 where E n is a constant and 𝜉 = (𝛽 2 /ghn )1/4 y. Using Eq. (6.32) and the eigensolutions (7.21), we can develop the following expressions for p′n and w′n . These are ( ) −ighn d 1∕4 2 (7.39) ei(kx+mz−𝜔t) e(−𝜉 ∕2) Fn (𝜉) p′n = En 𝜔 and

( ( 𝛾 )) w′n = En m𝛾G − i 1 − 2 ( ) ihn d 1∕4 i(kx+mz−𝜔t) (−𝜉 2 ∕2) e Fn (𝜉) e 𝛾 −1

where d = 𝛽 2 /ghn and

(7.40)

Hn+1 (𝜉) nHn−1 (𝜉) − 1 + kcn ∕𝜔 1 − kcn ∕𝜔

(7.41)

From Eqs. (7.39) and (7.40) we obtain the relationship ( ( 𝛾 )) ′ 𝜔 m𝛾G − i 1 − pn w′n = − (7.42) g(𝛾 − 1) 2 The product of Eqs. (7.36a and 7.36b) and (7.37a and 7.37b) defines vertical pressure work. Averaging over a wavelength we obtain p′n w′n = −

𝜔m𝛾G ′2 p g(𝛾 − 1) n

(7.43)

We can now seek conditions for which the pressure work p′n w′n averaged over a wavelength is positive. As we have defined frequency 𝜔 to be positive, a positive pressure work can only occur if m < 0 as G > 0 and 𝛾 >> 1. Therefore, in Eq. (7.36a) Bn must be zero as otherwise there would be a downward flux of energy toward the heat source. For z < z0 , there are also two waves, one propagating upwards and the other downwards. Above the heat source (z > z0 ), only one wave exists with a downward phase speed but an upward group speed and energy flux. Setting Bn to zero in Eqs. (7.36a) and (7.37a) to (7.37c) allows the constants An , C n , and Dn to be calculated: Dn = −

i𝜅 2

J0 exp(imz0 ) 2mG (im − 𝜅b∕G) i𝜅 2

J0 exp(−imz0 ) (im + 𝜅b∕G) 2mG i𝜅 2 J0 exp(−imz0 ) An = − 2mG ( ) (im − 𝜅b∕G) + exp(2imz0 ) × (im + 𝜅b∕G) Cn = −

(7.44)

The form of the solutions is shown in Figure 7.9c. Below the heat source there are two waves, one moving downwards and the other upwards, taking into account the lower boundary condition. Above the heat source there is only one wave with an upward group velocity. Note that if the heat source were at z = 0, there would just be one wave with a positive (upwards) group velocity.

7.3 Wave Characteristics in a Zonally Varying Basic State U = U(x) We now consider first the impact of longitudinal stretching deformation (U(x)) on equatorially trapped ER and MRG modes. We will then use a numerical nonlinear

169

170

7 Waves in Longitudinally and Vertically Varying Flows

model to test whether or not the ray-tracing techniques that we have used are valid, and, if so, use the same nonlinear model to consider more realistic flows. 7.3.1

Rays in the Longitudinal Plane

In Section 4.2.2, we found that if the basic state varied along a ray path the scale of the wave (defined by its longitudinal, vertical, and meridional wave numbers, expressed in Eqs. (4.14a) to (4.14c)) must change in order to conserve the intrinsic frequency of the mode. In addition, again depending on the form of the variability, wave energy (specifically wave energy density) may converge or diverge along a ray as described by Eq. (4.19). This possible local build-up of wave energy density is referred to as “wave energy accumulation.” The term “wave energy depletion” will be used to indicate the local divergence of wave energy density. These should not be surprising characteristics of the tropics as the group speed of a mode, with which energy is advected (Section 4.2.2), is a function of k, l, and m, all of which will change to some degree as a wave propagates through a spatially varying medium. Although the variation of the zonal basic flow along a ray is prerequisite for wave accumulation or depletion, it will turn out not to be a sufficient condition. It will also prove necessary that (i) the magnitude of the basic state and the group speed of the wave are sufficiently similar at some point along the ray and (ii) the basic flow changes sign at some location. In the equatorial regions these criteria are met in the slowly varying background climatological flow (e.g. Figure 1.8). Consequently, wave accumulation is possible for most classes and scales of waves except, perhaps, the Kelvin wave. Even if accumulation does not occur and a mode propagates around the equator, its wavenumber will still change as prescribed by Eqs. (4.14a) to (4.14c). It is convenient to view equatorial wave characteristics in a slightly different manner and one that allows us to make use of the important conclusion of ray tracing theory. This is that the intrinsic frequency of a mode along a ray is constant in a time-invariant flow. We will consider both ER and MRG modes. We now plot intrinsic frequency as functions of wavenumber ka and zonal velocity U using the dispersion relationships (6.25) and (6.32), noting that 𝜔 = 𝜔d + Uk. These diagrams appear in Figures 7.10a and 7.11a for the ER and MRG modes, respectively. It should be noted that we have made a change in the sign convention to that adopted previously. In Figures 7.10a and 7.11a, the solid and dotted lines represent, respectively, positive and negative frequencies. Hitherto, we have considered all frequencies to be positive and designated westward and eastward propagating

modes as ka < 0 and ka > 0 modes. However, as pointed out in Section 6.1.3, the choice of a sign convention is arbitrary as long as it is applied consistently. Here we allow the intrinsic frequency, 𝜔, to be either positive or negative and insist that wavenumber ka be positive, thus adopting the convention used in Figure 6.1b. We can now consider modes of some initial scale ka immersed in a background state with a specific U and follow their evolution. For example, we could consider a mode of scale ka = 10 located in a background flow of U = −4 m s−1 as referring to point C in Figure 7.10a and b(iii). With these initial conditions and a knowledge of the background wind speed, we can follow the wave “trajectory” that has to stay on an isopleth of constant intrinsic frequency (dotted lines < 0, solid > 0). Such a procedure will allow us to estimate whether the mode will propagate unattenuated around the equator (i.e. untrapped zonally), oscillate about a particular point, or accumulate in some region. To make these calculations we make use of Eq. (4.14a), which describes the variation of k within a U(x) field. The modal Doppler frequency relationships, Eqs. (4.11) and (4.14a), can be solved in tandem3 to calculate the characteristics of the horizontal propagation. We set cp at 63 m s−1 matching an equivalent depth of 400 m. Before proceeding, we must be most careful of interpreting Figures 7.10 and 7.11 too literally, especially at very small longitudinal scales. The abscissa of these two figures goes to extremely small longitudinal scales. Kinematic theory, as developed in Chapter 4, makes no distinction with respect to scales and merely says that relative to the kinematic rules (Eqs. (4.14a) to (4.14c)), in a region of energy accumulation ka will increase without limit and will have a corresponding increase in wave energy density (Eq. 4.19). Clearly, in reality, we cannot expect to find very large values of ka in accumulation zones that have the characteristics of coherent Rossby waves. In fact, kinematic theory cannot predict the behavior of waves in regions of energy accumulation. 7.3.1.1

Rossby Wave Characteristics

First, we consider the characteristics of the equatorially trapped Rossby waves (ER) as a function of ka in a background zonal field of U. The letters in Figure 7.10a refer to the initial locations of the mode relative to the background states shown in Figure 7.10b(i–iv). Short Rossby Waves Consider an ER having an initial

zonal wavenumber ka = 11. Assume the mode is excited at point A in Figure 7.10a at a longitude slightly west of where U= −10 m s−1 in a background flow that ranges 3 Webster and Chang (1988) and Chang and Webster (1990, 1995).

7.3 Wave Characteristics in a Zonally Varying Basic State U = U(x)

(a) ER ω-ka trajectories in U(x)-flow

20

Zonal wind speed (m s–1)

15

B: propagating A,A1,C,C1: trapped D: oscillatory

D

C1

(b) ER characteristics in U(x)-flow (i) Forward accumulation 10

10

A1 A (ii) Propagating mode B

1 5

0 U (m s–1) –10 2 0 U (m s–1) –2

0

B

0 –5

–2 –3

A1 A

–10

(iii) Forward and backward C1 1 C

–1

C

–4 –6

15 0 U (m s–1) –15

–8

(iv) Oscillation

–15

20 D

–20 1

11 21 31 41 Zonal wavenumber (ka)

D1

51 longitude

0 U (m s–1) –20

Figure 7.10 (a) The angular frequency (day−1 ) plotted as a function of zonal wavenumber, k, and zonal velocity, U(m s−1 ). Positive frequencies are shown as solid lines and negative as dashed lines. The chosen equivalent depth is 500 m. The letters denote starting or initialization points of the modes in (k, U) space. For example, point A refers to a k = 11 mode excited at a longitude where U = 9 m s−1 such that the angular frequency of the mode will be −2.5 day−1 . The arrows denote the extent of the wave trajectory in (k, U) space while the small bars denote the trajectory limit if it exists. If there is no horizontal bar, the mode will accumulate and approach zero group speed with increasing wavenumber. (b) Visualization of the Rossby wave characteristics for different basic flows. Heavy dashed arrows denote the wave path. Four examples are shown: (i) forward accumulation, (ii) propagation (no accumulation), (iii) forward and backward accumulation, and (iv) oscillation. The letters correspond to the cases shown in Figure 7.9a. Gray shading denotes regions of negative stretching deformation. Source: After Chang and Webster (1995).

(a) MRG ω-ka trajectories in U(x)-flow

(b) MRG characteristics in U(x)-flow

20

2 1

Zonal velocity (m/s)

10 5

(i) Backward accumulation

A: propagating B, C: trapped

15

x

C B

–5

A

0 U (m s–1)

B –5

0

0

(ii) Propagating mode

–1 –2 –3

5 x

–10

A

0 U (m s–1) –5

longitude

–15 –20

5

C

1

11 21 31 41 Zonal wavenumber (ka)

51

Figure 7.11 (a) Same as Figure 7.10a except for MRG waves. (b) Same as Figure 7.10b except for MRG waves. Two cases are shown: (i) backward accumulation and (ii) propagation. Source: After Chang and Webster (1994).

between ±10 m s−1 . From Figure 7.10b(i) we note that the initial location of the wave is at the eastern edge of the negative stretching deformation area (U x < 0: 𝜕U∕𝜕x < 0, shaded). The wave lies on the 𝜔 = −2.5 day–1 contour upon which it constrained to reside. The

initial group speed is small and to the east. However, the Doppler group speed is to the west, as described in Figure (7.2a). As the wave is advected into regions of larger U x < 0, the wave will achieve a higher wavenumber following rule (4.14a). Eventually, the mode will

171

172

7 Waves in Longitudinally and Vertically Varying Flows

be longitudinally trapped in regions of small negative values of U that match the decreasing positive values of the group speed or where the Doppler group speed goes to zero. This path may be seen as the blue arrows heading toward larger values of k to the right of point A in Figure 7.10a or as the large blue arrow starting at A in Figure 7.10b(i). From Eq. (4.19), the wave energy density 𝜉 will increase, producing a region of energy accumulation. Suppose now that a mode of the same scale is initially set at point A1 (Figure 7.10b(i)) where the initial stretching deformation field is positive. With westward propagation, the wave will decrease its wavenumber (lengthen its longitudinal scale) as U x > 0 until it reaches the longitude of maximum easterly basic flow (i.e. when = −10 m s−1 ). At that point, U x changes sign and the mode will start to decrease its longitudinal scale. Inevitably, however, this mode will become longitudinally trapped by the mean flow because, being constrained to follow the 𝜔 = −2 day–1 contour, the mode will never reach the region of westerly flow. Waves originating from either positions A or A1 will be trapped in the region of negative stretching deformation. As the group propagation is to the west, the process is referred to as forward accumulation.4 The sign convention chosen regarding accumulation is one that is in keeping with the westward phase velocity of a Rossby wave. Based on the results so far, we can come to a preliminary conclusion. Invariably, it would seem that, relative to the background zonal velocity fields encountered in the tropics, all short Rossby waves will be longitudinally trapped. Propagating Rossby Modes It is possible, though, under

certain conditions, for waves to propagate through the negative stretching deformation region and escape longitudinal trapping. This occurs only for rays that follow a constant 𝜔-contour that extends into both the positive and negative zonal velocity region. We will consider this exceptional case more thoroughly in Section 7.3.2.2. Consider a Rossby wave having an initial wavenumber k = 3/a, excited at the longitude where U = 1 m s−1 (point B in Figure 7.10b(ii)) and constrained to follow the 𝜔 = −1 day–1 frequency contour. Assume, also, that the basic state has a smaller amplitude variation of ±1 m s−1 . Irrespective of whether the mode is placed initially in a region of positive or negative stretching deformation, the wave will not be trapped as the wave will develop a group speed greater than the magnitude of the basic state and of the same sign. The propagation of the mode in this weak basic state is shown in Figure 7.10b(ii) as a green arrow. 4 Chang and Webster (1995).

Intermediate Rossby Waves Consider now a mode of

moderate scale such as ka = 7. The sign of U x at the initial location of the wave now becomes extremely important. If the initial point is located to the west of the westerly maximum (i.e. point C in Figure 7.10b(i)) where U x > 0, the mode will initially move eastward. However, with this movement, the wavelength and the westward group velocity will both increase. In some cases, the westward group velocity can become so strong that the mode starts to move westward before it reaches the westerly maximum. Here the mode will finally become trapped near the longitude where U ≈ 0 m s−1 . We refer to this as forward accumulation or accumulation to the west. If the mode is located at C1 to the east of the westerly maximum, the mode will move eastward with increasing wavenumber where it will be trapped at the same longitude as the wave starting off to the west of the maximum. This is referred to as backward accumulation or accumulation to the east. Figure 7.10b(iii) illustrates both the forward and backward accumulation processes where the evolving trajectory depends on the sign of the stretching deformation at the initial location of the mode. Oscillatory Transitions in More Complex Flows Depending

on the form and magnitude of the basic state, oscillatory ray paths are also possible for certain modal scales. Consider a basic state with two westerly maxima similar to that which exists in the boreal winter upper troposphere in the Pacific and Atlantic Oceans, bridged by a weaker band of westerlies (see Figure 1.8). Suppose the initial point of the mode is at point D lying to the west of the saddle point in a region where U x < 0 (Figure 7.10b(iv)). From Eq. (4.14a) the longitudinal wavenumber will increase and the group speed will become westerly so that the mode will move toward the saddle point. To the east of the saddle point, U x becomes positive and the wavelength increases and the group speed eventually becomes easterly. If the magnitude of the easterly group speed exceeds the magnitude of the westerly basic state (i.e. the scale of the mode is very large (k small) before the second maxima is reached) the mode will move westward and will execute oscillations between the two westerly maxima. 7.3.1.2

MRG Wave Characteristics

Figure 7.11a shows the isopleths of intrinsic frequency as a function of U and ka for the MRG mode. The most striking difference to the ER characteristics is that there are more 𝜔 isopleths that extend through both the positive and negative basic flow regimes. Thus, for a specific basic flow, it might be expected that the range of MRG waves that will be longitudinally trapped by the basic

7.3 Wave Characteristics in a Zonally Varying Basic State U = U(x)

7.3.2 Impact of Longitudinal Displacement of Wave Sources in a Zonally Varying Flow We will now explore whether or not the initial equatorial location of a wave source is important. We do this by initializing identical waves at different locations (A–D) around the equator spaced 60∘ in longitude apart as depicted in Figure 7.12. The forcing is placed within a zonal basic flow with easterlies and westerlies amplitudes of ±10 m s−1 (U upper panel). The phase of the basic state has been moved 90∘ to the east compared

(a) Equatorial Rossby: ka = –3, n = 1, hn = 400 m

U, cgx (m s–1)

20

(i) Zonal group speed

10

10 10

U(x)

0 –10

10

–20

40 30

B

10 20

A

C

D

–30 0°

60°

120°

180°

240°

300°

360°

0 A

5 ka

10

C

B

D

(ii) Wavenumber evolution

15 20 25

0°

60°

120°

180° longitude

240°

300°

360°

(b) Equatorial Rossby: ka = –7, n = 1, hn = 400 m U, cgx (m s–1)

20

(i) Zonal group speed

10 10

U(x)

0

20 30

10

B

10

A

–10 –20 –30

C

D

10 0°

60°

120°

180°

240°

300°

360°

0 5 ka

flow is less than for the Rossby wave. Because the longitudinal group velocity of the MRG wave is universally westerly (i.e. to the east), only an easterly mean flow can longitudinally trap the wave. Note, too, that the 𝜔-curves dip down to very high values of negative U. Thus, along a particular trajectory, the group speed of the wave has a good chance of exceeding the magnitude of the basic state and escaping longitudinal trapping. We now consider the longitudinal ray paths of three MRG modes in a sinusoidal basic state with an amplitude of ±5 m s−1 . The initial wavenumbers of the three modes are ka = 5 (at location A) in Figure 7.11a and b, ka = 6 (B), and ka = 7 (C), each of which has an initial eastward group speed. Each mode is located so that the wavelengths of the modes will all increase as they propagate according to Eq. (4.14a) and with an ever-slowing group speed (see Figure 7.3b and c). To avoid zonal trapping by the mean flow, a mode must have a group velocity larger (in this case) than 5 m s−1 when it reaches the easterly wind maximum position. Clearly, modes B and C cannot satisfy this criterion so they will be zonally trapped near U = 0 m s−1 , the same trapping region we found for the Rossby waves shown in Figure 7.10b. On the other hand, mode A will not be zonally trapped. Figure 7.11b shows, schematically, the ray paths of the modes relative to the basic state. Here we notice that accumulation, when it occurs, is backward. Figure 7.11a indicates that, initially, the group speed of all three MRG modes considered above are >5 m s−1 , the maximum speed of the background flow. Why then are modes B and C trapped whereas mode A is not? From the distribution of the basic wind field along the equator (Figure 7.11b) and the location of the initial points of the waves, it can be seen that modes B and C remain in the positive stretching deformation zone a much shorter time than does the mode first located at A. When modes B and C pass the westerly maximum into the negative stretching zone, their group velocity is not strong enough to overcome the effect of the shortening of the scale of the wave dictated by Eq. (4.14a) and the consequent decrease in group speed produced by the stretching deformation field of the zonal flow.

A 10 (ii) Wavenumber evolution 15

B

C

D

20 25 0°

60°

120°

180° longitude

240°

300°

360°

Figure 7.12 Simulations based on kinematic theory of (a) n = 1 long scale ER modes (ka = 3) in a zonally varying basic state U(x) initially located at one of four longitudes: (A) 90∘ W near the westerly maximum, (B) 90∘ E near the easterly maximum, and (C) 150∘ E and (D) 80∘ W. (i) Longitudinal trajectories with the numbers marked every 10 days following initialization. The solid black line depicts the zonal velocity. (ii) The change in wavenumber along a trajectory. (b) Same as (a) except for a medium scale ER; ka = 7. The trajectories are color coded for easier recognition. The equivalent depth is set at 400 m. The amplitude of the longitudinal variation U is ±10 m s−1 .

to Figures 7.10 and 7.11 so that the region of maximum negative stretching deformation is close to the center of the figure. In the upper panel, the local group speeds of the modes are plotted together with the background zonal fields. The curves in the middle diagram of each figure represent the local wavenumber of the wave along the ray path deduced from Eq. (4.14a). The actual ray paths of the modes emanating from the forcing points are shown in the upper panel as a function of longitude

173

7 Waves in Longitudinally and Vertically Varying Flows

7.3.2.1

Accumulating Modes

Consider the progress of a ka = 3, n = 1 mode forced, for example, at location A in Figure 7.12a (longitude 60∘ ). Initially, there is a westward propagation of the wave. This occurs because, locally, the initial group speed of the mode and the basic flow are of the same sign. Given the constant frequency constraint along a ray (Section 4.2.2(ii)), we see within the easterly regime that the wavelength is slightly longer than that at which it was forced. As the mode propagates westward through the region of negative longitudinal stretching deformation the wavelength starts to decrease rapidly according to Eq. (4.14a) and exhibits a corresponding decrease in group speed, as shown in Figure 7.2b, while approaching zero. Waves forced at other regions in the flow possess very different characteristics dependent upon their initial environment. For example, modes emanating from locations like D find themselves in an environment that immediately places their Doppler-shifted group speed near zero. All shorter modes, plotted in Figure 7.12b, eventually tend toward zero in their Doppler-shifted group speeds. That is, all ER modes have wave energy accumulation regions for the given basic flow along the equator. Figure 7.13 shows the theoretical trajectories for the MRG modes. Long waves (k = −3/a) with their large eastward group speeds can pass through the regions of negative stretching deformation depending on their initial Doppler-shifted group speed. Waves initiated from B and C propagate around the equator. However, modes starting at A and D accumulate. For smaller scales, all modes accumulate in a similar manner to the ER modes. The surprising aspect of the simple kinetic theory solutions is that the wave energy accumulation regions

(a) MGR: ka = –3, n = 0, hn = 400 m U, cgx (m s–1)

40

(i) Zonal group speed

30

10 10

A

10

C

20 10

D

20

U(x)

0

B

20

10

5

20

20

–10 0°

60°

120°

180°

240°

0

ka

300°

360°

B

5

A (ii) Wavenumber evolution

10 15

C

D

20 25

0°

60°

120°

180°

240°

300°

360°

longitude (b) MGR: ka = –3, n = 0, hn = 400 m

40 U, cgx (m s–1)

and time over a 40-day period. The equivalent depth, hn, is set at 400 m. In the following discussion, we concentrate on the ER and the MRG modes. For the Kelvin wave and with hn = 40 m, the propagation speed is >20 m s−1 , which is about the same as the observed upper-tropospheric westerlies in the equatorial zones. However, for hn = 100 and 400 m, the phase and group speed become greater than 30 and 60 m s−1 , respectively, well outside the observed limits. Thus, for realistic values of hn , the basic flow is not expected to impact the characteristics of the Kelvin wave and longitudinal energy accumulation will not be expected. From the discussion above, it would seem that except for the anomalous propagating mode (see below) and for the Kelvin wave, all low-frequency equatorial modes may be expected to “congregate” near the same longitude irrespective of their initiation latitude. This congregation will occur where U x < 0.

(i) Zonal group speed

30

30

20

10 5

10

40

B

20

10

A U(x)

0 –10

10

20

C

D

20

20

0°

60°

120°

180°

240°

300°

360°

0 (ii) Wavenumber evolution

5 ka

174

10

B

A

15

C

D

20 25

0°

60°

120°

180° longitude

240°

300°

360°

Figure 7.13 Same as Figure 7.12 except for (a) long MRG mode (ka = 3) and (b) medium scale MRG moderate (ka = 7).

for both the ER and MRG waves occur in the same longitude belt. That is, accumulation occurs to the east of the westerly wind maximum where the longitudinal stretching deformation is a negative maximum. For shorter ER and MRG waves, this may not seem surprising as the n = 0 MRG asymptotes toward the n = 1 ER mode. However, the similarity exists for the much longer waves as well. 7.3.2.2 Mode

Anomalous Non-accumulating Propagating

In Figure 7.13, there appear to be anomalous modes (originating at point C) that do not appear to possess cgxd zeros, are not longitudinally trapped, and hence propagate completely around the equator. In the ER case similar behavior occurs for smaller values of ka

7.3 Wave Characteristics in a Zonally Varying Basic State U = U(x)

k2 +

U 0 𝛽 2 (2n + 1)2 𝛽(2n + 1) − =0 c c2

(7.45)

We seek the wavenumber (i.e. k0 ) that will possess the slowest (or limiting) westward group speed that allows the mode to just pass through the westerly maximum; i.e. so that (cgx − U) ≤ 0 at x = 0. All modes for which k < k0 will pass through U (Figure 7.8). Equations (4.13a) and (4.14) combine to give dk = −k(dU∕dx)∕cgxd dx

(7.46)

Given the limiting value k0 , for a given n and U 0 as an initial condition, we can use Eq. (7.46) together with Eq. (7.27) to obtain values of k at all x for specific basic states defined by the value of the westerly maximum, U 0 . Figure 7.14a (for h = 400 m) plots the k-loci as a function of longitude. The intercept of the U 0 curves and the ordinate define the locus of all waves along the equator that will change their wavenumbers in moving through the U 0 basic state to k0 by the time they reach x = 0. For U 0 = 5 m s−1 , for example, k0 = 3.7. Following the U 0 = 5 m s−1 curve we can see that all waves with k < 2.2 originating at x1 will be untrapped. However, all k > 2.2 will be trapped, as by the time they have propagated westward and reach x = 0 their wavelength will have decreased beyond the critical value such that k > k0 . Similarly, waves originating at x2 will be untrapped by

critical zonal wavenumber (k0)

(a) n = 1, hn = 400 m –5

|k|>|k0| (trapped)

–4

U0 = 1

|k|=

–3

|k0 |

U0 = 3

–2 9

–1

5 7

|k| 0 at the easterly maximum. Noting the strength of the basic state and the magnitude of the longitudinal group speeds (e.g. Figure 7.2b), it is obvious that only the large-scale equatorially trapped Rossby and MRG waves can satisfy the cgx − U < 0 criteria. For the shorter-scale Rossby waves, cgx > U y (90∘ E) so that dl/dt(90∘ E) > > dl/dt(180∘ E). Therefore, modes will be more equatorially constrained in the easterlies than the westerlies. That is, equatorial modes may be expected to extend more poleward in the westerly duct than in the easterlies. Figure 7.18b shows the evolution of the perturbation velocity and height fields results of the numerical experiments for the first 19 days of integration with

177

7 Waves in Longitudinally and Vertically Varying Flows

forcing centered at 60∘ E. Initially, there is propagation of a Kelvin mode toward the east along the equator and a corresponding pair of equatorial off-equator modes moving slowly toward the west. The former mode continues to circle the globe but with decreasing intensity. It does not “accumulate,” as its group speed is far too

6

0

U (ms–1)

0

10

U (x,y = 0)

U (ms–1)

U (x,y = 0)

10

4

–10

PKE (m2 s–2)

–10

5 90 °E 150 °E 150 °W 90 °W

2

0

rapid. The slower modes continue to move westward and eventually stall near the longitude of maximum stretching deformation. There, amplitude grows and results in a pair of even modes that extend north and south about the equator. Emanation toward higher latitudes occurs in the region of equatorial westerlies.

PKE (m2 s–2): Forced at different longitudes. Mean days 10−25 (ii) Large-scale forcing (i) Small-scale forcing 15 10

PKE (m2 s–2)

Ux 0), but contract latitudinally because of the strong latitudinal shear. In the U x < 0 region the longitudinal scales decrease but the weakening latitudinal shear allows the mode to extend latitudinally, where they extend well into the extratropics.

7.5 Zonally Varying Basic State and the “Longwave Approximation”

7.5 Zonally Varying Basic State and the “Longwave Approximation”

5 E.g. Chang (1977), Liebman and Hendon (1990), Dickinson and Molinari (2002), Swann et al. (2006), among many others.

Rlw

0.4

n=1

2.5 n = –1

Frequency (day–1)

1.7

Period (day)

In Section 6.4 it was found that for equatorially trapped ocean waves, the response to mid-ocean forcing was accomplished by eastward propagating Kelvin waves or westerly propagating Rossby waves. To good approximation the large-scale Rossby waves, like the Kelvin wave, are non-dispersive. This is the “longwave approximation” developed in Section 6.4.3. It has served as a simplifying artifact for ocean dynamics if not over-simplifying the coastal reflection problem. Is this assumption useful for the tropical atmosphere? Given the smallness of the longitudinal group speeds of an initially short Rossby wave (Figure 7.2) it has been presumed that such a mode cannot move far away from the region in which it was forced. Thus, it is assumed (as in the ocean) that only Rossby waves of very long wavelengths (small ka) can provide a remote response to a regional forcing and produce physically significant global teleconnection patterns. The longwave approximation has been used extensively by Gill (1980) for the study of equatorial modes and stands as the keystone of a theory seeking a quasi-balanced dynamics for the tropics (e.g. Stevens et al. 1990). The dispersion relationship for dispersive Rossby waves and MRG modes are expressed in Eqs. (6.24a) and (6.32), respectively. If the longwave approximation is invoked for the motionless basic state U = 0, the dispersion relationship for equatorially trapped Rossby waves reduces to kc 𝜔=− (7.48) (2n + 1) √ where c = ghn . In essence, these waves are identical to the solutions found for the ocean except, of course, phase speeds are more rapid. As in the ocean case, the modified dispersion equation for the Rossby wave arises by setting dv/dt = 0 in the meridional momentum Eq. (5.8b), reducing Eq. (6.23) to Eq. (7.48) for small ka (i.e. long waves). The derivation, in essence, follows that for the Kelvin wave in Section 6.1.5.4, except with ka < 0 and with the opposite root being used to satisfy boundary conditions at large |y|. The same relationship exists between the phase speed of the Kelvin and long Rossby waves, as we found in Eq. (6.80). That is, the phase and group speeds of the resulting non-dispersive long Rossby waves are 1/3, 1/5, and 1/7 of the Kelvin wave for n = 1, 2, and 3, as described in Section 6.4.3. It should be re-emphasized that the longwave approximation completely eliminates the MRG wave, perhaps a serious omission as many studies5 have shown that the

0.6

n=2 n=3 0.2 MRG Xʹ 0 –20

5.0

K X R

–15 –10 –5 0 Zonal wavenumber (ka)

5

Figure 7.20 Dispersion curves for all equatorial modes from Eq. (6.53a) shown as dashed lines for hn = 400 m. Solid lines denote equatorial modes using the longwave approximation. All modes are now non-dispersive and the MRG mode has disappeared completely. The solid arrow (X − X′ ) shows the change in scale an initially dispersive long Rossby wave would undergo as it propagates through a zonally varying basic state such as shown in Figure 7.17a between 90∘ E and 120∘ W. Source: After Chang and Webster (1995). The usual Rossby curves (ER: transposed from Figure 6.1a and shown as dashed lines), rendered non-dispersive by the longwave approximation, appear as straight curves (ERLW : solid lines). The Kelvin mode is unaffected.

MRG wave is a critical component of the low-frequency, transient structure of the tropics. Furthermore, the longitudinal variation of the basic state (i.e.U = U (x)) may render the longwave approximation questionable in the tropics. The solid arrow on the dashed Rossby dispersion curves in Figure 7.20 shows the modification of an initially long Rossby wave at X as it moves through the idealized and zonally varying basic flow toward the accumulation zone. The scale of the wave changes from k = 3 to k = 15 during its propagation according to Eq. (4.14a). The transformation is shown in Figures 7.12 and 7.13, for example, as the mode travels from 90∘ E or 150∘ E into the negative stretching deformation region near 120∘ W. However, such modification of Rossby waves, apparent in both theory and modeling results, is completely absent from the equatorial dynamics governed by the longwave approximation. Thus, an atmospheric dynamic system based on the longwave approximation, aimed at providing greater insight into equatorial dynamics as well as computational efficiency, may lack generality and, thus perhaps, utility.

181

182

7 Waves in Longitudinally and Vertically Varying Flows

7.6 Vertical Trapping, Accumulation, and Lateral Emanation Earlier, we established that the time-averaged basic state of the tropics, besides possessing horizontal variations, also possesses strong vertical shear. Figures 7.2 to 7.4 also illustrate that, in general, the equatorial mode vertical group speed of equatorially trapped waves is a strong function of k. Consequently, through Eq. (4.14a), cgz should be dependent on the background basic state. Specifically, for a given equivalent depth, the vertical group speed for an equatorial Rossby wave will increase toward a maximum at k = −2 to −4 before slowly decreasing as k increases (Figure 7.2c). Thus, as a mode moves through a varying basic state where U = U(x), cgz will also vary. In Section 4.2.2, another basic state dependency was found. Equation (4.14b) suggests that if the zonal wind field contains vertical shear (i.e. U = U(z)) then the vertical wavenumber, m, will change along a ray with the sign of the shear (𝜕U∕𝜕z) determining whether m will increase or decrease in the vertical. By extension, the magnitude of cgz also depends on the vertical variation of the background state. These points raise further issues: (i) The influence of the basic state on the vertical group speed poses the question of whether transient modes will propagate vertically sufficiently rapidly to move out of the troposphere near a source region before they approach the accumulation regions near 𝜕U∕𝜕x < 0|max or whether they remain within the (a) Impact of U(z) z

troposphere as they propagate away from a source region. (ii) From Eq. (7.2) it is apparent that if m changes then hn will change, being inversely proportional to m2 . Thus, an alteration in m will alter the horizontal group velocity cgx and horizontal structure of a mode through changes in hn . These associations give us cause to examine the impact of U on the character of propagating waves more carefully. Figure 7.21a provides a schematic of the influence of a basic state on the modal structure of equatorially trapped waves. It shows the change in the vertical wavenumber, m, as a function of U. The inverse relationship between m and hn (i.e. Eq. 7.2) provides a proportionality between hn and Lz , the vertical length scale. Then, from Eq. (4.14b), if U z < 0, the vertical scale decreases (i.e. m increases) and the vertical group speed decreases. However, if U z > 0, the vertical scale increases (i.e. m decreases) and the group speed increases. These two configurations represent relative vertical trapping and propagation, respectively. The impact of U(x) on horizontal accumulation is shown in the schematic Figure 7.21b, essentially a depiction of Eq. (4.14a), albeit substantiated by the nonlinear calculations, and summarizes the results of the last section. Modes move toward the negative stretching deformation region from both the east and the west. Accumulation takes place in the longitude band where U x < 0.

(b) Impact of U(x) z

easterly

MG-R

ER

westerly U(x) westerly

–

zonal wind + Uz < 0 m increases with z cgz 0 vertical trapping

longitude

easterly − zonal wind + Uz > 0 m decreases with z cgz increases with z vertical propagation

Ux < 0 accumulation zone

Figure 7.21 Components of wave modification in a U(x, z) basic state. (a) Vertical character: with U z < 0, the wavenumber m increases (wavelength decreases) and cgz → 0, indicating a vertical trapping of the mode (region C in Figure 7.15). For U z > 0, m decreases so that cgz increases and the mode propagates upwards more rapidly (regions A and B in Figure 7.15 inferred from Eq. (4.14b) and Figure 7.11a). The longitudinal character is shown of both the equatorial Rossby wave and the mixed Rossby-gravity mode. If U x < 0, k increases and cgx → 0 and the mode is longitudinally trapped (accumulation). If U x > 0, k decreases and cgx increases. Arrows show the directions toward accumulation for the Rossby mode (forward accumulation) and the mixed Rossby-gravity mode (backward accumulation) inferred from Eq. (4.14a) and described in Figure 7.11b.

7.7 Quasi-Biennial Oscillation (QBO)

emanation to higher latitudes

emanation to higher latitudes vertical propagation vertical trapping

vertical propagation 200 mb emanation to higher latitudes

Figure 7.22 Composite schematic showing longitudinal, latitudinal, and vertical propagation and emanation. Regions of longitudinal accumulation are also regions of latitudinal emanation to higher latitudes.

Figure 7.22 attempts to capture the essence of the equatorial accumulation occurring in the vicinity of the equatorial ducts and wave emanation to higher latitudes also occurring in the same vicinity. The emanation patterns are essentially teleconnection patterns and are not completely controlled by the location of the heat sources but also by the structure of the background basic state. Thus, if the basic state were to change either from season to season or between phases of ENSO, we would expect there to be different trajectories in the vertical and, as the regions of upper-tropospheric westerlies change on these longer time scales, different emanation or teleconnection patterns between the equatorial regions and higher latitudes. These results offer at least a partial explanation of why the connections between the equator and the higher latitudes occur through specific corridors.

and the equatorial Rossby wave (ER) possessing easterly (westward) phase speeds. Both families propagated upwards as internal gravity waves. • Booker and Bretherton had described how an internal gravity wave propagating through a vertical shear flow (U(z)) may be attenuated if it encounters a critical level where the horizontal phase speed of the mode approaches the magnitude of U(z), i.e. where |U(z) − cp | → 0. Here cp is the horizontal phase speed of the wave. At such a location, wave momentum is transferred to the basic state. It was reasoned that at a critical height one set of modes would transfer easterly momentum to the basic state whereas the other would add westerly momentum. Importantly, Lindzen and Holton determined that the changes in the basic state induced by one modal family would restrict the vertical propagation and transfer of momentum by the other family. There are two parts to the theory. First, there is the effect that the basic state on the propagation of the waves and thus the transfer to the basic state of the momentum fluxes inherent to the modes. These effects are significant, as noted in Section 7.6. The second part of the feedback is the impact the momentum fluxes have on the background flow. The sequence is described in the schematic shown in Figure 7.23, following Plumb (1984) and Baldwin et al. (2001). Each mode propagates

Evolution of the QBO (a)

(b)

z U(z)

z U(z)

7.7 Quasi-Biennial Oscillation (QBO) The identification in the mid-twentieth century of a vertically propagating stratospheric quasi-biennial mode posed a number of problems. At the time that detailed descriptions of the oscillation were becoming available, the dynamics of the tropics were insufficiently developed to permit a reasonable physical examination. However, in 1968, a bold theory was proposed by Lindzen and Holton (1968)6 that melded the new equatorial wave dynamics of Matsuno (1966) with Booker and Bretherton’s (1967) critical layer absorption theory. • Matsuno had described two basic sets of equatorial tropospheric modes: the Kelvin wave possessing a westerly (eastward) phase speed whereas the MRG 6 Refined subsequently by Holton and Lindzen (1972).

cp0

0

(c)

cp0

0 z U(z)

U(z)

cp0

cp0

183

184

7 Waves in Longitudinally and Vertically Varying Flows

upward until its group speed is slowed, damping the wave as it encounters a region near the critical height. Consider vertically propagating equatorially modes in the basic state shown in Figure 7.23a that create a descending shear zone. With the descent, the negative shear region becomes sufficiently narrow that viscous effects become important, diminishing the westerly winds. This allows the eastward wave (the Kelvin wave) to propagate deeply into the stratosphere (Figure 7.23b). In turn, the transfer of momentum from the Kelvin wave builds up a new westerly (or eastward) regime high in the stratosphere. Continual absorption of momentum from the eastward wave causes the westerly maximum to descend (Figure 7.23c and d). Figure 7.23a to d represents a half cycle of the quasi-biennial oscillation (QBO) sequence. The subsequent descent of the westerly maximum above the easterly shear produces a lower-stratospheric easterly jet. Once this jet decays, a reciprocal profile of U(z) to

that in Figure 7.23b arises and the westward mode is free to propagate high into the stratosphere. Momentum transfer at the critical level will allow new easterly (i.e, westward) shear zone to form aloft. Continual momentum transfer provides a downward propagation of the easterly winds. The pioneering work of Lindzen and Holton (1968) and Holton and Lindzen (1972) and that of Plumb (1984) used zonally symmetric arguments. Yet Figure 7.22 suggests that the vertical propagation may occur in specific locations around the equator in accord with the variability of U(x, z) shown in Figure 1.11. Furthermore, the location of convection, and hence the source locations of vertically propagating modes, changes on interannual time scales. Whether these variations in space and time are important or whether they somehow rectify in some manner to produce a zonally symmetric forcing of the QBO is not known.

Notes I Airy functions, named after British astronomer George

II The concept of the “westerly duct” was introduced by

Biddell Airy (1801–1882), are the general solutions to Eq. (7.13) rendered into the canonical form yxx −xy = 0 as undertaken in Appendix F. See Gradshetyn et al. (2015) for details.

Webster and Holton (1982) to consider the possibility of extratropical waves propagating into the equatorial regions; these issues are expanded in more detail in Chapters 9 and 10.

185

8 Moist Processes and Large-Scale Tropical Dynamics In the last two chapters we have concentrated on the basic dynamics of near-equatorial waves and the manner in which their character depends on the basic flow. The background was found to alter the scale and group speeds of the modes and thus determine to at least some degree communication between different parts of the tropics and extratropics. We considered both “free” or unforced solutions including in Section 7.2.2 forced modes or solutions relative to some set vertical forcing distributions. Depending on the vertical structure of the background flow either and the location of the forcing function, either vertically propagating or attenuating solutions were found. We did not discuss, though, what determined the vertical distribution of forcing. Although it is generally conceded that a major component of the forcing of tropical motions is the release of latent heat, it is not immediately clear why, for example, there is a mid-tropospheric heating maximum and why clouds extend through the entire tropical troposphere. By the 1950s the vertical extent of deep convection was fairly well established. Scientists had known for centuries that there existed an upper-tropospheric return flow of the Hadley Circulation. In fact, Halley had suggested an “anti-trade” regime higher in the atmosphere based on observations of tropical clouds moving rapidly to the northeast in the subtropics (Persson 2006). These observations substantiated Halley’s surmise that the low-level equatorward flow associated with the trades must possess a return poleward flow somewhere above the surface. By the mid-twentieth century observations had established that the return flow of the Hadley Circulation took place in the upper troposphere. However, this raised another major hurdle of explaining how lower tropospheric mass made its way to the upper troposphere. Again, this was a question without an immediate answer. The potential temperature distributions of Figure 3.2 show an atmosphere where 𝜕𝜃/𝜕z is universally positive and thus statically stable. Clearly clouds and the release of latent heat were involved. Profiles of 𝜃 e (z), on the other hand, possessed a negative

gradient in the lower tropical troposphere, indicating the potential for convective instability. The issue of vertical transport through the depth of the troposphere was addressed in an iconic paper (RiehlI and MalkusII 1958) in which they suggested a distinct role for deep convection. We will return to this issue shortly. Other issues have arisen that appear to be connected to convection. The discovery of equatorially trapped modes in the 1960s allowed many features of the tropical atmosphere to be identified dynamically, showing modes with both eastward and westward phase and group speeds. Yet there were many features of the tropical atmosphere that eluded description. Generally, theoretical modes described in Chapters 6 and 7 were found to have significantly faster propagation speeds than were observed in nature as identified by propagation of organized convection such as appear in Figures 1.16 and 1.18. Waves with both Rossby and Kelvin wave characteristics can be identified, but with much slower phase speeds. The formal derivations of equatorially trapped modes were made without taking into account convection. Is it possible to reconcile these differences incorporating processes associated with convection in conjunction with the physics of equatorially trapped modes? We first consider the problem of vertical transport of mass and energy as discussed by Riehl and Malkus (1958). Some of the conclusions of the study have not received observational substantiation, but it provides a vehicle for discussion of an issue that still is not completely resolved. The study also allows us to consider a more complete view of tropical convection, one that is closer to observations containing not just a convective component but the substantial stratiform component that attends deep convection in the tropics. This allows us to speculate on the role of the stratiform region on determining the longevity of the life cycle of organized convection and what role it may play in the vertical transfer of heat and energy. Finally, we consider processes related to convection that are usually ignored in equatorial wave theory. Specically, we consider the impacts of convective dissipation

Dynamics of the Tropical Atmosphere and Oceans, First Edition. Peter J. Webster. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

8 Moist Processes and Large-Scale Tropical Dynamics

and reduced static stability on the character of equatorial waves.

that there is a seasonal migration of the equatorial trough of about 15–17∘ of latitude between the seasonal extremes, Riehl and Malkus determined that there is a net transfer of about 2.3 × 1015 J s−1 from the summer to the winter hemispheres, a quantity very similar to the difference in magnitude between the summer and winter lateral heat transports. From this observation, Riehl and Malkus were able to suggest that the trough position and its annual migration are controlled by the overall distribution of heat sources and sinks. Figure 8.1b plots a vertical profile of the meridional velocity component located about 10∘ on the winter side of the composite equatorial trough. As the net meridional heat transport through the trough is zero, the profile suggests strong convergence toward the trough in the lower troposphere and divergence aloft. This observation led to the important conclusion that the vertical heat and mass transport in the upward branch of the Hadley Circulation could only be accomplished by deep convection. The problems for vertical transports appear clearly in Figure 8.2, which shows the climatological vertical traces of 𝜃, 𝜃e , and 𝜃e∗ (respectively, potential, equivalent potential, and saturated equivalent potential temperature) in the tropics. Considering just dry processes, the

8.1 Convection and Large-Scale Budgets Over half a century ago, Riehl and Malkus suggested a preeminent role for tropical convection, one necessary for closing both the mass and energy budgets of the Hadley Circulation. Figure 8.1a shows Riehl and Malkus’s estimates of heat fluxes from the summer to the winter pole determined as a residual of estimates of radiative heating and cooling. This is the heat transport required to balance the larger net radiative loss to space occurring at higher latitudes, especially in the winter hemisphere. A “zero” in lateral transport is found near 12∘ of latitude in the summer hemisphere, corresponding to the mean location of the JJA equatorial trough. This suggested to Riehl and Malkus that there was no lateral heat flux, in a zonally averaged sense, across the trough. Maximum poleward transport in the summer hemisphere was computed to be about 2 × 1015 J s−1 compared to 4.5 × 1015 J s–1 in the winter hemisphere. Noting (a) Intergrated heat transport

(b) Meridional wind 10° from trough

90°

100

40° 300

30° 20°

latitude

10°

zero

transport 015 J s–1

0° 10° 20°

–5 –4 –3 –2 –1

1 2

3 4

5

pressure (hPa)

summer

200

winter

186

400 500 600 700

30° 40°

800 900

90°

1000 –2

–1

0 1 v (m s–1)

2

Figure 8.1 (a) Integrated heat transport as a function of latitude (1015 J s−1 ). The horizontal dashed line indicates the latitude of zero heat transport that coincides with the location of the equatorial trough. The diagram is separated into “winter” and “summer” hemispheres so that the heat flux distribution coincides with seasonal extreme (i.e. July or January). Seasonally, the zero transport latitude crosses the equator, indicating a change in the sign of the cross-equatorial flux. Source: Adapted from Riehl and Malkus (1958). (b) Vertical structure of mean meridional flow (m s−1 ) at a latitude 10∘ from the equatorial trough in the winter hemisphere as calculated by Riehl and Malkus (1958). The wind field shows convergence toward the equator in the lower troposphere and divergence in the upper troposphere.

8.1 Convection and Large-Scale Budgets

θ*e θe

200

C′

15 E

θ

12

400

D′

B′

600

R&M undilute ascent

6 5 4

height (km)

pressure (hPa)

9

3 C

800 1000 D 300

B

A′ observed cloud base

A H 320 340 360 380 potential temperature (K)

2 1 0

Figure 8.2 Vertical profiles of potential (𝜃), equivalent potential (𝜃 e ), and saturated equivalent potential (𝜃e∗ ) temperatures for the mean tropical atmosphere. For dry processes, the atmosphere is absolutely stable (𝜕𝜃/𝜕z > 0) no matter how far a parcel is raised (trajectory D–D′ ). However, if moist processes are taken into account the tropical atmosphere is potentially unstable to the mid-troposphere (∼ 600 hPa) with 𝜕𝜃 e /𝜕z < 0. Higher in the troposphere the atmosphere is stable (𝜕𝜃 e /𝜕z > 0). Trajectory A → A′ shows that an initially unsaturated surface parcel can achieve saturation at position A′ if it is raised vertically about 100–150 hPa. Environmental parcel B may just reach saturation at B′ but the required vertical displacement is greater than along trajectory A → A′ . Any parcel to the left of B (e.g. C) will not reach saturation (C–C′ ) irrespective of the degree of lifting. The left ordinate (km) shows that the amount of work against gravity necessary to raise a parcel and achieve saturation is greater for columns to the left of A → A′ . Trajectory A′ → E is the hypothetical undilute ascent (Riehl and Malkus 1958, discussed in the text).

tropical atmosphere is absolutely stable as 𝜕𝜃/𝜕z > 0 at all heights. Clearly, vertical transports in the Hadley cell could not occur through dry adiabatic processes. Transport via trajectory D → D′ would require large amounts of work for lifting. Riehl and Malkus were well aware that the lower part of the tropical atmosphere is conditionally unstable as (𝜕𝜃 e /𝜕z < 0). This reverse gradient can also be seen at low latitudes in Figure 3.2 for both JJA and DJF. Thus, if a surface parcel could be raised adiabatically from point A to A’, the parcel could attain buoyancy as saturation would be reached and a further displacement would result in the release of latent heat. In fact, any parcel on the environmental 𝜃 e curve between A and B can achieve saturation although the amount of work necessary to raise the parcel increases toward point B as the vertical distance to reach saturation increases. Any parcel to the left of B (say C) will never reach saturation no matter how far it is lifted. Riehl and Malkus foresaw an important problem. They noted that the mid-tropospheric minimum in 𝜃 e

would reduce the buoyancy of a cloud parcel through the entrainment of this relatively dry environmental air (low 𝜃 e ) into an ascending parcel. In the mid-troposphere the difference between the mean environmental and the saturated equivalent potential temperature (i.e. 𝜃e∗ (z) − 𝜃e (z)) is a maximum, suggesting the possibility of reducing buoyancy of an ascending cloud parcel. Thus, for clouds to reach the upper troposphere, this reduced buoyancy would have to be overcome or avoided. Clearly, another process was thought to be necessary. Riehl and Malkus proposed that within the convective towers were undilute cores or chimneys that would transport boundary layer air (see trajectory A → A′ → E in Figure 8.2) through the mid-troposphere without dilution. Based on heat and mass budget studies, Riehl and Malkus proposed that 1500–4000 cumulonimbus towers per day throughout the near-equatorial tropics, each with an undilute core, would be sufficient to take care of the required vertical transports. Such deep penetrative clouds developed the popular name “hot towers.” Satellite observations indicate that there are abundant deep convective elements in the tropics to satisfy the Riehl and Malkus vertical transport requirement if, within each system, there existed a conduit to avoid entrainment. However, observational evidence of undilute cores has proven elusive. As stated by Zipser (2003): … Recent decades of observations over tropical oceans – from 1974 (to the present) in the tropical Atlantic, … offshore of Taiwan, offshore of tropical Australia and the warm pool of the equatorial Pacific – have yielded data from aircraft penetrations of thousands of cumulonimbus clouds. (Observations) show great consistency … Undiluted updraft cores are not found. … Zipser (2003, p. 49). In agreement with Zipser there appears to be no maritime cloud, with the exception perhaps of those embedded in the eye wall of a hurricane, that possesses undilute cores as depicted by A′ → E. Yet, convection is observed to reach the upper troposphere, well above the 𝜃 e minimum, presumably in the absence of undilute cores. Zipser noted that there was a second water phase change occurring in the progression of water through tropical clouds. As isobaric or adiabatic cooling continues, the cloud parcel may cool to the point where ice crystals form with a consequent release of the latent heat of fusion. Thus, above about 600 hPa, buoyancy receives an additional heating boost. It should be remembered, though, that the latent heat of fusion is far smaller than that of the latent heat of condensation. In fact,

187

8 Moist Processes and Large-Scale Tropical Dynamics

from Eq. (2.4), the ratio LF ∕LE = 0.132! However, as summarized by Zipser (2003, p. 49): … The observations over tropical oceans can be hypothetically explained by assuming large dilution of updrafts by entrainment below about 500 hPa, followed by freezing of condensate. This freezing and subsequent ascent along an ice adiabat reinvigorates the updrafts and permits them to reach the tropical tropopause with the necessary high values of moist static energy, as the hot tower hypothesis requires …. Riehl and Malkus had not considered this second form of latent heating. However, both the Riehl and Malkus and the Zipser studies concentrated on vertical transport in the convective part of a tropical disturbance. More recent descriptions of convective clusters describe a broader entity containing a dynamically active stratiform component. In this component of a convective disturbance radiative heating destabilization may be important. Neither Riehl and Malkus (1958) nor Zipser considered a role for radiative heating. We will return to the problem of vertical transport after we review the structure of a tropical convective cluster and its attendant physical processes.

8.2 Emerging Perspective on Tropical Convection To delve deeper into the physics of tropical convective systems, it is important to acknowledge that the view of tropical cloud systems has changed considerably during the last 50 years.1 Prior to GATE, convection in the tropics was thought to consist of tall cumulonimbus clouds extending throughout the troposphere with a maximum release of latent heat in the mid-troposphere. In this view, the stratiform anvil was thought not to occupy a significant area, but the results of the many tropical field programs suggested otherwise. Tropical cloud systems are comprised of both cumulonimbus and stratiform components and these complexes account for much of the equatorial organized convection observed from space (e.g. Figure 3.3). Figure 8.3a shows a schematic diagram of a mesoscale convective system (MCS). There are three major components in the system. The first is the leading convection element (left-side “Conv”: column “A”) that

1 Houze (2018) presents a history of evolution of convective cloud physics during the last century.

(a) Schematic cross-section of an MCS A B C SW

Anvil

Conv

Ac

Strat

CT

HEIGHT

188

LW

Anvil

Csu

As LW

Ccu

0°C

Ecd

Esd

LW

Rc Rs A B C (b) Vertical profiles of latent and radiative heating A B C INT z z z z Rn Rd

Rn

Rd

Csu

Rn

Rd

Rn Rd

Ccu + Ecd

0°C

C+E

0°C

Esd

–

+

–

+ – + Relative heating

–

+

Figure 8.3 (a) Schematic cross-section of a tropical convective system with a convective region (Conv) extending through the entire troposphere, an extended stratiform deck (Strat) occupying the upper half of the troposphere, and a thinner stratiform region (Anvil) in the upper troposphere. The horizontal dotted blue line indicates the freezing level. Precipitation occurs in both CONV (RC ) and STRAT (RS ) of about equal magnitude, with attendant latent heat release of CCU and CSU , respectively. Evaporative cooling occurs in the convective region and also below the stratiform region as rainfall passes through the relatively dry sub-cloud layer. Advection of cloud material occurs between all three sectors of the MCS. In addition, in all three sectors, radiative cooling occurs at the top of the cloud (LW) although reduced by near-infrared absorption (SW) during the day. Source: Adapted from Houze (2018). (b) Vertical profiles of latent and radiative heating in sectors A, B, and C of the MCS. Black line denotes the total latent heating. Red line shows the radiative heating and cooling for day (solid) and night (dashed). The total heating integrated across the MCS is plotted in INT. Note, especially, the “top-heavy” latent heat profile signifying the contribution of the phase change in the stratiform heating. Heating units are relative as the absolute magnitude depends on the amount of precipitation in the system.

extends through the entire troposphere. There is also an extensive stratiform region (“Strat”: column “B”) occupying the troposphere above the freezing level and extending roughly 4–5 times the width of the convective zone (column “B”). Third, there is an extensive anvil cloud (“Anvil”: column “C”) at the rear of the stratiform region. Mass transports exist between the surface boundary layer through the deep convection and laterally into the stratiform region and eventually into the anvil. The rainfall of the entire MCS is shared about equally between the convection (RC ) and the stratiform regions (RS ).

8.2 Emerging Perspective on Tropical Convection

The distribution of heating and cooling across an MCS is quite complex. There are two major components: latent heating from the water phase change and radiative heating from the absorption and emission. The former heating tends to stabilize the atmospheric column while the latter acts to destabilize it. Figure 8.3b plots profiles of heating and cooling in the different sections of the MCS. Only relative units are used as the actual heating rates depend upon the overall precipitation and the spatial extent of the system components and on the stage of the life cycle of the MCS. The latent heating profiles combine condensational heating and evaporative cooling. For example, in the convective region, there is heating from latent heating of condensation (Ccu ) and a cooling as falling condensate evaporates (ECD ). The sum (CCU + ECD ) provides a net profile of heating with a maximum in the middle troposphere appearing as the solid black curve in section “A” of Figure 8.2b. Where is the source of the stratiform cloud material? The cross-section in Figure 8.3 indicates that the source is not from the environment around the system as the mid-troposphere is relatively dry. Stratiform cloud material comes from “Conv,” the convective component, and is advected into the middle troposphere (CT ). In turn, the source of moisture for the deep convection is the surface boundary layer. Latent heating of fusion and condensation occur in the stratiform region (CSU ) and evaporative cooling (ESD ) as rain falls through the dryer lower troposphere. Thus, net latent heating occurs aloft in the stratiform region with cooling below. Condensational heating in the stratiform region occurs in the melting layer near the freezing level. The descending air below the stratiform region, cooled by evaporation of precipitation, lowers the surface air temperature, causing an increase in heat exchange between the ocean and the atmosphere. Examples of the increased fluxes into the atmosphere would be similar to those listed in Table 2.4 for the TOGA COARE and JASMINE field surveys. Prior to GATE, the stratiform region was thought to be a passive entity and made up of “cloud debris” from the convective towers that, on spreading out, slowly dissipated through sublimation. However, the stratiform region turns out to be a much more dynamic component of the cluster than previously envisaged. The stratiform layers, made up largely of ice crystals, continue to rise and expand on time scales of many hours.2 The stratiform layer has the coldest cloud top temperatures and thus the highest clouds within the MCS, an indication of its buoyancy. It also accounts for about 50% of the total rainfall of a system, albeit spread 2 Houze (2014) provides a comprehensive review.

out over a much larger area, with latent heating taking place in the upper half of the troposphere. The result is a “top-heavy” latent heating profile far different from the pre-GATE assumption of a mid-tropospheric heating maximum. Mapes (2001) was the first to note the impact of stratiform diabatic effects on the integrated heating profile. This integrated latent heating profile (INT) appears on the right-hand section of Figure 8.3b . On the broader scale, the large integrated areal extent of the stratiform deck itself is a critical component of the gross radiative balance of the tropics, the latter of which is discussed in Section 2.5. However, radiative heating and cooling are also important on the MCS scale. Figure 8.3b shows the radiative components of heating in the column in different locations across the convective disturbance based on calculations with the radiative model of Stephens and Webster (1979) and Webster and Stephens (1980). Radiative heating rates are accentuated in the stratiform region and occur over a broad area. Radiative cooling to space occurs at the top of the stratiform deck but, during the day (solid red lines), is partially offset by the absorption of near-infrared solar radiation. At nighttime (dotted red lines), in the absence of solar heating, a diurnal maximum of net radiative cooling at the top of the stratiform layer occurs. At the base of the stratiform region, long-wave absorption occurs throughout the day and, as the sea-surface temperature (SST) is fairly constant, has little diurnal variation. Radiative model results suggested that there was a differential heating rate of about +10 K d−1 between the bottom and the top of the stratiform region during the day and possible doubling during the night (Webster and Stephens 1980). These conclusions were inspired by the radar data collected in Winter MONEX, where the diurnal cycle of organized convection was monitored for the first time. The radar was located on the northwest coast of Kalimantan at Bintulu (3∘ N, 113∘ E) viewing convection over land and the southeast South China Sea (Houze et al. 1981). The result is, either day or night, that radiative forcing creates an unstable heating profile across the stratiform layer, driving turbulence within the stratiform region. This induced turbulence is probably why the stratiform layer is a precipitating entity. It may be noted that the maximum growth rate of tropical convective complexes occurs at night. Arguably, the radiative destabilization is a critical mechanism in determining the longevity of the stratiform region and hence the time scale of the cluster itself. To our knowledge, the top-heavy heating profile, resulting from the sum latent heat release in the convective zone and the stratiform regions, has not been introduced into an operational weather or climate model. Heating schemes, instead, appear to be based

189

190

8 Moist Processes and Large-Scale Tropical Dynamics

Table 8.1 Comparison of characteristics of observed tropical zonal oscillations and those from frictionless, dry equatorial wave theory. The speed of background zonal flow is assumed to be −5 m s−1 . Observations MODE

Period days

k

Theory

−1

cp m s

−1

−1

Lz km

vms

cp m s

Lz km

v m s−1

Stratosphere Kelvin

10–15

1–2

25–45

17–26

0

30–40

17–29

0

MRG

4–5

3–4

25

17–26

2

25

17–29

2

Tropospheric MJO

40–50

1–2

4–8

15–30

0

15–20

8–9

0

Stationary circulations

∞

1–2

0

15–30

0

0

2–3

5

Troposphere

Source: Based on Table 1 of Chang (1977).

on the pre-GATE notion of “deep” convection (profile “A” in Figure 8.3). Cloud resolving models that do not depend on parameterization can predict MCSs with some degree of realism and therefore more closely represent atmospheric heating. Thus, eventually, when numerical prediction and climate models have a sufficiently high-resolution there will be a possibility that convective heating profiles may approach realism. Finally, returning to Figure 3.3, we note that about 50% of the tropics are covered by clouds, a large percentage of which are cold stratiform as described above. Yet, to date, radiative parameterization schemes are not coupled to convective schemes. Given that phase change and radiative effects are the two most critical heating or cooling components of the MCS, it is difficult to understand why this coupling of parameterizations is absent from weather and climate models.

8.3 Comparison of Observed Waves and Waves from Theory The waves considered in Chapters 6 and 7 were “dry waves” whose properties were examined in the absence of moist processes. The near-equatorial modes displayed in Figure 1.16b show propagation speeds that, for their longitudinal scale, are much slower than expected from theory. Similarly, the eastward propagating Madden–Julian oscillation (MJO) convective masses are far slower than either the theoretical Kelvin wave or MRG wave, appearing in composite form in Figure 1.18. Two immediate questions arise: Can the slow propagation of envelopes containing convection be understood using just the dynamics of dry waves? How does moist convection alter the character of equatorial modes? Can dry and moist waves exist simultaneously in the tropics?

Table 8.1 provides a comparison of the observed characteristics of equatorial waves and those predicted by “dry” theory. 8.3.1

Stratospheric Modes

The first mode listed in Table 8.1 is found in the tropical stratosphere3 and propagates rapidly to the east with phase speeds of +25 to +35 m s−1 . The vertical wavelength of this mode was estimated to be in the range 17–26 km and possessed a period of 10–15 days and a longitudinal scale of ka = 1–2. It has no meridional velocity component and closely resembles the structure of Matsuno’s Kelvin wave, described in Section 6.1.5.4. Maruyama and Yanai (1966) documented large-scale waves in the lower stratospheric. However, the characteristics of the Maruyama–Yanai (M-Y) waves are decidedly different to the stratospheric Kelvin wave. They possess a higher frequency with a finite meridional velocity component similar in magnitude to the zonal component. These modes also possess long vertical wavelengths and propagated toward the west with phase speeds of −25 m s−1 and been identified as a mixed-Rossby gravity (MRG) wave, described in Sections 6.1.4 and 6.1.5.3. The characteristics of both of these modes, the observed stratospheric Kelvin wave and the MRG wave, appear quite similar to the theoretical waves described in Section 6.1.5. Using the observed vertical structure and Eq. (5.44) we find equivalent depths of about 400 m. With these parameters, Figures 7.3 and 7.4 indicate that the waves occur at low frequency and possess large longitudinal scales. 8.3.2

Transient Tropospheric Modes

We now compare the observed structure of the Madden–Julian Oscillation (MJO, see Chapter 1.4.2, 3 Wallace and Kousky (1968).

8.4 Dry and Moist Modes in the Tropics

(a) MSLP (Pa) – OLR (Wm–2)

(b) MSLP (Pa) – 200 hPa geopotential (m2 s–2) 30

B′

20

20

10

10 A′

0 B

–10

A

–30

A′

0 –10

–20

C′

C

A

–20

0°

50°E 100°E 150°E 160°W 110°W 60°W 10°W

–30

Longitude

0°

50°E 100°E 150°E 160°W 110°W 60°W 10°W

120 100 80 60 40 20 0 –20 –40 –60 –80 –100 –120

Surface Pressure (Pa)

Composite Day

30

Longitude

Figure 8.4 Composite diagrams averaged between 5∘ N and 5∘ S of variance in the 20–90 day period band, from 30 days before to 30 days after maximum convection in the Indian Ocean. (a) Mean sea-level pressure (Pa: shading relative to scale to right) with cold colors representing low-pressure anomalies and warm colors high pressure and outgoing longwave radiation (OLR W m−2 ). Magenta contours represent negative anomalies (i.e. enhanced convection), black contours positive. Contour spacing is 2 W m−2 . (b) 200 hPa geopotential anomalies. Solid contours denote positive values and dashed, negative. Contour spacing is 10 m2 s−2 . Surface pressure anomalies are included to emphasize the differences in phase propagation. In both panels, the white dashed line denotes the propagation of the fast mode at about 25 m s−1 . The solid white line shows the slow mode (6 m s−1 ) associated with convection. The vertical dashed black lines denote the longitudes of the East African Highlands (about 50∘ E) and the Andes (70∘ W).

Figure 1.18a and b) with predictions from theory. From Table 8.1, the MJO has very similar horizontal characteristics to the Kelvin wave with the lack of a discernible meridional velocity component. It also propagates to the east with a phase speed of about 4–8 m s−1 with frequencies in the 30–50 day range. This is far slower in propagation speed and lower in frequency than its stratospheric counterpart. Its longitudinal scale is ka = 1–2 while the vertical scale is 15–30 km. Its eastwards phase speed at 4–8 m s−1 is far slower than the theoretical Kelvin wave of Figure 7.4. 8.3.3

Stationary Modes in the Tropics

In Chapter 1 we identified the macro-scale Walker Circulation. Initially, it was thought these slow circulation features could be described in terms of fundamental modes such as those developed in Chapters 6 and 7. Early numerical experiments of the response of the tropical atmosphere to specified forcing found modes reminiscent of the observed Kelvin and Rossby waves, which in combination produced these stationary features of the tropics with some fidelity. These studies were based on the concept that fundamental modes of the tropics could be “Doppler-shifted” to produce a stationary response.4 4 E.g. Webster (1972, 1981).

In summary, there is a clear mismatch between observations and theory. On one hand, there is a class of equatorial waves in the tropical stratosphere that possess theoretically predicted characteristics. On the other hand, there are equatorial waves that also occupy the entire depth of the tropical troposphere but possess small eastward and westward phase velocities, quite different from theoretical expectations. The question is, how can the characteristics of these two sets of waves be reconciled or can they coexist?

8.4 Dry and Moist Modes in the Tropics Figure 8.4a represents a composite of outgoing longwave radiation (OLR) and mean sea level pressure (MSLP) variability along the equator relative to development of convection (day 0) in the Indian Ocean in the 20–90 day period band. For a geographical context, the two vertical dashed lines denote the longitudes of the East African Highlands in the west and the Andes in the east. The composite extends from 30 days before (day −30) to 30 days after (day +30) maximum Indian Ocean convection. Day 0 matches phase “G” in Figure 1.18a or phase “P3” in Figure 1.18b. The background color shading represents the composite MSLP displaying a distinct eastward progression from the Indian Ocean with a phase speed of roughly 25 m s−1 .

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8 Moist Processes and Large-Scale Tropical Dynamics

.8

(a) Symmetric CLAUS Tb Spectrum

1.25

.7 1.54 n = 1 WIG

.5 25

.4

12 m

m

2.00 50 m

Frequency (cyles/day)

.6

2.50

3 days

.3 .2 .1

6 days

5.00

30 days

10.0 20.0

n = 1 ER MJO

0 –20 –15 –10 –5 0 5 10 Westward Zonal Wavenumber .8

3.33

KELVIN

period (days)

15 20 Eastward

(b) Antisymmetric CLAUS Tb Spectrum

1.25

.7 1.54 m

n = 2 WIG

25

m

50

.6

2.00

12

m

.5

n = 0 EIG

.4

2.50 3 days

3.33

.3 .2

MRG

6 days

5.00

30 days

10.0 20.0

.1 0 –20 –15 –10 –5 0 5 10 Westward Zonal Wavenumber

period (days)

The trajectory B–B′ possesses characteristics of the theoretical dry Kelvin wave. Although it is forced near day 0 in the region of maximum convective heating in the central equatorial Indian Ocean, its eastward progress is not accompanied by convection in the 20–90 day period band. The MSLP patterns appear to be well constrained between these two geographical features. Emanating from the same region in the equatorial Indian Ocean is a slower eastwardly propagating mode associated with strong convection (white dashed line marked B–B′ ). The propagation speed of this mode is about 7–10 m s−1 and eventually decreases in amplitude in the central equatorial Pacific Ocean in a similar fashion to the classical composites shown in Figures 1.18 at times “B” and “P8,” respectively. The same composite analysis but for the 200 hPa geopotential appears in Figure 8.4b. Line C–C′ shows the geopotential propagation to the east possessing similar characteristics to the OLR propagation. The propagation speed is also 7–10 m s−1 . However, there is a second fast signal in the 200 hPa that parallels A-A′ and extends past the Andes into the Atlantic Ocean. In summary, the composite analysis suggests that both dry and moist modes can exist within the tropical troposphere. A slow OLR signal moves slowly from the Indian Ocean to the central Pacific. The MSLP pattern propagating rapidly eastward appearing to lose its coherence near the Andes. In the upper troposphere the geopotential field possesses both the characteristics of the rapid MSLP signal and the slower OLR signal. An intriguing analysis is reproduced in Figure 8.5, providing insight into the structure of equatorial modes associated with convection. The diagram depicts the distribution of the infrared brightness temperature (T b ), as observed from space, expressed as a function of zonal wavenumber ka and frequency 𝜔. Both the characteristics of symmetric and antisymmetric modes about the equator are displayed. The period range encompasses the 20–90 day band used to define the MJO composite (Figure 8.4). The dispersion diagram convention from Figure 6.1a are used so that if ka > 0, the mode propagates to the east whilst if ka < 0 propagation is to the west. The contours show the ratio between actual OLR power and an estimate of the background red noise, averaged between 15∘ N and 15∘ S. Shaded contours indicate areas in wavenumber-frequency space that are at least 10% greater than the background red noise, which is statistically significant at the 95% level. Finally, the spectra are separated into symmetric (Figure 8.5a) and antisymmetric (Figure 8.5b) modes about the equator. Areas of variance in wavenumber-frequency space correspond to the major waveforms observed in the tropical atmosphere, as identified in Figure 6.1a. It is interesting to note that the modes are enclosed within

Frequency (cyles/day)

192

15 20 Eastward

Figure 8.5 Wave number–frequency spectral peaks as a function of zonal wavenumber for a long record of cloud-top temperature (Tb) data, separately for the (a) antisymmetric and (b) symmetric components about the equator between 15∘ N and 15∘ S for the period of July 1983 to July 2005. Contours show the ratio of the actual power relative to an estimate of the red-noise background power. A ratio of greater than 1.1 is statistically significant at the 95% level. Dispersion curves for the equatorial waves with equivalent depths of 12, 25, and 50 m appear as diagonal straight lines. Unit of frequency is cycles per day. Source: From Wheeler and Nguyen (2015).

8.5 Processes

equivalent depths of H = 12, 25, and 50 m that indicate slow moving modes. Strong symmetric signals coincide with the Kelvin wave, the large-scale equatorial Rossby wave, and at low frequencies, the MJO. Within the Kelvin wave space values of the ratio exceed 1.6. The latter mode is the only one that does not correspond to the “classical modes.” It occupies a very low frequency and large zonal scales with a slow eastward propagation. The strongest antisymmetric mode is the mixed Rossby-gravity. The correspondence of the spectral maxima in regions occupied by fundamental modes of the tropical atmosphere indicate that waves must play an important role in determining when, where, and how strong convection will be in the tropics. Also, it should be noted that the areas of maximum convection are located close to dispersion curves at very small equivalent depths, indicating relatively small phase speeds and that only those modes associated with convection appear in the analysis. Modes such as A–A′ in Figure 8.4 are not present. The modes of Figure 8.5 have become known as “convectively coupled equatorial waves”5 (CCEWs). Tacitly, it is assumed that somehow the differences between CCEWs and classical modes emerge through the interaction of equatorial modes and convection. In Section 3.1.5.1, we introduced the concept of the Rossby radius of deformation (R), the intrinsic scale at which rotational effects become as important as buoyancy effects. If a perturbation is made to a balanced geostrophic fluid with a scale of L < R, adjustment back to the previous balance will be accomplished by gravity waves. However, the problem is that at low latitudes as R ∝ 1∕f , then the scale of the forcing must be very large in order to produce the distributions of observed convection. In fact, from Figure 8.4, it appears that the majority of organized convection, especially associated with the ER waves, exists in scales of |ka| ∼ −1 to − 4. For both the MRG and the K waves, there is also strong variance at moderate scales of motion but this should not be surprising as they have characteristics of internal gravity waves. If the scale of forcing is > R, then it may be expected that a signal may project on to these modes during its geostrophic adjustment process. Between Figures 8.4 and 8.5 we find some interesting properties of equatorial modes. First, two Kelvin-like waves of differing phase speeds occur in the troposphere. Second, modes associated with convection possess much slower phase speeds than their “dry” counterparts and behave in a similar fashion to waves with much smaller 5 Wheeler et al. (2000), Wheeler and Kiladis (1999), Kiladis et al. (2009a,b), plus reviews by Kiladis et al. (2009a,2009b) and Wheeler and Nguyen (2015) are very useful.

equivalent depths. For example, the eastward phase speed of a theoretical Kelvin wave with an equivalent depth of 100 m is 2–3 times more rapid than a wave with H = 20 m. The problem is how to reconcile these differences. It is important to reemphasize that the spectra of Figure 8.5 is constructed from OLR variance. This means that variance that is not associated with convection (e.g. the rapidly propagating MSLP modes along the B–B′ axis or the rapid 200 hPa geopotential signal in Figure 8.4 b, also along the B–B′ axis) are not included in the analyses. That is, Figure 8.5 refers only to the convective modes and, unlike Figure 8.4, does not show the character of dual modes. However, in order to delve more deeply into the character of convective modes, we can ask what “essential” physics were not included in the study of dry equatorial waves in Chapters 6 and 7.

8.5 Processes There are two major processes that are not included, at least not explicitly, in classical studies of near-equatorial waves. (i) Dissipation specifically associated with convection: Dissipation associated with convection is generally omitted in equatorial wave theory. However, convection is a turbulent process and one expects a regional increase of dissipation in the vicinity of convection through the vertical mixing of horizontal momentum. In fact, in a very early study Houze (1973) found that dissipation within a convective system was of similar magnitude to other terms in the angular momentum budget. Early simulations of the tropical atmosphere noted that without large dissipation the upper tropospheric response to forcing was abnormally high. The anomalously large response was thought to be associated with smaller scale eddies. Later, a number of studies6 suggested the importance of friction associated with convection was important in tropical easterly waves. The concept of “cumulus friction” (i.e. the dissipation associated with turbulent convective motion and the vertical transfer of momentum) has become widely used over the last few decades in the parameterization of cumulus convection. (ii) Changes in vertical stability associated with convection: In Section 3.1.4.2, we noted that convection reduced the stability of an atmospheric column as originally suggested by Charney (1963). Specifically, it was suggested that the reduction of static 6 E.g. Holton and Colton (1972) and Colton (1973).

193

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8 Moist Processes and Large-Scale Tropical Dynamics

stability in the vicinity of convection allows an equatorial 106 m scale mode to be baroclinic rather than barotropic, thus permitting communication between the lower and upper tropospheres. (iii) Interactive surface fluxes: In examining the surface fluxes at the ocean–atmosphere interface (see, especially, in Section 2.6.4), a substantial enhancement of surface fluxes in the vicinity of large-scale convection was noted. Such an enhancement is not taken into account in classical wave dynamics although it has been included conceptually in the “Wind-induced Sea–Air heat exchange (WISHE)7 .” This concept was introduced as an alternative to “Convective Instability of the Second Kind (CISK)8 ” that attempted to relate frictional convergence to convective growth, discussed below.

Here, for clarity, we have omitted the tildes above the eigenvectors. F is a damping coefficient applied identically to momentum and thermal dissipation and Q̇ represents diabatic heating. In the real atmosphere dissipation will be more complicated than this simple single coefficient, but its inclusion even in this form will help elucidate the basic role of viscous effects. The static stability is represented by Γ. We have used the vertical coordinate Z = −G ln(p∕p0 ), where G is defined in Eq. (5.29d). The vertical velocity of the system is given by wZ = dZ/dt. We now introduce the inviscid Kelvin wave dispersion relationship c = 𝜔/k (from Eq. (6.42) in Section 6.1.5.4) into Eqs. (8.2a) and (8.2b). Then, between Eqs. (8.2a) and (8.2b) we obtain u, 𝜙(y, z) ∝ exp(−𝛽y2 ∕2̂c)

(8.3)

̂c = c + iF∕k

(8.4)

where 8.5.1

Convective Dissipation

Chang (1977) proposed that if viscosity is included in inviscid wave theory, two forms of the Kelvin wave are generated. The first mode (referred to as Mode I) is very close to the frictionless or inviscid case and the classical Kelvin wave solution. Mode II, influenced by dissipation, possesses a much slower Doppler-shifted phase speed and is closer to the observed slowly propagating MJO. In the following development, we will follow Chang (1977) to show how the two Kelvin modes, the “fast” and the “slow,” can occur simultaneously in the tropics. We commence with a system on an equatorial 𝛽–plane, expressed in equation set (5.30). Further, we assume solutions of the form: ̃n , w ̃ n (y, z) un , 𝜙n (x, y, z, t) = R(̃ un , 𝜙̃n , T × exp[ik(x − ct)]

𝜕 2 w′Z 𝜕Z2

+ m2 w′Z =

̂ Q ̂c2

(8.5)

where S 1 − 2 4H 2 ̂c R ⁀ ̂= Q Qe−Z∕2H Cp H m2 =

(8.6a) (8.6b)

and R Γ (8.6c) H Here, m is the vertical wavenumber while H is the atmospheric scale height (RT/g) with a typical value of 7 km or so. For any vertical wavelength 0 (see Section 7.1.2.3) so that ̂cr I > ̂cr II . If dissipation F is set to zero, then from Eq. (8.4) we obtain ̂cr I = c

MODE I

3 MODE II 4

20

5

10

MODE I

6

7 10 50

0

–2 –4 –6 –8 –10 Imaginary vertical wavenumber mi (10–4 m–1)

Figure 8.6 Estimates of the impact of convective dissipation on equatorial modes. (a) Real component of the Doppler-shifter phase speed as a function of vertical wavelength Lz (km) for different damping time scale (days). Two modes are apparent: Mode I is the fast inviscid mode and Mode II, the slow viscous mode. The two modes are separated at cr = ci , shown as the dashed line. (b) Imaginary vertical wavelength as a function of vertical wavelength for different damping time scales. As in Figure 8.5a, the two modes are separated at cr = ci . Source: Based on Chang (1977).

From Figure 8.6a and b it appears that, if a certain vertical wavelength is most efficiently excited by a given forcing function, both modes will be excited for a finite F. The regular mode will have a fast zonal phase speed and less vertical attenuation, while the viscous mode will have a slow zonal phase speed and is more trapped in the vertical away from the source region. We can now return to Table 8.1 and Figure 8.4a and see if some of the behavior of the MJO can be explained through Chang’s convective dissipation theory. First, for a given forcing, two modes will be produced: a rapidly propagating mode and a slower propagating mode. Evidence of a Mode I can be seen in the MSLP

195

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8 Moist Processes and Large-Scale Tropical Dynamics

pattern. On the other hand, a much slower eastward propagating Mode II follows the Mode I. This mode appears similar to the slow eastward propagation of the OLR in Figure 8.4a. Mode II modes appear in Figure 8.5. Chang and Lim (1982) extended their analysis to equatorially trapped Rossby waves and found the same scale duality resulting from convective dissipation. 8.5.2

Stability and Convection

Consider once again the plot of spectral variance of convection in terms of frequency and zonal wave number (Figure 8.5). The correspondence between the locations of the spectral maxima in zonal wavenumber-frequency space is quite remarkable. However, as we have noted, it is puzzling to find that the modes correspond to such small equivalent depths. These depths, of order 20–50 m, are much smaller than the equivalent depths normally associated with theoretical dry equatorial modes. At this stage, it is worth returning to some of the thoughts of Charney discussed in Chapter 3. Scale analysis suggests that modes of 106 m are barotropic despite observations indicating that modes of this scale are baroclinic and associated with convection. Charney (1963) suggested that, in regions of strong convection, the static stability would be sufficiently reduced to allow baroclinic modes and communication between the lower and the upper tropospheres. Kiladis et al. (2009) extended this argument by assuming that the ̇ in thermodynamic equation diabatic heating rate, Q, could be parameterized in terms of net upward motion in convective regions, such that Q̇ = 𝛼Q wp N2

(8.14)

where 𝛼 Q is an arbitrary constant and wp the vertical velocity in pressure coordinates. The thermodynamic equation can then be written as ( ) 𝜕 𝜕𝜙 (8.15) + wp (1 − 𝛼Q )N2 = 0 𝜕t 𝜕z where N2 is the Brunt–Väisälä frequency ((g(𝜕𝜃/𝜕z)/𝜃), as defined in Table 3.1). 𝛼 Q can be thought of as the fractional reduction in vertical stability by convection. If 𝛼 Q were found to be close to but 0 U 0(i.e. westerlies over easterlies) the vertical wavenumber m will increase this by decreasing the vertical scale of the mode. In regions of lateral shear (𝜕U∕𝜕y ≠ 0), the meridional wavenumber with increase or decrease depending on the sign of the shear. In the case of longitudinal basic state variability, integration of a nonlinear model verified the linear kinematics, as shown in Figure 7.17. In Section 7.42, the modification of the waves by the basic state was extended to consider more far-reaching implications. It was found, for example, that the modifications of the waveforms by the variable basic state created regions where emanation to higher latitudes or propagation in the vertical was either enhanced or constrained. These influences were illustrated in Figures 7.18 and 7.22. These conclusions referred to a collective impact of waves. What we did not ask at the time was what was the manner in which the basic states may influence individual transient motions and enhance convection? More specifically, is it possible that the variations of the basic state help with the transition of easterly waves or tropical depressions into tropical cyclones? Many studies have identified the importance of wave activity for tropical cyclone formation. Easterly waves dominate the summer lower troposphere in the Atlantic and Pacific Oceans resulting from instabilities of the ITCZ (see Section 11.3.3). In the Atlantic, Frank and Clark (1980) and Pasch et al. (1998) found associations with easterly waves accounting for at least half of tropical cyclones and almost all category 4 and 5 cyclones. Yet, there is less agreement on the specific mechanisms that promote the transition. Some information may be gathered from the association of the background basic state. Figure 1.17a and b shows the locations and frequency of tropical cyclones around the globe. They appear to follow the rules listed by Gray (1968). They form no closer than about 5∘ latitude from the equator, they exist in regions of low climatological vertical shear, a property of the mean state that can be inferred from Figures 1.7 and 1.8, they are located over regions of SST in excess of 26.5 ∘ C, and they occur in regions of least static stability and over regions where there is a deep ocean mixed layer. Are these sufficient conditions for cyclogenesis?

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8 Moist Processes and Large-Scale Tropical Dynamics

JJAS zonal wind, tropical storms and hurricanes 18 12

1976–2010

565

10°N

500

6

253

400

364

0

260

U

–6

5

0°

60°E

19 120°E

180°

300 200

20°N 120

–12

600

number

zonal wind speed U (m s–1)

200

120°W

30 60°W

100 0 0°

longitude Figure 8.9 Climatological distribution of tropical cyclones over the period 1976–2010 in the Northern Hemisphere summer with totals in the Indian Ocean, Western Pacific, Eastern Pacific, and Atlantic shown as bars. Total number of tropical storms are shown in blue. Violet bars indicate the number reaching hurricane strength. Solid and dashed lines show the 850 hPa zonal velocity at 20∘ N and 10∘ N, respectively.

There are two reasons to suggest that these conditions are not sufficient. First, within the tropics at any one time there are a plethora of MCSs yet only a relative few develop into tropical cyclones. Furthermore, throughout the cyclone season, storm genesis tends to form clusters over time. Therefore, it would seem that there may be another condition that is important for tropical cyclone genesis. An examination of the statistics of tropical cyclone incidence relative to the state of the climatological background flow helps identify such a condition. Figure 8.9 shows the June to September distribution of 850 hPa of the mean zonal wind at 10∘ N and 20∘ N for the 1976–2010 period. The total number of tropical cyclones and hurricanes in each of the four regions of high tropical cyclone incidence (eastern/central Indian, the Western Pacific, the Eastern Pacific and the Atlantic oceans) are shown as bars. In all four areas the stretching deformation 𝜕U∕𝜕x is negative. The coincidence of tropical cyclone formation and the sign of the stretching deformation has been noticed in a number of studies. Holland (1995) and Sobel and Bretherton (1999) noted the potential significance of Webster and Chang’s (1988) energy accumulation theory, reiterated above. Earlier, as discussed in Chapter 7, Farrell and Watterson (1985) had looked at the behavior of Rossby waves propagating in an opposing basis flow. Like Webster and Chang (1988), they also noted changes in the structure of the propagating mode but went further in suggesting that: Cyclogenesis due to zonal variation of the zonal velocity may be as important in the atmosphere as the more familiar growth of perturbations in shear and likely to be complementary Farrell and Watterson (1985, p. 1755)

Specific dynamical mechanisms for tropical cyclogenesis related to the background basic state have been addressed specifically by Done et al. (2011). They considered boundary layer vortex spin-up in a flow containing zonal stretching deformation from a theoretical and modeling perspective. The modification of a wave propagating westward through a constant background basic state with 𝜕U∕𝜕x = −0.8 × 10−6 s−1 , a value representative of the lower tropospheric zonal wind field across the Atlantic Ocean during JJA. These modifications are shown in Figure 8.10. Calculations were based on the equatorially trapped Rossby waves eigensolutions derived in Section 6.1.4. As the mode propagates eastward, it decreases in longitudinal extent, according to Eq. (4.14a), and extends latitudinally according to Eq. (4.14b). In addition, Done et al. (2011) noted that the “wave energy density” also increased in a commensurate manner with the contraction of the longitudinal structure of a westwardly propagating mode and its reduced group speed. This condition was expressed in Eq. (4.20). With the contraction, the relative vorticity increases substantially. Whilst wave energy density is conserved, relative vorticity is not. However, noting the changes in the relative vorticity is important as, following the earlier discussion of CISK theory, “Ekman pumpingIV ” out of the boundary layer is proportional to the boundary layer relative vorticity, i.e. from Eq. (8.16): 𝜔B ∝ 𝜁

(8.17)

Here we have assumed that the vertical velocity at the top of the boundary layer is proportional to the diver̃ . Thus, Done et al. (2011) propose that as an gence ∇•V easterly wave progresses to the west, Ekman pumping increases, providing an added mechanism for raising air to saturation. There is ancillary evidence to support the Done et al. hypothesis. Figure 8.11 presents a composite of the zonal velocity field and stretching deformation along the equator relative to maximum convection in the Indian Ocean during an MJO cycle. The figure is similar to the composites shown in Figure 8.4. Here yellow tones represent the westerly phase of the MJO. Blue tones represent the easterly phase. Dashed contours denote anomalous negative stretching deformation. From the arguments above, these periods would be conducive for the “spinning up” of westerly propagating regions of cyclonic vorticity. A number of studies have found a strong relationship between the phase of the MJO and tropical cyclone incidence in the Pacific domain.13 Specifically, 13 E. g. Liebmann and Hendon (1994) and Hartmann and Maloney (2001).

8.6 Synthesis

Easterly wave transformation in 𝜕U/𝜕x = –0.8 × 10–6 s–1 flow

30°N 0

–1

–2

–4

–6

–8

25°N 20°N 15°N 10°N 5°N Equ 0°

20° 30° 40° degrees longitude

50°

60°

1

2

70° 3

80° 4

90°

100°

5

cyclonic relative vorticity (10–6 s–1) Figure 8.10 Changes in the structure of a moving vortex migrating to the west as a function of time relative to a constant 850 hPa climatological stretching deformation (𝜕U∕𝜕x) of −0.8 × 0–6 s−1 , a typical value in JJA over the Atlantic Ocean. Note the increase in relative vorticity, the longitudinal contraction, and the slowing down of the westward propagation. Vertical lines denote days. Source: Based on Done et al. (2011).

Composite 850hPa U(x) and 𝜕U/𝜕x (5°N–5°S)

30

′

4 3

20

1

0

0

m s–1

composite day

2 10

–1

–10

–2 –20

𝜕U/𝜕x < 0 𝜕U/𝜕x > 0

–30 0°

–3

where KE = (u′2 + v′2 )/2 is conversion of the background kinetic energy to perturbation kinetic energy.14 Rather than an enhancement of cyclogenesis depending on the sign of the basic state, Eq. (8.18) suggests growth in regions where 𝜕U∕𝜕x < 0 or in regions of dashed contours in Figure 8.11. These ideas are substantiated to some degree in Figure 8.12, which shows observed tropical cyclogenesis in a field of deformation during 2005 as a function of longitude and time. Crosses mark the points of tropical cyclone genesis. During this period, cyclogenesis only takes place in regions of negative stretching deformation.

–4

50°E 100°E 150°E 160°W 110°W 60°W 10°W

longitude

8.6 Synthesis

Figure 8.11 Composite diagrams averaged between 5∘ N and 5∘ S of variance in the 20–90 day period band, from 30 days before to 30 days after maximum convection in the Indian Ocean. Same as Figure 8.4 except showing the anomalous zonal velocity component (background shading relative to left-hand scale: m s−1 ) and the zonal stretching deformation (𝜕U∕𝜕x: 10−6 s−1 ). Red contours are negative and black contours positive.

a reduced frequency of tropical cyclones in the eastern Pacific Ocean was noted during the easterly phase of the Madden-Julian oscillation and an increased incidence during the easterly phase. A careful examination by Hartmann and Maloney (2001) showed that the largest term converting background kinetic energy to perturbation kinetic energy was given by the term 𝜕KE ′ ∕𝜕t ≈ −u′ 2 𝜕U∕𝜕x

(8.18)

In this chapter we have attempted to address one of the major problems in tropical meteorology: What is the role of convection in the general circulation of the tropical atmosphere? We addressed three specific questions. What is the role of convection in the energy balance of the atmosphere? What form does convection take and what are the dominant physical processes that determine the scale and longevity of a convective element? How does convection interact with the fundamental dynamic modes of the tropics? Riehl and Malkus (1958) established an essential role for deep convective clouds in the large-scale atmospheric dynamics of the Hadley Circulation. They concluded that vertical transport had to take place within deep 14 Equation (5) of Hartmann and Maloney (2001).

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8 Moist Processes and Large-Scale Tropical Dynamics

11/27

Tropical cyclogenesis 15°N-20°N 2005

10/28 day/month 2005

202

9/28 8/29 7/30 6/30 5/31 5/1 100°W

80°W

–1.2

60°W

40°W

20°W

–1.0

–0.75

𝜕U/𝜕x

(10–6 s–1)

–0.5

–0.25

0°

20°E

0

Figure 8.12 Observed fields of negative stretching deformation (𝜕U∕𝜕x < 0) as a function of longitude along 15∘ N for 2005. Crosses indicate the genesis locations of tropical cyclones in the 15∘ N–20∘ N band. Source: Data from ERA-40 reanalysis.

tropical convection for the mass and energy balance of the general circulation to be accomplished. However, they also noted that the process was not straightforward because of the existence of the climatological mid-tropospheric minimum in equivalent potential, 𝜃 e . It was determined that the entrainment of this relatively dry air into clouds would reduce buoyancy and limit the vertical extent of convection. Riehl and Malkus proposed that convective transport between the tropical boundary layer and the upper troposphere had to take place within undilute cores that would minimize entrainment. However, a plethora of field experiments over the last few decades have not found undilute cores in maritime convection. Zipser suggested that deeper penetration could be invoked by accounting for the latent heat of freezing in convective updrafts in the upper troposphere. However, other processes may also be of importance in facilitating vertical transports. The Riehl and Malkus model was based on the pre-GATE thought that the tropics was populated mainly by deep convective towers. In this model, little of the cloud mass was assumed to be stratiform. In Section 8.2 we considered a more recent view of tropical convection, where organized convection consists of cumulus towers but with a significant stratiform cloud occupying the upper half of the troposphere. The areal extent of the stratiform component is about five times that of the deep convective towers. Whereas the most intense rainfall rate occurs in the cumulus region, almost half of the total rainfall of the system falls from the stratiform region.

The horizontal distribution of rainfall indicates that latent heating of the atmospheric column occurs throughout the MCS, yet the extent of the stratiform region suggests that other forms of heating may be of importance. At the base of the stratiform region there is an absorption of upwelling infrared radiation from the surface. At the top of the anvil there is a cooling to space. During the daytime, this cooling is offset somewhat by near-infrared solar radiation but there is a differential heating through the stratiform of about 10 K day−1 . During the nighttime, this differential can increase to 20 K day−1 , Thus, during the day and the nighttime, substantial radiative destabilization occurs, resulting in buoyancy in the stratiform layer. Neither the Riehl and Malkus nor the Zipser studies considered the role of radiation heating and cooling in enhancing buoyancy in an MCS. The MCS, described above, may be thought of as a basic convective element in the tropics. Areas of convection, such as those appearing in Figures 1.15 and 3.3, are conglomerations of these convective elements. Three particular questions were discussed. How are the convective elements integrated to become a synoptic-scale area of organized convection? Do these collective convective elements feed back on to the basic dynamics of the tropics? Finally, how are boundary layers parcels raised to saturation? With the elucidation of equatorial dynamics in the 1960s and early 1970s it was initially thought that convection were passengers on the large-scale dynamics occurring in the regions of ascent in the fundamental equatorial waves. To some extent this may be true as the horizontal structure of organized convection matches the structure of these modes. Yet waves associated with convection are observed to possess much slower propagation speeds than predicted from theory. Waves not associated with convection, for example as rapidly propagating Kelvin waves in the stratosphere and the troposphere, matched the more rapidly propagating theoretical phase speeds quite well. Theory, modified by the inclusion of processes associated with convection, such as dissipation and reduced vertical stability, predicted slower propagation speeds closer to observation. Finally, we returned to the problem faced by the Riehl and Malkus model (Figure 8.2b) in explaining how the transport of a non-saturated boundary layer parcel is raised to saturation along trajectory A → A’. Two theories were suggested: the CISK theory where the release of stored energy in the atmosphere is achieved by dynamic frictional convergence and subsequent feedbacks that deepen the convection. In this model, the ocean acts as a sink of momentum associated with the frictional convergence that, in turn, raises the parcel to

8.6 Synthesis

saturation. The second model, WISHE, notes the differential distribution of surface fluxes across a disturbance. The differential heating explains the eastward migration of a disturbance, but it does not explain specifically how the raising of parcels to saturation takes place. Finally, we proposed a third model to help explain the lifting of parcels to saturation. This involved the modification of equatorially trapped modes by specific aspects of the basic state. As a wave propagated westward through a region where 𝜕U∕𝜕x < 0, the lower tropospheric relative vorticity increases as the longitudinal

scale decreases, promoting increased Ekman pumping out of the boundary layer (Figure 8.10). It is noted that there is some suggestion that cyclogenesis takes place in regions where 𝜕U∕𝜕x < 0. These ideas possibly form a link between the thermodynamic necessary conditions suggested by Gray (1968) and equatorial dynamics. Noting that the stretching deformation changes with the phase of the MJO, it may help understanding of why there is a temporal clustering of cyclogenesis during the summer.

Notes I Herbert Riehl (1915–1997), sometimes referred to as

“the father of tropical meteorology,” investigated the heat and mass balances of the great circulations of the tropics, especially the Hadley Circulation and its surface component, the trade wind regime. He was German-born and emigrated to the USA in 1933. He eventually found meteorology and became a professor at the University of Chicago after WWII. He mentored a number of graduate students (T.-C. Yeh, Joanne Malkus (Simpson), Noel La Seur, M. Alaka, Charles Jordan, T. N. Krishnamurti, J. V. Colon, R. Simpson, and W. M. Gray), many of whom had distinguished careers in tropical meteorology. A full account of his research can be found in Lewis et al. (2012). II Joanne Simpson (1923–2010) was an American scientist of great renown and influence. She is most remembered for her work on large-scale aspects of the tropics and the manner in which vertical transports of heat and momentum occur. In addition to tropical circulation and cloud studies, she was the director of Project Stormfury, aimed at mitigating the intensity of hurricanes by seeding. She was a mentor to many

young scientists, especially women, and could relate to their professional problems based on her own difficult professional experiences. In our opinion Simpson (1973) should be a standard reference for all students in science, irrespective of discipline. III Initially proposed by Emanuel (1987) who referred to the process as WISHE, defined in the text, and Neelin et al. (1987) who referred to the process as “wind-evaporation” feedback. Subsequent versions of the hypothesis, with significant modifications have been proposed. See, especially, Sobel and Maloney (2013). IV In Section 3.2.5 we discussed the induced vertical motion in the ocean boundary layer by the curl of wind stress at the ocean surface. This is expressed in Eq. (3.79). A very similar situation exists in the boundary layer of the atmosphere. Consider a large-scale geostrophic flow in the lower atmosphere. Surface friction or drag moves the system away from balance and introduces a divergent component to the circulation, inducing vertical motion. This is the Ekman pumping or suction depending on the sign of the vorticity of the original flow.

203

205

9 Extratropical Influence on the Tropics One of the prevailing questions in tropical meteorology has been the degree of involvement of the extratropics in determining the state of the low-latitude atmosphere. Specifically, do the highly energetic extratropical transients influence the tropics? Do the extratropical disturbances of one hemisphere influence the extratropics of the adjacent hemisphere?

9.1 Lateral Wave Propagation in a Zonally Symmetric Basic State In Chapter 3 we discussed two classical papers of Charney’s that attempted to determine the cause of organized convection in the tropics. First, Charney (1963) performed a scale analysis noting that the observed scales of organized convection, motions were essentially barotropic, such that the lower and upper troposphere would be dynamically disconnected. The implication of this analysis was that it was unlikely that in situ mechanisms would provide the initiation of large-scale tropical convection. However, as mentioned in the last chapter, Charney noted that in regions of convection the reduction of static stability could render the atmosphere baroclinic so that connectivity between the lower and upper tropospheres was possible. Yet, for this reduction of stability to exist there would have to be preexisting large-scale convection. Charney (1969) proposed an alternative hypothesis whereby equatorial disturbances were forced by extratropical external influences. At the time of this surmise, a number of theoretical studies had developed the concept of the critical layer at which an incident wave is reflected or absorbed.1 A reflected wave will transfer incident energy to other regions of the domain. An absorbed wave may change the local structure of the background basic state. To consider the problem at hand we need to derive the governing quasi-geostrophic potential vorticity (PV), 1 E.g. Dickinson (1970), Geisler and Dickinson (1974), Beland (1976), and many others.

qg , on the mid-latitude β-plane. We commence with the mid-latitudes because we are interested in the properties of waves originating in extratropics as distinct from the equatorially- trapped modes considered in Chapters 6 and 7. The derivation of a quasi-geostrophic potential vorticity equation can be found in Appendix H, leading to Eq. (H.8). To study the propagation of equatorial modes toward the equator in a zonal symmetric basic state we can split the stream function, 𝜓, and horizontal geostrophic ̃g , as velocity, V 𝜓 = 𝜓(y) + 𝜓 ′ (x, y, z, t)

(9.1)

̃g = U g + Vg′ V

(9.2)

and

where U g is the mean geostrophic zonal wind. With Eqs. (9.1) and (9.2), the linearized perturbation potential vorticity Eq. (H.8) can be written as ) ( 𝜕 𝜕 q′ + 𝛽v′ + Ug 𝜕t 𝜕x g [ ( )] ) ( f02 𝜕 𝜌s 𝜕𝜓 ′ 𝜕 𝜕 2 ′ ∇𝜓 + = + Ug 𝜕t 𝜕x 𝜌s 𝜕z N2 𝜕z +𝛽

𝜕𝜓 ′ =0 𝜕x

(9.3)

where primed quantities denote deviations from the mean state. Assume, now, that 𝜓 ′ (x, y, x, t) = 𝛾(z)Ψ(y, z) exp ik(x − ct)

(9.4)

Here c is the zonal phase speed of a wave and 𝛾(z) = 𝜌s exp(−z∕2G)∕N2 , which allows an algebraic simplification. With Eq. (9.4), Eq. (9.3) becomes ) ( f02 𝜕 2 Ψ f02 𝛽 𝜕2Ψ 2 Ψ=0 −k − + 2 2 + 2 𝜕y2 N 𝜕z Ug − c 4G N2 (9.5) Equation (9.5) possesses solutions of the form Ψ(y, z) ∝ exp(ly − mz). In order for there to be oscillatory

Dynamics of the Tropical Atmosphere and Oceans, First Edition. Peter J. Webster. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

9 Extratropical Influence on the Tropics

solutions (i.e. l and m real), the following conditional must be met: ) ( f02 2 f02 𝛽0 2 2 + >0 l + 2m = − k + N U −c 4GN2

Longitudinal phase speed ω/ k (m s–1)

30 25

g

(9.6) That is: 0 < U g − c < 𝛽0 ∕(k2 + f02 ∕4GN2 )

(9.7a)

Equation (9.7a) states that midlatitude energy can propagate into a tropical area only if the phase speed, c, exceeds the local Rossby phase velocity: U g = −𝛽0 ∕(k2 + f02 ∕4GN2 )

(9.7b)

At the same time, the phase speed must be more negative than U g In other words, in an easterly basic flow, the propagation of waves from extratropics to tropics is possible only if c is more easterly than the basic flow, U g . These findings are summarized in Figure 9.1, which shows the Rossby wave longitudinal phase speed as a function of k at 30∘ N and the wave period. The solid contours denote the zonally averaged U g -field. Assume that the background basic state is 10 m s−1 . For a low-frequency wave of a 25-day period, only those waves longer than about ka = −2 can propagate through the basic state. For a 15-day period wave, this limit has increased to −3. For a 5-day wave, scales longer than ka = −8 would seem to be able to propagate through a −10 m s−1 flow. However, this conclusion requires further consideration. In Chapter 4 we examined the impact of a background basic state on the kinematics of waves. We can use kinematic relationships (4.13a) and (4.13c) to further our understanding of equatorward wave propagation but first need an expression for the meridional group velocity cgy . Consider the dispersion relationship for a Rossby wave in a barotropic flow: 𝜔 = Uk −

(k2

𝛽 + + f02 ∕c2 ) l2

(9.8)

where l is the meridional wavenumber and c2 = gH, where H is the equivalent depth of the shallow fluid. To obtain the meridional and zonal group speeds, we differentiate Eq. (9.8) by l and k, obtaining, respectively: cgy =

2𝛽kl 𝜕𝜔 = 𝜕l (k2 + l 2 + f02 ∕c2 )

cpx =

𝛽 𝜔 =U− 2 2 k (k + l + f02 ∕c2 )

(9.9)

and (9.10)

By our convention, k is a negative number for an extratropical Rossby wave and 𝜔 > 0. Transporting energy

period (dys)

206

–5

20

–10

15

–20

10 5 –10

–9

–8

–7 –6 –5 –4 wavenumber (ka)

–30

–50

–3

–2

–1

Figure 9.1 Longitudinal phase speed (cgx = 𝜔∕k) as a function of wavenumber, ka, and period for different values of the basic state, U (m s−1 ). For a basic state of −10 m s−1 , long waves of greater scale than ka = −3 with a period of 15 days may propagate into the easterlies, according to Eq. (9.7a). For lower-frequency modes (e.g. period 25 days), only scales longer than about ka =−2 can penetrate easterlies of 10 m s−1 . At higher frequencies waves longer than ka = −8 would appear to be able to penetrate.

from the extratropics to the equatorial regions requires that cgy < 0. Solving Eq. (9.10) for l 2 gives 𝛽

l2 =

(U − cpx )

− k2 − f02 ∕c2

(9.11)

and with Eq. (9.9) we obtain cgy =

2kl(U − cpx )2 𝛽

(9.12)

For further insights we return to the kinematic relationships relating stretching deformation to changes in k along a ray (Eq. (4.14a)). As we are considering a zonally symmetric basic state (𝜕U∕𝜕x = 0), Eq. (4.14a) states that zonal wavenumber k will not change along a ray. However, from Eq. (4.14c), the meridional wave number, l, will change if U = U(y). As k is negative, we can rewrite Eq. (4.14c) as dU dl = |k| dt dy

(9.13)

If the zonally symmetric basic state possesses westerlies in the extratropics and easterlies at low latitudes (i.e. dU∕dy > 0), then l will increase in magnitude toward the equator. Conversely, if the basic state were such that dU∕dy < 0, then l will decrease toward the equator. We can now determine the impacts of the changes in l on the meridional group speed. From Eq. (9.9) we know that cgy → 0 if l increases as predicted by Eq. (9.11)

9.1 Lateral Wave Propagation in a Zonally Symmetric Basic State

encroach into the equatorial easterly domain, as pointed out in Figure 9.1. Clearly, (U-cpx ) is positive. However, for those modes in Eq. (4.13c) cgy must decrease in a positive shear regime. Mak (1969) tested the Charney hypothesis using a two-level numerical model with a zonally symmetric basic flow (see Figure 9.2a) with stochastic forcing derived from observations imposed at 45∘ N, with U 1 (y) and U 2 (y) representing the mean zonally symmetric basic flow at 250 and 750 hPa, respectively. Bennett and Young (1971) also considered lateral influence on the tropics from a more basic perspective. Both studies considered zonally symmetric basic states and both noted that the Doppler-shifting of modes and the impact of

if dU∕dy > 0. Thus, irrespective of the value of k, an extratropical mode will not propagate far into the tropics. (i) From Eq. (9.12), as long as U > cpx , meridional dispersion of energy is possible. (ii) As U → cpx , Eq. (9.12) states that cgy → 0. Thus the meridional energy flux should also cease at the critical latitude. (iii) As the wave approaches a critical latitude (i.e. cpx → U), Eq. (9.11) states that l 2 must increase. We can now return to the issue of smaller-scale higherfrequency modes. As stated above, Eqs. (9.7) can be interpreted to suggest that transient modes may Two-Layer steady state model

Zonally symmetric basic state

ω=0

30 m s–1

0 U1, ϕ1, u1, v1, ϕ1

250

1

γ, S

500

U2, ϕ2, u2, v2, ϕ2

2

hPa U2(y)

10

750

ω=0

U1(y)

20

1000 45°S

0°

45°N

30° 23° 18° 6° latitude south

Ω

6° –10

18° 23° 30° latitude north

(b) Perturbation kinetic energy response f(v

) v1 ) f(

200 m2 s–2

45°N

2

1 2

PKE1

0°

100 PKE2

45°S 30° 23° 18° 12° 6° latitude south (a)

6° 12° 18° 23° 30° latitude north (c)

Figure 9.2 (a) Two-level steady state model with zonally symmetric basic state defined by U i in geostrophic balance with 𝜙i . The two levels are 250 hPa (i = 1) and 750 hPa (i = 2) at which variables ui , vi , and 𝜙i are defined. The perturbation vertical velocity (wp = dp∕dt) is defined at 500 hPa. S represents the static stability of the background state. The vertical velocity is set to zero at the top and bottom of the atmosphere. Forcing of the tropics is applied at 40∘ N using a forcing function (f (vi )) derived from the observed mean longitudinal distribution of the meridional velocity component. (b) The zonally symmetric basic state: U 1 at 250 hPa and U 2 at 750 hPa (m s−1 ). (c) Distribution of perturbation kinetic energy (PKE) in the upper troposphere (PKE 1 ) and the lower troposphere (PKE 2 ) (units: m2 s−2 ). Note the rapid reduction of the kinetic energy where U i approaches zero. These latitudes are indicated as solid (dashed) vertical lines along the abscissa for the upper (lower) troposphere. The model is described in Appendix J.

207

208

9 Extratropical Influence on the Tropics

Westerly symmetric basic state 30 m s–1

Easterly symmetric basic state 30 m s–1 20 U1 = U2 = –5 m s–1

30°S

18°

20 U1 = U2 = 5 m s–1

10

6°

6°

18°

30°N

30°S

18°

10

6°

–10

6°

18°

30°N

–10

Perturbation kinetic energy response

Perturbation kinetic energy response

200 m2 s–2

200 m2 s–2

PKE1

100

100

PKE1

PKE2

PKE2 30°S

18°

6°

6°

18°

30°N

latitude (a)

30°S

18°

6°

6°

18°

30°N

latitude (b)

Figure 9.3 Results of two experiments with the steady state two-level model described in Figure 9.2. (a) Constant westerly zonally averaged basic state where U 1 = U 2 = −5 m s−1 and (b) a constant easterly basic state where U 1 = U 2 = +5 m s−1 . The latitudinal distribution of the perturbation kinetic energy at both levels are shown in the lower panels. The model is described in Appendix J.

shear on the background flow inhibited propagation of wave disturbances into the tropics. Figure 9.2b shows the results of a steady (i.e. 𝜕/𝜕t = 0) Mak-like model2 (Webster 1973) forced at the lateral boundaries with observed mean forcing f (vi ). The upper panel shows the basic state. The lower panel provides a plot of the perturbation kinetic energy (PKE i ) at the two model levels (i = 1, 2). At both levels PKE decreases rapidly equatorward of the latitude at which U i = 0. The results of further hypothetical experiments using the steady state model hint at how the extratropics may influence the tropics. Lateral white noise spatial forcing was introduced at 40∘ N for a constant easterly basic state (Figure 9.3a) and a constant westerly basic state (Figure 9.3b). In the easterly case, the tropics appear impervious to extratropical forcing. On the other hand, if the tropical domain contains westerlies, there is a sizeable response both in the tropics and the other hemisphere. The question may be raised as whether in a zonal state that varied longitudinally, such as in Figure 1.8, and contained regions of near-equatorial westerlies, 2 The steady state model is described briefly in Appendix J.

would allow regional incursions of energy into the tropics and, perhaps, from one hemisphere to the other?

9.2 Equatorial Wave Propagation in a Zonally Varying Basic State Hints of extratropical influences on the deep tropics had been noted well before the Charney papers. For example, Riehl (1954) observed that: … The intermittent appearance of high tropospheric westerlies on the equator … is a foreign thought in classical views of the general circulation. Yet … (there is evidence) … that they do occur. … Since flow in the high levels is so unsteady, coupling at high altitudes between the circulations of the northern and southern hemispheres promises to provide an important link in the understanding of the fluctuations of the general circulation…. Riehl (1954, p. 254)

9.2 Equatorial Wave Propagation in a Zonally Varying Basic State

which led him to make a prescient conclusion. He noted that there was considerable evidence of:

Riehl’s surmise was made at a time of very little upper tropospheric data in the tropics. However, in the 1970s as the density of tropical observations increased, it became apparent that there was a large longitudinal variability in the mean state of the tropical atmosphere deviating from the zonally symmetric, as apparent in Figures 1.8 and 1.11. The westerly part of the variability became known as westerly wind ducts (WDs) and tropical upper-tropospheric troughs (TUTTs). Although the physical nature of the WDs and TUTTs and the physics of their creation would have to wait a couple of decades, their coexistence with the high-frequency activity became apparent in the 1970s. Figure 9.4 plots the 200 hPa distribution PKE, along with contours of the background basic state U(x, y) = 0. Here PKE is defined as (u*2 + v*2 )/2, where u* and v* represent temporal deviations from longer-term averages. These were the first tropics-wide diagrams to show Distribution of PKE (m2 s–2) (i) January 1971

latitude

30°N

20 10 5

0°

10

20

10

U8 8 2

B′

>8

equ low-trop monsoon flow

10°S 30°E

45°E

60°E (a)

75°E

(i) 15°N JJA Section A-A′ 400

335 325

600

320

600 315

800

305

1000 30°E

300 40°E

50°E 60°E longitude

335 330 325

320 315

310

800

90°E

(ii) Diagonal JJA Section B-B′ 400

330

pressure (hPa)

254

70°E

300 1000 31°N/ 26°N/ 40°E 45°E

80°E

310 305 300

21°N/ 16°N/ 11°N/ 6°N/ 50°E 55°E 60°E 65°E latitude/longitude

(b) Figure 12.7 (a) Schematic of the impact of the outflow of the Arabian heat low (brown arrows). Smith (1986a, 1986b) referred to the outflow as the desert “lateral exhaust” that occurs over the summer monsoon boundary layer flow (bold blue arrows) in the Arabian Sea. Mean JJA isohyets (units: mm day–1 ) are shown as red dotted contours. Below the desert outflow region there is a minimum of rainfall. Source: Based on Figure 15 from Smith, (1986b). (b) Isentropic analysis along the sections marked A–A′ and B–B′ in 12.7(a). Both sections show isentropes sloping downwards toward the Arabian Sea. As the flow is largely adiabatic, flow (red arrows) will be along the isentropes promoting a strong inversion over the eastern Arabian Sea across the southwest monsoon flow.

Depending on the time of the year, the boundary layer could undergo an almost a complete inertial oscillation by dawn. 12.5.3

A Subsiding Lateral Exhaust

From Figure 12.5 it is clear that the middle and lower troposphere undergoes a net energy gain due to diabatic processes. There is only one mechanism that can balance this energy gain and that is a lateral transport of energy out of the region.9 For the Arabian heat low, the

9 Smith (1986b).

main lateral transport is toward and over the western Arabian Sea, where the descending warm air serves to maintain an inversion in the West Arabian Sea. In turn, the inversion traps water vapor in the lower troposphere as it is carried toward India in the strong southwest summer monsoon flow (Figure 12.7a). Cross-sections of isentropes along the sections A–A′ and B–B′ sloping downwards toward the east appear in Figure 12.7b. As the flow out of the deserts is largely adiabatic the flow will be downward along the isentropes. This stable descending air creates an inversion that is largely responsible for the cloudlessness and the suppression of rainfall in the western Arabian Sea. Smith (1986b, p. 1101) concluded that:

12.6 Desert–Monsoon Relationships

… There would appear to be scales of influence (of the desert regions) embedded within the larger domain in which the deserts manifest themselves as local energy sources to processes that feed back to the larger scale – in particular, feed back to processes that influence the distribution of moisture which the deserts themselves lack….

50°N

2

30°N

6

4 –10

–8

15°N latitude

In essence, the “lateral exhaust” from the desert, as Smith called it, appears to be an influence by the desert regions on near-equatorial and monsoon rainfall.

Monsoon forcing: wp response (10–3 N m–2 s–1) 90°N

–4 –2

0° 0 0

15°S

12.6 Desert–Monsoon Relationships Rodwell and Hoskins noted that the major premise of Charney’s theory was that the radiation loss and high surface albedo enhanced the zonally symmetric Hadley Circulation in the subtropics. This led to the comments that: … The mechanism for desertification proposed here is that the Asian monsoon heating induces a Rossby-wave pattern to its west. Integral with the Rossby pattern is descent and a thermal structure with isentropes sloping down into the monsoon region. Air on the southern flank of the mid-latitude westerlies interacts with the warm structure and, even in the absence of diabatic effects, descends as it glides down the sloping isentropes. Rodwell and Hoskins (1996, p. 1399) This isentropic “gliding” may be seen in sections A–A′ and B–B′ in Figure 12.7b(i) and (ii). Hoskins and Rodwell had noted that many earlier studies had assumed that deserts were the result of subtropical subsidence from the Hadley Circulation. However, maximum Hadley subtropical subsidence occurred during the boreal winter (Figure 10.5c). During JJA, the zonally averaged subsidence is much weaker so that it was concluded that additional subsidence was necessary to maintain the desert regions. Rodwell and Hoskins suggested that the large-scale Rossby wave associated with monsoon heating provided sufficient additional subsidence to maintain the desert climate. Figure 12.8 offers some support to the hypothesis. Forcing, denoted by the colored shading, is placed at 25∘ N, close to the center of JJA monsoon precipitation, as shown in Figure 1.6. The steady state model is a linear, baroclinic spherical primitive equation model with a symmetric basic state with horizontal and vertical shear. The model is described in Section 9.1. An iterative heating scheme was used to calculate the total diabatic heating resulting from the initial imposed heating plus

0 30°S 50°S 90°S 0°

45°E

90°E longitude

135°E

180°E

Figure 12.8 Steady state solution of a simple linear model relative to heating (colored ellipse) imposed at 25∘ N at the central longitude in a background flow with vertical and latitudinal flow. Contours denote vertical velocity distribution (wp : 10−3 N m−2 s−1 ) at 500 hPa. Note the region of descent (red contours) to the west of rising motion. The two-level state steady is described in detail in Appendix J and Figure 9.2, together with the basic state. The model extends from pole-to-pole. Source: Based on Webster (1981).

a vertical velocity response resulting from feedbacks between the heating and the dynamic response of the system.10 The final heating is shown in Figure 12.8 and the vertical velocity response with black contors denoting ascent and red subsidence. Two major vertical velocity regimes may be seen, one over the initial heating and a second a broad scale area of subsidence. In essence, Rodwell and Hoskins were proposing links between regions of strong diabatic heating over the monsoon regions and its associated subsidence to the west and thus to the maintenance of deserts. In concert, the results of Smith (1986b) and Rodwell and Hoskins suggest that there is a strong two-way interaction between a monsoon and an adjacent desert. At the same time, the same mechanism may come into play with monsoon heating intensifying the ocean surface anticyclones either adjacent to the convective regions or across the equator.11 There is a further implication of the descending flow from the desert regions over the western equatorial and 10 The iterative heating scheme is discussed at length in Section 3 of Webster (1981). 11 Hoskins and Wang (2006).

255

256

12 Arid and Desert Climates

northwest Indian Ocean. During the northern summer, the descent causes a regional stabilization of the lower troposphere. This will turn out to have important influences on the form of cross-equatorial flow and its stability. We will address this issue in Chapter 13. Thus two factors conspire for a funnellng of moisture to India and South Asia. The first is the boundary flow trapped to the east of the East African Highlands producing the Somali Jet. The second is the subsiding lateral exhaust that reduces convection and precipitation in the western Indian Ocean and the Arabian Sea (Slingo et al. 2005)

Finally, one other question arises. Why does the monsoon not form over some other region within the subtropics? In Chapter 16, when we discuss the physical structure of the macroscale monsoon circulation, we will argue that the elevated heating associated with the Himalayan-Tibetan Plateau (HTP) anchors the South Asian monsoon heating, making it a unique location for the major subtropical monsoon. By extension, then, one could argue that the location of the major desert region, extending across Southwest Asia, the Middle East, and North Africa, ultimately depends upon the location and scale HTP.

Notes I An idea of what the climate of the Sahara may have

been some 7000 years ago is suggested by the wall paintings of the Caves of the Swimmers in Southwest Egypt. The Neolithic paintings show humans swimming and a far larger range of animals than exist there today. The cave was discovered by the Hungarian explorer László Almásy in 1933, who appears prominently in the novel “The English Patient”

by Michael Ondaatje, which became the basis of the film of the same name. II Self-induction is a term borrowed from electronics where an electromagnetic field (EMF) is induced in a circuit when the current in that circuit varies. This is distinct from induction where changes in the circuit are produced by an externally applied EMF.

257

13 Near-Equatorial Precipitation The tropics are characterized by regions of relatively intense precipitation and, often in close proximity, arid, dry desert regions as shown in Figure 1.6 and discussed in the last chapter. The major pluvial regions occur in the monsoon systems of Asia, Africa, Australia, and the Americas, in addition to the Indo-Pacific warm pool, the zonal bands across the tropical oceans, the ITCZ, and the diagonally oriented Great Cloud Bands (GCBs), extending across the southern oceans. The drier regions are associated with large-scale quasi-stationary summer low-pressure zones over North Africa, the Arabian Peninsula, Southwest Asia, and the Southwest of North America in the NH and over Southern Africa and central Australia in the SH. Oceanic precipitation minima usually coincide with the subtropical high-pressure zones. Therefore, paradoxically, the driest regions of the tropics are associated with high surface pressure over the ocean and low surface pressure over land. However, also paradoxically, low pressure over land is also associated with some of the most copious rainfall on the planet, such as in the South Asian region. In the last chapter, the basic physics of arid and desert regions was considered and also the possible dynamic interaction between dry and pluvial regions of the tropics. Here, we consider the near-equatorial regions with a high precipitation rate, in particular the ITCZ and the GCBs of the SH, all easily identifiable in Figure 1.6. We postpone a discussion of monsoon climates until Chapter 16. The problem of understanding tropical precipitation distributions may be illustrated by the following example. For a number of decades, a question often raised in tropical meteorology, often contentiously, is why precipitation maxima lie on the equator in some longitudes but off the equator in others. This question has been confounded by confusion between the location of troughs and zones of convergence, such as the ITCZ and association of each with sea-surface temperature (SST) maxima. This question has tended to be avoided by using the simple explanation that the ITCZ coincides with the maximum SST also supposedly coinciding, as well, with the location of the maximum convergence of the trade wind regimes and the lowest surface pressure.

For example, Philander (1990, p. 17) makes a simple association that: … moist air rises where sea surface temperatures are highest and subsides where the surface waters are cold … This connection may not be the case, at least not in general. Hastenrath (1994, Section 6.7.2) points out a more complicated structure, one in which embedded within the oceanic low-pressure trough are regions of wind convergence, axes of confluence between the trade-wind regimes of the two hemispheres, and bands of maximum precipitation. Furthermore, Hastenrath also indicated that there is a general misconception that all of these features, often referred to collectively as the ITCZ, are of similar scale and are necessarily collocated in latitudes. Careful regional analysis of research aircraft data such as there was at that time (e.g. Ramage 1974) showed that collocation of maximum SST and minimum surface pressure rarely occurs, pointing out that over the ocean the lowest surface pressure occurred in clear skies, with a separation between the trough and maximum precipitation of about 100–200 km, the former lying poleward of the latter. Ramage raised the question that if maximum convection were simply tied to the maximum in the SST, how could this extreme SST be maintained in what would be a region of decreased insolation and maximum upwelling? The identification of the ITCZ over land is perhaps even more troublesome. For example, Nicholson (2018) lists a number of definitions used in literature discussing African rainfall: the ITCZ is referred to separately as the location of low-level wind convergence, the zone of maximum rainfall, the region of minimum surface pressure, and the loci of propagating disturbances. S. E. Nicholson1 prefers the use of the term “tropical rainfall belt” over land. The multiple and seemingly muddled definitions of relative locations of these low-latitude phenomena over 1 Personal communication.

Dynamics of the Tropical Atmosphere and Oceans, First Edition. Peter J. Webster. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

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13 Near-Equatorial Precipitation

the ocean and the areas are not just a matter of the use of different nomenclature or being arrived at through the use of different data. Rather, the various definitions have tended to obscure basic physical understanding of large-scale tropical phenomena. Here, we attempt to outline the basic differences of these low-latitude phenomena and determine their various roles in the maintenance of the large-scale structure of the tropics.

13.1 Near-Equatorial Distributions of Precipitation We noted in Chapter 1 that the climatological ITCZ appears as a narrow band of precipitation located on one side of the equator or occasionally on both sides simultaneously, depending on the ocean basin and the season. Its character can be seen vividly in Figure 1.6, especially in the Pacific Ocean zone. Here we explore the physics of the ITCZ. Noting the vertical distribution of vapor pressure plotted in Figure 1.2, eventually near-equatorial precipitation must be the result of horizontal convergence of moisture. In fact, the almost zonally oriented bands define the region of confluence of the trade-wind regimes and are the point of coalescence, in a mean sense, of the local Hadley Circulations. Whereas the importance of the deep tropical convection is readily accepted, there still appears to be considerable argument and confusion regarding the nature of the ITCZ and the mechanisms that define its location. Viewed on time scales of weeks to months, the zonally oriented convective phenomenon appears as more or less a coherent entity girdling the planet. On shorter time scales, the ITCZ is made up of a series of propagating disturbances and, in the major warm pools, aperiodic conglomerations of mesoscale convective clusters (Figure 1.16a). During the boreal summer, the regions of organized convection are generally found in the NH displaced 4∘ N–13∘ N from the equator in certain regions, especially in the eastern sections of the Atlantic and Pacific Oceans. Over the Indian Ocean, organization is distinctly to the north of the equator in the boreal summer and to the south during the boreal winter. The collocation of the mean ITCZ and the 2–10 day high-frequency variability in all four seasons can be seen clearly in Figure 1.15. 13.1.1 Relationships Between Convection, MSLP, and SST Figure 13.1 displays the mean SST (∘ C) for both JJA and DJF together with contours of outgoing longwave radiation (OLR: W m−2 ). Maximum precipitation

(minimum OLR) generally occurs in the summer hemisphere located to the north or along the equator. For later reference, regions of strong and weak surface cross-equatorial pressure gradient, determined from Figure 1.7, are noted along the equator in each diagram as “strong,” or “weak.” The bold yellow line represents the location of the 925 hPa zero absolute vorticity contour (AV: 𝜂 = f + 𝜁 = 0). We note that there are two major regimes defined by the relative position of 𝜂 = 0 and the OLR. Either the 𝜂 = 0 contour is equatorward of OLR minima (e.g. eastern Pacific Ocean) or, alternatively, the zero contour bisects the OLR minima as in, for example, the Indo-Pacific warm pool. Figure 13.2 examines these relationships further in terms of mean seasonal cross-sections of OLR, SST, and mean sea-level pressure (MSLP) averaged across 30∘ longitude sectors through the Indian Ocean, the East Pacific Ocean, and the Africa-Atlantic Ocean. During the boreal winter, in the Indian Ocean sector, the maximum SST occurs as a broad band just south of the equator with two local maxima near 4∘ S and 14∘ S. Convection is deepest (minimum in OLR) at 8∘ S. A near-surface cross-equatorial pressure gradient (CEPG) is directed into the SH. During the boreal summer, the Indian Ocean maximum SST is nearly centered over the equator. The CEPG has reversed since winter and is now directed northwards toward South Asia. There are two regions of maximum convection, near the equator and between 15∘ N and 20∘ N, the latter occurring equatorward of the low-pressure minimum. We have already noted in Figure 1.6 that the boreal winter ITCZ in the eastern Pacific Ocean is weaker than its boreal summer counterpart. Convection, with reduced magnitude, is also closer to the equator. In the boreal summer it is located near 8∘ N–10∘ N, equatorward of the maximum SST. Between Figure 13.2a and b (i) and (ii), we see that the CEPG during winter is also weaker than in JJA. There is a similar contrast between winter and summer in the African sector (Figure 13.2c). During DJF, maximum convection, albeit relatively weak, lies just north of the equator. A weak CEPG extends into the NH, reflecting the cooler SSTs in the SH (Figure 1.6). During the boreal summer, the pressure gradient associated with the African monsoon has increased substantially. The OLR minimum now occurs at about 10∘ N, well equatorward of the minimum surface pressure. Collectively, Figures 13.1 and 13.2 suggest that: (i) The equatorial trough (minimum MSLP), convection (minimum OLR), and the SST maxima appear collocated when there is little or no cross-equatorial pressure gradient. These collocations appear in the eastern Pacific and eastern Atlantic during DJF and in the warm pool region of the tropical western Pacific.

13.1 Near-Equatorial Distributions of Precipitation

SST (°C), OLR (W m–2), η = 0

30

30°N (a) DJF 15°N

mod

0°

29

weak

15°S 30°S 120°E

28 150°E

180°E

150°E

120°W

90°W

60°W

30°N 15°N

(b) DJF

0°

27

strong

weak

26

15°S 30°S 60°W

30°W

0°

30°E

60°E

90°E

120°E

25

30°N 15°N

SST (°C)

Figure 13.1 Sea-surface temperature (∘ C: right-hand scale) and outgoing longwave radiation denoting deep convection (black contours: 0 A

η/f < 0

B EQU

D C

η0 F EQU

E η/f < 0

H η/f < 0 G

15°S 60°W

30°W Jan

0°E Feb

30°E Jun

η> 𝜁 . However, close to the equator, as f → 0, it is possible that 𝜂 will change sign such that 𝜂/f < 0. This can occur if fields of 𝜂

are advected across the equator under the action of a CEPG. “Contra-signed 𝜂” can be seen in all of the cross-sections in Figure 13.2 as the gray-shaded bar on one side of the equator or the other. These regions exist where there is a lower pressure poleward in the summer hemisphere and a higher pressure in the winter hemisphere, resulting in a divergent wind toward the summer hemisphere and the advection of AV across the equator, producing a region of 𝜂/f < 0. The CEPG, in turn, is associated with the SST gradient (or the existence of a land mass such as Africa, South Asia, or North Australia), and thus varies slowly over time. In the following paragraphs we elaborate on the dynamic theory posed above. While the theory is not a panacea, it does go some distance in explaining many features of the ITCZ. Eventually, we will meld this theory with that of Holton et al. (1971), incorporating propagating equatorially trapped disturbances.

9 Tomas and Webster (1997), Tomas et al. (1999), and Toma and Webster (2010a, b).

Figure 13.3 describes the 1979–2015 mean monthly positions of the 925 hPa 𝜂 = 0 contour for the months

13.2.1

Distributions of Absolute Vorticity

263

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13 Near-Equatorial Precipitation

of December–February and June–August. A number of important characteristics appear: (i) For the most part, is AV is negative in the SH and positive in the NH, increasing in magnitude away from the equator. This general change in the magnitude of 𝜂 with latitude is best seen in Figure 13.2. However, Figure 13.3 shows that this “rule” is broken over wide regions where the 𝜂 = 0 contour strays either southward or northward a few degrees into the summer hemisphere. For example, in the eastern Pacific in all months, the zero AV contour is displaced northward of the equator. Maximum displacement occurs during July and August (A) and minimum displacement in the winter months (D). (ii) In the western Pacific Ocean to the north of Papua-New Guinea, there is a large northward displacement of the 𝜂 = 0 contour associated with displacements of the monsoon trough (B). Displacements into the SH in the western Pacific occur during December–February (C) in the period of maximum amplitude of the South Pacific Convergence Zone (SPCZ). (iii) In the Atlantic Ocean the zero contour remains north of the equator during all months but further poleward in the boreal summer (F) than in the winter (H). (iv) The largest off-equator displacements of the zero AV contour occur in the Indian Ocean. During the austral summer, the zero AV contour extends to 8∘ S in the western Indian Ocean (G), moving closer to the equator in the east. In August, the zero AV contour assumes its most northerly position near 10∘ N in the northwestern Indian Ocean (E). By comparing the locations of the zero 𝜂 and the location of convection (e.g. Figure 13.1b) some important inferences can be made. In the presence of a CEPG, the zero 𝜂 contour moves from the equator in the direction of the pressure gradient. In between the equator and the zero 𝜂 contour is a region in which 𝜁 > f , defining a region of negative 𝜂 in the NH and positive in the SH. These regions occur as a result of the advection of 𝜂 across the equator by the CEPG. Finally, returning to Figure 13.1a, we note that the zero 𝜂 contour, when it is located away from the equator, is on the equatorward side of the convection. We will now suggest that these features are signatures of inertial instability. Inertial instability finds its roots in mechanics. As machines advanced in sophistication through the industrial revolution, it became important to understand the development of turbulence in rotating axisymmetric coaxial pipes. Under some certain circumstances the pipes vibrated drastically. The problem was tackled by some of the leading fluid dynamicists of the day.III Through these efforts, the concept of centrifugal

instability was developed. Extensions of these concepts to flow on a rotating planet lead to the concept of inertial instability that appears extensively in geophysics. Consider a parcel located some distance r from the axis of rotation in a rapidly rotating cylinder of fluid (Figure 13.4). If the rotation rate of the system is very large, the radially directed centrifugal force10 will exceed gravity substantially. This allows us to neglect gravity and consider the balance of forces between the (i)

Ω2

Ω1

A

B

g

D r1

C r2 r

(ii) A

Ω1r12

B

r

D r = r1

r+δr

Ω2r22

C r = r2

Figure 13.4 Centrifugal and inertial instability. (i) Schematic diagram of centrifugal instability between two differentially rotating cylinders. If the rotation rate of the inner cylinder (Ω1 radians s−1 ) is less than the rotation rate of the outer cylinder (Ω2 ), the angular momentum Ma increases outward (i.e. dMA ∕dr > 0), which is inertially stable. However, if Ω2 < Ω1 and MA increases outward (i.e. dMA ∕dr < 0), the flow will be unstable. Then, the lateral movement of a parcel will cause energy to be liberated and instability to ensue. Panel (ii) presents a detail of displacements in the plane ABCD of panel (i). 10 “Centrifugal” is a combination word coming from the Latin “centrum” for center and “fugere” to flee. The term was introduced by Sir Isaac Newton in Principia in 1686.

13.2 Dynamic Instabilities Associated with a Cross-Equatorial Pressure Gradient

centrifugal force, directed away from the axis of rotation, and the pressure gradient force directed toward it. If the angular velocity of the parcel is 𝜛, then its angular momentum, MA , is 𝜛r2 . MA will be conserved if the parcel is moved laterally, providing there are no tangential forces present. Consider now a lateral movement of the parcel to location (r + 𝛿r) where it attains a new angular velocity (𝜛 + 𝛿𝜛). From the conservation of angular momentum principle we can write MA = (𝜛 + 𝛿𝜛)(r + 𝛿r)2 𝜛 + 𝛿𝜛 = MA ∕(r + 𝛿r)

or

2

(13.1)

The centrifugal force acting on the displaced parcel can be expressed as: (𝜛 + 𝛿𝜛)2 (r + 𝛿r) = MA2 ∕(r + 𝛿r)3 or (𝜛 + 𝛿𝜛)(r + 𝛿r)2 = MA2

(13.2)

which is exactly balanced by the outward-directed centrifugal force. If the parcel arrives at its new location with a smaller centrifugal force than the environment, then the pressure gradient will apply a restoring force and the parcel will be returned to its initial location. If, on the other hand, the parcel finds itself with a larger centrifugal force, then it will accelerate away from its perturbed location. We now have a set of criteria for the state of inertial stability: (i) If the LHS > RHS in Eq. (13.2) and angular momentum increases outwards, there is inertial stability. This is the equivalent of the outside pipe in Figure 13.4 rotating faster than the internal pipe. This is referred to as the Rayleigh stability criterion (Strutt 1916) where stability is assured if MA (r) < MA (r + 𝛿r) . (ii) If the RHS = LHS, the situation is neutral, the parcel and the environment are identical, and the parcel remains in the position to where it is displaced. This occurs if the pipes rotate with the same angular velocity. (iii) If, however, the RHS < LHS (i.e. the inside pipe rotates faster than the outside pipe), then the angular momentum decreases outwards and there is inertial instability. Simply, the angular momentum of a fluid must decrease away from the axis of rotation for stability.IV

13.2.2

Geophysical Context for Inertial Instability

It may seem that vibrations in fluids between rotating coaxial pipes have little to do with the tropical atmosphere and ocean, but the connection is rather straightforward and very important. In fact, the relevance of inertial instability has a long history in

geophysical sciences. Here we concentrate on the inertial instability of the atmosphere. Solberg (1933) introduced the concept of dynamic instability (now more frequently referred to as symmetric instability) into meteorology. He noted that instability existed if the anticyclonic wind shear along an isentrope exceeded the Coriolis parameter. The name, symmetric instability, is derived from the observation that circulations resulting from the instability are symmetric when viewed along the direction of the basic-state flow (e.g. Emanuel 1988). Solberg made use of the parcel method, allowing symmetric instability to be seen as the result of two forces that are nearly perpendicular to each other, one acting in the horizontal direction (the imbalance between the pressure gradient and Coriolis forces) and the other in the vertical direction (the imbalance between the vertical pressure gradient and gravitational forces, defining buoyancy). Inertial instability is closely related to symmetric instability but involves only horizontal forces. For motion along a constant potential temperature surface, the buoyancy force is zero and symmetric instability reduces to inertial instability. For motion along a surface of constant angular momentum, symmetric instability reduces to convective instability (e.g. Emanuel 1988). Given the near parallel nature of isobaric and isentropic surfaces in the tropics (see Figures 1.2 and 3.1), inertial and symmetric instabilities are effectively synonymous for large-scale tropical flows. The relevance of symmetric instability to meteorological problems was addressed first by Sawyer (1949) a little over a decade after Solberg’s work was published. Sawyer examined the possible role of symmetric instability in the formation of extratropical cyclones, in determining limits on wind shear on the equatorial side of jet streams, in the formation of tropical cyclones and frontal structures, and the maximum pressure attainable in an anticyclone. He concluded that, although it may not be important for the formation of extratropical cyclones (the cause célèbre of the time), it possessed relevance in explaining many geophysical phenomena. Through the 1950s and 1960s, the importance of inertial instability appeared to lose its luster compared to the success of baroclinic instability and the emergence of the quasi-geostrophic balance approach. Part of the reason was that the theory of baroclinic instability (Charney 1947; Eady 1949) offered a longitudinal scale for the most unstable mode that matched the scale of the observed fastest growing extratropical modes. Inertial instability, more of a zonally symmetric parcel concept, offered only a frequency range for induced motion, depending at which latitude the instability occurred. During the last two decades, there has been somewhat of a resurgence of interest in inertial instability.

265

266

13 Near-Equatorial Precipitation

Today, the concept arises in a wide range of subjects: the dynamics of mesoscale convection and monsoons, wave generation and breaking in the stratosphere and mesosphere, and the maintenance of jets in planetary atmospheres and equatorial oceanography. Its importance in low-latitude dynamics has been addressed frequently in recent years.V Following Tomas and Webster (1997) and Toma and Webster (2010a, b) , we add to this list its role in determining the latitude of the ITCZ.

13.2.3

Concept of “Perpetual” Instability

Inertial instability results from an imbalance between the environmental pressure gradient and the inertial terms in the equations of motion. Consider a geostrophic basic state where U g (y) = −

1 𝜕p f 𝜌 𝜕y

(13.3)

where U g (y) is the mean geostrophic zonal wind. Neglecting friction and assuming there is no zonal pressure gradient, we can write the momentum equations as dy du = fv = f dt dt dv = f (U g − u) dt

( f−

𝜕U g

) 𝛿y = −f

𝜕y

𝜕Ma 𝛿y 𝜕y (13.7)

d 2 𝛿y dt

2

+f

𝜕Ma 𝛿y = 0 𝜕y

(13.8)

where the angular momentum Ma = fy − U g . There are three possible solutions to Eq. (13.8): (i) If the coefficient of the second term is negative (i.e. Ma decreases away from the equator), the solution to Eq. (13.8) states that if there is a poleward displacement of a parcel, then there will be exponential growth of the displacement 𝛿y. The fluid is then said to be inertially unstable. Such growth occurs in regions where 𝜂/f < 0, as displayed in Figure 13.5a. Following displacement, the parcel finds itself at a speed less than geostrophic (i.e. subgeostrophic: u < U g ) and the displaced parcel (a) Inertially unstable

(13.4a) dMA

(13.4b)

Here u and v are the zonal and meridional horizontal velocity components. The distribution of U g (y) depends on the meridional pressure gradient. The question we ask is whether for a given geostrophic basic state U g (y) will a parcel displaced laterally be forced to return to its original position and remain in its displaced position or accelerate away in the direction of the displacement. This is the same question that was asked about displacements in the rotating annulus considered in the last section. Using Eq. (13.3), consider an initial poleward displacement of a parcel so that u(y0 + 𝛿y) = U g (y0 ) + 𝛿u = U g (y0 ) + f 𝛿y

U g (y0 + 𝛿y) = U g (y0 ) +

𝜕U g 𝜕y

𝜕u v dt 𝜕y

dy

0

η/f > 0

>0

Figure 13.5 Schematic diagram of inertial instability near the equator. The situation is almost identical to that portrayed in Figure 13.13. However, a fluid ring moving closer to the axis of rotation will follow a spherical path. If the absolute angular momentum of the fluid decreases toward the poles (i.e. (i) dMA ∕dy < 0), then there will be inertial instability. If, however, the absolute angular momentum increases toward the poles (i.e. (iii) dMA ∕dy > 0), the flow will be inertially stable. Panel (iii) describes the inertially neutral state where there is no gradient of angular momentum.

13.2 Dynamic Instabilities Associated with a Cross-Equatorial Pressure Gradient

finds that the pressure-gradient force directed northwards exceeds the Coriolis force. The parcel then accelerates toward the north. Note that this is an analogous relationship to the classical rotating cylinder where, if angular momentum decreases outward, there is centrifugal instability. (ii) If Ma increases with latitude, Eq. (13.8) has the form of a wave equation and the parcel will oscillate about its initial location. This situation is portrayed in Figure 13.13c, where it can be noted that 𝜂/f > 0. The frequency of the oscillation can be found by seeking wave-like solutions in Eq. (13.8) such as 𝛿y = A exp(i𝜔t)

(13.9)

where 𝜔 is frequency of the oscillation and A an arbitrary constant. Between Eqs. (13.8) and (13.9), the frequency of the oscillation is ( ) dMA 𝜔=f (13.10) = f (f + 𝜁g ) dy Thus, if MA increases toward the poles, there will be an oscillation about the initial point. Physically, the displaced parcel finds itself at a speed greater than the local geostrophic flow (I.e. u > ug or supergeostrophic). Then the Coriolis force is greater than the pressure gradient force and the net force returns the parcel toward its initial location. In essence, this is a geophysical statement of the Rayleigh stability criterion, stating that there is stability if MA increases between the equator and the poles. In summary, in this simple system, inertial stability is assured if 𝜂/f > 0. This situation is shown in Figure 13.5c. (iii) If MA is constant with latitude, the parcel, moved to its new location, will remain in place. This is referred to as inertial neutrality, as illustrated in Figure 13.5b . Here 𝜂/f = 0. It is interesting to note that Eq. (13.8) is of the same form as an equation describing the vertical displacement of a parcel in a stratified atmosphere. Depending on the static stability, a parcel will either accelerate northward, be neutrally displaced, or oscillate about its initial position. The combination of the horizontal inertial and static stability, when isentropes and isobaric surfaces intersect, results in symmetric stability (e.g. Emanuel 1979). In the equatorial regions, p- and 𝜃-surfaces are essentially coincident, as discussed in Section 3.1.2. Stevens (1983) investigated the consequences of inertial instability in the tropics. He noted that in a zonally averaged perspective, cross-equatorial wind shear was absent or weak. The distribution of the zonally averaged geostrophic wind U g is given by the background basic-state pressure gradient dp∕dy,

which in near-equatorial regions is very small (i.e. dp∕dy ≈ 0). Stevens concluded that inertial or symmetric instability is a major homogenizer of flow near the equator by producing rapid horizontal mixing. Although his study presents a sound review of the importance of inertial-symmetric instability in the zonally averaged tropics, the restriction to zonally averaged flows precluded the consideration of strong regional cross-equatorial flows, such as in the monsoon regions, where substantial and persistent CEPGs exist. Observations from field experiments became available that were at odds with Stevens’ conclusion. Perhaps the first observations of a near-equatorial region where 𝜂/f < 0 were made by Mapes and Houze (1992) during the Equatorial Mesoscale Experiment (EMEX 11 ), held in Northern Australia in January and February, 1987. Mapes and Houze made three important observations. They first noted a large-scale incursion of NH air (i.e. with 𝜂 > 0) advected across the equator over Indonesia. The second was that the incursion was associated with organized convection in the form of a mesoscale convective system (MCS) located on the poleward side (i.e. to the south) of the 𝜂 = 0 contour. Third, there was a rapid westward acceleration of surface winds, again poleward of the 𝜂 = 0 contour. As we will show, each of these three observations turn out to be signatures of near-equatorial inertial instability. Perhaps of greatest significance, Stevens did not consider the constant or very low frequency variability of cross-equatorial forcing and the production of a perpetual instability. As long as a central equatorial pressure gradient (CEPG) is maintained locally by a slowly varying SST gradient and persists on time scales greater than the inertial frequency, a necessary condition for inertial instability is maintained. Such regions appear in Figure 13.3, where mean monthly locations of the 𝜂 = 0 contour are found well off the equator. There are other examples of perpetual instability. One example is the conditional or gravitational instability close to the surface of the planet. Consider the boundary layer on a hot day, perhaps over a roadway. The visual shimmying is an optical distortion caused by rapidly ascending and descending eddies that attempt to transform the super-adiabatic lapse rate close to the ground to a neutral adiabatic lapse rate. The outcome is that above the skin layer the average lapse rate of the boundary layer is adiabatic. However, the thermal eddies continue as long as the surface is heated and sensible heat is transferred to the atmosphere, and eddies continue to transfer heat in the vertical. The instability remains until the road cools into the evening. The important point is that the time scale of the surface heating (hours) is far 11 See Webster and Houze (1991) for an overview of EMEX.

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13 Near-Equatorial Precipitation

greater than the time scale of the ameliorating eddies (seconds). A second example occurs in the subtropics/extratropics. Baroclinic eddies created in the vicinity of the westerly jet streams have time scales of days whereas the latitudinal temperature gradient, induced by radiative imbalance between the tropics and higher latitudes, has a seasonal time scale. The heating gradient will create an unstable situation and eddies are produced that feed on the basic rendering of the system temporally stable through efficient latitudinal heat and momentum transports. However, the radiational heating gradient is continual and an unstable gradient will be recreated, producing new set-stabilizing eddies and so on. 13.2.4

Near-Equatorial Inertial Instability

The ideas discussed in the last section are applied to near-equatorial regions where the cross-equatorial pressure gradient is substantial and the local 𝜂 is the opposite sign to the Coriolis parameter f so that 𝜂/f < 0. We defer discussion of regions with weaker cross-equatorial gradients until later. We utilize the “parcel method” definition of inertial instability in a background state that possesses a sufficiently large CEPG to overcome surface dissipation so that local 𝜂 is anticyclonic, produced by the advection of 𝜂 across the equator. In Figure 13.6, P and C represent the pressure gradient and Coriolis forces, respectively,

and R represents the residual force or the imbalance between P and C. The axes on the diagram represent time (abscissa) and latitude (ordinate). The panel is oriented appropriately for northern summer conditions with low pressure to the north of the equator and high pressure to the south. Consider a parcel at time t = to that is located just north of the equator in a region in the NH where 𝜂 is negative. Let the parcel be displaced northward at time t = t1 . The zonal velocity of the parcel changes as a result of this displacement as dictated by Eq. (13.7) but, because the environment is inertially unstable, the new zonal velocity of the parcel is subgeostrophic (R > 0 directed northward). The unbalanced pressure gradient force, R, acts to accelerate the parcel northward. At time t = t2 , the parcel reaches the zero 𝜂. At this stage, the acceleration is zero (R = 0). Further northward motion (time t = t3 ) results in the parcel experiencing a southward restoring force (case III: R < 0). Thus, when the zero 𝜂 contour is found some distance from the equator, the wind will be accelerated on the equatorward side and decelerated on the poleward side of the 𝜂 = 0 contour. If this interpretation is applicable to the real atmospheric system, the signal of these processes should appear in the divergent wind field. Specifically, a local maximum in the divergent wind field should be centered on the 𝜂 = 0 contour, with divergence on the equatorial side of the 𝜂 = 0 contour and convergence on

Cross-equatorial pressure gradient and the potential for inertial instability LOW PRESSURE P

P η=0 latitude

268

P

R

t = t2

t = t1 R

P tt=t = t00

EQU

C

R C

R0

t = t3

CONV η=0

C

R>0 η0 η 0 is everywhere in the NH or < 0 in the SH so that 𝜂/f > 0 everywhere. The only way to produce areas of negative 𝜂 regions in the NH (positive in the SH) is by cross-equatorial advection of AV. The first two terms on the right-hand side of Eq. (13.11) are capable of making negative contributions to the local tendency of 𝜂 in the NH. However, in the absence of cross-equatorial vorticity advection, they can contribute only to cyclonic vorticity generation so that without cross-equatorial advection, the advection term acts only to redistribute cyclonic AV. The divergence term (or vortex stretching term) can only make a locally anticyclonic contribution to the 𝜂 tendency if there is divergence. However, as the 𝜂 approaches zero, this mechanism becomes less and less efficient; that is, as ̃ → 0. 𝜂 → 0, then 𝜂∇ ⋅ V The general collocation of divergent wind maxima and the zero AV contour away from the equator is precisely the distribution expected to result from inertial instability as depicted in Figure 13.6. That is, the divergent wind is being accelerated on the equatorward side of the zero AV contour and decelerated on the poleward side. Returning to the mechanism proposed by Walisser and Somerville, described earlier, one would expect to observe the maximum divergent wind speeds at the equator, where the Coriolis force is zero, and not at the latitude of 𝜂 = 0 unless there is no CEPG. There appears to be, then, a dual role for the divergent part of the wind field: 12 See Holton (2004), Section 4.5.2, Eq. (4.28).

(i) The divergent wind is responsible for creating the conditions that lead to near-equatorial inertial instability via the interhemispheric advection of “counter-signed” absolute vorticity and (ii) The divergent wind accelerates locally in an effort to mitigate the inertially unstable conditions caused by the advection of absolute vorticity across the equator. An example of this dual role can be seen in the JJA fields in the Pacific Ocean (Figure 13.7c). In regions where 𝜂 is locally anticyclonic (e.g. between the equator and the 𝜂 = 0 contour in the NH), divergence is located on the equatorward side of the 𝜂 = 0 contour, whereas convergence is located on the poleward side, resulting from the acceleration of the divergent wind across the region where 𝜂 is locally anticyclonic, thus acting to reduce the local inertial instability. The relaxation of the instability is accomplished by making a locally cyclonic contribution to the 𝜂 tendency via the divergence generation term in Eq. (13.11). We noted in the discussion of Figure 13.1 that strong convection generally occurs on the poleward side of the 𝜂 = 0 contour. This convection is a direct consequence of the convergence of moist static energy in the boundary layer and the upward vertical velocity accompanying the divergence. That is, the ITCZ off-equator convection is the result of a stabilizing secondary circulation attempting to neutralize the inertial instability. A dual role for the divergence–convergence doublet in mitigating the instability may have some merit and its role is also relatively straightforward. However, it comes at the cost of ignoring the vertical structure and thermodynamics existing in a more realistic atmosphere. To include these aspects, it is useful to expand the arguments presented thus far in several ways, including rephrasing the problem in terms of potential vorticity. When this is done, one must reconcile the idea of processes that modify the 𝜂 distribution with processes that modify the PV distribution. For example, divergence and convergence alone do nothing to change the PV distribution; PV is modified only by diabatic processes such as friction and vertical gradients in heating. With the latter point in mind, one is led to conclude that the arguments concerning the secondary circulations mitigating the instability of the basic state may be spurious. However, we now show that the divergence–convergence doublet is associated with vertical circulations in the latitude–height plane, which in turn are associated with vertical gradients in the heating that act to modify the PV distribution in a way that reduces the instability, thus reconciling the 𝜂 and PV viewpoints. In fact, consistency between the dynamics and thermodynamics requires that the divergence and

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13 Near-Equatorial Precipitation

convergence modifying the 𝜂 distribution are associated with vertical gradients in heating that change the PV distribution in a similar way. 13.2.6

and convergence to poleward. In the upper troposphere, the zero 𝜂 contour moves to the winter hemisphere and is attended by a reversed divergence–convergence doublet. Mean upwards vertical motion, with a maximum in the mid-troposphere, lies above the low-level convergence. Where there is not a strong CEPG (e.g. the east Atlantic-Africa section in DJF or the central Pacific during both seasons), the vertical variation of the zero AV contour is less, lying closer to the equator at all heights (e.g. in the central Pacific Ocean: Figure 13.8c). Of particular interest are the JJA Indian Ocean sections. At this time of the year, the strongest CEPG exists in the Indian Ocean, yet the sections appear to have little resemblance to the other strong CEPG cases. Clearly, as we have discussed previously, the Indian Ocean requires special attention. As noted above, the divergent flow in the upper troposphere is directed from the summer hemisphere to the winter hemisphere, the reverse of that in the lower troposphere. Commensurate with this change in direction

Vertical and Latitudinal Structures

Figure 13.8 presents four latitude–height cross-sections, averaged across 30∘ longitude bands, in the eastern Atlantic, the central Indian Ocean, the central Pacific Ocean, and the eastern Pacific Ocean, for both DJF and JJA. In each figure, there are three panels depicting sections of the divergent wind field (vd ), vertical velocity ̃h ) (wp : dp/dt), and the horizontal divergence field (∇ ⋅ V ∘ ∘ between 30 S and 30 N. A similar structure can be found in sections where there is a strong CEPG, such as in JJA through the eastern Atlantic-Africa, both seasonal sections through the Indian Ocean, and the JJA section through the eastern Pacific. These sections possess a zero 𝜂 contour that lies away from the equator in the summer hemisphere, low-level divergence on its equatorward side, (a) Atlantic Ocean (0°–30°E) (ii) JJA

(i) DJF

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vd (m s–1) wp(10–2 Pa s–1) div(10–6 s–1) Figure 13.8 Latitude–height cross-sections of circulations: northward divergent wind vd (m s−1 ), topmost panels; vertical velocity (wp , dp∕dt: 10−2 Pa s−1 ), middle panels; divergence (div: 10−6 s−1 ) lowest panels for (a) Atlantic Ocean (0–30 ∘ E); (b) Indian Ocean, 55∘ E to 85∘ E; (c) the central Pacific Ocean (170∘ E–170∘ W: section “B,” Figure 13.7); and (d) eastern Pacific Ocean (130∘ W to 90∘ W: section “A,” Figure 13.7) for DJF (left columns) and JJA (right columns). The bold black line denotes the mean location of the zero absolute vorticity (𝜂 = 𝜁 + f = 0). The dashed vertical line represents the equator. The magnitudes of the three quantities are given by the common color code at the base of the figures.

13.2 Dynamic Instabilities Associated with a Cross-Equatorial Pressure Gradient

(c) Central Pacific Ocean (170°E–170°W) (i) DJF (ii) JJA vd

200

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vd (m s–1) wp(10–2 Pa s–1) div(10–6 s–1) Figure 13.8 (Continued)

of flow, there is a reversal in the advection of absolute vorticity and the 𝜂 = 0 contour is found in the winter hemisphere. In combination, the vertical and horizontal flows define strong latitude–height circulations, with the ascending branches tightly bounded just polewards of the low-level 𝜂 = 0 contour and coincident with regions of low OLR. Using the same arguments, the upper troposphere may also be inertially unstable. Yet the pattern of divergence in the upper troposphere does not seem to be quite the same as in the lower troposphere. Instead of a tight local convergence-maximum attending the 𝜂 = 0 contour, the convergence is more spread out, with weaker local extrema. The probable reason for the difference is that the upper troposphere is statically very stable, so that vertical motions are inhibited. Also, there are no mechanisms whereby, if secondary circulations were to develop, they could produce the necessary thermodynamic response to counter this potential instability, as occurs in the lower troposphere through latent-heat generation. Radiative cooling occurs in the stratosphere but this is a relatively slow process, with an e-folding time scale >10 days. The absence of heating processes, and the stronger static stability, result in a much broader descending region, extending as far polewards as 30∘ of latitude.

A careful inspection of the cross-sections hints at the existence of an additional shallower mean divergent circulation in the lower troposphere. This is especially evident in the eastern Atlantic (Figure 13.8a(i)) and the eastern Pacific (Figure 13.8d(ii)). A schematic of these two divergent circulations is presented in Figure 13.9.13 In Section 13.4 we will argue that the existence of these dual meridional circulations appearing in the climatological fields is the result of the destabilization–stabilization oscillation of the ITCZ. 13.2.7 Is the Existence of a CEPG a Sufficient Condition for Inertial Instability? Tomas et al. (1999) developed a simple numerical model to test whether or not the existence of a CEPG is a sufficient condition for the existence of inertial instability. Although a CEPG meets the parcel criterion for parcel instability, the stabilizing effects of dissipation and a finiteness of vertical extent forced by boundary layer stability need to be examined. 13 This dual circulation was described by Tomas and Webster (1997) and Toma and Webster (2010a, b). Trenberth (2000) and Zhang et al. (2004) also found a second mode in divergent circulations similar to the shallow circulation of Figure 13.9.

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13 Near-Equatorial Precipitation

100

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Figure 13.10 Schematic of the zonally symmetric shallow fluid model used to represent the tropical boundary layer. Mean depth of layer is HB with a constant potential temperature 𝜃. The boundary layer is capped by a potential temperature jump, or inversion, of magnitude 𝛿𝜃(y). The temperature difference induces a pressure gradient producing horizontal shear about the equator. Source: Based on Tomas et al. (1999).

η=0 700

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200

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274

conv EQU

10°N

Figure 13.9 Schematic diagram showing the circulation found in the latitude–height plane in regions where conditions of near-equatorial inertial instability exist. In a conditionally unstable atmosphere, a deeper convective circulation occurs. The bold curve denotes zero absolute vorticity (𝜂 = 𝜁 + f = 0), with negative absolute vorticity to the left (south). A second shallower circulation is also evident. Blue and pink squares denote maximum divergence and convergence, respectively. Yellow shows the maximum divergent wind bisected by the 𝜂 = 0 contour. Source: Based on Figure 11, Tomas and Webster (1997).

Over large parts of the tropical ocean the planetary boundary layer is generally well mixed, extending in altitude to approximately 1–2 km in which the potential temperature (𝜃) is constant. The boundary layer is usually capped with a temperature inversion with a potential temperature jump of 𝛿𝜃. Here we adopt a slab model developed by Battisti et al. (1999) that formulates these basic characteristics of the tropical atmosphere. The model is assumed to be zonally symmetric. Importantly, the model allows the determination of the impact of stability on inertial instability. A schematic version of the model is shown in Figure 13.10. ̃ = (ui , vj ) is assumed to be The horizontal velocity V ∼

∼

homogeneous throughout the depth of the boundary layer. The model includes a latitudinal pressure gradient created by a potential temperature gradient and the varying depth of the boundary layer, a background zonal wind U(y), proportional to the background pressure gradient, a measure of the static stability given by 𝛿𝜃 and a dissipation assumed to be proportional to the wind field. The governing equations, linearized about

the basic state U(y), are 𝜕u + v(U y − 𝛽y) + 𝛼u = 0 𝜕t 𝜕𝜙 𝜕v + 𝛽yu + + 𝛼v = 0 𝜕t 𝜕y 𝜕𝜙 𝜕v + CB2 + 𝜀𝜙 = 0 𝜕t 𝜕y

(13.12a) (13.12b) (13.12c)

gHB , where HB is the mean depth of the Here CB2 = ̃ g = g𝛿𝜃∕𝜃, fluid and ̃ g is the reduced gravity,14 given by ̃ where 𝛿𝜃 𝛽CB

(13.18)

Therefore, for instability the CEPG (here measured by the magnitude of the basic wind shear) overcomes the impact of vertical stability contained within C B . Equation (13.18) suggests the following: inertial instability occurs only if the shear of the zonal wind is sufficiently large to overcome the impact of static stability. Using values of observed static stability, we can test whether inertial instability is a viable process in ITCZ physics in different regions of the tropics: (i) Eastern Pacific Ocean: Figure 13.11a shows a plot of the mean boreal summer vertical profiles of potential temperature (𝜃) and equivalent potential temperature (𝜃 e ) from ERA data (dashed line) and from atmospheric soundings from the NOAA Research Ship Ron Brown cruise from the East Pacific Investigation of Climate (EPIC) 2001 field campaign (which appear as solid red).15 Analyses close to 1800 UTC are presented for 2∘ S, 5∘ N, and 8∘ N, along 95∘ W. The character of the atmospheric boundary layer changes across the equator. At 2∘ S there is a potential temperature cap at the top of the atmospheric boundary layer of magnitude 𝛿𝜃 = 3–6 K. At 5∘ N the atmosphere seems to be neutrally stratified without a stable cap; that is, 𝛿𝜃≈ 0. When moisture is considered, it is apparent from both Figure 13.11a(i) and (ii) that the profiles at 5∘ N are conditionally unstable as 𝜕𝜃 e /𝜕z < 0. The linear criterion for inertial instability is met for any positive shear of the zonal wind (i.e. 𝛾 c > 0). (ii) Western Indian Ocean during JJA: Figure 13.11b plots the vertical profiles of 𝜃 and 𝜃 e for positions north of the equator in the western Indian Ocean 15 From Raymond et al. (2004); Raymond, et al. (2006).

during JJA. This is the region of greatest poleward excursion of the zero absolute contour vorticity during JJA (Location “E,” Figure 13.3). It is also a region where the potential temperature inversion at the top of the boundary layer is large, with a magnitude of 4–5 K. Using Eq. (13.18), the zonal wind shear would have to be 50 m s−1 across the equator for inertial instability to occur. Even though the cross-equatorial shear is large, the criterion for instability is not met. Thus, the cross-equatorial flow in the East Indian Ocean is inertially stable. Here we see a clear example of the influence of the desert regions of North Africa and the Arabian Peninsular on the South Asian monsoon. Figure 12.7 showed the eastward descending air from the desert regions. This is associated with Smith’s lateral exhaust described in Section 12.5.3. This descent (Figure 12.7b) creates a strong inversion in the western Indian Ocean, rendering the boundary layer inertially stable and restricting the development of convection. It is not until the monsoon flow reaches the vicinity of India, further to the east, that substantial precipitation is found. We speculate that if it were not for the influence of the North African deserts, a near-equatorial ITCZ would exist in the western Indian Ocean, probably meaning less summer rainfall over South Asia. 13.2.8 Dynamic Estimate of the Latitude of the Mean ITCZ in Regions of Strong CEPG An optimal latitude for the formation of an ITCZ can be determined based on the physical reasoning we have developed above. Using the absolute vorticity Eq. (13.11), we have argued that if there is a CEPG then a cross-equatorial advection of absolute vorticity occurs. This flow becomes inertially unstable if there is no inhibition by vertical stability. Assume now that the system is in steady state and consideration of the oscillation of the system is left until later. In an inviscid steady state, Eq. (13.11) provides the following balance: ̃ .∇𝜂 − 𝜂∇ ⋅ V ̃ =0 −V

or

− vd

𝜕v 𝜕𝜂 −𝜂 d =0 𝜕y 𝜕y (13.19)

where, for the second expression, it is assumed that there is no longitudinal variability. Here, vd is the meridional component of the divergent wind. Noting that 𝜂 = f + 𝜁 = f − 𝜕ug /𝜕y and that as the latitude increases the planetary vorticity increasingly dominates, then 𝜕f 𝜕𝜂 −vd (13.20a) → −vd → −vd 𝛽 𝜕y 𝜕y and ( ) 𝜕ug 𝜕vd 𝜕v 𝜕vd = f− →f d (13.20b) −𝜂 𝜕y 𝜕y 𝜕y 𝜕y

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13 Near-Equatorial Precipitation

(a) θ and θe along 95°W 4

4 Oct ERA-40

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Figure 13.11 Vertical profiles of potential temperature (black) and equivalent potential temperature (red) for JJA (1981–2000) in (a) the eastern Pacific ITCZ at 2∘ S, 95∘ W (near location “D,” Figure 13.3: from the EPIC experiment) and 5∘ N, 95∘ W (between locations “A” and “D”) dashed lines. Solid lines denote mean October ERA profiles for the same location as the EPIC profiles. (b) Vertical profiles of potential temperature (𝜃) and equivalent potential temperature (𝜃 e ) in the Indian Ocean for JJA (1981–2000) at 60∘ E, 15∘ N (red) and 80∘ E, 15∘ N (blue). Source: Data from ERA-40. Near location “E” in Figure 13.3. Units: K.

Now, Eq. (13.20a) decreases with latitude following 𝛽, whereas Eq. (13.20b) increases with latitude following f . Therefore, for a steady state there must be a latitude where these two terms in Eq. (13.19) are equal. This is the latitude where the destabilizing advection of absolute vorticity across the equator equals the stabilization factor: the production of opposite-signed vorticity by vortex tube stretching. Between Eqs. (13.20a) and (13.20b) and with Eq. (13.19), and noting that tan 𝜑 ≈ 𝜑 as 𝜑 → 0, we find a mean equilibrium latitude for the ITCZ, 𝜑e , to be 𝜑e ≈ vd ∕a

𝜕vd 𝜕y

(13.21)

where the divergent meridional wind is given by vd = (𝜕p/𝜕y/𝜌 − fu)/𝛼, where 𝜕p/𝜕y is the CEPG, the cross-equatorial pressure gradient.

Figure 13.12 plots the equilibrium latitudes of the ITCZ as a function of divergent wind speed and divergence. Observed values of 925 hPa divergence and a divergent wind region for the eastern Pacific Ocean leads to an equilibrium latitude between 8∘ N and 10∘ N (the shaded rectangle in Figure 13.12). Overall, the theoretical estimates of the climatological location of ITCZ convection appear to match observations for JJA quite well. The exception, of course, is in the anomalous North Indian Ocean during the boreal summer, where an ITCZ is not expected because of vertical stability constraints. 13.2.9 Low-Level Near-Equatorial Westerlies in Regions of Strong CEPG In Section 1.3.3, we commented on a band of weak westerlies extending on the poleward side of convection

13.2 Dynamic Instabilities Associated with a Cross-Equatorial Pressure Gradient

the steady state 𝜂- equation written at the latitude where 𝜂 = 0, becomes

Equilibrium latitude of the ITCZ

horizontal divergence (10–6 s–1)

10 1°

8

̃𝜓 ⋅ ∇𝜂 = 𝛼𝜁 |𝜂=0 ̃𝜒 ⋅ ∇𝜂 − V −V 3°

̃𝜓 We note that, as shown in panel (ii) of Figure 13.14, V ̃ turns eastward, becoming orthogonal to ∇𝜂 so that V𝜓 ⋅ ∇𝜂 → 0 as the 𝜂 = 0 contour is approached. Therefore, only the divergent part of the wind field is responsible for the advection of AV across the equator. Thus, at the location where 𝜂 = 0:

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Figure 13.12 Equilibrium latitude (𝜑e ) of the mean ITCZ as a function of the divergent meridional wind (vd ) and the horizontal ̃ ) based on Eq. (13.31). Observed values of the divergence (∇ ⋅ V divergent meridional velocity component and divergence at 925 hPa are shown as shaded areas. Interception occurs near 8–10∘ latitude, matching the observed location of the ITCZ in regions of substantial CEPG.

in the SH (Figure 1.13a(ii)). We suggested that these westerlies were not the result of Coriolis turning, but indicative of some other process. Here we examine these anomalous winds more thoroughly. To help determine why these westerlies exist, we first look in detail at the JJA 925 hPa winds displayed in the eastern Pacific Ocean (Figure 13.13a), the DJF winds in the Indian Ocean (Figure 13.13b) and over the Eastern Atlantic and equatorial Africa during JJA (Figure 13.13c). All three cases of shallow westerlies, contained within the red rectangles, occur in the presence of a strong or a moderate CEPG. Here we argue that the low-level westerlies on the poleward side of the ITCZ is a signature of inertial instability. Figure 13.14 provides a schematic of the processes leading to the low-level westerly wind maximum.16 The first panel plots the distribution of the divergence field relative to the zero AV contour. We note that in the ̃ → 0 so that the term vicinity of the 𝜂 = 0 contour, ∇ ⋅ V ̃ → 0. This distribution is consistent with the 𝜂∇ ⋅ V divergence–convergence doublet located around 𝜂 = 0 found in Figures 13.8 and 13.9. It is useful to break down the velocity vector into its divergent and rotã𝜓 . Then Eq. (13.11), ̃ =V ̃𝜒 + V tional parts, such that V 16 Following Tomas and Webster (1997) and Toma and Webster (2010a).

(13.23)

̃𝜒 is a maximum at 𝜂 = 0, then 𝜁 must also Now, as V be a local maximum to achieve balance. In a zonally axisymmetric state (no longitudinal variation), this relative vorticity maximum must result from a maximum in zonal shear; that is, as 𝜂 → 0, |𝜕U∕𝜕y| must be a maximum. ̃𝜒 ⋅ ∇𝜂 > 0 so that Consider the NH case. At 𝜂 = 0, −V 𝜁 |max < 0 such that 𝜕U∕𝜕y|max > 0. Typically, easterlies are observed in the tropics. However, in regions of a strong cross-equatorial pressure gradient, the shear in the zonal wind, necessary to balance the advection of 𝜂, is sufficient to turn the wind toward the east. Furthermore, westerly winds must increase in strength to the north of the 𝜂 = 0 contour (panel iii in Figu13.14). In the SH, similar arguments produce a low-level westerly maximum on the poleward side of the 𝜂 = 0 contour. We can calculate the momentum balance by integrating Eq. (13.23) with respect to latitude, assum̃𝜒 = v𝜒 = constant and ing via axisymmetry that V 𝜁 = − 𝜕u𝜓 /𝜕y, leading to v𝜒 = 𝛼u𝜓

(13.24)

For the case of a northward-directed v𝜒 maximum, bisected by the 𝜂 = 0 contour, this balance requires easterlies to the south of the 𝜂 = 0 contour, zero zonal velocity at 𝜂 = 0, and westerlies to the north. The results of this analysis are consistent with the observations made during the EMEX, discussed in Section 13.2.3, where it was found that following an incursion of PV from the NH there was convection to the south of the zero 𝜂 contour and westerlies further south. Signs have to be reversed to explain these observations of the westerly acceleration, but theory and observations appear to match. The existence of anomalous westerlies may also have some significance for climate variability. For example, in the western Indian Ocean, the confluence of the anomalous near-equatorial westerlies and the southeast trades creates an upwelling over the Seychelles-Chagos thermocline ridge, a region considered climatically

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(a) Americas/East Pacific JJA 925 hPa winds (m s–1) 40°N 30°N

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Figure 13.13 Mean July–August near-surface winds (925 hPa). (a) Eastern Pacific and the Americas for JJA, (b) Indian Ocean and South Asia during DJF, and (c) Africa and East Atlantic for JJA. Bold blue line represents the location of the 𝜂 = 0 contour. Bold black dashed in represents the equator. Red boxes enclose regions of anomalous surface westerlies on the poleward side of the 𝜂 = 0 contour. Winds are shown as vectors relative to key bottom right of panels. Source: Data from ER-40 reanalysis.

sensitive. Schott and McCreary (2001) argue that the curl of the wind stress induces Ekman divergence and a local cooling of the upper ocean. 13.2.10 A Potential Vorticity View of Ameliorating Secondary Circulations We now consider the interaction between the motion and moisture fields occurring in association with the

subsiding branch of the secondary circulations in regions of strong CEPG. Consider the cross-sections displayed in Figure 13.8 where secondary circulations are easy to identify. We hypothesize that the sinking motion on the equatorward side of the convection brings dry air downwards, resulting in a humidity profile that is conducive to low- and mid-level radiative cooling. The adiabatic warming balances radiative cooling in the subsiding branch of the circulation.

13.2 Dynamic Instabilities Associated with a Cross-Equatorial Pressure Gradient

pressure level, the atmosphere is much drier to the south of the 𝜂 = 0 contour than to the north in agreement with the subsidence to the south of the zero 𝜂 contour and rising motion to the north. The subsiding dry air to the south of the 𝜂 = 0 contour and the absence of cloud permit efficient radiative cooling of the lower troposphere, as seen in the mean material change in potential temperature shown in Figure 13.15b. Away from the lower boundary there is cooling, with a maximum located near 825 hPa. North of the 𝜂 = 0 contour, the heating field is positive, dominated by extremely strong latent heating that increases in magnitude abruptly into the mid-troposphere. The vertical gradient of the heating (Figure 13.15c) is important for determining the PV balance in the vicinity of the ITCZ, as its product with the local 𝜂 is proportional to the material source of PV. For large-scale flows, the vertical components dominate the PV equation, so that ( ) dqp 𝜕𝜃 𝜕Fy 𝜕Fx 𝜕 𝜃̇ −g = − g𝜂 − (13.25) dt 𝜕p 𝜕p 𝜕x 𝜕y

(c) East Atlantic/Africa JJA 925 hPa winds (m s–1) 40°N 30°N

10°N 0° 10°S 20°S 30°S 20°W

0°

20°E

longitude

40°E

60°E

15 m s–1

Figure 13.13 (Continued)

To investigate this hypothesis, we construct crosssectional diagrams of the humidity mixing ratio (w), the material tendency of the potential temperature ̇ and the partial derivative with respect to (d𝜃/dt or 𝜃), ̇ p (i.e. 𝜕 𝜃∕𝜕p) for mean JJA conditions in the eastern Pacific at the same location as the cross-section of Figure 13.8d(ii). These fields are shown in Figure 13.15. The vectors in Figure 13.15a display the vertical profile of the meridional wind field and the vertical velocity as well as the mixing-ratio field (Figure 13.15a). At any

cross-equatorial pressure gradient

(i) Divergent part

Equation (13.25) is derived from first principles in Appendix I. Here qp (= −g𝜂𝜕𝜃∕𝜕p) is the potential vorticity on an isobaric surface, g is the acceleration due ̇ to gravity, 𝜕 𝜃∕𝜕p is the material tendency of potential temperature with respect to pressure, 𝜕𝜃/𝜕p is the partial derivative of potential temperature with respect to pressure, and (𝜕F y /𝜕x − 𝜕F x /𝜕y) represents the curl of the frictional force. The boxed term on the right-hand side

(ii) Rotational part

(iii) Relative vorticity & zonal wind

observed location of low-level westerlies

ζ(y)

u(y)

convergence η=0

∇·Vχ = 0 Vχ (max)

η=0

equ

equ

∼

Vψ · ∇η = 0

latitude

latitude

20°N

η=0

equ ζ(y),u(y)

divergence

Figure 13.14 Schematic diagram showing the balance of forces in the vicinity of the zero absolute vorticity contour. (i) The divergent part of the wind field showing a maximum coincident with the 𝜂 = 0 contour. The maximum in the divergent wind coincides with the 𝜂 = 0 contour (red line). (ii) The rotational part of the wind field where the advection of the gradient of absolute vorticity by the rotational part of the wind field is zero where 𝜂 = 0. The observed location of the low-level westerly jet is shown on the poleward side of the 𝜂 = 0. (iii) The distribution of relative vorticity and the resultant zonal wind field. Note the zonal wind maximum on the poleward side of 𝜂 = 0. Source: Adapted from Tomas and Webster (1997).

279

13 Near-Equatorial Precipitation

(a) Humidity mixing ratio and divergent velocity 25

500

w, Vd 20 600

hPa

g kg–1

15 700 10 800 5

900 1000 20S 15S 10N 5S

0

5N 10N 15N 20N

0

(b) Tendency of potential temperature 500

· θ

6 4

600

hPa

700

0 −2

800

10–5 K s–1

2

−4 900 −6 1000 20S 15S 10N 5S

500

0

5N 10N 15N 20N

(c) Vertical gradient of the material tendency of potential temperature · θp

8 6

600

700

0 −2

800

−4 −6

900 1000 20S 15S 10N 5S

10–9 K s–1 Pa–1

4 2

hPa

280

−8 0

5N 10N 15N 20N

latitude Figure 13.15 Mean latitude–height cross-sections of moisture and thermodynamic quantities over the eastern Pacific Ocean 130∘ W–90∘ W for July: (a) humidity mixing ratio or specific humidity w (g kg−1 ) and wind vectors denoting the divergent part of the circulation (the vertical component of each vector is scaled by a factor of 50 relative to its horizontal component); (b) material tendency of the potential temperature 𝜃,̇ indicating regions of diabatic heating (red) and cooling (blue); and (c) partial derivative of the material tendency of the potential temperature with ̇ respect to pressure 𝜕 𝜃∕𝜕p (K s−1 Pa−1 ). The bold black line denotes zero absolute vorticity 𝜂 = 0.

of Eq. (13.25) is the diabatic source of qp . The second term represents the frictional dissipation of qp . Below the 850 hPa level, the partial derivative of heating with respect to pressure is positive to the south of the zero 𝜂 contour and negative to the north (Figure 13.15c). Conversely, absolute vorticity is negative to the south of the zero contour and positive to the north. Thus, the diabatic source term, which is the product of these two quantities and also a negative definite quantity (minus the gravitational acceleration), is positive everywhere and the diabatic source acts to oppose the low-level advection of potential vorticity, thus reducing the instability. This is a negative feedback on inertial instability near the equator. Other aspects of Figure 13.15c merit further discussion. The large negative maximum in 𝜃̇ p centered near 7.5∘ N at 875 hPa is associated with the increase of latent heating with height. To the south, there is another maximum lying between 925 and 850 hPa, extending from the equator to southward of 20∘ S. There is an even stronger maximum, although smaller in lateral extent, centered near 2∘ N, at 1000 hPa. The large magnitude of this near-equatorial maximium results from surface warming as air moves off the equatorial cold tongue. Thus, from the PV perspective, the secondary circulations act to mitigate the symmetrically unstable large-scale flow by inducing the necessary heating distributions through interaction with the moisture field. This is analogous to the 𝜂 perspective, where the secondary circulations mitigate the inertially unstable large-scale flow by producing the necessary divergence and convergence distributions that result in subsequent vortex tube stretching. This agreement is not surprising since, in the tropics, heating tends to be balanced to a large degree by vertical motion. Thus, through continuity, an increase of heating with height is associated with convergence and an increase of cooling with height is associated with divergence (cf. Figures. 13.15c and 13.8d(iii)). It is reasonable to expect that frictional dissipation opposes the local relative vorticity, so that the dissipation term also acts to oppose the cross-equatorial advection of PV. This can be seen by noting that, for linear dissipation, the distribution of the dissipation mirrors the distribution of relative vorticity, i.e. −(𝜕F x /𝜕y − 𝜕F y /𝜕x) = − 𝛾𝜁 . Thus, whereas diabatic processes are important in opposing advection to the north and south of the 𝜂 = 0 contour, frictional dissipation is also of potential importance.

13.3 Transient States of the Intertropical Convergence Zone In previous sections we have posited criteria for the location of the mean seasonal ITCZ, yet during our survey

13.3 Transient States of the Intertropical Convergence Zone

(a) 120°W–110°W JJA (section A) (i) Mass streamfunction and relative humidity

(b) 170°W–180°W JJA (section B) (i) Mass streamfunction and relative humidity 0

600

P

–3.5

45

M

800 1000 30S

10S

20S 20

0

30

40

10N

50

60

20N

1

600

0

800 1000 30S 30N

0

1000

2

30S

20S –8

hPa

hPa

400

800

0

4

0 latitude

–6 –4 –2

0

2

10N 4

20N

30N

6 ms–1

0

30

40

10N

50

60

20N

30N

70 %

(ii) Divergence and relative humidity 0

0

0

2

0

400

0

600 800

10S

0

10S

200

4 2

0

0

20S 20

0

600

–1

70 %

(ii) Divergence and relative humidity 200

1

400

2 3

–3 –1 0 1

–1

0

–4

2

1 0 –1 –2

hPa

400

1

– –3 2

200 –2

0

hPa

200

1000

0

30S

0

–2

2

20S

10S

0 latitude

–8

–6 –4 –2

0

10N 2

4

20N

30N

6 ms–1

Figure 13.16 Characteristics of the mean 1981–2000 summer (JJA) meridional circulation in the east and central Pacific Ocean averaged between (a) 130∘ W–110∘ W (section “A,” Figure 13.7) and (b) 170∘ W–180∘ W (section “B”). Panels (i) show the mass streamfunction (10 kg s−1 ) and relative humidity (shading relative to bar below figure: %) The lower panels show the horizontal wind divergence (10−6 s−1 with red contours positive, black contours negative) and meridional wind (shaded contours m s−1 , bottom scale). Locations “P” and “M” are discussed in the text and refer, respectively, to the protrusion of the lower tropospheric 𝜂 = 0 and the lower tropospheric shallow meridional circulation. Source: From Figure 4, Toma and Webster (2010a).

of transients in Section 1.4, we noted that the ITCZ is far from being a steady phenomenon but rather characterized by considerable high-frequency variability. This was most evident in Figure 1.16, where systems propagated from east to west occurred across the equatorial Pacific Ocean. As indicated in the spectral variance distributions (Figure 1.15), these waves appear to have periods similar to the inertial periods at the latitudes of the mean ITCZ.17 13.3.1

Character of the Transients

Here we concentrate on the off-equator convection in the Pacific Ocean. We focus on the Pacific since there is an anomalous band of convection that extends across the Pacific even though, in the Central Pacific Ocean, the zero 𝜂-contour lies close to the equator. Thus, we 17 From Eq. (3.40), inertial periods at 10 and 5∘ of latitude are 2.9 and 6.5 days, respectively.

have the opportunity of testing the inertial instability hypothesis and assessing whether we require an additional mechanism for the western Pacific off-equator convection. Consider the long-term average meridional circulation fields in two longitude belts: 130–110∘ W and 180–170∘ W in Figure 13.8. These belts are labeled A and B, respectively, in Figure 13.7c. In the first longitudinal belt, the CEPG is strong and the zero 𝜂-contour is located well to the north of the equator. The zero 𝜂-contour in the second section lies much closer to the equator. The upper panel of Figure 13.16a shows the mean JJA meridional circulation in the eastern Pacific for the period 1981–2000 in terms of the mass stream function, 𝜓, derived by downward integration of the meridional mass flux. There is a broad band of rising motion located between 6 and 13∘ N with subsidence to the south extending to 20∘ S. The major ascending region corresponds to a vertical incursion of moisture into the upper troposphere (shading). The lower

281

13 Near-Equatorial Precipitation

20°N

Time sections of 925 hPa absolute vorticty (b) 180°W-170°W (section B) 20°N

(a) 120°W-110°( section A)

10°N latitude

282

10°N

0°

0°

10°S

10°S

20°S 150

170

190 210 days (1996)

220

–100 –80 –60 –40 –20 0

240

20°S

150

20 40 60 80 100 –25

OLR anomaly (W m–2)

170

–15

190 210 days (1996)

220

240

–5 0 +5 +15 +25 η(10–6 s–1)

Figure 13.17 Time–latitude evolution of daily values of anomalous OLR (shaded, bottom scale W m−2 ) and absolute vorticity (𝜂 10−6 s−1 ) at 925 hPa in (a) 130∘ W–110∘ W longitude sector (“A”, Figure 13.7) and (b) 180∘ W–170∘ W (section “B”) for a period in the NH summer of 1996. Commencing and ending dates in the sections are May 29 (day 150) and August 27 (day 240) for 1996. Contours of 𝜂 intervals every 5 × 10−6 s−1 between ±25 × 10−6 s−1 . Colored bold contours (relative to bottom scale below) show the 𝜂 = +15, 0, and −15 × 10−6 s−1 contours. Source: Figure 5, Toma and Webster (2010a).

panel shows the horizontal divergence field (contours) and the meridional wind component (shading). There are boundary layer cross-equatorial winds of magnitude of about 6 m s−1 with a return flow aloft with speeds exceeding 8 m s−1 . A second weaker southward 2 m s−1 meridional velocity maximum (location “M”) resides between 700 and 600 hPa, also over the equator. This lower circulation is located near the southward protrusion of 𝜂 = 0 at 850 hPa (location “P”) and represents the shallow cell described initially by Trenberth (2000) and Zhang et al. (2004) and discussed in Section 13.2.6. The zero absolute vorticity contour in the eastern Pacific section bisects the meridional wind maximum in the lower troposphere. The northward winds accelerate equatorward of the zero contour, whilst on the poleward side there is a region of rapid deceleration corresponding in location to the rising arm of the meridional cell. The result is a boundary layer divergence-convergence doublet centered on the 𝜂 = 0 contour, as noted in cross-sections of Figures 13.8. In the upper troposphere there is a strong region of divergence located directly above the boundary layer convergence, with strong southerly winds and a southward displacement of the 𝜂 = 0 contour into the SH. Figure 13.16b is the counterpart of Figure 13.16a except for a section 8000 km to the west (section B, Figure 13.7), located in a region of weak cross-equatorial pressure gradient. A completely different meridional structure is found. Near the equator, the magnitude of 𝜓

has decreased by a factor of four compared to section A. Instead of a strong cross-equatorial meridional circulation, weak ascent is confined to the NH collocated with the maximum SST. Boundary layer convergence is also weak and the 𝜂 = 0 contour is aligned with the equator throughout the troposphere. In this location, no evidence of the presence of a shallow circulation is present. Figure 13.17 describes the latitude–time behavior of OLR anomalies and the 925 hPa 𝜂-field averaged in the longitudinal sections A and B. Periods of enhanced (negative OLR values: blue) and reduced (positive OLR: red) convection exist in both regions, with changes in sign occurring every few days. In the eastern section (Figure 13.17a), convective activity is confined between 5 and 13∘ N to the north of the zero AV contour. By contrast, the convection in the western section (Figure 13.17b) appears less spatially restricted, with convective events occurring on either side of the equator. There are also distinct differences in the absolute vorticity fields between the two sections. The eastern section (panel a) shows that the absolute vorticity is generally anticyclonic south of about 5∘ N, but with large excursions of the zero contour both to the north of 5∘ N and back toward the equator every few days. North of the equator, the northward extent of the +15 × 10−6 s−1 contour (bold red) varies in both space and time. Regions of counter-AV extend northwards and then retreat back toward the equator. Here, there is an anticyclonic AV field extending to at least 5∘ N and

13.3 Transient States of the Intertropical Convergence Zone

4–8 day filtered latitude time sections (a) 120°W–110°W (section A)

(b) 180°W–170°W (section B)

(i) OLR and 925 hPa δη

(i) OLR and 925 hPa δη

20°N

20°N

15°N

15°N

10°N

10°N

5°N

5°N

0° 203

210

220

230

240

0°

246

20°N

15°N

15°N

10°N

10°N

5°N

5°N 210

220

230

240

0°

246

(iii) 250 hPa ∇·V 20°N

15°N

15°N

10°N

10°N

5°N

5°N 210

203

210

220

230

240

246

0°

203

210

230

240

246

220

230

240

246

230

240

246

220

days

days η (10–6s–1)

OLR (Wm–2) –50

220

(iii) 250 hPa ∇·V

20°N

0° 203

210

(ii) 925 hPa ∇·V

(ii) 925 hPa ∇·V 20°N

0° 203

203

–30 –10 +10 +30 +50 –25

–15

–5 0 +5

∇·V 10–6 s–1 +15

+25

–8

–4

0

+4

+8

Figure 13.18 Time–latitude plots of the 4–8 day filtered data for 44-day period during the summer of 1996 between the equator and 20∘ N for two bands: (a) 120∘ W–110∘ W (section “A”) and (b) 180∘ W–170∘ W (section “B”). Panel (i) OLR anomalies (W m−2 , shading lower scale) and anomalies of the absolute vorticity field 𝜂 (10−6 s−1 , contours, lower scale). Bold black contour denotes the zero 𝜂 anomaly. Panels (ii) and (iii) show divergence at 925 and 250 hPa, respectively, relative to the contour scale at the bottom of the figure.

sometimes as far north as 8∘ N. In the western section (Figure 13.17b), the variations of 𝜂 = 0 are confined relatively close to the equator. A four to eight day band-pass filter was used to analyze the latitude–time series of Figure 13.17. Results of the analysis are displayed in Figure 11.18. The uppermost panels show the band-passed OLR and 925 hPa 𝜂-fields, while the lower two panels show the corresponding divergence fields at 925 and 250 hPa. In the eastern section (Figure 13.18a), OLR anomalies are

negatively correlated with absolute vorticity and show a tendency for northward propagation from south of the equator, culminating in the region of maximum convective near 10∘ N. The low-level divergence (panel ii) correlates negatively with convection and, in general, is out-of-phase with the 250 hPa divergence field (panel iii). The overall configuration of the fields suggests a systematic northward propagation of alternatively signed anomalies across the equator into the convective region of the mean ITCZ near 10∘ N.

283

284

13 Near-Equatorial Precipitation

̃ for the western The distributions of 𝜂 and ∇ ⋅ V section (Figure 13.18b) are dramatically different from those found farther east. The magnitudes of all fields are smaller and there is less evidence of systematic association between OLR and AV. The upper and lower level divergence patterns have a similar out-of-phase relationship but the extrema occur much closer to the equator. It may be argued that the alternation of convection and absolute vorticity is the result of the propagation of troughs and ridges past a point. However, a complicating factor is that there appears to be a coherent poleward propagation from the near-equatorial SH to latitudes poleward of 10∘ N, taking roughly four to six days in transit. The question of an in situ or remote origin of Pacific waves will be considered in a modeling study described in Section 13.3.4. 13.3.2

Transient Composites

Figure 13.18 suggests that there are two major extreme states in the four to eight day period band. The first is dominated by anticyclonic absolute vorticity (Δ𝜂 < 0) and shallow convection (ΔOLR > 0). The second extreme is dominated by cyclonic absolute vorticity (Δ𝜂 > 0) and deep convection (ΔOLR −20 W m−2 ; analyzing some 40 cases found in the period 1981–2000. These were used to define the state of the ITCZ at days −3, −2, …, +2, +3 relative to day 0 between 30∘ S and 30∘ N. In essence, these composite circulations can be thought of as transient anomalies that are superimposed upon the long-term mean circulation displayed in Figure 13.16. Figure 13.19 shows the latitude–height structure of the composite meridional circulation along section A (120∘ W–110∘ W) from day −3 to day +3, with the long-term mean circulation subtracted. Figure 13.19a shows the mass stream function (𝜓) and relative humidity (RH %), whilst Figure 13.19b shows the meridional wind speed (v m s−1 : shaded lower scale) and diver̃ , 10−6 s−1 ). It is apparent that gence contours (∇ ⋅ V changes in the circulation, throughout the composite period, extend over the entire 30∘ S and 30∘ N latitudinal domain. Following the composite sequence, we note: Day −3: A broad meridional circulation extends from 30∘ S to 10∘ N with subsidence from 10∘ S to 10∘ N collocated with boundary layer divergence. A negative mid-tropospheric relative humidity anomaly is located north of the equator. Upper tropospheric meridional winds converge near 10∘ N. Anomalous rising motion is apparent in the SH and corresponds to the relatively weak negative anomalies in OLR.

Day −2: A shallow cell has formed in the region between the equator and 10∘ N, with weak rising motion to the north. Low tropospheric convergence has developed to the south of 10∘ N. Convergence still exists in the upper troposphere, but a region of lower–middle tropospheric divergence with a southward flow accompanies the development of the shallow cell. Day −1: Large-scale changes have occurred throughout the domain. The shallow cell has extended into the upper troposphere although remnants still exist, but with a weakened southerly flow in the middle troposphere. Relative humidity has started to increase in the middle troposphere equatorward of 10∘ N. Strong northerly cross-equatorial flow has developed, producing strong convergence near 10∘ N. In the upper troposphere, the flow has become more divergent and cross-equatorial northerly winds have developed. Day 0: The OLR has reached its most negative value. The meridional circulation has increased in magnitude by a factor of nearly two along with the relative humidity growth throughout the near-equatorial NH. The lower tropospheric shallow circulation has disappeared entirely and the tropospheric column to the north of the equator is dominated by strong low-level convergence and upper-level divergence. The southward cross-equatorial upper-tropospheric flow has reached its strongest magnitude. The low-level convergence, near 10∘ N, strongest at day −1, has weakened and a small region of divergence has developed near 6∘ N. Day +1: The overall circulation has weakened in magnitude by almost 50%. A shallow cell has developed, accompanied by strong boundary layer divergence near 6∘ N and convergence near 700 hPa. A two-cell structure similar to Day −2, although reversed in sign, has developed. Day +2: The shallow circulation has extended vertically and now occupies the entire troposphere, producing anomalous subsidence and drying over the NH tropics. Day +3: The pattern returns to a state similar to that of day −3. The magnitudes of the band-passed filtered transient anomalies displayed in Figure 13.19 are about a factor of two smaller than the mean circulation mass stream function (Figure 13.16). Superimposing the composite circulations on the mean state produces a meridional circulation and convection that oscillates in intensity and also displays a changing latitudinal structure on four to eight day time scales. The evolving composite state of the total meridional circulation (mean plus anomaly) during a disturbance is plotted in Figure 13.20. The left-hand column shows

13.3 Transient States of the Intertropical Convergence Zone

(a) Streamfunction and relative humidity anomaly along section A: (120°W–110°W) day +1

day –3

0

400

–0.3 –0.6

600

0 0.9

800

1.5 1.2

600

–0.9 –0.3

–0.6

0

–0.6 –0.3

0

0.3 0.6

–0.3

10N

20N

1000 30S

20S

10S

0

10N

20N

30S

30N

day –2

20S

0.6

400

hPa

600

0.3

800

0 –0.3

10S

–0.6

0

10N

0 0.3 –0.6

0 –0.6

–0.3

30N

–0.3

0.3

–0.6

600 –0.3

0.9

0.3

1000 30S 20S day +3

10S

0

10N

20N

30N

200 0

hPa

400

20N

0

0.6

800

1000

–0.9

600 –1.2

800

30N

0

400

0

200

0

0

200

30S 20S day –1

10S

day +2

200 hPa

0.3

400

800

0.6 0.3

1000

hPa

0

200 hPa

hPa

200

0

400

1.5 –0.3

600 1.2

800

0.9 0.6

0.3

1000

1000 30S

20S

10S

0

10N

20N

30N

30S

20S

10S

0

10N

20N

30N

latitude

day 0

hPa

200 –0.9

400

0.3

RH

–0.6 –1.8 –1.5

600

–1.2

800

% –10

0

–5

0

+5

+10

–0.3

–0.6

1000 30S

20S

10S

0 10N latitude

20N

30N

Figure 13.19 (a) Composites of the anomalous circulation in the height–latitude plane between 30∘ S and 30∘ N averaged between 130∘ W and 110∘ W (section “A”: Figure 13.7) for day −3 to day +3 relative to the occurrence of maximum convection at 10∘ N. All diagrams are constructed from the 4–8 day band-passed fields. Day 0 of the composites is defined as days in which Δ OLR ≤ 20 W m−2 at 7.5∘ N in the 4–8 day band. A total of 40-day zeros were so defined in the June–September period from 1981 to 2000. The left-hand panels (a) show the mass stream function (1011 kg s−1 ) and the relative humidity (% shaded, bottom scale) and the right-hand panels. (b) Same as (a) except for the meridional wind component (m s−1 , shading, bottom scale) and the horizontal divergence (10−6 s−1 ). For clarity, the zero divergence contour is omitted. Source: After Figure 8 from Toma and Webster (2010a).

285

13 Near-Equatorial Precipitation

(b) Divergence and meridional wind anomaly along section A: (120°W–110°W) day +1

day –3

400

1000

0.6 1.2

30S

20S

10S

0

10N

400

800 0.6

20N

30N

1000 30S

–0.6

hPa

hPa

0.6

400

0.6

600 1.2

800

10S

–0.6

20S

10S

0

10N

20N

30N

–0.6

400

–0.6

600

10N

20N

30N

1000

0.6 1.8

30S

20S

10S

0

10N

20N

30N

20N

30N

day +3

200

200

0.6

–0.6

hPa

0.6

400

0.6

800

–1.8

10S

0

10N

–1.2

600 0.6

800

–0.6

20S

–0.6

0.6

400

20N

30N

1000

1.2

30S

20S

10S

0

10N

latitude

day 0 200

1.8 –0.6

1.2 0.6

400

v

–0.6 –1.2

800 30S

20S

10S

Figure 13.19 (Continued)

0 10N latitude

(ms–1) –2.4 –1.8 –1.2 –0.6

600

1000

0

–0.6

800

day –1

hPa

20S

200

–1.8

1000 30S

–0.6 –1.2

day +2

200

600

–0.6

600

day –2

1000 30S

0.6

–0.6

600 800

1.8

200

–1.2 –0.6

hPa

hPa

200

hPa

286

20N

30N

0

+0.6 +1.2 +1.8

13.3 Transient States of the Intertropical Convergence Zone

Composities of total circulation and heating rates along section A (120°–110°W) (a) Streamfunction and RH day –2

1

4 2 1 0 –1

–2

400 –2

400

hPa

hPa

0

0

200

200

0 1

(b) Heating rate (K day–1) day –2

3

600

–2

600 0

10°S

0°

10°N

20°N

200 –2

1

400

400

–4

10°S

0°

10°N

20°N

200

0

–1

–3 –1 –2

10°S

0°

10°N

0

5

–2

400

20°N

day +2

400

1

hPa

hPa

31

–2

0

200

20°N

11

1000 20°S

day +2 0

5

–2

–2

600 800

5

0

–2

10°N

–1

0

1000 20°S

0°

day 0

600 800

0

0

1

hPa

hPa

200

10°S

0

0

2 6

0 1

1000 20°S

day 0

–1

–3

–1 0

3 1

–2

1

1000 20°S

800 –2

5

3

800

2 0 3 –1

4

600 –2

0

600 3 5

800

1000 20°S

–2

3

800

1

10°S

0°

1000 20°S

1

0

10°N

–2 –1

20°N

–3 –1 0

1

10°S

0°

Relative Humidity

2

3

10°N

20°N

% 10

20

30

40

50

60

70

80

90

Figure 13.20 (a) Height–latitude sections of the total (mean plus anomaly) mass stream function (1011 kg s−1 ) circulations between 30∘ S and 30∘ N for composite days −2, 0, and + 2 (left-hand column). (b) Corresponding height–latitude sections (20∘ S–20∘ N) of material tendency of potential temperature (K day–1 , gray shading represents positive heating values) are shown in the right-hand column. Heating rate contours >5 K day–1 are in red. Source: After Toma and Webster (2010a).

287

13 Near-Equatorial Precipitation

(a) 120°W – 110°W; 10°S – 8°S

Time section of composite heating (b) 120°W – 110°W; 8°N – 10°N 200 pressure hPa

200 pressure hPa

288

400 600 800

–2

–2 –4

–2 0 +2 Composite Day –4

+4 –2

1000

+6 0

+11

600 +6

+7

800

1000 –6

400

2

4

θ (K/day)

–6

–4

6

8

–2 0 +2 Composite Day

+4

+6

10

Figure 13.21 Composite of height composite sequence of the material tendency of the potential temperature 𝜃̇ (K day–1 ) for two regions: (a) 10∘ S–8∘ S and (b) 8∘ N–10∘ N along the 130∘ W–110∘ W meridian. Source: From Toma and Webster (2010a).

latitude–height distributions of the total circulation for composite days −2, 0, and + 2. The right-hand column shows distributions of total heating (the material diffeṙ as discussed in Section ential of potential temperature, 𝜃, 13.3.4.10). At all times, the region to the south of the 𝜂 = 0 contour is relatively dry and strongly subsident, leading to efficient radiative cooling in the middle and low tropospheres throughout the composite sequence. Over the ocean cool tongue, between 4∘ S and the equator, there is a narrow band of boundary-layer cooling. To the north of the 𝜂 = 0 contour, the vertical velocity is positive and moisture extends throughout the entire column. This is a region of intense latent heating that oscillates throughout the composite cycle. During the maximum convective phase (day 0), the latent heating has increased at 500 hPa from 6 K day−1 at day −2 to 13 K day−1 at day 0. The vertical heating structure evolves in a complicated fashion. To examine this behavior, composite time–height distributions of total heating are shown in Figure 13.21 for the bands 10∘ S–8∘ S and 8∘ N–10∘ N in the eastern Pacific. Prior to day 0, the heating is strongest in the lower and middle troposphere. As the convection intensifies, the heating expands upwards, forming a new and stronger maximum in the middle upper troposphere. The lower maximum remains and increases in magnitude but does not reach the levels of the more elevated heating maximum. In summary, the ITCZ in the eastern Pacific oscillates between a highly convective state (day 0) and a period of reduced convection (day −3 or +3), cycling through periods of enhanced and reduced heating over the NH equatorial regions. In a sense, this shows an oscillation between successive periods when the system is rendered unstable by the advection of absolute vorticity across

the equator and when the system is returned to stability by vortex tune stretching. The shallow meridional circulation occurring on the transitional days −2 and +2 has many of the characteristics of the lower meridional circulation shown schematically in Figure 13.9. Overall, the sequence of relative heating and cooling, occurring in accord with the evolution of the unstable circulations and the ameliorating secondary circulations is termed the “inertial oscillator”18 that operates in locations of strong CEPG. 13.3.3

Diagnostics of ITCZ Transients

To determine the physical mechanisms that determine the transient state of the ITCZ, we follow the evolution of the terms in the absolute vorticity equation expressed in Eq. (13.11). Figure 13.22 shows two composite transient cycles (i.e. from day −6 to day +6) as a function of latitude from 5∘ S to 20∘ N in longitude band 130–110∘ W. Panel (i) shows the evolution of the OLR (W m−2 ) and the 925 hPa 𝜂-field (10−6 s−1 ). Against a background of negative anomalous OLR, the fields oscillate between deep and shallower convection. Deep convection occurs with the growth of cyclonic AV and the extension of the 𝜂 = 0 contour away from the equator. This association is seen more easily in terms of OLR and 𝜂 anomalies (panel ii), except that a strong northward propagation is now apparent. The 925 hPa meridional wind (panel iii) shows a clear cross-equatorial flux across the equator at least a day ahead of the OLR maxima anomalies near 10∘ N. Panel (iv) shows the consistent generation of convergence and divergence ahead of the convective maxima and minima. Finally, panel (v) describes the evolution of the vortex stretching and divergence terms 18 Toma and Webster (2010a).

13.3 Transient States of the Intertropical Convergence Zone

Dynamic evolution of ITCZ Oscillation (i) OLR (shading) and 925 hPa η (lines)

(iv) V-field (vectors), ∇·V(lines) 20N

20N

3 ms–1

latitude

50

15N

40

10N

30 5

0

0.8

5N

0

–10

–6 –4 –2 0 +2 +4 (ii) δOLR, 925 hPa δη (lines), 4–8 days 20N

+6

latitude

2

10N 1

–4

–4

–2

0

0

+4

+6

5S

latitude

–2 –4

0

0 =

0

–6

–4

+4

–2 0 +2 composite day

200

230

260

δOLR

Wm–2 –20 –10 0 +10

+20 10–12 s–2

−V·∇η –0.4 0

0.6 0

–4 –3 –2 –1

–0.4

0.4

0.2

0

+6

Wm–2 170

–0.6

5N

5S

2

OLR

10N

0.4

+6

0

–4

0

0

0

0

+4

2

(iii) RH (500hPa), v(lines), 4–8 days

–0.6

+2

–2 2

20N 15N

0

5N 0

+2

–2

–2

2

0

0

–6

–6

0

15N

3

1

0 5S

–1

–2

5N

5S

0

15N –2

0

(v) –V·∇η (shading), η∇·V (lines) 20N

0

–1

–0.8

0.8

0

5S

10N

1.6

0

–5

0

–0.8 –1.6

10N

20

10

5N

15N

0

+1 +2 %

RH –7.5 –2.5 0 +2.5 +7.5

–6

–4

–2 0 +2 composite day

+4

+6

Figure 13.22 Dynamic balances during the transitions of the ITCZ between composite days −6 to +6: (a) time–latitude sequence of the OLR (shaded: scale bottom right: W m−2 ) and the total absolute vorticity at 925 hPa (𝜂; contours 10−6 s−1 ); (b) 925 hPa band-passed absolute vorticity and band-passed OLR anomaly; (c) band-passed meridional wind v (contours m s−1 ) and 500 hPa relative humidity ̃ (m s−1 , vector scale top right) and 925 hPa divergence (shaded bar scale below); (d) band-passed 925 hPa horizontal velocity vector V −6 −1 ̃ ̃ ̃; (∇ ⋅ V ; contours: 10 s ); and (e) absolute vorticity advection −V .∇𝜂; shading scale below (10−13 s−2 ) and vortex stretching term (𝜂∇ ⋅ V contours 10−13 s−2 ). Terms defined in Eq. (13.21). Source: Adapted from Toma and Webster (2010a, b).

of the absolute vorticity equation. In the region of maximum convection on day 0, the two terms almost completely balance. 13.3.4

Origin of “Easterly Waves”

The prevailing thought for many years was that westward propagating equatorial waves or “easterly waves” originated in the African region and propagated across

the Atlantic and into the Pacific Ocean. Frank (1969) offered the idea that westward propagating waves formed over Africa and propagated along the equator more or less globally. Burpee (1972) suggested that the state of the lower troposphere over North Equatorial Africa was unstable, producing waves of four to five day periodicity. These became known as African Easterly Waves.

289

13 Near-Equatorial Precipitation

400 330 500

pressure (hPa)

290

25.0 700

SFC

310

6.25

850

320

E

12.5

E

W 300

0°

5°N

10°N

15°N

20°N

25°N

30°N

latitude (along 5°E) Figure 13.23 Latitude-height cross-section along 5∘ E across the Gulf of Guinea and equatorial West Africa. Color shading denotes the zonal velocity component. Blue shade denotes westerlies ( − 5 m s−1 and −10 m s−1 . Isentropes (∘ K) appear as dashed blue lines. Red contours show the isentropic potential vorticity field (units PVU: 1 PVU = 10−6 m2 s−1 K kg-1) . Note the change in the gradient of q along an isentrope to the south of the low-level easterly jet. Source: Based on the analysis of Burpee (1972).

Burpee argued that the very strong surface heating gradient between the desert regions and the Gulf of Guinea (see Figure 13.13c) created an easterly zonal maximum in the lower troposphere with wind strengths of order 10–15 m s−1 between the 700 and 600 hPa levels, shown clearly in the meridional cross-section along 5∘ E shown in Figure 13.23. The maximum arises from the influence of the strong surface heating gradient between the Gulf of Guinea and the West African land mass. In Figure 3.2a, the potential temperature field is flat or slightly negative between the equator and 30∘ N, but, along 5∘ E, the horizontal gradient is positive until nearly 25∘ N. Burpee calculated the zonal component of the isentropic potential vorticity distribution which, from Eq. (9.14) is given by ) ( 𝜕𝜃 𝜕U 𝜕𝜃 = −g − |𝜃 + f (13.26) q = −g𝜂𝜃 𝜕p 𝜕y 𝜕p Contours of q are shown in red in Figure 13.23. Charney and Stern (1962), in developing a generalized theory for the instability of an internal jet, had shown that the vanishing of the gradient 𝜕q∕𝜕y along an isentrope is a necessary condition for instability, allowing the growth of perturbations at the expense of the background energy. It is clear from Figure 13.23 that 𝜕q∕𝜕y on the equatorward side of the maximum easterlies 700 and 500 hPa possesses zeros. In a more extensive analysis than shown here, Burpee showed that the lower troposphere was unstable west of about 30∘ E and south of about 15∘ N. Questions have been raised about North Africa being the source of Pacific Ocean easterly waves by a number

of careful diagnostic studies,19 which found that the majority of Pacific easterly waves formed in the eastern part of the East Pacific Ocean north of the equator. This is an important observation, as it suggests other regions within the tropics may also be genesis regions of easterly waves. To investigate the role of African waves forcing those in the Pacific, a series of controlled numerical experiments (Toma and Webster 2010b) used the NCAR WRF model20 to determine the relative importance of Atlantic easterly waves in forcing Pacific Ocean easterly waves, versus the importance of in situ development of inertial instability in the large CEPG region in the eastern Pacific. These experiments employed observed flux conditions at the eastern boundary of a Pacific Ocean domain to allow a potential Atlantic influence. Then a no-flux condition was placed at the eastern boundary of the model domain to prevent an Atlantic influence. In all of the simulations, westward-propagating waves formed in the region to the west of South and Central America, where the CEPG is large and the zero AV contour was well removed from the equator. In summary, influence from the Atlantic was not a factor in the generation of near-equatorial waves in the Pacific. We can now return to the issue region of off-equator divergence and convection that exists in the centralwestern Pacific Ocean, where in JJA the zero AV contour lies close to the equator (Figure 13.7c). Here, local inertial instability cannot explain the location of convection. However, the westward propagation of waves generated remotely in the eastern Pacific, as the result of inertial instability, may account for convection removed from the equator. In this manner, the theory of Holton et al. (1971) comes into play (Section 13.1.2.2), which stated that the ITCZ is the locus of propagating waves. Together, the inertial instability mechanism in the eastern Pacific and Holton et al.’s (1971) wave propagation may contribute significantly to our understanding of transient variance of easterly waves across the Pacific Ocean.

13.4 The Great Cloud Bands In Section 1.2.2(b), we identified a number of notable features in the climatology of tropical rainfall. The first was the rainfall associated with the summer monsoon circulations (e.g. Figure 1.13a to c). The second feature is the oceanic ITCZ, discussed in detail in the last chapter. 19 E.g. Serra et al. (2008). 20 Weather Research and Forecasting regional model: version 2.2 (Skamarock et al. 2005), with 28 levels in the vertical. Details of the model set-up can be found in Toma and Webster (2010b).

13.4 The Great Cloud Bands

A third feature is the extensive precipitation occurring in the “Great Cloud Bands” (GCBs), three of which occur in the SH and, possibly, a fourth in the NH. Except perhaps for the ITCZ, the GCBs are the most extensive precipitating phenomena on the planet. The GCBs are of interest for a number of reasons: first because of their prominence and second, in the SH, because they constitute areas of greatest total rainfall amounts either by rate or volume. It may be recalled from Table 2.1 that the total rainfall by volume per year is almost exactly the same in both hemispheres, yet the distribution of rainfall between the hemispheres is very different. For example, the ratio of land rainfall versus ocean rainfall is 38% in the NH but only 21% in the SH. In fact, almost all of the SH ocean precipitation takes place in the GCBs. Except in the central and eastern Indian Ocean, there is little evidence of the ITCZ in the SH. Third, the orientation of the SH bands, sloping poleward from the northwest to the southeast, is intriguing. Finally, the poleward ends of the bands are located in the vicinity of the westerly duct troughs. It is interesting to speculate whether the location and orientation of the GCBs are involved with tropical–extratropical interaction, as discussed in Chapter 11, or perhaps even interhemispheric interaction. 13.4.1

Climatology of the GCBs

Figure 13.24 shows the mean OLR (W m−2 ) for DJF and JJA with the GCBs marked by solid red lines. The bands were noted early in the satellite era by Saha (1973) and Streten (1973, 1975). As noted above, the GCBs appear to stretch diagonally across the tropical oceans oriented on a general northwest–southeast axis. For example, the Pacific Ocean band of low OLR extends from the equatorial Maritime Continent to the extratropics south of 35∘ S. This feature has become known as the South Pacific Convergence Zone (SPCZ) and is the most prominent of the three SH bands. A similarly oriented band, the South Atlantic Convergence Zone (SACZ), extends out of Brazil and into the South Atlantic. A third band is located in the South Indian Convergence Zone (SICZ) and, although weaker than the other two, extends from Central Africa to the southern Indian Ocean. These bands were not recognized prior to the satellite era and their global scope until GARP.21 Figure 13.24 also suggests that, during the JJA, there is an NH cloud band extending out from South Asia across the North Pacific Ocean in a generally northeasterly direction. This is the so-called “Mei-yu-Baiu 21 Vincent (1994) provides a comprehensive review of the history of the SPCZ including early theories about its formation.

Frontal Zone” (MBFZ). 22 Unlike the SH cloud bands, the MBFZ influences a large percentage of humankind with its association with summer precipitation over China and Japan. Kodama (1992, 1993) provides an interesting comparison of the properties of the SH cloud bands and the NH MBFZ. 13.4.2

Variability within the GCBs

We consider the SPCZ first. Figure 13.25 describes background DJF climatology across the Pacific Ocean basin constructed from a 1982–2008 database. In each section, the bold blue contour encloses the area where OLR < 240 W m−2 , which we define arbitrarily as the delimiter of the convection within the SPCZ. The three panels show, respectively, the 200 hPa vector wind field and geopotential (panel a), the magnitude of the 200 hPa zonal wind field, U, together with the zonal stretching deformation (dU∕dx) (panel b), and the SST (panel c). A strong trough occupies the upper troposphere of the eastern Pacific Ocean. This is collocated in longitude with an NH trough that, collectively, constitutes the Pacific Westerly Duct discussed in Chapters 9 and 11. This is the region through which disturbances in the extratropics can infiltrate the tropics and where one hemisphere can potentially influence the other. On the western flank of the SH trough is a region of strong negative stretching deformation. Also, the SPCZ extends across the weak SST gradient of the western Pacific warm pool and then southeastwards across much cooler water. Figure 13.26 shows the normalized Fourier power spectra of OLR for the equatorial, tropical, and subtropical regions (identified as E, T, and ST in Figure 13.25a). Daily OLR anomalies for the period November through March (NDJFM) were used to develop 27 seasonal time series for each region, providing sufficient temporal resolution and length of data for analyzing synoptic (3–6 day) to intraseasonal (30–60 day) variability.23 The data series provides 26 degrees of freedom. In the tropical portion of the SPCZ, variability shows peaks near two weeks and between 30 and 60 days, with variance decreasing markedly toward shorter time scales. Overall, the spectra suggests that equatorward of 20∘ S the SPCZ is strongly influenced by intraseasonal variability (e.g. the Madden-Julian oscillation) and oscillations occurring on biweekly time scales. South of 20∘ S, most of the significant OLR variability appears 22 E.g. Kodama (1992). 23 Red-noise spectra (lower dashed lines) for mean lag-1 autocorrelations of r = 0.90 (a), 0.78 (b), and 0.70 (c) along with the corresponding 95% confidence spectra (upper dashed lines) are calculated using the methods of Torrence and Compo (1998) with the assumption that each season is independent.

291

292

13 Near-Equatorial Precipitation

Distribution of seasonally OLR (W m–2)

(a) DJF 45°N 30°N 15°N 0°

SP

CZ

SI

15°S

CZ

30°S 45°S 0° (b) JJA

60°E

120°E

180°W

120°W

60°W

0°W

120°W

60°W

0°W

45°N 30°N

MBFZ

15°N 0°

SP

CZ

15°S 30°S 45°S 0°

60°E

120°E

180°W

W m–2 195

205 215 225 235 245 255

265 275

285 295

Figure 13.24 Global mean distribution of OLR (W m−2 ) for the years 1982–2008: (a) DJF, (b) MAM, (c) JJA, and (d) SON. A 240 W m−2 OLR contour, indicating deep convection, outlined by blue lines in each panel. Red lines indicate the GCBs: the South Pacific Convergence Zone (SPCZ), the South Atlantic Convergence Zone (SACZ), the South Indian Zone (SICZ), and the Meiyu-Baiu Frontal Zone (MBFZ).

to be contained in synoptic time scales with periods 28 ∘ C). Red boxes represent the “Equatorial (E)” (7.5∘ N–7.5∘ S, 135∘ E–165∘ E), “Tropical (T)” (5∘ S–20∘ S, 165∘ E–165∘ W), and “Subtropical (ST)” (20∘ S–35∘ S, 165∘ W–135∘ W) regions used for calculating OLR time series. Source: Adapted from Widlansky et al. (2011).

2

(c) Sub-tropical (ST)

1.5 1.0 0 64 32 16 8 4 period (days)

model basic state is shown in Figure 13.27a and the steady state response to equatorial heating appears in Figure 13.27b.24 The maximum vertical velocity response occurs at the equator, where the diabatic ̇ is a maximum and is in phase with the heating (Q) geopotential perturbation. Further poleward, where the magnitude of the westerly basic state winds increase, the vertical velocity maximum moves to the east whilst temperature fields move west. The orientation of the vertical velocity latitudinal profile possesses a similar orientation to that of the GCBs.

2

Figure 13.26 Normalized Fourier power spectra of NDJFM (1982–2008) OLR in the equatorial (E), tropical (T), and subtropical (ST) regions of the SPCZ, of Figure 11.30. Lower dashed lines are the red-noise spectra for mean lag-1 autocorrelations of r = 0.90 (a), 0.78 (b), and 0.70 (c). Upper dashed lines denote the 95% confidence limit. Source: From Widlansky et al. (2011).

Consider the steady state thermodynamic Eq. (J.1c) that, with (J.2) and the thermal wind equation, gives Q̇ v pf 𝜕U −U 𝜕 (T) − + wp S = − a cos 𝜑 𝜕𝜆 a cos 𝜑 R 𝜕p Cp (A)

24 Following Webster (1981).

4

1.0

0.5 25

30°S 120°E

2

1.5

2.0

(c) SST (°C)

4

(b) Tropical (T)

0.5 0 64 32 16 8

10

20

30°S

1.0

2.0

20

15°N

1.5

0.5 0 64 32 16 8

1.18 1.16

45°S 120°E

(a) Equatorial (E)

(B)

(W)

(C) (13.27)

where we have ignored dissipative effects. The vastly different latitudinal structures of the geopotential and the vertical velocity fields produced by a low-latitude SST anomaly arise from the transition from a “diabatic limit” balance in the vicinity of the anomaly to an “advective limit” at higher latitudes. The first limit arises from balance between (W) and (C); that is, the vertical velocity is determined almost solely by the distribution of diabatic heating. The second limit, even if Q̇ is sizeable, has the vertical velocity determined by advective events. For this limit to be approached, U would have to be large. These two limits were introduced by Webster (1981) in an attempt to explain why tropical responses

293

13 Near-Equatorial Precipitation

(a) Zonal Wind

(b) Vertical velocity (wd) and geopotential (ϕ) 1.0

1.0

90°N 60°N

–

0.8

+

45°N

0.6 750 hPa

0.5

250 hPa

+

15°N

–5

10

0.2

0°

0

–

–0.2

15°S

+ –0.5

0.4

–0.4

30°S –0.6 45°S

–

60°S –1.0

90°S 90°E

Figure 13.27 Steady state model results. (a) Zonal wind U = U(𝜑, p) used in the model containing both latitudinal and vertical shear. (b) Steady state response to equatorial forcing. Black ellipse shows the location of the e-folding magnitude relative to the central value. Contours denote vertical velocity (red) and geopotential (blue). (c) Schematic of the relative locations of the diabatic heating, vertical velocity, and temperature. Results from a series of experiments where the heating was moved progressively poleward into stronger and stronger westerlies. Shaded region denotes upward vertical velocity. Bold contours denote zeros. Source: Based on Webster (1981).

–0.8

+

0°

sin (latitude)

U (φ) 20

30°N

latitude

294

180° longitude

90°W

–1.0 0°

(c) Q

(i) Equatorial w

U∼ 0

T longitude

U>0

(ii) Subtropics

Q w

λ T longitude

U≫0

Q

(iii) Mid-latitudes w

T longitude

to SST anomalies appeared to be much larger than their extratropical counterparts for a given SST anomaly. In addition, the two limits were used to explain the curiosity that in the tropics the heating and vertical velocity responses are in phase whereas at higher latitudes the vertical velocity response leads the heating in longitude. Returning to Eq. (13.27), (W) balances (C) at low latitudes, a balance expressed earlier in Eq. (3.33), with the vertical velocity field, the heating, and the geopotential extrema in phase. From the scaling arguments in Chapter 3, the temperature perturbation is small

along the equator so that (A) and (B) are small as f → 0. At higher latitudes, the magnitude of (C), the diabatic heating, rapidly decreases in the example shown in Figure 13.27b. The vertical velocity term (W) is then determined by the advective terms (A) and (B). As a result, the vertical velocity maxima are found further and further downstream of the heating as U increases in magnitude with longitude. Actually, the magnitude of Q̇ with latitude is not important in determining the phase of the response. Consider Figure 13.27c. In the tropical diabatic limit,

13.4 The Great Cloud Bands

advection is very small and terms (A) and (B) do not come into play. As wp and Q̇ are in phase, the ensuing circulation is thermodynamically direct. At higher latitudes with an increasing U and a larger f , (A) and (B) approach parity with (C). As a result, the vertical velocity field becomes more and more out of phase with (C) and the vertical velocity raises cooler air. In the advective limit, the circulation is decreasingly direct or even indirect. Irrespective of the magnitude of the residual value of Q̇ at higher latitudes, the displacement of the vertical velocity field is similar to the orientation of the GCBs. 13.4.3.3

Continental and Orographic Forcing

Saha (1973) suggested that the SST distribution was tied to the locations of the GCBs. Other theories suggested that the influence of the three SH continents25 during the austral summer force standing waves producing the SH cloud bands. Numerical experiments were conducted, in which Australia and South America were selectively removed and replaced by set SST distributions. Results suggested that the presence of South America and the equatorial Pacific upwelling zone do not appear to be crucial to the location and orientation of the SPCZ, but the removal of the Australian continent destroys the southern hemisphere monsoon and substantially weakens the western part of the SPCZ. These results were interpreted to suggest that the northwest–southeast orientation of the SPCZ during the southern summer is more dependent on interactions with the midlatitude westerlies over the South Pacific than on the distribution of SST and land distribution over the SH. A further set of numerical experiments was conducted to test suggestions that the basic structure of the SPCZ was either orographically forced or related to the slowly varying background SST distributions.26 These more recent experiments suggested that a reduced magnitude of continental elevation had little impact on the climatological basic state. Similar theories27 have been advanced for the MBFZ. There the subtropical jet stream, presumably forced by the monsoonal heating, is deviated by the Himalayan-Tibetan complex to produce a trough over East Asia and Japan. Together with the poleward convergence of moisture in the lower troposphere, a cloud band is produced. In the paragraphs below another hypothesis will arise where the position of the SPCZ, for example, involves transients and the divergent flow set up by tropical heating in the manner suggested in Section 11.4.

25 Kiladis et al. (1989) and Kalnay et al. (1986). 26 See Widlansky et al. (2011) for details of the experiments. 27 Kodama (1992, 1993).

13.4.4

High-Frequency Variance in the GCBs

Figure 13.26 described a change in the nature of the variance along the SPCZ. A clue to this variability may be imbedded in the observations of Trenberth (1976). After describing the extratropical character of the SPCZ, he noted that the SPCZ tends to form a “graveyard” for fronts moving from the southwest in the troughs of low pressure between the migratory anticyclones across Australia and New Zealand. Trenberth subsequently observed28 that there was an association between the exit, or diffluent, region of the SH subtropical jet stream and the location of the SPCZ (Figure 13.24a). If this association is correct, then it would seem that the essence of an explanation of the diagonal section of the southern portion of the SPCZ might reside in the interactions of transient modes with the more slowly varying higher latitude background flow. The Trenberth hypothesis is supported by the wave kinematics discussed in Chapter 4 and in Section 7.3, where we discussed in detail the influence of the sign of U x on the group speed of Rossby waves. To reiterate, along a ray, if 𝜕U∕𝜕xx < 0, the longitudinal wavenumber decreases, following the kinematic rule of Eq. (4.4a), so that the longitudinal group speed decreases. Further, wave energy (actually wave energy density) will accumulate regionally, such as prescribed by Eq. (4.19). The following hypothesis was proposed by Widlansky et al. (2011). Eastward propagating midlatitude disturbances encounter a region of negative zonal stretching deformation in the upper troposphere, where their longitudinal extent shrinks, group speed is reduced, and wave energy accumulates. The U x < 0 region corresponds to the exit region of the jet extending across the south of Australia and to the east. This results in a highly energetic convective region that is oriented diagonally away from the equator in the South Pacific To assess the Widlansky et al. hypothesis, consider the transient structure of the diagonal portion of the SPCZ. The period 1 December 2005 through to 28 February 2006 is chosen arbitrarily for the case study, as representative of typical ENSO-neutral conditions. Figure 13.28a plots the longitude–time distribution of unfiltered OLR and negative zonal stretching deformation at 200 hPa, averaged between 20 and 35∘ S around the entire SH. A striking feature of the section is that the shaded OLR regions ( 240 W m–2

(b) Climate vs 2005–6 DJF (20°S–35°S)

280

5 4

270

3

260

2

250

0

1

–1

240

–2 –3

230 220

–4 0°

45°E 90°E 135°E 180°W 135°W 90°W 45°W

Climatology

–5 0°

Stretching deformation (10–6 s–1)

local surface moisture to promote convection. Near 130∘ W, between the central and eastern South Pacific, there exists clear demarcations in both the OLR and U x fields, showing that the eastern border of the SPCZ corresponds spatially and temporally to the sign change of zonal stretching deformation. Within the region of negative U x , zonal wave propagation speeds slow down compared to phase speeds upstream of the cloud bands in the positive U x areas. Outside of negative stretching deformation zones, which often correspond to jet stream exit regions, any disturbance that develops is observed to propagate eastward with a reduction in speed, as proposed in Section 7.3. Figure 13.28a appears to be representative of other years as well. Figure 13.28b compares the 2005–2006 DJF OLR and 200 hPa U x fields with the 27-year climatology. Except for subtle differences, the mean fields reflect the characteristics of the 2005–2006 case study quite well. Three pronounced areas occur in the 20–30 ∘ S band, where U x coincides with OLR minima, each corresponding to a major SH convergence zone plus the fourth weaker region near 135∘ E. On the other hand, Figure 13.28b indicates that regions where U x > 0 are associated with limited or absent deep convection. To investigate the impact of the basic state on the scale of waves further, we consider two zones of differing stretching deformation: a low convective region in the southeastern Indian Ocean (Figure 13.29a) and the convective South Pacific Ocean (Figure 13.29b). The results of a regression analysis of the OLR time series in the two regions is summarized in Table 13.1, showing differences in wavelength (km), phase speed (m s−1 ), and period (days) of synoptic waves. The OLR time series were passed through a three to six day Lanczos filter with 241 weights (following Duchon 1979) and then scaled by the standard deviation of the respective time series to compute OLR and U x linear regressions. Widlansky et al. (2011) reasoned that these regressions track synoptic-scale Rossby waves more clearly depicted than the longitude–time diagram of unfiltered data. In Figure 13.29a, fast waves are observed over the southeastern Indian Ocean with phase speeds of about 11 m s−1 and wavelengths of 4000 km. Regressions for the SPCZ base point (Figure 13.29b) depict slow waves with phase speeds and wavelengths of about 8 m s−1 and 3000 km. Composite analyses were constructed, following the methodology described by Serra et al. (2008), in order to examine more closely changes in the scale and propagation characteristics of waves moving into the diagonal SPCZ region. The regressions of unfiltered OLR and U x in Figure 13.29, relative to a base point at 30∘ S, consider the same 27 DJF seasons of daily

OLR (W m–2)

296

2005–2006

Figure 13.28 (a) Hövmoller (longitude–time) of OLR (shading depicts values less than 240 W m−2 ) and 200 hPa zonal stretching deformation (s−1 , contours) during the period 1 December 2005–28 February 2006. Zonal stretching deformation contour interval: −4 × 10−6 s−1 ; starting at -2 × 10−6 s−1 contour. Averaging is over a 15∘ latitude band (20∘ S–35 ∘ S). (b) 2005–2006 DJF (dashed lines) and 27 season climatology (solid lines) of OLR (blue) and zonal stretching deformation (red). Gray shading depicts regions of negative climatological zonal stretching deformation. Source: From Widlansky et al. (2011).

data as used to construct the climatologies appearing in Figures 13.25. The 135∘ W regressions (Figure 13.30a) show propagating convective anomalies with a period of three to five days. Set in a background of climatological negative zonal stretching deformation Ux , anomalies

13.4 The Great Cloud Bands

(a) SE IO: (30°S,105°E)

lag (days)

Figure 13.29 (a) Hövmoller (longitude–lag) of OLR (W m−2 , shading, − 1 W m−2 contour outlined) and 200 hPa zonal stretching deformation (s−1 , contours) regressions at 30∘ S for a 3–6 day filtered OLR base point at 30∘ S, 105∘ E (Southeast Indian Ocean). Zonal stretching deformation contour interval: 2 × 10−7 s−1 ; starting at ±4 × 10−7 s−1 . Solid (dashed) lines depict negative (positive) anomalies. (b) Same as (a) but for base points at 30∘ S, 135∘ W (SPCZ). Source: From Widlansky et al. (2011).

(b) SPCZ: 30°S, 135°W)

–8

–8

–4

–4

0

0

4

4

8

8 0°

40°E

80°E 120°E 160°E

120°E 160°E 160°W 120°W 80°W W m–2

–6

–5

Table 13.1 Synoptic wave characteristics for the Southeast Indian Ocean (30∘ S, 105∘ E) and the SPCZ (30∘ S, 135∘ W).

Base point

30∘ S, 105∘ E 30∘ S, 135∘ W

Phase speed (m s−1 )

Wavelength (km)

–4

–3

–2

–1

4000

4.3

7.8

3000

4.4

Source: After Widlansky et al. (2011).

1

2

3

4

5

6

Table 13.2 Synoptic-wave characteristics for a disturbance propagating through the SPCZ at lags of −2, 0, and +2 days. Base point (30∘ S, 135∘ W).

Period (days)

10.8

0

Base point

Phase speed (m s−1 )

Wavelength (km)

Period (days)

Lag = −2 days

15.2

5800

4.4

Lag = 0 days

7.0

2700

4.4

Lag = +2 days

3.5

1300

4.4

Source: After Widlansky et al. (2011).

lead the OLR negative anomalies by about one day or 5∘ of longitude. Trenberth (1991) had noted similar zonal wind anomalies around 30∘ S ahead of vertical motion perturbations. By tracking the negative OLR signal westward from the base point (lag 0 days), anomalies exceeding −1 W m−2 are observed to 170∘ W (lag −5 days). The same anomalous OLR, growing to −5 W m−2 , continues eastward from the base point for about another five days, although only reaching 130∘ W. The composite disturbance propagates 35∘ of longitude during the first five days, then slows and amplifies near 135∘ W, finally dissipating near the eastern SPCZ boundary. Phase speeds become increasingly slower as disturbances propagate eastward through the SPCZ, responding to the background location of negative zonal stretching deformation. Table 13.2 compares the wave characteristics (indicated by U x anomalies in Figure 13.31b to d) at different lags within the SPCZ. At lag 0 days, wavelength and phase speed are comparable to those derived from OLR anomalies in Figure 13.29b. The table confirms that wave characteristics change as disturbances propagate through the SPCZ. Wavelengths shrink from about

5800 to 1300 km, while the phase speed decreases from 15.2 to 3.5 m s−1 between lags of −2 and 2 days. Observations presented in Figure 13.30 confirm that longitude bands of slow wave propagation (i.e. low OLR) are correlated spatially (r = 0.57) with the climatological locations of U x . Equation (4.19) states that wave energy density (𝜉) will grow rapidly in regions where U x < 0. In fact, the magnitude of OLR anomalies shows a tendency to become more enhanced as cgd decreases. Regression anomalies exceeding −6 W m−2 are more common with the slower, and shorter, synoptic disturbances in the subtropical SPCZ compared to over the Southeast Indian Ocean (Figure 13.30 and Table 13.1). In summary, the diagonal orientation of the southern SPCZ may owe it existence to the modifications of the propagation characteristics of transient Rossby waves by the background basic state. On the other hand, the tropical portion of the SPCZ is determined by the slow propagation of equatorially trapped convective modes discussed in Chapter 8. The circulation patterns described above suggest strong linkages between the tropics and the extratropics and a commonality with processes in the NH discussed

297

13 Near-Equatorial Precipitation

–10

(a) Hövmoller (long-lag) along 30°S

(b) Lag = –2days

–8

15°N 0° 15°S

–6

30°S 45°S

–4

120°E 150°E 180°W 150°W 120°W 90°W (c) Lag = 0 days

2 lag (days)

298

15°N 0° 15°S

0

30°S 2

45°S 120°E 150°E 180°W 150°W 120°W 90°W (d) Lag = +2 days

4

15°N 0° 15°S

6 8

30°S

10 120°E 150°E 180°W 150°W 120°W 90°W

45°S 120°E 150°E 180°W 150°W 120°W 90°W W m–2

–6 –5 –4 –3 –2 –1

0

1

2

3

4

5

6

Figure 13.30 (a) Hövmoller (longitude–lag) of OLR (W m−2 , shading, −1 W m−2 contour outlined) and 200 hPa zonal stretching deformation (s−1 , contours) regressions at 30∘ S for a 3–6 day filtered base point (−20 W m−2 threshold) at 30∘ S, 135∘ W. Zonal stretching deformation contour interval: 2 × 10–7 s−1 ; starting at ±4 × 10–7 s−1 . Solid (dashed) lines depict negative (positive) anomalies. Panels (b), (c), and (d) are the same as (a) but for longitude–latitude regressions at lags −2, 0, and 2 days, respectively. The 240 W m−2 OLR contour (DJF climatology) is outlined by bold blue contour. Source: From Widlansky et al. (2011).

in Chapter 11. First, in the Pacific sector, the source of the eastward propagating transients is the jet stream over and to the south of Australia. Second, the jet stream owes its existence to the divergent circulation created by monsoonal heating over Indonesia and North Australia. Third, in Figure 13.24, we noted that the large-scale upper tropospheric trough located in the SH Pacific region was directly south of the NH upper tropospheric trough. In between these two troughs are the strong equatorial westerlies. In Chapter 11 we concluded that the climatological NH upper-tropospheric trough was the result of recursive Rossby wave breaking, resulting in the equatorward advection of potential vorticity substances and a necessary condition for the balanced interaction between the tropics and higher latitudes. Similar basic state distorting fields exist in the SH. In essence, the SPCZ (and probably the SACZ) in terms of potential vorticity substance conservation and a mirror image of the Figure 11.11 for the SH. This should not be a surprise as the impermeability theory of Haynes and McIntyre (1987) is a global theory and

speaks to the global conservation of potential vorticity substances.

13.5 Some Conclusions In this chapter, with the dynamics of the desert in mind (Chapter 12), we have addressed some of the physical mechanisms that determine the major arid and pluvial regions on the planet, deciding to leave the discussion of the monsoon until Chapter 16. A few conclusions can be drawn. (i) The ITCZ may lie off the equator in regions of cross-equatorial pressure gradients that force the advection of “wrong-signed absolute vorticity” across the equator. The pressure gradient is the result of slowly varying SST differences. Such an intrusion is inertially (symmetric) unstable. The CEPG is not sufficient as static stability on the warm side of the equator must not be too great. Such a stipulation renders the cross-equatorial

13.5 Some Conclusions

monsoon flow inertially stable, even though the region possesses the strongest CEPG on the planet. (ii) The GCBs are the predominant precipitation features of the SH. There is some indication that the NH also contains a cloud band with similar features to those in the SH. The equatorial part of the bands appear “anchored” in a warm pool region or a location of monsoon rainfall extending toward higher latitudes. Simple first-order dynamical arguments were used to explain the orientation as the thermodynamical balance changes from a diabatic limit in low latitudes to an advective limit as the zonal winds increase in higher latitudes. (iii) High-frequency variance changes from 20 to 40 days in the equatorial regions of the bands to high-frequency multiday variance at higher latitudes. The extratropical variance appeared to be strongly modified by the longitudinal variability of the basic flow, showing contraction of a

longitudinal scale and a slowing down of phase and group speeds. (iv) In Chapter 11 we developed an argument to suggest that within the westerly duct or tropical upper-tropospheric trough (TUTT) potential vorticity was funneled toward the equator through reoccurring breaking of Rossby waves that were generated in the unstable extratropical jet stream and propagated eastward in the outflow region of the jet. The jet itself is the result of the poleward advection of potential vorticity by the tropical and subtropical heat sources. The reoccurring high-frequency waves complete the potential vorticity balance by advecting potential vorticity back toward the equator. In essence, the ducts and TUTTs completed the PV balance circuit.

Notes I James Reed Holton (1938-2004): Holton was a theorist

who specialized in the dynamics of the atmosphere. He spent most of his career in the Department of Atmospheric Sciences at the University of Washington, Seattle, WA. He is author of the authoritative text book “An Introduction to Dynamic Meteorology” that was extended to five editions. He received many prestigious awards and was a Fellow of the United States Academy of Sciences. II “Shoaling” or shallowing of the thermocline is analogous to the depth of the ocean becoming shallower over a sand shoal or decreases in depth of the ocean in the near coastal waters. Waves may propagate along the thermocline and may break when the wave enters shallow water or when the thermocline is sufficiently thin. As mentioned in the text, the thermoclines of the equatorial Pacific and Atlantic Oceans generally shoal to the east. See, for example, Figure 3.22. However, as we will note in Section 13.2, there is considerable interannual variability in the degree of shoaling, lessening with the advent of an El Niño and increasing with a La Niño. Also, the sign of the Indian Ocean thermocline shoaling may change between phases of the Indian Ocean dipole to be discussed subsequentially. III Including John William Strutt (1842–1919), 3rd Lord of Rayleigh: Strutt (1916). He was a British scientist who made considerable contributions to experimental and theoretical physics and was a resident at the

University of Cambridge. He performed work important to a wide range of geophysical problems, developing what became known as the Rayleigh Number associated with convection and the concept of the Rayleigh–Taylor instability between two fluids of different densities. He received the 1909 Nobel Prize in Physics. IV Imagine you are mixing together the ingredients of a cake. What you want to do is generate as much turbulence (mixing) as you can with as little effort as possible. If the outer side of the bowl is stationary and the mixer is rotated rapidly in the center of the bowl, turbulence is generated and mixing occurs. However, if the mixer is held stationary in the center of the bowl and the bowl is rotated, turbulence is not generated even if the same amount of work is expended. The first case is inertially unstable while the second is inertially stable. Mechanical mixers, either in the kitchen or industry, can take advantage of the instability by rotating the container in the opposite direction to the mixing blades. V E.g. Nicholson and Webster (2007), Nicholson (2009), and Cook (2015) for the African monsoon, Fortuin et al. (2003) relative to the South American monsoon, Dickinson and Molinari (2002) for the North Australian monsoon, and the stability of upper-tropospheric outflow regions of tropical cyclones (Molinari and Vollaro, 2014).

299

301

14 Large-Scale, Low-Frequency Coupled Ocean–Atmosphere Systems In Chapter 2, we discussed the physical processes that help determine the local tropical sea-surface temperature (SST) and the structure of the upper ocean. Using data collected in the equatorial Pacific Ocean during TOGA COARE in 1992–1993 and in the Bay of Bengal in the JASMINE expedition in 1999, we concluded in Section 2.3.4 that the net heating rate of the upper ocean depended strongly on atmospheric processes such as latent heat fluxes, net radiative heating of the ocean surface, wind strength and direction, and fresh water flux (P − E). Each of these variables is a strong function of atmospheric activity. Yet, in turn, the atmospheric activity depends on the state of the ocean surface and the gradients of SST. In addition, we discussed the importance of wind-driven Sverdrup and Ekman transports (Sections 3.2.3 and 3.2.4) as important components of heat transport required to satisfy the heat budget of the planet. Thus, the tropical ocean and atmosphere are strongly coupled with the buoyancy flux of one system depending on the state of the other. Historically, it was thought that ocean transports were relatively constant from year to year with most variability occurring on seasonal time scales. In the early part of the twentieth century, as discussed earlier in Section 1.4.3, Walker and colleagues found a large-scale oscillation in surface pressure possessing a multi-year time scale. The primary oscillation, essentially a seesaw in surface pressure between the eastern and western hemispheres, was called the “Southern Oscillation” (SO). The designation “southern” was used to differentiate it from other oscillations discovered by Walker. It does not signify that it is restricted to the SH and, as can be seen in Figure 1.19, straddles the equator and the eastern and western hemispheres. Much later, it became apparent that this oscillation had its roots in the dual interaction of the ocean and the atmosphere and that the central and eastern Pacific Ocean warmed or cooled in harmony with the SO. This joint ocean–atmosphere system was called the El Niño-Southern Oscillation (ENSO). We will explore its physical basis in Section 14.2.

In Section 14.2, we also provide a detailed description of the SO. We discuss the early observations in the Pacific Ocean and the suggestions that the variability of the SO has its roots in coupled ocean–atmosphere processes. A number of theories, each based on coupled processes, will be discussed. Finally, we will consider the predictability of ENSO and the role of the annual cycle. Strong interannual variability has also been found in the Indian Ocean. It had been known for some time that the basin-wide Indian Ocean SST tended to oscillate in tandem with the phase of ENSO with higher SSTs during an El Niño and decreased SSTs with a La Niña. However, during 1997–1998, a different situation emerged. A high amplitude dipole mode in SST between the eastern and the western equatorial Indian Ocean was identified. The oscillation, occurring on interannual time scales, possesses some regularity but with sporadic amplitude variations. We will discuss the coupled ocean–atmosphere physics of this Indian Ocean Dipole (IOD) in Section 14.3. We will consider the relationship between the magnitude of SST and the degree of convection in Section 18.1. During the last century SSTs have waxed and waned with a general upward trend. We will pose the question: Are there natural limits to the SST in a changing climate? Answering this interesting question allows a basic character of the coupled system to emerge. We will find that although the SST has risen, the SST isotherm enclosing deep and organized convection has also increased and that the area of net heating of the tropical atmosphere has remained about the same. Within this increasing threshold isotherm, convective intensity has increased. In the following sections, we examine coupled phenomena that dominate the structure and variability of the tropics. In particular: (i) The along-equator Walker Circulation: We use a simple model to examine its physical structure. (ii) The SO and ENSO: We discuss a number of hypotheses that have been posed to explain the oscillatory phenomenon.

Dynamics of the Tropical Atmosphere and Oceans, First Edition. Peter J. Webster. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

14 Large-Scale, Low-Frequency Coupled Ocean–Atmosphere Systems

(iii) The predictability of ENSO: The phases of ENSO (El Niño and La Niña) impart varying degrees of influence on the weather through local changes in circulation or in the driving of teleconnection patterns. In the degree to which ENSO can be predicted we note the existence of the boreal spring predictability barrier. (iv) Interannual variability in the Indian Ocean that is commonly called the IOD. (v) The tropical warm pool: The Indian Ocean–western Pacific warm pool (Figure 1.5) is central to the existence of large-scale phenomena such as the Walker Circulation and ENSO. Here we examine its nature, maintenance, variability, and evolution and how it, and its accompanying convection, has changed over the last century.

14.1 The Walker Circulation During the 1920s, Walker and colleagues1 described a strong coherence between the surface pressure variance in the eastern and western hemispheres oscillating on a time scale of years. This was the SO. Walker’s main motivation, determining precursors for seasonal monsoon rainfall, will be discussed at some length in Chapters 16 and 17. Here we are interested in the atmospheric circulation associated with the SO and its evolution.

troposphere … and that this flows from the negative area (of the SO and hence anomalously low surface pressure) across the Pacific to its central and eastern regions. It is compensated by a return flow near the surface from the eastern Pacific to the negative area; this is a component of the frictional ageostrophic flow between high pressure over the eastern Pacific and low pressure over the negative area. There is descent in this circulation over the central and eastern Pacific, ascent over the negative area and adjacent regions and a toroidal circulation in a vertical and zonal plane is set up …. Troup (1965, p. 502) Figure 14.1 presents a schematic representation of Troup’s circulation for the two phases of the SO. This circulation became known later as the “Walker Circulation” coined by Bjerknes (1969). Troup made another important deduction. He noted that in these zonal circulations, warm air was lifted while colder air subsided. Thus, they were thermodynamically direct, converting potential energy to kinetic energy. Walker had described a slow large-scale oscillation in surface pressure between the eastern and western hemispheres. Troup was able to “close” the mass balance of

0 200

Early Depictions

The early studies, mentioned above, showed an awareness of the spatial scale and period of the SO. However, little was known about its vertical structure. Clearly, if the surface pressure varied over large areas in an out-of-phase manner (e.g. Figure 1.19a), then there must be transfer of mass between the eastern and western hemispheres. How and where the mass transfer took place, however, was not known. In the 1960s, a number of surveys were made using the relatively scarce upper atmosphere sounding data that existed at the time in the Pacific-Indian ocean sectors. In particular, Troup (1961, 1965) drew attention to interannual anomalies in the Indian and Pacific upper tropospheric wind field. These anomalies appeared to be out of phase with the lower tropospheric zonal wind anomalies associated with the SO. His analysis allowed the following deduction: … For simplicity we shall consider that there is a single drift of most importance in the upper 1 Walker (1923, 1924a,b, 1928) and Walker and Bliss (1932).

(a) Normal/Cold event Walker Circulation Co Dw Ca 55

6

3

Lm 4

400

hPa

14.1.1

600 800

5

1000 90°E

0 200

3

3

18 L warm

120°E

150°E

H cold

180°E

150°W

120°W

90°W 60°W

(b) Warm Event event Walker Circulation Co Dw Ca 20

3

3

2

Lm 11

11

400

hPa

302

600 800

1000 90°E

15

2

H cold 120°E

150°E

180°E

L warm 150°W

120°W

90°W 60°W

longitude Figure 14.1 Schematic representation of the Walker Circulation, based on the initial findings of Troup (1961, 1965). Vectors show the departures from average (tenths of m s−1 ) at 200 and 850 hPa at four stations: Cocus Island (Co), Darwin (Dw), Canton Island (Ca), and Lima (Lm), which were all that were available at the time. Source: Based on Julian and Chervin (1978).

14.1 The Walker Circulation

the oscillation showing that it extended deeply through the troposphere with a return flow near the tropopause. Noting the similarity of Figures 14.1 and 1.12, Troup’s depiction of equatorial zonal circulations is supported by many recent observations. 14.1.2

Nature of Zonal Circulations

To understand the physical processes that drive these zonal circulations we return to the scaling of the equations of motion presented in Section 3.1.2. Thus, observed convective disturbances (∼106 m) should be barotropic so that dynamical communication between the lower and upper troposphere, at least theoretically, would be difficult (see Figure 3.4b). As a way of avoiding this scaling problem a number of studies invoked a “weak temperature gradient” approximation (Section 3.1.4.2). This approximation reduces the thermodynamic equation to a balance between the adiabatic and diabatic heating (see Eq. (3.33)). However, from Eq. (3.34) we note that the Rossby radius of deformation (R) at 5∘ N is less than the scale of the Walker Circulation (∼107 m). On this basis, we argued in Section 3.1.4 that macro-scale divergent circulations, such as the Walker Circulation, are baroclinic systems and, in regions of strong surface and heating, allow communication in the vertical. Thus, a weak temperature gradient approximation may not be necessary to explain macro-scale tropical circulations. The location and amplitude of equatorial Walker cells change from season to season and from year to year. Figure 1.12c shows an example of the differences in the zonal circulation seasonal differences between a weak La Niña year of DJF 1984–1985 and a moderately intense El Niño year (1986–1987). In 1984–1985, the maximum rising motion was found over Indonesia and the eastern equatorial Indian Ocean. During the 1986–1987 El Niño, the region of maximum ascent had shifted 60∘ to the east. There were also other, more subtle changes that may be tied to impacts of the extremes of ENSO. For the duration of El Niño, the circulation over equatorial South America is more subsident than during La Niña and strong subsidence has replaced the net ascent over the eastern Indian Ocean and western Indonesia. Also, there is enhanced ascent over Africa. Of course, these differences relate to just two ENSO events. 14.1.3 Simple Model Simulations of Zonal Circulations Simple linear models2 have been useful in increasing our understanding of how the atmosphere responds 2 E.g. Webster (1972), Gill (1980), and Webster (1981, 1982).

to heating associated with SST anomalies. They have allowed responses to forcing to be understood in terms of fundamental modes such as those described in Section 6.1. The basic state employed here is a zonally symmetric flow (U = U(y, p)) shown in Figure 9.2, chosen to represent an annual averaged zonal wind similar to the observed flow (Figure 1.4). Westerly maxima occur in both the upper and lower troposphere with maxima near 40∘ N and 40∘ S. The forcing function is spatially identical to that plotted in Figure 6.12a, i.e.: ̇ Q(𝜆, 𝜑) = A sin 𝜆 exp (−B(1 − sin2 𝜑) − sin 𝜑0 )2 (14.1) and is applied at the midlevel of the model (p = 500 hPa). A and B are arbitrary constants. Figure 14.2a shows the response of the steady state (i.e. 𝜕/𝜕t = 0) two-level model3 to forcing applied at the midlevel of the model (500 hPa) representing heating located at the equator (𝜑0 = 0). Figure 14.2b shows the model response to forcing located in the subtropics (𝜑0 =23∘ N, 𝜇 = 0.4). The 250 and 750 hPa distributions of perturbation wind vectors and perturbation height field are plotted at each pressure level as a function of latitude and longitude. With equatorial forcing there are two major responses appearing in Figure 14.2a. The first response is lower-tropospheric convergence into the location of the heating with maximum ascent near its center. In the upper troposphere there is divergence along the equator away from the heat source to both the east and west. The second response occurs to the northeast and northwest of the forcing as two lower-tropospheric troughs that stretch toward higher latitudes. Matching ridges occur in the upper troposphere. The scale of the equatorial response is >107 m and girdles the tropics. The total solution, similar to that found by Gill (1980), possesses the elementary structure of the Walker Circulation. To understand this time-independent circulation (often called “stationary”), it helps to return to Figure 6.12b, which showed the transient response (𝜕/𝜕t ≠ 0) to the same spatially constructed equatorial forcing described above but within a quiescent basic state and simulated in a nonlinear model. There, a Kelvin wave moves rapidly to the east whilst a pair of Rossby waves drifts slowly westward. However, with the addition of a non-zero basic state, as in the present case, certain modes may possess zero Doppler-shifted frequencies and hence zero Doppler-shifted phase speeds. This suggests that there is a relatively simple explanation for 3 Model is described briefly in Appendix J. Differences and similarities between the Webster and the Gill models are also discussed in Appendix J.

303

14 Large-Scale, Low-Frequency Coupled Ocean–Atmosphere Systems

(a) Background basic state U(y,z)

30

(c) Forcing at 25°N 90°N

250 hPa

25 30°N

20

m s–1

304

15

250 hPa

10 5

0°

750 hPa

30°S

0 –5 90°S

45°S

30°S 20°S 10°S 0° 10°N 20°N 30°N 45°N

90°N

latitude

90°S 0° 90°N

180°

90°W

90°E

180°

90°W

0°

750 hPa

(b) Forcing at equator 90°N

90°E

30°N

250 hPa 30°N

0°

0° 30°S

30°S 90°S

90°S

0°

0°

90°E

180°

90°W

0°

20 m s–1

0°

90°N

750 hPa

Figure 14.2 (Continued)

30°N 0° 30°S 90°S

0°

90°E

180°

90°W

0°

20 m s–1 Figure 14.2 Linear, steady state simulations of the response of the tropical atmosphere to steady forcing. Colored background shading represents the forcing that possesses the same spatial structure expressed in Eq. (14.1) and Figure 6.12a. (a) The zonally averaged background basic state (U m s−1 ). Steady state linear model response to heat sources located at (b) the equator and (c) at 25∘ N. Upper panels refer to flow at 250 hPa, bottom panels, 750 hPa. Solid contours represent geopotential (𝜙 m2 s−2 ). Vector magnitude key bottom right. The two-level model, originating in Webster (1972), is described in Appendix J.

the near-equatorial Walker Circulation. Consider the dispersion relations for equatorially trapped modes displayed in Figure 6.1. All modes to the right of 𝜔 = 0 have phase velocities to the east. The Doppler-shifted frequency is given by 𝜔d = 𝜔 − kU. For a stationary mode 𝜔d = 0 then 𝜔 = kU. These are diagonal lines of slope U emanating √ out of the origin. As the phase speed of a Kelvin is gH (from Eq. (6.43)), where H is the equivalent depth, there is always a Kelvin wave at the equator that will be Doppler-shifted to zero in a given basic state. The simple Kelvin wave theory explains the alongequatorial response. However, it does not explain the

twin vortices to the west of the forcing nor the poleward deflection of the winds in the lower levels to their east. The existence and location of the twin vortices may be explained in the following manner. The rising motion in the vicinity of the heat source stretches the vortex tubes creating cyclonic vorticity and increases the absolute vorticity of a parcel. Parcels then develop meridional motion to change their latitude to a location where their absolute vorticity is the same as the background flow. Thus, parcels must move poleward in each hemisphere in the lower troposphere and equatorward aloft, forming the pair of vortices about the equator (Gill 1980). The same physics explains the twin vortices found in the transient response to equatorial forcing described in Figure 6.12b. When the same forcing is moved to 23∘ N (Figure 14.2b), the response is very different. The major feature is a Rossby wave oriented southwest to northeast, the slope being determined to some degree by the latitudinal shear of the basic state and rotation. For the same scale and magnitude of forcing, the near-equatorial response is at least the same size as when the forcing was set at the equator. There is also a substantial response in the subtropical low troposphere. 14.1.4

Role of an Interactive Ocean

It is not difficult to anticipate the potential impact of the low-level flow of the model Walker Circulation on the upper ocean. In Sections 3.2.5 and 3.2.6 we discussed

14.2 The Southern Oscillation, El Niño and La Niña

how wind-driven ocean Ekman transports contribute to the observed SST distribution. The impact is substantial and the magnitude of Ekman upwelling per year in the eastern equatorial Pacific >100 m year−1 (Figure 3.14). Consider, now, the impact of the lower tropospheric winds from the steady state model displayed in Figure 14.2a and b. Remembering that the Ekman transports are to the right of the wind in the northern hemisphere and to the left in the southern hemisphere, the converging easterly winds eastward of the heat source would cool the upper ocean by generating a greater upwelling along the equator, producing a shoaling of the thermocline. To the west of the heat source, on the other hand, the converging westerlies would create a convergence of mass, reducing Ekman upwelling and deepening the thermocline. One might expect that if the model were fully coupled that the maximum in SST, and the associated columnar heating, would drift to the east. Furthermore, if the winds changed, the distribution of upwelling and downwelling and the SST pattern would also change. In this sense, it is difficult to conceive of a non-migrating steady atmospheric Walker Circulation solution in a coupled ocean–atmosphere system. In fact, the non-stationarity of the Walker phenomena forms the basis of a number of theories on the formation of an El Niño or a La Niña in the Pacific and how transitions between these extremes may occur.

14.2 The Southern Oscillation, El Niño and La Niña Scientists in the late nineteenth and the early twentieth centuries noted that the SO variability may be cyclic and of a very large scale. For example, seasonally averaged surface pressure variations in Sydney, Australia, were found to be out of phase with the surface pressure variations in Buenos Aires, Argentina,4 even though the stations were separated by nearly 12 000 km. As the number of stations increased, the oscillation was found to be spatially coherent and near global in scale.5 Figure 14.3 shows the seasonal correlations of variations of the Darwin surface pressure with surface pressure variability elsewhere. The seasonal patterns are very similar to the annual anomalies displayed in Figure 1.19 although with subtle differences from season to season. The correlations appear strongest in DJF and weakest in MAM. Following the early excitement in the early twentieth century of finding a phenomenon that pointed toward longer-term predictability, questions were raised about 4 Hildebranson (1891). 5 Lockyer and Lockyer (1902, 1904) and later Lockyer (1906).

the statistical non-stationarityI of the SO. Scientists found that many of the correlations found earlier by Walker had become weaker and some had even become decorrelated. Troup (1965) compared Walker’s correlations, made between station pairs from 1882 to 1921, with data from 1921 to 1960. For example, the correlation between Darwin, Australia, and southeast South America was calculated by Walker to be −0.46 and between Darwin and Honolulu, −0.66, both of which were significant at the 99% level. These correlations then dropped to −0.31 and −0.12 for the 1921–1960 period. However, correlations between Samoa and Honolulu and between Colombo, Sri Lanka (then Ceylon), and Darwin retained both their sign and statistical significance. By the mid-1960s, interest in the SO appeared to have waned. Despite the early revelations by Walker, Troup (1965, p. 490) noted: … At present the phenomenon known as the “southern oscillation” receives little attention . . . . The impact of finding former climate precursors weakening and had a demoralizing effect on many scientists. What had seemed to be robust relationships between Australian rainfall and the SO, for example, had diminished. Treloar and Grant (1953) and Grant (1953) went so far as to conclude that the secular reduction of correlations threw considerable doubt on the direct use of correlation relationships in forecasting. Despite these changes in the statistics, Troup persevered, and stated that the SO: … has intrinsic interest as a complex of interactions over a very large area. Moreover, there is promise of its use in seasonal forecasting or “foreshadowing” and it may be said that some regions affected by it are those most favoured for success in this field. Hence increased understanding of its nature and mode of operation should have valuable practical results.… Troup (1961, p. 490) Troup noted that the correlation charts (e.g. Figure 14.3) indicate that seasonal anomalies are dominated by the SO. He was also able to determine when the commencement of anomalous conditions occurred, discovering, in essence, the phase locking of the SO with the annual cycle. He noted for the Southern Hemisphere that: … There are important lag correlations between certain elements affected by the Southern Oscillation. These are strongest across southern spring

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14 Large-Scale, Low-Frequency Coupled Ocean–Atmosphere Systems

Figure 14.3 The seasonal cycle of the Southern Oscillation expressed by correlation patterns between the Darwin, Australia (12 27′ ∘ S, 130.50′ ∘ E), MSLP and the MSLP elsewhere for (a) DJF, (b) MAM, (c) JJA, and (d) SON.

Seasonal correlations of Darwin MSLP

(a) DJF 40°N 20°N 0° 20°S 40°S 0°

60°E

120°E

180°W

120°W

60°W

0°

60°E

120°E

180°W

120°W

60°W

0°

60°E

120°E

180°W

120°W

60°W

0°

60°E

120°E

180°W

120°W

60°W

0°

–0.4

0 Correlation

(b) MAM 40°N 20°N 0° 20°S 40°S 0° (c) JJA 40°N 20°N 0° 20°S 40°S 0° (d) SON 40°N 20°N 0° 20°S 40°S 0°

–0.8

0.4

and weakest across southern autumn and the “year” of the oscillation begins in late autumn (boreal spring) or early winter (boreal summer) …. Troup (1965, p. 505) Troup produced maps of the regression of surface pressure on the SOI. These, he reasoned, provide more physical meaning by delineating the areas where the

0.8

mass exchange takes place during an oscillation. He then inferred the variations of the winds associated with the mass transfer. These results confirmed Troup’s thought that the SO was an interchange of mass between the eastern and western hemispheres and, further, allowed him to suggest that the trade winds undergo a strengthening or weakening depending on the sign of the SO. He then argued that variations in the strength of the trade winds would change the degree of upwelling in

14.2 The Southern Oscillation, El Niño and La Niña

the central Pacific Ocean as well as along the South American coast. Thus, Troup had reached an important conclusion that the modification of air–sea interaction, produced by the oscillating winds of the SO, is a critical component in producing the phenomenon itself! In essence, Troup was laying out the physical components of the SO as a coupled ocean–atmosphere system. Observations made during the International Geophysical Year (IGY) suggested that the SST distribution in the interior of the Pacific Ocean differed greatly from what was thought to be the climatological norm. Anomalously warm SSTs were found in the central and eastern Pacific from late 1957 through 1958. In fact, oceanographers, as would later become apparent, were observing a strong El Niño. It was soon realized that the 1957–1958 event was not isolated but one that recurred every few years. Following the IGY, there was a renewed interest in the low-frequency variation of sea-level pressure between the eastern and western hemispheres. Figure 14.4a provides a compilation of observations made in the 1950s and 1960s that led to the thought that the SO was a coupled oceanic–atmospheric phenomenon.6 Figure 14.4b maps the locations of the observation stations. At first glance the data appears disparate but certain patterns do emerge. (i) SST anomalies averaged in the areas 5∘ N–5∘ S and 180∘ W–80∘ W (red curve). The blue curve shows the SST measured at Puerto Chicama, Peru (8∘ S, 79∘ W). (ii) Two measures of the SO. The Santiago, Chile (33∘ S, 71∘ W)–Darwin (12∘ S, 131∘ E) surface pressure difference (blue curve) and the anomalous surface pressure difference between Easter Island, Chile (27 ∘ S, 109 ∘ W), and Darwin (blue curve) (from Quinn 1974). (iii) Strength of the South Equatorial Current calculated by Wyrtki (1974). Magnitude of the current was computed from relative sea-surface height data at a number of stations in the equatorial Pacific Ocean. (iv) Zonal wind anomalies at 850 hPa (blue) and 250 hPa (red) over Canton Island (3∘ S, 172∘ W). (v) A measure of the convergence of the meridional component of surface trade winds defined as vT = (vSH − vNH )/2, where vSH and vNH are anomalies of the meridional velocity components averaged in boxes (vSH : 1∘ S–15∘ S, 125∘ E–75∘ E and vNH : 5∘ N–19∘ N, 125 ∘ E–95∘ W). In essence, positive values of vT (blue curves) represent an enhanced convergence of the meridional wind in the equatorial central Pacific while negative values 6 Thus confirming Troup's hypothesis.

represent anomalous divergence (Reiter 1978a,b). The red curve shows the “Line Island Precipitation Index” defined by Meisner (1983). Meisner noted that the mean precipitation varied substantially from one island to the next. One option is to use the long-term mean of the total Line Island precipitation and look at year to year anomalies. Meisner chose to use the deviation from the mean for each island and construct an index based on the sum of these individual deviations. In this manner, bias toward the wettest of the stations is eliminated. The Line Islands, part of the country of Kirbati, lie roughly between 7∘ N and 11∘ S spanning the equator and the meridians 162∘ W and 150∘ W. If we view the variability of indices in Figure 14.4a collectively, relative to the central Pacific SST, an interesting set of relationships emerges. During cold phases (La Niñas: e.g. 1955–1956), a measure of the SO, plotted in Figure 14.4a(ii), is positive, reflecting an increase in the longitudinal surface pressure gradient. At such times, the Walker Circulation is strongest, as suggested by the increase in vertical shear (panel (iv)). At the same time vT is a minimum, as is the Line Island precipitation index (panel (v)). During a warm phase in the central Pacific Ocean (El Niños: e.g. 1957–1958 and 1965–1966) the South Equatorial Current is at a minimum (panel (iii)) while the SO is negative, showing a lessening of the longitudinal pressure gradient. Also, the vertical shear in the Central Pacific is at a minimum, possibly indicating an eastward migration of the Walker Cell. Finally, it is interesting to note that the trade wind convergence index (vT , panel (v)) correlates well with the Line Island precipitation and the SST anomaly in the eastern Pacific Ocean. Together, these early observations provided compelling evidence of a coupled ocean–atmosphere system with spatial scales of tens of thousands of kilometers and periods of years. The issue that arose was how to relate these seemingly disparate time series and construct a coherent model of the variability. Figure 14.5 shows a wavelet analysisII of the SST in the central Pacific Ocean (see Figure 1.19a for location of the Niño-3 area: 5∘ N–5∘ S, 120∘ W–170∘ W). The right-hand diagram shows the average spectra over the entire 119-year period. The spectra is very similar to that of the SOI with a broad band of statistically significant variance across the 3–6 year period band shown in Figure 1.19c. In fact, the time series of the SOI and the Niño-3 SST correlate at >0.8. The upper panel provides the wavelet variance by period and time. The contours indicate the magnitude of the variance with the bold lines indicating variance >95% significance level. The bottom panel of Figure 14.5 provides a time series of

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14 Large-Scale, Low-Frequency Coupled Ocean–Atmosphere Systems

the Niño-3 SST anomalies and also the variance in the two to eight year band (red line). There are two periods with high amplitudes: before 1910 and after 1960. In between these two periods, there is a prolonged period of low variance. This matches the period of low variance and diminished correlations identified by Troup. The wavelet analysis provides an interesting example of problems associated with analysis of time series. Let us assume that instead of a wavelet analysis, we had used a Fast Fourier Transform (FFT) analysis of the Niño 3

time series. The results of such an analysis appears on the right-hand side of Figure 14.5. It shows a statistically significant peak of 3–4 years. Unlike the wavelet analysis, it does not provide information that there was an almost disappearance of variance in the 2–8-year band for a 50-year period (red curve, bottom panel of Figure 14.5). An FFT analysis may be prone to undue influence of a small number of high-amplitude events that occur for a limited time during the period of analysis. Such influence, and when it occurs, is exposed by wavelet analysis.

(a) Compliation of early observations

Figure 14.4 (a) Compilation of observations made in the equatorial Pacific region that suggested to scientists that the SO was a coupled ocean–atmosphere phenomenon and a large-scale dynamical system. Time sections of: (i) SST anomalies (∘ C) in the area 5∘ N–5∘ S and 180∘ W–80∘ W (red curve) and Puerto Chicama, Peru (blue curve). Data from the solid red rectangle. Source: Based on the analysis of Allison et al. (1972). (ii) The Southern Oscillation computed from the surface pressure difference between Santiago, Chile (red curve), and Easter Island, Chile (blue curve), and Darwin, Australia. Based on the analysis of Quinn (1974). (iii) Strength of the South Equatorial Current based on sea-level anomalies at Canton Island and La Libertad, Talara, Callao, Peru, and the Galapagos Islands, Ecuador. Source: Based on the calculations of Wyrtki (1974). (iv) Canton Island zonal wind anomalies at 250 hPa (red) and 850 hPa (blue). Source: Based on Krueger and Gray (1969). (v) A measure of the convergence of the trade winds, v T (blue contour), in the eastern Pacific Ocean according to Eq. (2.1). The components vNH and vSH were calculated in the dashed rectangles shown in (b). Source: Based on Reiter (1978b). Red curve illustrates an index of precipitation for the Line Islands from Meisner (1983). Vertical dotted lines refer to five-year periods. Source: Based on the summary of Julian and Chervin (1978). (b) Geographic locations of stations referred to in (a) and in the text.

(i) δSST (5°N-5°S: 80°W-180°), Puerto Chicama, Peru °C

+1.0 0 –1.0 (ii) SO: Santiago-Darwin δp, Easter Is-Darwin δp

13 12 hPa 11 10 9 8

2 1 0 –1 –2 (iii) Strength of the South Equatorial Current

cm/sec

30 25 20 (iv) δ(Zonal wind): Canton Island 200 hPa 850 hPa

m s–1

8 4 0 –4 –8 (v) Trade “convergence” (vT) & Line Islands Precipitation Index

m s–1

0.4 0.3 0.2 0.1 0 –0.1 –0.2 –0.3 –0.4

80 60 40 20

1950

55

60

65 year

70

73

14.2 The Southern Oscillation, El Niño and La Niña

(b) Geographical locations of stations and areas 40°N Honolulu

20°N

vNH

Canton δSST

Colombo

0°

Djakarta (Batavia)

20°S

Line Islands

Darwin Sydney

Galapagao

vSH Tahiti Easter Island

40°S 60°E

0°

120°E

180°

120°W

Santiago

La Libertad Talara Puerto Chicama Callao Buenos Aires

60W

0°

Figure 14.4 (Continued)

14.2.1

Evolution

Figure 14.6 is a plot of the Niño 3.4 SST anomaly (average SST in 5∘ S–5∘ N and 170∘ W–120∘ W: see Figure 1.19a) for the period 1950 through mid-2017. The anomaly is rarely zero. Rather arbitrary definitions of whether an anomaly represents an El Niño or a La Niña have been developed. For example, the NOAA definition takes into account both longevity and magnitude.7 The anomaly must span at least five overlapping three-month periods. In addition, magnitudes must be |SST| >0.5 ∘ C during this period. A weak event is defined if |SST| is in the range 0.5–0.9 ∘ C. Limits for moderate, strong, and very strong events are set for the ranges 1.0–1.4, 1.5–1.9, and ≥2 ∘ C. We observe that: (i) Since 1950 there have been 9 weak, 6 moderate, 3 strong, and 3 very strong El Niño events, according to the definition. During the same period there were 10 weak, 6 moderate, and 3 strong La Niña events. (ii) The reoccurrence period of ENSO events (La Niña or El Niño) is between 3 and 7 years, albeit with magnitudes that are quite irregular. (iii) El Niño tends to form during or after the boreal spring, reaching a maximum amplitude in the following boreal fall and winter, as discovered by Troup (1965). (iv) The demise of a warm event usually occurs in the following boreal spring. Four of the strong El Niños were followed by La Niñas (1972–1973, 1987–1988, 1997–1998, and 2009–2010), but other strong events were followed by periods of relatively neutral conditions or small negative anomalies. In this short record, there were no La Niñas that started after a period of neutral conditions. In the 7 See http://www.cpc.noaa.gov/products/analysis_monitoring/ ensostuff/ensoyears.shtml.

four La Niña events listed above, a rapid reversal from warm to cold conditions occurred. When we examine the predictability of ENSO in Section 14.2.4, we will see that the phase locking of ENSO with the annual cycle has significant implications. Niño 3.4 SST anomalies occurring prior to the boreal spring possess almost no persistence for the following year, indicating a distinct lack of predictability of the onset of an El Niño.8 On the other hand, once a warm episode exceeds the El Niño criterion of +0.5 ∘ C during the boreal spring, there is a high probability that the El Niño will continue to grow during the forthcoming calendar year. The demise of an El Niño is less predictable. A phase locking of El Niño with the annual cycle is very apparent in Figure 14.7a. The first diagram plots the evolution of the Niño 3.4 warm events during the 1950–2017 period against the calendar year. All events commence in the boreal spring (Year 0) and reach maximum amplitude in the following January (Year +1). The decay of the event occurs through the following spring, reversing to either a La Niña event or relatively neutral conditions. Figure 14.7b shows the evolution of the La Niña during the same period. Like El Niños, cold events commence in the boreal spring and decay during the following year. The major difference is that, prior to the La Niña (January Year 0), the SSTs are often associated with a warm event. Prior to an El Niño, the SST may be slightly negative or neutral. 14.2.2

Annual Cycle of the Upper Pacific Ocean

A logical place to start in the unraveling of the ENSO-annual cycle phase-locking issue is by describing 8 This phenomenon is referred to as the “spring predictability barrier” by Webster and Yang (1992), discussed at length in Section 14.2.6.

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14 Large-Scale, Low-Frequency Coupled Ocean–Atmosphere Systems

1

1

2

2

4

4

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8

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16 32

32 1880

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0 2000

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2–8 yr var.

(°C)

period (years)

period (years)

Wavelet Analysis Niño-3 SST

2.5 0.0 Avg Var

Figure 14.5 Wavelet power spectrum (using the Morlet wavelet: refer to Torrence and Compo 1998) of the Niño 3.4 SST (area, 5∘ N–5∘ S, 170∘ W–120∘ W). The upper panel shows the evolving wavelet amplitude as a function of time from 1871 to 1996. Shaded contours indicate statistical significance with the solid black contour indicating >95% confidence. Cross-hatched regions indicate the “cone-of-influence,” where zero padding has reduced the variance. Lower panel shows time series of the Niño 3.4 SST with the red contour indicating a 20-year running average of variance in the 2–8-year period band. Panel to right indicates the average spectra over the entire period computed by two methods: Fast-Fourier Transform of the entire data series and the summed wavelet modulus. The solid red curve on the periodogram indicates the 95% statistical significance level.

Variation of the Niño-3.4 SST anomaly 1950–2017 2.0

57/58

δSST Niño 3.4 (°C)

82/83

72/73

65/66 63/64

76/77

68/69

1.0 0

–1.0

–2.0 1950

70-71

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60

65

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80

year 97/98

2.0 δSST Niño 3.4 (°C)

310

87/88

15/16

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02/03

09/10

94/95 1.0 0

–1.0 –2.0

85

07/08

98/00

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95

2000

05

10/11 10

15

year Figure 14.6 Detailed time series of the Niño 3.4 SST for the period 1950–2017. Major El Niños and La Ninãs are indicated by years of occurrence in red and blue lettering, respectively.

14.2 The Southern Oscillation, El Niño and La Niña

(a) Composite El Niño 1950–2015

27

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26 °C

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−1

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Mean annual cycle Pacific Ocean (a) SST (°C) 29

1957−58 1963−64 1965−66 1968−69 1972−73 1976−77 1982−83 1986−87 1991−92 1994−95 1997−98 2002−03 2004−05 2006−07 2009−10 2014−15 Avg

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D 160°E 180 160°W 140°W 120°W 100°W

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m

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50

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1955−56 1970−71 1973−75 1975−76 1988−89 1998−99 1999−00 2007−08 2010−11 Avg

17

110

3

Month

(b) Composite La Niña 1950–2015

22

(b) Depth of the 20 °C isotherm (m) 150

J F M A

130

Jl(+1)

130

Ja(+1) Jl(0) relative month

190

−3 Jl(−1) Ja(0)

60 40

D

20 160°E 180 160°W 140°W 120°W 100°W

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D

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the annual cycle itself. In climate studies we often concentrate on deviations from the annual cycle. Consequently, the physics of the annual cycle itself are often obscured. Figure 14.8a displays the mean annual cycle of a Pacific Ocean SST as a function of longitude averaged between 2∘ N and 2∘ S. Throughout the year there is a persistent gradient of SST with warmer temperatures in the west and cooler in the east. A relaxation of the longitudinal gradient occurs in the boreal spring whereas

S O N

1

10 09

1010

Figure 14.7 Composite evolution of (a) El Niños (extension of McPhaden 2015) and (b) La Niñas, in the period 1950–2017 from Figure 14.6. Events are color coded by year relative to the time of maximum amplitude. For example, Jl(0) refers to the July value of the Niño 3.4 in the year of maximum amplitude. Jl(−1) refers to the value in the year before maximum amplitude. Bold red lines indicate average El Niño and La Niña. Bold black and blue lines in (a) plot the evolution of the 1997–1998 El Niño. Bold gray line in (b) describes the evolution of the prolonged La Niña that persisted from 1999 through 2001.

M J J A

1011

1008

10

Jl(+1)

(c) Mean sea-level pressure (hPa)

9

Jl(0) Ja(+1) relative month

Month

−3 Jl(−1) Ja(0)

J F M A

100

−2

1006

160°E 180 160°W 140°W 120°W 100°W

Longitude Figure 14.8 Mean annual cycle averaged between 5∘ N–5∘ S in the Pacific Ocean as a function of longitude of (a) SST (∘ C), (b) depth of the 20 ∘ C isotherm (m), and (c) mean sea level surface pressure (hPa). Scales are to the right of each panel.

in the boreal fall there is a steepening of the gradient. During the boreal spring, the eastern Pacific SST is 4 ∘ C warmer than in the fall. Figure 14.8b shows the annual cycle of the depth of the 20 ∘ C isotherm, a measure of the total heat content of the upper oceanIII where the

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14 Large-Scale, Low-Frequency Coupled Ocean–Atmosphere Systems

Annual cycle of upper Pacific Ocean (a) Annual mean T(x,z) 5°N-5°S Depth (m)

greater the depth of the isotherm, the greater the heat content of the ocean column. The depth of the isotherm remains relatively uniform in longitude throughout the year, with only a small deepening occurring in the west during the boreal spring and a shallowing in the east in the early boreal fall. Finally, Figure 14.9c displays the along-equator MSLP. During the boreal spring, the equatorial pressure gradient is a minimum. Taken together, these figures suggest that there are strong variations in the annual cycle in the equatorial upper Pacific Ocean, but they are mainly restricted to the near-surface layers. Figure 14.9a to e display vertical cross-sections of the mean annual, January, April, July, and October temperature along the equator averaged between 2∘ N and 2∘ S to a depth of 260 m. At first glance, the mean monthly sections may seem very similar to the annual mean (Figure 14.10a), but there are subtle and important differences. Major changes occur between January and April when the entire upper 50–100 m layer warms. From Figure 14.8c we note that this occurs at a time of decreasing longitudinal pressure gradient. During the boreal summer the eastern Pacific surface layer starts to cool. This may be seen by following the location of the 20 ∘ C isotherm, which during the boreal spring is at a depth of 50 m in the eastern Pacific during the boreal autumn. The role of the atmosphere in determining the annual cycle of the upper ocean thermodynamical structure becomes apparent when the wind fields are taken into account. Figure 14.10a and b show the annual cycles of surface zonal and meridional wind components averaged between 2.5∘ N and 2.5∘ S. During the boreal winter, northerly cross-equatorial winds dominate the western and central Pacific Ocean. The westward flow extends across the Pacific. Together, the two components suggest a period of strong northwesterly trade winds. During the boreal spring (MAM) the trades relax across the entire Pacific Ocean, responding to a minimum of convective heating gradient across the Pacific. With the reduction of the wind strength along the equator, upwelling decreases, allowing the SST to warm. With the approach of the boreal summer the southwesterly trades are replaced by strong southeasterly trade winds. These are driven by the development of the northern hemisphere ITCZ to the north of the equator in the eastern Pacific. The growth of cross-equatorial flow is especially evident in the eastern Pacific, where it promotes coastal upwelling and SST cooling. Reiter (1978a,b) showed that the interannual variability of the convergence of the trade winds (vT , Figure 14.4(v)) coincided with variations of Line Island precipitation. In turn, the variations of vT correlated with changes in the central Pacific SST (Figure 14.4(i)).

0 50 100 150 200 250

140°E

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(b) January mean T(x,z) 5°N-5°S

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(c) April mean T(x,z) 5°N-5°S

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(d) July mean T(x,z) 5°N-5°S

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It would seem that interannual variability in the coupled ocean–atmosphere system is very similar to the annual cycle. Further, we also note that the magnitude of the annual cycle anomalies Niño-3 region is about +1.5–2 ∘ C, while further to the east it is nearer to +4 ∘ C. These mean seasonal anomalies are of the same magnitude as those associated with a moderate to a strong El Niño.

14.2 The Southern Oscillation, El Niño and La Niña

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14.2.3

Interannual Variability of the Annual Cycle

We can now look at interannual variations of the annual cycle. Figure 14.11a to c provides time-sections of SST, the zonal wind, and the upper ocean heat content anomalies from the long-term mean. These quantities are plotted as a function of longitude from January 1996 through December 2000. The period was chosen because it includes one of the two strongest El Niños on record, the second occurring in 2015–2016. Also, the 1997–1998 event was followed by a prolonged La Niña, lasting through 2000. The strong El Niño developed in the late boreal spring of 1997 (Figure 14.11a). Anomalous temperatures in the eastern Pacific, to the east of the Niño 3.4 region, reached +4.5 ∘ C. It should be noted that with the annual cycle included (Figure 14.8a), the total (mean plus anomaly) along-equator SST gradient was flat. The growth of the subsequent La Niña was also rapid with anomalies in the same eastern location of > − 3 ∘ C. During the 1997 boreal spring, anomalous westerly surface zonal winds extended across the equatorial Pacific Ocean (Figure 14.11b), tending to lead the warming of the central and eastern ocean consistent with a reduction of equatorial upwelling. In the following boreal spring, the westerlies were replaced by enhanced easterlies and an increase in upwelling. Accompanying the El Niño were changes in the anomalous heat content (Figure 14.11c), with decreases in the west and increases in the east. During the following La Niña, the heat content anomaly pattern reversed.

Clearly, changes were occurring not just at the surface but throughout the upper ocean. Examples of the change in structure along the equator between ENSO extremes appear in Figure 14.12. Mean monthly distributions of T(x, z) are shown for January 1998 and January 1999, at the times of the peak intensity of the El Niño and the La Niña. Almost all of the contributions to changes in heat content occur in the upper 200 m. Furthermore, the greatest changes occur along the thermocline. Such changes may be the result of waves propagating along the thermocline. In summary, a comparison of the annual and interannual cycles reveals an interesting similarity. The period of the two cycles is very different, of course, but the sequence that both follow is very similar. Such systems are called “self-similar.” In many respects, the oscillation of the thermocline during a year matches its oscillation during an ENSO cycle. Formally, self-similarity is the property of having a substructure analogous or identical to an overall structure. Majda (2007) points out the existence of self-similarity between convection with clusters and between clusters and larger-scale convective conglomerates such as the Madden–Julian Oscillation (MJO). 14.2.4

Large-Scale Signals of El Niño and La Niña

Many studies have attempted to relate ENSO with variations of weather and climate around the globe.9 9 Rasmusson and Carpenter (1982, 1983) and Ropelewski and Halpert (1987), among many others.

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Ropelewski and Halpert (1987) was perhaps the most thorough study and used a much longer station data set. The statistical technique they used is rather complicated. Regions of common correlation coefficients with ENSO extremes were identified. Because stations have different total rainfall, the time series at a station was normalized. In this manner stations with low amplitude but possessing a strong phasic relationship with ENSO could be identified. A total of 26 ENSO events were identified in the period 1877–1982. Year 0 corresponds to the year of the largest SST anomaly in the eastern Pacific Ocean. The actual peak occurs in the boreal spring of Year 0 or February to April of Year 0 (i.e. Feb0 to Apr0).10 Precipitation data from a multitude of stations was categorized from the July before the event (Jul-) until the June following the event (Jun+). The white contours in Figure 14.13 provide a summary of the Ropelewski–Halpert correlations. Solid contours enclose drier than average areas and dashed lines wetter than average. The period of extreme conditions relative to the typical year is displayed at the bottom of the figure. Capital letters denote locations of drier conditions while lower case denote wetter. For example, region “a” in the central Pacific, corresponding to “May(0)–Apr(+)” in the key, indicates that anomalously wet conditions tend to exist in the spring of year 0 and extend through the summer to April of 10 Note that the ENSO year used by Ropelewski and Halpert (1996) is different from that used by McPhaden (2015) to develop Figure 14.7a and b. Jan(+1) corresponds to Ropelewski and Halpert’s Jan(0).

Year(+). Thus, the display indicates both the sign of the anomaly and when it occurred during the composite calendar year. The station data analysis indicated increases in precipitation in the central Pacific and along the coast of South America and in South America to the east of the Andes during an El Niño. In addition, there is a small area of positive values over Sri Lanka. Despite these positive regions, the analysis indicates that the major impact is one of less rainfall during El Niño. Most of Australia, Indonesia, India, and the western Pacific, in addition to northern South America, exhibit drier than average conditions with El Niño. The Ropelewski–Halpert study provided the first global analysis of precipitation anomalies associated with ENSO. It was based on the available station data that included a large number of ENSO events. However, the reliance on station data does have its weaknesses as the available data are often widely separated. Further, the determination of statistical significance is problematic. Similar analyses using satellite data have been undertaken in recent years.11 Whereas satellite data is global, it is also relatively short in duration compared to station data. The background color shading in Figure 14.13 shows differences in normalized precipitation between El Niño and La Niña composites over the period 1979–2001. Solid black lines refer to anomalies at the 95% significance level. Overall, the patterns are remarkably 11 E.g. Curtis and Adler (2003).

14.2 The Southern Oscillation, El Niño and La Niña

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similar to those found by Ropelewski–Halpert. The satellite data allows the filling in of data gaps existing in the analyses between meteorological stations. The strongest positive anomalies lie along the equator over the Pacific Ocean east of 150∘ E extending to the South American coast. A clear result of the satellite analysis is that the anomalies extend coherently over vast distances. Both positive and negative anomalies appear to be connected across the oceans in a pair of macro-scale “boomerang” patterns that are significant at the 95% level. The “elbows” of the positive and negative boomerangs are located in the central and western Pacific, respectively. Wet anomalies extend across the Pacific southeastward, across Chile and Argentina, and into the South Atlantic Ocean. In the Northern Hemisphere, a weaker counterpart pattern crosses the southern US and Atlantic Ocean into Europe. Further

to the west an area of statistically significant negative departure extends southeastward from the Maritime Continent across the South Pacific and the southern tip of South America. The Northern Hemisphere arm of the boomerang crosses the North Pacific and southern Canada. Finally, something resembling a third boomerang of enhanced precipitation extends from the Horn of Africa (west Arabian Sea) northward to central Asia and southward into the southern Indian Ocean. As may be expected, climatological differences between El Niño and La Niña exist in other meteorological fields as well. Figure 14.14a and b provides a plot of the mean boreal winter (DJF) 200 hPa zonal velocity anomalies and the boreal summer (JJA) for both El Niño and La Niña. The along-equator anomalies are consistent with the relocation of the Walker circulation and its associated divergence and convergence patterns

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Figure 14.13 Correlations of rainfall with extremes of ENSO. The white contours represent the anomalies in rainfall (dashed above average rainfall, solid below) between El Niño and La Niña found in meteorological station data as determined by Ropelewski and Halpert (1987). The letters denote the periods within the ENSO year where the anomalies are maximized: upper case dry anomalies, lower case wet. These contours are superimposed on the background El Niño minus La Niña composites of normalized precipitation obtained from satellite-determined OLR (from Curtis and Adler 2003). Cold colors denote rainfall deficits and warm colors excesses. The satellite data matches the Ropelewski–Halpert correlation patterns quite well but extends them to a wider geographical domain where station data was not available for the earlier Ropelewski–Halpert analysis.

in each season. In the subtropics of the Pacific, easterlies are about 10 m s−1 greater during a warm phase than during the cold phase, more so in DJF than in JJA. The anomalies over the Atlantic are quite different and, in fact, possess an opposite polarity. Over the equator stronger easterlies are found together with a decrease in the subtropical easterlies. A near-equatorial maximum in U > 0 (200 hPa) occurs in the Pacific Ocean during La Niña and in the Atlantic Ocean during El Niño. Following arguments developed in Chapter 7, the communication between the tropics and higher latitudes would also change with the phase of ENSO. Most importantly, these changes alter the zonal wind stretching deformation and thus the manner in which transients formed with the tropics will move into and influence higher latitudes (see Figure 7.22). At the same time, the changes in U x will alter the region where wave breaking occurs, thus altering as well potential communication from the extratropics to the tropics. Specifically, the corridor within which potential vorticity (PV) is returned to lower latitudes changes between phases of ENSO, as suggested by the arguments in Section 11.3.

If it were possible to forecast El Niño/La Niña (see Section 14.2.6 below), a knowledge of canonicalIV patterns associated with ENSO extrema would be very useful. However, we should be cautious about making very general conclusions from the results of the data studies discussed above. (i) The time series describing the SO and ENSO may not be stationary in a statistical sense. This may be of concern, as in Figure 14.6 where we found two periods with sustained high amplitude SST anomalies in the central Pacific (before 1910 and after 1960) but, in between, a prolonged period of low variance. This matched the period of low variance of the SO discussed by Troup (1965). (ii) We have noted in Figures 14.6 and 14.7 that each El Niño and La Niña event is different. They possess variable amplitude and duration even though the commencement of an ENSO event tends to start in the boreal spring. However, the transition from an El Niño to La Niña is very variable, sometimes occurring after a strong El Niño but sometimes not. (iii) Also, are teleconnections patterns associated with ENSO extrema statistically stationary? Do the

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impacts of an event relate to the magnitude of the event? That is, if we expect an El Niño for a particular magnitude, can we expect it to have the same impact on Indian rainfall? The 1982 and 1987 events, both El Niño years, coincided with diminished rainfall over India. The 1997–1998 event was one of the strongest El Niños ever recorded, yet rainfall in the summer of 1997 over the Indian subcontinent was normal. Also, a severe drought accompanied the moderate El Niño of 2002–2003. Other factors come into play. Can they be identified? (iv) Recent research has suggested that there may be a different form of El Niño that has been termed a Modoki El Niño, where maximum changes in SST take place more in the Niño 4 region that in the Niño 3 or 3.4 regions (see Figure 1.19a for the location of the regions). The word modoki comes from the Japanese word meaning “almost but not quite,” signifying an “almost” El Niño.12 Have we yet identified all the subspecies of ENSO? Alternatively, are these different patterns of ENSO variability expected of a highly nonlinear phenomenon or are different influences at play? A useful goal would be the determination of the probability of an impact given that an ENSO episode has been forecast or is underway. Ropelewski and Halpert (1996) attempted to gauge such probabilities for the regions where coherent correlations are identified in Figure 14.13. For low SOI (warm events), neutral, and high SOI (cold events) the variation of the median precipitation was calculated in these particular regions. There was some overlap of the three categories but the information about the variation of the median rainfall for a given event allowed a quantification of impact. This appears to be the first study to quantify the probability of a particular impact relative to an ENSO extrema. We will consider the ENSO predictability in Section 14.2.3. It will turn out that it is difficult to predict the state of ENSO across the boreal spring. This makes the creation of “useful” forecasts for India difficult because of a limited lead time. Yet once an ENSO event is underway, an event appears more predictable. So perhaps the probability of certain teleconnection patterns, within the confines defined by Ropelewski and Halpert (1996), may be more useful at some times of the year than others.

12 See Larkin and Harrison (2005) and Ashok et al. (2007). Others, too, have spoken about “different flavors” of El Nino. Trenberth and Smith (2009) discuss variability in El Niño large-scale patterns.

14.2.5

ENSO Theories

Figure 14.15 presents a schematic diagram of state of the upper ocean and anomalous surface winds in the equatorial Pacific for three ENSO periods. These are: (i) A “neutral period” (Figure 14.15a), where the easterly anomalous wind stress tilts the ocean surface toward the west. The pressure gradient force, so created, must be balanced by a force raising the thermocline upwards toward the east. In Section 6.4.1 we were able to show that if the ocean surface was forced to slope upwards longitudinally, then thermocline must slope in the opposite direction. Furthermore, the magnitude of the slope of the former is two orders of magnitude less than the latter. (ii) An El Niño period where if the surface wind is weaker (Figure 14.15b) the wind stress will be overcome by the longitudinal ocean pressure gradient and the ocean slope will be reduced, resulting in a much flatter thermocline, deeper in the west and shallower in the east. (iii) A La Niña period where if the westward wind along the equator is stronger than average (Figure 14.15c), the increased wind stress will tilt the surface to a greater extent than during the neutral period. Increased upwelling in the eastern ocean and increased downwelling in the west will increase the slope of the thermocline, warming the west and cooling the east. The strengthening and weakening of the nearequatorial winds are closely associated with the change in location of the rising motion of the Walker Circulation, which, in turn, is tied to the SST distribution. The basic physics of these balances were discussed in Section 6.4.1. A complete theory of ENSO evolution must address a number of questions: (i) How does the coupled system transform itself from a neutral state to an El Niño or La Niña state? In the ENSO collage (Figure 14.7) we found that most El Niños in the short data record emerge from relatively neutral conditions. At the same time, we have noted that most La Niñas follow an El Niño and usually not from neutral conditions. How does this seemingly selective transformation take place? (ii) What process, or state of the coupled system, determines the phase locking of ENSO with the calendar year, in particular the growth of an episode during the boreal spring, as apparent in both Figures 14.6 and 14.7? The first theory we discuss depends on a positive feedback between the ocean and the atmosphere. In essence,

14.2 The Southern Oscillation, El Niño and La Niña

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it follows on from some of the ideas of Troup who noted the importance of variation of the trade winds and their impact on near-equatorial SST. This first theory is called the “Bjerknes Hypothesis” and provides a physical path between normal conditions to an El Niño or the transition of normal conditions to a La Niña. However, it does not allow for the transition between the two ENSO extremes. It does not possess a negative feedback.

Through the pressure variations associated with the SO, Bjerknes noted a correspondence between changes in the wind patterns (determined from the surface pressure oscillation) and the SST. He proposed a coupling between the atmosphere and the ocean whereby changes in the wind patterns induced changes in the SST distribution and these, in turn, modified the wind patterns, producing anomalies in the east–west overturnings, or Walker Circulations. Bjerknes posed the first coherent explanation of the atmosphere–ocean coupling that led to the transition between a climatological normal state and extrema of the ENSO phenomena (either El Niño or La Niña). His theory was based on the observations transcribed in Figure 14.4, all that were available to him at the time. At the same time, we should not forget the earlier work of Troup (1965), who suggested that variations of the trades accompanying the SO changed the near-equatorial upwelling patterns, altered the SST patterns, and subsequently changed the sign of the SO. Bjerknes, unfortunately, appeared to be unaware of Troup’s work. Bjerknes’ hypothesis, in simple terms, is outlined below. Consider the Pacific Ocean in a neutral phase (Figure 14.15a). Assume that there is an anomalous change in intensity of the trades to the north and south of the equator, as appearing, for example, in the plot of vT in Figure 14.4a: (i) If the trade winds become anomalously strong, easterly surface winds along the equator will increase, invoking greater Ekman divergence and a decrease of SST in the eastern equatorial Pacific. The resultant downwelling in the west and the deepening of the thermocline intensifies the Walker Circulation. (ii) If the trades lessen in intensity, Ekman upwelling along the equator will decrease and the thermocline will become shallower and warm water will move eastward, resulting in a Walker Circulation that is displaced eastward and a reduction in the slope of the thermocline. What changes the strength of the winds in the equatorial region? Early thoughts were that the changes in equatorial convection accompanying SST changes associated with anomalous Ekman upwelling and downwelling enhanced or relaxed equatorial winds. A number of studies,13 though, have noticed that during an El Niño there are central Pacific accelerations called “westerly wind bursts” (WWBs). We will discuss these 13 E.g., Moore and Kleeman (1999).

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eastward mode is similar to the Bjerknes-type positive feedback mode. It should be emphasized that both the Bjerknes and the Lau hypotheses possess a common major shortcoming. Changes in the atmospheric wind forcing may invoke either an El Niño or a La Niña but there is no mechanism to reverse the sign of an ENSO extrema. The question then becomes, once an El Niño (or La Niña) is produced by the Bjerknes mechanism, how does it make the transition to the opposite phase? Clearly, a negative feedback is needed. 14.2.5.2

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Figure 14.16 Schematic flow diagram of the Bjerknes positive feedback leading to (a) a cold phase (La Niña) and (b) a warm phase (El Niño). As discussed in the text, the feedback sequences are positive, leading to either extreme ENSO event. A negative feedback was not included in the original Bjerknes hypothesis that would lead from one extreme to another.

ENSO Theories with Negative Feedbacks

Nearly two decades were to pass before insights were developed into why the ENSO system was oscillatory. Two theories provide a negative feedback to the Bjerknes hypothesis. These are the “delayed oscillator” (or “delayed action oscillator”) and the “recharge-discharge oscillator” theories. We will also mention more briefly the “advective-reflective” theory. The Delayed Oscillator15 Consider the following represen-

events in Chapter 15 where we concentrate on intraseasonal variability. These events, lasting a week or so, are capable of producing eastward propagating ocean Kelvin waves. Whether or not the WWBs alter the state of El Niño, or are in fact the result of a more conducive state of the ocean during an El Niño, is still a subject of some discussion.14 A schematic of Bjerknes’s positive feedback mechanism appears in Figure 14.16 for the transition from a neutral state to a La Niña (i.e. the development of a cold phase of ENSO) and for the development of a warm or El Niño phase from a neutral state. It shows the one-way positive feedback between alterations in the strength of the trade winds, their impact on equatorial upwelling, and the transition of a neutral state to either an El Niño or a La Niña. Although it was possible for the Bjerknes hypothesis to explain a full El Niño–La Niña sequence, it provided a basis for subsequent understanding of ENSO as a coupled ocean–atmosphere phenomena. Lau (1981) added some basic dynamics to the Bjerknes hypothesis. He developed a simple coupled model made up of two shallow water systems, one of the atmosphere and the second for the ocean, the latter using a reduced gravity formulation as described in Section 3.2.2. Together, the simple coupled ocean–atmosphere system produced two atmospheric Kelvin modes: a rapid easterly propagating mode appearing as the atmosphere responded to the imposition of the initial anomalous ocean forcing and a second, much slower, mode tied to the evolving ocean. In essence, Lau’s slow 14 E.g. Eisenman et al. (2005).

tation of the ENSO system: dT = aT − bT(t − d) − 𝛾T 3 dt

(14.2)

Here, T is some representation of ENSO, perhaps the Niño-3 SST, t is time, a and b are positive constants, d is a delay time scale and 𝛾 represents a dissipation coefficient. Thus, the time rate of change of T is the balance between the positive feedback, aT, and a negative feedback bT(t − d). If b = 0, Eq. (14.2) describes the Bjerknes hypothesis, where a change in the strength of the trade winds creates an increase in the along-equator SST gradient that in turn drives even stronger surface winds, and so on. However, if the negative feedback term is non-zero (b ≠ 0) then it is possible for T to oscillate with a period depending on the magnitude of “d.”16 The question is whether or not it exists within the coupled ocean–atmosphere system and, if so, what is the magnitude of the delay constant d? Does it produce the observed time scales of ENSO variability? The basis of the theory is the manner in which near-equatorial waves generated by atmospheric forcing behave in an ocean basin restricted by boundaries (see Section 6.4.4). Initially, there is a balance between the wind stress and the slope of the thermocline along the equator in any of the states represented in Figure 14.15. Suppose a change in the wind field occurs, upsetting 15 Schopf and Suarez (1988) and Battisti and Hirst (1989). 16 It turns out that Eq. (14.2) is a very rich equation with many solutions, depending on the parameters used. Power (2011) explores the behavior of the equation over a large parameter range.

14.2 The Southern Oscillation, El Niño and La Niña

the force balance. The ocean will adjust in the manner described in Section 3.2.2, with a response of two families of low-frequency modes: eastward propagating Kelvin waves and westward propagating Rossby waves. The Rossby wave phase speed is about −0.6 to −0.8 m s−1 . The phase speed of the ocean Kelvin wave is about 2–3 m s−1 to the east at a rate an order of magnitude slower than a non-convective atmospheric Kelvin wave. An example of a propagating Kelvin wave crossing the Pacific Ocean appears in Figure 3.13. Waves on the thermocline are referred to as either upwelling or downwelling waves depending on their impact on the local thermocline. Waves that tend to deepen the thermocline are referred to as downwelling waves and tend to warm the upper ocean. Waves that reduce the depth of the thermocline are referred to as upwelling waves, tending to cause cooling. These waves carry both energy and momentum obtained from the surface wind stress and constitute the system memory throughout the year and also from one year to the next. As suggested in Section 6.4, the difference in the wave propagation speeds between the atmosphere and the ocean modes turns out to be a very important factor as it determines, to large degree, the relative time scales of the two spheres. The imbalance introduced into the atmosphere by an SST anomaly will adjust very quickly: perhaps in 1–2 weeks. On the other hand, the slow propagation speeds of ocean waves forced by a wind stress anomaly means that the ocean adjustment to a wind stress anomaly will be much slower, taking months. This means that we may assume that the atmosphere is in continual statistical equilibrium, with the slower evolving ocean as suggested by Lau (1981). The assumption has been put to good use in formulating early ENSO prediction models.17 Figure 14.17a provides a schematic depiction of the delayed oscillator. Consider a neutral situation such as depicted in Figure 14.15a, although the chosen initial state is arbitrary. If the system is perturbed, both eastward propagating Kelvin waves and westward propagating Rossby waves will be generated as the system seeks a new equilibrium. As the Kelvin wave moves eastward it tends to deepen the thermocline, creating the configuration shown in Figure 14.16b, and an SST warming to the east of the forcing. The Rossby wave, on the other hand, moves westward, shallowing the thermocline and cooling the western Pacific. In this manner, the neutral state slowly changes toward an El Niño state. In the meantime, the Rossby waves reach the western boundary and are reflected as Kelvin waves, which then propagate to the east. These downwelling waves tend to deepen the thermocline in the west. In

the east, the reflected Rossby waves tend to shallow the thermocline. Through this sequence, the system changes from El Niño to La Niña. It should be noted that a continual oscillation of reflections will not be perpetual. As we have argued in Section 6.4.3, there will be leakage of energy to higher latitudes at the coastal boundaries. Thus, the oscillation will be damped and the ocean will move toward some final quasi-equilibrium state. Observations lend some support to the delayed oscillator theory. First, the delay time matches the time scale between ENSO extremes. Second, Kelvin waves have been observed in the far western equatorial Pacific Ocean during a developing warm event.18

17 See Section 14.2.4.

18 Mantua and Battisti (1994).

The “Recharge–Discharge” Oscillator To develop a con-

ceptual model, Jin (1997) started with a simple reduced gravity 1 1/2 shallow-water model (see Section 5.2), where the basic state of the thermocline can be expressed as hE = hW + 𝜏

(14.3)

Here, hE and hW are the anomaly depths of the thermocline in the eastern and western Pacific, respectively, and 𝜏 is proportional to the zonally integrated wind stress. Equation (14.3) is valid within one ocean Rossby radius about the equator (see Section 3.2.2) and states that the difference in the anomaly depth of the eastern and western parts of the basin is proportional to the integrated wind stress. That is, the principle ocean dynamical balance in the equatorial band is between the pressure gradient, proportional to (hE − hW ), and the wind stress 𝜏. There are two components to the Jin theory. The first is essentially the delayed oscillator mechanism, described above, whereby Kelvin and waves generated by wind anomalies slowly propagate along the equator, altering the SST through increases or decreases in upwelling, and then reflect (partially or completely) off the boundaries. In this case, one of the two unknowns in Eq. (14.3) can be related to the boundary conditions that determine wave reflections. In essence, in this class of theories, the evolution of ENSO is controlled by the near-equatorial waveguide physics. The second component of the theory arises by noting that changes in wind fields also induce changes outside the equatorial waveguide and these may have important implications for the low-frequency variability of the Pacific. Given the broadness of the atmospheric equatorial wind system, Coriolis effects increase in importance off the equator. With this broader wind system, a latitudinal Sverdrup transport (see Section 3.3.2) either into or out of the equatorial zone will develop, depending on the sign of the zonal wind. The zonally integrated

321

322

14 Large-Scale, Low-Frequency Coupled Ocean–Atmosphere Systems

(a) Initial conditions and domain 20N

(i) Wind stress (eastward)

(ii) Initial response 20N 10N

10N Eq

Eq

10S

10S

20S

20S 140E

160E

180

140E

160W 140W 120W 100W 80W

downwelling Kelvin

upwelling Rossby 160E

180

longitude

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longitude (b) Evolution of equatorial response

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(i) 25 days

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160W 140W 120W 100W 80W

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(iii) 75 days

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Eq

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(iv)100 days

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160E

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160W 140W 120W 100W 80W

180

160W 140W 120W 100W 80W

20N (vii) 225 days 10N

160E

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(vi) 175 days

140E

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(v) 125 days

180

160W 140W 120W 100W 80W

longitude

160E

(viii) 275 days

140E

160E

180

160W 140W 120W 100W 80W

longitude

Figure 14.17 Schematic diagram of the delayed oscillator theory offered by Schopf and Suarez (1988) and Battisti and Hirst (1989). Here, boundary reflected waves (both the eastwardly propagating Kelvin and westwardly propagating Rossby waves) change from net upwelling (downwelling) waves to downwelling (upwelling) waves, respectively, thus changing the sign of the anomalous upper ocean temperature.

14.2 The Southern Oscillation, El Niño and La Niña

impact of the Sverdrup transport will be a transport of mass and heat content resulting in a shoaling or shallowing of the western Pacific thermocline. Although the changes in the tilt of the thermocline occurs relatively rapidly, the thermocline depth takes some time to adjust to the zonally integrated meridional transport, being related to both the wind stress and its curl, as described in Eq. (3.61). Collectively, these two components form the basis of the “recharge–discharge” mechanism. We now consider processes associated with the adjustment of the thermocline, described above. Jin (1997) writes the symbolic relationship: dhW (14.4) = −rhW − F𝜏 dt showing that there are two major groups of processes that change hW . The first term on the right–hand side represents the ocean adjustment to wind forcing, the damping due to mixing, and energy loss such as latitudinal seepage at the boundaries of the basin. Here r is a dissipation coefficient. The second term on the right represents wind forcing, where F𝜏 = 𝛼𝜏. The curl of 𝜏 determines the Sverdrup transport in or out of the basin, as described in Section 3.2.3. The negative sign of the wind forcing term comes from the fact that a westerly wind anomaly will lead to a decrease in hW or a shallower thermocline in the west. On the other hand, an easterly wind anomaly, consistent with the strengthening of the trades, will lead to a thickening of the western thermocline and a build-up of the western Pacific warm pool. Equations (14.3) and (14.4) provide a broad-scale description of the basin-wide equatorial oceanic adjustment resulting from anomalous wind stress forcing. Following Jin (1997), we can describe the evolution of the SST (T E ) in the equatorial eastern Pacific in terms of three forcing factors: dT E (14.5) = −cT E + 𝛾hE + 𝛿s 𝜏E dt The first term on the right represents the relaxation of the SST anomaly occurring at some rate c. Such a damping is affected by the mean climatological upwelling and the heat exchange between the ocean and the atmosphere. The second term represents thermocline upwelling occurring at rate 𝛾. The third term refers to advective feedbacks via Ekman pumping occurring at rate 𝛿 and 𝜏 E is the wind stress averaged over the area of the SST anomaly. Consider a warm SST anomaly located in the central/eastern Pacific Ocean. The atmospheric response is a westerly wind anomaly to its west and an easterly anomaly to its east. Averaged across the entire tropical belt there is an overall westerly anomaly. If the temperature anomaly is negative, the anomaly will be easterly.

However, to the east of the SST anomaly the response will be weaker. This thought allowed Jin (1997) to make the following assumptions that greatly simplify the equations. He set: 𝜏 = bT E

(14.6a)

𝜏E = b TE

(14.6b)

′

allowing the system to be closed. Here, b and b′ are constants and where, for reasons mentioned above, b > b′ allowing a further simplification. Substituting Eq. (14.6a) into Eqs. (14.4) and (14.5) leads to equations for the time rate of change of the eastern thermocline depth and the eastern SST anomaly. These are dhE = −rhW − 𝛼bT E dt

(14.7a)

dT E (14.7b) = RT E + 𝛾hW dt ′ where R = 𝛾b + 𝛿 S (b T E ) − c ≈ 𝛾b − c. Figure 14.18 describes Jin’s recharge–discharge hypothesis. We start with a small positive SST anomaly in the eastern ocean (i.e. T E > 0). The sequence, according to Eqs. (14.7a) and (14.7b), may be followed in Figure 14.18 such that: (I) A small positive T E will lead to basin averaged positive wind stress anomalies as expressed in Eqs. (14.6), leads, in turn, to a deeper thermocline in the east and a shallower thermocline in the west according to Eq. (14.7a) and a positive growth of the anomaly T E in Eq. (14.7b). To a large degree, this growth of the warm phase is identical to the positive feedback in the Bjerknes mechanism. The evolution of the warm anomaly and the thermocline distribution appears schematically in Figure 14.18a. (II) However, the wind distribution at low latitudes drives a meridional Sverdrup transport out of the equatorial regions (Figure 14.18a). Thus, the wind fields create a net export of mean zonal heat content to higher latitudes. This is referred to as the heat discharge stage. (III) The discharge of heat out of the equatorial zone leads to a reduction of T E , producing a weaker zonal wind stress and, consistently, the tilt of the anomaly depth thermocline reduces toward zero. As a result of the discharge, the net thermocline depth across the entire near-equatorial basin decreases (Figure 14.18b). A consequence of the discharge of heat from the tropics is a negative temperature anomaly in the east. This may be thought of as the warm to cold phase transition. (IV) From Eq. (14.7a) and (14.7b), T W will increase while T E decreases, leading to a mature cold phase. At this stage, the resulting Sverdrup transport will

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14 Large-Scale, Low-Frequency Coupled Ocean–Atmosphere Systems

(a) Discharge phase

(b) Cold phase

τE > 0

δTE > 0 Sv S

W τE ∼ 0

δTE ∼ 0

δTE ∼ 0

Equ

S

E N Equ

S

τE ∼ 0

depth anomaly

Equ

E N

depth anomaly

Sv

W

E N

W τE < 0

(d) Warm phase

Sv δTE < 0 Sv

E N Equ

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W

depth anomaly

324

(c) Recharge phase

Figure 14.18 Schematic diagram of the recharge-discharge theory of Jin (1997). The theory consists of two components: (i) a near-equatorial waveguide mechanism such as the delayed-oscillator described in Figure 14.17 and (ii) an importation of mass (recharge) into the near-equatorial regions by Sverdrup transports induced by the broader scale equatorial winds and an export of mass (discharge), again induced by a Sverdrup transport. The figure describes the sequence of the recharge–discharge cycle with (a) the discharge, (b) the cold phase, (c) the recharge phase, and (d) the warm phase, as described in the text. “Sv” refers to Sverdrup transport. The bold blue line below the equatorial plane denotes the anomalous distribution of the thermocline depth hE. . The dashed line refers to the average depth of the thermocline. Based on Jin (1997).

lead to a net heat transport into the equatorial regions. This is the heat recharge phase of the cycle appearing in Figure 14.18c. As described in (III) above, but now with the signs reversed, the seeds of the demise of the cold phase have been planted and the cold to warm phase transition follows.

wave dynamics (delayed oscillator) can also coexist with the large scale and slow discharge–recharge oscillator. Perhaps, in tandem, the combination of these theories may help explain the irregularity of the ENSO process as seen, for example, in Figure 14.6.

Over a wide range of parameters, Jin (1997) estimates that the system may oscillate on a 3–5 year period. There is also observational evidence19 to support the discharge–recharge hypothesis. The volume of warm water within the 1980–1999 period was noted to change between El Niño and La Niña in accord with the theory that the equatorial Pacific contained more warm water during the former epoch than the latter. Note that the recharge–discharge hypothesis does not preclude either the Bjerknes hypothesis or the delayed oscillator mechanism. The Bjerknes positive feedback is still considered to be very prominent but may be curtailed and reversed by discharge and recharge. Near-equatorial

theory emphasized processes in the eastern Pacific Ocean. However, observations show that ENSO displays both eastern and western Pacific interannual anomaly patterns as described in Section 14.2.3. Earlier, in Figure 14.2a we observed that a heat source in the central Pacific will induce a pair of off-equatorial cyclones and a westerly wind anomaly over the eastern Pacific. The westerlies produce Ekman convergence and a deepening of the thermocline to the west of the heating. This is a positive feedback that causes the anomaly to grow, warming the central-eastern Pacific Ocean. However, the twin cyclonic systems have a further impact. The winds induce Ekman divergence and a shallowing of the

19 Meinen and McPhaden (2000).

The Western Pacific Oscillator20 The Delayed Oscillator

20 Weisberg and Wang (1997).

14.2 The Southern Oscillation, El Niño and La Niña

thermocline to the north and south of the equator. It is theorized that, in time, this shallow thermocline will expand across the western Pacific invoking a drop in SST and an increase in surface atmospheric pressure. In turn, the western Pacific SST anomalies will produce westerly winds. As the convection moves eastward, as a result of the positive feedback described above, the spreading of the equatorial easterlies will produce upwelling and an encroaching cooler SST. This is a negative feedback that allows an interannual oscillation. The Advective–Reflective Oscillator In the discussion of

the discharge–recharge hypothesis, we noted the important role of the lateral advection of water mass by Sverdrup transports. Advection is also important in the so-called Advective–Reflective theory (Picaut and du Penhoat, 1997).V The Bjerknes positive feedback and eastward extension of warm water is countered by negative feedbacks that push the warm pool back to the western Pacific. These are related to the zonal currents associated with modes reflected at both the eastern and western boundaries, in effect producing a coupled oscillating system.VI Stochastic Forcing Previously, we have commented on two particular characteristics of ENSO. First, the extrema of ENSO from one event to another vary in duration, amplitude, and longevity (e.g. Figure 14.6). Also, extrema of ENSO occur over a relatively wide period range of 2–5 years. However, despite the difference in detail from one event to another, an El Niño or a La Niña tends to develop during the boreal spring and early summer. In an attempt to account for the irregularity, Lau (1985) suggested that El Niño and La Niña were two unstable states of the Pacific coupled ocean–atmosphere system. However, for the instability theory to be useful one would need to show that the system is more susceptible to instability growth at certain times of the year to explain the annual cycle phase locking of ENSO. What triggers these unstable modes? Lau (1985) suggests that stochastic processes external to the coupled system force are responsible. “Stochastic” and “random” are often considered synonyms, but the usage of the term stochastic in geophysics has evolved. A stochastic signal may be used to describe a non-deterministic signal within which there is some uncertainty. On the other hand, a random signal is a stochastic signal possessing complete uncertainty. Lau listed many candidates associated with stochastic forcing, such as high-frequency WWBs, extratropical incursions into the tropics, or subseasonal variability like the MJO. However, this still leaves open the question of why ENSO development is tied so strongly to the annual cycle.

Let us assume that the amplitude of the highfrequency variability is constant throughout the year. If this assumption is true, why would stochastic influences trigger the unstable mode preferentially during the boreal spring? In Section 14.2.2 we considered the relationship between equatorial wind strength and the thermocline slope during El Niño, neutral, and La Niña periods. There, we referred to balances between wind stress and the anomalous winds associated with ENSO extrema. Now, though, we need to look at shorter-term variations. Consider the annual cycle of climatological MSLP together with ± 1 SD at Darwin and Tahiti (Figure 14.19a). During the boreal spring (rectangle), the longitudinal pressure gradient is weakest and, in fact, Darwin pressure occasionally overlaps the Tahiti pressure. As the difference in the surface pressures between Tahiti and Darwin represents the SOI, the contours represent its annual cycle. These differences (Tahiti minus Darwin) are plotted in Figure 14.19b. During the boreal spring, the pressure difference (or SOI) is a factor of two smaller than during the boreal winter. In fact, the MSLP, less 1 SD, is near zero. This means that in about 16% of boreal springs, the SOI will be below zero. On the other hand, there will be a similar probability that the MSLP difference will be large, in fact with values near the boreal winter mean! We also note that along the equator, U s ∝ 𝜕p∕𝜕y and that the surface stress is given by 𝜏 ∝ U s , where U s is the surface zonal wind speed along the equator. Then, during the boreal springtime, when the surface pressure gradient is a minimum, the zonal surface wind and the wind stress will also be a minimum. Here we have assumed that the Darwin–Tahiti pressure difference represents the equatorial pressure difference. This is not strictly true but is, perhaps, a fair approximation. Prior to the spring, the MSLP gradient is stronger and during the next month or two it relaxes, consistent with a decrease in the along-equator SST gradient (Figure 14.8a). In that time, as surface forcing decreases, the couple system has to adjust, as now the slope of the thermocline is no longer in balance with the wind stress 𝜏. During this adjustment period, where the forcing (the signal) is weak, high-frequency variability (even if at the same magnitude as at other times of the year) may have a greater impact as the system adjusts from strong to weak surface forcing. In other words, during the boreal spring, the signal to noise ratio is small! This time of the year has been referred to as the “spring frailty”21 period. It is a time of the year where stochastic external influences may possess their greatest impact. We note, though, from Figure 14.19, that there is no similar period of frailty (i.e. a period where the signal 21 Webster and Yang (1992) and Webster (1995).

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14 Large-Scale, Low-Frequency Coupled Ocean–Atmosphere Systems

Figure 14.19 (a) Annual cycles of the MSLP at Darwin, Australia (12∘ 27′ S, 130∘ 50′ E) and Papeete, Tahiti (17∘ 52’S, 149∘ 56’W) plotted as a function of month. The difference between these two surface pressures defines the SOI. Units: hPa. (b) Differences in the mean monthly surface pressure between Darwin and Papeete (hPa) as a function of month. Dashed lines denote ±1 SD.

(a) Annual cycle of MSLP at Darwin and Tahiti

surface pressure (hPa)

1015

+1sd

Tahiti

1013 1011

–1sd

1009

+1sd

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1005 J

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to noise ratio may be small) during the boreal fall. Although symmetry may not be expected as during the fall, the along-equator SST gradient is substantial and the relative importance of noise may be less. In summary, stochastic influences, such as those suggested by Lau (1985), are more likely to grow during the boreal spring, forcing the slow eastward propagating joint ocean–atmosphere mode. Eventually, negative feedbacks occur, as described above, limiting the growth of the ensuing warm phase and setting up interannual oscillations. 14.2.6

Predictability of ENSO

Here we enter into a brief discussion on the predictability of ENSO extrema. Given the near global impact of ENSO (see Figure 14.13), it would be extremely useful to forecast its commencement and its subsequent growth as far in advance as possible. Given the number of theories that speak to the basic physics of ENSO, it is interesting to know if they are useful in its prediction.

There are a number of characteristics of ENSO that would suggest it should be highly predictable. The spatial scale of an ENSO event is very large and evolves slowly in time. Furthermore, although each event is somewhat different from the previous (e.g. Figure 14.6), they are sufficiently similar that their general character may be described, at least approximately, by a composite such as those appearing in Figure 14.7a and b. Once an event starts it follows a rather predictable evolution and an event always commences and decays at roughly the same time of the year. Yet, difficult questions remain. In a given year will an El Niño develop at all in a particular boreal spring period and, if it forms, will it decay during the following year and return to a neutral state or develop into a La Niña? 14.2.6.1 Annual Cycle of Persistence and the Boreal Springtime “Persistence Barrier”

The basic problem of predicting the initiation of an ENSO event is illustrated in Figure 14.20. The first panel (a) shows lagged correlations between the mean

14.2 The Southern Oscillation, El Niño and La Niña

22 Torrence and Webster (1998).

Figure 14.20 Annual cycle of persistence. (a) Lagged correlations of the monthly Southern Oscillation Index (SOI). Letters indicate the anchor or starting month of the correlation. The lagged correlation plots have been offset so that the correlations for the same month are lined up along the abscissa. Note that the lagged correlations also decrease abruptly in the boreal spring (box). Data are monthly mean values of the SOI from 1935 to 1990. Source: Based on Webster and Yang (1992). (b) SST persistence, Pm (i), between (i) JJA and DJF and (ii) DJF and JJA. The fixed phase persistence, Pm (i), is defined as the correlation of values for one month m with a future month m + i. Note the strong 6-month persistence in the eastern Pacific between JJA and DJF and the substantial reduction between DJF and JJA. (c) Annual cycle of persistence of SST along the equator through two annual cycles.

running average. December persistence (Figure 14.21a) is limited from three to five months (i.e. March–May), after which a catastrophic reduction takes place. The March persistence is even more limited to about two to three months. It is not until the boreal summer that strong persistence extends to six to nine months. Persistence is also high in September, extending to the boreal spring. In addition to the annual cycle in persistence, there is long-term variability in the limits of persistence. For example, during the 1920s to the 1950s, persistence was lower and probably matched the period of Troup’s concern, noted earlier. In summary, a persistence barrier has been found in numerous ENSO indices. The barrier is phase locked with the annual cycle and occurs at the time when the across-Pacific SST gradient is at a minimum and the Walker Circulation its weakest. In this state, the system is most easily influenced by stochastic processes such as weather events that at other times of the year would perhaps have a minimal effect.

14.2.6.2

The Boreal Springtime “Predictability Barrier”

ENSO variability has been predicted by many different types of models, ranging from statistical persistence models, simple ocean-atmosphere coupled models (both linear and non-linear), to high-resolution coupled ocean–atmosphere climate models. Statistical hybrid models have been developed based on a set of predictors determined from data analysis allowing the creation of regression equations. Future values of the predictors are obtained from evolving atmosphere and (a) Lagged correlations of the SOI 1.0

1.0 0.8

Correlation Coefficient

monthly values of the SOI. The anchor or starting point of the correlation is denoted by month. Note that the plot is offset so that the correlations of the same month are lined up along the abscissa. The rectangle indicates the boreal late-spring early-summer period during which time there is a precipitous drop in lagged correlations irrespective of the starting point of the lagged correlations. Panel (b) plots the six-month persistence of SST starting in JJA (upper panel) and DJF (lower panel).22 The fixed phase persistence, Pm (i), is defined as the correlation of values for one month m with a future month m + i. If we have information about the JJA SST anomaly patterns in the central and eastern Pacific Ocean, then we can infer >60% of the SST anomaly six months later. On the other hand, knowledge of the SST distribution in DJF offers little information of even the sign of the JJA SST anomaly. It is interesting to note, for future reference, that in the equatorial Indian Ocean persistence exists at about the 40% level for other lags. Panel (c) plots the annual cycle of the six-month persistence along the equator over a two-year period. The figure emphasizes the rapid growth of persistence in the eastern Pacific Ocean following MAM. Furthermore, this panel illustrates that persistence in the equatorial Indian Ocean continues from the boreal fall through the next boreal summer. Figure 14.21 describes the long-term variability of persistence of the Niño 3.4 SST between 1870 and 2005 starting at four different months, December, March, June, and September. Persistence is plotted from a 0 to 18 month lead with the application of a six-month

0.8

Sep

Jul

Jan

Nov

0.6

0.6

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May 0.4

0.4

0.2

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0 J

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D 6

F 8

A 10

J 12

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A 14

O 16

D 18

F 20

–0.2 22

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14 Large-Scale, Low-Frequency Coupled Ocean–Atmosphere Systems

Figure 14.20 (Continued)

(b) 6-month persistence (x10) of SST anomalies 60°N

(i) JJA

DJF

latitude

30°N EQU 30°S 60°S 60°W

0°

60°E

60°N

(ii) DJF

120°E longitude

180°

120°W

60°W

180°

120°W

60°W

JJA

latitude

30°N EQU 30°S 60°S 60°W

DJF

0°

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120°E longitude

(c) Annual cycle of 6-month persistence of SST anomalies Atlantic Indian Pacific

SON JJA Africa

South America

MAM Starting season

328

DJF SON JJA MAM DJF 60°W

0°

60°E

120°E longitude

180°

ocean models.23 The first coupled dynamical prediction models24 were essentially coupled shallow water models similar to those described earlier in the text. These required a parameterization to facilitate coupling between the ocean and the atmosphere. All of these models, including the coupled climate models, were relatively successful in forecasting the large-scale features of the evolving ENSO phenomena, providing predictions in the 3–10 month range depending upon the time of initialization of the models. All models, though, lost 23 E.g. Latif and Graeme (1994). 24 E.g. Zebiak and Cane (1987).

120°W

60°W

predictive skill through the boreal spring. In effect, the persistence barrier became a “predictability barrier.” 14.2.6.3

Real-Time Forecasts of ENSO Variability

Figure 14.22 presents the results of two early but completely different models: the Latif and Graham (1992) statistical model and the Zebiak and Cane (1987) numerical model. The predictions are evaluated by comparing the forecast fields with observations. The analysis of the Latif-Graham model is based on one year of integration whereas Zebiak-Cane forecasts are provided for a much longer period. Using the same comparative method deployed in Figure 14.20a, the

14.2 The Southern Oscillation, El Niño and La Niña

(a) December persistence

Lead month

18 15

Mr

12

De

9

Sp

6

Jn

3

Mr 1856

0.8 0.4 0 –0.4

1876

1896

1916

1936

1956

1976

1996

2014

(b) March persistence 0.8

Lead month

18 15

Jn

12

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Figure 14.21 Interdecadal variability of persistence out to 18 months over a 120-year period of the Niño 3.4 with starting months (a) December, (b) March, (c) June, and (d) September. Throughout the period persistence is weakest across the boreal spring period. However, the degree of persistence varies from decade to decade. A 10-year running mean has been applied. Letters on ordinates signify calendar month of lead. Data used described in Kaplan et al. (1998) and Reynolds (2002). Source: Data available at: https://iridl.ldeo .columbia.edu.

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14 Large-Scale, Low-Frequency Coupled Ocean–Atmosphere Systems

(a) Latif-Graham model 1.0

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Month Figure 14.22 Annual cycle of predictability of the variation of correlations between observations and predictions of the sea surface temperature of the coupled system of the Pacific Ocean occurring in two early intermediate models. (a) The Latif and Graham model (Latif and Graham 1992) and (b) the Zebiak-Cane model (Zebiak and Cane 1987). Lines denote correlations for forecasts commencing at four different times of the year: April, July, October, and January, offset as in Figure 14.20a. Irrespective of when the prediction was commenced, the correlation coefficients decrease rapidly between April and June (boxed area). (b) Same as (a) except for the Zebiak and Cane model (1987).

forecast-observation correlations are offset in time in order to compare the skill of the forecasts for the same month. For both forecast models, the forecast skill decreases very rapidly across the boreal spring. We now consider models at the other end of the complexity spectrum: global coupled ocean–atmosphere models used to make seasonal forecasts. The number of degrees of freedom of these prediction models is many orders of magnitude greater than either the Latif-Graham or the Zebiak-Cane systems. The European Centre for Medium Range Weather Forecasts (ECMWF)25 system, for example, consists of an ocean 25 See https://www.ecmwf.int/en/forecasts/documentation-andsupport/long-range/seasonal-forecast-documentation/user-guide/

and atmosphere assimilation scheme to integrate vast amounts of disparate data sets in order to estimate the initial state of the ocean, a global coupled ocean–atmosphere general circulation model to calculate the evolution of the ocean and atmosphere, and a post-processing suite to create forecast products from the raw numerical output. In all of the leading weather and climate centers, these global models are run in ensemble mode many times per week with forecast horizons of months. Each ensemble represents an integration with a slightly different set of initial conditions in order to account for the growth of inherent errors.26 Figure 14.23 provides a summary of observed and predicted Niño-3 (90∘ W–150∘ W and 5∘ S–5∘ N) and Niño-4 SST (160∘ E–150∘ W, 5∘ N–5∘ S) SST anomalies obtained from the ECMWF seasonal forecast system. As in Figure 14.20, the forecast initialization month appears on the ordinate and the forecast horizon in months appears on the abscissa. Consider a forecast of Niño-3 SST made for April (month 4). Proceeding horizontally, the forecast skill (measured in terms of correlation) drops off rapidly on a lead time of 2–3 months. Yet a forecast made just two months later (month 6 or June) has a skillful forecast horizon of many months longer. The Niño-4 forecasts, for reasons not understood, appear to have less of a springtime predictability barrier compared to Niño-3.4. In summary, observations, simple models, and sophisticated coupled ocean–atmosphere models all show a rapid decline in persistence and predictability across the boreal spring. CaneVII (1991) remarked: … The degree of forecasting skill obtained, despite the crudeness of the model, is teIling. It suggests that the mechanism responsible for the generation of EI Niño events and, by extension, the entire ENSO cycle, is large-scale, robust and simple: if it were complex, delicate or dependent on small-scale details, this model would not succeed.… Cane (1991, p. 363) However, at the same time, the general demise of predictability across the boreal spring is also “…robust and simple ….” It occurs at the time of the year when the ENSO signal is weakest, when the magnitude of the Walker Circulation is at an ebb, and the signal to noise ratio is lowest. And also, models of all complexity perform quite similarly. Once the coupled ocean–atmosphere model “stiffens” following the boreal spring, all models do remarkably well. seasonal-forecasting-system for details on the seasonal forecasting system. 26 E.g. Palmer (2016).

14.2 The Southern Oscillation, El Niño and La Niña

(a) Niño-3 SST 1

(b) Niño-4 1

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Figure 14.23 Annual cycle of predictability in the equatorial Pacific in the European Centre for Medium Range Weather Forecasts (ECMWFs) coupled ocean–atmosphere–land operational seasonal climate model for (a) Niño-3 SST (90∘ W–150∘ W and 5∘ S–5∘ N) and (b) Niño-4 (160∘ E–150∘ W and 5∘ S–5 N) areas. Contours of correlation coefficients are plotted as a function of the month of the forecast initiation (ordinate) versus lead-time in months. Source: Model described at http://www.ecmwf.int/services/dissemination/3.1/Seasonal_ Forecasting_System_3.html. From Kim et al. (2009).

14.2.6.4 Can Forecasts of ENSO Extrema and Their Impacts Be Improved?

All models, irrespective of complexity, appear to possess a spring predictability barrier. All models correctly assess the spring predictability barrier, thus, as a property of the coupled ocean–atmosphere system in the Pacific Ocean. This similarity would suggest that, in the absence of new predictors not currently included in the models, any increases in predictability across the boreal may be marginal at best. As discussed previously, the coupled ocean–atmosphere system is remarkably frail during the boreal spring (MAM) and perhaps may be nudged by a range of stochastic weather events originating in either the tropics or in the extratropics.VIII For example, in Chapters 10 and 11 examples are given of how Rossby waves breaking in the westerly ducts can allow propagation of extratropical influences into the deep tropics. The evolution of such a system at this time in the annual cycle is essentially unpredictable. However, the coupled system does appear to contain predictable elements following the boreal spring. Once the system has been “nudged” toward its new climate trajectory, an ENSO event develops in a more or less recurrent fashion. Figure 14.7 shows that following springtime, either an El Niño or a La Niña grows in intensity for the remainder of the calendar year, achieving maximum intensity in the early boreal

winter before diminishing in the springtime, but we also note that in the limited number of events sampled in Figure 14.7, each ENSO event is different both in amplitude and phase of maximum intensity. In such a complex system, other factors may still nudge ENSO development, making it stronger or weaker, earlier or later, than a canonical ENSO event. Furthermore, the remote impact of an ENSO event may be different between two seemingly similar episodes. Statistically, rainfall is reduced in JJA over India during an El Niño. However, during the especially strong 1997–1998 event, rainfall was normal, while during the relatively minor El Niño of 2002–2003, the country faced one of the worst droughts in its history. If there are improvements to be made in ENSO prediction, they will probably occur in the post-spring period and relate to how a particular ENSO event transmits its influence on the surface climate. There are two general approaches to ENSO prediction. Empirical Forecasts The basis of empirical forecasting

is using past associations to infer future states. A first method (e.g. Ropelewski and Halpert 1987, 1996) takes a measure of ENSO variability (e.g. the Niño-3 or Niño-3.4 SST index) and looks for distributions of correlations relative to a predictand, usually regional precipitation or surface temperature. Such a technique

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14 Large-Scale, Low-Frequency Coupled Ocean–Atmosphere Systems

provided the correlation patterns superimposed on Figure 14.13. A second empirical method employs a family of predictors to create a regression equation to derive a future state of a predictand. These predictors may be derived from observational relationships or theoretical considerations. In essence, this is an application of Bayes TheoremIX , which helps determine the probability of an event based on conditions that are related to the predictand. These conditions are referred to as “Bayesian priors.” The resultant probability of the repeatability of the event is called “conditional probability” and is the basis of Bayesian Statistics. Webster and Hoyos (2004), for example, developed such a Bayesian scheme for the prediction of North Indian rainfall. Unfortunately, empirical methods depend on both having a data base that is sufficiently long to include a full statistical description of the phenomena, its variability and the environment within which it exists. Lorenz (1969a,b) estimated that for analogue techniques to be used for the forecasting of extratropical variability on the timescale of days , 150 years of daily data of the entire troposphere would be required, far longer than existed in any data base. To compound the issue further for the slowly varying ENSO signal, Figure 14.5 shows that there appears to be long-term non-stationarity in the SST data from the Central Pacific. It is difficult to determine how long a data set would need to be to include these multidecadal variations. Ensemble Forecasts of the Couple Ocean–Atmosphere System

Over the last few decades, numerical weather prediction has come to depend on “ensemble” techniques, whereby a model is run multiple times per day with slightly different initial conditions. This is done to assess the influence of an exponential growth of errors in the initial data and also uncertainties of the representativeness of the model or its physical parameterizations.X Ensemble forecasting is essentially a Monte Carlo process that determines a probability density function of possible outcomes. As computer power expands, the resolution and number of ensemble members increases and there is at least the potential of gains in the predictability of ENSO and its impacts. However, it is unlikely that a forecast of the onset of an ENSO event will improve if we are correct in suggesting that the “predictability barrier” is the property of the real coupled ocean–atmosphere system.

early nineteenth century, Schove and Berlage (1965) found evidence of basin scale interannual and decadal variations across the Indian Ocean region. Besides computing annual anomalies, they also considered half-year variations finding indications of association with the SO. Figures 1.19 and 14.3 suggest that with more recent data that this association would seem plausible. The SOI extends over both the Pacific and Indian oceans and, over an SO cycle, a reversing anomalous surface pressure gradient would occur across the Indian Ocean, with the largest changes in the east. Furthermore, these variations were found to be associated with strong convective anomalies.27 Because of the relative scarcity of data, conclusions about variability in the Indian Ocean were based on a few case studies. Once a greater density of tropical SST data became available, it became evident that the Indian Ocean did indeed undergo significant interannual variability. Occasionally, the eastern Indian Ocean warmed while the western basin cooled, followed often in the next year or so by a reversal of this anomalous SST dipole. The absolute magnitudes of the anomalies were smaller than the extremes of El Niño and La Niña found in the Pacific Ocean but, given the relative sizes of the basin, the changes in the absolute longitudinal gradient of SST were larger. Two events in particular, one in 1961 and the other in 1997, suggested that it might be worthwhile to look beyond just the influence of ENSO in considering climate variability in the Indian Ocean. 14.3.1

The 1961 Event

Using an extended data set based on ship observations from 1954 to 1976, Reverdin et al. (1986) catalogued a number of observations:

14.3 Indian Ocean Interannual Oscillations

(i) During 1961, changes were found in the along-equator SST with a distinctive displacement of the climatological maximum in the eastern Indian Ocean moving toward the west. (ii) The changes in SST were related to rainfall anomalies. The warm SST anomaly in the western Indian Ocean coincided with catastrophic rainfall over tropical eastern Africa, especially in the boreal fall of 1961.28 (iii) Anomalies in the near-equatorial wind field tended to grow to a maximum in the October–November 1961 period. (iv) Changes in the wind field were associated with a redistribution of rainfall along the equator with strong anomalous easterly winds extending across the equatorial Indian Ocean.

In a careful analysis of barometric pressure data, with some records extending back to the late eighteenth and

27 E.g. Barnett (1983) and Cadet (1985). 28 Described in detail by Flohn (1987) and Kapala et al. (1994).

14.3 Indian Ocean Interannual Oscillations

(v) Within the period of analysis, year 1961 proved to be the most anomalous of all years found in the data set. (vi) In some years, the equatorial Indian Ocean variability occurred in phase with the phase of ENSO, but not in 1961. Using a much more complete data set, Clark et al. (2003) were able to verify the general results of Reverdin et al. (1986). 14.3.2

The 1997–1998 Event

Between January 1997 and March 1998 SST anomalies in the equatorial Indian Ocean were far from normal. During this period, SST anomalies in the western Indian Ocean changed from −2 to +2 ∘ C, while in the east the near-equatorial SST cooled by over 4 ∘ C. Throughout the spring and early summer anomalous equatorial easterly surface winds spanned the entire equatorial Indian Ocean. As in 1961, the changes in SST were also accompanied by a relocation of maximum precipitation from the eastern to the western Indian Ocean. At this time, precipitation, SST, and wind data were available throughout the Indian Ocean basin and allowed the evolution of the event to be examined in detail. In many respects, anomalies across the Indian Ocean mirrored those identified in 1961. Figure 14.24 describes the state of the Indian Ocean during November 1997, showing the spatial distribution of the deviations of SST (𝛿SST ∘ C) from the long term November mean, the outgoing longwave radiation anomaly (𝛿OLR W m−2 , used as a proxy for rainfall), the anomalous surface zonal wind velocity (𝛿u m s−1 ), and the sea level height (𝛿SSH cm). There is a clear spatial structure in the anomalous SST fields, with extreme values in the eastern and western regions of the basin and a “boomerang-shaped” positive SST anomaly pattern about the equator. Near-surface zonal wind fields were strong and easterly (>6 m s−1 ) with a core located just south of the equator between 60∘ E and 100∘ E. Usually, the zonal winds in autumn (September–November) are weak westerlies (Figure 1.8b). Fields of OLR depict greater than average rainfall in the western and northwestern Indian Ocean and diminished rainfall in the east, again completely out of phase with normal rainfall distributions (Figure 1.6). Along the equator there is an OLR gradient of ∼70 W m−2 , roughly the same magnitude as between the western and eastern Pacific Ocean but occurring over only half the horizontal distance and, hence, producing a zonal heating gradient of twice the magnitude. This gradient is consistent with an atmospheric circulation with strong rising motion above the western Indian Ocean, strong surface low

pressure to the west, and subsidence in the east. The SSH field shows depressed heights in the east, especially near Sumatra, and an elevation in the west. The zonal SSH gradient is substantial. Between the Sumatra coast and the East African coast, the SSH rises by 80 cm. To the south of the equator and in the central and western Indian Ocean there is an elongated “ridge” some 30 cm above adjacent ocean regions. The location and magnitude of the ridge is consistent with Southern Hemisphere Ekman mass transports to the left of the surface equatorial easterlies and westerlies near 10∘ S, as discussed in Section 3.2.4. As discussed earlier, the existence of interannual variability in the Indian Ocean SST has been known for decades and thought, generally, to be a basin-scale warming forced by changes in surface winds related to ENSO. Warmer than average SSTs were associated with El Niño and cooler temperatures with La Niña. A number of studies, though, had suggested that there existed an autonomous variability in the Indian Ocean,29 perhaps independent of ENSO. It was unclear, however, how internal coupled ocean–atmosphere dynamics would instigate variability within the Indian Ocean, or if coupled dynamics were involved at all. The 1997/98 event afforded an excellent opportunity to investigate what appeared to be a phenomenon of some interest. Figure 14.25 shows longitude-time sections of daily values of SST (∘ C) and 925 hPa zonal wind (m s−1 ) averaged between 5∘ N and 5∘ S for the period 1 January 1997 through 1 July 1998. The positive dipole pattern (warm in the west, cool in the east) appears as early as July 1997. The rapid decrease in eastern Indian Ocean SST occurs in September and the rapid rise in SST in the west even later. Easterly wind anomalies (Figure 14.25b) formed in July, consistent with the SST gradient, before accelerating in October 1997 as the gradient SST increased. These features were examined in a number of studiesXI and a general conclusion was that there exists an interannual mode relating the western and eastern tropical Indian Ocean. This phenomenon became known as the Indian Ocean Dipole (IOD). The Dipole Mode Index (DMI) is a measure of the IOD and is defined as the anomalous zonal SST gradient across the equatorial Indian Ocean. Specifically, it is the difference between the magnitude of the SST anomaly in a western box (60∘ E–80∘ E, 10∘ S–10∘ N) and an eastern box (90∘ E–110∘ E, 10∘ S–0∘ S). A time series of the DMI appears in Figure 14.26 between 1950 and 2000 (black line) together with the Niño-3.4 SST where positive values indicate an El Niño (red line). The black line represents the October–November rainfall in equatorial East Africa. These data indicate that: 29 E.g. Reverdin et al. (1986), as discussed above, Nicholls (1983, 1995), and Meehl (1994a,b, 1997).

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14 Large-Scale, Low-Frequency Coupled Ocean–Atmosphere Systems

(a) δSST (°C) Nov 1997

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Figure 14.24 Anomalous state of the Indian Ocean during November 1997. Panels show anomalies from the long-term November averages of: (a) 𝛿SST (∘ C), (b) 𝛿OLR (W m−2 ), (c) zonal wind anomaly (𝛿u: m s−1 ) at 925 hPa, and (d) sea level height anomaly (𝛿SSH: cm) obtained from the TOPEX/POSEIDON satellite. Positions C and D are for later reference. Source: Adapted from Figure 2, Webster et al. (1999).

(i) A positive DMI is only sometimes associated with an ENSO event, with coincidence between the two phenomena occurring in 1951, 1972, 1983, 1987, and 1997–1998 (Figure 14.26a). These are usually associated with above-average precipitation over East Africa. (ii) Other positive DMI years were apparently associated with ENSO, such as in 1961, 1967–1968, 1977–1978, and 1994 (Figure 14.26a). These years were also associated with anomalously high precipitation in East Africa. (iii) The zonal SST gradient during a positive IOD drives anomalous easterly zonal winds along the equator. In a negative IOD period the anomalous winds are westerly. To the south of these near-equatorial winds is an anomaly of the opposite sign. These two maxima are important as they determine the distribution of the Ekman transports. (iv) The IOD (of either sign) appears to be phase locked with the annual cycle. An IOD forms in the boreal summer remaining until the late northern spring/early summer. This is collaborated in Figure 14.20c, where we note that there is extended persistence of SST in the Indian Ocean. Specifically,

the SST anomaly will persist at the 40% level from JJA until DJF or DJF to JJA. (v) There appears to be a biennial period within the DMI. It is clear that the 1997–1998 dipole event was not isolated. In fact, over the last half-century there has been a sequence of positive and negative episodes of the IOD. Analysis of Indian Ocean SST data reveals that the 1997–1998 and 1961 events were members of a sequence of oscillations of varying magnitudes involving longitudinal SST anomalies across the tropical Indian Ocean. 14.3.3

Association of the IOD with ENSO

The association of the IOD with the El Niño of 1997–1998, or with the ENSO phenomena in general, remains unclear. Saji et al. (1999) states that there is little or no relationship between ENSO and the dipole. This conclusion, made using correlations with annual ENSO parameters, is weakened when seasonal parameters are used. Based on correlations between ENSO and the dipole, Reason et al. (2000) argue that the dipole is simply an extension of the ENSO influence in the Indian Ocean. However, Webster et al. (1999) state that

14.3 Indian Ocean Interannual Oscillations

(a) 5°N-5°S δSST (°C)

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Figure 14.25 Time–longitude sections of (a) the 𝛿SST and (b) the 925 hPa zonal wind anomaly (𝛿u) averaged between 5∘ N and 5∘ S for the period January 1997 through July 1998. Units are ∘ C and m s−1 , respectively.

the dipole is “… arguably independent of ENSO …” based on the differences of the 1997–1998 event relative to what are thought to be normal influences of El Niño on the Indian Ocean. During 1961–1962, the SST gradient across the Indian Ocean reversed, with substantial warming in the western basin. In all, between 1950 and 1998 there were 16 years in which the equatorial SST gradient reversed for at least a month. Of these years, only three were El Niño years. None coincided with a La Niña. Figure 14.26a illustrates this point, showing the association of Kenyan rainfall, the DMI, and Niño-3.4 SST. The figure shows the rainfall in the Kenyan region for the period under discussion. Whereas the associations between SST and rainfall during the 1997–1998 event are of the same sign as the normal excursions occurring during El Niño, they are very much larger in magnitude.

Furthermore, during the 1997–1998 period, the climate patterns around the Indian Ocean rim were very different from those normally associated with El Niño. Although the 1997–1998 El Niño was the strongest ENSO event in the century, monsoon rains were normal in South Asia and North Australia when, in a canonical sense, drought may have been expected. Arguably, in 1997–1998, climate anomalies around the basin could be more associated with the anomalous conditions in the Indian Ocean rather than in the Pacific. Figure 14.26b provides a plot of the mean monthly Kenyan rainfall. The three curves represent the 1997–1998 rainfall, the long-term mean rainfall, and the rainfall normally associated with an El Niño, the latter showing little difference from the long-term mean. In 1997–1998, the “short” rains (October–December) were well above average. The earlier “long rains”

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(a) JAS DMI, OND rainfall and JAS Niño 3.4 SST 3

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wavelet technique. Three correlation curves are plotted, representing relationships between the DMI and East African rainfall, Niño-3.4 SST and the DMI, and Niño-3.4 SST and the East African rainfall. Prior to 1966, the DMI:JAS Niño-3.4 was not statistically correlated. After that time it hovered at the 95% significance level. Prior to 1980, the DMI:rainfall relationship was highly significant before decreasing rapidly for a 10 year period, even changing sign. Similarly, the relationship between the Niño-3.4 and the DMI also collapses. From 1990 onwards, all correlations return to their former significance. Of course, the data set is short and a more extended analysis is required, but it is clear that the statistics are non-stationary. As we concluded regarding the SOI and ENSO, one must be very cautious about examining short periods of high correlation and making conclusions of casuality. 14.3.4 What Produced the 1997–1998 IOD Episode?

10 0 Oct 96 Jan 97 Apr 97 Jul 97 Oct 97 Jan 98 Apr 98 month/year (c) 10-year sliding correlations 1.0

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year Figure 14.26 (a) Time series of the JAS dipole index (DMI), the JAS Niño 3.4, and the OND Kenyan rainfall, (b) 10-year running correlations, and (c) detail of the Kenyan rainfall for 1997–1998 from Clark et al. (2003) compared to the long-term mean and the average during an El Niño.

(March–May) normally unaffected by ENSO were little affected by the 1997–1998 event. Figure 14.26c suggests a more complicated picture. The diagram shows the waxing and waning of correlations over the years between the DMI, rainfall in East Africa, and the Niño-3.4 SST using a sliding or running window correlation technique in essence a less-formal

We commence by examining the SSH deviations through the 1997–1998 period plotted in Figure 14.27a to c as a function of time and longitude. In each panel, propagation of SSH anomalies are marked by arrows. It appears that the IOD event in 1997–1998 (and possibly, by extension, to other IOD events) is associated with dynamic modes that propagate and extend across the entire basin. Two forms of propagating modes are evident. The first mode is a westward propagating Rossby wave appearing with different phase speeds in all three latitudinal sections. Faster phase speeds occur near the equator and become progressively faster with increasing latitude, which is consistent with the discussion on Rossby wave propagation speeds in Section 6.4.2. The second mode eastward propagating mode is a Kelvin wave. For 100 m thermocline depth, the observed phase speed of the Kelvin wave is about 2.4 m s−1 , similar to the Kelvin waves observed in the eastern Pacific Ocean (Figure 3.12). Figure 14.28 provides an indication of the scale of the ocean modes based on TOPEX/POSEIDON satellite reconstructions of SSH across the Indian Ocean, with five 10-day average periods from mid-November 1997 (bottom panel) to early January 1998 (top panel). The letters “C” and “D” show features of the surface topography identified in the mean November field (Figure 14.24d). Clear westward propagations of elevated sea-level height occur throughout the period. The ridge to the south of the equator is consistent with Ekman transports associated with the wind field anomalies shown in Figure 14.24c. The core of the anomalous easterlies is to the south of the equator. Further south there are anomalous westerlies. Ekman

14.3 Indian Ocean Interannual Oscillations

Sea-surface height sections across the Indian Ocean (cm) (a) 1°N-1°S

(b) 4°S-6°S

(c) 11°S-13°S 7/1/98

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20

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80°E

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Figure 14.27 Time-longitude sections of the sea-surface height anomalies (units: cm) between (a) 1∘ N and 1∘ S, (b) 4∘ S and 6∘ S, and (c) 11∘ S and 13∘ S, for the period using TOPEX/POSEIDON altimeter data. Black lines indicate the propagation speeds of westward propagating anomalies. The latitudinal dependence of phase speed indicates that the anomalies are associated with westward propagating and downwelling Rossby waves and eastward propagating Kelvin waves. Source: Adapted from Webster et al. (1999).

transports from these two streams converge mass, producing the sea-level ridge. Further evidence of the involvement of large-scale coupled modes can be seen in the rapid eastward propagation of a mode in mid-spring of 1997 following the arrival of the Rossby wave at the west coast (Figure 14.27a). Starting in the western Indian Ocean at the beginning of April 1997, the positive height anomaly propagated across the entire basin in about two months. This is an example of an eastward propagating upwelling Kelvin wave, first identified in the Indian Ocean by Knox (1976). The reflection of a Kelvin wave at the western boundary of the North Indian Ocean is consistent with the discussion in Section 6.4.4, where we noted that a westwardly propagating Rossby wave would be reflected as a Kelvin wave with the same frequency. The southern flank of the equatorial Kelvin wave is present in the 4∘ S–6∘ S band and is identified by the dashed line in Figure 14.27b. Besides the relationship between the IOD and Kenyan rainfall, there is some evidence of a wider scale influence of the interannual variability in the Indian Ocean. A significant correlation between the phase of the IOD and drought in the southern half of Australia has been identified, particularly in the southeast of the continent.

Every major southern Australian drought since 1889 has coincided with positive-neutral IOD fluctuations including the 1895–1902, 1937–1945, and 1995–2009 droughts.30 14.3.5

Composite Structure of the IOD

Figure 14.29a and b show composites of SST computed from the ensemble of positive and negative dipole events identified between 1950 and 2002. Recognizable signatures of the dipole events commence in the early-mid summer for both composite positive and the negative phases of the dipole. Positive events (Figure 14.29a) become evident in June and both phases reach maximum amplitude in the mid- to late-boreal fall. The signatures of both phases disappear by the following June. Once triggered, the dipole appears to be a selfmaintaining coupled ocean–atmosphere instability. Analyses suggest that the initial cooling of the eastern Indian Ocean sets up an east-to-west SST gradient that drives near-equatorial anomalous easterlies. In turn, these winds change the SSH to tilt upwards to the west. Relaxation of SSH anomalies, in the form of westward 30 Ummenhofer et al. (2009).

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Sequence of 10-day mean SSH anomalies across the Indian Ocean 12/30/1997–1/9/1998 equ 40

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propagating and downwelling ocean Rossby waves, depresses the thermocline in the west and enhances the warming of the western Indian Ocean. The slow propagation of these modes (1–2 m s−1 ) and the manner in which they maintain the warm water by deepening the thermocline ensures that the dipole is a slowly evolving phenomenon. Figure 14.30 provides a schematic description of the evolution of the positive phase IOD based upon observations and inferences from the 1997–1998 event. The figure depicts the IOD as a distinctly coupled ocean–atmosphere phenomenon. The sequence of events in the Indian Ocean occurring once the east–west equatorial SST gradient is established is shown in Figure 14.30b to d. With a source of moist air from the

Indian Ocean, strong convection developed over the heated land mass of east Africa during autumn of 1997, enhancing the equatorial easterly flow (Figure 14.30b). Ekman transports, driven by the easterly anomalies, produce the Ekman ridge just south of the equator. In seeking equilibrium, ocean Rossby wave dynamics cause the ridge to propagate westward which, in turn, deepens the thermocline to the west of the source of the Ekman convergence (Figure 14.30c). With a deeper thermocline and reduced upwelling, the western Indian Ocean continues to warm, thus maintaining the driving force of the anomalous easterly surface winds. In this manner, the Ekman ridge is sustained, allowing for a continual eastward propagation of the equatorial ocean Rossby waves. We propose that this self-sustaining

14.3 Indian Ocean Interannual Oscillations

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system prolonged the warming in the western basin in 1997–1998 by a number of months. A warming period may end, or diminish, in a number of ways. For example, a warm West Indian Ocean is often associated with a strong monsoon. This means that the summer monsoon following the Indian Ocean

warming would have stronger winds in the western basin that would induce greater mixing, greater Ekman transports forcing coastal upwelling, and greater evaporation, all of which would contribute to rapid cooling. Also, as the El Niño weakens (as it did during the spring of 1998) and the locus of convection moves

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(b) Composite Evolution of SST during negative IOD July

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14.3 Indian Ocean Interannual Oscillations

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Figure 14.30 Schematic of the sequence of events in 1997–1998. (a) The climatological alongshore winds off Sumatra (E) and the east African coast (F). The winds observed in the late summers and early autumns are denoted by G and H, respectively. The right-hand panel shows the effect at the equator on the upper ocean induced by increased upwelling in the east and decreased upwelling in the west. Wind into and out of the plane of the page are denoted by the bull’s-eye and cross-hair symbols, respectively. (b) Distribution of the winds resulting from the anomalous SST gradient along the Equator and the changes in the SSH distribution. (c) Formation of the Ekman ridge in the central Indian Ocean and the forcing of westward-propagating downwelling equatorial Rossby waves to the west. The right-hand panel shows the effect on the upper ocean near 5∘ S. (d) Subsequent cooling of the western Indian Ocean through enhanced mixing and coastal Ekman transports from stronger than average monsoon winds and through circulation changes associated with the weakening of the 1997–1998 El Niño. Source: Adapted from Webster et al. (1999).

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back toward Indonesia, the equatorial wind patterns in the Indian Ocean revert to westerlies (Figure 14.30d). The resulting relaxation of the SSH fields would force eastward-propagating and downwelling Kelvin waves

to deepen the eastern mixed layer and return the system to a normal configuration. These changes may be seen in the spring and early summer of 1998 in Figure 14.27.

Notes I A stationary time series is one whose statistical

II

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properties, such as mean, variance, autocorrelation, etc., are constant over time. That is, the autocorrelation function depends on lag alone and does not change with the time at which the function was calculated. If statistical properties vary through the length of the time series it would be termed non-stationary. This was the nature of the SO Troup (1965) found in analysis of the SO for the two periods 1882–1921 and 1921–1960. A wavelet modulus provides a history of when, in the data record, certain periodicities are dominant and when they are not. A wavelet analysis may be thought of as an evolving periodogram advancing through the data record from the beginning to the end. A cross-wavelet analysis emphasizes when two time series possess similar periodicities. Lau and Weng (1995) and Torrence and Compo (1998) provide detailed summaries of the use of a wavelet analysis. The latter study is particularly useful as it provides a method of calculating significance levels of the wavelet moduli and also gives access to modules for the calculation of statistics. Strictly, the heat content ocean layer, hc (y, t) 300m (J m−2 ), given by 𝜌Cp ∫0 𝛿T(y, z, t)dz, where 𝛿T(y, z, t) = T(y, z, t) − T(y, z, t) with T(y, z, t) representing the annual cycle of temperature at a point. “Canonical,” strictly a provision of canon law or a regulation decreed by a church council, refers to conformation to a general rule. In geophysics a canonical signal refers to specific pathways or signals. With respect to El Niño, a canonical signal would be anomalous warmer SSTs in the eastern Pacific Ocean, cooler in the west, persisting for many months commencing in the boreal spring ir early summer. Many of the observed features of the ENSO cycle appear in the theories described above. Wang and Weisberg (2000) and Wang (2001) developed the Unified Oscillator Theory within which, inside a range of parameters, each of the previous theories emerge as special cases.

VI See https://www.ecmwf.int/en/forecasts/

VII

VIII

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X

documentation-and-support/long-range/seasonalforecast-documentation/user-guide/seasonalforecasting-system for details on the seasonal forecasting system. Mark A. Cane is the G. Unger Vetlesen Professor of Earth and Climate Sciences at Columbia University and the Lamont Doherty Earth Observatory. His advisor was Prof. Jule Charney of MIT and he has conducted pioneering work on tropical oceanography, climate modeling, paleoclimate, impacts of climate on society, and the El Niño and the Southern Oscillation (ENSO) prediction. The Law was first expressed in the book by Hofstadter (1980), “Gödel, Escher, Bach: An Eternal Golden Braid,” which explores common threads in the lives of artist M. C. Escher, composer J. S. Bach, and logician K. F. Gödel. From this book emerges keen insights into geophysical predictability. Bayes Law or Theorem was probably developed by the English theologian and mathematician Reverend Thomas Bayes (1702–1761). The concept of probability had been introduced by the French mathematicians Fermat and Pascal in 1554. In 1705, Sir Edmund Halley (of comet and monsoon fame) produced the first actuary tables to determine the life expectancy (and hence the probability) of a person of a particular age living to a certain age (Halley 1693). From his “Table of Life” the costs of annuities could be determined for the first time. See contemporary discussions by Ciecka (2008) and Bacaër (2011). Bayes Law was published posthumously in 1763 as a generalized calculation of conditional probability and formed the basis of “Bayesian Inference.” There is some mystery about whether Bayes really developed the theorem. For the statistically minded, read Stigler (1983) who used Bayesian Statistics to determine the most probable discoverer of Bayes theorem! It is quite a “thriller.” Palmer (2016) provides a detailed history of the use of ensemble techniques in numerical weather and

14.3 Indian Ocean Interannual Oscillations

climate models. Ensemble prediction schemes depend on “the Monte Carlo method”, which is a class of statistical techniques that rely on random sampling. They are particularly useful in systems with a large number of degrees of freedom. XI Webster et al. (1999), Saji et al. (1999), and Yu and Rienecker (1999, 2000). The first paper referred to

the phenomena as the Indian Ocean Zonal Mode (IOZM). The second paper called it the Indian Ocean Dipole (IOD). We adopt the IOD terminology. The latter two studies lean toward a more important role for El Niño than the first two studies.

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15 Intraseasonal Variability in the Tropical Atmosphere 15.1 Introduction In Chapter 1, Section 1.1, we commented on a number of key discoveries that led to an expansion of our knowledge across a broad range of tropical phenomena. One such phenomenon is a low-frequency nearequatorial large-scale mode with an eastward phase speed of about 5 m s−1 . This discovery arose from a careful analysis of what was then a relatively sparse surface and upper air data across the tropics.I The period of the oscillation varied within the 20–60 day range, occupying the entire depth of the troposphere with a wavelength of ∼3–5 × 103 km. The relative variance in this spectral band, compared to other bands, is illustrated in Figure 1.18 and also Figure 8.3. The phenomenon has become known as the Madden-Julian Oscillation (MJO) after its discoverers. In particular, its discovery was important as it filled the “spectral gap” between higher-frequency synoptic convective events (e.g. easterly waves) and longer period interannual ENSO variability, thus offering, perhaps, the prospect for weather prediction on subseasonal time scales. Even though variance in the 20–60 day period band is generically referred to as the MJO, there are a variety of phenomena that exist with similar frequencies. These modes also propagate along the equator but have somewhat different characteristics with different geographical regions of influence. Furthermore, as discussed in Chapters 10 and 11, the equatorial Pacific possesses an extremely noisy environment through which low-frequency equatorial modes have to propagate. Such influences may be especially strong in the regions of the westerly ducts where breaking Rossby waves infiltrate the tropics from higher latitudes quite regularly. Thus, empirical and numerical models alike have to contend with higher-frequency and high-amplitude “noise” that may decrease the degree of predictability. Figure 1.15 displays the seasonal 20–60 day OLR variance for both JJA and DJF. Both seasons have a number of common properties although the background surface wind fields are quite different (Figure

1.7). Although the low-frequency mode tends to move eastward in each season it tends to propagate northward in the boreal summer as well. These northward propagations create “active” (enhanced rainfall) and “break” (reduced rainfall) periods in the South Asian monsoon.1 Here we will concentrate on the boreal winter intraseasonal variance (ISV) when the time variance is spread across the Indo-Pacific warm pool and the southern portions of the Indian and the Pacific Oceans, especially the SPCZ discussed Section 13.4 .

15.2 Structure of the Austral Summer ISV 15.2.1

Early Constructions

Figure 1.18a displays Madden and Julian’s (1972) original schematic of the intraseasonal oscillation, based on the variance spectra of surface pressure at stations located across the tropics. The sparse network of stations was sufficient to allow the construction of time series of wind profiles as well as other meteorological variables. The original spectral analysis of surface pressure for this period is reproduced in Figure 15.1a. The yellow bars highlight the mean sea level pressure variance (MSLP: hPa2 day) in the 40–50 day period band. Madden and Julian found statistically significant spectra from the central-eastern Indian Ocean, across Indonesia and the eastern Pacific by matching the power distributions of spectra shown in Figure 1.15 (panel iii). In all four seasons there are extensive areas of subseasonal variability. During JJA and DJF maximum variance occurs in the Indio-Pacific warm pool and within the monsoon regions. Overall maximum 20–60 day variability occurs in the same location as higher-frequency variability (panels i and ii). Only equatorial Africa appears not to follow this rule, as there is little 20–60 day variability. Most of the variance appears restricted to the synoptic 1 We will discuss these boreal summer intraseasonal variations later in Section 16.5 in some detail. There they will be referred to as Monsoon Intraseasonal Oscillations or MISOs.

Dynamics of the Tropical Atmosphere and Oceans, First Edition. Peter J. Webster. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

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15 Intraseasonal Variability in the Tropical Atmosphere

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2–10 day band. Similar associations appear to exist in DJF and MAM. The main locus of 20–60 day variability occurs in the Southern near-equatorial regions. There is also a maximum in South Africa, at least in DJF. Madden and Julian also found strong coherence across the Indian Ocean and Pacific Ocean between the different time series in the 40–50 day period band.II This allowed relative phases of the oscillation to be calculated. Figure 15.1b (from Madden and Julian 1994) provides a plot of the mean phase angle (degree) of the oscillation relative to Canton Island (1∘ 46′ S, 169∘ 09′ W: now Kanton). The statistics suggest a coherent propagation of an MSLP pulse along the equator with a phase speed out of the East Indian Ocean of about 20 m s−1 and 30 m s−1 in the eastern Pacific. This is far more rapid than the eastward propagation of convective anomalies suggested in Figure 1.18. There is also a slow

meridional extension to the north and south of the equator in both hemispheres. 15.2.2

More Recent Analyses

A more recent version of Madden and Julian’s diagram appears in Figure 8.3 showing a composite of MSLP, convection (OLR), and 200 hPa geopotential throughout a composite MJO cycle. Deep convection reaches the greatest amplitude in the equatorial Indian Ocean (day “0” of the composite) and rapidly loses power to the east of the dateline. Sea level pressure variability on ISV time scales continues eastward but becomes increasingly less correlated with convection. In Figure 15.1b the propagation of the signal in the surface pressure wave is very rapid along the equator, well ahead of the convective signal. In Section 8.5

15.2 Structure of the Austral Summer ISV

we discussed how convective processes, specifically convective destabilization and dissipative effects, might produce such a slower mode. Madden and Julian pointed out that there was little amplitude in the ISV spectral band over Africa and concluded that the phenomenon formed over the equatorial Indian Ocean. It is worth noting that Figure 8.3 also indicates the lack of an MSLP precursor over Africa prior to the development of the Indian Ocean convection. Furthermore, at 200 hPa, up-stream upper-tropospheric precursors are also absent. This raises the question of what in situ or perhaps remote laterally located physical processes exist in the Indian Ocean region (or surrounds) that produces the formation of an intraseasonal event. It is tempting to conclude from Figure 1.18a and b that the MJO is a two-dimensional entity in the zonal-height plane propagating along the equator as a longitudinal waveform. However, Figure 8.3a suggests that the zonal average anomalous surface pressure throughout the composite cycle is non-zero, suggesting that the MJO includes a net exchange of mass between the tropics and higher latitudes. This is reemphasized in Figure 15.2, where the zonally averaged MSLP is plotted as a function of the composite time lag. The variability is roughly ±0.5 hPa. In fact, Madden and Julian (1972, p. 1119) had noted that … The fact that (surface pressure) anomalies do not always add to zero along the equator indicates that mass flows in and out of the equatorial belt with the oscillation…. Composite zonally-averaged MSLP variation 40

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During the most convective period of the MJO (days −5 to + 10 of the composite), the zonally averaged MSLP is lower near the equator, which is consistent with a strengthening of the zonally averaged Hadley Circulation. If there were a mass flow in or out of the tropics representing a latitudinal redistribution of mass, as suggested above, it should be reflected in the total angular momentum of the planet on the time scale of the MJO. Madden and Julian (1994) did show that the total angular momentum varied on intraseasonal time scales, indicating that a net mass transfer between the tropics and the extratropics occurs throughout the MJO cycle. That is, there is an interaction between the large longitudinal form of the MJO and the zonally averaged flow. Finally, it interesting to look in greater detail at the structure of the overlapping modes mentioned previously. We noted earlier in Figure 1.20 that where variance in the ISV band was large, the amplitude of the biweekly, synoptic, and diurnal variability was also large. As mentioned earlier, McBride (1983) had noted that the seemingly complex conglomeration of near-equatorial convection possessed an organizational structure of eastward and westward propagating modes. Nakazawa (1988) also commented on an example of spectral overlap he observed in the Pacific within a propagating MJO envelope. The propagating MJO contained two additional higher-frequency convective components. He referred to these as “super cloud clusters” (SCCs). The large-scale MJO, with a longitudinal scale of about 9000 km, is shown outlined in longitude and time by the heavy black rectangle between 90∘ E and 180∘ E in Figure 15.3a propagating eastward at about 5 m s−1 . • Within this broad zone of convection are four convective entities labeled A–D, each moving eastward with the same phase speed as the ISV. These convective components have a much shorter longitudinal wavelength, perhaps 2–4000 km. • Within each of these super-clusters are counterpropagating entities, moving in an absolute sense to the east but to the west relative to the MJO. These westward moving cloud clusters, which may be easily identified in the high-resolution diagram of the ISV shown in Figure 12.3b, continually develop in the east section of the super-cluster and decay toward the west. Each of these cloud clusters lasts for about one to two days. Using data from TOGA COARE, Chen and Houze (1997) noted that the two-day cycle could be explained in terms of surface-cloud-radiation interaction, referred to as “diurnal dancing.” This conclusion rested on the observation that the initiation of the

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“counter-propagating” convection commenced in the late afternoon, reaching a maximum before dawn and extending into the next day. The cloud-cluster was thus initiated and driven by the diurnal variation of radiative forcing described in Section 8.2. It can be argued that these two-day waves are indeed manifestations of classical wave theory such as presented in Chapter 6. Haertel and Kiladis (2004) argue quite convincingly that these 2 day waves are actually Matsuno’s n = 1 westward inertio-gravity waves, which happen to phase lock with the diurnal cycle at times.

15.3 Variability of Austral Summer ISVs The ISV composites, such as those displayed in Figures 1.18 and 8.3, are often referred to as the “canonical”

MJO, yet an important fraction of the total ISV variance of tropical convection over the Indian and West Pacific Oceans deviate substantially from this structure, exhibiting large differences from event to event. The largest class fits the description of the canonical MJO. Often if ISV variability does not fit this canonical form, it is ignored. Although we do not challenge the generic MJO definition, only about half of the ISV systems adhere to this definition. We feel that defining ISV strictly to the canonical MJO mold reduces focus and hinders a broader phenomenological understanding of the ISV, thus limiting the potential predictability of a significant fraction of tropical ISV variability. Hirata et al. (2013) adopted a statistical definition for tropical ISV events that closely follows the arguments of Kessler (2001), consistently identifying large-scale slow eastward propagating ISV patterns over the warm

15.3 Variability of Austral Summer ISVs

pools. Band-passed (20–90 days) OLR data averaged between 5∘ N and 5∘ S during the austral summer season (October–March, from 1979 to 2011) were decomposed to simple empirical orthogonal functions (EOFs).III (Figure 15.4a). The first two EOF modes were found to be significantly correlated with each other, with a maximum lag correlation of 0.65 when EOF1 principal component time series (PC1) led the EOF2 series (PC2) by approximately 11 days. Together, the two EOFs explain most of the ISV over the Indian Ocean, the Maritime Continent, and the West Pacific Ocean. These two modes were then used to describe ISV variations of large-scale convection and the ensuing propagation over the tropics of the Eastern Hemisphere. A total of 95 ISV events were identified. Three sets of distinct large-scale ISV convective events were defined as combinations of patterns associated with EOF1 and EOF2 (Figure 15.4b and c). Figure 15.4d to f provide composite Hovemöller diagrams of the three modes averaged between 5∘ N and 5∘ S, respectively. Figure 15.5a and b provide spatial distributions of OLR and sea-surface temperature (SST) and surface pressure and lower tropospheric wind fields,

10

Patterns of Intraseasonal Variability (ISV) in the Indo-Pacific Ocean 30°N (b) (a)

30 20

0°

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0 (c)

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respectively, at day −14 and day 0 of the composite. These figures emphasize the character of the development phase of the three modes. (i) First mode (MJO): The first mode is the “canonical” MJO, containing the largest fraction of planetaryscale ISV variance. The formation of convection at day 0 in the Indian Ocean (70–90∘ E) is preceded by a prolonged period of anomalously warm SST across the equatorial Indian Ocean. Figure 15.5a(i) indicates that the warming is largely contained between 5∘ N and 10∘ S in the early part of the composite (day −14) and is associated with above-average surface pressure anomalies. The 850 hPa velocity shows strong divergence out of the warming region at this time (Figure 15.5b(i)). At this stage, cool SST anomalies exist across the eastern Maritime Continent and the western Pacific warm pool. With the development of convection, SSTs fall dramatically across the East Indian Ocean and remain so for the next 20–30 days. Surface pressure also has fallen over a broad region. The largest fall coincides with the location of the maximum convection but it extends northward over an extensive area including South Asia and into the southeast Indian Ocean (Figure 15.5a(i), day 0). Strong

–0.4 90°E

120°E 150°E

60°E

90°E

120°E 150°E

60°E

90°E

120°E 150°E

longitude Figure 15.4 Patterns of ISV variability: (a) EOF modes for the austral summer averaged between 5∘ N and 5∘ S of intraseasonal OLR (W m−2 ). (b) Composite day 0 for PC1 events with a minimum below −1 standard deviation as per (a). (c) Composite day 0 for PC2 events with a minimum below −1 standard deviation as per (a). Longitude–time diagrams of composite life cycles for (d) MJO, (e) eastward decaying (ED) mode, and (f ) eastward intensifying (EI) events, respectively. Shading and contours represent band-passed SST (∘ C) and OLR (W m−2 ) anomalies, respectively, averaged between 5∘ N and 10∘ S to highlight stronger SST anomalies south of the equator in the surroundings of the maritime continent. Source: From Hirata et al. (2013).

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15 Intraseasonal Variability in the Tropical Atmosphere

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(b) Surface pressure (Pa) and 850 hPa velocity anomaly Figure 15.5 (a) Composite snapshots at days −14 and 0 of the three identified modes of ISV: (i) MJO (left column), (ii) ED (middle), and (iii) EI (right). Shading represents significant SST anomalies relative to the lower bar. Negative OLR anomalies (5 W m−2 intervals) appear as magenta contours, positive anomalies in green. Source: Adapted from Hirata et al. (2013). (b) Same as (a) except that background shading represents surface pressure anomalies (blue negative, yellow positive) relative to the low bar. Vectors represent the anomalous 850 hPa wind field. Maximum magnitude of 5 m s−1 on MJO, day 0. Source: Adapted from Hirata et al. (2013).

convergence occurs into the convective zone, especially from the western Pacific Ocean (Figure 15.5a(i),day 0). Figure 15.4d and Figure 15.5a(i) show the characteristic eastward propagation of convection to the east following development. This first mode accounts for about 60% of the total ISV variance. (ii) Second mode (ED): The second set of ISV events is identified whenever PC1 presented a minimum below

1 standard deviation (this minimum is referred to as composite day 0 for this cycle), but a subsequent minimum below 1 standard deviation is not observed on PC2 within a period of 25 days. Events matching this criterion are referred to as eastward decaying (ED) events due to the observed weakening of convective activity over the Maritime Continent (Figure 15.4e). The anomalous SST in the eastern Indian Ocean is

15.4 Mechanisms

weaker than the first mode, as is the OLR anomaly. Having reached a state of maximum convection over the eastern Indian Ocean and Indonesia, only a weak remnant propagates to the east. The ED explains about 25% of the variance. (iii) Third mode (EI): The third category comprises ISV events in which PC2 falls below 1 standard deviation (the composite day 0 for this cycle), but it is not preceded by a minimum on PC1 below 1 standard deviation. This last category describes weak (even absent) convection over the tropical eastern Indian Ocean that later strengthens over the Maritime Continent (Figure 15.4f). These are referred to as eastward intensifying (EI) events. The EI mode explains about 15% of the variance. There are similarities and differences among these three modes of ISV: all three occupy the entire depth of the troposphere, but their propagation characteristics differ, as do their intensity and their domain. We now attempt to determine if there exists a common physical basis in their formation.

15.4 Mechanisms The existence of strong ISV in the 20–60 day period band has been documented for nearly half a century but is now categorized with the identification of new IS modes, as outlined above. Yet there are many aspects of the phenomenon (or phenomena) that are still uncertain. Paramount among this list is a credible theory for the formation of an IS event or the common genesis location in the equatorial Indian Ocean. In Chapter 8, we addressed the slower eastward propagation speeds of convective modes compared to “dry modes.” A number of possibilities were considered including the role of convective dissipation, changes in the vertical stability associated with convection itself and the role of ocean–atmosphere interaction. These ideas provide potential mechanisms for the slow eastward propagation of an established mode but each depends on the preexistence of a large-scale equatorial mode. In the following paragraphs we will pose mechanisms that may initiate these large-scale disturbances. 15.4.1 A Local Instability Mechanism for the Initiation of an ISV Event The TOGA COARE2 data collected over a four-month period provided an unprecedented view of the structure of the western Pacific warm pool from the top of the 2 The TOGA COARE was briefly described in Section 1.1. For more details see Webster and Lukas (1993) and preliminary results (Godfrey et al. 1998).

atmosphere through the upper layers of the ocean. During TOGA COARE there were two major ISV events. Of particular interest was that, prior to the development of an eastward propagating event, the SST reached a maximum over a broad region of the Indo-Pacific warm pool. Associated with the SST warming were changes in the vertical stability of the atmosphere, the vertical distribution specific humidity and the temperature of the atmospheric column. Figure 15.6a provides a record of SST at the center of the Intensive Flux Array (IFA) measured by the improved meteorological (IMET) buoyIV during TOGA COARE. Two surface warming–cooling episodes were observed during the intensive observing period (IOP), with each cycle having a period within the 20–40 day range. A maximum in SST preceded the onset of wide-scale deep convection by one to two weeks (Figure 15.6e). The first warming phase occurred during the latter part of November and early December when the SSTs reached a maximum in excess of 30 ∘ C, which coincided with a period of light winds (Figure 15.6b). This warm phase was followed by a period of marked cooling for the remainder of December that extended into early January. During this cooling period, the winds became stronger with widespread precipitation. A second cycle of warming persisted throughout January and into February. A characteristic of the warming periods and the low surface winds is the large diurnal variation of SST with an amplitude ∼1−2 ∘ C. These variations are most pronounced in the high-insolation, low-wind speed phases of the oscillation (Figure 15.6b). Similar diurnal variations of SST were noted in WEPOCS (Figure 2.22), suggesting that they are a characteristic of the western Pacific Ocean warm pool and probably the equatorial Indian Ocean as well (Figure 2.27) during quiescent periods. The upper tropospheric temperature (Figure 15.6c) also varied during the IOP with the same period as the surface oscillations. Figure 15.6d shows the difference between the temperature anomalies at 200 hPa and the SST providing a crude estimate of vertical stability. During quiescent periods, the difference is negative, indicating a growing vertical instability. During periods of low SST, when the region is disturbed and highly convective, the vertical gradient relaxes and the system tends toward increased stability. Thus, the warming phase of SST is associated with a destabilization of the atmospheric column. A stabilization of the atmospheric column, on the other hand, occurs with ocean cooling. The evolution of the water vapor (Figure 15.6f and g) indicates a direct response of upper- and lower-tropospheric water vapor to the cycle of ISV convection. The lower troposphere generally moistens

351

15 Intraseasonal Variability in the Tropical Atmosphere

Intraseasonal variability during TOGA COARE

Dec 1

Jan1/1993

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Mar 1

s–1)

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°C

352

1.2 0.8 0.4 “stable” 0 –0.4 –0.8 –1.2 “unstable” Nov 1/1992 Dec 1

Jan1/1993

Feb 1

Mar 1

Figure 15.6 Time series of (a) hourly and 24 hours average SST from surface buoy (IMET) centered in the intensive flux array (IFA) of TOGA COARE; (b) mean 850 and 200-hPa zonal wind speeds over the IFA; (c) deviation of the 200-hPa atmospheric temperature from the average during the TOGA COARE period; (d) difference between the 200-hPa temperature anomaly and the SST anomaly (∘ C), which acts as a simple proxy for convective instability; (e) average IFA precipitation from a budget analysis of the amount of deep convection, and the amounts of high cirrus and cirrostratus over the IFA, as determined by ISCCP; (f ) vertically integrated water vapor amount from the surface to 3 km, obtained from the IFA sounding data; and (g) MLS 200-hPa water vapor mixing ratios provided by the Microwave Limb Sounder (MLS) instrument flown on the Upper-Air Research Satellite (UARS). Source: Adapted from Stephens et al. (2004) and Johnson and Ciesielski (2000).

throughout most of November and into early December during the quiescent period, mainly as a result of evaporation of shallow cumulus clouds that develop during that period.3 At the same time, there is a drying of the upper troposphere occurring in a generally subsiding environment that removes moist air from the upper levels at a rate determined by the large-scale radiative cooling. This water vapor cycle is reversed with the outbreak of widespread convection similar to that which occurred during December. Water vapor condenses into cloud water and is removed by precipitation. At the same time, moisture is lofted by deep clouds to upper levels. The lower levels of the atmosphere dry throughout this stage (Figure 15.6f) and the upper troposphere moistens by the sublimation and evaporation of the ice crystals and droplets detrained from the convective 3 Inferred from radar observations reported in Johnson et al. (1999).

systems in the form of widespread upper-level anvil clouds. TOGA COARE was designed to address one of the outstanding problems in tropical meteorology and climate at the time: an accurate determination of ocean–atmosphere fluxes over the tropical warm pools. These fluxes were needed for coupled ocean-atmosphere modeling and were critical for understanding the maintenance of the tropical warm pools. Figure 15.7 shows time series of surface fluxes averaged over the inner dense network of observations of TOGA COARE. This was referred to as the intensive flux array (IFA). The largest variations in surface fluxes occurred in both the fluxes of latent heat (Figure 15.7a) and solar radiation (Figure 15.7b). The freshening of the low-level winds over the IFA during late December (Figure 15.6b) produced a factor of three enhancements (or an

15.4 Mechanisms

TOGA COARE IFA surface fluxes (W m–2) (a) Latent and sensible surface heat flux

W m–2

180 120

latent sensible

60 0 Nov 1/1992

Dec 1

Jan 1/1993

Feb 1

Mar 1

(b) Downward shortwave surface flux 300 225 W m–2

Figure 15.7 Time series of (a) the IFA mean surface sensible and latent heat, (b) the incoming daily averaged surface solar radiation, (c) the daily averaged net surface longwave flux, and (d) the net radiative heating of the entire atmospheric column from the surface to the tropopause. All units are W m−2 . Sensible and latent heat derived from measurements of four moored buoys in the IFA. Radiative fluxes obtained from the IMET mooring (solid line) and the Moana Wave (dashed line). Net heating of the atmospheric column is derived from budget studies using TOGA COARE data. Source: Lin and Johnson (1996) and Johnson and Ciesielski (2000).

150 75 0 Nov 1/1992

Dec 1

Jan 1/1993

Feb 1

Mar 1

Feb 1

Mar 1

Feb 1

Mar 1

(c) Net longwave surface flux

W m–2

0 –25 –50 –75 –100 Nov 1/1992

Dec 1

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W m–2

(d) Net surface radiation 100 50 0 –50 –100 –150 –200 Nov 1/1992

increase of about 100 W m−2 ) of latent heat flux into the atmosphere. The decrease of more than 100 W m−2 between late November and late December is related to increasing amounts of upper-level clouds over the IFA associated with the widespread convection during the period (Figure 15.6e). The variation of the net radiative heating (Figure 15.7d) has a magnitude of 100 W m−2 . The time series of the TOGA COARE IFA fluxes have common properties with other parts of the Indo-Pacific Warm pool. In fact, in Section 2.6.4 we compared TOGA COARE surface fluxes with those obtained in the Bay of Bengal some years later in the JASMINE campaign and noted the similarity of the ocean–atmospheric flux differences between disturbed and undisturbed conditions.

Dec 1

Jan 1/1993

The composites allow a basic understanding of the nature of the coupling between the ocean, moist atmospheric thermodynamics, radiation, and the dynamics of the tropical warm pool regions. The observations allowed Stephens et al. (2004) and Agudelo et al. (2006) to propose a base feedback that regulates, in unison, the SSTs and hydrological cycle in the Indo-Pacific warm pool. The feedback system may be divided into three phases, which are presented schematically in Figure 15.9.

15.4.1.1

Destabilization Phase

This phase occurs during the period of SST warming, such as during late November and early December in

353

15 Intraseasonal Variability in the Tropical Atmosphere

(a) Western Pacific Ocean (5°N–5°S, 165°E–155°E) 30.0 SST (°C)

29.9 29.8 29.7 29.6 29.5 –28

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50 UTH (%)

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190

+28

Figure 15.8 Composite of the upper-tropospheric relative humidity (UTH: dotted line), rainfall (red), OLR (blue), and surface wind speed (black) plotted relative to the period of the SST cycle (histogram). The time of maximum SST associated with the latter cycle is designated as day 0 and all observations are composited relative to that time. Data are shown for a region: (a) the tropical western Pacific region between 25∘ N–5∘ N and 165∘ E–155∘ E and (bottom) over the tropical Indian Ocean between 25∘ N–5∘ N and 85∘ E–95∘ E. The composites each represent more than 20 events for each region over the period of 1991–1995. Source: Adapted from Stephens et al. (2004).

the TOGA COARE region (Figure 15.6a) and in the broader composite analysis of Figure 15.8. The SST warming phase coincides with quiescent conditions of relatively calm winds, resulting in relatively low but

non-negligible evaporation and relatively clear skies. These are conditions that favor strong solar heating of the upper ocean exceeding 50 W m−2 and a large diurnal cycle (Figures 15.6a and b and 15.7a).

15.4 Mechanisms

15.4.1.2

Convective Phase

The convective phase follows the destabilization, at which time deep convection occurs over the large area where SST has increased. This phase is shown schematically in Figure 15.9b. The region of destabilization determines the scale of the forcing of the ensuing ISV event and the subsequent dynamic response discussed in Section 8.5. The convection is organized into large-scale cloud clusters or super clusters. The convective period is marked by a strengthening of the lower-level winds, a subsequent increase in surface evaporation and also the upward mixing of colder subsurface water, and a reduced solar heating of the upper ocean, each contributing to a decrease of SST. These effects lead to heavy precipitation, a drying of the lower layers of the atmosphere, and a substantial moistening of the upper troposphere. The envelope of deep convection eventually projects on to dynamic moist equatorially trapped Kelvin waves that move eastward. At the same time, a trailing pair of Rossby waves moves slowly toward the west like that portrayed in Figure 6.12b. 4 See Chapter 7.2, Curry and Webster (1999).

Stages in the development of an ISV (b) Convective

(c) Restoration

SST

(a) Destabilization

time

dry

moist

cool

warm slight drying

warm

weakning winds evaporation

strong winds

weak winds ΔSST > 0

slight warming

cool

dry

moist

ΔSST < 0

ΔSST ~ 0

Figure 15.9 A schematic of the thermodynamic feedback showing its three principal phases. The cycle of SST (top) is a simple depiction of the SST variation as a function of time and is used as a reference for how the atmospheric temperature (red profiles) and moisture (blue) change during these phases defined with respect to the SST cycle (middle). Changes of cloud conditions and associated wind field and heating changes are indicated (bottom). Source: From Stephens et al. (2004). Evolution of CAPE during an ISV in the Western Pacific warm pool 3.0 CAPE (103 J kg–1)

Simple arguments can explain the rise in SST. In Section 2.72 we calculated the rate of upper-ocean heating in both undisturbed and disturbed conditions. The results are summarized in Table 2.4. During undisturbed periods the mean net heat flux into the ocean was roughly 100 W m−2 in both the eastern Indian Ocean and the western Pacific. Using Eq. (2.35) and assuming that the heat flux is spread through the mixed layer of the warm pool (e.g. upper 50 m: see Figure 2.11), the upper ocean would heat at a rate of about 1−1.5 K month–1 , matching quite well the observed changes in the TOGA COARE SST. The combination of SST warming and enhanced radiative cooling of the upper atmosphere associated with the lack of high clouds favors a destabilization of the atmosphere, which is associated with a steady rise in convective available potential energy (CAPE): a measure of the amount of energy available for the upward acceleration of a parcel.4 At this stage of the cycle, the amount of shallow convection begins to increase, resulting in a steady low-level moistening (Figure 15.6f). This moistening also conditions the atmosphere for deep convection as, for a given temperature profile, CAPE will increase with an increase of low-level moisture. Figure 15.10 shows a time series of CAPE during the first ISV destabilization period of TOGA COARE. Starting in late November, CAPE increases through mid-December, at which stage convection occurs. The increase of SST occurs over a broad area (Figure 15.6a) so it can be presumed that the destabilization also occurs over a similarly large area.

2.5 2.0 1.5

destabilization

1.0 26/11/92

2/12/92

convective 9/12/92 date

16/12/92

24/12/92

Figure 15.10 Time series of CAPE during the destabilization and convective phases of the first ISV event during TOGA COARE IOP. Source: Based on Agudelo et al. (2006).

15.4.1.3

Restoration Phase

The restorative phase (Figure 15.9c) commences with the eastward propagation of the large-scale convective envelope, leaving in its wake high-level clouds associated with the residual upper-tropospheric moisture.

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15 Intraseasonal Variability in the Tropical Atmosphere

These upper-level clouds reduce the radiative cooling of the atmosphere through absorption of the infrared radiation emitted upward from the warmer atmosphere and surface below (Figure 15.7c). SSTs continue to cool, owing to the solar radiation reduction by the high clouds (Figure 15.7b). The combination of SST cooling and the heating of the upper atmosphere by high clouds promote an increase of atmospheric stability and a decrease of CAPE. As the winds weaken and the thick upper-level clouds dissipate to thinner cirrus clouds, evaporation from the ocean surface is reduced. At this stage solar heating of the upper-ocean mixed layer begins to increase, bringing the cooling of the SSTs to an end and returning to the warming phase with the return of a strong diurnal cycle in the SST, noted in Figure 15.6a. What determines the time scale of the three phases of the instability mechanism described above? It is difficult to quantify but it must be related to the time scale of the heating of the upper ocean during the convective phase and the subsequent build-up of CAPE in the atmospheric column and its subsequent release. We have already speculated that the scale of the region of SST warming determines the spatial scale of the forcing. Similarly, the time scale may be related to the period it takes for the CAPE to reach some critical value. For this to occur it would seem that the near-equatorial warm pool region remains undisturbed for a sufficiently long period for CAPE to increase over the broad region. In the following paragraphs we consider some dynamic processes that may promote longevity of the quiescent period. 15.4.2 The Indian Ocean as an ISV Generation Region It was pointed out earlier in this chapter that the convection associated with an ISV occurs in the Indian Ocean – in fact, only about 50% of the total number. The remainder originate over the Maritime Continent or the western Pacific Ocean. Here we concentrate on the Indian Ocean generation region. The instability described in the last section provides, at best, a framework for the initiation of an ISV event. It states that a quiescent period is needed to allow SST to warm and a build-up of CAPE. The Indian Ocean is well suited for the steady increase of SST as the climatological easterly surface winds promote Ekman transport toward the equator in both hemispheres, thus minimizing surface cooling by equatorial upwelling. In addition, Figure 1.7b and d indicate that the equatorial Indian Ocean is relatively climatologically quiescent at the surface during the equinoctial seasons SON and MAM. Even during the boreal winter (Figure 1.7c), the winds along the equator, whilst stronger, are to

the east, again minimizing equatorial upwelling. Thus, the climatological state near-equatorial Indian Ocean would allow SST to increase, conducive to the build-up of CAPE. Early thoughts centered on the MJO being a circumnavigating equatorial mode with the development of an intraseasonal event in the Indian Ocean associated with its passage. However, these ideas were dispelled to a large degree by Hendon and Salby (1994) and Yano et al. (2004), who came to describe the MJO as a “pulse-like” feature with a decorrelation time scaleV that was far less in period than a typical MJO cycle. More recently, these doubts were supported by observational and modeling studies by Zhao et al. (2013) and Li et al. (2015), producing similar evidence to that shown in Figure 8.4. There are currently two main directions in an effort to understand the initialization of an MJO. The first is understanding the meteorology of the equatorial Indian Ocean that would produce conditions conducive to convective formation. The second is the influence of remote events, such as by extratropical systems, that force conditions over the tropical Indian Ocean leading to the formation of convection (e.g., Mattthews and Kiladis, 1999, Ray and Zhang, 2010). 15.4.2.1

Feedbacks from Wave Dynamics

The predominant theory for the generation of an ISV event is the release of CAPE, as described in the instability theory of Section 15.4.1 above and the projection of this heating on to a Kelvin-Rossby wave doublet that propagates to the east. In this theory, the latter part of the doublet provides a subsiding wake and encourages the next warming period. The modes are moist with hydrological processes modifying the speed of propagation of the convective part of the disturbance, as discussed in Chapter 8. However, the instability mechanism described above requires an undisturbed or quiescent period in the Indian Ocean to build up sufficient CAPE during the destabilization phase. The cycles of the canonical MJO and easterly decaying (ED) ISV (see Figure 15.4) event begin over the Indian Ocean initiation region, where subsidence generates positive OLR anomalies and surface anticyclones straddling the equator as Rossby waves (Figure 15.11a). A dry Kelvin wave propagates eastward away from this region, damping convection to the east. These conditions allow sea surface warming by a combination of lower evaporative cooling and radiative heating related to minimum cloud cover. Convection is initiated by the warm SST anomalies, generating a Rossby–Kelvin wave response to heating to the west of the damped region of convection (Figure 15.11b). Once the convective phase is active, the entire system propagates to the east,

15.4 Mechanisms

(a) MJO (i) Convective destabilization

(ii) Convective-propagating

z

z N

N H

W

E H

H

L

W

E H

L

(b) EI (i) Convective destabilization

(ii) Convective-propagating

z

z N

W

E

L

W

E

L

Figure 15.11 Schematic diagram. (a) MJO: (i) convective destabilization period, (ii) convection and projection on to equatorial modes. (b) Same but for the EI mode. Horizontal dotted lines represent the equator at 850 hPa. Red and blue areas denote positive and negative SST anomalies. Dashed arrows represent subsidence and ascent between 850 and 200 hPa. Black arrows represent the direction of wind anomalies. Red arrows along the equator indicate 850 hPa winds associated with Kelvin waves. L(H) indicates low-level negative (positive) lower tropospheric pressure anomalies. Based on Hirata et al. (2013).

consistent with the excitation of a convective Kelvin wave and/or the propagation because of surface flux WISHE feedbacks, discussed in Section 8.5.3. Generally, it is assumed that there is a subsiding Rossby wave associated with the previous ISV that precedes the new convective phase. For the MJO, the subsidence is stronger and more persistent, resulting in warmer SST anomalies in the eastern Indian Ocean and over the Maritime Continent. These warmer SST anomalies combine with the Kelvin wave propagation to favor the eastward propagation of the entire system across Indonesia. This subsiding wave is weaker in ED events, resulting in lower or absent positive SST anomalies over the Maritime Continent and lack of eastward propagation. We have described above a self-regulation or self-induction mechanism for the maintenance of these convective cycles. Central to these ideas is the existence and longevity of a convective break. This break needs to be of sufficient duration to build up CAPE and for the development of a convective phase. However, it is unclear whether the sufficiently long break duration bears a relationship with a previous ISV. This raises an issue that the previous ISV convection may not a good predictor of the subsequent break phase, although the longevity of the break phase seems to be a good indicator of the following active phase. It is conceivable that the longevity of the break phase is driven by

stochastic processes. A source for such influences may be the extratropics. 15.4.2.2

Extratropical Influence

One of the conclusions from Chapters 10 and 11 is that there was a strong extratropical influence on the equatorial regions, especially in the region of the westerly ducts. There we were particularly concerned with the recursive influence of breaking Rossby waves forming and penetrating toward the equator through the westerly ducts. In essence, the extratropical influences create a rather “noisy” environment for the development of an ISV event. Ray and Zhang (2011) point out the importance of initial conditions including penetrating mid-latitude events in order to model ISV properly. Furthermore, they suggest that extratropical events, besides influencing the development and propagation of an ISV event, may play an important role in its initiation. Through the use of a zonal momentum budget for a number of case studies, it was suggested that equatorial advection by meridional winds is important in the initiation of an ISV event. Furthermore, this influence occurred in both the lower and upper tropospheres. What fraction of MJO events are affected by the extratropical influences described by Ray and Zhang? Matthews (2008) took advantage of the sporadic nature of the ISV by allowing a basic segregation of type. Two classes of MJO events were defined, primary

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and secondary, using data from September 1974 to September 2005. The former event is one in which there is no evidence of the influence of a preceding near-equatorial ISV event. The latter event appears to have an association with a preceding event. Of all events, 40% were found to be primary and 60% secondary. It was also observed that the successive events are systematically influenced by the extratropics, whereas no such extratropical influence was found for primary events. Consistent with Ray and Zhang, Matthews found that the source appeared to be the southern hemisphere (SH) extratropics. Why the extratropical influence appeared to be selective between primary and secondary events remains obscure. 15.4.2.3

Impact of Climatological State

In Chapter 7 we examined the role of the background basic state on the characteristics of propagating waves. In fact, the climatological Indian Ocean surface winds undergo significant, if spasmodic, biennial variability. In Section 14.3 we introduced the Indian Ocean Dipole (IOD), the quasi-biennial feature of the equatorial Indian Ocean that alters the near-equatorial longitudinal SST gradient. During positive IOD events (the SST increases to the west) the central and eastern Indian Ocean cools due to upwelling induced by the equatorial easterlies associated with the mode. This configuration results in lower tropospheric humidity anomalies across the eastern Indian Ocean. During a negative IOD event (warm SST in the east, cool in the west) anomalous surface westerlies prevail and positive lower tropospheric humidity fields exist across the eastern Indian Ocean. In terms of the instability hypothesis described above,

during a positive IOD event CAPE will have a greater difficulty increasing across the entire equatorial Indian Ocean. On the other hand, a negative IOD event would be conducive to the broad-scale development of CAPE. There is some supporting evidence of the influence of the phase of the IOD and ISV generation.5 The eastern Indian Ocean and Maritime Continent regions appear less favorable for ISV development during a positive IOD than during weak or negative IOD episodes. Overall, it appears that there is a slightly stronger than normal MJO propagation through the region during negative IOD and the relatively weak propagation during positive IOD. That is, the subseasonal variability is reduced during a positive IOD event (Kug et al. 2009).

15.5 Conclusions In summary, the results presented in this chapter indicate that the full nature of ISV physics is not fully understood. Within the tropics there are different forms of ISV, some appearing to grow in situ while others are influenced by preceding events. Some forms appear more likely than others to be influenced by extratropical events, others less so. It should also be noted that the background basic state both in the tropics and extratropics change with the phase of ENSO (and the IOD), as discussed in Section 14.2.3, so that the location of instabilities due to the tropical thermodynamic state as well as the regions influenced by extratropical forcing will change from epoch to epoch and within any one year. Determining when and where these various influences occur is still a major challenge.

Notes I Madden and Julian (1971, 1972). Madden and Julian

(1994) compiled a review of low-frequency variability in the tropics. A historical perspective is provided by Madden and Julian (2012). We also note the earlier work of Xie et al. (1963), that remianed relatively unknown until the recent paper by Li et al. (2018). Xie et al.’s work did not appear in the comprehensive reviews of Zhang (2005) and Lau and Waliser (2012), which we recommend for their thoroughness. We are also of the belief that the Xie et al. (1963) paper does not distract from the substantial study of Madden and Julian (1971). II In time series analysis “coherence” is used to describe the strength of association between two time series 5 Wilson E. A, et al. (2013), Shinoda and Han (2005), and Kug et al. (2009).

where the possible dependence between the two series is not limited to simultaneous values but may include leading or lagged relationships. Overall, coherence reflects the degree to which the data and information from a single time series can be related with time series at another location. For example, a sine wave moving through a medium (e.g. Figure 4.1a) would be such that at any two points there would be unity correlation separated by some lag, depending on the phase speed of the wave. The time series taken at two points would be fully coherent. It is a powerful tool in data analysis. See Everitt and Skrondal (2010).

15.5 Conclusions

III In climate studies, Empirical Orthogonal Function

analysis is used to study possible spatial modes of variability, such as those associated with ISVs or ENSO, and how their amplitudes wax and wane in time. The technique may be powerful but it also has its weaknesses. Mathematically, each EOF is orthogonal and hence independent. However, physical independence of each EOF is not guaranteed. A brief introduction into EOFs may be found at https://climatedataguide.ucar.edu/climatedata-tools-and-analysis/empirical-orthogonalfunction-eof-analysis-and-rotated-eof-analysis. IV The IMET buoy was designed to make continual measurements of meteorological variable, including

wind velocity, barometric pressure, incoming solar and longwave radiation, air temperature, sea-surface temperature, humidity, and precipitation. In TOGA COARE, the IMET buoy was moored at location 1.76 ∘ S, 156 ∘ E at the center of the intensive flux array. Weller and Anderson (1996) provide a full description of the instrument package and the measurements taken during TOGA COARE. V A decorrelation time scale can be defined as the period over which a phenomenon becomes fully decorrelated. That is, the antecedent response becomes uncoupled statistically from an initial signal.

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16 Dynamics of the Large-Scale Monsoon Monsoons influence the lives of more humans on the planet than any other climate phenomenon (Ramage 1971, Webster et al. 1998). Monsoons force lives, customs, and economies to be divided into two distinct phases: the “wet” and the “dry.” The wet phase refers to the rainy season during which warm and moist winds blow from warm tropical oceans toward land areas. The dry phase is characterized by an outpouring of colder and drier air from the northern continents and, in the case of Australia, from the cool wintertime southern oceans. These distinct annual cycles occur over the subtropics of Asia, Australia, West Africa, and the Americas. In some locations (e.g. in the Asia–Australia sector), dry winter air crosses the equator, picking up moisture from the warm tropical oceans to become the wet monsoon of the summer continent. In this manner the “dry” of the winter monsoon is tied to the “wet” of the summer monsoon, and vice versa. Vagaries of the monsoon are consequential. A failure of summer monsoon rains to arrive on time with sufficient quantity may lead to famine. An overabundance of rainfall may bring floods. High-amplitude spatial and temporal variability occurs throughout the year, especially during summer. For very obvious reasons, the prediction of monsoon variability has been a preeminent objective for climatologists, meteorologists and humankind since the beginning of time. A monsoon is a complex system depending on interactions between the atmosphere, ocean, and the land surface. In this chapter we will explore the physics of the mean monsoon.

16.1 Overview In the broadest sense, the annual cycle of the monsoon is very predictable. One can forecast with some certainty that in the next year there will be a wet and a dry season. But each wet season is different in terms of the total amount of precipitation, when rainfall commences, when it ceases, and where it falls. Forecasting these variations is the challenge. Even when the monsoon

rains have commenced, there are periods of reduced or enhanced rainfall that sometimes last for weeks. During winter, surges of cold high latitude air penetrate the tropical landmasses, bringing periods of strong winds and often deadly cold spells. Agricultural practices in monsoon regions have long been tied to an assumption of a relatively regular annual cycle of monsoon activity. Whereas the regularity and repeatability of at least the gross features of the monsoon cycle would seem to be ideal for agricultural success, the expectation of an invariable annual cycle can be costly to agricultural yield. Thus, small variations in the timing and quantity of rainfall from one region to another can be associated with significant societal consequences. The complexity of the South Asian monsoon is described rather poignantly by J. A. Young (1987, p. 211) … physically, the monsoon system is a complex of seemingly disparate parts: two fluids, the mobile air and the changing ocean below; deserts and areas of torrential rain; the dramatic seasonal progression of winter to summer; winter and summer hemispheres linked by strong wind and ocean currents across the equator; mountain complexes that assist and inhibit rising air motion; water evaporation and condensation; clouds that alter most properties, from wind to radiation; cyclones that grow over the ocean and drive deadly surges…. Whereas Young was referring to South Asia, he is describing the complexity described of all of the monsoon systems displayed in Figure 1.13a to c. A conceptual organization of the monsoon is presented in the monsoon-centric model of Lau et al. (2000). It attempts to categorize both the temporal and spatial structure of the monsoon and the interplay between external forcing and internal dynamics. The model is comprised of four components that represent three time scales of variability plus a component of interaction with surrounding climate circulations.

Dynamics of the Tropical Atmosphere and Oceans, First Edition. Peter J. Webster. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

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16.1.1

Slow Component (Months to Years)

In Section 1.3.3, we described the basic structure of regional monsoon systems in terms of the distribution of precipitation rate and low-level winds. Here we consider year-to-year variability of the regional monsoons through an examination of regional mean rainfall. Figure 16.1 shows time series of the mean summer rainfall of North Australia and India using a compilation of data by Lavery et al. (1997) and the All India Rainfall Index (AIRI) developed initially by Mooley and Parthasarathy (1984). The AIRI is an area-weighted mean of summer monsoon rainfall, based on a homogeneous rainfall data set of 306 rain gauges over the subcontinent, although not including Bhutan, Pakistan, or Bangladesh. The AIRI area is shown in Figure 1.13a(ii). The North Australian data set, referring to meteorological stations equatorward of 26 ∘ S, was compiled in a similar fashion to the AIRI, with the identification of reliable observing stations going back to the early twentieth century (Figure 1.13a(i)). Distinct interannual variations appear in both the Indian and North Australian rainfall time series. The mean and standard deviations of the two series are 852 ± 84 mm for India and 735 ± 161 mm for North Australia. Together, the time series hint that positive extremes in the Indian monsoon rainfall are followed some months later by a positive North Australian monsoon. Conversely, a weak South Asian monsoon is followed by an anomalously weak Australian monsoon, suggesting a biennial variability in the overall Asian–Australian monsoon system (e.g., Meehl 1994a, Fasulko 2004). It is interesting to note that the standard deviation of the AIRI is relatively small, suggesting that on the scale of the Indian subcontinent a relative constancy of summer rainfall occurs from year to year. Thus, there is about a 68% probability of rainfall lying between 768 and 936 mm. Locally, however, the deviation is much greater, but the small deviation does raise the question of what maintains the relative constancy in the Indian monsoon rainfall. Figure 16.1b describes the interannual variability of the mean May to October summer rainfall for the West African monsoon system. Rainfall is shown for two regions: along the Guinea coast and in a band further to the north in the Sahel. These zones are marked in Figure 1.13c. The Guinea Coast rainfall possesses relatively little interannual variability with a relatively small standard deviation about a large mean similar in magnitude to the Indian rainfall. The mean Sahel rainfall has an annual mean about half of that of the Guinea coast, but it shows a very large interdecadal variability that was absent in the Guinea Coast time

series. Between 1920 and 1960 the mean rainfall was about 600 mm yr−1 , decreasing to less than 400 mm yr−1 in subsequent decades. In Chapter 12 we noted that the multidecadal drought across the Sahel drew international attention with the tentative conclusion that the reduction in precipitation was likely the result of teleconnection patterns emerging out of Atlantic Ocean oscillations rather than local surface feedbacks.1 However, in Section 13.2 we argued that the cross-equatorial sea surface temperature (SST) gradient was important in the location of near-equatorial precipitation. Consequently, variability of the SST in the Gulf of Guinea may also impact the rainfall of the Guinea coast and the Sahel.

16.1.2 Intermediate Component (Weeks to Months) Figure 1.15a and b (panels (ii)) indicated the existence of large amplitude variance in the intraseasonal 20–40 day band in both the South Asia in JJA and North Australia/Indonesia during DJF. The term monsoon intraseasonal oscillation (MISO) is often used to describe this variability. However, throughout the year there is very little variability on these time scales in the West African region. Most likely this absence is because intraseasonal variance (ISV) events tend to form in the tropical Indian Ocean and move eastward, as discussed in Chapter 15. In effect, the lack of variability on the intermediate time scale over Africa may be additional evidence that the Madden–Julian oscillation (MJO) is not a circumnavigating mode.

16.1.3

Faster Components (2–15 Days)

This component incorporates quasi-biweekly waves, synoptic scale variability, and monsoon depressions associated with instabilities of the macroscale monsoon circulation. Concentrated variance occurs over the South Asian and North Australian monsoon regions (Figure 1.15a and b, panels (iii)). Note that the scale magnitude of the variance is twice that of the other periodicities. It should also be mentioned that there is a high-frequency variance associated with the Asian winter monsoon. Given that these are associated with cold air outpourings from Siberia and Central Asia, they are not directly associated with convection and are absent from the outgoing longwave radiation (OLR) maps of Figure 1.16. In West Africa the OLR is dominated by variance in the 2–10 day period band. 1 E.g. Marshall et al. (2001) and Nicholson (2001, 2011).

16.1 Overview

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Figure 16.1 Time-sequence of mean annual precipitation for (a) India (solid line) represented by the “All-India” rainfall index (AIRI: Parthasarathy et al. 1992) and a North Australia Index north of 25∘ S (red dashed line, Lavery et al. 1997) and (b) West Africa for the Guinea Coast and the Sahel. In general, the All-Indian rainfall is less variable from year-to-year (mean 852 ± 84 mm) than the North Australian rainfall (735 ± 161 mm). The Guinea Coast shows little long-term interannual variability (1097 ± 104 mm), similar to the AIRI. On the other hand, the Sahel (465 ± 110 mm) shows a large interdecadal variability. Areas used for averages shown in Figure 1.13a(ii) and c(ii). Source: North Australian data courtesy of N. Nicholls (Monash University, Australia). African data courtesy of S. Nicholson and D. Klotter (Florida State University, US).

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16.1.4

Connective Components (All Timescales)

The Lau et al. model acknowledges that tropics and the extratropics are dynamically connected by either modulations of the Hadley cell or through wave trains such as described in Chapter 10. The connectivity also acknowledges that phenomena such as the El Niño-Southern Oscillation (ENSO) induce alterations of the Hadley and Walker circulations and hence the monsoon, and perhaps vice versa.

sortie into the Indian Ocean illustrated that direct ocean trade routes between Europe and the East were possible, thus significantly decreasing the costs of goods and changing forever the balance of Asian and European commerce.3 However, in the late 1970s, dhows, the traditional lateen-rigged sailing vessel of the North Indian Ocean, were still in use in great numbers at the time of the Global Atmospheric Research Program (GARP) Monsoon Experiment (MONEX). 16.2.1.1

Sir Edmund Halley’s Tropical Wind Climatology

Scientific investigations aimed at understanding the basic physics of a monsoon commenced over 300 years ago. When the Europeans finally rounded the Cape of Good Hope in 1497, four ships under the command of Portuguese Captain Vasco de Gama found a flourishing sea-borne trading system between India, Southeast Asia, the Middle East, and East Africa. Arab traders who used the seasonally reversing monsoon winds in the Indian Ocean to great effect, trading between Africa and South Asia during the summer monsoon season and the reverse during winter. They had done so for thousands of years. Goods transported by Arab dhows eventually found their way to Egypt, Greece, Rome, and Western Europe. In fact, Greek trading enclaves were established along the western Indian coast, probably following the invasion of India by Alexander the Great c. 330 BCE. Trading routes also served as conduits between east and west for Chinese trade as well. However, the final sections of the trade routes between the East and Europe were over land and the Portuguese

In the sixteenth century, European mariners became acutely aware of the seasonally reversing monsoon winds just as they became aware of the doldrums and the terrors of being becalmed.I Arriving in the Indian Ocean in the wrong season could be disastrous. Arriving in the correct season would lead to speedy passages and greater trading profits. Climatologies of surface winds were thus extremely valuable to the mariners of Europe, as they attempted to optimize trade in remote parts of the planet. Based on an exhaustive survey of British ship logs, Halley (1686)II compiled a near-global climatology of surface winds (Figure 16.2a). These maps provided a remarkably accurate depiction of surface winds in the tropical oceans and, to a large degree, have stood the test of time.III It is clear from Halley’s map that he was aware that the monsoon was an interhemispheric phenomenon, with low-level winds crossing the equator from the winter to the summer hemispheres. The great trade wind regimes of the three oceans basins are quite evident and depicted with unidirectional arrows, indicating that the winds are relatively constant in direction through summer and winter. These regimes are the northeast trades of the Atlantic Ocean and the western Pacific Ocean (marked “A”) and, in the southern hemisphere (SH), the Atlantic and Indian Ocean southeasterly trades (“B”). The trade winds of both hemispheres are shown correctly converging toward the equator. Halley identified regions of seasonally reversing winds and included them in his atlas, such as region “D” near western Africa and “E” over the northern and northeastern Indian Ocean. They are depicted by sets of vectors alternating in direction from one diagonal row to the next. As these are difficult to see in the original figure they are reproduced here for an area to the south of the Bay of Bengal (Figure 16.2b). Such areas are particularly evident in the North Indian Ocean, the

2 For a thorough account of the history of the monsoon we recommend Warren (1987) and Kutzbach (1987a and b). The first paper refers to ancient and medieval accounts describing the monsoon. The second paper discusses pre-twentieth century perspectives on monsoon physics.

3 Holland and England were especially active trading nations in the Indian Ocean in the sixteenth and seventeenth centuries. An excellent history of the spice trade is given by Giles Milton (2000) in “Nathaniel’s Nutmeg: How One Man’s Courage Changed the Course of History.”

16.2 Theories of the Monsoon and Its Variability In Section 10.1, we noted that an explanation of the westward and counter-rotation direction of the trade winds was a question that fascinated a generation of scientists. Halley thoughts were that the trades were caused by the “apparent” diurnal rotation of the sun around the planet, forcing a westward motion. Unfortunately, his theory was more of an explanation of thermal tides that constitute a much smaller percentage of the observed variance of the total wind field than the trade wind regime. However, he was much more successful in determining the fundamental physics of the monsoon. 16.2.1

Early Descriptions2

16.2 Theories of the Monsoon and Its Variability

(a) Halley’s Surface Wind Climatology (1686) A

(b) Monsoon annual cycle

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Figure 16.2 Halley’s surface wind climatology (Halley 1686) in the Indian and Atlantic oceans. The map accurately represents the constancy of the trade winds throughout the year in the North and South Atlantic Ocean (A: northeast trades, B: southeast trades) and the South Indian Ocean (B: southeast trades). It also depicts where the winds reverse seasonally off the coast of West Africa (D: African monsoon) and in the North Indian Ocean (E: Indian monsoon). Monsoon regimes are tinted green. Reversing wind regimes are associated with the annual migration of the oceanic ITCZ and are tinted pink. (b) Halley’s depiction of the reversing winds in the Indian Ocean Winter winds (gold) and summer winds (blue). It may be noticed that the South Atlantic Ocean is referred to as the Aethiopic Ocean or Ethiopean Ocean, which was its classical name. Source: From Halley (1686).

South China Sea, and equatorward of West Africa. Even to the north of Australia, Halley described winds as seasonally reversing. He also managed to define regions where the ITCZ moved latitudinally between summer and winter (“T”), resulting in a near-equatorial surface wind reversal. Again, these are shown as rows of alternately directed vectors confined within dashed lines. These are most notable in the northern Indian Ocean, the eastern Pacific, and the equatorial Atlantic. The painstaking exercise of compiling the observations from ships’ logs proved to be an extremely useful guide for mariners. 16.2.1.2

Halley’s Differential Buoyancy Hypothesis

The composite maritime observations allowed Halley to formulate a physically based theory of the seasonal reversibility of monsoon winds. As a starting point Halley noted that in the summer hemisphere in the vicinity of land masses, low-level winds tended to be onshore, whereas in the winter hemisphere they were generally offshore. Halley argued that differential heating between the land and ocean surface heating produced the seasonally reversing monsoon winds: … action of the Suns Beams upon the air and water … [that] … according to the Laws of Staticks, air which is less rarified or expanded by heat … must have a Motion towards those parts …, which are more rarefied … to bring it to an Equilibrium …. (Halley 1686) Essentially, Halley had described the impact of differential heating of air parcels over regions of differing temperature allowing the hypothesis that: … as the cold and dense Air, by reason of its greater Gravity, presses on the hot and rarified

air must ascend … and being ascended it must disperse to preserve the Equilibrium by a contrary current which must move from those parts where the greatest heat is: So that by a kind of Circulation, the North-East Trade below, will be attended by a South-Westerly above, and the South-Easterly with a North-West Wind above…. (Halley 1686) Although he did not incorporate the role of Earth’s rotation in his theory, Halley’s ideas of differential buoyancy of the lower atmosphere induced by surface heating gradients were prescient. Also, it should be noted that included within his theory is the concept that the monsoons must possess a return poleward flow aloft: … for reasons of equilibrium the North East trades below will be attended by South West trades above and South Easterly with North Westerlies above …. (Halley 1686) 16.2.1.3

Determining the Origin of Monsoon Flow

There were early questions of whether or not the monsoons were interhemispheric phenomena. Halley’s charts do suggest an interhemispheric origin of monsoons and it was also noted that the strength of the winds in the western equatorial Indian Ocean were much stronger than in the east. These observations indicated that the overall structure of the Indian Ocean was a basin-wide interhemispheric gyre, with the strongest cross-equatorial flow likely located in the west.4 Although Halley had surmised that there must be a higher-level return flow of the monsoon, there was some confusion about at what height this reversal occurred. Observations obtained from elevated stations in South 4 Blandford (1874) and Seewarte (1892).

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Asia suggested that during winter there was a shallow antimonsoon return flow (e.g. Blandford 1874). However, during the summer, it was also noted that this antimonsoon changed direction in opposition to the surface flow, with a depth three times larger than during winter. However, a full determination of the depth of monsoon flow would have to wait a further 75 years until the availability of upper air observations. To a significant degree, these pioneers of monsoon meteorology described the basic features of the broad-scale monsoon systems. They identified the locations of the monsoons and offered physical explanations of their climatological structure. However, for the traders, there appeared to be little thought about monsoon variability or whether the next monsoon would be weaker of stronger than the last, which was of great importance to those who had to cope with the vagaries of the monsoon. In the late eighteenth century meteorologists did begin to look for processes that would indicate whether the subsequent monsoon would be strong or weak. The first suggestion of external influences was that anomalously extensive springtime snows over Eurasia would cause a delay in the onset of the monsoon (Blandford 1884). The physical basis for this idea was that late snowfalls would limit the heating of the continental surface so that, in essence, Halley’s “differential buoyancy” would be reduced and the subsequent monsoon would arrive late and be weaker. After some initial success, Blandford’s forecasts offered little predictability. Later empirical studies5 using extensive satellite data have found only a weak relationship between springtime snow coverage and a rather non-stationary connection between snow extent and subsequent Indian rainfall. 16.2.2 Attempts to Determine Remote Influences on the Monsoon The early decades of the twentieth century saw the first attempts at relating regional climate to remote, slowly evolving global scale circulations. These were initiated by scientists such as the British statistician and physicist Sir Gilbert Walker, who hypothesized that remote influences were being brought to bear on the Indian monsoon rainfall through the near-global large-scale oscillation of pressure patterns such as the southern oscillation (SO).6 Eventually, these influences would be recognized, first by Troup (1965) and Bjerknes (1969), as part of the coupled ocean–atmosphere ENSO, discussed at length in Chapter 12. 5 E.g. Hahn and Shukla (1976), Dickson (1984), and Bamzai and Shukla (1999). 6 Walker (1923, 1924b) and Walker and Bliss (1932) and many other reports.

16.2.2.1

Walker’s Surmise

In an attempt to anticipate the variability of monsoon rainfall, Walker (1924a,b) sought general relationships between the large-scale global oscillations, found previously in global surface pressure data records, and the variability of the Indian summer monsoon. The hypothesis was that Indian monsoon rainfall, occurring over a relatively small region, would be influenced by the larger scale influence of Walker’s SO. Specifically, it was hypothesized that monsoon rainfall would be inhibited in the descending air of one phase of the SO but enhanced during its opposite phase. The annual signal of the SO is plotted in Figure 1.19a together with a time series of the SO (Figure 1.19b) from 1876 to 2015. With the aid of such slowly varying circulations, there was the hope that the variability of the Indian monsoon could be “foreshadowed” through relationships with other aspects of the general circulation, which were termed: … strategic points of world weather . . . . Walker (1923b, p. 320) Walker’s slow rhythms, most notably the SO, suggested the possibility of forecasting climate variations, which, if successful, would be of singular importance to the agrarian society of India. Yet the ability to predict the future course of these rhythms was severely hampered by vagaries of observed relationships. The process was also hampered by an inability to identify underlying physical processes that might have fostered a greater comprehension of the relationships. To make use of the observations of large-scale coherent rhythms for predicting the intensity of the subsequent monsoon, Walker would have had to satisfy four basic criteria:7 (i) The precursor circulation (here the SO) should possess a spatial scale, which encompasses the circulation or the phenomenon (the monsoon) that is to be predicted; (ii) The precursor should possess a time scale that is very much longer than that of the phenomenon being predicted in order to provide a sufficient lead time for the prediction to be useful and also to be significantly different from the time scale of the predictand (here monsoon rainfall) to allow separation of “cause and effect;” (iii) The precursor circulation should be the “active” circulation and the circulation to be forecasted should be “passive.” That is, an obvious cause and effect relationship should be apparent; and 7 Suggested by Webster and Yang (1992).

16.2 Theories of the Monsoon and Its Variability

(iv) The statistical relationships would need to be both stationary (not varying in time) and significant. As we will see, not all of these criteria are met by a hypothesis that states that the Southern Oscillation Index (SOI) (or its later incarnation: ENSO) forces variability in the South Asian monsoon. From the spatial and temporal scales of the SO, it would appear that the first two criteria are met. The pattern resembles a dipole, with a maximum spanning the Indian and the Pacific Oceans. The general 3–4 year period of the SO (or ENSO) is far greater than the intrinsic annual period of the monsoon. That is, the spatial and temporal scales of the SO appear to easily encompass the scales of the South Asian monsoon. 16.2.2.2 The Demise of the Walker Relationships: A Mid-century Conundrum

The third criterion proves much more difficult to satisfy. Normand (1953) suggested that the Indian summer rainfall, while weakly correlated with pressure variations some months earlier in locations as far away as South America, was more strongly correlated with subsequent events. Second, there was a general demise in the amplitude of AIRI variability during the 1920–1960 period (Figure 16.1a) and also of the SO. These occurrences led to a questioning of the original Walker hypothesis. Specifically: … To my mind, the most remarkable of Walker’s results was his discovery of the control that the Southern Oscillation seemingly exerted upon subsequent events and in particular of the fact that the index of the Southern Oscillation as a whole for the summer quarter June–August had a correlation coefficient of +0.8 with the same index for the following winter quarter though of only –0.2 with the previous winter quarter …. Normand (1953, p. 469) In fact, Normand reached a conclusion quite contrary to Walker regarding the relationship of the SO to the Indian summer monsoon: … It is quite in keeping with this (the correlations) that the Indian monsoon rainfall has its connections with later rather than earlier events.… Unfortunately for India, the Southern Oscillation in June–August, at the height of the monsoon, has many significant correlations with later events and relatively few with earlier events .... The Indian monsoon therefore stands out as an active, not a passive feature in world weather, more efficient as a broadcasting tool than an event to be forecast . . . .

On the whole, Walker’s worldwide survey ended offering promise for the prediction of events in other regions rather than in India.… Normand (1953, p. 469) There are numerous corollaries to Normand’s speculations. Contemporaneous relationships appear to exist between the AIRI and the SOI, but a prior knowledge of the SOI in the previous winter or spring does not seem to help in forecasting the strength of the subsequent summer monsoon. This is because of the lack of persistence of the SOI through the boreal spring, as discussed at length in Section 14.2.4a However, the persistence of the SOl for the six- to nine-month period following the boreal summer suggests that perhaps the austral summer monsoon and precipitation in Indonesia and North Australia are more highly predictable than over South Asia. Finally, the fourth criterion, the necessity for statistical stationarity of the SO time series and its relationships with other indices, is questionable. In the mid-twentieth century, examination of the Walker relationships revealed that the strong variance in both the Pacific SST record and the AIRI diminished. Walker’s correlations were reexamined by stratifying the deviations of surface pressure by season at the extremes of the Southern Oscillation from the long-term means. Planetary-scale regions of negative and positive sea level pressure are clearly evident in all seasons (Figure 14.3a to d). In the boreal summer the zero-pressure anomaly isopleth passes just east of the Indian subcontinent. That is, at the time of maximum precipitation in the annual cycle, India is located near a node in the SO pressure pattern and well away from the regions of anomaly extrema. While the seasonal SO diagrams provide an explanation for the contemporaneous correlations of Indian rainfall and the SOI, they present a consistent picture of the vagueness of the relationship and why forecasting variations of the monsoon using just the SOI is difficult. Small displacements of the node would place India in either a weak positive or weak negative pressure regime relative to the long-term summer average. Thus, even with a perfect forecast of the SOI, a discriminating foreshadowing of the summer monsoon over India may be rather difficult. 16.2.2.3 Series

Relationships Revisited Using a Longer Data

Data exists to describe both the Indian summer monsoon rainfall and the mid-Pacific SST from 1877 to the present. Table 16.1 summarizes the observed relationship between the gross magnitude of the Indian summer rainfall (i.e. the AIRI) and the phase of ENSO. Over the 123 year period, of the 53 below average rainfall seasons, about half occurred during an El Niño event and only

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Table 16.1 Relationship between the mean summer season rainfall over India. Rainfall

Below average

All India summer

North Australian summer

Total

El Niño

La Niña

53

24

2

Total

El Niño

La Niña

49

20

4

Above average

71

4

19

58

5

17

Deficient (1 SD)

18

0

7

17

2

5

A “deficient rainfall” season is defined as possessing a total of at least a standard deviation below the average while a “heavy rainfall” season has rainfall at least one standard deviation above average. Source: All-India rainfall 1871–1994: from Parthasarathy et al. (1992) and the North Australian summer rainfall 1886–1993: from Lavery et al. (1997).

two occurred during a La Niña. Of the 71 years of above average years, 19 were La Niña years and 4 El Niño years. Roughly the same proportionality exists for more extreme rainfall variations such as during a deficient rainfall summer (defined >1 standard deviation (SD) below the mean) and abundant rainfall summer (>1 SD above the mean). Thus, roughly half of the anomalous monsoons are associated contemporaneously with ENSO events. For completeness, the statistics for North Australian summer rainfall are also listed in Table 16.1. Like the Indian statistics, about half of the below and above average monsoons are associated with El Niño and La Niña, respectively. Although the relationships between monsoon rainfall and ENSO are not perfect by any means, they are sufficiently seductive to suggest that if the phase of ENSO can be forecast then at least the probability of an anomalous monsoon rainfall could be sensed. However, a number of issues suggest that we should delve a little more deeply: (i) As discussed above, forecasting the sign of the ENSO anomaly for the boreal summer early in the calendar year is a most difficult problem, largely because of the “springtime predictability barrier” (Section 14.2.6.2). Thus, it is difficult to know whether the Pacific will be in El Niño, La Niña, or a neutral state until the late boreal spring or early summer. (ii) The use of an ENSO–monsoon intensity relationship requires that the statistics are stationary over the entirety of the data record. (iii) Normand’s suggestion that the variability of the monsoon may lead ENSO must be considered. To address these questions we use a wavelet analysis of the Niño 3.4 SST (Figure 16.3a) and the AIRI (Figure 16.3b). Further, we examine relationships between the two time series by creating a cross-wavelet modulus (Figure 16.3c). A cross-wavelet analysis shows when certain periodicities are common to both time

series. The bold black contours enclose significance levels >95% level. The wavelet analyses used here consider periods longer than one year. Below each wavelet analysis in Figure 16.3 are time series of the Niño 3.4 SST and the AIRI and the percentage of the total variance in the two to eight year period band throughout the period of the analysis explained by the power at a particular time. In essence, these curves provide a running mean of the variance in the band. On the right-hand side of each wavelet analysis is a fast Fourier transform (FFT8 ) or periodogram, of the time series, of the entire data time series and the average wavelet modulus as a function of period. The FFT provides the total power in a particular frequency band averaged over the entire data record. However, as discussed in Section 14.2, there is extra information in a wavelet analysis in terms of an evolving spectra. Furthermore, an FFT of the entire data set will not give any information about the stationarity of the time series. With these thoughts in mind, the results of the analysis allow the following conclusions: (i) The Niño 3.4 SST possesses a broad spectral peak between three and five years (right-hand graph, Figure 16.3a) averaged over the entire data period. The peak is statistically significant at the 95% confidence level. The AIRI, on the other hand, has a broad peak in the two to three year period band (right-hand graph, Figure 16.3b). Averaged across the entire data period, it is only marginally statistically significant. The cross-spectra (Figure 16.3c) illustrates a common broad covariability between the AIRI and ENSO between two and five years. (ii) The time series of both the Pacific Ocean SST and the AIRI are not statistically stationary, whereby the statistical characteristics of both time series 8 An FFT is an algorithm that computes the discrete Fourier modes of a series. The analysis converts a signal from its spatial or temporal domain to a representation in the frequency domain and vice versa.

16.2 Theories of the Monsoon and Its Variability

2

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8 16

16 1900

1920

1940

1960

1980

2000

3

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1

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–1 1880

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16 32

(mm)

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8

40 0 –40 –80

1880

1880

1900

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16 1960

1960

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150 100 50 0 2000

2-8 yr Var.

Period (years)

(b) All-India rainfall 1

2 1 0 Avg Var

Period (years)

4

Period (years)

1

2

32

(°C)

(a) Niño-3 SST

95%

Period (years)

1

32

(c) (Niño-3) X (All-Indian Rainfall) Period (years)

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8

16

16

32 2-8 yr Var.

95%

6 4 2 0

1880

1900

1920

1940

1960

1980

2000

1880

1900

1920

1940

1960

1980

2000

2

1 0 Avg Var

Period (years)

1

1

32

Figure 16.3 Wavelet modulus analyses of climate time series of (a) Pacific Ocean sea surface temperature (SST) (1875–1992) in the Niño 3 region, (b) the all-India rainfall index (AIRI) (1875–1992), and (c) the cross-wavelet coherency modulus between the AIRI and the SST. The shaded regions denote statistical significance within the shaded area >90% significant. Innermost bold contour denotes >99% significance. The total and 2–8 year band variance are shown below each wavelet analysis as black and red time series. Contours indicate the percent of total variance at a particular frequency explained at a particular time in the data record. The dashed areas on the right and left of the moduli distributions indicate the limitations of the data to define variance given the length of the data set. On the right-hand side of each modulus are periodograms of the summed wavelet modulus. Source: Based on Torrence and Webster (1999).

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vary throughout the data record. There are at least three distinct regimes in the time series. A relatively large variance existed in both the AIRI and Niño-3.4 SST before 1920, correlating at levels near r = −0.8 compared to an overall average correlation of r = −0.64; these were the high correlations that excited Walker and his colleagues. Between 1920 and 1960, there is a second regime with little interannual variability in either time series and a correlation between the two time series has dropped to mean + 1 SD (U WS >0), the monsoon is called “strong,” while if U WS > mean − 1 SD (U WS −1 SD. South Asian strong monsoons 1985, 1990, 1999, 2000, 2001, 2005, 2010, 2011, and 2013. For North Australia/Indonesia strong monsoon years were 1981, 1984, 1986, 1987, 1991, 2001, 2007, 2008, and 2016. Here, for example, 2008 refers to November 2007–2008. Weak South Asian monsoons: 1979, 1983, 1992, 1997, 2002, and 2016. Weak North Australian/Indonesian monsoon years: 1979, 1982, 1983, 1989, 1990, 1992, 1994, 1995, 2002, and 2006. Source: Updated and adapted from Webster and Yang (1992).

implying a strengthening or a weakening of the Walker Circulation across the Pacific, thus producing increased longitudinal convergence in the western Pacific during a strong monsoon and less convergence during a weak monsoon. 16.2.3.2 Impact of an Anomalous Monsoon on the Indian Ocean SST

How do strong and weak monsoons impact the upper Indian Ocean? Figure 16.6 displays the anomalous 925 hPa vector winds and SSTs in the Indian Ocean region between the monsoon extremes. The east and west equatorial Indian Ocean are impacted differently. During strong monsoons there is an enhancement of the cross-equatorial winds off the Somalia coast and a corresponding increase of the coastal upwelling and a cooling of the Arabian Sea. In the east, increased

on-shore winds near Sumatra reduce upwelling and a corresponding positive SST anomaly is apparent. During weak phases of the monsoon, reduced upwelling occurs off the coast of Africa and enhanced upwelling off Sumatra. Thus, strong and weak monsoons produce a dipole in SST across the equatorial Indian Ocean similar to the Indian Ocean Dipole (IOD) patterns discussed in Section 14.3. 16.2.3.3 Indian Ocean SST Anomalies and Monsoon Precipitation

We noted earlier that most empirical monsoon forecast schemes have attempted to take advantage of an ENSO–monsoon rainfall relationship. A recent series of empirical studies have shown that the relationships between Indian Ocean SST and Indian rainfall are stronger than had been thought, with correlations

16.2 Theories of the Monsoon and Its Variability

Figure 16.6 Vector differences in the 925 hPa winds between strong and weak South Asian monsoons defined in Figure 16.5. Changes in SST between strong and weak South Asian monsoons are reflected in the colored contours (0.1 ∘ C intervals). In the Arabian Sea north of the equator, stronger winds occur along the East African coast that appear to be associated with enhanced upwelling during a strong monsoon. In the eastern equatorial Indian Ocean onshore winds toward Sumatra are conducive to reduced coastal upwelling and hence anomalous warming. Note the longitudinal dipole pattern in SST.

(Strong-Weak) δSST, δ925 hPa winds 30°N 20°N 10°N 0° 10°S 20°S 30°S 40°S 30°E

60°E

90°E

120°E

3 m s–1 –5

as high as +0.8 occurring between equatorial Indian Ocean SSTs in the winter prior to the monsoon wet season and Indian precipitation.12 A combined Indian Ocean SST index retained an overall correlation of 0.68 for the period 1945–1994, after the removal of ENSO influence. Between 1977 and 1995, the correlation rises to 0.84, again after the removal of ENSO influence. In addition, the so-called Indian Ocean Dipole (Section 14.3) appears to have strong coupled ocean–atmosphere signatures across the basin, especially in the equinoctial periods, exerting a significant influence on the autumnal solstitial rainfall over East Africa.13 The empirical relationships found between the Indian Ocean SST and monsoon variability are important because they suggest that other basin-wide relationships may also be relatively independent of ENSO, and thus inherent to the Indian Ocean–monsoon system. We have noted earlier in Figure 14.20b(ii) that there is SST persistence of >40% between the DJF and JJA Indian Ocean SSTs, in contrast to the almost zero persistence in the central and East Pacific. A similar persistence value also exists between JJA and DJF in Figure 14.20b and c. The relatively strong Indian Ocean persistence between DJF and JJA is in contrast to the Pacific Ocean, which has a persistence minimum across the boreal spring. 12 Sadhuram (1997), Harzallah and Sadourny (1997), and Clark et al. (2000). 13 See Webster et al. (1999), Saji et al. (1999), Yu and Rienecker (1999, 2000), and Clark et al. (2000).

16.2.3.4 Winds

–4

–3

–2

–1 0 1 δSST (x10 °C)

2

3

4

5

Annual Cycle of Anomalous 850 and 200 hPa

At what time in the annual cycle do anomalously strong and weak characteristics of the monsoon seasons emerge? To help answer this question, mean monthly circulation fields were composited for the weak and strong monsoon years. The upper and lower tropospheric zonal wind fields in the south Asian sector for the composite annual cycle of the strong and weak monsoons are shown in Figure 16.7. At the time when the anomalous nature of the monsoon is defined (July), both the low-level westerlies and the upper-level easterlies are considerably stronger for strong monsoon years than for weak years, consistent with Figure 16.5. It is striking that the anomalous signal of upper-level westerlies during the strong years extends back until the previous winter, with 5–6 m s−1 weaker westerlies during strong years. However, in the lower troposphere the difference between strong and weak years occurs only in the late spring and summer. Thus, there is a suggestion that the anomalies indicate external and broader-scale influences on the monsoon system. Generally, enhanced upper tropospheric winds will be accompanied by enhanced lower tropospheric flow of the opposite sign. However, this is clearly not the case prior to the strong monsoon, suggesting that the modulation of the upper troposphere probably results from remote influences. This is troublesome to understand if variations of the monsoon are assumed to be simply subject to ENSO variability. The ENSO cycle that is

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16 Dynamics of the Large-Scale Monsoon

16.3.1

Annual cycle of NIO 850 and 200 hPa between strong and weak South Asian monsoons 25

strong weak

20 Zonal wind speed (m s–1)

374

15 10

U850

5 0 −5

−10

U200

−15 −20 −25 −6 −5 −4 −3 −2 −1

July 0

1

2

3

4

5

Lag month Figure 16.7 A further depiction of the differences between South Asian strong and weak monsoons for the 850 and 200 hPa zonal wind components over the entire year. The anomalous monsoon is defined at month zero (July). The curves indicate a different circulation structure in the North Indian Ocean region prior to strong and weak summer monsoons up to two seasons ahead. Source: Updated following Webster and Yang (1992).

phase-locked with the annual cycle and an El Niño, for example, develops in the boreal spring. Figure 16.7 suggests that there are precursors to anomalous monsoons that develop before the growth of an ENSO event. Identifying the precursor event may suggest a pathway to predict the anomalous state of the incipient South Asian summer monsoon.

16.3 Macroscale Structure of the Summer Monsoon To make headway in predicting monsoon behavior, we now look at the very large-scale circulations occurring during both the NH summer and winter. It is convenient to consider the distribution of potential vorticity (PV) q on an isentropic surface (i.e. isentropic PV or Ertel PV) as discussed in Section 9.3.1 and defined in Eq. (9.14). The zonal distribution of isentropic surfaces averaged between 75∘ E and 85∘ E are plotted as a function of pressure in Figure 16.8 across the Himalayan–Tibetan Plateau (HTP) for the two solstitial months January and July. Isentropes greater than 300 K do not intercept the HTP. We choose the 370 K surface (approximately 250–150 hPa) upon which to calculate q.

Mean Seasonal PV Distributions

Figure 16.9a and b display the mean boreal winter and summer 370 K PV fields. During the northern winter, the PV distribution is nearly zonal with the exception of two major troughs, one over the eastern Pacific (marked “A”) and the other over the Atlantic (“B”). These correspond to regions of equatorial upper tropospheric westerly ducts (Chapter 9). These troughs are also regions within the upper troposphere that allow deep penetration of extratropical disturbances into the tropics.14 Likewise, the troughs are regions of equatorial wave emanation that allow tropical modes to influence the extratropics (Figure 7.22). In addition, there are three major PV ridges: one over North Australia (“C”), the second over Brazil (“D”), and also a strong SH trough (“E”) that was previously associated with the SPCZ (Section 13.4). During the boreal summer (Figure 16.9b), the PV contours in the SH become more zonal. The NH, however, is dominated by an extremely large-scale anticyclonic gyre (“F”). This isentropic anticyclone (here referred to as the Asian Monsoon Gyre: AMG) is centered near 75∘ E and 30∘ N and extends eastward to 140∘ E and westward to the prime meridian. It is arguably the largest asymmetric circulation feature on the planet and encompasses the HTP, outlined by the bold dashed contour, with westerlies to the north and easterlies on the southern side. The latter winds are associated with the Tropical Easterly Jet that extends across Africa, and is important in determining the climatological patterns of summer rainfall over West Africa and its seasonal and higher-frequency variability.15 To the east of the AMG is a deep and a high-amplitude ridge (“G”) that extends from the central Pacific Ocean southwest across the equatorial Indian Ocean and onwards across Africa. Over the Eastern Pacific and the Rockies, there is a strong ridge (“H”) but not a closed anticyclonic circulation such as that occurring over Asia. The central Pacific trough (“I”) is the TUTT, discussed in Chapter 10. It has been argued16 that the AMG is unstable and the source of extratropical disturbances propagating southward into the tropics and then westward, north of the equator and across the Indian Ocean. It is useful to compare the location of the ridges and anticyclones of the upper troposphere PV field between the boreal winter and summer. The latitudinal 14 E.g. Webster and Holton (1982), Tomas and Webster (1994), Ambrizzi et al. (1995), Kiladis (1998), Funatsu and Waugh (2008), and Ortega et al. (2017, 2018). 15 E.g. Nicholson and Flohn (1980), Webster (1987b), and Cook (1999). 16 Hsu and Plumb (2000), Popovic and Plumb (2001) and Liu et al. (2007).

20

20

15

10

10

0

–10

5 0

PVU

Height (km)

16.3 Macroscale Structure of the Summer Monsoon

–20 0°

20°N

40°N

0°

(a) January climatology

20°N

40°N

(b) July climatology

Figure 16.8 Climatology of potential temperature, 𝜃 (black contours: K), and potential vorticity, q (right-hand colored scale: PVU, 10−6 m−2 s−1 K kg−1 ) averaged from 75∘ E to 85∘ E for (a) January and (b) July. The bold dashed line marks the zero line of potential vorticity. The heavy green line corresponds to the 200 K isentrope and mostly follows the surface. A scale height of 8 km was assumed to compute the approximate height. The 370 K 𝜃-surface corresponds to about p = 200–150 hPa or z ∼ 18 km. Source: Data from the ERA-interim reanalysis (from Ortega et al. 2017).

Figure 16.9 Seasonal means of potential vorticity, PV, on the 370 K isentrope defined in Eq. (16.2) for (a) DJF and (b) JJA. The boundary between easterlies and westerlies is separated by the bold contour. In each panel, the Himalayan-Tibetan Plateau is shown as a bold dashed contour. Letters refer to locations mentioned in the text. Note the change of the PV gradient in the summer south of Asia and the strong trough to the east. Units: PVU (10−6 K m−2 kg−1 s−1 ). Source: From Ortega et al. (2017).

(a) DJF 370K PV climatology 40°N 20°N

B

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0°

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20°S

(b) JJA 370K PV climatology 40°N

G

E

F

20°N 0° 20°S 0°

60°E

120°E

180°E

120°W

60°W PVU

–8

location of the boreal summer eastern hemisphere anticyclone is well poleward of the locations of PV ridges over South America, North America, and North Australia. This anomalously located AMG and the associated convection separate out the Asian monsoon system from others around the planet. Explaining its location is an important objective of this chapter. 16.3.2

Annual Cycle of PV Fields

Figure 16.10 shows plots of the mean monthly fields of 370 K isentropic PV for the boreal spring to

–4

0

4

50 m

0° s–1

8

early-summer period. The April distribution (panel (i)) is very similar to the mean winter PV field of Figure 16.9a. However, from May to June (panels (ii) and (iii)) the anticyclonic PV maximum increases its intensity and moves northward. Simultaneously, the trough to the east over the Pacific Ocean deepens and a closed anticyclonic develops with its center over the HTP (panel (iii)), accompanied by a reversed PV gradient just north of the equator. During October and November (not shown), the PV distribution returns to its winter distribution (Figure 16.9a). During this seasonal progression a ridge also builds up in the eastern

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16 Dynamics of the Large-Scale Monsoon

Evolution of the Asian Monsoon Gyre (370 K PV) 60°N

Figure 16.10 Cycle of the 370 K PV field for (i) April, (ii) May, and (iii) June. Format and color shading are the same as in Figure 16.9.

10 5 0

equ

PVU

30°N

–5 (i) Apr –10

30°S 0

60°E

120°E

180°

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60°W

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equ

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Pacific, presumably associated with the elevated heating from the Rockies and Sierra Madre. In summary, the discriminating characteristic of the NH summer circulation is the size, location, and magnitude of the AMG. Weaker counterparts of the AMG exist elsewhere in the tropics.

16.3.3 A Physical Basis for the Character of the Macroscale Monsoon We commence by noting that during the boreal summer, the HTP is enclosed by the AMG (Figure 16.9b). As a working hypothesis we follow the German climatologist Herman Flohn,IV who stated that the HTP acts as an elevated heat source. Specifically, referring to a series of studies: … One of the remarkable results obtained was that the valleys of southern Tibet, at altitudes between 3500 and 3700 m, enjoy higher summer temperatures than … "Hill Stations" in the Himalayas, at altitudes of 2100–2300 m. At least during the

60°W

0°

warm season, the surface temperatures are definitely higher than the latitudinal average in the free atmosphere at similar heights. Furthermore, a large frequency of showers and thunderstorms, often with snowfall and hail … lasting during the whole warm season, i.e. without being limited to the time period of the Indian summer monsoon…. (Flohn 1968) It was thus proposed that elevated atmospheric heating by the HTP created a reversed north–south uppertropospheric pressure gradient that serves as the foundation of the AMG. The following points are offered in support of Flohn’s hypothesis. 16.3.3.1 Anomalous Location of South Asian Monsoon Precipitation

Figure 16.11 shows the latitudinal profiles of the precipitation rate (P mm day−1 ) and mean sea-level pressure (MSLP, hPa) for both the boreal summer and winter for each of the monsoon regions. The black dashed rectangle indicates the latitudinal extent of the South Asian monsoon boreal summer precipitation maxima

16.3 Macroscale Structure of the Summer Monsoon

(a) Boreal summer (JJA) (i) Australian sector

(ii) African sector

12 MSLP

1020 1010

8 6

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1000 EQ

20°N 40°N

40°S

(iii) South Asia sector

20°S

2 0 20°N 40°N

EQ

(iv) Americas sector

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10

MSLP

mm/day

mm/day

1020

mslp (hPa)

MSLP mslp (hPa)

14

mm/day

mslp (hPa)

1030 mm/day

Figure 16.11 Latitudinal traces of precipitation (mm day–1 ) and MSLP (hPa) for (a) JJA and (b) DJF for the following sectors: (i) Australian sector (120∘ E–140∘ E), (ii) Africa (0∘ E–30∘ E), (iii) South Asia (70∘ E–100∘ E), and (iv) Americas (75∘ W–45∘ W). The dashed black rectangle depicts the location of the South Asian boreal summer rainfall and MSLP distribution between 10∘ N and 30∘ N. It is reproduced in all sections to indicate how much more poleward the South Asian precipitation distribution lies relative to other monsoon regions.

2 EQ

20°N 40°N

40°S

20°S

0 20°N 40°N

EQ

(b) Austral summer (DJF) (i) Australian sector

(ii) African sector

1030

12

1010

P

mslp (hPa)

MSLP

8 6 4 P

1000 40°S

20°S

EQ

20°N 40°N

40°S

(iii) South Asia sector 1030

20°S

EQ

2 0 20°N 40°N

(iv) Americas sector

P

40°S

20°S EQ latitude

from Figure 16.11b(iii), allowing a comparison with the location of the precipitation of other monsoon regions. In all sectors, monsoon precipitation is found to be well equatorward of precipitation in the South Asian sector. The Australian, South American, North American, and African summer monsoons have maxima close to the equator-side coasts. In Africa the rainfall diminishes

8 6 4

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mslp (hPa)

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14

2 20°N 40°N

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0 20°N 40°N

toward the Sub-Saharan Sahel and the Australian monsoon section is a near-mirror image of the African system. Extensive desert regions extend poleward of the rainfall maxima. We have already noted in Figure 1.7a that in the western Indian Ocean, the gradient in surface pressure between the Arabian Sea and the countries of Pakistan,

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Iran, etc., is just as great as between the Indian Ocean and India and South Asia. However, Figure 1.6a indicates there is limited regional rainfall to the west of India. There are perhaps two reasons for these differences. First, Figure 12.7a indicates that the Arabian Sea is located in the subsiding lateral exhaust from the major desert regions. Second, although there is very strong surface heating across West Asia, it occurs well away from the equator and it is not elevated. The major orographic maxima occur to the north of India and Southeast Asia. 16.3.3.2 Seasonal Distribution of Mean Upper Tropospheric Temperature and Specific Humidity

The horizontal distribution of the mean upper tropospheric temperature averaged between 200 and 500 hPa is shown in Figure 16.12 for both the boreal winter and summer. The colored area demarks mean temperatures > –26 ∘ C. Both seasons show a warm band extending around the tropics, consistent with the zonally averaged temperature field (Figure 1.2), but it is the mean temperatures in the vicinity of the monsoon regions that are of particular interest.

During the boreal winter (Figure 16.12a) the upper troposphere over northern Australia is anomalously warm, with maximum mean temperatures near –23 ∘ C. There is also a relative maximum of –24 ∘ C in the western hemisphere over Brazil. Both maxima are associated with the convective heating, as apparent from Figure 1.6. The mean boreal summer upper-tropospheric temperature over Asia is strikingly different from other regions (Figure 16.12b). The South Asian upper troposphere is dominated by an extremely warm air mass centered over the HTP, with mean temperatures of > –21 ∘ C with both strong zonal and meridional temperature gradients. A warm ridge extends over Mexico and southwest North America although its magnitude is far smaller than over Asia. In addition, the boreal summer maximum is located in the subtropics, much farther poleward than the weaker boreal summer and winter counterparts. The off-equator temperature maximum and the reversed zonal temperature gradient (𝜕T∕𝜕y > 0 in the NH and 𝜕T∕𝜕y < 0 in the SH) appearing in Figure 16.12b are consistent with an upper tropospheric easterly zonal wind maximum on the equatorward side of the HTP, designated by “E” in Figure 1.8a. The Figure 16.12 Mean upper tropospheric (200–500 hPa) temperature (∘ C) for (a) the boreal winter (DJF) and (b) the boreal summer (JJA), averaged between 1979 and 1992. The boreal summer plot is based on calculations first made by Li and Yanai (1996). Mean columnar temperatures warmer than –25 ∘ C are colored relative to the scale.

(a) DJF Mean Anom Temp (°C) 500–200 hPa 60°N 40°N 20°N 0° 20°S 40°S 60°S

0

40°E

80°E

120°E

160°E

160°W

120°W

80°W

40°W

0°

40°W

0°

(b) JJA Mean Anom Temp (°C) 500–200 hPa 60°N 40°N 20°N 0° 20°S 40°S 60°S 0

40°E

80°E

120°E

160°E

160°W

120°W

80°W

°C

16.3 Macroscale Structure of the Summer Monsoon

Figure 16.13 Same as Figure 16.12 except for anomalous specific humidity (gm kg–1 ). Warm and cool colors denote below and above anomalous moisture values, respectively.

(a) DJF Specific Humidity Anom (g/kg) 500–300 hPa 60°N 40°N 20°N 0° 20°S 40°S 60°S 0

40°E

80°E

120°E

160°E

160°W

120°W

80°W

40°W

0°

40°W

0°

(b) JJA Specific Humidity Anom (g/kg) 500-300 hPa 60°N 40°N 20°N 0° 20°S 40°S 60°S 0

40°E

80°E

120°E

160°E

160°W

120°W

80°W

g/kg

strongest upper tropospheric easterly winds occur in the boreal summer between East Asia and West Africa (Figure 1.8a), with wind speeds exceeding 20 m s−1 . During the boreal winter, an easterly wind maximum may be found over North Australia, although it is much weaker than its South Asian counterpart. The corresponding anomalous upper tropospheric water vapor concentration relative to the annual average distribution is plotted in Figure 16.13. Overall, the upper troposphere of the summer hemisphere is anomalously moister than the winter hemisphere, consistent with Figure 1.2. Furthermore, anomalies in the specific humidity tend to be negative in the boreal winter (Figure 16.13a) and positive in the summer (Figure 16.13b). In addition to the upper tropospheric column over the Tibetan Plateau being warmer than its surroundings during the summer, it is also much moister (Figure 16.13b). This anomalously abundant water content in the atmosphere above the HTP is why the HTP has been referred to as the “World Water Tower” (Xu et al. 2008). The HTP is the third largest repository of continental ice on the planet behind Antarctica and

Greenland, and has the added importance of supplying fresh water to most of Asia. Much smaller anomalies in exist within the other convective regions of Africa and South America. 16.3.3.3

Mean Monthly Geopotential Sections

Figure 16.4a plots the May to August mean monthly cross-sections of the 200, 500, and 850 hPa geopotential through the South Asian summer monsoon (80∘ E). The same quantities are plotted in Figure 16.14b but along 130∘ E, through the maximum convection of the DJF Australian summer monsoon. During May, the geopotential gradient across Asia is relatively flat between the equator and 30∘ N. As summer advances, the latitudinal gradient rapidly intensifies, supporting the strong easterly upper tropospheric jet stream. At 500 and 850 hPa, the geopotential decreases in magnitude toward the Himalayas, consistent with lower tropospheric monsoonal westerlies that also intensify during summer. The austral summer 130∘ E profiles (Figure 16.14b) show much smaller latitudinal gradients in all months, whereas a small, reversed gradient

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16 Dynamics of the Large-Scale Monsoon

(a) Geopotential (m) 86°E

(b) Geopotential (m) 160°E 12600

12600

200 hPa

12500

12500

12400

12400 (m)

(m)

200 hPa

12300

12300 12200

12200 12100

40°S 20°S

0

12100

20°N 40°N

0

20°N 40°N

0

20°N 40°N

0

20°N 40°N

500 hPa

500 hPa 5800 (m)

5800 (m)

40°S 20°S

Figure 16.14 Latitudinal distribution of mean monthly geopotential along (i) 86∘ E for May to August at 200, 500, and 850 hPa, (ii) 160∘ E between November and February. The lowest panel on the left depicts the orography (m) along 86∘ E. The color scale (lower right) refers to the month in (a) and (b).

5900

5900

5700

5700 5600

5600 40°S 20°S

0

20°N 40°N

40°S 20°S 1600 850 hPa

850 hPa

1600 1500

(m)

(m)

1500

1400

1400

1300

1300 40°S 20°S

5000 (m)

380

0

20°N 40°N

40°S 20°S

(a)

3000 1000 0

JUL

DEC (b)

JAN FEB

20°N 40°N

develops between the equator and 20∘ S during the austral summer, when the magnitude is 25 m per 20∘ latitude, almost an order of magnitude smaller than across South Asia during the boreal summer. Also, the exaggerated summer geopotential gradients in the NH summer are located equatorward of the latitudes of the HTP, identified in the lower left panel. 16.3.3.4

JUN AUG

40°S 20°S

NOV

MAY

orography

Comparison of Surface Heat Fluxes

Figure 16.15 summarizes the annual cycle of the components of the surface energy balance, defined in Section 2.5, for a number of locations: the plains of India, the Bay of Bengal, East Asia, the western equatorial Pacific

Ocean, the Tibetan Plateau, north Central Australia, and the Saharan region. The two monsoon continental regions (panels (a) and (e)) are quite similar. During the winter and spring, SH ≫ LH but tends toward parity at the end of spring and early summer when SH ≈ LE . In summer, the proportionality reverses and the latent heat flux dominates. The North Australian total turbulent heating remains relatively constant through the year, whereas the Gangetic Plain possesses a wintertime minimum. This difference can be accounted for by noting the 20∘ latitude difference between locations and the warmer wintertime surface temperatures over Australia. The western Pacific

16.3 Macroscale Structure of the Summer Monsoon

(a) Ganges Valley

200

0

–200

–200 J F M A M J J A S O N D

J F M A M J J A S O N D (d) TOGA COARE

200

J F M A M J J A S O N D

(e) North Australia

200

100 Wm–2

100 0

0

J F M A M J J A S O N D SS

LH

0

–100

–200

–200

(f) Sahara

100

–100

–100

0

–100

–100

–200

(c) Bay of Bengal

100 Wm–2

Wm–2

Wm–2

0

–100

Wm–2

200

100

100

200

(b) Tibetan Plateau

Wm–2

200

–200 J F M A M J J A S O N D month SS + LH

LW + SW

J F M A M J J A S O N D SS + LH + LW + SW

Figure 16.15 Annual cycle of surface energy budgets for (a) Ganges Valley (25∘ N–27∘ N, 73∘ E–100∘ E), (b) Tibetan Plateau (27∘ N–40∘ N, 80∘ E–100∘ E), (c) Bay of Bengal (6∘ N–15∘ N, 80∘ E–100∘ E), (d) Western Pacific warm pool: TOGA COARE (10∘ S–10∘ N, 147∘ E-180∘ E), (e) North Australia (15∘ S–10∘ S, 130∘ E–135∘ E), and (f ) the Sahara (15∘ N–35∘ N, 17∘ W–60∘ E). The curves (relative to the color code) denote the turbulent transfers of sensible heat (SH : red), latent heat (LH : blue), the total turbulent transfer (SH + LH : green), total net radiation (gray), and total energy flux into the surface (black dashed). Units: W m−2 .

section (panel d, labeled TOGA COARE)17 is typical of the tropical warm pool, with small variations in heating accompanying the small annual cycle of SST and a lag with the solar cycle by about two months. At all times, SH ≪LH . In the Bay of Bengal (panel (c)) there is a relative minimum in LH in winter and a corresponding relative maximum in SH . These extrema are associated with the cooler and dryer winter monsoon flow and a winter minimum in SST. The heat budget over the HTP (panel (b)) shows a different annual cycle than over the Ganges Valley to the south. Radiative heating at the surface is low during the winter because of a high albedo and reduced insolation but a maximum during summer. Latent and sensible heat fluxes are much the same throughout summer (in contrast to the Indian monsoon region), although they sum to provide similar total heating values to those of the Gangetic Plains. Total radiative 17 E.g. Godfrey et al. (1998).

heating of the surface is at a maximum in late spring and early summer. These estimates are similar to those observed at a number of field stations on the Plateau. Bian et al. (2012) analyzed data from eight stations, six of which were at elevations greater than 4000 m and two at about 2500 m, which showed great similarity in character. Significantly, surface albedo at each of the sites averaged about 0.2 between May and October, not too dissimilar from the ground albedo over the Ganges plains. The conversion of radiant energy to sensible and latent heat fluxes and the convergence of moisture creates an elevated heat source, the implications of which we examine in the next few paragraphs. 16.3.3.5 Comparison of Vertical Temperature Profiles over the Gangetic Plains and the HTP

A first step in assessing the possible role of the HTP as being an important entity in driving the Asian monsoon is the determination of the vertical temperature

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16 Dynamics of the Large-Scale Monsoon

profiles above the plateau relative to its surroundings. Figure 16.16 shows vertical temperature profiles based on the computations of Molnar and Emanuel’s (1999) radiative convective model calculations above the mean sea level (z = z0 ) and above an elevated region (z = zm ). The temperature profile BCF is the assumed temperature profile above the Ganges Plain. For reference, ADE is an equivalent profile to the north of the HTP. MN represents the temperature profile above the Plateau. Based on observations, the temperature of the surface of the HTP is assumed to be T B , the same temperature as the surface of the Ganges Plain at z = z0 . We are interested in the relative temperature differences as a function of height between columns CF and MN. If the temperature lapse rate above zm is the same as above the plain (i.e. CF), the temperature at some height z1 will be N or N′ (the latter if latent heat release takes place over the plateau), thus increasing the temperature gradient (and hence the pressure gradient). Thus, the temperature difference between the column above the Plateau and over the Plain continues to grow with height so that at z = z1 , T N > T F . That is, between F and N there is a reversed pressure gradient so that from the thermal wind equation, the vertical shear of the zonal wind is negative producing, thus giving the easterly upper-tropospheric maximum on the equatorward of the HTP. If there were no HTP then the latitudinal temperature gradient would be negative and westerlies would increase with height. It is possible that the lapse rate between the plateau and height z1 is greater than the lapse rate between B and C. Yet, given the magnitude of the turbulent fluxes (Figure 16.15b) and the temperature of the plateau’s surface, such an increase in lapse rate is unlikely. In addition, as there is evidence of precipitation over the plateau as early as spring, it is to be expected that there will be an elevated latent height release above the plateau. 16.3.3.6 Precipitation over the HTP and the Evolution of the Elevated Surface Cyclonic Vortex

If the AMG is associated with elevated heating, then there should be evidence of heating of the atmospheric column above the HTP and the formation of an HTP-surface cyclone (i.e. at roughly 500 hPa). This would occur in spring and early summer, at the same time as the formation of the upper tropospheric anticyclone. A detailed reanalysis data set18 shows this to be the case. Figure 16.17a displays the annual cycle of precipitation rate (mm month–1 ) for each calendar month. The area covered is roughly 25∘ N–40∘ N and 70∘ E–100∘ E. Widespread light precipitation extends 18 Data from the High Altitude Reanalysis data set: Maussion et al. (2014).

Comparison of temperature profiles above HTP and plains to the south

z1

E

F

N N′

D

CM

z1

Height

382

zM

A

z0

M′

B

zM

z0

Temperature Figure 16.16 Profile of temperature T(z) for a cooler boreal winter column (ADE), for reference, and a warmer boreal summer column (BCF), both above the Ganges Plain at z = z0 . Profile (MN) depicts a column above elevated terrain located at height z = zm , assuming the same surface temperature as at z = z0 . The temperature at the surface of the Ganges plain (z = z0 ) is assumed to be T B and the temperature at the surface of the HTP (z = zm ) is also assumed to be the same temperature T B . The yellow region contains all possible temperature profiles above the Plateau depending on the assumed lapse rate above the HTP. If the lapse rate is identical to the atmosphere above the plain for all but unphysical assumptions, the temperature difference FN at z = z1 will be positive, signifying an enhanced pressure gradient between the Plateau, the plains to the south, and the equator.

across most of the HTP by April, with the most substantial rainfall occurring on the eastern plateau. The analysis supports the observations of Flohn (1957), who noted substantial springtime precipitation over the Assam region in the southwest area of the plateau. Figure 16.17b provides latitude–longitude plots of the 500 hPa mean monthly wind field for April, May, and June. In effect, these are surface winds at the HTP level. A weak cyclonic surface circulation starts to form in April at the same time as the incipient AMG (Figure 16.10). The development of the low-level circulation is consistent with the fields of precipitation surface fluxes that increase over the HTP shown in Figure 16.15b. 16.3.3.7 A Heating Threshold for a Subtropical Meridional Circulation

The existence of an off-equator heat source is not sufficient for the forcing of a meridional circulation that is needed for South Asian monsoon circulation, with rising motion over South Asia.19 For such a circulation 19 Plumb and Hou (1992).

16.3 Macroscale Structure of the Summer Monsoon

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

40°N 35°N 30°N 25°N 40°N 35°N 30°N 25°N

40°N 35°N 30°N 25°N 40°N 35°N 30°N 25°N 70°E

80°E

90°E

100°E

70°E

80°E

90°E

100°E

70°E

80°E

90°E

100°E

Longitude

0

5

01

15 20 25 30 35 Precipitation (mm/month)

40

45

50

(a) Mean monthly precipitaion rate Himalayan-Tibetan Plateau (ii) May

(iii) June

35°N 30°N 25°N 80°E

90°E

100°E

80°E

90°E 100°E 80°E Longitude (b) 500 hPa streamlines and precipitation rate (% annual)

90°E

100°E

35 30 25 20 15 10 5 0

% annual precipitation

(i) April

40°N

Figure 16.17 Analyses on the Himalaya–Tibetan Plateau: (a) Annual cycle of precipitation rates (mm month–1 ) by month. Widespread light precipitation covers the eastern Plateau as early as March. Through springtime, the area of rainfall increases and by May has rates of >30 mm month−1 . In summer, rates increase to >40 mm month−1 . (b) Mean monthly wind field and streamlines relative to vorticity at 500 hPa near the mean surface of the HTP. In April and May, a cyclonic surface circulation forms, which is fully established in June. Color scale on right denotes the percent of total annual precipitation during that month. In April 10% of the total Plateau precipitation occurs, rising to 25–30% during June. Source: Data from The High Altitude Reanalysis data set (Maussion et al. 2014).

383

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16 Dynamics of the Large-Scale Monsoon

to be forced, a minimum threshold in the heating has to be exceeded. Furthermore, the more poleward the location of the heat source, the more difficult it is for a direct meridional circulation to be created. As will be developed, the limiting threshold of the required heating increases toward the poles. An astute interpretation of paleo-sedimentary data from the great rivers of Asia20 allowed an understanding that led to a possible explanation of why the elevated heating of the Tibetan Plateau forces a subtropical meridional circulation, whereas the non-elevated Australian and African deserts do not. It was noted that the Tibetan Plateau engaged in rapid uplift during the last few million years. Specifically: … the climate of the equatorial Indian ocean and southern Asia changed at about 6–9 Ma (million years ago): monsoonal winds apparently strengthening, northern Pakistan becoming more arid, but the weathering of rock in the eastern Himalaya apparently increased …. Molnar et al. (1993, p. 357) During the early part of the HTP uplift, when the Tibetan Plateau was closer to sea level, South Asia was arid and perhaps similar to the current North African and central Australian climate, with little precipitation and run-off. Molnar et al. (1993) posed a hypothesis that when the Plateau reached a certain height a subtropical meridional cell developed and with it an increase in precipitation. In Figure 16.16, when the temperature differences MN and CF become large enough, a meridional circulation could develop that placed the region on the equatorward side of CF in subsidence. If the subtropical circulation is strong enough it may reduce the intensity of or even eliminate the near-equatorial cell, leading to the vertical velocity distribution shown in Figure 1.12a(iii). By inference, heat sources over the subtropical continents of Africa and Australia are not sufficiently large to create a subtropical cell that would subdue the near-equatorial circulation and their monsoon precipitation would lie closer to the equator than over South Asia. From the results of Plumb and Hou (1992), Molnar et al. (1993) were able to calculate a required threshold heating for the existence of a subtropical meridional circulation as a function of latitude. Following Molnar et al. we write the steady state momentum equation on a sphere as ̃ = − 1 ∇p 2Ω sin 𝜙 k ×V ∼ 𝜌 20 Molnar et al. (1993).

(16.2)

Assuming zonal symmetry (i.e. 𝜕/𝜕𝜆 = 0), Eq. (16.2) reduces to U(z) = −

gz 𝜕T faT 𝜕𝜑

(16.3)

where the equation of state and hydrostatic balance has been used and zonal velocity set to zero at the surface. T(𝜑) represents the latitudinally varying vertically average temperature of the atmospheric column. A process that will change the temperature of an atmospheric column will distort the pressure fields, the distribution of the latitudinal pressure gradient, and hence the zonal wind field. The response to a heat source is easily understood in terms of the conservation of angular momentum. The intrinsic angular momentum (per unit mass) is defined as MA = a cos 𝜑(Ωa cos 𝜑 + U)

(16.4)

where Ω (s−1 ) is the rotation rate of the planet Conservation of angular momentum requires that dMA 𝜕MA ̃ ⋅ ∇MA = 0 = 0 or +V dt 𝜕t so that in the steady state ̃ ⋅ ∇MA = 0 V

(16.5)

(16.6)

signifying that the velocity vector is perpendicular to ̃ = ṽj + w̃ the gradient of angular momentum. Here V k, where v and w represent the meridional and vertical velocity components. First consider a situation where no heat source exists. This case is represented in Figure 16.18a. Surfaces of constant MA will be vertical and the gradient of MA will be horizontal and point toward a maximum at the equator. No vertical shear will be induced. Consider now a localized heat source of magnitude Q̇ located in the subtropics (Figure 16.18b) with a negative gradient of T(𝜑) to the north and positive gradient to the south of the heat source. From Eq. (16.3), the thermal wind relationship, easterly winds increase with height on the equatorward side of the heating, while westerlies increase with height on the northern flank. The generation of westerlies to the north of the heat source and easterlies on the equatorward side is consistent with the conservation of angular momentum, M. From Eq. (16.6), MA will increase with height on the poleward side of the heat source and decrease on the equatorward side, producing differential slopes of the lines of constant angular momentum. Note that the changes in the zonal wind about the heat source are equal and opposite in order to conserve angular momentum. As the heat source increases in magnitude, the changes of angular momentum are magnified and the gradient

16.3 Macroscale Structure of the Summer Monsoon

MA1

MA2

MA4

MA3

∇MA horizontal

P

p2

∇MA

p1

∇p = 0 north

equator (a) No heat source ∇MA decreases! ∇p increases! ∇MA U > 0, Uz > 0

p2 p1

U < 0, Uz < 0

north

equator (b) Weak axisymmetric heat source

∇MA U>0 U > 0, Uz > 0

U 0 (north of the heat source), surfaces of MA bend toward the pole, and the gradient of MA points upward toward the equator, consistent with the development of negative vertical shear. (c) Strong heat source in the subtropics associated with the elevated HTP. South of a strong axisymmetric heat source, wind speeds increase rapidly with height, causing MA to decrease rapidly with height. Surfaces of constant MA are bent sufficiently that the gradient of M points downward. Closed surfaces of MA form annuli surrounding the Earth. Because wind velocity vectors lie within surfaces of constant MA , a closed meridional circulation is possible. Source: Based on Figure 26, Molnar et al. (1993).

of angular momentum points increasingly downward (Figure 16.18b). Plumb and Hou (1992) note that a meridional circulation cannot occur until the gradient of MA points vertically downward – that is, until a contour of constant MA is parallel to the surface of the planet (Figure 16.18c). At that stage, the circulation becomes unstable and a

horizontal component of relative vorticity is induced in the latitude–height plane, now with non-zero fields of v and w. Thus, a threshold criterion for the onset of a meridional circulation is when the latitudinal gradient of MA vanishes. Using Eq. (16.4) we obtain 𝜕MA 𝜕U = 2Ωa2 cos 𝜑 sin 𝜑 + ua sin 𝜑 − a cos 𝜑 𝜕𝜑 𝜕𝜑 =0 (16.7) Scaling the equations (neglecting the second term) leads to the limiting criterion for the formation of a meridional circulation: 1 𝜕U = 2Ω sin 𝜑 or 𝜁 = f a 𝜕𝜑

(16.8)

where 𝜁 is the zonally symmetric relative vorticity. If 𝜁 > f , then a meridional circulation can ensue. As 𝜁 depends completely on the distribution of U, the existence of a meridional circulation depends solely on ̇ As the instability the magnitude and distribution of Q. is driven by the existence of a heat source that has a seasonal time scale, the system will be perpetually unstable on the time scale of a season and a steady state circulation will develop with an ascending motion within the subtropical heat source and descent over latitudes on the equatorial side of the heat source. Equations (16.3) and (16.8) can be used to produce an expression for the required temperature gradient induced by a heat source for the existence of a meridional circulation; i.e. ( ) 4TΩ2 a2 sin 𝜑 1 𝜕T 𝜕 >− for v ≠ 0 𝜕𝜑 sin 𝜑 𝜕𝜑 zg (16.9) From Eq. (16.9), the limiting temperature gradient between the mountains and the Plains to the south of the HTP is, at z = 15 km, between 5 and 10 ∘ C. The limit appears to be less than that found in the mean distribution of the mean upper tropospheric temperature to the south of Asia, displayed in Figure 16.12b. However, on the other hand, from the same figure, it is clear that the critical temperature gradient required for a forced subtropical circulation is far greater than the gradients occurring between the subtropics and the equator over North Australia, Africa, or North and South America! In the paragraphs above we have presented observational, diagnostic, and theoretical arguments that suggest that the large-scale AMG is the result of elevated heating. In Section 13.2 we formulated a theory for the generation of near-equatorial convection that depends on the existence of a cross-equatorial pressure gradient that will produce a near-equatorial meridional circulation with convection 5–10∘ poleward of the

385

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16 Dynamics of the Large-Scale Monsoon

equator in the summer hemisphere (e.g. the African and North Australian summer monsoon or the SH summer convection in the South Indian Ocean). Now, with the theory developed here, we can state that only with strong elevated heating can 𝜁 > f , so that the South Asian monsoon precipitation can extend well northward into the summer subtropics of the NH. This is similar to the criterion for symmetric or inertial instability for the determination of the location of off-equator precipitation.

16.3.4

West African Summer Monsoon

There is a general tendency to concentrate on the Asian monsoon. This is easy to understand. It is the largest of the monsoon systems and its influence is widespread. Here we briefly consider the dynamics of the summer West African monsoon shown in Figure 1.13c(i). Unlike the Asian monsoon, but similar to North Australia, North Africa does not possess a major topographic feature that compares in scale with the Himalayan–Tibetan Plateau. However, there is a distinctive cross-equatorial surface pressure gradient (CEPG: Figure 16.11a(ii)) and West African monsoon rainfall generally occurs near the coast and reduces in intensity rapidly inland. In addition, the West Africa region may be affected by the large-scale dynamics of the Asian monsoon as well as SST variations associated with both Pacific and Atlantic modes. For example, Figure 16.9b shows that the AMG (feature “F”), a product of the Asian monsoonal heating, extends as a distinct ridge from near the date line westward across Africa and into the Atlantic Ocean. Of particular interest is the easterly jet on the southern flank of the AMG that passes over Africa. The lateral extent of the jet is more easily seen in Figure 16.19a. The influence of the easterly jet stream on West African precipitation can be assessed from the following simple development. We rewrite Eq. (3.1a) on a beta-plane centered at latitude 𝜑 = 𝜑0 as ̃ 𝜕V ̃ = −∇𝜙 − ̃ ̃ ̃ •∇V k × (f0 + 𝛽y)V +V 𝜕t

(16.10)

Splitting the horizontal velocity into geostrophic and ̃a ), Eq. (3.1a) ̃ =V ̃g + V ageostrophic components (i.e. V becomes ̃g dg V dt

=

̃g 𝜕V

̃ g + f0 V ̃a ̃g = −∇𝜙 − ̃ ̃ •∇V k × (f0 V +V 𝜕t ̃g + V ̃a )) + 𝛽y(V

By definition, geostrophic balance is given by ̃g = ̃ f0 V k × ∇𝜙

̃a , we obtain from Eq. (16.10): ̃g >> V and, as V ̃g dg V

̃a − ̃ ̃g k × 𝛽yV (16.11) ≈ −∇𝜙 − ̃ k × f0 V dt For our purposes here, it is sufficient to reduce the system to an f -plane with 𝛽 = 0. Solving for the ageostrophic component gives ̃g dg V ̃a = 1 ̃ V k× f0 dt

(16.12)

which states that if there is an acceleration of the geostrophic wind, there will be a compensatory generation of an ageostrophic wind that attempts to reestablish the geostrophic balance. Consider the westward flow through a easterly jet such as depicted in Figure 16.19b. Higher pressures exist to the north of the jet and lower pressures toward the equator. Parcels entering the jet from the east, initially in geostrophic balance, will accelerate, promoting a southward ageostrophic wind field (bold vector labeled ̃a ). On exiting the jet, the geostrophic wind decelerates V and the ageostrophic wind field will be toward the north. Together, these ageostrophic flows will promote ascent in the left-hand southerly quadrant and the right-hand northerly quadrant. At the same time subsidence is promoted in the right-hand northerly and left-hand southerly quadrants. Figure 16.19c shows the resultant secondary circulations. The easterly jet hypothesis was first introduced in the context of Africa by Nicholson and Flohn (1980) to help decipher the role of the South Asian monsoon in determining changes in the North African climate during the Pleistocene and the Holocene periods. Webster and Fasullo (2002) suggested that the secondary circulations associated with the easterly jet would promote wetter conditions over the Bay of Bengal but drier conditions over northern Africa and the Middle East. In West Africa the secondary circulation appears consistent with the rapid decrease of precipitation rate toward the north away from the coast. Other factors are at play as well. We noted in Section 13.2 that to the south of the Guinea coast that there was a CEPG resulting from the cooler waters south of the equator and warmer waters to the north. Further, Figure 13.8b(ii) showed signatures of an inertially unstable regime including low-level westerly winds (Figure 13.13c). In fact, Grist and Nicholson (2001) had noted that precipitation over the Sahel was directly related to the strength of the low-level westerlies with distinct differences between Sahel wet years (1948–1969) and dry years (1970–2004) consistent with the multidecadal variability occurring in Figure 14.1b. Figure 16.20a shows a strong relationship between the location of

16.3 Macroscale Structure of the Summer Monsoon

30°N 15°N

10 20 30 40 50

0° 15°S 30°S 50°W

0°

50°E

100°E

U (m s–1)

–30 –20 –10

150°E

(a) Mean JJAS 200 hPa winds (1948-2000) B

D

High

p3

conv

div

Va

Vg

–40 –30 –20

div

Low

A

p2

Va conv p1 C

(b) Ageostrophic flow at exit/entrance of Easterly Jet

z

A

B

C

D

250 hPa

surface EQU

y mslp

fall

N-Africa

EQU

N B of B

rise

rise

fall

(c) Secondary circulations at exit/entrance of Easterly Jet Figure 16.19 Impacts of the large monsoon heating over South Asia. (a) The upper tropospheric circulation. Note the extent of the Easterly Jet that commences over the north-eastern Indian Ocean and extends out across the Atlantic Ocean. Magnitude of the westerly wind component is color coded. (b) The secondary circulation associated with the entrance and exit regions of an easterly jet stream. Note that the ageostrophic flow produces a northward deflection of the winds that is consistent with subsidence on the northward side of the jet and rising surface pressure. Lettering denotes cross-sections through the entrance and exit regions of the jet and “div” and “conv” denote upper tropospheric divergence and convergence from the ageostrophic winds. (c) Schematic diagram of the induced secondary ̃ and V ̃ represent the geostrophic and ageostrophic components of velocity, respectively, and circulations along sections A–B and C–D. V g a pi denotes pressure.

the latitude of the zero absolute vorticity and the strength of the westerlies, consistent with Eq. (13.23). There is a further characterization between wet and dry years in terms of the CEPG to the south of the Guinea coast (Figure 16.20b), with the stronger gradient matching wet years and vice versa. The two relationships are highly correlated at 0.70 and 0.84, respectively. Changes in the CEPG would appear to be associated with SST pattern modulations, especially with the weakening or strengthening of the cold tongue in the eastern Atlantic, sometimes referred to as the

Atlantic Niño (e.g. Covey and Hastenrath 1978; Zebiak 1993). However, the statistics indicate more complicated relationships. What is clear, though, is that when the 𝜂 = 0 contour is further north, the dynamic structure associated with African easterly wave generation (Section 13.3.4) changes as the AEJ strengthens and moves northward so that the locus of rainfall producing easterly waves migrates across the Sahel. Hints at shorter-term variability have been found by Nicholson and Grist (2001), with relationships between extremes in Sahel and Guinea coast rainfall where a maximum in one is coupled to a minimum in the other.

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16 Dynamics of the Large-Scale Monsoon

U850hPa (m s–1)

20

(a) Max U850hPa versus η = 0 latitude

15 10 5 0 2°N

r = 0.70

3°N

4°N

5°N

6°N

Latitude

20

(b) Max U850hPa versus CEPG

15 U850hPa (m s–1)

388

10 5

0 2°N

r = 0.84

3°N

4°N Latitude

5°N

6°N

Wet Sahel (1948-1969) Dry Sahel (1970-2004) Figure 16.20 Relationship of wet (1948–1969) and dry (1970–2004) Sahel rainfall years. (a) Strength of the 850 hPa westerly winds (m s−1 ) north of the equator plotted as a function of the location of the zero absolute vorticity contour (∘ N). (b) Strength of the 850 hPa westerly winds (m s−1 ) north of the equator plotted as a function of the magnitude of the cross-equatorial pressure gradient (10−3 hPa km−1 ). Red and black circles denote dry and wet years. Correlations are highly statistically significant at (a) 0.70 and (b) 0.84, respectively. Source: Based on Nicholson and Webster (2007).

16.4 Macroscale Structure of the Winter Monsoon The winter monsoon circulations of West Africa and the Americas coincide with a weakening and a southward migration of the ITCZ toward, but not across, the equator. As described in Chapter 13, the regional cross-equatorial pressure gradient (CEPG) in the African and Atlantic sectors, although weaker, retains the same sign throughout the year as the NH

near-equatorial SST and remains warmer than that of the SH. The seasonal variation of the Asian–Australian monsoon (Figure 1.13a) is markedly different from other monsoon regions. The location of maximum precipitation changes hemispheres, from a summer location centered in South Asia to then lie across Madagascar, Indonesia, and North Australia during the boreal winter. In accord, the CEPG changes sign seasonally. During winter, high pressure dominates North and Central Asia with outflow through East and South Asia crossing into the SH in the western Indian Ocean (Figure 1.13a). A detailed view of the mean DJF Asian surface pressure appears in Figure 16.21a. The center of the high-pressure system, usually called the Siberian High, is centered at roughly 50∘ N and 90∘ E poleward of the Tibetan Plateau. The central pressure of the Siberian High is far from constant and varies rapidly on the time scale of days between 1020 and 1050 hPa (Figure 16.21b), with an average DJF central pressure of about 1030 hPa. The anticyclonic flow is also far from constant and rises and falls in magnitude-reflecting changes in the central pressure. This anticyclonic flow is made up of a series of pulses called “cold surges” that bring cold spells, often associated with strong surface winds, to the eastern and southern half of the continent, near-equatorial rainfall episodes, and along-the-equator wind surges across both the Indian and Pacific Oceans. Cold surges out of winter anticyclones also occur in other locations around the globe. During the boreal winter cold surges are often found to the east of the Appalachians (e.g. Bell and Bosart 1988) that may extend to the tropics. Similarly, equatorward cold surges occur to the east of the Andes in South America following the development of a winter high-pressure system (Lupo et al. 2001). Colloquially, these bring about periods of “friagem” (Portuguese for “cold weather”) to Brazil. The Australian winter monsoon also possesses anticyclonic outflow that recurves across the equator toward South Asia. Associated with this outflow are surges across North Australia but are weaker than their Asian counterparts. It turns out that the same physics apply to surges in the Australian winter monsoon as in the Asian winter monsoon, but the surface temperature differences between the oceans to the south of Australia and the tropics are far less than equivalent differences in Asia and as a consequence the Australian cold surges are less dramatic. Cold surges are interesting phenomenologically as they involve the dynamics of cold anticyclones, the influence of orography, and westward-propagating upper-tropospheric midlatitude disturbances. Furthermore, observations suggest that the propagation

16.4 Macroscale Structure of the Winter Monsoon

3

10

19

1010

10

16

101

75°N 10 07

1022

1

2

9 101 6 1 0 1

102

1031

1028

10

13

1010

1 1 0 2 1022 5 1016

10

16

1022 1019 1016

HK

1013 1013

10

1013

1010

10

10

0°

1016

102 5

10

25°N

10

1010

40°E

07

1013

DW 10

10

10

25°S

100

1016

80°E

1004

1007 1010 1013

8

102 16

19 10

19

7

1001

025

22

100

6

1034

SA

1

50°N 10

1025 103

1031

1

1028

2

025

1019

9 101

1022

1025

3

6 101

10

102

2

101

101

102

019

1016

Figure 16.21 (a) The climatological MSLP distribution (red contours: hPa) over Central, North and East Asia during DJF. North Asia: the background shading denotes surface elevation. Shaded contours represent elevations >1000 m. The darkest shading denotes elevations >4000 m. SA and HK represent the locations of the climatological center of Siberian High and Hong Kong. (b) Variation of the central pressure and surface temperature of the Siberian High (location SH) and Hong Kong (HK) from 1 October 1992 to 14 January 1993. The peak magnitude varies rapidly from about 1020 to 1050 hPa about a mean central pressure of 1030 hPa. Mean HK MSLP is 1012 hPa with variations 8–10 hPa. Variations of SH MSLP and surface temperature lead variations of HK. (c) Hovmöller diagram of surface pressure anomalies along 110∘ E for the same period as (b). Anomalies were computed after subtracting a three-harmonic annual cycle from the original time series, as described in detail by Compo et al. (1998). Large amplitude MSLP anomalies starting over North Asia, with values 20–30 hPa, propagate rapidly toward the equator at speeds ranging from 10 to 25 m s−1 . Anomalies near the equator are of order 5 hPa. Source: After Compo et al. (1998).

7

1010

120°E

160°E m

0

1000

2000

3000 4000 orography (m)

5000

6000

7000

(a) DJF MSLP (hPa) and orography (m)

of the signal is extremely fast so that within a few days the pressure pulse arrives in the tropics far more rapidly that could occur by advection (Figure 16.21c). Consequently, it would seem that cold surges possess properties of gravity waves and edge waves.21

16.4.1

Siberian Cold Anticyclone

We start by considering the dynamics of anticyclones with special emphasis on the North Asian or Siberian high cold anticyclone. We first discuss the dominant physics that produce the surface anticyclone and determine its amplitude. We then consider why it varies so rapidly from day to day. Fundamentally, a cold anticyclone is created through intense radiative cooling of the surface in a relatively clear atmosphere. In the Asian case the rising surface pressure commences in the autumn and continues 21 Compo et al. (1998).

through the boreal winter. Depending upon the radiative model chosen, the increase in surface pressure can be on the order of 1–1.5 hPa day−1 . Curry (1987) used a variety of radiative schemes in a numerical model to determine the degree of radiational cooling of the surface. Yet the surface pressure rarely exceeds 1050 hPa and ranges between 1020 and 1050 hPa (Figure 16.21b) before rising and plummeting by 15–20 hPa on the time scale of days. Thus, both the rapid rise and the fall of MSLP exceeds what can be explained by radiative processes alone. Therefore other processes must also be occurring that, first, limit the magnitude of the surface pressure and, second, produce variability of the central pressure. 16.4.2

Limitations on Central Pressure

One dynamic constraint on the central pressure is the maximum outward directed pressure gradient that can exist in an anticyclonic circulation. For balanced motions, where the pressure gradient is orthogonal to

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16 Dynamics of the Large-Scale Monsoon

Figure 16.21 (Continued)

MSLP (hPa)

1060 SA

1040 1020

HK 1000 300

2m temperature (K)

390

280 273

HK

260

SA

240 Oct (1992)

Nov

Dec

Jan (1993)

(b) Siberian and Hong Kong MSLP and Tsfc hPa B

60oN

30 25

40oN 20oN EQU

35

B 22 m s–1 A 13 m s–1

Oct (1992)

B

22 m s–1

20 15

A 13 m s–1

10 5

Nov

Dec

Jan (1993)

0

(c) Surface pressure anomaly along 100oE

the isobars, we have the balance equation22 𝜕𝜙 1 𝜕p v2 + fv = − = Rc 𝜕n 𝜌 𝜕n

(16.13)

where Rc is the radius of curvature (negative for anticyclonic flow, positive for cyclonic flow) and ̃ n is the unit vector pointing orthogonally to the velocity vector. In Eq. (16.13), 𝜙 and p represent geopotential and pressure, respectively, and v is the parcel velocity. For balanced flow around a high-pressure system, 𝜕p/𝜕n > 0, whilst (16.15) 𝜕n 4 As Rc decreases toward the center of the anticyclone, the required limiting pressure gradient becomes smaller, explaining why the pressure gradient in the center of an anticyclone is slack compared to a cyclone. Why is the mean central pressure of the Siberian high so much larger than those over the other winter continents? During DJF, for example, the climatological surface pressure over North Asia is >1030 hPa (Figure 1.7c). Over North America, the continental high pressure is much less, 1020–1022 hPa, similar to the high pressure over the Sahara. The SH winter surface pressure patterns are quite different. Maximum pressure occurs over the three oceans near 30∘ S: the South Indian Ocean, with a climatological central pressure of about 1026 hPa, the Southeast Pacific and South Atlantic, each with similar surface pressures. Over Australia there is a relative maximum surface pressure of about 1024 hPa. North Asia is different,

16.5 Subseasonal Summer Monsoon Variability

perhaps because of its geography. Referring back to Figure 16.21a, the orography of northern Asia may restrict the expansion of the shallow Siberian anticyclone to the south. Over Mongolia, the orography rises toward the Tibetan Plateau, a demarking of the southern extension of the high-pressure system. If the extension of the high-pressure system is restricted and if radiational cooling continues, pressure will continue to rise until the condition expressed in Eq. (16.15) is exceeded. At this stage, geostrophic adjustment (Section 3.1.5) occurs through a radiating family of gravity waves. These waves follow a distinct path around the Asian landmass with maximum amplitude close to the surface and against the orography rising to the west. The structure is very similar to that of an edge or Kelvin wave (see Section 6.1.5.4). These waves, moving rapidly from high to low latitudes across East Asia, bring cold conditions to the equatorial regions. These are referred to as East Asian cold surges. The physics of the transient winter monsoon will be discussed in Section 16.6.2.

the equatorial Indian Ocean and systematically propagated slowly northwards, bringing periods of rainfall over South Asia with intervening periods of reduced rainfall.23 Over time, these oscillations became known generically as Monsoon Intra-Seasonal Oscillations (MISOs). Figure 16.22 plots the northward propagation of the MCZ found by Sikka and Gadgil (1980) for the boreal summer of years 1971–1975 (Also, Gadgil 2003). A major result from this early work was noting that the MCZ forms near the equator and moves northward as a zonally oriented trough, leaving in its wake a convection-free zone or ridge. Further, once the MCZ reaches the continental region, moving to the northern part of India, another MCZ often forms near the equator where it restarts its own northward journey. With respect to the nature of the MCZ and the subseasonal variability of the monsoon, Figure 16.22 may suggest a seesaw between two attractors. In fact, such a hypothesis was posed by Sikka and Gadgil (1980): … The presence of two ITCZs – a continental one in the north on a majority of the days and (sometimes simultaneously) an oceanic one in the equatorial belt of the NH . . . . Epochs of the oceanic ITCZ are characterized by long life spans only when the continental ITCZ is absent…. Sikka and Gadgil (1980, p. 1850)

16.5 Subseasonal Summer Monsoon Variability Although subseasonal variability of the monsoon rains has long been recognized as part of a monsoon season, it is only relatively recently that it has been identified as a coherent, low-frequency, and large-scale mode. Indian scientists have been aware that large areas of India and surrounding countries undergo extended periods of enhanced rainfall (“active” periods of the monsoon) or drought (“break” monsoon) during the summer with periods that could sometimes last for weeks. It was well known, too, that the cessation of rainfall over the great plains of India coincides with an increase in the Himalayan foothills. Breaks in the monsoon were thought to be periods of cessation of the overall monsoon rainy season as distinct from an oscillation of the overall monsoon system between wet and dry phases (e.g. Asnani 1993). Thus, for a long period the emphasis was more on finding what caused the monsoon rainfall to “cease” rather than attempting to understand the active-break sequence as a physical mode. 16.5.1 Identification of Propagating Intraseasonal Signals Satellite data allowed major advances in describing intraseasonal monsoon variability, indicating a systematic northward propagation of convection, referred to then as the “maximum cloudiness zone” (MCZ) with time scales of 20–40 days. The zone formed in

and … Thus, the atmosphere is faced with the choice of forming an ITCZ in the warm oceanic regions near the equator or over the heated continent to the north or a combination of both. It is seen that the choice made consists of a combination of both, with the continental ITCZ being dominant most of the summer…. Sikka and Gadgil (1980, pp. 1850–1851) The fundamental contribution of these early studies is that they showed that the dynamic domain of monsoon variability includes both the equatorial oceans and the subtropical continents. It also raised the prospect that the monsoon could be interpreted as an oceanic ITCZ impacted by the presence of the continents24 and, in South Asia, by the elevated terrain of the HTP. Importantly, it suggests that the active-break sequences were part of a long period of oscillation extending from south of the equator to the Himalayas and from west of India to nearly the date line. Furthermore, it was realized that the generation of the oscillation was continual 23 Especially Yasunari (1979) and Sikka and Gadgil (1980). 24 E.g. Chao and Chen (2001) and Gadgil (2003).

391

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16 Dynamics of the Large-Scale Monsoon

oN

Propagation of Maximum Cloudiness Zone along 90oE

30

1977

15 0 30

1976

Figure 16.22 The latitudinal locus of the maximum cloud zone (MCZ) obtained from satellite imagery along three meridians 90∘ E for the years 1971–1975. The red contour depicts the center of the MCZ, the salmon colored area denotes the latitudinal extent of the MCZ and the dashed blue line the 700 hPa trough. Source: Transcribed from Sikka and Gadgil (1980).

15 0 30

1975

15 0 30

1974

15 0 30

1973

15 0

Apr

May

Jun

Jul

Aug

Sep

Oct

throughout the summer. In essence, these observational studies suggested that the intraseasonal oscillations of the monsoon are manifestations of basin-wide processes that form at very low latitudes within the Indian Ocean before propagating to the land masses of South and Southeastern Asia. 16.5.2 Impacts of Monsoon Intraseasonal Oscillations (MISOs) Figure 16.23a illustrates the vagaries of a subseasonal monsoon between years in plots of the Central India rainfall rate25 (see map inset). First, the magnitude of the area-integrated rainfall is different each year; the annual cycle is modulated by a relatively small interannual variability. It should be noted, though, that the year-to-year variation of monsoon rainfall is far greater in a region than over the entire Indian subcontinent. The essential variability, though, is on the intraseasonal time scale with high-amplitude peaks and valleys in precipitation rate. Figure 16.23b describes details of the rainfall rate for four years from 1999 to 2002. The bold blue curve shows the long-term climatological annual cycle of precipitation plotted for the boreal summer. Within each summer season there are large variations of rainfall on time scales of 20–40 days. Each year exhibits peaks (“active” monsoon periods) and valleys (“breaks” in the monsoon) of rainfall rate. The smoothness of the long-term averaged annual cycle of precipitation (blue curve) suggests that there is no preferred timing for the initial or subsequent episodes of 25 Rainfall rate derived from GOES Precipitation Product (GPI): http://disc.sci.gsfc.nasa.gov/DATASET_DOCS/arkin_gpcp_gpi_ dataset.html.

intraseasonal variability and that they occur randomly throughout the summer within the broad 20–40 day period band. Light-blue and white background bands indicate months (“a” April, “m” May, etc.). In 2002 the monsoon began auspiciously with widespread rainfall across most of India. However, this period was followed by a prolonged break in the monsoon rains, which produced a substantial decrease in crop yields nationwide. The prolonged “break period” produced the fifth driest summer over India in over 120 years. Commenting on the disastrous year, it was noted that: ……The minimum length of time of a forecast that will allow a farming community to respond and take meaningful remedial actions … is about 10 days although 3 weeks would be optimal.… Assuming (such forecasts) were available by the third week of June 2002 … farmers could have been motivated to postpone agricultural operations saving investments worth billions of dollars … water resource managers could have introduced water budgeting measures .… Sri A. R. Subbiah, personal communication noted in Webster and Hoyos (2004).V Subbiah’s comment underlines an important principle of forecasting severe events regarding a necessary lead time. In the forecasting of floods, for example, it is necessary to have a lead-time sufficient for the slowest members of a society, perhaps a farmer with cattle, to move safely to higher ground. This requires at least a seven-day notice of impending flood for example.26 Changing agricultural practices is a much slower process 26 E.g. Hopson and Webster (2010) and Webster et al. (2010).

16.5 Subseasonal Summer Monsoon Variability

GPI rainfall (mm/day)

40 period of detail 30

20

10

0 1986

1990

1995

2000

(a) Central India pentad rainfall 1986-2002 1999 a

j

j

a

s

30

o

20 10 0 20

30

40

50

a

m

j

j

a

s

o mm day–1

10 0 20

30

40 pentad

50

60

m

j

j

a

2000 s o

20 10

30

20

a

0 20

60 2001

30 mm day–1

m

mm day–1

mm day–1

30

30 a

40

m

j

j

50 a

60

2002 s o

20 10 0 20

40 pentad

30

50

60

(b) Central India pentad rainfall 1999-2002 Figure 16.23 Intraseasonal variability in Central India (see inset for region). (a) Satellite-derived GOES Precipitation Index (GPI: mm day−1 ) for the central India region for the period 1986–2002. (b) Pentad (five-day average) rainfall rate for four years: 1999, 2000, 2001, and 2002. Blue curve is the climatological rainfall rate through summer. Source: Data from Joyce and Arkin (1997) and Webster and Hoyos (2004).

and hence the 2–3 week lead-time suggested by Subbiah. Clearly, forecasts of these subseasonal oscillations of the monsoon would be of great benefit to all societies, especially agrarian ones. 16.5.3

Inter-event Variability

A more exhaustive study of the Central Indian rainfall with a longer data period shows that each MISO, like the MJO and its variants, appears to have its own time scale within a broad 20–50 day envelope. If the annual monsoon is separated into weak, average, or neutral (see

Section 16.2.3), there is a tendency for more active/break periods during a weak year and less in a strong year, perhaps reflecting that a prolonged active period will be associated with greater rainfall than shorter active periods. Although the majority of MISOs propagate northwards across South Asia, there are distinct family members that have common and distinct properties. Figure 16.24 describes the composite behavior of 58 summer MISOs from 20 days before and 20 days after maximum precipitation occurs at the equator near 75–80∘ E (day 0). The upper row shows longitude–time

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16 Dynamics of the Large-Scale Monsoon

Propagation characteristics of the composite MISO (i) All MISOs (58)

(ii) Eastward MISOs (21)

75°E–80°E

75°E–80°E

30°N

30°N

(iii) In situ MISOs (27)

20°N

20°N

10°N

10°N

10°N

Equ

Equ

Equ

10°S

10°S

10°S

0 10 lag days

20

20°S –20 –10

30

0°–5°N

–20

0 10 lag days

20

20°S –20 –10

30

0°–5°N

–20

10

0 10 20

20 30 40°E

75°E 110°E 145°E 180°

–25

–20

30

0 10 20

30 40°E

–15

20

–10 lag (days)

lag (days)

0

0 10 lag days 0°5°N

–20

–10

–10

75°E–80°E

30°N

20°N

20°S –20 –10

lag (days)

394

75°E 110°E 145°E 180°

–10

–5

5

10

15

30 40°E

75°E 110°E 145°E 180°

20 W m–2

Figure 16.24 Lag-latitude sections along 75∘ E–80∘ E and lag-longitude sections along 0∘ N–5∘ N of composited OLR. From left to right: composite categories of (i) all MISOs; (ii) eastward-propagating MISOs; and (iii) quasi-stationary MISOs, as defined in the text. Source: Adapted from Lawrence and Webster (2002a).

sections of composite OLR. The bottom row of panels displays latitude–time sections of OLR (W m−2 ) as a function of lag relative to maximum precipitation at the equator. The left-hand pair of diagrams (column (i)) illustrates the mean composite behavior of all 58 MISOs. The second and third columns show MISOs that have characteristics of eastward propagation along the equator (21 of 58% or 36%) or those that remained roughly at the same longitude along the equator (27 of 58% or 46%). In time, both the eastward propagating and the stationary MISOs extend northwards (upper panels), providing rainfall to the South Asian land mass and the Bay of Bengal. In both of these categories the rainfall maximum (active monsoon) is replaced over South Asia by a rainfall minimum (i.e. the break period of the monsoon). 16.5.4

Composite Structure of the MISO

In Figure 16.22 we noted an apparent out-of-phase relationship between convection at the equator and its

absence in the North India Ocean and vice versa. We can examine this phasic relationship more thoroughly by constructing composites of 25–80 day band-passed daily rainfall, 925-hPa winds and sea-level pressure (MSLP), and SST, relative to maximum convection over the Bay of Bengal (day 0). We use a suite of satellite data (for precipitation rate, surface winds, SST, and cloud liquid water) using TRMM satellite data. The composites appear in Figure 16.25: (i) The composite sequence corresponds to an intraseasonal event that starts as a positive precipitation anomaly (cold colors) in the equatorial Indian Ocean (day −15) followed by an eastward propagation that bifurcates with poleward extensions in both hemispheres. (ii) The event develops (day −10) over warm surface waters (pink) and low pressure (blue) at the equator. Around the same time, the Bay of Bengal is under break (dry and suppressed) conditions associated with the previous MISO and high surface pressure

16.5 Subseasonal Summer Monsoon Variability

Figure 16.25 Composites of 25–80 day band-passed (a) GPCP daily rainfall, (b) 925 hPa winds and MSLP from the NCEP–NCAR reanalysis, and (c) TMI SST for 16 canonical MISO events over BoB. Day −15 to Day 15; day 0 corresponds to the maximum MISO rainfall over the BoB. Source: From Hoyos and Webster (2007).

Spatial structure of the composite MISO (a) Rainfall

(b) Surface pressure

ISO Rainfall (mm)

SLP (hPa)

(c) SST

–15

–10

lag days

–5

0

+5

+10

+15

–6 –4 –2 0

with a corresponding anticyclonic circulation. The high-pressure system also extends across India and the Arabian Sea. These relatively cloud-free conditions favor SST increases over the Bay of Bengal and are integral parts of the MISO cycle.27 (iii) The equatorial low-level surface pressure and associated convection propagates to the east, accompanied by surface cooling from freshening winds and increased surface evaporation. By the time the anomaly reaches the west coast of Sumatra (day −10), the waters at the north of the Bay have 27 Stephens et al. (2004), Wang et al. (2005), and Agudelo et al. (2006).

2

4

6 –12 –8 –4 0

4

3.00 m/s

SST (°C/10)

8 12 –6 –4 –2 0

2

4

6

warmed under the prior anticyclonic conditions, inducing the northward propagation of the surface low pressure from the equator, with positive anomalies of rainfall. (iv) As the MISO evolves, the surface low-pressure center is located over India (day 0), driving moist air toward the eastern side of the Bay and the Burma mountains where orographic lifting enhances precipitation, resulting in the precipitation maximum aligned parallel to the mountain range. (v) By day +5, reductions in insolation and the freshening of winds cool the surface waters of the Bay of Bengal (BoB), acting as a negative feedback on

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16 Dynamics of the Large-Scale Monsoon

MISO convection. Also, the following break phase starts to develop in the equatorial Indian Ocean, induced largely by the subsidence of air and subsequent SST warming caused by convection further north.28 If we define an active phase as a time interval where the rainfall anomalies in the 25–80-day band are positive on average in the period 1997–2004, 70% of the summer BoB precipitation falls during these active phases. In fact, fluctuations with a dominant period of 20–50 days account for the majority of the total variance in the time series. The ratio of the square root of variance between the 20–50 day filtered and daily time series is 60, 74, and 74% for 2000, 2001, and 2003, respectively.

16.5.5 Theories Regarding the Genesis and Maintenance of the MISO There are a number of different theories regarding the genesis of a MISO. One class of theories suggests that the oscillation is the result of external forcing from higher latitudes. Another class considers the MISO as an oscillation resulting from the internal dynamics of the monsoon. In the following paragraphs we consider both sets of theories although, in some ways, it is difficult to differentiate between the MJO and the MISO. Both have similar time scales and members of both phenomena (but not all) propagate eastward along the equator. The MISO, on the other hand, has a distinct northward propagation.

and the ENSO cycle except during extremely large warm ENSO events (e.g. 200229 ) when MJO activity appears suppressed. Hendon (2000) suggested that most of the year-to-year variability of the MJO is internally generated, independent of any slowly varying boundary forcing. In addition, Lawrence and Webster (2001) found that MISO activity is relatively uncorrelated with ENSO, except for a weakly positive correlation at the beginning of the Southeast Asian monsoon season. As a consequence, it is necessary to consider the distribution of the interannual internal variability in order to explain some of the spatial features of JJAS rainfall anomalies. It is quite unlikely that the phase of the IOD has much influence on the sequence of MISOs during the summer monsoon. The annual cycle of the IOD, discussed in Section 14.3, is such that it peaks in the months following the summer monsoon. Its influence, though, on the MJO in SON and DJF requires investigation. Relationship with the Extratropical Variability Given the

such as ENSO play a role in at least modulating a MISO. Yet, little statistically significant evidence has been found to exist between seasonal MISO activity

scale of the Asian monsoon system, and other monsoon systems as well, it could be a simplification to consider that just the influences of the tropics on the monsoon are important. The South Asian monsoon can be thought of as a series of interlocking components, some located at higher latitudes and remote from the regions of monsoon rainfall described earlier in Section 16.1. Krishnamurti and Bhalme (1976) considered the Mascarene High, which spans the South Indian Ocean at roughly 30∘ S, as an integral component. This system lies in the subsident region of the mean monsoon meridional cell (see Figure 1.12a(iii)) and the outflow of the Mascarene High forms the cross-equatorial Somali Jet off the coast of Africa (Figure 1.7a), which, in turn, feeds into the North Indian Ocean lower-tropospheric monsoon gyre. Rodwell (1997) proposed that an increase in magnitude of the Mascarene High, resulting from propagating extratropical disturbances in the SH, will inject an anomalous surge of negative PV into the NH. Normally such a surge would result in inertial instability. However, it may be remembered that in Section 13.2.8(h) the cross-equatorial flow in the western Indian Ocean is inertially stable due principally to the subsiding “desert exhaust” (see Figure 12.7). Rodwell also argues that the surge of negative PV alters the low-level flow north of the equator. This injection, so unhindered by inertial instability, produces a coherent turning of the NH low-level winds southward so that the flow tends to avoid India and return to the SH. In this manner, it is proposed that rainfall is reduced during the period following a negative PV injection. A schematic of Rodwell’s hypothesis appears in Figure 16.26a.

28 Wang et al. (2005).

29 As described by Gadgil (2003), for example.

16.5.5.1

External Forcing of the MISO

The Circumnavigating Upper-Tropospheric ISO In Section

15.4.2, we discussed a prevailing theory identifying the MJO as a circumnavigating divergent upper tropospheric mode that “reignites” convection as it crosses the Indian Ocean. This theory was discussed in Section 15.4.2. Similar theories also exist for the MISO. However, as stated earlier, the MJO decorrelation time scale (defined in Section 15.4) is less than the time scale of one MJO cycle. If a MISO is not related to a previous zonally propagating MJO event, what promotes its formation? We are faced with the familiar problem of why, during both summer and winter, the ISV tends to form quite often in the Indian Ocean and not elsewhere. Influence of Large-Scale External Forcing such as ENSO and the IOD A second possibility is that large-scale entities

16.5 Subseasonal Summer Monsoon Variability

30°N

The numerical experiments used to support this hypothesis are rather convincing but there does seem to be some question about the time scale. Spectra of the components of the monsoon show biweekly bands principally over the North Indian Ocean.30 A longer period 40-day mode associated with the active and break sequence, apparent in the rainfall statistics over South Asia, appeared more confined to the equatorial Indian Ocean where it bifurcates about the equator and moves eastward.

abundant rainfall

20°N 10°N PV flux 0

SE Trades

10°S 20°S

16.5.5.2 MISO as an Internal Mode: Self-Induction and Self-Regulation

High

30°S

40°S 30°E 60°E 90°E 120°E (i) Active monsoon: moisture flux across South Asia (a) Trade-wind surge MISO theory 30°N deficient rainfall

20°N 10°N

strong PV flux

0 Surge SE Trades

10°S 20°S 30°S 40°S 30°E

High

60°E 90°E 120°E (ii) Break monsoon: Moisture flux diverted south

30°N 5 20°N 4

10°N 0

1

3 2 3’

10°S

4’

20°S 30°S 30°E

60°E 90°E (b) Self-induction MISO theory

120°E

Figure 16.26 Schematics of MISO hypotheses. (a) Influence of extratropical disturbances. Induced changes in the monsoon triggered by extratropical effects (based on Rodwell 1997). (b) A self-induction process whereby Rossby waves “shed” by the convection that is moving slowly eastward with the convective Kelvin wave. The Rossby waves drift polewards on either side of the equator. The arrival of the Rossby wave over South Asia produces an active period in the monsoons. Numbers indicate successive locations of the trailing Rossby wave. Source: From Wang and Xie (1997).

Some hypotheses31 suggest that the summer intraseasonal activity is an inherent instability of the monsoon system that is dependent on the background state. As unstable modes MISOs would occur during the summer monsoon without preferred timing. If the MISO is indeed an internal mode it would have consequences for longer-term prediction of the initiation of the mode, which may be difficult. However, once the MISO has formed there is a fairly predictable sequence of development and progression even though there are differences from one MISO to the next. The life cycle of the composite MISO (Figure 16.25) may suggest a self-induction mechanism32 where interactions within the MISO system promote a further MISO. This is different from an induced mechanism whereby some factor external to the monsoon gyre (e.g. the surge of negative PV) produces a MISO. The surface wind convergence in the western-central basin and the ocean surface warming in the central basin, both critical for breeding a new convective cycle, could each be induced by the anomalous conditions occurring during the peak phase of the previous cycle. The hypothesis is complementary to the self-regulation of the MJO (discussed in Section 15.4.1) that may set the time scale between oscillations. In this sense, an MISO would be primarily an internal oscillation or instability of the monsoon system itself. An extension of the self-induction–self-regulation mechanism is offered by Wang et al. (2005). Once convection forms along the equator, there will be a dynamic response to the symmetric equatorial forcing, such as that found in the simple model response described in Figure 6.12b. An eastward propagating Kelvin wave and a pair of trailing Rossby waves are produced. The slow poleward migration of the Rossby modes is hypothesized to explain the convective zone of the MISO, such as shown in Figure 16.26b. However, why do the Rossby waves migrate poleward seemingly becoming less-equatorially trapped? This is discussed in Section 16.5.5.4 below. 30 Annamalai and Slingo (2001). 31 E.g. Palmer (1994), Webster et al. (1998), Hendon et al. (1999), and Slingo et al. (1996, 1999). 32 See Section 11.2.1 for a definition.

397

398

16 Dynamics of the Large-Scale Monsoon

16.5.5.3

Extensions of the Internal Mode Theory

One interpretation of the Sikka-Gadgil MCZ analysis (Figure 16.22) is that there are two favorable locations for the formation of the ITCZ in the Indian Ocean sector: one over the warm waters of the equatorial Indian Ocean and the other over the heated continent in the vicinity of the seasonal continental trough. In all ocean basins where there exists a surface cross-equatorial pressure gradient (CEPG), the oceanic ITCZ lies off the equator in the direction of the surface pressure gradient and equatorward of the maximum SST. Thus, if there were no continents in the northern Indian Ocean, it might be expected that the Indian Ocean ITCZ would lie north of the equator in the boreal summer and to the south of the equator in the austral summer. However, there does exist a landmass to the north and it is elevated. We have argued extensively that the elevation of the HTP is important when differentiating the South Asian region from other tropical ocean/land configurations producing a climatological precipitation maximum far poleward of the other monsoon regions. In the simplest sense, there are two possible solutions that may exist in the South Asian monsoon sector. The first is a meridional circulation with near-equatorial convection associated with the near-equatorial SST maximum, as discussed in Section 10.1. Such a situation may be similar to Day 0, Figure 16.25. Maximum upward velocity would exist near the equator with subsidence elsewhere to the north and south. The second possibility is a meridional circulation associated with the elevated heat source where rising motion would occur near or over the HTP with subsidence to the south. Is it possible for these two circulations to exist in tandem or for there to be an oscillation between two “attractors” or states toward which a system tends to evolve even with a wide range of initial conditions? Here we have two attractors: two meridional circulations, one near the equator and the other over the heated continent. Figure 16.22 seems to suggest that only one state (indicated by the OLR minimum) exists at any one time or perhaps, occasionally, two states separated by great distances latitudinally. This may be because one cell would enhance or diminish the other through the location of associated subsidence. As suggested earlier, Figure 16.22 could suggest a seesaw between two attractors. The problem of explaining the northward propagation of the ITCZ or the MCZ remains. Another mechanism is needed. 16.5.5.4 MISO

Northward Propagation of the Boreal Summer

Here we discuss two hypotheses, one relating to the role of surface hydrology over the land areas of the monsoon system and the other relating to the role of

strong vertical shear of the background state on Rossby wave propagation. Interactive Land-Hydrology During an active monsoon

period, copious rainfall moistens the land surface. During a break period, surface temperatures increase substantially as the land surface dries. These two states of the land surface produce different sets of surface fluxes. Latent heating dominates during the active phase whilst sensible heating increases during the break phase. However, there will be a latitudinal gradient of heating during an active phase. The total column heating will be dominated by latent heating within the precipitating band, but by surface sensible heating ahead of the band. Relative to the position of land ITCZ or MCZ (Figure 16.22), the total heating will be placed slightly ahead of the cloud band or slightly poleward of the maximum vertical velocity promoting a slow northward march of the ITCZ. This is because the vertical velocity maximum occurs where the total heating of the column is a maximum (see Eq. (3.33)). In this theory, the surface hydrologically does not cause a MISO over land but may be thought of as an “encouragement” for the band to move poleward continually. The same mechanism has been proposed for the northward creep of the East Asian summer rainfall, often called the Mei-Yu or “plum rains.” The Mei-Yu starts as a quasi-stationary band of rainfall in spring and early summer and eventually moves northward. The mean location of the band Mei-Yu, referred to as the Mei-Yu Baiu frontal zone (MYFZ), depicted in Figure 11.24. Murakami et al. (1984) proposed that hydrological surface feedbacks move the band slowly northwards during the summer. The surface hydrology hypothesis was tested by numerical experimentation using a coupled zonally symmetric ocean–land–atmosphere primitive equation model without orography.33 A “continent” was introduced as a cap northward of 14∘ N. Using reasonable values of water storage, which govern the magnitude and longevity of evaporation at the surfaces, and hence surface temperature, slow northward propagations were found across the model continent with roughly the observed time scale. Figure 16.27a presents a schematic of the processes involved in the northward propagation. Noting the phase lead of the vertical velocity ahead of the convective band, this process can perhaps be thought of as a “land-WISHE”, an extension to that discussed for the ocean in Section 8.5(c). 33 The model is described in detail in Webster (1983). Srinivasan et al. (1993) improved ground hydrology in the model. Whereas the model appears to do well over the land areas, an oceanic ITCZ is absent or weak. This is probably due to the simplicity of the ocean model and its zonal symmetry. More complex models (e.g. Shin and Huang 2016) find similar time scales over land and an oceanic ITCZ.

16.5 Subseasonal Summer Monsoon Variability

OCEAN

Rossby Waves in a Strong Vertical Shear Here we consider

CONTINENT

(i) circulation

(ii) surface temperature

(iii) ground water content

(iv) surface albedo

(v) evaporation

(vi) total heating and vertical velocity

propagation QT

w

S

N equ

coast

(a) Surface hydrology propagation feedback z propagation U1 Uz < 0 w ζB < 0 U2

ζB > 0 CONV

CONV

PBL

y

x (b) Vertical shear and propagation Figure 16.27 Schematics of theories for the northward propagation of MISO. (a) The role of land-surface hydrological processes in “encouraging” the northward propagation of the MISO. Heavy dashed line denotes land area. Latitudinal profiles of (i) a typical mean daily meridional profile of circulation; (ii) surface temperature; (iii) ground water content; (iv) surface albedo; (v) surface evaporation; and (vi) the vertical velocity profile (w) and the total columnar heating (QT ) are depicted. Note that the total heating leads to the vertical velocity signifying a northward propagation. Source: From Srinivasan et al. (1993) based on the model results of Webster (1983). (b) Mechanisms by which easterly negative vertical shear of the mean flow (low-level westerlies, upper-level easterlies) generates a northward propagation of moist Rossby waves. Source: After Wang (2005) and Hoskins and Wang (2006).

what may happen to the Rossby waves, once excited, in a realistic background monsoon basic state. To the north of the equator, the vertical shear of the background possesses strong monsoonal flow easterly vertical shear (i.e. 𝜕U∕𝜕z < 0), as is clear between Figures 1.7 and 1.8. The vertical shear in the monsoon regions is strongly negative (𝜕U∕𝜕z < 0), the strongest anywhere in the tropics. Does this strong shear invoke a northward propagation of Rossby waves? In Section 7.5 we considered the impact of vertical shear on the vertical wavenumber and vertical group speed of equatorial Rossby waves. Within such a sheared environment the vertical group speed decreases as the vertical wavenumber increases and the waves are trapped in the vertical, but are there more basic processes at work? Analysis of the NCAR/NCEP reanalysis data set placed the planetary boundary layer convergence leading the MISO convection by about 3∘ of latitude (Wang 2005). This was observed to be associated with a tilting of the horizontal vorticity vector by the vertical shear. In Section 3.1.3 we argued that the tipping effect, contained within the term in Eq. (3.28), was considered to be vanishingly small in the tropics. However, the vertical shear over the Indian Ocean is exceptionally large. To study this factor, a two-dimensional barotropic vorticity equation was derivedVI that relates changes in the barotropic vorticity to vertical shear (U z ) in a non-divergent zonally symmetric atmosphere, i.e.: ( ) 𝜕wp 𝜕𝜁B ∝ −U z (16.16) 𝜕t 𝜕y where 𝜁 B is the barotropic component of vorticity and wp is the vertical velocity in pressure coordinates. The mean flow with easterly shear has a large equatorward horizontal relative voracity of order 10−3 s−1 . The right-hand side of Eq. (16.16) represents the tipping term where a horizontal vortex tube (resulting from the vertical shear) projects on to the vertical component of vorticity, 𝜁 B , when tipped by a latitudinal gradient of vertical velocity. Given the magnitude of U z the contribution of the tipping of the vortex tubes by the vertical shear to 𝜁 B may be substantial. Consider, now, an MISO located at some latitude north of the equator coinciding with a maximum in 𝜔. To the south of the MISO the gradient of 𝜔 will be negative whilst to the north of the MISO the gradient is positive. As U z < 0 across the entire domain, anticyclonic vorticity is generated behind the convective part of the MISO while cyclonic vorticity is generated ahead. The induced convergence/divergence in the boundary layer provides a continual destabilization poleward of the MISO that as a consequence propagates northward. The northward propagation mechanism is illustrated schematically in Figure 16.27b.

399

400

16 Dynamics of the Large-Scale Monsoon

In summary, we have posed two self-induction complementary mechanisms for the MISO, each providing an impetus for a northward migration of convection. The first provides migration by modifying heating ahead of the MISO through land–atmospheric interaction. The second mechanism generates cyclonic vorticity poleward of the MISO and anticyclonic vorticity behind.

16.6 Higher-Frequency Monsoon Variability We now consider, at high frequency ( 0

0

25

25 26 27 28

23 0

JJA31 29

15°S

260 270

28

23

30

15°N

270 -5

5

30°N

25 24

0 24

270

2276

30°S

30

latitude

15°S

230

0

0°

240 250 26700 2

260

21 22

MAM

0270

30°N

280

0 23

-5

15°S 30°S

2300 24 2500 26

27600 0 2 25

29

29

210 220

-5

28

0°

270

DJF 280

0 -5

15°N

2425 267 2

DJF

22 0

20

DJF

1.0 m/s >

0

0

19

2 40

30°N

0

-5

600 227 280

–5

19

18

80°E

280

2254

23

30°S 22 21 20 40°E 60°E

0

290

15°S 270

100°E 120°E 40°E

60°E

80°E

100°E 120°E 40°E

60°E

80°E

100°E 120°E

longitude Figure 17.1 Mean seasonal climatologies of SST (∘ C) (left-hand column), near-surface (925 hPa) wind vectors (m s−1 ) (middle column), and outgoing long-wave radiation (W m−2 ) (right-hand column) for DJF, MAM, JJA, and SON. SST isotherms every 1 ∘ C. Bold contours indicate 20, 25, and 28 ∘ C. Wind speeds >5 m s−1 shaded yellow, >10 m s−1 , red. OLR contours every 10 W m−2 . Bold black contour encloses 200 W m−2 during the spring months and 181 W m−2 over the entire year, compared to an annual average of about 145 W m−2 for the western Pacific (Webster et al.

1998). The annual net surface flux averaged over the entire North Indian Ocean is about +50 W m−2 or at least a factor of two or three larger than in the western Pacific warm pool. Overall, it is relatively clear that over a year there is a strong flux of heat into the North Indian Ocean that is greater than into the western Pacific Ocean. Furthermore, there is a far larger seasonality in this flux than in the western Pacific Ocean, with maximum values occurring in spring and early summer.

409

410

17 The Coupled Monsoon System

Table 17.1 The long-term average components of the surface heat balance for the North Indian Ocean (a) north of the equator, (b) from the equator to 10∘ N, and (c) north of 10∘ N. Units are W m−2 . The net solar radiation, net longwave radiation, latent heat flux, sensible heat flux, and the net flux at the surface are denoted by S, LW, LH, SH, and NET, respectively. Surface flux (W m−2 )

S

dSST/dt (year−1 )

LW

LH

SH

NET

Model

Obs

(a) Entire North Indian Ocean DJF

125

−57

−103

−4

15

2.2

MAM

223

−49

−82

JJA

190

−39

−117

SON

187

−48

Annual

181

−48

0

−2

90

13.4

2

0

34

5.1

−2

−88

−3

50

7.5

1

−98

−2

47

7.1

∼0

24

(b) North Indian Ocean: 0∘ N–10∘ N DJF

181

−49

−96

−4

MAM

202

−47

−88

−3

65

JJA

178

−42

−116

−2

17

SON

186

−46

−87

−4

50

−105

−5

3

−1

112

0

44

−2

42

(c) North Indian Ocean: north of 10∘ N DJF

174

−61

MAM

232

−48

−74

JJA

191

−35

−113

SON

179

−47

−88

Source: Data from COADS (Oberhuber 1988). Heating rates of a 50 m layer for the entire North Indian Ocean are shown in the right-hand column (K year−1 )

(a) 850hPa zonal wind component (m s–1)

(b) Heat flux into North Indian Ocean (W m–2)

2

20 20°N

20°N

40 –40

10°N

6 2

20 + 40

10°N –2

–

20

4

20

0°

0°

Figure 17.2 Time–latitude sections of the annual cycle of (a) 850 hPa zonal wind component (m s−1 ) at 15∘ N and (b) the heat flux (W m−2 ) into the North Indian Ocean averaged from the equator to 20∘ N. Bold black lines denote zeros. Source: Data from NCEP/NCAR reanalyses.

+

60 +

80 20

10°S

–6

10°S –8

60

40

–10 20°S

20°S

–80

–6 J F M A M J J A S O N D month

–40

20

J F M A M J J A S O N D month

Figure 17.2 presents estimates of the annual cycle of the 850 hPa zonal wind component and the heat flux into the North Indian Ocean along 90∘ E. Of primary importance is the relatively weak magnitude of the

lower-tropospheric winds in the boreal spring in the 0∘ N–20∘ N band accompanied by the very strong heat flux into the ocean. It is straightforward to estimate the changes in SST resulting from the observed net fluxes

17.2 Processes Determining the Indian Ocean SST

where F net is the net heat flux into the upper ocean. Heating rates, assuming a 50 m surface layer with zero heat flux at that level, are listed in the right-hand column of Table 17.1. These calculations suggest a very different behavior in the evolution of SST in the North Indian Ocean compared to that observed. Figure 17.1 displays a rather gradual change in SST from one season to the next, whereas the fluxes, listed in Table 17.1, suggest that the North Indian Ocean would be continually warming at an annual rate > 7 ∘ C year−1 . Alternatively, one may use a sophisticated one-dimensional mixed-layer model to make the same calculation. Webster et al. (1998) used the model of Kantha and Clayson (1994), finding results quite similar to those listed in Table 17.1.

17.2 Processes Determining the Indian Ocean SST

Meridional Ocean Heat Transport (1014 W) (a) Hsiung et al. (1989)

latitude

20 10 0 –10 10 –20 J

0

–10

0 –20

F

M

A

M

J

J

A

S

O

N

D

O

N

D

(b) Hasterath and Greishar (1993) 20

latitude

listed Table 17.1. The simplest method is to assume that the net flux is spread through an upper ocean layer of some defined depth for which the heating rate may be written as 1 dF net dT =− (17.1) dt 𝜌Cp dz

0

0

10 0 –10 –20

–10

-5 J

F

M

–10

A

M

J J month

A

S

Figure 17.3 Comparison of the seasonal cycle of zonally averaged northward heat transport in the Indian Ocean: (a) Hsiung et al. (1989) and (b) Hastenrath and Greischer (1993). The two annual cycles are quite similar with broad northward cross-equatorial ocean transports of heat during the fall and southward transports during the spring and summer and early fall.

Figure 17.3 displays two empirical estimates of the annual cycle of the zonally averaged meridional ocean heat transport plotted as a function of latitude.5 They are quite similar. Between spring and the fall the cross-equatorial heat flux is southward with a maximum amplitude of −2 PW. During the boreal winter the ocean heat flux reverses with magnitudes of about +1.5 PW. Furthermore, both estimates show a maximum heat transport on the south side of the equator between 10∘ S and 15∘ S. These results allow speculation that during the boreal winter cooling of the northern Indian Ocean is offset by a northward transport of heat, whereas during spring and summer, excessive heating is balanced by a southward transport. Levitus (1987) proposed that the transport was the result of Ekman drift (see Section 3.2.4). Levitus’ hypothesis has gained support by subsequent theoretical and modeling studies.

In order to examine the dynamics of ocean heat transport, we now look at the results of a numerical model described in some detail in Appendix K.II Figure 17.4 shows computations of the annual cycle of meridional ocean heat transport and heat storage changes in the North Indian Ocean forced by the annual cycle of climatological surface winds and heating. The net surface flux into the North Indian Ocean appears to have a semiannual variation, perhaps due to a combination of net solar heating and cooling by evaporation. Evaporative cooling is largest in the summer and is associated with the strong monsoon winds. There is also a second maximum in winter associated with the winter monsoon. Solar heating also has two minima: in summer where cloudiness has increased with the onset of the monsoon and in winter because of solar declination. Together these two competing fluxes provide the double maximum evident in Figure 17.4a. The major components of the heat balance are the lateral heat transport and the storage terms. A very strong southward flux of heat occurs during the spring and early summer, and a reverse flux is evident during the winter, when the north Indian Ocean is losing considerable amounts of heat by both evaporation and vertical turbulence mixing that entrains colder water from below the thermocline. The net heat flux across the equator is made up of opposing flows in the upper and lower layers of the model and may be thought of as a seasonally reversing meridional ocean circulation.6 Figure 17.4b

5 From Hsiung et al. (1989) and Hastenrath and Greicher (1993).

6 See McCreary et al. (1993).

If the surface of the ocean column for a given surface flux is observed to heat less than theoretical estimates, then there must be a significant role for dynamics. There are only two possible processes. These are the horizontal advection of heat by ocean currents and the change of heat storage in the ocean column through vertical mixing of heat, or a combination of both processes. 17.2.1

Ocean Heat Transport

411

17 The Coupled Monsoon System

(a) Heat balance of the North Indian Ocean (PW) 2.0

Meridional atmos latent heat transport

1.0 PW

412

Heat flux into NIO

0

–1.5

–2.0

Meridional ocean heat flux (Qv) Jan

Feb

Total heat storage change (Qt) Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

(b) Meridional ocean heat flux (PW) 20N 0

10N +1.0

0

+0.5 –1.5

+1.5

10S –0.5

–1.0

May

Jun

0

20S Jan

Feb

Mar

Apr

Jul

Aug

Sep

Oct

Nov

Dec

(c) Meridional atmospheric latent heat flux (PW) 20N 10N 0

–0.5

0

+1.5

–1.0

–0.5

+1.0

0

+0.5

10S 20S Jan

Feb

Mar

Apr

May

–1.5 –1.0 –.5

Jun 0

Jul

Aug

Sep

Oct

Nov

Dec

+.5 +1.0 +1.5

PW Figure 17.4 Heat transport in the Indian Ocean: (a) annual cycles of northward ocean heat transport across the equator (Eq. (17.2a)), the net flux of heat into the North Indian Ocean and the changes in heat storage (Eq. (17.2b)) in the North Indian Ocean. (b) Latitude–time section of the annual cycle of northward ocean heat transport in the Indian Ocean. (c) Latitude–time section of the annual cycle of the northward flux of latent heat averaged throughout the troposphere. In the two lower panels, dashed lines indicate negative (southward) transport. Units in PW. Source: Based on the results of Chirokova and Webster (2006).

presents a latitude–time section of the annual cycle of the climatological meridional oceanic heat flux averaged across the basin. The year is divided into a period of northward heat flux in winter and spring, and a slightly stronger southward heat flux between late spring and early fall. Maximum transport occurs at all seasons close to 10∘ S near the zone of maximum SST gradient. Figure 17.4c shows a similar plot but for the northward

flux of latent heat throughout the troposphere. The latent heat flux was calculated from the data used to force the ocean model. Comparing Figures 17.4b and c, we find a remarkable feature. The magnitude of the dynamic heat transport in the ocean is almost equal and opposite to that of atmospheric latent heat flux. That is, the oceans are transporting as much heat from the summer hemisphere

17.2 Processes Determining the Indian Ocean SST

to the winter hemisphere as the atmosphere is transporting from the winter hemisphere to the summer hemisphere. Thus, to at least first order, the heat balance of the coupled monsoon system in the Indian Ocean appears closed. 17.2.2

Changes in Ocean Heat Storage

Greater insight can be gained by exploring details of the heat storage represented by Eq. (K.6). The storage term has two components, one relating to the change in depth of a layer of a certain temperature and one related to the change of temperature of a layer of fixed depth. Rewriting Eq. (K.6), we find these two components, integrated in the vertical through the ocean column, may be written as 𝜕T(t) H(t) dx dy (17.2a) QtΔT (t) = 𝜌w Cw ∫∫ 𝜕t and QtΔh (t) = 𝜌w Cw

∫∫

T(t)

𝜕H(t) dx dy 𝜕t

(17.2b)

representing general forms of the terms on the right-hand side of Eq. (K.6). Figure 17.5 shows the annual cycle of the two components of the storage terms plotted together with the net surface heat flux and the northward meridional heat transport. During the winter, QtΔT is matched by QtΔh . During spring, the polarity changes with QtΔT outweighing QtΔh . This seems reasonable as the net heat flux into the North Indian Ocean is now large

Annual cycle of heat storage in the NIO 3 2

Heat storage temp-change (QtΔT)

PW

1

Heat flux into NIO

0 –1 –2

Heat storage depth-change(QtΔh)

Meridional ocean heatflux Qv

Total heat storage change

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Figure 17.5 Annual cycle of the energy balance of the North Indian Ocean with the rate of heat storage broken down into storage associated with temperature change with a constant depth of the mixed layer and that associated with changes in depth of the mixed layer at constant temperature, as expressed in Eqs. (17.2a) and (17.2b). The figure shows clearly that heat storage in the basin can change either through a deepening of the upper layers and/or by increasing temperature. Units in PW. Source: From Sahami (2003).

and positive. However, the surface winds are light, thus minimizing changes in the depth of the mixed layer. Only during the summer do the two storage terms have the same sign. The relationship between the meridional transport of heat and the change storage with temperature of the North Indian Ocean is complicated. During the spring and summer, the two track each other very well. However, the relationship is opposite during the fall and winter when the meridional heat transport and QtΔh tend to covary. Thus, although the transport of heat in and out of the North Indian Ocean matches the total heat content of the North Indian Ocean, when the surface heat flux is taken into account the changes in heat content are not always manifested as changes in surface temperature. 17.2.3 Spatial and Temporal Variability of Ocean Heat Flux The results presented above provide a basic summary of the gross behavior of the Indian Ocean heat balance from a zonally averaged perspective. However, it would be a mistake to think of the zonally averaged heat flux and storage terms as representing the flow at all longitudes. The longitudinal structure of the cross-equatorial ocean heat transport (Figure 17.6) shows a rich variability across the Indian Ocean. The model was forced with a long-term climatological annual cycle of winds and surface flux data. The computed heat flux calculated along the equator appears in Figure 17.6a. There are three distinct regimes of advection: narrow regions of boundary advection and counter-boundary advection in the west and a lower amplitude drift over the remainder of the interior ocean. At the equator, a strong heat flux occurs within the Somalia Current. The flux is initially southward before changing to a northward flux during April for the rest of the calendar year. The flux during both of these periods exceeds values greater than 2.5 PW but is reduced by the counter-heat transport. During the summer and fall there is a southward heat flux close to the western boundary. To the east, throughout the interior of the Indian Ocean, there is a widespread but far weaker heat flux of the opposite sign to the western boundary flux. As it turns out, the structure boundary counter-current is consistent with the dynamical interpretation of Anderson and Moore (1979), to which the interested reader should refer. The annual cycle of the longitudinally integrated heat flux along the equator (Figure 17.6a(iii)) exhibits maximum southward values (∼ −2 PW) during the boreal summer and fall and roughly the same magnitude but with opposite sign during the winter and spring. This

413

17 The Coupled Monsoon System

(a)Zonal heat transport (PW) equator

(b) Zonal heat transport (PW) 10°S

(iii)

(i)

(iii)

300

300

300

200

200

200

200

100

100

100

100

0 0

0

0

40

60

80 longitude

100

–2 –1 0 +1

zonal heat transport (PW)

(ii)

2 1

Net annual zonal transport: –0.253 PW

0 –1

–2

40

60 –1

80

100 0

+1

+2 PW

Julian day

300

merid. heat trans (PW)

Julian day

(i)

merid. heat trans (PW)

414

40

60

80 longitude

0

100

8

–3 –2 –1 0 +1 +2

(ii)

4

Net annual zonal transport: –0.493 PW

0 –4 40 –2

zonal heat transport (PW)

60 –1

80 0

100 +1

+2 PW

Figure 17.6 Meridional ocean heat transport across (a) the equator and (b) 10∘ S forced by climatological reanalysis products. Panel (i): time–longitude distribution of heat transport with warm colors indicating a northward transport and cold colors a southward transport relative to the color bar. Panel (ii) shows the annual average of the longitudinally averaged cross-latitude heat flux. Panel (iii) shows the zonally averaged time integral of the ocean heat flux. The net annual transport across the equator is −0.253 PW compared with −0.493 PW across 10∘ S. NCEP-NCAR reanalysis wind and flux data was used to drive the ocean model. Units are PW. Source: From Sahami (2003).

estimate is in concert with the empirical and model estimates (Figures 17.3 and 17.4, respectively). Integrated through the entire year (Figure 17.6a(iii)) the overall flux is weakly southward with magnitude −0.25 PW. The overall northward flux in the western Indian Ocean is matched by the weaker but more spatially extensive southward drift in the eastern Indian Ocean. Figure 17.6b shows the same analysis but along 10∘ S. The northward transport along the coast is associated with the East African Coastal current, or Zambezi current, which remains northward throughout the year. The zonally averaged heat flux (panel ii) is similar to the equatorial sector, except for a slight delay of the change to northward transport in the early boreal winter. Within the interior of the ocean the drift is to the south throughout the year. The latitude is on the equatorial side of the persistent southeasterly trade winds. Finally, the annually averaged fluxes (panel iii) are larger in each sector, summing to nearly double the values at the equator. From the calculations above, it is clear that without ocean transport across the equator and changes

in the heat storage in the North Indian Ocean, the cross-equatorial buoyancy gradient during early summer, induced solely by the surface fluxes, would be much larger than observed. Yet the processes that accomplish the cross-equatorial transport of heat in the ocean are essentially wind driven. In turn, the atmospheric circulation is driven by surface fluxes, each a strong function of surface temperature, and heating gradients associated with the induced buoyancy gradient and the configuration of ocean and land areas. We recall from Figure 17.4 that the latitudinal atmospheric flux of heat nearly balances the ocean heat flux. Clearly, the annual cycle in the Indian Ocean is a result of coupled ocean–atmosphere dynamics. A critical element of the meridional heat transport is that it is accomplished to a large degree by Ekman processes, an important conclusion of Levitus (1987). In invoking Ekman transport, though, we have to account for possible singularities at the equator as the factor (1/f ) enters into the dynamics (Section 3.2.4) where f is the Coriois parameter. We will consider this issue shortly.

17.3 Do Ocean Heat Fluxes Regulate the Annual Cycle of the Monsoon?

17.3 Do Ocean Heat Fluxes Regulate the Annual Cycle of the Monsoon?

latitude

20°N

warmer

EQU

20°S

cooler 40°S 30°E

60°E

20°N

cooler latitude

Figure 17.7 presents a very simple schematic diagram representing the regulation of the annual cycle in the Indian Ocean monsoon system. The two panels represent summer and winter with relatively warm water (shaded) in the northern basin during summer and in the southern basin during winter. Surface winds, similar to those shown in Figure 17.2, are superimposed as solid contours. In each season there is a strong flow from the winter to the summer hemisphere with a characteristic monsoon “swirl.” The broad blue arrows to the right of the panels represent the ocean Ekman transports associated with the surface wind forcing. Irrespective of the season, the Ekman transports are from the summer hemisphere to the winter hemisphere. Only in the southern hemisphere south of 15∘ S is the Ekman drift unidirectional. This is because of the constancy of the southern hemisphere Southeast trade winds, which ensure that Ekman drift is always to the south. The total effect of the feedback is to cool the upper ocean of the summer hemisphere and warm the winter hemisphere, thus reducing the SST gradient across the equator at all times of the year. These transports are sufficiently large to be responsible for reducing the heating of the upper layers of the summer hemisphere to values less than the computational estimates listed Table 17.1. In summary, the amplitudes of the seasonal cycle of the monsoon are modulated through the negative feedbacks between the ocean and the atmosphere. Later we will argue that an anomalous monsoon (strong or weak, for whatever reason) will introduce a biennial variability to the system through a similar negative feedback system.

150°E

EQU

20°S

net Ekman heat transport

Balanced Interhemispheric Heat Fluxes

90°E 120°E longitude

(b) Boreal Winter

40°N

17.3.1

net Ekman heat transport

Clearly, the oceanic transport of heat by wind-driven ocean currents is critically important in determining the state of the Indian Ocean. However, the winds that produce the ocean currents are also part of the monsoon system. It should also be remembered that the ocean transport of heat is almost exactly equal and opposite to that of latent heat in the atmosphere (Figure 17.4). It is an obvious extension to realize that the monsoon is a coupled ocean–atmosphere system. We now attempt to determine the degree to which feedbacks are responsible for the annual cycle of the monsoon or, in fact, its interannual variability.

(a) Boreal Summer

40°N

warmer 40°S 30°E

60°E

90°E 120°E longitude

150°E

Figure 17.7 Heat fluxes in the upper Indian Ocean and the regulation of the seasonal cycle of the Indian Ocean for (a) the boreal summer (June–September) and (b) the boreal winter (December–February). Curved solid lines indicate near-surface winds forced by the large-scale pressure gradient associated with the cross-equatorial heating gradient responding to the zones of “warmer” and “cooler” SST. Dashed blue arrows denote the wind-forced Ekman drift within the ocean away from the boundaries and the direction of the associated heat flux. The large blue arrow on the right denotes the sense and magnitude of the seasonal net zonally averaged heat flux. Overall, the wind-driven southward flux of heat in the boreal summer tends to cool the North Indian Ocean, while the northward flux during the winter tends to heat the North Indian Ocean, thereby reducing the SST gradient at all times of the year. The coupled ocean–atmosphere interaction described in the figure imposes a strong negative feedback on the system regulating the seasonal extrema of the monsoon. Source: From Loschnigg and Webster (2000).

17.3.2 Cross-Equatorial Ocean Ekman Heat Transport In Section 3.2.4, we noted that wind-driven Ekman transport was an important process moving mass from one location to another and, if there is a gradient of a quantity (e.g. temperature), the advection of that

415

416

17 The Coupled Monsoon System

quantity. The vertically averaged Ekman transport is to the right of the wind in the NH and to the left south of the equator, yet the Ekman transport appears singular at the equator, being proportional to 1/f . However, studies have indicated that the wind forcing, defined by 𝜏 x /f , is well defined at the equator and that a singularity in Ekman transport does not exist, arguing that mass transport and the response to the pressure gradient explain cross-equatorial flow (and hence mass and heat transport) rather than adjustment to wind forcing. We follow Miyama et al. (2003),III who argues that Ekman transports at the equator are not singular and that cross-equatorial Ekman transport is a well-behaved viable solution. Consider a simple 1 1/2-layer ocean model where the equations of motion may be written as 1 −fhv + ̃ g𝜕h2 ∕𝜕x = 𝜏 x 2 1 g𝜕h2 ∕𝜕y = 0 fhu + ̃ 2 𝜕h∕𝜕t + 𝜕hu∕𝜕x + 𝜕hv∕𝜕y = 0

(17.3a) (17.3b) (17.3c)

Here h is the depth of the upper layer and ̃ g is the reduced gravity introduced in Section 3.2.2; 𝜏 x is the zonal wind stress. Consider, now, the response of this system with an anti-symmetric wind stress (e.g. westerlies to the north of the equator, easterlies to the south, or vice versa) about the equator, similar to observations, i.e. y (17.4) 𝜏 x = 𝜏0 X (x) P(t) L where 𝜏 0 is a constant, L is some latitudinal scale (say 5∘ latitude), X (x) is an arbitrary longitudinal structure and P(t) is a slowly varying time scale. Differentiating Eqs. (17.3a) and (17.3b) with respect to latitude and longitude, respectively, and substitution into Eq. (17.3c) leads to ( ) 𝛽 𝜕 𝜏x 𝜕h ghhx = − − 2̃ (17.5) 𝜕t f 𝜕y f where 𝛽 = f y . For the form of wind forcing used, the second and third terms of Eq. (17.5) are independent and thus ( ) 𝜕 𝜏x =0 (17.6) 𝜕y f Substituting the wind stress Eq. (17.4) in Eq. (17.5), the Ekman pumping velocity from (3.80) we = curl(𝜏/f ) vanishes at the equator. Miyama et al. assume that the initial state is at rest so that h = H for all time. Thence, Eq. (17.6) implies that h does not change, resulting in the solution that 𝜏 1 h = H, u = 0, and v = − x = − 𝜏y x (17.7) fh 𝛽h Expressions (17.6) and (17.7) indicate that no pressure gradients are generated by this wind field so that the

perturbation and the flow field are composed entirely of Ekman drift. In Eq. (17.7), it appears that the Ekman transport (−𝜏 x /f ) is equal to the Sverdrup transport (−𝜏 y x /𝛽). In summary, Miyama et al. presented a physical explanation of how there is Ekman transport at the equator in the interior of the Indian Ocean. If true, we can assume that there is a seamless heat and mass transport between the North and South Indian Ocean across the equator. There is one further aspect of cross-equatorial transport to consider and for which there is observational evidence. As the Ekman time scale goes as 1/f , the Ekman spin-up time is infinite at the equator. Figure 17.8 shows a series of vertical current sections from a cruise in the eastern equatorial Indian Ocean during the boreal summer of 1964.7 Data from five stations are reproduced. All of the stations are in the NH. Within the Somalia Current, off the coast of East Africa, Swallow and Bruce found that south of about 5∘ N the Ekman deflection was to the left, showing SH Ekman layer characteristics (red arrows). North of 5∘ N the characteristics are distinctly those of a Northern Hemispheric Ekman layer with a deflection of the subsurface currents to the right of the surface wind (blue). Thus, it appears that the SH Ekman layer is advected across the equator, indicating, from a second perspective, that the equator is not a singular point. This is reasonable if one remembers that the spin-up (or spin-down) of an Ekman layer goes as 1/f . In other words, the spin-up time of the Ekman layer is far slower than the advective time scale. Eventually, as the latitude increases, the spin-up time shortens and the local winds change the structure of the Ekman layer north of 4∘ N, as shown in Figure 17.8. The observations of Swallow and Bruce support the theoretical development of the Miyama et al. (2003) study.

17.4 Variability Within the Coupled Monsoon System In previous sections we have noted some indication of atmospheric variability of the monsoon on a variety of timescales (e.g. Figure 16.3). Here we consider how coherent variability may occur in the coupled ocean–atmosphere system. 17.4.1

Biennial Variability

We have remarked on the strong bienniality in the spectra of South Asian rainfall, implying that above-average monsoon rainfall is followed, one year later, by below average rainfall. A similar variability has been 7 Documented by Swallow and Bruce (1966).

17.4 Variability Within the Coupled Monsoon System

Ekman transport in the Somalia Current 5535

currents (colored solid arrows)

5538

0 10 20 30 40 50 cm/s wind scale (bold black arrows) 0

5

10

10m

15

60 m

m/s

10m

40m

40m

100m

70m

50oE

45oE

55oE

5530

10m

10oN

10oN

40m 5549

East Africa

10m 40m

5535

5oN

5525

5530

5527 10m 40m

5oN

5538 5542

EQU

5525 5527

45oE

EQU

50oE

55oE

Figure 17.8 Vertical sections of ocean currents obtained from the RRS Discovery of the British National Institute of Oceanography and the RV Argo of the University of California during the International Indian Ocean Expedition. Sections taken in the Somalia Current during the established southwest monsoon. Five vertical sections in the NH are shown. The heavy solid arrows indicate the surface wind at the station relative to bar on the left. The lighter arrows indicate measured currents at 10 m, 40 m, and 60 m. The dashed circles indicate estimates of the uncertainty in the measurements. Stations between the equator and 4∘ N show current deflections (red) to the left of the surface wind. North of 4∘ N the deflections are to the right (blue). Source: Data from Swallow and Bruce (1966).

noted in Indonesia, North Australia, and East Asia.8 Lower-tropospheric wind fields associated with the bienniality in the SST fields were shown to possess an out-of-phase relation between the Indian Ocean and the Pacific Ocean basins, with an eastward phase propagation from the Indian Ocean toward the Pacific Ocean providing possible links between monsoon variability and low-frequency processes in the Pacific Ocean. Many early studies were aware of these connections.9 In the following paragraphs we briefly discuss a number of theories on the biennial nature of the monsoon. 17.4.1.1

Ocean-Atmosphere Feedbacks

For the quasi-bienniality observed in North Australian rainfall Nicholls (1978) proposed the first ocean–atmosphere interaction theory. Simple empirical 8 E.g. Tian and Yasunari (1992) and Shen and Lau (1995). 9 E.g. Kutsuwada (1988), Rasmusson et al. (1990), Ropelewski et al. (1992), Shen and Lau (1995), Yasunari and Seki (1992), Clarke et al. (1998), and Tomita et al. (2004).

relationships between the tendencies of SST and surface pressure were set up for both summer and winter, producing an out-of-phase relationship between surface pressure and SST on a biennial time scale. Specifically, it was hypothesized that: … throughout the year, above-normal equatorial SST will tend to decrease the atmospheric pressure of the tropics and subtropics. It is proposed that the anomalous pressure pattern thus produced will cause variations in the original SST anomaly. The sense of the variation (increase or decrease) in the SST anomaly will depend on the direction of the prevailing wind which itself varies seasonally. Thus, during part of the year, a positive pressure anomaly will be associated with an increase in SST, while during the remainder of the year the same pressure anomaly would be associated with a decrease in SST…. Nicholls (1978, p. 1507)

417

418

17 The Coupled Monsoon System

Meehl theory of monsoon biennial variability (a)

DJF(0)

JJA(0) strong monsoon

H decreased warm snow

L

strong E. African rainfall

(b)

warm SST

cool SST weak Australian monsoon

(c)

warm SST

DJF(+1)

warm SST

cool SST

(d)

cool SST

JJA(+1)

H L

H cold weak E. African rainfall

weak monsoon

increased snow

cool SST

warm SST

strong Australian monsoon

cool SST

warm SST

cool SST

warm SST

Figure 17.9 Meehl’s Theory of the biennial component of the monsoon circulation showing the progression from (a) the winter before a strong Asian monsoon through (b) the strong monsoon season to (c) the northern winter after the strong monsoon before a weak monsoon to (d) the following weak monsoon. Here, the bienniality in the magnitude of the monsoon relies on the changes in SSTs in northern equatorial SSTs induced by strong and weak monsoons. Source: From Meehl (1994a).

The proposed system was simple and not particularly different from the more complex theories discussed below. However, Nicholls’ acknowledgment that the bienniality of the tropical system depends on the coupled nature of the ocean and the atmosphere was prescient! 17.4.1.2

Pacific Warm Pool Seasonal Cycle Instability

It has been suggested that the biennial oscillation may be produced by an air–sea interaction instability involving the mean seasonal wind cycle and evaporation.10 It was argued that similar instabilities are not possible in the Indian Ocean and that Indian Ocean oscillations are the result of Pacific instabilities. However, this seems inconsistent with our observations of strong bienniality in the Indian Ocean and in the South Asian rainfall. 17.4.1.3

Indian Ocean Feedbacks I, The Meehl Theory

It has also been proposed that biennial oscillations occur as a natural variability of the monsoon coupled ocean–atmosphere–land monsoon system. In this view, the source of the biennial oscillation resides in feedbacks residing in the Indian Ocean11 and not the Pacific 10 Clarke et al. (1998). 11 Meehl (1994a,b, 1997).

Ocean. A schematic diagram of the four phases of the biennial monsoon system is shown in Figure 17.9. The panel (a) represents the winter season (DJF: 0) prior to the first summer monsoon season (JJA: 0) showing anomalously warm SST in the central and western Indian Ocean and cooler SSTs in the eastern Indian Ocean and the Indonesian seas. Anomalously, warm SSTs in the Indian Ocean herald a stronger monsoon, supposedly by a heightened surface hydrological cycle (panel b), consistent with the hypothesis posed in (i), above. A stronger monsoon is accompanied by stronger wind mixing and evaporation, which leads to cooler SSTs in the central and eastern Indian Ocean. A reversal of the east–west SST gradient produces a stronger North Australian monsoon (panel c). In turn, the colder-than-normal Indian Ocean leads to a weak Indian summer monsoon (panel d). The theory also notes that a strong South Asian monsoon is preceded by a strong East African monsoon and a weak South Asian monsoon by a weak East African monsoon. Presumably, the oscillation of the East African monsoon is associated with the change of the longitudinal SST gradient or, perhaps, the Indian Ocean Zonal Mode.

17.4 Variability Within the Coupled Monsoon System

Indian Ocean Feedbacks II: A Dynamic Ocean

ENSO and Internal Dynamics

Figure 17.11 displays the variability of components of the heat balance (cross-equatorial heat transports, heat storage change and net heat flux) integrated over the North Indian Ocean for three periods (1969–1973, 1984–1988, and 1994–1998) extracted from the 40-year integration. The total period contains nine El Niño and eight La Niña periods. Each component of the heat balance shows large interannual variability with amplitudes of the variability being nearly as great as the climatological annual cycle. Also, variability occurs in all three quantities at all times throughout the year. The time series were stratified relative to years exhibiting ENSO extrema and also relative to strong and weak

warmer

latitude

20°N EQU 20°S cooler 40°S 30°E 60°E

90°E 120°E longitude

150°E

(b) Weak Boreal Summer monsoon 40°N cooler 20°N EQU 20°S warmer 40°S 30°E 60°E

90°E

120°E

net Ekman heat transport

17.4.1.5

(a) Strong Boreal Summer monsoon 40°N net Ekman heat transport

The sequence of SST displayed in Figure 17.9 follows observations quite closely, especially the oscillation of precipitation from year to year hinted at in Figure (16.1). Whereas it would seem that an oscillation of the SST in the Indian Ocean should be tied to the variability of the monsoon rains, there are two problems with the theory. It is difficult to account for the persistence of SST anomalies for the length of time between the end of one summer season and the start of the next. It must be presumed that thermodynamical processes alone accomplish this longevity. However, this is unlikely! The e-folding time of 50 m mixed layer with a 1 K anomaly at 303 K is between 40 and 60 days. Thus, it would seem that there must be dynamical ocean processes at work in the Indian Ocean in order to increase the persistence of the anomalies. Such dynamical ocean processes can be found in negative feedbacks used earlier to argue for a regulated annual cycle (see Figure 17.7). How the negative feedback may come into play on interannual time scales is shown schematically in Figure 17.10. Assume, for example, that the North Indian Ocean SSTs were warmer than normal in a particular boreal summer (top panel). The ensuing stronger monsoon flow would produce greater Ekman drifts and thus increases in fluxes of heat toward the winter hemisphere. If the North Indian Ocean were cooler than average for some reason (bottom panel), then one would expect a reversal in compensation through a reduced Ekman drift due to lighter surface winds accompanying the reduced cross-equatorial pressure gradient. In a sense, the interannual regulation described here is very similar to that in the Meehl theory shown in Figure 17.9, but with a dynamic coupled component. Like the Meehl theory, the natural time scale of the negative feedback oscillation is biennial. In addition, because ocean heat transport is an integral part of the theory, it provides a dynamic element to Meehl’s theory.

latitude

17.4.1.4

150°E

longitude Figure 17.10 Same as Figure 17.7 except for the regulation of the interannual monsoon system and the impact of an anomalously strong and weak monsoon on the Indian Ocean. A strong monsoon drives an enhanced southward Ekman heat transport, leading to a cooler North Indian Ocean that, in turn, leads to a weak monsoon. A weak monsoon, though, is associated with a reduced Ekman heat transport. Following the reasoning of Meehl (1997; see Figure 17.9), the tendency will be to produce an anomalous monsoon of the opposite sign the following year and introduce a strong biennial component into the system, thus producing a regulation of the interannual variability of the monsoon. This may be thought of as the “modified” Meehl hypothesis but with wind-driven heat transports taken into account. Source: Based on Webster et al. (2002).

monsoons. Figure 17.12 shows the differences in heat fluxes between El Niño and La Niña and between strong and weak monsoons, the latter being depicted earlier in Figure 16.5. The differences are large and systematically correlate well with the annual cycle. During a strong monsoon season, the anomalous flux is southward in the late spring/early boreal summer and negative during the boreal fall. This enhanced southward transport during spring is consistent with stronger surface winds associated with the strong monsoon driving a stronger Ekman drift. The El Niño–La Niña composite

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17 The Coupled Monsoon System

NIO heat balance anomalies

PW

1

(i) 1969–1973

0

–1 1969

PW

1

1970

1971

1972

1973

1985

1986

1987

1988

1995

1996

1997

1998

(ii) 1984–1988

0

–1 1984

1 PW

420

(iii) 1994–1998

0

–1 1994 heat storage

net surface heating

cross-equatorial heat transport

Figure 17.11 Time series of the components of the North Indian Ocean heat budget for the three periods 1969–1973, 1984–1988, and 1994–1998. The seasonal cycle has been removed and a nine-point smoothing has been applied. Curves denote net heat flux (blue line), rate of change of heat storage (red), and meridional ocean cross-equatorial heat transport (black). Interannual anomalies occur at all times of the year and are of a similar magnitude to the mean annual cycle shown in Figure 17.4. Units in PW. Source: After Chirokova and Webster (2004).

is also consistent with the fact that an El Niño is normally associated with a weak monsoon, lighter surface winds, and reduced southward Ekman transport of heat across the equator. The reversed fluxes that occur for the ENSO and the monsoon cases are interesting and may be associated with a weakening (strengthening) of the monsoon induced by the changes in SST associated with the anomalous fluxes earlier in the season.

The biennial nature of the coupled Indian Ocean system is quite evident in the time series of the annual heat balance plotted in Figure 17.13a. The mean annual cross-equatorial flux is about −0.2 PW but oscillates from year to year by ±0.2 PW. The rate of change of heat storage in the North Indian Ocean is also plotted. Rather than having a zero change from year to year, the North Indian Ocean shows an ability to store heat in one year and lose it in another. For example, in 1965,

17.5 An Holistic View of the Monsoon System

Difference in IO meridional heat transport between ENSO and monsoon extremes (EL Niño-La Niña) northward heat flux transport anomaly (PW)

0.3

(strong - weak monsoon)

0.2 0.1 0 –0.1 –0.2 –0.3 J

F M A M J J A S O N D J month

Figure 17.12 Composites of the annual cycle of differences in northward cross-equatorial ocean heat transports between El Niño and La Niña (solid curve) and strong and weak monsoons (dashed curve). Units in PW.

1973, and 1987, the net storage increased, while in 1960, 1972, and 1988, the heat storage decreased. The net annual heat flux into the ocean, on the other hand, is much more constant. Thus, there appears to be some evidence of changes in the dynamic state of the Indian Ocean. Figure 17.13b shows annual averages of heat fluxes plotted as a function of latitude over the same time period. The largest zonally averaged heat transports occur near 15∘ S. At this latitude, the heat fluxes range in value from −0.1 to −0.6 PW. A maximum at 15∘ S appears to occur over a wider latitude band; that is, the entire tropical Indian Ocean appears to be consistently anomalous. To check this, the long-term mean heat flux at each latitude is removed and the resulting latitude time series plotted in Figure 17.13c. In general, coherent variations between 15∘ N and 20∘ S appear from year to year. One of the interesting features of this figure is that the anomalies extend across the equator between 15∘ N and 20∘ S. Thus, the entire low-latitude Indian Ocean behaves dynamically in unison from year to year. Figure 17.14 presents the power spectra of the cross-equatorial heat transport for periods >1 year. The biennial nature of the cross-equatorial ocean heat flux is very clear, surpassing the statistical 99% confidence level. A broader peak in the 3–5 year band surpasses the 95% level, possibly indicating the influence of ENSO.

17.4.2 Intraseasonal Variability in the Indian Ocean Are there changes in the heat balance occurring on subseasonal time scales? Here we choose two specific years, 1987 and 1988, that exhibit very different characteristics representing “strong” and a “weak” monsoon forcing, respectively.12 Heat transport is plotted as a function of latitude and time for each year in Figure 17.15b and c relative to climatology (Figure 17.15a). A general annual cycle of heat transport is apparent in both years with northward transport (“warm” colors) during the winter and early spring and southward transport (“cold” colors) during summer. Both plots are broadly reminiscent of Figure 17.4 except that the transport is punctuated by a series of pulses of the same sign as the background climatology. The overall SST anomaly associated with a MISO is not very large. Following the first MISO of the boreal summer season, the SST in the North Indian Ocean may drop by over 1 ∘ C. The signal associated with subsequent MISOs occurring later in the season is much smaller, perhaps half in magnitude (Figure 16.23c).

17.5 An Holistic View of the Monsoon System In Section 17.4.1.4, Meehl’s monsoon bienniality theory was modified by adding a wind-driven dynamic component wherein anomalous monsoon winds associated with an anomalous monsoon will induce ocean heat transports that, in turn, reverse the sign of the monsoon anomaly. This negative feedback system theory included both changes in basin heat storage due to anomalous heating and mixing at the surface, and lateral transfer of heat. The advection of heat simplifies the problem of having to account for the persistence of anomalies longer than radiative cooling rates of the upper ocean. However, there are still a number of issues regarding the modified Meehl hypothesis that require consideration. In particular, the regulation theories, either the Meehl theory or the modified-Meehl theory, do not involve the Indian Ocean dipole (IOD), the phenomenon described in Section 12.3. It is possible, of course, that the dipole is an independent phenomenon. However, the similarity of the basic time scale of the dipole to that of monsoon variability (essentially biennial) and the fact that the dipole emerges during the boreal summer monsoon suggests some degree of interdependence. The problem, though, is how to incorporate an essentially zonal mode (the IOD) and a meridional phenomenon (the oceanic heat transport) 12 E.g. Han et al. (2004).

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17 The Coupled Monsoon System

(a) Annual means of heat balance of the NIO 0.4

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58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 year Figure 17.13 Interannual variability of the heat balance of the Indian Ocean for the period 1958–1998. (a) Annually averaged heat budget for the North Indian Ocean showing the cross-equatorial heat flux (black contour), the net heat flux into the NIO (blue), the rate of change of heat storage (black dashed), and the sum of the cross-equatorial flux and the heat transport (red). Units in PW; (b) latitude–time distributions of the annually averaged meridional cross-equatorial heat transports including the long-term seasonal mean. Pink shading denotes heat transport > − 0.3 PW; and (c) same as (b) but with the annual average removed. Anomalous southward transport in pink (red arrows) and northward in blue (blue arrows). Data from the ocean model integrated from 1958 to 1998. Note the strong biennial tendency of the heat transport and also that the anomalies tend to have the same sign on both sides of the equator, extending across much of the basin. A nine-point running average has been used for all three components and the annual cycle has been removed. Units in PW. Source: Based on Chirokova and Webster (2006).

into a general theory of the coupled monsoon. Further, how does ENSO become involved in these feedbacks? 17.5.1

Indian Ocean Sector

An attempt at an overall theory of the variability of the monsoon is sketched out in Figure 17.16, in which

some of the problems inherent to the initial hypotheses are addressed. The original theory of overall monsoon variability was conceived by Loschnigg et al. (2003), based initially on the results of Loschnigg and Webster (2000). A series of experiments with a coupled ocean–atmosphere general circulation model, added the influence of the Pacific Ocean variability. The theory

17.5 An Holistic View of the Monsoon System

shown in the second panel of the row. At that state, the NH becomes anomalously warm compared to the 41 10 5 3.3 2.5 2 1.7 1.4 1.3 1.11 1 12 average. If the anomaly persists through to the second power spectra spring (MAM:2), it will lead to a strong monsoon red-noise spectrum 10 99% significance level in the second summer (JJA:2). Enhanced southward 8 transports and reduced net heating, associated with 6 the strong monsoon, will, in turn, lead a North Indian Ocean depleted of heat and thus a weak monsoon in 4 JJA:3, and so on. 2 (ii) Stronger and weaker summer monsoons also influence 0 the ocean system in other ways. As we have noted, 0.0 0.2 0.4 0.6 0.8 1.0 the anomalous monsoon circulation (Figure 16.6) will frequency (year–4) influence the upwelling patterns in two major areas: along the East Africa coast north of the equator and Figure 17.14 Power spectra of the cross-equatorial heat transport along the western coast of Sumatra and the eastern for periods >1 year. The annual cycle has been removed by subtracting the first four harmonics of the annual cycle. The 99% equatorial Indian Ocean. Thus, during the weak monconfidence level is calculated from the chi-square distribution, soon of JJA:1 the changes in the monsoon circulation assuming a theoretical red-noise spectrum (Gilman et al. 1963). will create anomalously warm water in the west and The spectrum has been smoothed by a five-point running colder water in the east Indian Ocean. On the other average. Each point has 10 degrees of freedom. Note the very hand, the circulation associated with the strong monstrong biennial period that exceeds the 99.9% confidence limit. Source: Based on the results of Chirokova and Webster (2006). soon in JJA:2 will produce cooler water in the western basin (resulting from enhanced southwesterlies) and warmer water in the east (onshore winds along the advanced here is an extension and a conglomeration of Sumatra coast). In summary, changes in the monsoon the two hypotheses. winds between strong and weak monsoons can create Figure 17.16 displays a schematic sequence through zonal anomalies in the SST distribution. two monsoon seasons, starting (arbitrarily) in the (iii) The east–west SST gradients caused by the anomalous boreal spring of year 1 referring to processes occurring monsoon intensities lead to enhancements of the zonal within and around the Indian Ocean. The first three SST gradients by coupled ocean–atmosphere instacolumns represent the anomalous meridional oceanic bilities, as summarized in Section 14.3. Simply, the heat transports (column 1), the influence of the anomaSST gradients force zonal wind anomalies that change lous monsoon circulation on the ocean (column 2), and the distribution of low-latitude sea-level height distrithe evolution of the Indian Ocean Dipole (column 3). bution. During JJA:1, the sea-level height will slope The fourth column refers to oscillations in the Pacific upwards to the west while during JJA:2 it will slope Ocean, in particular ENSO. upwards to the east. Relaxation of the sea-level height The ordinate in Figure 17.16 represents time, marked is adjusted by equatorial modes. For example, during off in seasons. During a two-year sector, the monsoon the period JJA:1 through DJF:1 the relaxation will goes through both weak and strong phases linked be in the form of downwelling Rossby waves. Besides together by the biennial negative feedback system dishaving a slow westward propagation, the downwelling cussed above. At the same time, the IOD progresses deepens and warms the western Indian Ocean. In through a positive and negative phase. We will now turn, the enhanced SST gradient will produce stronger argue that the evolution of the dipole and the strength easterly winds that will continue to maintain the of the monsoon are intimately related, proposing the east-to-west slope of the surface. Between JJA:2 and following sequence: DJF:2 the onshore winds toward Sumatra will deepen (i) The left-hand column describes essentially the regulathe thermocline and enhance the zonal west-to-east tion theory discussed in Section 17.4.1.2. The sequence SST gradient. In turn, responding to an increasing SST may start anywhere so we arbitrarily commence with gradient, the winds will be enhanced. The important an anomalously cold North Indian Ocean in the boreal aspect of the dipole is that it introduces slow dynamics spring (March–May of the first year: MAM:1), which into the system. often precedes a weak monsoon in the summer of the (iv) Careful inspection of Figure 17.16 shows that the first year (JJA:1), consistent with the initial Meehl impact of the dipole is to enhance the SST distribienniality theory. A weak monsoon is associated butions associated with meridional heat transports with a reduced southward heat transport, leading to appearing in the first column. For example, the dipole the SST distribution in the first boreal fall (SON:1), that develops in the period JJA:1 through DJF:1 will power spectrum (10–4)

period (years)

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17 The Coupled Monsoon System

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Figure 17.15 Zonally integrated heat flux averaged across the Indian Ocean for 1987 and 1988 that were weak and strong monsoon seasons, respectively. Data from the integrations of Loschnigg and Webster (2000) using the intermediate ocean model of McCreary et al. (1993) forced by 5-day average winds and net surface heat flux rom NCEP/NCAR reanalyses. Rather than the smooth patterns found in Figure 17.4b, each year shows strong intraseasonal variability. Units in PW.

increase the SST in the northwest equatorial Indian Ocean. This SST enhancement can be seen by following the sequence “a” through “e” in Figure 17.16. On the other hand, the opposite signed dipole will cool the SST in the same location. This is the region established by Sadhuram (1997), Harzallah and Sadourny (1997), and Clark et al. (2000) for correlations between winter SST and rainfall during the following summer. Thus, the role of the dipole is to enhance and prolong the SST patterns necessary to regulate the subsequent intensity of the monsoon system. In summary, the conjunction of the components of the coupled system of the Indian Ocean sector appears capable of producing a strong biennial oscillation through a series of coupled ocean–atmosphere negative feedbacks. The question, though, is what is the role of ENSO in

the variability of the monsoon or, conversely, does the monsoon play a part in ENSO variability?

17.5.2 Speculations on the Interaction of the Indian and Pacific Ocean Sectors We have noted many times that the prevailing theory of monsoon variability is controlled by ENSO and, following this surmise, forecasting of the monsoon is predicated on being able to forecast ENSO. However, there are three points that need further thought: (i) The South Asian–North Australian monsoon system possesses a strong biennial oscillation that can, arguably, be explained in terms of regional negative ocean–atmosphere feedbacks. Thus, the elements of

17.5 An Holistic View of the Monsoon System

Anom meridional heat transport (a)

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Figure 17.16 Schematic of a general theory of an ocean–atmosphere regulation system for the monsoon and the Indian Ocean. Each column indicates a set of processes. The first column shows modulation of the monsoon variability by changes in the heat transport induced by the monsoon winds. In essence, this sequence represents the Meehl (1997) biennial oscillation mechanism but with ocean dynamics. The second column shows the impact of the strong and weak monsoons on the upwelling regions of the ocean basin, as suggested in Figure 16.6. The third column represents the development of the Indian Ocean dipole relative to the upwelling patterns developed by the anomalous monsoon wind fields. Growth of the dipole anomaly is assumed to follow the coupled ocean–atmosphere instability described by Webster et al. (1999). The fourth column indicates the change in the upper ocean structure along the equator as the monsoon changes from a weak to a strong monsoon. Taken as a whole, the figure suggests that there are multiple collaborative components that regulate the monsoon. One important role of the dipole (either positive or negative) is to provide slow dynamics (or memory) to the SST anomalies induced by the strong or weak monsoons. For example, the sequence (a) to (e) helps perpetuate the NH anomalously warm temperatures created by the weak monsoon during the previous summer. Source: Adapted from Webster et al. (2002) and Loschnigg et al. (2003).

a monsoonal biennial oscillation in monsoon amplitude rest within the internal dynamics of the coupled ocean–atmosphere dynamic system. (ii) The regression analysis between El Niño 3 SST variability and the Indian monsoon rainfall (Figure 16.4) shows that monsoon anomaly occurs before the peak in central Pacific SST. That is, the anomalous South Asian monsoon is manifested before the ENSO anomaly is set. One could argue, then, that the anomalous monsoon winds force variability in the Pacific Ocean. Indeed, stronger easterly lower-tropospheric winds over the Pacific (stronger Walker Cell) attend strong South Asian monsoons, whereas a weaker Walker Cell is associated with weaker monsoons. As discussed in Chapter 14, these variations of the Walker Cell (or the trade wind regime) are precursors of La Niña and El Niño, respectively. Importantly, the monsoon-induced anomalies in the Pacific Ocean related to ENSO

occur during the boreal spring, referred to earlier as the Pacific Ocean “springtime frailty” period. These points allow us to speculate on the influence of the Asian–Australian monsoon in the coupled ocean– atmosphere low-frequency variability in the Pacific Ocean. The schematics displayed in Figure 17.17 posit a sequence of events that may support (i) and (ii) above. Panel (a) illustrates the changes in the Indian Ocean and Pacific Ocean winds associated with an anomalously strong and weak monsoon. Stronger trade winds coincide with a strong monsoon and weaker trades with a weak monsoon. These two states are conducive to the formation of a La Niña and an El Niño, as discussed extensively in Chapter 14. Panel (b) illustrates the relative timing of the growth of convection in the monsoon regions of South Asia along 14∘ N as a function of time of year and longitude. The section is dominated by the rapid growth of South Asian convection in May–June

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17 The Coupled Monsoon System

around 90∘ E. Changes in convection along the equator occur almost in quadrature with the monsoon convection; that is, when the Walker Circulation is at its annual minimum, and also on a time of minimum longitudinal pressure gradient along the equator, the monsoon convection (and the associated anomalous trade winds) are approaching their maximum. Panel (c) illustrates the timing of the possible monsoon influence in terms on the physical state of the coupled Pacific Ocean

system. During the spring, the along-equator pressure gradient is at a minimum and the surface winds are the weakest. At that stage, the signal to noise ratio is at its smallest annual value and susceptible to external forcing, such as trade wind anomalies associated with the monsoon. The hypothesis described here is in keeping with Normand’s suggestion (Section 16.2.2.2) made nearly 70 years ago that monsoon variability actually

(a) Trade winds associated with anomalous South Asian monsoon g on str es d tra

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Figure 17.17 Schematics representing the interaction of the anomalous monsoon with the coupled ocean–atmosphere system of the Pacific Ocean. (a) Schematic of the gross surface wind variability associated with a strong monsoon (strong Pacific trades) and weak monsoon (weak Pacific trades). These are consistent with the boreal springtime preconditions for La Niña and El Niño, respectively. (b) The growth and decay of convection associated with the monsoon and the Walker Cell plotted as a function of longitude. Consistent with the regression analysis shown Figure 16.4, the growth of the monsoon convection precedes the development of the equatorial convection. (c) The relative timing of the influence of the anomalous monsoon on the coupled Pacific Ocean system. Anomalous Pacific trade winds occur at the time of minimum longitudinal SST gradient along the equator.

17.5 An Holistic View of the Monsoon System

Figure 17.17 (Continued)

(c) Possible influences of monsoon anomalies on ENSO OTHER EXTERNAL INFLUENCES (C)

External influences & forcing

Climate variability in the PO basin

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foreshadows other large-scale variations of the climate system. Normand, of course, was not aware of either of the physics of El Niño or La Niña but was aware of the timing of the SO transitions and variation of the strength of the monsoon. If the speculation described in the paragraphs above is accurate, one may ask the question of why every anomalous monsoon is not associated with a subsequent El Niño or La Niña. Perhaps there are two answers. First, there are varying degrees of the intensity of the monsoon ranging from very strong to very weak. Here it is useful to think of the larger scale monsoon and not just its Indian component. In this broader scale context, the associated trade wind anomalies may vary in intensity. Second, there may be other factors that influence the state of the Pacific Ocean at the time of the springtime frailty. Another question is what produces an anomalous monsoon in the first place? We have argued that once there is an anomalous monsoon, negative feedbacks will produce a monsoon of the opposite polarity on a biennial time scale. However, the processes that promote a monsoon anomaly in the first place are harder to understand. We do know, though, from Figure 16.7 that anomalies in the large-scale wind fields occur well before the monsoon season itself, making it unlikely that anomalies are just the result of ENSO. This would seem to be a potentially fertile area of research.

Jan

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New joint O/A trajectory

One of the problems that emerged in earlier theories of monsoon amplitude regulation (e.g. Meehl 1997) is that it is difficult to understand how an SST anomaly pattern produced by an anomalous monsoon persisted from the summer to the following spring. By involving ocean dynamics in the regulation process we have managed to introduce mechanisms that allow SST anomalies to persist from one year to the next. This is accomplished by noting that the IOD is also parented by an anomalous monsoon through the generation of zonal temperature gradients between anomalous upwelling regions. The slow dynamics of the dipole act to enhance the zonal SST gradient initiated by the anomalous monsoon irrespective of the sign of the initial perturbation. As the dipole grows, the SST anomalies so produced occur in locations that are conducive to the generation of a reverse anomaly in the following summer monsoon. In other words, the IOD provides slow dynamical processes needed in the Meehl theory. There is a further problem though. That is regarding the interdecadal variability of the biennial nature of the climate system. We noted with reference to Figures 16.1 and 16.3 that amplitudes of variability have waxed and waned over time. For example, between about 1920 and 1955, variance of monsoon intensity and the mid-Pacific SST lost most of its power in the 2–8 year band. A close inspection of the wavelet analysis indicates that the bienniality also lessens at roughly the same time. Careful numerical experiments with a coupled climate

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model suggest that the base state changes, and with it the spectrum of higher-frequency modulations of the monsoon and ENSO.13 A clear and important conclusion is that the ocean involves itself in a dynamic manner in the evolution of the monsoon on time scales at least longer than seasonal and perhaps even intraseasonal times. Although the ocean is responding to forcing from the atmosphere, the response is such that a strong return feedback to the atmosphere is produced. This feedback governs the amplitude and phase of the annual cycle and also modulates interannual variability; that is, considered holistically the monsoon system is self-regulating. However, keeping the interdecadal variability in mind, the self-regulation may be a statistically non-stationary feature. Whereas some predictability may exist in the biennial nature of the monsoon and the manner in which it

interacts with ENSO, we are of the opinion that useful predictability lies in the intraseasonal band. Variability of the mean Indian precipitation from year to year is rather small, with a standard deviation of roughly 10%. The ability to forecast seasonal monsoon precipitation is poor and determining which part of India will receive more or less than average precipitation is even worse. As we pointed out earlier, a precipitation forecast of 20–30 days would be very useful, especially as it would give assessments of regional precipitation in both amount and timing. It is an essential target and it is doable. The prediction of small variations in year-to-year broad-scale precipitation, a preoccupation for over a century, has shown little success to date and, even if possible, may not provide useful and usable information.

Notes I COADS represents the “The Comprehensive

Ocean-Atmosphere Data Set.” The original data were obtained from many sources, including merchant vessels, research ships, and moored and drifting buoys. Parameters include: atmospheric pressure, surface pressure, air temperature, sea surface temperature, humidity, water vapor, surface winds, cloud information (amount, height, and type), and “present weather.” The scientists assembling COADS have attempted to integrate all available digitized, directly sensed surface-marine data sets that would contribute information of reasonable quality (Oberhuer 1988). II The model used is a version of the McCreary et al. (1993) 2 1/2 Indian Ocean model with a mixed layer imbedded within the upper layer and a 55-km grid resolution. The model was developed by McCreary et al. (1993) and used in several studies (e.g. Loschnigg and Webster 2000 and Chirokova and Webster 2006), where details of the model are presented. It is also

13 Meehl and Arblaster (2011).

assumed in our configuration of the model that there is no Indonesian Through Flow (ITF). The ITF is a current of some importance, flowing though the complex Indonesian archipelago, providing a connection between the tropical Pacific and the Indian Ocean. The mass transport is roughly 10 Sv or about half of the Gulf Stream transport (see Godfrey 1995). The assumption that the ITF is zero is questionable although the largest impact of the ITF would be in the South Indian Ocean (Godfrey 1995 and Chirokova and Webster 2006). Longer period variability remained much the same over the basin irrespective of the inclusion of even a variable ITF. III The Miyama et al. (2003) paper is highly recommended for those interested in near-equatorial ocean dynamics. It presents a convincing set of numerical and theoretical arguments that lay out clearly how cross-equatorial transports take place in a monsoon climate.

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18 The Changing Tropics

18.1 Tropical Warm Pool In the description of the tropics in Chapter 1, we identified the warm and relatively fresh surface water in the western Pacific and eastern Indian Oceans as the ocean warm pool (OWP). The extent of the warm pool (here arbitrarily defined by an SST > 28 ∘ C) is seen clearly in Figure 1.5. The collocation of the climatological OWP with very high precipitation rates may be seen by comparing SST with the climatological precipitation rate maps shown in Figure 1.6. In Chapter 2, we noted that the OWPs are a stable feature of climate. Zonal SST gradients drive surface wind fields that produce a deep mixed layer in the western Pacific Ocean and in

Surface temperature anomaly (°C) relative to 20th century average +0.6 +0.4 °C

There is robust evidence that the surface of the planet has warmed during the last century. Figure 18.1 shows that the increase in surface temperature has taken place in stops and starts. During the latter period of the nineteenth century and into the early part of the twentieth century, cooling of about 0.2 ∘ C occurred, followed by a substantial increase of 0.6 ∘ C until 1945. After 1945, the global temperature remained essentially steady until the mid-1970s, after which it warmed. Overall, the global surface temperature has increased by about 1 ∘ C from 1900 until the present. Figure 18.1 says nothing about the regionality of temperature changes and whether there have been corresponding changes in the precipitation amount or rate, or where precipitation may have changed. Also, there is no information on whether or not the intensity of circulation features such as the Hadley Cell, the Walker Circulation, or the monsoons have also changed as the planet warmed. Has the area of the ocean warm pools and their associated convection changed as well? Furthermore, during this period of global warming has the character of large-scale interannual variability such as the El Niño-Southern Oscillation changed its character? These questions provide the focus for this chapter.

+0.2 0 –0.2 –0.4 1880

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the eastern Indian Ocean. Stability is maintained by a positive buoyancy flux produced largely by upper ocean heating with contributions from a positive fresh water flux as described in Section 2.6. Despite the stability of these large-scale fields, we noted in Section 14.2.3 that small long-period oscillations of the position and magnitude of the Pacific warm pool leads to large-scale changes in tropical precipitation patterns. The sensitivity of precipitation rate to SST magnitude was attributed to the ClausiusClapeyron thermodynamics (Section 2.3.1), where small changes in temperature, where the ocean temperature is warmest, invoke large changes in surface fluxes and, thus, the coupling between the ocean and the atmosphere. One question might be then, has tropical precipitation reacted to the Figure 18.1 changes in global temperature? 18.1.1 Changes in the Ocean Warm Pool During Last Century Figure 18.2a provides a plot of the evolution of the tropical SST averaged between 20∘ N and 20∘ S from 1915 and 2005 for each ocean basin as well as the interbasin global average. Overall, throughout the twentieth century, the tropical Indian Ocean has been the warmest basin followed by the Pacific and the Atlantic. During the twentieth century the average tropical SST between

Dynamics of the Tropical Atmosphere and Oceans, First Edition. Peter J. Webster. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

18 The Changing Tropics

(a) Pentad SST

Figure 18.2 (a) Evolution of SST for the tropical, Pacific, Indian, and Atlantic basins in 5-year (pentad) bins from 1910 to 2004 in the 20∘ N–20∘ S band. Global SST (∘ C) appears as the red line. (b) Area in each basin where SST >28∘ C. Units 1012 m2 . Source: NOAA Extended Reconstructed SST v2 data.

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2000

1920

20∘ N and 20∘ S has increased by about 0.8 ∘ C, with each basin having a very similar trend. As with the global surface temperature, the tropical sea surface temperatures have warmed in a number of steps. Tropical SSTs increased through the early part of the twentieth century before reaching a peak in 1940–1945. A plateau of slightly cooler temperatures occurred for the next 20–25 years. During the 1970s and into the present era, tropical SSTs have generally increased.I Thus, the surface temperature of the tropical oceans appear to have changed in concert with global temperatures. Figure 18.2b shows the change in area of the OWP in each ocean basin, here defined by the 28 ∘ C SST isotherm. The overall 0.8 ∘ C increase in tropical SST since 1910 has been accompanied by a 70% expansion of the area of the global OWP (SST > 28 ∘ C), with changes in area of roughly 40–68 × 1021 m2 . The behavior of each ocean basin is similar to the tropics-wide variation with the area of the Indian and Pacific OWPs increasing by about 65 and 40%, respectively. In the Atlantic, the smallest and coolest of the tropical ocean basins, the OWP has increased by 350%. Overall, the global OWP area in the 2000–2004 pentad is ∼20% larger than in 1980. To provide a context for the magnitudes of these changes, during a typical El Niño event the area of the Pacific OWP increases by about 15%. The annual cycle of SST variability does not appear to have changed since the early twentieth century. Overall, contributions to the yearly OWP increase are spread almost equally throughout the year. There are two maxima in OWP area, each following an equinox. The larger of the two maxima occurring in the boreal spring probably reflects the spring warming of both the

1940 1960 1980 pentad year

2000

northern near-equatorial SSTs occurring in the Pacific (Figure 14.8) and the Indian Oceans (Figure 16.23), both occurring when the surface wind is weakest. 18.1.2

The Mid-Twentieth Century SST Plateau

There have been different interpretations of the 1930–1940 maximum and the broad cooler valley between 1950 and 1970. Some argue that the pattern is the result of uncorrected instrumental biases in the SST record, reflecting a general change from the use of bucket temperature measurements to engine intake measurements.1 If this argument is correct then the decrease in SSTs starting in the mid-1940s was artificial and a disruption of what would be a generally warming trend from the beginning of the twentieth century until the present. There are a number of careful comparisons of SST measurements by different techniques (e.g. Kent and Taylor 2006) but, being interested specifically in the trends and patterns rather than absolute values, we adopt a much simpler technique. Our testing of the uncorrected instrumentation error hypothesis is straightforward. It is accomplished by comparing times series of SST and similarly located nearby land-based surface temperatures. Land station temperature measurements have been made by the same method at least for the last 100 years. Figure 18.3a shows tropical land temperatures grouped by the World Meteorological Organization region (Figure 18.3a) for the period 1930–1950. Land stations near the coast or island land stations were chosen. Land surface temperatures universally increased until the early 1940s and then declined, matching the characteristics of the SST time 1 E.g. Thompson et al. (2008).

18.1 Tropical Warm Pool

(a) WMO Meteorological Regions used for land surface temperature 30°N 20°N

58

42

62

91

Eq

84

69

10°S

61

78 80/81

43

10°N

83

20°S 94/95

30°S 40°S 0°

60°E

120°E

180°

120°W

60°W

0°

(b) Land surface temperature time series for different WMO regions +0.3 Arctic > 60°N

58

+0.2 94

91

+0.1

62

78

42

83

43

0 –0.1

46 95

84

–0.2 –0.3 1930

1935

1940 Year

1945

1950

1930

1935

1940 Year

1945

1950

Figure 18.3 (a) World Meteorological Organization (WMO) meteorological regions used to calculate an average land surface temperature. (b) Time series of annually averaged tropical land surface temperatures in World Meteorological Organization regions (e.g. #94 or #61) for the period 1930–1950. Left panel shows time series of land surface temperatures for meteorological regions 42, 43, 46, 58, 62, and 78. The right-hand panel shows time series for regions 83, 83, 91, 94, and 95. Contours show deviations (∘ C) from a long-term average. Principal features are a cooling prior to 1935 followed by a general warming until the early 1940s. The anomalous surface temperatures of land surface stations north of 60∘ N, computed by Polyakov et al. (2003) for the 1930–1950 period, are shown as the bold dashed line. The arctic curve shows similar characteristics to the more tropical regions. Source: Data for the WMO regions kindly supplied by Professor P. Jones, Climate Research Center, of the University of East Anglia, UK.

series of Figure 18.2a. In addition, each tropical land time series appears to follow other land temperature time series such as the Arctic land record compiled by Polyakov et al. (2003) for the land surface north of 60∘ N (heavy dashed line, Figure 18.2b). Thus, it is fairly clear that the rise in surface temperatures occurred globally and that the increase and the subsequent decrease occurred over land as well as the ocean. Quite likely, then, the character of the SST record from 1945 onwards is certainly valid and not an artifact of an observation technique.

18.1.3

Longer-Term Changes in the Tropical SST

Proxy paleoclimate data records2 suggest that tropical SSTs have changed on century and millennial time scales. The last glacial maximum (LGM), occurring 25–16 ky BP (16 000 years before the present) with the coldest period near 21 ky BP, is an epoch during which 2 The term “proxy data” refers to inferred quantities when direct measurements are not available. For example, SST can be inferred from the ratio of oxygen isotopes 16 O and 18 O found in corals. For example, see Gagan et al. (2004), Mayewski et al. (2004), and Furtado et al. (2009).

431

18 The Changing Tropics

(i) OWP temperatures during the LGM were estimated to be 2–4 ∘ C cooler than present, especially when estimates of land-based temperatures were taken into account (e.g. Webster and Streten 1978). These estimates proved to be consistent with coupled climate model simulations of LGM climate (e.g. Pinot et al. 1999 and Otto-Bliesner et al. 2009) (ii) Some studies suggest that during the mid-Holocene (∼6000 BP) SSTs were very similar to present-day values (Rahmstorf 2002 and Mayewski et al. 2004). Other studies suggest that SSTs in the Indo-Pacific OWP temperatures may have been 1 K higher than present values (Gagan et al. 2004). 18.1.4 Relationship Between SST and Convection in the OWP Figure 18.4 plots the distribution of deep organized convection in the form of OLR (W m−2 ) against SST compiled for the 1975–1987 period in the 20∘ N–20∘ S equatorial channel. It is indicative of a series of studies3 in the late 1980s and early 1990s suggesting that tropical SSTs have an upper limit near 30–31 ∘ C. However, the deepest and most frequent convection (minimum OLR) was found near 29 ∘ C (point B, Figure 18.4) with the warmest SSTs coinciding with relatively clear skies (maximum OLR). Specifically, the intensity (or frequency) of organized convection rises sharply as SSTs increase from 26.5 (point A) to 29.0 ∘ C, reaches a maximum at 29–29.5 ∘ C (point B), and then declines very rapidly with higher SSTs (point C). The second curve (red dashed) shows the number of 2∘ × 2∘ pixels found at a particular SST. The ascending side of the SST-OLR relationship (A-B) has been the subject of considerable discussion. Waliser and Graham (1993) suggest that the sharpness of the increase reflects that large-scale SSTs above 27–28 ∘ C are required to provide the necessary moist static energy to the near-surface layers in order for saturated air parcels to ascend into the high troposphere. This assentation raises interesting questions. It is quite possible that during the LGM the mean SST of the warm pool may have been 28 ∘ C, >29 ∘ C, >30 ∘ C, and > 31 ∘ C in the Table 18.1 Percent area occupied by SST > 28 ∘ C, >29 ∘ C, >30 ∘ C, and > 31 ∘ C in (a) the Indian ocean in the region (20∘ S–20∘ N, 59∘ E–101∘ E) and (b) the Pacific Ocean (20∘ S–20∘ N, 129E–169∘ W). (a) Indian Ocean: 20∘ S–20∘ N, 59∘ E–101∘ E >28 ∘ C

>29 ∘ C

>30 ∘ C

>31 ∘ C

1903–1922

32.0

6.3

0.8

0

1933–1952

46.3

14.5

3.0

0.3

1963–1981

52.1

19.3

4.0

0.5

1998–2017

65.0

34.5

9.9

1.6

(b) Pacific Ocean: 20∘ S–25∘ N, 129E–169∘ W >28 ∘ C

>29 ∘ C

>30 ∘ C

>31 ∘ C

1903–1922

38.5

10.8

1.9

0.1

1933–1952

54.3

22.8

5.3

0.5

1963–1981

61.0

23.8

6.4

1.0

1998–2017

71.5

37.4

13.3

1.4

Source: Data from the 1∘ × 1∘ International Comprehensive Ocean–Atmosphere Data Set (ICOADS) available at [email protected]. The data are sorted into 20-year blocks, 1903–1922, 1933–1952, 1963–1981, and 1998–2017.

18.1 Tropical Warm Pool

equatorial Indian and Pacific Oceans. Consistent with Figure 18.1a, the area of the OWP (SST >28 ∘ C) has grown substantially over the last century. The percentage occupied by SST in the next two categories has also increased. Yet the warmest category (>31 ∘ C) has shown only incremental growth in absolute values, However, in terms of percentage it is similar to increases in the cooler categories. A number of theories of SST regulation have been put forward, which are considered below. 18.1.4.1

Surface Energy Balance Regulation

Newell (1979) put forward a thermodynamic “thermostat” theory that attempted to explain both the growth of convection with SST and its rapid demise at higher temperatures. Convection was thought to increase with SST and the increased cloudiness caused a reduction of solar heating of the ocean surface. Newell argued using Clausius-Clapeyron arguments that as SST increased, evaporation would increase exponentially, thus placing an upper bound on SST. Further work by Hartmann and Michelson (1993) appeared to support the Newell hypothesis. 18.1.4.2

Cloud-Radiation Feedbacks

Ramanathan and Collins (1991) argued that there existed a “super greenhouse” in the tropical warm pool, where increased water vapor in the atmosphere would increase the downwelling longwave radiation (LWR) to the surface. This argument also included a Clausius-Clapeyron component, with evaporation increasing at the top end of the SST distribution. Increased deep cloud would reduce the loss of radiation to space. Collectively, these two processes would cause unlimited heating of the tropical atmosphere. On the other hand, increased cloudiness would lead to a greater overall albedo and hence increased reflection. A second hypothesis, based on an observation that the amount of cirrus diminished as SST increased led Fu et al. (1992) Lindzen et al. (2001) to suggest that there existed negative radiative feedback that limits SST. It was proposed that as SST increased the cirrus coverage decreased, allowing a greater loss of infrared radiation to space. This is referred to as the “adaptive iris,” whereby it is hypothesized that as SST increases, the infrared cooling of the planet increases as well. 18.1.4.3

Ocean Feedbacks

Clement et al. (1996) suggested the existence of an ocean thermostat where upwelling counteracts changes in external forcing. In the context of the Cane–Zebiak coupled model, changes in the external forcing are balanced almost equally by a change in the heat flux out of the ocean and by vertical advection of heat in

the ocean by anomalous equatorial ocean upwelling. The generation of upwelling is considered to be the key mechanism of regulating SST. This proposed coupled ocean–atmosphere process argues that ocean dynamics cannot be ignored in the discussion of SST regulation. 18.1.5

SST and Column Integrated Heating (CIH)

̇ is The three-dimensional atmospheric heating rate,Q, calculated as a residual of the approximate form of the thermodynamic Eq. (3.15e), in Eq. (3.33), as Q̇ ≈ Cp T0 wS

(18.1)

Here, w is the vertical velocity, T 0 temperature, S the static stability, and C p the specific heat of air at constant pressure. This near-equatorial approximation comes from Charney (1969), as discussed in Section 3.1.4. We use Eq. (18.1) to calculate column integrated heating (Q̇ or CIH) using data climate models and reanalysis data sets. Figures 2.17 and 2.18 showed the mean spatial climatologies of the total columnar heating. Here we calculate its variation with time. Figure 18.5a shows the evolution of total tropical (20∘ S–20∘ N) CIH from 1950 to 2004 as a function of SST. The CIH profiles shift to higher SSTs so that as the average tropical SST increases, the threshold temperature T H (the temperature where net column cooling changes to net column heating) also increases (Figure 18.5b). In the last 50 years, T H has increased about as much as the average tropical SST has increased (i.e. ∼0.5 ∘ C from 26.6 ∘ C to 27.1 ∘ C). In addition, there is a ∼15% increase in net heating within the tropics during the same period. Similar results are obtained using ECMWF ERA-40 reanalysis. There are some differences between the NCEP and ERA-40 estimations although the progression of T H and the trends in net heating are very similar.II It would be useful to be able to project SST-CIH relationships in the future using climate models. The degree to which these model simulations are useful can be gauged to some degree by how well the same models reflect changes during the last century when the simulations can be checked by observations. Figure 18.6 describes the evolution of the uncorrected tropical OWP area from 1920 to 2000 as simulated by four different ocean–atmosphere CMIP3III coupled models. Scatter plots of observed versus simulated OWP area relative to their corresponding long-term means, including the model-observation cross-correlations, appear in the left-hand column. While the magnitude of the OWP area differs slightly between models, owing to their different spatial resolutions, all models simulated the observed OWP expansion during the twentieth century quite well, with cross-correlations ranging from

433

18 The Changing Tropics

(a) Pentad CIH versus SST 1950–2004

2

(b) Pentad TH & net heating 27.1

2000/04

Threshhold SST (TH)

1

Threshold SST TH

0

columnar cooling

–1 22

24

columnar heating 26 SST (°C)

28

30

27.0

NCEP ECMWF

2.4 2.2

26.9 2.0 26.8 1.8 26.7

Net heating (PW)

1950/54 CIH (PW)

434

1.6 26.6

1960 1970 1980 1990 2000 year

Figure 18.5 (a) Evolution of the total tropical columnar integrated heating (CIH) binned every 0.5 ∘ C SST in pentads from 1950–1954 to 2000–2004. Different colors represent different pentads: cold colors represent pentads closely to 1950–1954 (purple) with warm colors representing pentads closer to 2000–2004 (red). The dotted box highlights the changes of the SST-threshold temperature T H at which net cooling columnar changes to net columnar heating. (b) Pentad evolution of T H (black line) and net heating in the tropics (red line) from 1950–1954 to 2000–2004. Data: NCEP-NCAR and ECMWF ERA-40 reanalyses and the NOAA Extended Reconstructed SST v2. Source: From Hoyos and Webster (2011).

0.68 and 0.86. Correlations between the observed and the simulated area of SST > 26.5 ∘ C are also high (∼0.8). To obtain an unbiased and unified multi-model estimation of the changes of the OWP, the simulated OWP and area for SST > 26.5 ∘ C were corrected using the twentieth century observation-model comparisons appearing in Figure 18.6. Specifically, Hoyos and Webster (2011) used linear least square regressions between the observed and the simulated OWP area for each model during the twentieth century. These biases were used to adjust statistically the magnitude of model biases during the twentieth and twenty-first centuries from the CMIP5 model integrations.IV Figure 18.7 displays the bias-corrected areal changes of SST > 26.5 ∘ C and > 28 ∘ C from the CMIP3 simulations for different emission scenarios. The figure includes the multi-model bias-corrected ensemble mean and the variability estimated as ±1 SD of the different bias-corrected model simulations. Projecting into the future, the OWP and the area of SST > 26.5 ∘ C for the COMMIT scenarioV stabilizes in 30–40 years at about 10–20% greater than the 2000–2004 area. Under scenarios A1B and A2, the OWP area is expected to increase during the twenty-first century to a multi-model average of +70% and +90% by 2100, respectively, relative to the 2000–2004 value. For the area of SST > 26.5 ∘ C, the increase is +30 and + 40%, respectively, for scenarios A1B and A2. Figure 18.8a compares the area of the OWP (SST >28 ∘ C) with the area of convection (positive CIH). Hoyos and Webster (2011) called the area of positive CIH the dynamic warm pool (DWP). The figure shows the area of positive CIH during the twentieth century and IPCC A1B (twenty-first century) emission scenarios

averaged across the results of the four different CMIP3 models. In spite of the expansion of the areas above different static SST thresholds, the area of CIH > 0 does not show any significant long-term trends, covering about 25% of the global ocean or about 60% of the ocean area between 20∘ S and 20∘ N. The area of the OWP (>28 ∘ C) is plotted for comparison. Thus, while the SST and the OWP areas increase sharply with time, the area of net heating in the tropics, the DWP, remains remarkably constant over the range of climate observed in the twentieth century and simulated for the twenty-first century. Rather than a static definition set by some constant temperature, the climatically active warm pool (the DWP) produced dynamically by the large-scale coupled ocean–atmosphere system. Specifically it corresponds to the area where SST > T H enclosing the area of positive CIH. Dutton et al., (2000) also found evidence of a creep upwards of T H . Figure 18.8b shows the change in CIH relative to the year 2000 value in each of the models. Despite the near constancy of the area of positive CIH, the magnitude of the columnar heating increases substantially in the twentieth and twenty-first century simulations. The increase of heating during the twentieth century is similar to that observed from the reanalysis data sets (Figure 18.5a). The change in magnitude of the net heating during the twenty-first century shows a 20% increase relative to the simulated 1995–2000 values. Such a future projected increase in CIH, together with a constant area of positive heating, would suggest a more vigorous circulation, with intensified moist convection in tropical areas and enhanced subsidence outside of the DWP.

18.1 Tropical Warm Pool

15

(a) NCAR CCSM3 (0.78) 95 90

5 area

model

10 0 –5

–10 –10 –5 0 5 10 15 observations 15

80 1920 1940 1960 1980 2000 years

(b) NOAA GFDL (0.79) 100

10 5 0

95 area

model

85

90 85

–5

80 –10 –10 –5 0 5 10 15 1920 1940 1960 1980 2000 observations years (c) NCAR PCM1 (0.75) 10 100

0

95 area

model

5

–5

90 85

–10 80 –10 –5 0 5 10 15 1920 1940 1960 1980 2000 observations years (d) NASA GISS_B (0.80) 110 10

0

area

model

5 105

–5

–10 –10 –5 0 5 10 15 observations

100 1920 1940 1960 1980 2000 years

Figure 18.6 Observed versus modeled twentieth century changes in area of SST > 28 ∘ C between 1920 and 2000. Models used are: (a) NCAR CCSM3, (b) NCAR PCM1, (c) NOAA GFDL, and (d) NASA GISS. Scatter plots show model/observation comparisons together with their correlation (parentheses). Second panels plot the evolution of CIH between 1920 and 2000. Source: Adapted from Hoyos and Webster (2011).

18.1.6 Why Is the Area of Organized Convection Relatively Constant? If the area of the positive CIH remains constant in an environment of increasing SST, the increased heating within the DWP must be balanced by an increased cooling in regions where SST < T H . By extrapolation, in a cooler environment (e.g. during the LGM) the region of CIH > 0 should possess the same area. Thus,

a similar compensation must occur. We investigate this hypothesis using two simple models. Consider a simple “two-box” model of the climate system (Figure 18.9a). The first box corresponds to the area where CIH >0, or the DWP region. This region is dominated by net rising motion and convective latent heat release. The second box encompasses the surrounding colder region, where air is descending and cooling, resulting from radiative loss to space. In the cooling box the vertical convergence of heat is given by the difference between the LWR entering the column at the surface and the LWR space at the top of the atmosphere (TOA). The LWR is given by the Stephan–Boltzman law: Fnet = 𝜀𝜎T 4

(18.2a)

where 𝜀 is the longwave emissivity and 𝜎 is the Stefan–Boltzman constant. In the warm box the vertical convergence of heat is given by the difference between LWR at the surface and at TOA, plus the atmospheric warming from condensation (QE ), where QE = 𝜌a LCDE |Vs |q∗s × [(1 − RH ) + RH Le (T − Ta )∕Rv T 2 ] (18.2b) assuming that all evaporation from the warming and cooling boxes condenses within the warming box. In Eqs. (18.2a) and (18.2b), 𝜌a , Le , C DE , U, q∗s , RH , Rv , and T a represent the air density, the latent heat of vaporization, the aerodynamic transfer coefficient for water vapor, the surface wind speed |V s | (about 5–10 m s−1 ), the saturation specific humidity at the surface, the relative humidity (∼70%), the gas constant for water vapor, and the air temperature, respectively. In Eq. (18.2a) we have used Hartmann (1994) formulation. Assume now that the two boxes are in equilibrium, with convection and heating within the warm box balanced by radiative cooling in the colder box. The radiative loss is also nonlinear, proportional to the fourth power of temperature through the StefanBoltzmann Law. Over a range of SST between 27 and 32 ∘ C, enclosing the LGM, mid-Holocene, pre-industrial, and the present climate, as well as what may be expected reasonably in the future, the area of the DWP (the convective box) ranges from 21 to 23.5% of the surface of the idealized planet (Figure 18.9b) for a homogeneous SST increase across both the heating and cooling boxes. There is some justification for assuming a constant SST change across the entire domain. In Figure 18.2, SST changes are relatively constant over the entire domain as a function of time. This simple model provides similar results to the estimates of a relatively constant ∼25% coverage of convection over an ocean found in the CMIP3 models. Thus, for a homogeneous SST increase the DWP area will essentially remain

435

18 The Changing Tropics

140

Area (1012 m2)

120

Multi-model

160

obs vs. model

30 20 10 0 –10 –20 –30

Figure 18.7 Changes in area of SST > 28 ∘ C and 26.5 ∘ C from observations (black) and from selected CMIP3 coupled models for the twentieth century runs (blue) and for the following scenarios: COMMIT (green), A1B (red), and A2 (orange) scenarios. Shading denotes intermodal variability. The inset shows the relationship between the observed twentieth century changes in area of SST > 28 ∘ C versus the relative area changes in the multi-model ensemble mean of the selected WCRP CMIP3 models for the twentieth century for the 16 pentads between 1920 and 2000. Quantities shown are deviations from long-term mean. Source: After Hoyos and Webster (2011).

Observations

R = 0.82 (28°C)

20th century Commit SRES A1B SRES A2

–30 –20–10 0 10 20 30

Observations

100

area SST > 26.5°C

80 60 40

area SST > 28.0°C 1950

2000

2050

2100

Year

(a) DWP and OWP areas

(b) Relative net heating

30 20

100

10

DWP 80

0

60 40

–10

OWP 1950

2000 Year

2050

2100

constant, with a tendency to decrease slightly as SST increases. To first order, the near-constancy of the area of CIH > 0 suggests that the sensitivity of radiative cooling in space to changes in SST (i.e. dF net /d(SST) from Eq. (18.2)) must be almost equal (and opposite) to the sensitivity of latent heat release within a column to changes in SST (i.e. dQE /d(SST) from Eq. (18.2a)). To explore this finding further we introduce a relatively simple zonally symmetric aqua-planet model.4 The nonlinear model has two levels and a full hydrology cycle. The atmospheric component is a primitive equation model extending from pole to pole, with explicit calculation of latent heating due to regional 4 Specifically Webster and Chou (1980a,1980b) and Webster (1983).

1950 2000 2050 Year

–20

Percent relative to 1995–2000

120 Area (1012 m2)

436

Figure 18.8 (a) Evolution of the area of the dynamic warm pool (DWP) area in the twentieth century and the twenty-first century for the IPCC A1B scenario in the NCAR CCSM3, NCAR PCM1, NOAA GFDL 2.1, and NASA GISS models. The area of the OWP area is also shown for the same scenarios for comparison purposes. (b) Net heating over the tropical ocean for the twentieth century and the A1B CMIP3 scenarios for the twenty-first century. Net heating is shown as a percentage relative to 1995–1999 values. Source: After Hoyos and Webster (2011).

convective effects and large-scale convection depending on moisture availability and equivalent potential temperature, bulk aerodynamic formulation for estimation of surface fluxes, and a cloud-radiation scheme. The atmospheric component of the model was run in an uncoupled mode to equilibrium for different sinusoidal SST configurations, as depicted in Figure 18.10a. The SST distributions have the same latitudinal distribution so that there is no change in gradient but possess different magnitudes due to a homogeneous increase of surface temperature incremented in steps of 1 ∘ C from 26 to 38 ∘ C. Figure 18.10b shows the resulting vertical velocity for each of the SST configurations. The model simulations result in a zone of narrow convection at the equator and subsidence in both hemispheres

18.1 Tropical Warm Pool

40 30

TH

TH subsidence

subsidence

convection

Increased Shear TH + ΔTH

enhanced subsidence

low SST

10 0 –10

SST + ΔSST

enhanced convection

–20

TH + DTH

–90°

–60°

–30°

0°

30°

60°

90°

(a) SST forcing

enhanced subsidence

6

(a) Dynamic warm pool with increasing SST

23

high SST

4 10–4 m s–1

24

DWP area (%)

high SST

20

SST (°C )

SST

2

low SST

0

22 21

Convective area change –1%/1.5°C SST increase

–2 –90°

–60°

–30°

0° latitude

30°

60°

90°

(b) Vertical velocity response

20 27

28

29 30 31 SST (°C) (b) Sensivity of DWP area to SST increase

Figure 18.9 (a) Schematic diagram of the simple “two-box” model containing a convective and a subsidence region. The top panel represents the current state of the climate where it is assumed that the convective (SST > T H ) and subsidence regions are in dynamic and thermodynamic balance. The bottom panel represents the two-box model after a homogeneous increase in SST in both boxes. Here, the heating threshold SST increases (T H + ΔT H ), but the area of the convective region remains almost constant. The balance and constancy of the climate state is achieved by both enhanced convection and subsidence. (b) Dynamic warm pool (convective region) size relative to the surface area of the tropics in the simple two-box model. Source: Adapted from Hoyos and Webster (2011).

poleward of about 15∘ N and S. Equatorial convection and off-equatorial subsidence increase as the SST forcing increases in magnitude, enhancing the meridional circulation (Figure 18.10b) and, hence, precipitation. Careful examination of the vertical velocity response will show a small contraction of the region of ascent as SST increases. Figure 18.11 summarizes the results from the box and zonally symmetric models. The simulations indicate that

Figure 18.10 (a) Configurations of SST forcing for the zonally symmetric nonlinear global primitive equation model varying the equatorial temperature in steps of 1∘ C from 26 to 38 ∘ C, (b) simulated vertical velocity, (c) meridional distribution of simulated vertical wind shear, and (d) 250 hPa temperature for each of the SST configurations. The colors in all four panels correspond to the same experiment. Source: From Hoyos and Webster (2011).

the size of the convective area in the zonally symmetric aqua-planet behaves in a remarkably similar way to the two-box model, staying relatively constant, with a tendency to slightly decrease with increasing equatorial SST (continuous and dashed black lines, Figure 18.11). In addition, results indicate that the total precipitation, the magnitude of the average vertical velocity in the convective region, and the vertical wind shear at the edge of the DWP increase considerably with increasing SST (red and yellow lines). These simple model results are consistent with the CMIP3 simulations discussed earlier. Finally, Hoyos and Webster (2011) used the PMIP5 model results to calculate the CIH as a function of SST for the LGM, the Mid-Holocene period, and the Post-Industrial Period. These results are shown in Figure 18.12. During the LGM, the transition temperature T H was 2 ∘ C cooler. During the latter two periods the DWP appeared almost the same as in the 5 PMIP: Paleoclimatological Modeling Intercomparison Project.

437

18 The Changing Tropics

24

20 Box model (% DWP)

23

22

10

21

20 20

0

Dynamical model (% DWP) % change precipitation % change vertical velocity % change wind shear 28

29 30 SST (°C)

31

–10

Relative parameter change (%)

Dynamic Warm Pool (DWP) area (%)

438

–20 32

Figure 18.11 Size of the DWP relative to the surface area of the planet in a simple two-box model and in the zonally symmetric aqua-planet (continuous black line). Changes of total precipitation (blue line), magnitude of average vertical velocity within the convective region (red line), and wind shear at the edge of the convective region (orange line) are plotted relative to the present (SST∼29 ∘ C). Source: From Hoyos and Webster (2011).

present climate. Figure 18.12 provides an evolving probability density function of tropical CIH in terms of SST between 20∘ S and 20∘ N for the LGM, the Mid-Holocene and the Pre-Industrial period. One inference is that during the LGM, the OWP was probably smaller than at present. However, the area of convection (i.e. the DWP) would have been the same and within the DWP the convection, measured by CIH, is less intense. With respect to relationships between convection and SST, results from both simple models appear to be in agreement with observations. CMIP3 and PMIP model results indicate that increased heating within the DWP (area of positive CIH) is in near balance with increased cooling outside, following an increase in global SST. The results provide a nearly constant DWP area.

18.2 Circulation Changes A number of studies6 based on satellite data and climate model projection have suggested that during the period of warming in the last 50 years or so, the tropics have been expanding, with the consequences of moving the jet streams poleward and bringing drier conditions to 6 E.g. Quan et al. (2014), Lucas et al. (2014), and Turton (2017). The third study concentrates on a worst-case scenario of the social and biological consequences of a tropical band expansion.

the subtropics. In addition, model results have indicated that the tropical circulation is weakening.7 In model simulations the latitude of P-E < 0 is often chosen as the indictor of the latitude of the expansion. Results suggest a slow poleward creep of the subtropical dry zone with different explanations, including increases in anthropogenic gases and natural variability. A comprehensive study using the CMIP3 climate models suggested that increased greenhouse gases induce a weakening in the zonally asymmetric circulation calculation and a lesser extent in the zonally symmetric Hadley Circulation. The study concluded that global warming renders the Pacific Ocean to a more ENSO-like pattern. The Asian monsoon is not considered explicitly in the study but there is a wide range of speculation in the literature, suggesting possible increases or decreases in Asian monsoon rainfall.8 By contrast, in the analyses presented here the expectation is of a slight contraction of the tropical dry zone, as suggested by Figure 18.10b. Clearly, this discrepancy among different analyses is an issue of consequence, given the social impact of potential changes in rainfall in perhaps the most populated regions of the planet. We will address this issue more thoroughly below while concentrating on monsoon variability. 18.2.1 Definition of a Broad-Scale Monsoon System One of the problems in determining the variability of a monsoon system is the identification of a monsoon region itself. It is usual for the various monsoon climates (e.g. South and East Asia, West Africa, North and South America, and North Australia) to be studied individually and treated as separate entities. In addition, time series of precipitation usually concentrate on a monsoon region, the demarcation of which is not just physical but often political. The All-India Rainfall Index (AIRI: shown in Figure 16.1a) is understandable from a practical point of view because each system impacts local populations and the forecasting of their variability is done by national or regional entities. However, the large-scale summer monsoon Asian Monsoon Gyre (AMG) (Figure 16.9) encloses the regional monsoons of South Asia, East Asia, and Africa. Is it possible that the geographical domination of the AMG imposes common long-period variability on a more broadly defined, more global monsoon? To test this proposition, we define a Northern Hemisphere Monsoon System (NHMS). For the purpose of seeking common long-term variability among the collective monsoon systems, we 7 E.g. Vecchi and Soden (2007), 8 See Turner and Annamalai (2012) for a review.

18.2 Circulation Changes

(a) SST versus CIH Ensemble Average Area (1012 m2)

100

0.5

0

–0.5

LGM M-HP P-IP

–1.0 –6

(b) Relative OWP, DWP area and TH 26.5 DWP

80

TH

–2

0

2

25.5

>26°C

60 25.0 40

24.5 >27°C

20

>28°C

0 –4

26.0

4

P-IP

M-HP (6ka)

Relative SST (°C)

SST Threshold (TH °C)

Standardized average CIH

1.0

24.0 23.5 LGM 21 ka)

Epochs

Figure 18.12 Summary of analyses of the PMIP II models (NCAR CCSM, CNRM CM33, Hadley Center CM3M2, LASG FGOALS, and MRI CGCM2.3.4) for the Last Glacial Maximum (LGM), the mid-Holecen period (M-HP), and the pre-Industrial period (P-IP). (a) SST versus average CIH in 0.5 ∘ C bins for the LGM (blue), M-HP (orange), and P-IP (red). The average CIH is presented in a standardized manner and the SST is shown relative to the corresponding T H in the P-IP. (b) Area of SST >26 ∘ C, >27 ∘ C and > 28 ∘ C as well as the DWP area for the three different periods (black line). The diagram also shows the SST thresholds T H (dashed black line).

60°N

Climatological mean JJA minus DJF precipitation and (V850-V200) shear

60 ms–1

40°N 20°N 0° 20°S 40°S 180°

120°W –9

60°W –7

–5

0° –3

–1

60°E 1

3

5

120°E 7

180° mm day–1

9

Figure 18.13 NH monsoon rainfall system (NHMS: outlined by thick green curves), climatological mean JJA minus DJF rainfall (shading) and vertical wind shear vectors (850 hPa winds minus 200 hPa winds). The monsoon rainfall domain is defined by the summer (MJJAS) minus winter (NDJFM) precipitation rate exceeding 2.0 mm day−1 and the summer rainfall exceeding 55% of the total annual rainfall. The boxed area (0–20∘ N, 120∘ W–120∘ E) indicates the region where the NH monsoon vertical shear index is defined. Source: From Wang et al. (2013).

define a monsoon region generically as one where the summer to winter precipitation rate differences are >5 mm day-1 . Also, the summer rainfall must be >55% of the total annual precipitation.9 The green contours in Figure 18.13 define regions that meet this criterion. The vectors within the NHSM region show strong negative vertical shear of the total horizontal wind between 850 and 200 hPa over the entire NHMS.

9 As defined by Wang et al. (2013).

18.2.2 Variability and Trends of the Northern Hemisphere Monsoon System Figure 18.14 displays the variability and trends of the NHMS averaged between May and September for the period 1979–2011. The vertical zonal wind shear, V WS , is used to gauge the intensity of the NHMS similar to the definition (16.1) but averaged over the NHSM area. Plots of the NHSM monsoon intensity index, V WS , are shown in addition to the average precipitation rate (mm day−1 ), together with a measure of the boreal summer Hadley and Walker circulations, for the period

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18 The Changing Tropics

mm day–1

7.0

(a) NHSM precipitation

6.5

6.0

ms–1

10.0

(b) NHSM circulation (VWS)

7.0

4.0

1010kgs–1

25

(c) Boreal summer Hadley Circulation

22

19 –2.0 ms–1

440

(d) Boreal summer Walker Circulation

–5.0

–8.0 1980

1985

1990

1995

2000

2005

2010

Figure 18.14 Variability and trends of the NHSM system, the Hadley Circulation, and the Walker Circulation. Time series of (a) NH summer (MJJAS) mean monsoon rainfall rate (mm day−1 ) (defined in Eq. (13.2)) averaged over the entire NH monsoon rainfall domain, (b) vertical shear of zonal wind (VWS) averaged over 0∘ –20∘ N, 120∘ W–120∘ E, (c) MJJAS mean Hadley Circulation intensity measured by the maximum absolute value of the mean meridional mass streamfunction (1010 kg s−1 ), and (d) Walker Circulation intensity measured by the vertical ∘ shear of zonal winds (850 hPa zonal wind minus that at 200 hPa) averaged over the equatorial Pacific (10∘ S–10 N, 140∘ E–120∘ W). The ERA interim and GPCP v2.2 data were used for the period 1979–2011. Source: From Wang et al. (2013).

1979–2011. The intensity of the Hadley Circulation is estimated by the strength of the cross-equatorial mass stream function. The Walker Circulation intensity is measured by the 850 hPa zonal wind averaged across the Pacific Ocean (10∘ S–10∘ N, 140∘ E–120∘ W). Figure 18.13 shows the following: (i) A steady increase in the mean NHSM precipitation at the rate of about 0.08 mm day−1 per decade (Figure 18.14a) between 1979 and 2011. By these calculations, precipitation has increased at the

rate of about 9.5% per degree of global surface temperature increase. (ii) An increase of NHSM intensity (Figure 18.14b) that correlates strongly with the precipitation rate (r = 0.85). (iii) The boreal summer Hadley Circulation has increased in magnitude by about 10% over the period, the Walker Cell by nearly 20%. Trends in these indices and their statistical significance are listed in Table 18.2. All indices possess high statistical significance.

18.2 Circulation Changes

60°N

SST, T2m(shading), MSLP (contour) and 850 hPa winds 10

30

–30 –10

1 ms–1 10

–10

40°N –30 –10 30

–30

20°N

50

0°

10 70

20°S 40°S

–30

0°

60°E

180°

120°E

120°W

60°W °C

–0.5 –0.4 –0.3 –0.2 –0.1

0

0.1

0.2

0.3

0.4

0.5

Figure 18.15 Climate anomalies associated with the NHSM circulation index. Regressed 2 m air temperature anomalies over land and SST anomalies over ocean (shading: degrees Celsius), sea-level pressure anomalies (contours: Pa), and 850-hPa wind anomalies (vector) with respect to the NHSM circulation (VWS) index for the period 1979–2010. Wind vectors are significant at 95% confidence level by the Student t-test. The blue lines outline the eastern Pacific triangle and western Pacific K-shape regions where the mega-ENSO index is defined. The HadISST and ERAI reanalysis data were used. Source: From Wang et al. (2013). Table 18.2 Trends (1979–2011) of the NHSM (May through September, MJJAS) precipitation and circulation indices in addition to the Hadley and Walker intensities and the mega-ENSO index. 1979–2011

Trend per decade

TH %

MK %

NHSM precipitation

+0.08 mm day−1

98

98

NHSM circulation intensity +0.85 m s−1

99

99

Hadley intensity

+0.73 × 1010 kg s−1

99

99

Walker intensity

−0.44 m s−1 +0.20 ∘ C

95

95

98

99

Mega-ENSO index

Source: Given the definition of the Walker Circulation, a negative trend infers an increase in intensity of the longitudinal cell. Increasing surface easterlies infers a negative trend. Two statistical tests were used: TN, trend-noise ratio and (MK) Mann-Kendall rank. Source: From Wang et al. (2013).

Thermodynamical arguments, in accord with simulations by the Coupled Model Intercomparison Project,10 have suggested that with global warming both the Hadley and Walker cells should decrease in intensity as the planet warms.11 We noted above that the Vecchi–Soden study suggested a decrease in both circulations, especially the Walker Cell. If the model results are correct one can presume that the changes in the intensity of the monsoon and Hadley and Walker systems, as observed, must arise from internal feedback processes within the climate system not dealt with by 10 CMIP5: Taylor et al. (2012). 11 E.g. Held and Soden (2006) and Vecchi et al. (2006).

the models or that the climate models do not deal well with slow interdecadal modes.12 18.2.3 Why Has the Northern Hemisphere Monsoon System Intensified? Wang et al. (2013) showed that long-term variability of the NHMS monsoon is associated with coherent spatial patterns of lower boundary anomalies. Figure 18.15 shows the regressions of the SST anomalies and the 2 m land temperature against the NHSM index together with anomalous surface pressure (hPa) and the 850 hPa anomalous wind vectors. The SST pattern portrays a generally cooler eastern and southeastern Pacific Ocean. Overall, a greater cooling has occurred in the SH along with anomalous higher surface pressures similar to an ENSO pattern but with a larger meridional extent and occurring over a longer time scale. This pattern is referred to as a mega-ENSO. There is also an interhemispheric difference in temperature measured by a thermal contrast index (HTC), defined as the mean surface temperature difference between 0∘ and 60∘ N and 0∘ and 60∘ S. Longer-term time series are shown in Figure 18.15. The mega-ENSO has a well-defined temporal evolution (Figure 14.22). The long-term variation of the NHSM circulation index in the period 1958–2011 is well correlated with the mega-ENSO index (r = 0.77) and the hemispheric thermal contrast (HTC) (r = 0.63). The heavy black line denotes the 3-year running average 12 E.g. Essex and Tsonis (2018).

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18 The Changing Tropics

(a)

VWS

Mega ENSO

r = 0.77

(b)

VWS

AMO

r = 0.44

(c)

VWS

HTC

r = 0.63

2 1 0 –1 –2

2 1 0 –1 –2

2 1 0 –1 –2

1960

1965

1970

1975

1980

1985

1990 1995

2000

2005

2010

Figure 18.16 NHSM circulation index in (V WS ), the mega-ENSO hemispheric thermal contrast (HTC) that correlate, respectively, at 0.77 and 0.63 over the period 1958–2011. The merged ERA-40 (1958–1978) and ERAI (1979–2011) reanalysis datasets were used. Source: After Wang et al. (2013).

of the NHMS index. These significant correlations were confirmed for the period 1871–2010 by using the twentieth century reanalysis data set13 (r = 0.62). Physically, the eastern Pacific cooling and the western Pacific warming are consistent with a strengthening of the Pacific subtropical highs in the two hemispheres and their associated changes in the trade winds, causing moisture to converge into the Asian and African monsoon regions and thus contributing to the intensification of NHSM rainfall. Therefore, the recent trend in NHSM is partly driven by a “mega-ENSO.” In addition, Wang et al. (2013) showed that the Atlantic Multidecadal Oscillation (AMO) was also correlated with the NHSM index at the 0.44 level. Although the mega-ENSO and AMO are primary sources of the interdecadal variations of the NHSM, one cannot rule out the influence of the global warming trend. Figure 18.16 shows that the NH 2-m air 13 Compo et al. (2011).

temperature has warmed more than the Southern Hemisphere (SH) counterpart by 0.36 ∘ C over the past 32 years. The NHSM intensity is linked to the HTC defined by the 2-m air temperature difference between the NH (0∘ –60∘ N) and Southern Hemisphere (60∘ S–0∘ ) (r = 0.63; Figure 18.15c). Dynamically, the enhanced HTC can generate meridional pressure gradients that drive low-level cross-equatorial flows from the SH to the NH (Figure 18.14c) and converge into the NHSM trough regions. We note that the “NH warming faster than the SH” or “warm NH–cold SH” pattern is a characteristic of the projected warming under increasing greenhouse gases forcing, as found by Wang et al. (2013).

18.3 Summary and Conclusions Analysis of the evolution of CIH in the tropics during the last 50 years reveals a non-stationary relationship

18.3 Summary and Conclusions

with SST as areas containing organized deep convection shifting to higher values of SST. CMIP3 climate model simulations show a similar behavior to observed variability in the twentieth century observations and project a movement to even higher threshold temperatures during the twenty-first century occupying about 25% of the global ocean irrespective of the background mean SST. Overall, as well as the mid-Holocene and the LGM. In essence, with the warming of SST during the twentieth century, the observational and model results point to an intensification of the hydrological cycle as suggested by Trenberth (1999), with intensified latent heat release within the DWP. These concepts provide an intersting extension of the Lindzen and Nigam (1987) results for a changing climate. Lindzen and Nigam pointed out the importance of the gradient of SST in driving the tropical circulation. However, even if the gradient of SST were to remain the same as SST universally increases, one would expect an increase in moisture convergence from ClausiusClapeyron effects that are consistent with the simple models presented above. Intensification of the hydrological cycle could have important thermodynamical and dynamical implications in tropical and global climate, ranging from the location and amount of convection and rainfall to the vigor of teleconnection patterns between the tropics and higher latitudes. The near constancy in DWP area, together with the intensified CIH and vertical shear may provide insights into a future shift in the hurricane distributions toward more intense, yet fewer storms (e.g. Bender et al. 2010 and Knutson et al. 2010). Overall, the increasing magnitude of the atmospheric heating within the DWP portends an increase in the average intensity of tropical cyclones, as disturbances have a higher likelihood of intensification in large-scale moist-convective environments characterized by high CIH (e.g. Hoyos et al. 2006). However, accompanying the increased heating within the DWP is an increase of mass flux into and out of the region of net heating, increasing the vertical wind shear at the peripheries of the dynamics warm pool (Figures 18.9 and 18.11). This increase in vertical wind shear may provide a constraint on cyclone formation, potentially limiting the number of tropical cyclones, even as their average intensity increases. Paleoclimate simulations (PMIP I and II) for the LGM and the mid-Holocene are in agreement with the results from CMIP3 simulations of the twentieth century matching observations, with a change in the size of the OWP as temperature increases relatively from one period to another, and a shift in the convective

SST threshold that renders the DWP almost constant in area. Specifically, Figure 18.12 illustrates that the transitional temperature T H is about 2 ∘ C cooler during the LGM than during the Mid-Holocene or the pre-Industrial period compared to a current transitional T H of about 27 ∘ C (Figure 18.5b). The relative areas of the OWP and the DWP during these climate epochs were shown in Figure 18.12b. The OWP during the mid-Holocene and pre-Industrial periods appears considerably larger than during the LGM. However, the LGM convective threshold is about 2 ∘ C lower than either the mid-Holocene or pre-Industrial period while maintaining a nearly constant DWP area between epochs. With a lower threshold temperature, the integrated heating in the LGM the DWP would be considerably smaller than the other periods considered. The associated reduction in the vigor of the hydrology cycle is consistent with the apparent relative aridity occurring in the tropical regions during the LGM (Webster and Streten 1978 and Pinot et al. 1999). However, a number of PMIP studies have suggested that the result may seem counterintuitive as the number of tropical cyclones in the LGM period was much the same as at present, with similar genesis locations. Since strong vertical wind shear is not conducive to the formation of tropical cyclones (e.g. Gray 1968), the reduced shear associated within convective areas during the LGM may have allowed the formation of tropical cyclones within the DWP region with SSTs that are lower than present values. Finally, we considered changes in tropical circulations during the periods of global warming, especially in the last century particularly during the last 40 years, for which credible global precipitation data was archived. This examination was facilitated by defining an NH-wide monsoon index as well as reanalysis data. It was shown that since 1980, the major tropical circulations had increased in magnitude. This analysis is in stark contrast to the climate model analysis of Vecchi and Soden (2007). This was surprising as we had found that the same models had replicated our distributions of CIH during the present era quite well, although they were statistically adjusted somewhat. It could well be that the climate models are speaking to a longer time scale and that the Wang et al. (2013) pertain to interdecadal modes. However, irrespective of this, it would seem that the diagnostics do point toward a predictive potential of the larger-scale monsoon. In fact, the same predictors have been used by Wang et al. (2018) to produce outlooks on the decadal time scale for the monsoon regions.

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Notes I Consistent with the detailed analysis of Deser et al.

(2010). II Chan and Nigam (2009) also found differences between the two estimates of CIH due principally to different estimates across Indonesia. The biggest difference in Figure 14.34 occurs around 1970, a period that unfortunately was not contained within the Chan-Nigam analysis. III CMIP refers to the “Climate Model Intercomparison Project” of the World Climate Research Program. Version 3, CMIP3, is described by Meehl et al., (2007). Model results shown in Figure 14.35 are: the NCAR CCSM3 and PCM1, the NOAA GFDL-CM2.1, and the NASA GISS. Hoyos and Webster (2011) show

observation-model comparisons for a further four models. IV It is worth reemphasizing that all models discussed here reproduced the observed increase in OWP reasonably well before the bias correction (see Figure 14.35). All corrections introduced by the linear fitting equations only account for the magnitude of the increase and not for the sign of the tendencies. The multi-model observation correlation with observations is 0.82. V IPPC emission scenarios: COMMIT, CO2 emissions at 2000-year level. A1B and A2 rapid industrial and population growth, the former more integrated.

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19 Some Concluding Remarks When I entered graduate school in the late 1960s, my advisor at the time had just completed a treatise entitled “Some remaining problems in numerical weather prediction” (Charney 1967). Five problems were described and referred to as: . . . . certain islands of resistance which seem to hold out stubbornly in the face of all attacks.… Charney (1967, p. 61) Broadly speaking these obstacles were: • What is the relationship between the turbulent boundary layer and synoptic scale variability? • How can steep gradients associated with fronts and topographic features be handled in models? • Do models correctly handle the cascade of energy between scales of motion? • How are convective processes and large-scale tropical circulations related? • What determines the structure, variability, and location of such preeminent tropical features as the ITCZ organized and maintained? Charney’s paper deeply depressed me. Here I was, a brand new student in graduate school embarking on a career in tropical meteorology surrounded by bright and eager graduate students, all of whom seemed to know what they were doing, a faculty that was acknowledged as the world leaders in atmospheric science, and only a handful of questions remained. Clearly all of these would be answered by the end of the semester. This made me wonder whether I really should have gone to medical school after all. Now, over 45 years later, many new questions regarding the tropical system have arisen. It is interesting though to determine what progress has been made in solving the Charney problems and how we have approached their solution. Understanding complex natural phenomena has generally been undertaken by following a reductionist approach, whereby a phenomenon’s complex nature is reduced into individual components that are assumed to

work together to produce observed structures. Reductionism evolved from Rene Descartes’I “mechanical philosophy,” whereby the universe is thought of as a complicated machine made up of identifiable components. Essentially, reductionism aims to understand the nature of complex things by reducing them to the interactions of their parts, or to simpler or more fundamental components. This implies that a complex system is nothing but the sum of its parts. Chapters 6 and 7 adopted a reductionist approach in attempting to explain tropical variability in terms of fundamental linear equatorially trapped modes. This modal structure formed the basic explanations of intraseasonal variability (Chapter 15) and the coupled ocean–atmosphere ENSO and Indian Ocean Dipole (Chapter 14). And, to some degree, the behavior of many of these large scale systems circulation systems may be identified as combinations of fundamental components of the ocean–atmosphere system. However, when it comes to predicting the state of a complex system, we find that the reductionist approach does not help in the prediction of emergent (or unforeseen) phenomena. If the climate system were purely a combination of linear modes, then prediction would be a much simpler endeavor, with the main challenge being to reduce the impact of inaccuracies in the initial fields that may introduce uncertainty into a prediction. Yet extended weather and climate prediction has proven to be universally difficult. For example, even though in Chapter 14 we pointed out that the basic components of ocean–atmosphere interaction are basically understood, each ENSO event (the supposed sum of these parts or components) is very different both in timing, duration, and amplitude. Predictions of whether or not an El Niño or La Niña event will develop following the boreal spring show little skill. Also, it is difficult to assess whether or not a La Niña will be followed by an El Niño or vice versa. Once an ENSO event develops in the early boreal summer, it tends to follow its own trajectory, which may be similar or somewhat different to other El Niño or La Niña events.

Dynamics of the Tropical Atmosphere and Oceans, First Edition. Peter J. Webster. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

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19 Some Concluding Remarks

Similar differences between the characteristics of individual circulation events exist on subseasonal time scales as well. As with ENSO, much of the structure of intraseasonal events features can be recognized in terms of fundamental equatorial modes whereby convection has been accounted for in some manner. Each MJO has an individual character. Even the canonical MJO, forming in the equatorial Indian Ocean, varies from case to case. In Chapter 11 (also Chapters 13 to 18), it was argued that a more holistic approach was necessary in order to understand the complex nature of the Earth system. Holism1 claims that complex systems are inherently irreducible and are more than the sum of their parts, owing to chaos and nonlinearities. Emergent behavior may arise from complex systems that cannot be deduced from consideration of the components of the system alone. Holism leads to “systems thinking” and possesses derivatives such as chaos and complexity. This discussion grew from attempts to understand interactions between the extratropics and the tropics. In Chapter 9 it was argued that extratropical waves had difficulty propagating through zonally symmetric easterlies toward the equator so that more complex explanations were necessary to explain extratropical–tropical interactions. In Chapter 10, it was found that a zonally symmetric Hadley Circulation could not explain fully the influence of the tropics on the extratropics or vice versa. Yet progress was made toward understanding the interaction between the tropics and the extratropics by noting the interaction of two nonlinear systems, one being the divergent circulations transporting PV poleward and the other, the Rossby wave regime, returning the PV to the tropics. The predictability of complex systems can be described using concepts introduced by Hofstadter (1980).II Simply stated, system predictive skill depends on its degree of complexity. Three hierarchies of organization and disorganization are suggested: simple, complex, and tangled, where the simpler the complexity the greater the potential predictability. By extension, the more complex the system, the less predictability the system possesses. (i) A simple system possesses two components, A forcing B or two interacting bodies as in the classical “two-body” problem involving the interaction of a planet and its moon. Variability of the predictive outcome arises only from the uncertainty of describing either A and B. An example of a simple and highly predictable system is the lunar forcing of ocean tides. 1 Holism emerges from the Greek hólos meaning all, whole, entire, total . . . .

(ii) The introduction of a third component (C) produces a complex system and introduces uncertainty into how the three components (A, B, and C) interact. First, initial conditions require a description of the system now extended to three components instead of two thus adding greater uncertainty. Further, the system trajectory may be very different depending on the initial scale of each of the three components or its initial magnitude. (iii) The most complex system, tangled, may have multiple interacting components (C, D, …, etc.). The climate system itself is such an example, with interacting oceans, atmosphere, cryosphere, and land systems. Specific circulation patterns in the climate system may be complex or tangled. The existence of some predictability of an ENSO extrema, once it is initiated, suggests that the system is complex and probably not tangled. Similarly, the wider influence of ENSO is a complex system since some predictability is retained after initiation. However, the lack of persistence or predictability across the boreal spring suggests that at longer time horizons, or at certain times of the year, the ENSO system would appear to be tangled and unpredictable. Over the course of an annual cycle, though, the system moves from tangled to complex as it changes from frail to more robust, as discussed in Chapter 14. Hence predictability of a system depends on a number of factors: (i) The degree to which the initial conditions of the system are known. (ii) How well A and B are understood physically and represented by a model (either theoretical, empirical or numerical). (iii) How large and variable is component C (or D and E …)? Are they stochastic? Does one element or one process dominate over all others? For example, the solar system is a complex multi-body system but the Sun’s gravitation makes it (almost) a stable system as it represents 98% of the mass of the solar system and, thus, chaotic motion of the planets is rare. Given these points, it is tempting to adopt a holistic approach to the prediction of tropical phenomena by resorting to complicated coupled ocean–atmosphere– land models. However, given model formulation and initial data uncertainties, it is necessary to use a probabilistic approach in which the model is perturbed many times to produce an ensemble of forecasts. It is also clear that a hierarchy of methods are needed to increase understanding and predictive capability, including both holistic and reductionist approaches. If the components of a complex system can be identified and it can be

Some Concluding Remarks

determined that component C, for example, is more dominant at some stage of the prediction than another, one may be able to anticipate confidence in the results of the probabilistic forecast. As discussed above, understanding of the interaction of the tropics and extratropics could only have been developed through a holistic or system approach. The behavior of an individual component (the collective divergent circulations or the recurring Rossby waves families) could not have led to a determination of the synergies between the tropics and extratropics as discussed in Chapter 11. Instead, we gained an understanding by applying the Haynes and McIntyre impermeability theorem that constrained the advection of a potential vorticity substance between the tropics and the extratropics or, specifically, across a latitude circle. It could be that other difficult problems, such as why there is little difference in annual precipitation (rate or volume) of each hemisphere, will be understood through similar system constraints. So, what can we say now about Charney’s obstacles laid out in 1967? There has been substantial progress in the first two problems. In 1967, the grid point resolution of the earliest numerical weather models was hundreds of kilometers. Now it is closer to 10 km and will possess

greater resolutions and become cloud resolving in the near future. The number of vertical levels has increased as well from only a few to over 50 in some operational models. Topographic relief is incorporated directly through use of the 𝜎-coordinate system.2 However, Charney’s fourth and fifth problems remain “islands of resistance” to this day. Simply, we still are uncertain about how equatorial dynamics and convection interact and the degree of their mutual dependency. With respect to the ITCZ, Section 13.1 offered six theories regarding the location of equatorial convection. Although some are stronger than others, their number is an indication that closure on the issue has not yet been reached. In addition, we have unearthed many new mysteries. One is the discovery of enclaves of disturbances existing within tropics made up of families of convection ranging from diurnal through synoptic and biweekly to intraseasonal. In retrospect, Charney’s tropical problems were not solved by the end of the semester, nor by the end of the decade, and not even in the present time. In fact, investigations of these problems have spawned many new exciting problems. It seems that I was needlessly depressed in 1967 about the future challenges in tropical meteorology.

Notes I René Descartes (1596–1650): French philosopher,

mathematician, and scientist. II D. R. Hofstadter (1945–) is a Professor of Cognitive Science at the University of Indiana. He is widely known for Hofstadter’s Law, which states: “It always takes longer than you expect, even when you take into account Hofstadter’s Law.” The Law was first

2 The pressure height of a surface is scaled by the surface pressure, thus taking into account the variability of surface topography.

expressed in the book by Hofstadter (1979), “Gödel, Escher, Bach: An Eternal Golden Braid,” which explores common threads in the lives of artist M. C. Escher, composer J. S. Bach, and logician K. F. Gödel. From this book and his works have emerged keen insights into geophysical predictability.

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Appendix A Thermal Wind Relationship From Figure 1.2 it is clear that along a line of constant pressure there is a latitudinal gradient of atmospheric temperature. Furthermore, the gradient of temperature is a maximum in the winter hemisphere, reflecting the greater radiational cooling occurring in the winter higher latitudes. Figure A.1 shows a schematic version of Figure 1.2 depicting pressure surfaces as a function of height. Now, the geostrophic wind is proportional to the spatial gradients of pressure. The geostrophic wind will increase with height because the upper tropospheric pressure surface is more inclined than the lower. We can use the hydrostatic equation in simple form to show 𝜌g = −𝜕p∕𝜕z

(A.1a)

Thus the change in geostrophic wind with height (i.e. the geostrophic shear) must be related to the horizontal temperature gradient. We can approach this problem more formally using the scalar geostrophic wind equations:

p = 𝜌RT

(A.1c)

𝜌fvg = −𝜕p∕𝜕x

(A.2b)

together with the hydrostatic Eq. (A.1a) and the equation of state (A.1c). In Eqs. (A.2), u and v represent the zonal and meridional components of horizontal velocity. We now use the equation of state to eliminate 𝜌 in the geostrophic and hydrostatic equations, giving

(A.1b)

In Figure A.1, the pressure differences between p and p − 𝛿p in the two columns A and B is the same (𝛿p) so 𝜌A > 𝜌B . As average pressure in the two columns is the same, then the temperature in column B must be greater than A. This follows from Eq. (A.1a) using the equation of state:

(A.2a)

and

or Δz ≈ Δp∕𝜌g

𝜌fug = −𝜕p∕𝜕y

fvg = RT𝜕 ln p∕𝜕x

(A.3a)

fug = −RT𝜕 ln p∕𝜕y

(A.3b)

g = −RT𝜕 ln p∕𝜕x

(A.3c)

and

We now eliminate all reference to pressure through a series of cross-differentiations. First, we differentiate Eqs. (A.3a) and (A.3c) with respect to z and x, sequentially followed by a cross-differentiation of Eqs. (A.3b) and (A.3c), with respect to z and y. We then have two equations in u, v, and T: 𝜕ug 𝜕z

=−

g 𝜕T ug 𝜕T + fT 𝜕y T 𝜕z

(A.4a)

and 𝜕vg

p-δp

𝜕z

z p A

B x,y

Figure A.1 Cross-section of two differentially tilted isobaric surfaces, p and p + 𝛿p. A and B represent two columns between the isobaric surfaces.

=

g 𝜕T vg 𝜕T + fT 𝜕x T 𝜕z

(A.4b)

These are the complete thermal wind equations expressed differentially. Simple scaling analysis leads to the conclusion that the second terms on the right-hand side in each equation are considerably smaller than the first. For example, from observations, the vertical shear (𝜕ug /𝜕z) varies in the vertical by about 25% per km, whereas the vertical stratification varies by less than 5% per km. Further, ug /T and vg /T are both ≪1. Therefore

Dynamics of the Tropical Atmosphere and Oceans, First Edition. Peter J. Webster. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

450

Appendix A Thermal Wind Relationship

the first terms on the right-hand side of the thermal wind equations must account for most of the vertical wind shear. Thus, to good approximation: 𝜕ug 𝜕z

≈−

g 𝜕T fT 𝜕y

(A.5a)

and 𝜕vg 𝜕z

≈

g 𝜕T fT 𝜕x

(A.5b)

We see later in Section 3.1 that the tropical atmosphere is largely barotropic so that surfaces of pressure and density tend to coincide. That is, 𝜌 = 𝜌(p) and T = T(p) only. This has interesting consequences to the shear of the geostrophic flow. For a barotropic fluid: 𝜕T dT 𝜕p dT = = −𝜌fug 𝜕y dp 𝜕y dp

(A.6a)

and dT 𝜕p dT 𝜕T = = −𝜌g 𝜕z dp 𝜕z dp

(A.6b)

Inserting these two expressions into Eqs. (A.4a) and (A.4b) leads to 𝜕ug 𝜌gug dT 𝜌gug dT = − = 0 and 𝜕z T dp T dp 𝜕vg =···=0 (A.7) 𝜕z Therefore in a barotropic atmosphere, the geostrophic wind cannot vary with height. Equation (A.7) has consequences in the ocean as well, and states that there will be no vertical variation of an ocean current if the ocean is barotropic and has a density that does not vary horizontally.

451

Appendix B Stokes’ Theorem The circulation of the atmosphere in the latitude–height plane may be thought of as “vortex tubes” defined by fields of [𝜔] and [v] such as shown in Figure B.1. Stokes’ theorem allows the conversion of an area integral to a line integral. Formally, Stokes’ theorem is written as ∮Li

̃ •d̃l = V

∫ ∫A

̃ )•̃ (∇ × V n da

(B.1)

The theorem states that the sum of the vorticity associated with the vortices within area A can be represented as a line integral around a circuit Li . The choice of the line Li is arbitrary. Here, the theorem allows the integration of all vortex tubes around a set path to produce a circulation. Thus, a contour of [𝜓(𝜑, z)] (say 6 × 1010 kg s−1 in Figure 1.4d(ii)) represents the integral of mass transport of all of the vortex tubes within the space defined by the contour.

A

B

C

D

Figure B.1 A fluid may be thought of as a collection of vortex ′ tubes, each having its own circulation c . Stokes’ theorem states that the integral of all of the subcirculations, C, can be expressed as the line integral through the fluid. Note that within the fluid there is a cancelation of the circulations between vortices.

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453

Appendix C Dry and Moist Thermodynamical Stability Figure 3.2 described the mean thermodynamical state of the tropics in terms of potential temperature (𝜃), the environmental equivalent potential temperature (𝜃 e ), and the environmental saturated equivalent potential temperature (𝜃e∗ ). It is useful to define these fundamental properties of the tropical atmosphere and describe how they influence vertical stability. The definition of potential temperature 𝜃 is the temperature a parcel located at some pressure p would obtain if it were raised or lowered adiabatically to some standard pressure p0 , usually 1000 hPa. The equivalent potential temperature, 𝜃 e , is the temperature of a parcel with qv < qvs that is raised to saturation and then further to a level where qv = 0, its latent energy expended, and adiabatically lowered to some standard pressure p0 . Saturated equivalent potential temperature 𝜃e∗ is the temperature of a saturated parcel qv = qvs at some level p that is raised until qv = 0 and lowered adiabatically to some standard level. 𝜃e∗ represents the theoretical limit of 𝜃 e assuming saturation. ̇ s−1 ) of a saturated Consider now the heating rate Q(J parcel undergoing pseudo-adiabatic ascent, such that dq Q̇ = −LE vs (C.1) dt Here LE is the latent heat of condensation and qs the saturated vapor pressure. The first law can then be written in the form:1 ) ( L dqvs LE d d ln 𝜃 (C.2) =− E ≈− qvs dt Cp T dt dt Cp T where we have taken advantage of noting that the rate of change in qs , following the parcel during 1 We have assumed that the reader of this text is well-versed in basic thermodynamics of dry and moist air. Otherwise, the fundamental text of Curry and Webster (1999) may be useful toward understanding the thermodynamical properties of atmospheres and oceans.

pseudo-adiabatic ascent, is much larger than either the change in temperature or latent heating coefficient. Noting that the saturated vapor pressure will only change with ascent, we can write: | 𝜕qvs ,w > 0 dqvs || (C.3) = | 𝜕t | dt |0, w ≤ 0 | Integrating Eq. (C.2) with respect to time leads to the definition of equivalent potential temperature: 𝜃e = 𝜃 exp(LE qs ∕Cp T)

(C.4)

where 𝜃 = T(p0 /p)𝜅 and 𝜅 = R/C p . Equation (C.4) is the temperature of a moist parcel with q < qs at some reference level that is raised to saturation and further until qs = 0 and then lowered adiabatically to some reference level (usually 1000 hPa). In Eq. (C.4) the saturated vapor pressure is that which a parcel will attain on ascent. Thus the drier the parcel, the less the value of qs . This may be seen from Figure 3.2. The first law can then be written as ( ) 𝜕 (C.5) + V .∇ 𝜃 + wΓe = 0 𝜕t where Γe is the equivalent moist static stability replacing the dry static stability and defined by | 𝜕 ln 𝜃e | for qv ≥ qvs and w > 0 | Γe = || 𝜕z | 𝜕 ln 𝜃 for q < q and w < 0 | v vs | 𝜕z (C.6) The development above points toward a particular problem. Returning to Figure 3.2, we note that the further away from saturation (i.e. the larger the difference between 𝜃 e and 𝜃e∗ (or qv and qvs ) at a particular level), the greater the work necessary to lift a parcel to saturation and to buoyancy.

Dynamics of the Tropical Atmosphere and Oceans, First Edition. Peter J. Webster. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

455

Appendix D Derivation of the Wave Equation (5.11) First, u is eliminated from equation set (5.10) by multiplying Eq. (5.10a) by 𝛽y and Eq. (5.10b) by i(𝜔 − kU); adding the resultant equations produces a first equation in v and h: ] ) [ ( dU 2 − 𝛽y + (𝜔 − kU) v 𝛽y dy dgh + ik𝛽y(gh) + i(𝜔 − kU) =0 (D.1) dy Here U = U(y) represents the background basis state. We multiply Eq. (5.10a) by kc2 and Eq. (5.10c) by (𝜔 − kU) and obtain the second equation in v and h: ] [ ) ( dU 2 kc − 𝛽y − (𝜔 − kU)𝛽yU v dy dv + (𝜔 − kU)c2 + i[k2 c2 − (𝜔 − kU)2 ]gh = 0 dy (D.2) Solving for gh we find that −1 igh = 2 2 [k c − (𝜔 − kU)2 ] ] ([ ( ) dU 2 × kc − 𝛽y − (𝜔 − kU)𝛽yU v dy ) 2 dv + (𝜔 − kU)c (D.3) dy To obtain an expression dgh/dy we differentiate Eq. (D.2) with respect to y to give, after some manipulation: dv (𝜔 − kU)c2 2 dy [ ( ) ] dU dv 2 2 dU + kc − 𝛽y − (𝜔 − kU)𝛽yU − kc dy dy dy ( [ ) d2U 2 + kc − 𝛽 − (𝜔 − kU) dy2 ] ( ) dU dU × 𝛽U + 𝛽y +k 𝛽yU v dy dy

dU (gh) dy d(gh) =0 (D.4) + i[k2 c2 − (𝜔 − kU)2 ] dy We then substitute Eq. (D.3) into Eq. (D.4) and, again after much manipulation, obtain ] [ 2k𝜔d dU 𝛽yU dv d2v + − 2 dy (𝜔d 2 − k2 c2 ) dy c dy2 ( ) ( ) ⎡ k ⎤ 𝛽y dU d2U − 𝛽 + − 𝛽y ⎢ ⎥ 2 2 dy c dy ⎢ 𝜔d ⎥ ⎢+ 1 (𝜔2 − k2 c2 ) ⎥ 2 ⎢ c d ⎥ ( ) ⎢ ⎥ dU dU 2k2 − 𝛽y + ⎥v = 0 + ⎢ (𝜔2 − k2 c2 ) dy dy d ( ⎢ )⎥ ) ( ⎢ ⎥ 1 dU dU ⎢− 𝜔 c2 𝜔d 𝛽U + 𝛽y dy − k dy 𝛽yU ⎥ ⎢ d ⎥ ⎢ k𝛽 2 y2 U dU 2𝜔d k𝛽yU ⎥ ⎢− 𝜔 c2 − dy 2 2 ⎥ 2 2 c (𝜔d − k c ) ⎣ ⎦ d (D.5) + 2i(𝜔 − kU)k

In the general form of the wave Eq. (5.11), the coefficients are 2k𝜔d dU 𝛽yU (D.6a) − 2 p(y) = (𝜔d 2 − k2 c2 ) dy c and ( ) ( ) 𝛽y dU k d2U q(y) = − 𝛽 + − 𝛽y 𝜔d dy c2 dy2 1 + 2 (𝜔2d − k2 c2 ) c ( ) 2k2 dU dU + 2 − 𝛽y dy (𝜔d − k2 c2 ) dy ) ( ( ) 1 dU dU − + 𝛽y 𝜔 𝛽U − k 𝛽yU d dy dy 𝜔d c2

Dynamics of the Tropical Atmosphere and Oceans, First Edition. Peter J. Webster. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

−

k𝛽 2 y2 U dU 2𝜔d k𝛽y − dy c2 (𝜔2d − k2 c2 ) 𝜔d c2

(D.6b)

456

Appendix D Derivation of the Wave Equation (5.11)

If U= 0, Eq. (D.5) reduces to Matsuno’s (1966) classical equation, which is be discussed in Section 6.1.1. We note, too, that the introduction of even a constant zonal wind increases the complexity of the system and

even more so if U = U(y). The majority of the terms are non-Doppler. Pure Doppler effects are contained within 𝜔d .

457

Appendix E Conservation of Potential Vorticity of Shallow Water System From Section 5.1, we can write the nonlinear governing equations of a shallow water system in an adiabatic and frictionless system: 𝜕u 𝜕u 𝜕h 𝜕u +u + v − fv + g 𝜕t 𝜕x 𝜕y 𝜕x 𝜕v 𝜕v 𝜕v 𝜕h +u + v + fu + g 𝜕t 𝜕x 𝜕y 𝜕y ( 𝜕h 𝜕h 𝜕h 𝜕u +u +v +h + 𝜕t 𝜕x 𝜕y 𝜕x

=0

(E.1)

=0 ) 𝜕u =0 𝜕y

(E.2) (E.3)

Differentiating Eq. (E.2) with respect to x and Eq. (E.1) with respect to y we obtain ( ) 𝜕𝜁 𝜕𝜁 𝜕𝜁 𝜕u 𝜕v +u +v +𝜁 + 𝜕t 𝜕x 𝜕y 𝜕x 𝜕y ( ) 𝜕f 𝜕u 𝜕v +v +f + =0 (E.4) 𝜕y 𝜕x 𝜕y where 𝜁=

𝜕v 𝜕u − 𝜕x 𝜕y

is the relative vorticity. Noting that f = f(y) only, Eq. (E.4) becomes

( + (𝜁 + f )

𝜕u 𝜕v + 𝜕x 𝜕y

) =0

(E.5)

Performing the operation ((E.5)/h − (𝜁 + f )(E.3)/h2 ), we obtain 1 𝜕(𝜁 + f ) (𝜁 + f ) 𝜕h − h 𝜕t( h2 𝜕t ) 1 𝜕(𝜁 + f ) (𝜁 + f ) 𝜕h +u − h 𝜕x h2 𝜕x ( ) 1 𝜕(𝜁 + f ) (𝜁 + f ) 𝜕h +v − =0 (E.6) h 𝜕y h2 𝜕y Finally, we define potential vorticity as 𝜁 +f h and Eq. (E.6) becomes qh =

(E.7)

𝜕q 𝜕q dq 𝜕qh +u h +v h = h =0 (E.8) 𝜕t 𝜕x 𝜕t dt In this shallow water system potential vorticity is conserved.

𝜕(𝜁 + f ) 𝜕(𝜁 + f ) 𝜕(𝜁 + f ) +u +v 𝜕t 𝜕x 𝜕y

Dynamics of the Tropical Atmosphere and Oceans, First Edition. Peter J. Webster. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

459

Appendix F Solutions to the Vertical Structure Equation for a Constant Lapse Rate Atmosphere The use of an isothermal atmosphere approximation is still, of course, a considerable simplification of the real world. A constant lapse rate atmosphere (i.e. dT/dz = constant) is closer to reality and a reasonable approximation for at least the troposphere, as can be noted in Figure 7.1. Also, analytic solutions to Eq. (7.1) exist for this atmosphere, where the stability, S(z), takes on a linear form in z such as S(z) = az + b

(F.1)

where a = (ST − SB )∕zT and b = SB where ST and SB are the stabilities at some height zT and z = 0, respectively. Thus, Eq. (7.1) becomes ( ) d2Z 1 1 + (az + b) − Z=0 (F.2) gh 4 dz2 where a and b are constants. This is a more complicated equation to solve than for the isothermal atmosphere as the coefficient of Z is a function of z. However, there are standard solutions for equations of the same form as Eq. (F.2). An equation of the form: d2Z − 𝜂Z = 0 d𝜂 2 has solutions Z(𝜂) = aA(𝜂) + bB(𝜂)

(F.3)

(F.4)

where A and B are the Airy functions and a and b are constants. The task is to find the conditions for which Eq. (F.3) has solutions of the form (F.4). Noting that the coefficient of Z is linear in z, we try the transformation: 𝜂 = pz + q

where p and q are constants. Substituting Eq. (F.5) into Eq. (F.3) yields [ ) )] (( aq 1 a b 1 d2Z + 3 + − Z=0 − d𝜂 2 p gh p2 p3 gh 4p2 (F.6) Matching coefficients gives ) ) (( aq 1 b a 1 − =0 = −1 and − p3 gh p2 p3 gh 4p2 (F.7a) so that: (gh)2∕3 a1∕3 b , q = − 2∕3 , (gh)1∕3 4a2∕3 a (gh)2∕3 ( ) a1∕3 b gh 𝜂=− z+ − (F.7b) 1∕3 a 4a (gh) p=−

With these coefficients, Eq. (F.3) has solutions given by Eq. (F.4). Like the functions for an isothermal atmosphere, Airy functions are either oscillatory or evanescent depending on the sign of 𝜂 in (F.5) as plotted in Figure 7.8. For negative values of 𝜂 the solutions are oscillatory. For positive values the solutions either grow with increasing 𝜂 (the B(𝜂) solution) or decay exponentially (the A(𝜂) solution). The B(𝜂) solution may be neglected for 𝜂 > 0 to satisfy the boundary condition of finiteness as 𝜂 (and therefore z) becomes large.

(F.5)

Dynamics of the Tropical Atmosphere and Oceans, First Edition. Peter J. Webster. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

461

Appendix G Nonlinear Numerical Model The numerical model developed here will be utilized in a number of sections of the book. Thus, the model will be described here in some detail. To overcome the very obvious scale separation problems that exist with the WKBJ approximations, we will now utilize a numerical model that can handle wave propagation through an inhomogeneous basic flow without scale separation approximations and that includes wave–basic state and wave–wave nonlinear interactions. Although numerical eigenvalue–eigenfunction techniques could be employed to find the solutions to the steady linear problem, we prefer to utilize a numerical initial value technique in order to study the evolution of the equatorial response to a transient equatorial forcing function within very complicated basic flows.

Here, the m0 represent map factors chosen for either Mercator (Webster and Holton 1982) or a sphere (Chang and Webster 1995). In Eq. (G.1), M(x, y, t) is a specified mass source-sink function, which will be used to develop a specific basic flow and also to act as the transient energy source,1 u and v represent the eastward and northward components of velocity, and 𝛾 is the Rayleigh dissipation rate (s−1 ) set to give a 10-day e-folding time scale. The sets (Eqs.(G.1) and (G.2)) represent the full nonlinear model. A companion linear system can be created about an arbitrary nonlinear two-dimensional basic state u, v, h, where the dependent variables have been decomposed by setting, for example: u(x, y, t) = U(x, y) + u′ (x, y, t)

(G.4)

so that

( ) 𝜕u′ ′ 𝜕U ′ 𝜕U I (u) = m0 U +u +v 𝜕x 𝜕x 𝜕y

G.1 Model

′

We choose a global shallow fluid mode that uses a spectral transform method. In generic form, the equations representing a divergent shallow fluid are written as 𝜕H 𝜕u − fv + I(u) = −gm0 − 𝛾u 𝜕t 𝜕x 𝜕H 𝜕v − fu + I(v) = −gm0 − 𝛾v 𝜕t 𝜕y 𝜕H + I(H) = M 𝜕t where I are the nonlinear operators: ( ) 𝜕u 𝜕u I(u) = m0 u +v 𝜕x 𝜕y ( ) 𝜕v 𝜕v I(u) = m0 u +v 𝜕x 𝜕y [ ] d𝜇 𝜕uH 𝜕vH I(H) = m0 + − Hv 𝜕x 𝜕y dy

(G.1)

(G.2)

H represents the total depth of the shallow fluid that can be resolved into a mean depth H 0 and a deviation about the mean h(x, y, t) such that H = H0 + h(x, y, t)

(G.3)

(G.5)

G.2 Method of Solution A semi-spectral method of solution is utilized from Webster and Holton (1982) where the longitudinal structure of the variables is expanded in terms of sines and cosines while a grid-point representation is used in latitude. The nonlinear terms (the I-functions of Eq. (G.2)) are evaluated using a transform method of Orzag (1970) that uses an exact grid representation of the Fourier coefficient recomposition computing the I function efficiently and exactly on a Gaussian grid. These resultant grid quantities are then expanded in longitudinal eigenfunctions. Using a semi-implicit time differencing scheme (Holton 1976) and a staggered space-difference scheme in latitude, sets of linear simultaneous equations evolve that are readily solved by matrix inversion. 1 From a dynamical similitude perspective between a multilevel model and a divergent one-layer barotropic model, a mass sink is equivalent to net heating in a column while a source is equivalent to net cooling. Thus, a specified distribution of M is equivalent to specifying a heating distribution in a baroclinic system.

Dynamics of the Tropical Atmosphere and Oceans, First Edition. Peter J. Webster. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

462

Appendix G Nonlinear Numerical Model

G.3 Generation of the Nonlinear Basic States To generate basic states that are similar to that observed flow we separate the flow into zonally symmetric and asymmetric components. In Eq. (G.3) we write H0 (y) = H00 + Hb (y)

(G.6)

where H 00 is the mean depth and H b (y) defines the zonal geostrophic flow. In a more complicated system, a symmetric basic state would be maintained by a slowly varying radiational heating gradient. As this is not possible we assume that the zonal wind relaxes back to a set function using a Rayleigh friction formulation with a damping rate of b (s−1 ). The zonal wind equation (for example) would take the form: 𝜕H 𝜕u − fv + I(u) = −gm0 − 𝛾u − 𝛿bu (G.7) 𝜕t 𝜕x where 𝛿 is a Dirac delta function that is zero except for the zonally symmetric part of the flow. We now need to introduce a longitudinal or asymmetric component of the background basic flow. To do this we specify a particular form to the mass source-sink

system M that matches the stationary Kelvin wave-like structure, which is consistent with both the observations and theory for quasi-stationary flow at low latitudes (e.g. Webster 1972; Gill 1980). We let M = Mb (x, y) + M′ (x, y, t)

(G.8)

The first term generates the equatorial basic shear and the second the transient forcing. These two terms are defined by Mb = C4 exp(−y2 ∕y1 ) ⋅ cos kx M′ = E J(x, y)𝜏(t)

(G.9)

Here, C 4 is a constant and y1 is chosen to provide an amplitude e-folding scale about the equator. E is an amplitude function set at 5 × 10−3 . The spatial and temporal functions are defined as J(x) = exp(−|x − x0 |∕2) exp(−|y − y0 |2 ∕4) 𝜏(t) = (t3 ∕(2Λ3 )) exp(−t∕Λ)

(G.10)

The spatial and temporal scales of the forcing can be altered through choosing values of x0 , y0 , and Λ. The spatial function comes from Webster and Holton (1982) and the time dependency from Lim and Chang (1983). The forcing functions are shown in Figure 7.16.

463

Appendix H Derivation of the Potential Vorticity Equation on an Extratropical 𝜷-Plane We set:

(

p = ps (z)

+ p⁀(x, y, z, t)

(H.1a)

𝜌 = 𝜌s (z)

+ 𝜌⁀(x, y, z, t)

(H.1b)

⁀

𝜃 = 𝜃s (z) + 𝜃 (x, y, z, t)

(H.1c)

𝜌s = 𝜌(0) exp(−z∕G)

(H.1d)

Here G is the atmospheric scale height RT∕g. We separate the velocity components into geostrophic and ageostrophic parts such that 1 𝜕p⁀ + ua and f0 𝜌s 𝜕y 1 𝜕p⁀ + va v(x, y, z, t) = vg + va = f0 𝜌s 𝜕x u(x, y, z, t) = ug + ua = −

(H.2)

where ug ≫ ua and vg ≫ va . On a mid-latitude 𝛽-plane, the vorticity equation is written as ( ) 𝜕 𝜕 𝜕 + ug + vg (𝜍g + 𝛽y) 𝜕t 𝜕x 𝜕y ( ) 𝜕u 𝜕v + f0 + =0 (H.3) 𝜕x 𝜕y where the vorticity of the geostrophic flow, 𝜍 g , is defined as ( 2 ) 1 𝜕2 𝜕 𝜍g = + (H.4) (p⁀) f0 𝜌s 𝜕x2 𝜕y2 The hydrostatic, continuity, and thermodynamic equations may be written as ( ) 𝜕 p⁀ 𝜃⁀ (H.5a) =g 𝜕z 𝜌s 𝜃s ( ) 𝜕ua 𝜕va 𝜌s (H.5b) + + (𝜌s w) = 0 𝜕x 𝜕y

𝜕 𝜕 𝜕 + ug + vg 𝜕t 𝜕x 𝜕y

)( ⁀ ) d ln 𝜃s 𝜃 =0 +w 𝜃s dz (H.5c)

We define a stream function of geostrophic wind as 𝜓=

1 p⁀ f0 𝜌s

(H.6)

allowing us to rewrite thermodynamic (H.5c) and vorticity (H.3) equations as )( ) ( 𝜕𝜓 N2 𝜕 𝜕 𝜕 w = 0 (H.7a) + ug + vg + 𝜕t 𝜕x 𝜕y 𝜕z f0 )( ) ( 𝜕𝜓 𝜕𝜓 𝜕 𝜕 𝜕 + ug + vg + + 𝛽y 𝜕t 𝜕x 𝜕y 𝜕x 𝜕y ( ) 𝜕u 𝜕v + f0 + =0 (H.7b) 𝜕x 𝜕y where N2 = d ln 𝜃 s /dz is the Brunt–Väisälä frequency. We now eliminate the divergence term between Eqs. (H.5b) and (H.7b). We eliminate w using Eq. (H.7a), which leads to the quasi-geostrophic potential vorticity equation in a mid-latitude 𝛽-plane as ( ) 𝜕 ̃g ⋅ ∇ qg = 0 (H.8a) +V 𝜕t where ( ) f02 𝜕 𝜌s 𝜕𝜓 𝜕2𝜓 𝜕2𝜓 qg = + + 𝛽y + (H.8b) 𝜌s 𝜕z N2 𝜕z 𝜕x2 𝜕y2 For simplicity, we assume that values of the Brunt– Väisälä frequency N2 and scale height G are constants.

Dynamics of the Tropical Atmosphere and Oceans, First Edition. Peter J. Webster. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

465

Appendix I Derivation of the Barotropic Potential Vorticity Equation (13.25) with Friction and Heating The equations of motion on a pressure coordinate of barotropic atmosphere are 𝜕𝜙 𝜕u 𝜕u 𝜕u +u + v − fv + = Fx 𝜕t 𝜕x 𝜕y 𝜕x

(I.1)

𝜕𝜙 𝜕v 𝜕v 𝜕v +u + v + fu + = Fy . 𝜕t 𝜕x 𝜕y 𝜕y

(I.2)

where F x and F y are frictional forces in the longitudinal and latitudinal directions, respectively. We define vorticity and divergence in the usual manner as 𝜕v 𝜕u − 𝜕x 𝜕y

(I.3)

𝜕u 𝜕v Dv = + 𝜕x 𝜕y

(I.4)

𝜁=

Then, differentiating Eqs. (I.1) and (I.2) with respect to y and x, respectively, leads to 𝜕Fy 𝜕Fx 𝜕𝜂 𝜕𝜂 𝜕𝜂 +u + v + 𝜂Dv = − 𝜕t 𝜕x 𝜕y 𝜕x 𝜕y

(I.5)

where we have noted f is a function of y only and the absolute vorticity may be written as 𝜂 = (𝜁 + f ). The equation of continuity is 𝜕wp 𝜕u 𝜕v 𝜕wp + + = Dv + =0 𝜕x 𝜕y 𝜕p 𝜕p

(I.6)

where the vertical velocity in pressure coordinates is given by wp = dp/dt. Inserting Eq. (I.6) into Eq. (1.5) becomes: 𝜕wp 𝜕Fy 𝜕Fx 𝜕𝜂 𝜕𝜂 𝜕𝜂 𝜕𝜂 +u + v + wp −𝜂 = − 𝜕t 𝜕x 𝜕y 𝜕p 𝜕p 𝜕x 𝜕y (I.7) The thermodynamic equation is 𝜕𝜃 𝜕𝜃 𝜃 ̇ 𝜕𝜃 𝜕𝜃 Q +u + v + Dv = 𝜕t 𝜕x 𝜕y 𝜕p Cp T

(I.8)

where 𝜃 is potential temperature, T temperature, C p the heat capacity of air at constant pressure, and Q̇ is the diabatic heating rate. Differentiating Eq. (I.8) with

respect to p and multiplying by the absolute vorticity leads to ) ( 𝜕 𝜕 𝜕 𝜕𝜃 𝜕 +u + v + wp 𝜂 𝜕t 𝜕x 𝜕y 𝜕p 𝜕p ) ( ) ( 𝜕wp 𝜕𝜃 𝜕 𝜃 ̇ Q (I.9) +𝜂 =𝜂 𝜕p 𝜕p 𝜕p cp T Between Eqs. (I.7) and (I.9) we obtain1 ) ( ) ( ( ) 𝜕𝜃 𝜕Fy 𝜕Fx 𝜕𝜃 𝜕 𝜃 ̇ d Q + 𝜂 =𝜂 − dt 𝜕p 𝜕p cp T 𝜕p 𝜕x 𝜕y (I.10) where the substantial derivative is defined as d 𝜕 𝜕 𝜕 𝜕 = +u + v + wp dt 𝜕t 𝜕x 𝜕y 𝜕p By inserting Eq. (I.8) into Eq. (I.10) and defining the PV on an isobaric surface as qp = −g𝜂𝜕𝜃∕𝜕p we obtain, after some manipulation: ( ) dqp 𝜕 𝜃̇ 𝜕𝜃 𝜕Fy 𝜕Fx = −g𝜂 −g − (I.11) dt 𝜕p 𝜕p 𝜕x 𝜕y which is identical to Eq. (13.25). The same procedure can be followed for a baroclinic atmosphere, although the algebra is somewhat more tedious. For a baroclinic atmosphere, extra terms enter Eq. (I.11). For example, we would find that the righthand side of Eq. (I.11) becomes ( ) ( ) ( ) d 𝜕𝜃 𝜕 d 𝜃̇ 𝜕𝜃 𝜕Fy 𝜕Fx 𝜂 =𝜂 + − dt 𝜕p 𝜕p dt 𝜕p 𝜕x 𝜕y ) ( 𝜕𝜃 𝜕Fx 𝜕u 𝜕 𝜃 ̇ Q + + 𝜕y 𝜕p 𝜕p 𝜕y cp T ) ( 𝜕𝜃 𝜕Fy 𝜕v 𝜕 𝜃 ̇ − Q (I.12) − 𝜕x 𝜕p 𝜕p 𝜕x cp T One can argue, returning to the scaling in the tropical atmosphere (Section 3.1.2), that terms involving horizontal gradients of thermodynamical quantities are small. 1 Specifically differentiating Eq. (I.7) with respect to p and adding Eq. (I.9).

Dynamics of the Tropical Atmosphere and Oceans, First Edition. Peter J. Webster. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

467

Appendix J Steady State Model of the Tropics Webster (1972) developed a spherical steady state (𝜕/𝜕t = 0) model linearized about a zonally symmetric basic state U = U(𝜑, p) based Mak (1969). The governing equations are 1 U u + (Uv cos 𝜑)𝜑 + (Uwp )p a cos 𝜑 𝜆 a cos 𝜑 ( tan 𝜑 ) 1 =− U + F1 𝜓p + v f + a cos 𝜑 a

(J.1a)

2uU tan 𝜑 1 U v𝜆 = − 𝜓𝜑 − fu + + F2 (J.1b) a cos 𝜑 a a Q̇ v U T 𝜑 = wp S + + F3 (J.1c) T𝜆 + a cos 𝜑 a cos 𝜑 Cp 𝜕wp 1 1 + (v cos 𝜑)𝜑 + u = 0 (J.1d) 𝜕p a cos 𝜑 a cos 𝜑 𝜆 RT 𝜓p = − (J.1e) p representing, sequentially, the zonal and meridional momentum Eqs. (J.1a) and (J.1b), the first law of thermodynamics (J.1c), the conservation of mass (J.1d), and the equation of state (J.1e). Here, the dependent variables u, v, 𝜓, T, and wp represent the perturbation zonal and meridional velocities, the geopotential, the temperature, and vertical velocity (dp/dt), respectively. The stability is defined as S = RT∕Cp p − 𝜕T∕𝜕p. The set (J.1) is the linearized version of the equation set (5.30). The model is linearized about a basic state defined by 2 tan 𝜑 1 =0 − Ψ𝜑 + f U + U a a RT Ψp = − (J.2) p

The dissipative terms are given by 𝜕u 𝜕v F1 = −K1 − K2 , F2 = −K1 − K2 , 𝜕p 𝜕p F3 = −K3 T

(J.3)

where K 1 represents surface friction (set at a decay rate of 6 days), K 2 a vertical mixing coefficient (25 days), and K 3 a Newtonian cooling coefficient (40 days). The decay rates refer to the time it would take for the kinetic energy to fall by 1/e if all other factors (e.g. solar heating) were non-existent. A schematic of the model structure appears in Figure 9.2a. The method of solution is explained in detail in Webster (1982). This paper also improves the model to allow an iterative calculation of heating. The Gill (1980) model is simpler than the spherical model described above and is based on the Matsuno (1966) model. It is also a steady state model (𝜕/𝜕t = 0) but with U = 0. The steady state solutions of the Gill model are, in essence, the equilibrium solutions between the imposed heating and the dissipation terms. Whereas this leads to simpler analytic solutions it does not differentiate between the role of advection and direct heating. For example, Eq. (J.1a), with U = 0, reduces to a balance equation. Near the equator, where U is small and the gradients of thermodynamic quantities are also small, the advective terms are not important. Webster (1981) refers to this as the “diabatic limit.” This is handled well by the Gill model. However, as the latitude increases and U becomes larger, the atmospheric response to the same forcing decreases in amplitude. This reduction in amplitude occurs through an increase in the magnitude of the advective terms. Webster (1981) referred to this as the “advective limit.” The Gill model, with a zero basic state, will always reach a diabatic limit irrespective of latitude.

Dynamics of the Tropical Atmosphere and Oceans, First Edition. Peter J. Webster. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

469

Appendix K Intermediate Ocean Model The temperatures of the two active model layers (T 1 and T 2 ; see Figure K.1 for a schematic diagram of the model) may vary in space and time. Furthermore, the lowest layer is assumed to be inert with a set temperature of 4 ∘ C. In all, the model vertical structure mimics the observed vertical shown in Figure 2.11. In the simplest configuration, this entails layers 2, 3, . . . . Entrainment and detrainment of water account for the exchange of mass, momentum, and heat between layers. The model develops its own turbulent fluxes of sensible and latent heat from the imposed atmospheric forcing and the state of the upper layers of the ocean model. The mixed layer entrainment is governed by a mixed layer model1 where mixing is maintained by turbulence generated at the surface by wind stirring and cooling. The southern boundary of the model does not correspond to any real boundary of the Indian Ocean, and the zero-gradient, open-boundary conditions are applied there.

Intermediate ocean model

OCEAN h1>h1min

Qnet

Vwind

ATMOS

h1

V1 T1

w1

layer 1

layer 2 h2>h2min

h2

V2 T2

w2 layer 3

T3 V3 =0 Figure K.1 Schematic diagram of the intermediate 2 1/2 level ocean model. Layers 1 and 2 are defined by three parameters: hi , V i , and T i (x, y, t) representing the layer thickness, horizontal current velocity, and temperature, respectively. The third layer is assumed to be inert (V 3 = 0) and held at a constant temperature of 277 K. The ocean is forced by a wind field V s (x, y, t) and a net heat flux Qnet (x, y, t) into the ocean. Changes in heat storage may occur in each layer through changes in h and T through horizontal advection of heat from one latitude to another, and through the entrainment of heat between layers (wi ≠ 0). Source: Based on McCreary et al. (1993).

For the closed region as the NIO used in the model, the model heat budget for a volume of the ocean is Qt = Qv + Qs + diffusive terms

(K.1)

where Qt is the change in heat storage in a column, Qv is the heat transport into the volume, and Qs is the total net surface flux into the ocean. Diffusive terms are much smaller than other terms and are neglected. The rate of change of heat storage is defined as Qt (t) = 𝜌Cp

∫∫∫

𝜕T dx dy dz 𝜕t

(K.2)

the surface heat flux into the ocean is Qs (t) =

∫∫

(Qsw + Qlw + Qlh + Qsh ) dx dy (K.3)

and the meridional heat transport is Qv (t) = 𝜌Cp

∫∫

vT dx dz

(K.4)

where Qsw is the solar radiation, Qlw is the outgoing longwave radiation, Qlh is the latent heat flux, Qsh is the sensible heat flux, and v and T are the meridional velocity and temperature, respectively. Longitude, latitude, and depth are represented by x, y, and z, respectively. For the definition of the heat transport to be valid, there must be a zero net mass transport across the meridional section. The model formulation includes the requirement of the conservation of the total mass in the basin; however, the mass of the water north of any given latitude inside the basin is constantly changing. To ensure conservation of mass, it is assumed1 that the mass transport in the abyssal layer compensates for the mass transport in two upper layers; that is, ∫

v3 H3 dx = −

∫

(v1 H1 + v2 H2 ) dx

(K.5)

where vi (i = 1, 2, 3) represents the meridional velocity component and H i represents the depth of the layers. For the layers having temperatures T i and the deep 1 Following Hall and Bryden (1982) and Loschnigg and Webster (2000).

Dynamics of the Tropical Atmosphere and Oceans, First Edition. Peter J. Webster. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

470

Appendix K Intermediate Ocean Model

ocean temperature is T 3 (assumed constant), the total heat transport is given by 3

Qv (t) = 𝜌Cp = 𝜌Cp

∫i=1

∫

vi Hi Ti dx

[v1 H1 (T1 − T3 ) + v2 H2 (T2 − T3 )] dx (K.6)

We note that the heat storage within a column can vary through a change in depth of a layer or a change in temperature. The atmospheric fields and surface fluxes were obtained from the NCAR-NCEP reanalysis fields and the ocean force between 1969 and 1998.

471

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Zebiak, S. E., 1993: Air-sea interaction in the equatorial Atlantic region. J. Clim., 6, 1567–1586. Zebiak, S. E., and Cane, M. A., 1987: A model El Niño-Southern Oscillation. Mon. Wea. Rev., 115(10), 2262–2278. Zhang, C. 2005: Madden-Julian oscillation. Rev. Geophys., 43(18), 2441–2459. Zhang, C. and Ling, J. 2011: Potential vorticity of the Madden–Julian oscillation. J. Atmos. Sci., 69, 65–78. Zhang, C. and Webster, P. J., 1989: Effects of zonal flows on the equatorially trapped waves. J. Atmos. Sci., 46, 3632–3652. Zhang, C. and Zhang, B., 2018: QBO-MJO connection. J. Geophys. Res. (Atmos.), 123, 2957–1968. Zhang C., McGauley, M., and Bond, N. A., 2004: Shallow meridional circulation in the tropical eastern Pacific. J. Clim. 17,133–139. Zhao, C.-B., Li, T., and Zhou, T., 2013: Precursor signals and processes associated with MJO initiation over the tropical Indian Ocean. J. Clim., 26, 291–307. Zipser, E. J., 1970: The line Island experiment. Its place in tropical meteorology and the rise of the fourth school of thought. Bull. Amer. Meteor. Soc. 51, 1136–1146. Zipser, E. 2003: Some views on “hot towers” after 50 years of tropical field programs and two years of TRMM data. Meteor. Monogr. 51, 49–58. Zwillinger, D, 1965: Special Integrals of Gradshteyn and Ryzhik: the Proofs - Volume I (Chapman & Hall/CRC Monographs and Research Notes in Mathematics)

493

Index a Absorptivity critical layer absorption (wave) 183 in shear flow 184, 188 long-wave radiation ocean 63 stratiform cloud 188, 189 solar absorption ocean near-infrared 63, 189 ocean heating 63 spectral dependence 63 solar radiation atmosphere 42 hemispheric difference 51 stratosphere 71 Albedo of cloud 42 cloud thickness dependency 42 Earth or planetary albedo 41 primitive (effective temperature) 41 impact of moisture 399 interhemispheric symmetry 11, 39 of land surface 51 desert 246, 249 dry land versus vegetated 245 of ocean surface 51, 248, 381 of phases of water 37 planetary 41, 49 relationship to cloud emissivity 41–42 single scattering albedo 45 Arid and desert climates area of desert 245 Charney’s self-induction theory of desert growth 246 definition of desert climates 245 dynamics of deserts: domination of diurnal cycle atmospheric divergence 250 dynamics of nocturnal jet stream 251–254

inertial versus diurnal clocks 253 net columnar heating 247 net heating of surface 249 nocturnal stabilization of boundary layer 251–254 surface heat budget 248 field surveys in empty quarter, Saudi Arabia 246 remote influences of deserts 254–255 impact of desert regions on monsoon rainfall 255–256 isentropic “gliding” 254–255 Rodwell-Hoskins theory 248–255 subsiding lateral exhaust 254 substantiation with numerical model 255, 467 Atmospheric climatology convective variance by spectral band 24–25 land versus ocean precipitation 6, 39, 40 precipitation rate by season 6, 39 precipitation volume by season 6, 39 sea-surface temperature (SST) 4, 5 by season 9–10 surface high pressure systems Bermuda 11 Mascarene 396 Siberian 21, 388–389 sub-tropical band 12 variability along equator with height divergent wind and vertical velocity along equator 19–20 along 25∘ N, 19–20

Dynamics of the Tropical Atmosphere and Oceans, First Edition. Peter J. Webster. © 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.

temperature and specific humidity 16 zonal and meridional wind fields along equator 17 zonal averaged meridional wind by season and height 7 zonal averaged seasonal temperature with height 5–6, 52 equatorial minimum 5 equatorial minimum explanation 67–68 zonal averaged zonal wind by season and height 7 Atmospheric dynamics: fundamental 79–90 barotropic tropics 85 basic spherical equations of motion 79 convection from reduced static stability 86–87 dry and moist thermo dynamical stability 453 early interpretations of scaling Ro ∼ 1 84 geostrophic adjustment at low latitudes 87–89 hydrostatic balance in the tropics 81–82 Lagrangian or total time derivative 79 Rossby radius of deformation 87 two definitions 88 scaling the atmospheric tropical motions 83–84 as a function of vertical scale 88 reduced gravity: effective static stability 196 thermodynamic balances in tropics 83

494

Index

Atmospheric dynamics: fundamental (contd.) vorticity equation at low latitudes early interpretations 85 L ∼ 106 m Ro ∼ 1 non-divergent 85 L ∼ 107 m Ro < 1 divergent 85 traditional approximation and spurious KE generation 82–83 tropics driven by extratropical forcing 86

b Baroclinic instability 53, 79, 85, 265 Barometric equation 53 Barotropic instability 56 Bayesian priors 332 and conditional predictability 332 ENSO 332 North Indian rainfall 332, 393 Bayesian Law/theorem 332, 342 Bayesian statistics 75, 125 Buoyancy: atmospheric compensation with ocean buoyancy 68 definition 50 differential buoyancy 16, 50 gradients 45 induced by radiative forcing 51 production of body forces 50–51 SST and buoyancy 51 Buoyancy: ocean 57–58 compensation with atmospheric buoyancy 68 components 58–59 fresh water flux 59, 61 net heating 58, 61 distribution 60, 61 tropical approximation 61

c Charney’s five “islands of resistance” 445 current status 447 Clausius–Clapeyron 39–41 Clausius–Clapeyron and planetary evolution 41 “Goldilocks”, Earth and the triple point of water 41, 42, 74 Goody–Walker planetary evolution theory 41 equation 40 implications 40

dry and moist adiabatic lapse rates 45–46 precipitation and SST 39, 49 saturation vapor pressure and temperature 41, 43, 44, 68, 78, 187–196, 435, 453 vertical convective penetration 68–69 vertical profiles of temperature and water vapor 42, 44 water vapor and liquid ice distribution 45 partial pressure 40 phases of water 39 phase transitions 39 latent heats of transition 40, 41 standard boiling point 41 standard melting point 41 water triple point 39, 40 Clouds cloud albedo and emissivity relationships 43 cloud-radiation feedbacks 433 proportion convective to stratiform area 29, 202 precipitation 3 mesoscale convective systems (MCSs) 188 radiative destabilization 188, 189 stratiform characteristics 3, 188 diurnal variation 189 radiative forcing distribution 188, 189 source of anvil material 189 top-heavy convective heating 189 Cold surges as component of monsoon 361 edge wave characteristics 141, 389 oscillations of Siberian High 404 and Pacific westerly wind bursts 388, 404 regional East Asian 361, 388, 390–391, 402, 404 North American 388 North Australian 388 South American friagem: Brazilian cold surges 388 Columnar integrated heating (CIH): atmospheric definition 53 differential buoyancy 52

gradients and monsoon forcing 19, 384 gradients by season along equator 19, 67 along 25∘ N, 19, 67 along 90∘ E, 19, 67 latitudinal variation by season 55 role of surface fluxes 56 Coupled horizontal and vertical structure equation 120 derivation of coupled equations 120–121 forced and free solutions 121 separation coefficient between components 122 Coupled monsoon system annual cycle in Indian Ocean region 408–409 Ekman ocean heat transport 410 SST, 925 hPa wind, OLR 409 surface energy balance 410 zonal heat transports 414 biennial modulation of Indian Ocean interannual variability coupled biennial SST theory 419 general theory of bienniality 423–424 heat transport between extremes of ENSO 421, 422 Meehl’s biennial SST theory 418 power spectra of cross-equatorial heat transport 423 strong/weak monsoon negative feedback 419 cross-equatorial Ekman transport: theory 415 heat storage formulation 413 influence of anomalous Indian Ocean 426–428 interaction between the Indian and Pacific sectors 424–428 negative coupled feedback on SST 415 annual cycle 415 interannual variability 419 observations Ekman transport Somali Current 417 Ocean and atmospheric transports: negative correlation 412

Index

Paradox: theoretical/observed in boreal spring Indian Ocean SST 408–410 springtime “frailty” in the Pacific Ocean 426–428

d Dirac delta (𝛿) function definition 113 properties 113 Dispersive versus non-dispersive waves 111 Diurnal variability: tropical oceans convective line propagation in Bay of Bengal 64–65 diurnal cycle in South China sea convection 189 ocean heating with depth 63 phase locking with equatorially trapped waves 348 diurnal convective “dancing” 347 SST 62 surface energy balance 63–66 Diurnal variability: land surface See Arid and desert climates

e Easterly waves African easterly waves versus Pacific easterly waves 289 origin 289 Burpee’s instability theory: instability of African easterly jet 290 propagating ITCZ transients 283–285 Ekman mass transport orthogonality with surface wind 96, 97–98 Emissivity: radiational albedo-emissivity compensation 43 black body 18 cloud albedo-emissivity relationship 43 definition 18 determination 45 gray body 42 impact on surface temperature 42 mean global emissivity 49 in Stefan–Boltzman law 18 ENSO: El Niño-Southern Oscillation 305–331

annual cycle in longitude along equator Pacific 309–313 depth of 20∘ C isotherm 311 mean sea-level pressure 311 ocean temperature with depth along equator 312 surface meridional winds 313 surface zonal winds 313 anomalies associated 1996–2000 213, 218 SST 311 zonal wind, depth of 20∘ C isotherm 314 correlations of precipitation with ENSO extremes 116 differences between El Niño and La Niña 313–318 T(z) and anomalies strong El Niño January, 1998 315 T(z) and anomalies strong La Niña January, 2000 315 zonal 250 hPa wind for ENSO extremes 317 measures of ENSO central Pacific SSTs: the “Niño-indices” 309 composite of Niño 3.4 SST for El Niño and La Niña 311 phases of ENSO: schematic 319 time series of Niño 3.4 SST 1950-2017 310 wavelet analysis of Niño-3 SST index 310 ENSO predictability annual cycle of SST persistence 328 annual cycle of predictability 327 boreal springtime reduced predictability 327, 329, 331 interdecadal variability of predictability 329 possible improvements in prediction of ENSO 331–332 empirical modeling 331 ensemble modeling 332 predictability barrier in different models 330 ENSO theories 318–326 negative feedback theories advective–reflective oscillator 325 delayed oscillator 320, 322

recharge-discharge oscillator 321–324 stochastic forcing instigating unstable states 325 western Pacific oscillator 324–225 requirements of an ENSO theory 318 positive feedback theories Bjerknes theory 319 Equatorial waves: atmospheric 125–150 Doppler versus non-Doppler impacts 133 definition of Doppler shifted frequency 126 dispersion and dispersion diagram 112, 127 equatorial modes in U = constant basic flow 127–141 dispersion diagrams in terms of H 129 equatorial Rossby (ER) 129–137 anomalous small scale behavior for U> 0 136–137 dispersion relationship 129, 130 Doppler eigenfrequencies 130 Doppler phase speeds 130 eigensolutions 135 equatorial Rossby radius 132, 133 frequency domain 129 turning latitudes 132, 133 zonal Doppler group speeds 131, 132 inertia-gravity (IG) 137–139 dispersion relationship 137 latitudinal structure 138 Kelvin 141–144 eigensolutions 144 equatorial edge wave 143 dispersion relationship 143 phase and group speeds 143 waves at a boundary 141–143 general dispersion relationship 126 mixed Rossby-gravity (MRG) 139–141 dispersion relationship 139 Doppler phase and group speeds 139, 140 eastward and westward modes 139 equatorial modes in shear flow U = U(y) 144–146

495

496

Index

Equatorial waves: atmospheric (contd.) eigenfrequencies in different shear regimes 146 regions of shear in subtropics and tropics 144–145 governing equations basic state with shear (U=U(y)) 144 constant basic state (U=constant) 126 motionless basic state (U=0) 125 regions of strong/weak U=U(y) 145 numerical evolution of forced K-and ER-waves 141–142 model description 462 modal evolution 142 physics of equatorial trapping 146–150 degrees of trapping by basic state 148–151 restoring agent: induced relative vorticity 148 simple potential vorticity arguments 147, 457 ubiquity of K-and MGR-waves 128 wave resonance and an “oenophilic” example 128, 158 Equatorial waves: ocean 151–156 constancy of frequency and nature of reflecting modes 155 dispersion curves for ocean modes 154 energy dispersion with long-wave approximation 156 elimination of the ocean MGR mode: implications 156–157 impact of boundaries on near-equatorial modes 153 constraint of constant frequency 155 implications 156–157 intermediate ocean model 424, 469–470 limits on reflecting modes 155 long-wave approximation 156–157 rotational ocean modes 152, 153 simple ocean model 151, 152

f

i

Fluid state baroclinic or barotropic tropics? 77, 85

Impermeability 234–237 conservation of qs 237 Haynes-McIntyre “impermeability theorem” 237–238 consequences on zonally symmetric Hadley models 236 constraints on theorem 238 zero zonally averaged potential vorticity flux 236 shallow fluid experiments equatorial heating 238–239 subtropical heating 238–239 Indian Ocean interannual oscillations 322 Indian Ocean Dipole (IOD) definition 333 Australian and Kenyan rainfall anomalies 337 composite annual structure of the IOD 337–340 positive phase 339 negative phase 340 IOD and ENSO relationships 334–336 physical hypothesis for generation of positive IOD 337–340 quasi-biweekly PV variability 401 breaking Rossby waves 401 induced bi-weekly precipitation variability 401 and South Asian precipitation variability 401 PV incursions around monsoon gyre 400–401 1961 West Indian ocean warming 332–333 1997-1998 West Indian warming event 333–334 anomalous convection and precipitation 334 ocean Rossby and Kelvin signatures 336–337 sea level height 334, 337, 338 surface wind fields 334 Inertial and rotational motion 88–91 definition versus rotational modes 89 inertial or rotational modes? 85–86, 88 inertial scale and period 88

g Great Cloud Bands 290–298 anchoring and orientation of the GCBs 291 global distribution in terms of OLR 10, 292 South Atlantic Convergence Zone (SACZ) 9 South Indian Ocean Convergence zone (SICZ) 9 South Pacific Convergence Zone (SPCZ) 9, 30 migrations with ENSO 292 seasonal variability 11 Meiyu-Baiu Frontal Zone (MBFZ) 291, 292 locations relative to westerly dusts 298 power spectra components 293 theories 291–298 continental and orographic forcing 295 diabatic and advective limits 293–294 synoptic graveyard and dU/dx < 0 295–298 variance within the GCBs 293 Greenhouse effect 27, 42, 43 and area of the ocean warm pool 436 Earth and the water triple point 41 Mars’ truncated greenhouse 41 Venus’s “runaway” greenhouse 41 dry stratosphere impact 72 greenhouse gases 17, 41 model projections 438 and planetary evolution 42 quantification of impact 43 SST with increased greenhouse forcing 438 “super” greenhouse 433 Gulf stream 23, 56, 75, 99, 105

h Holism versus reductionism 446

445,

Index

time scales and the Foucault pendulum 88 Inertial instability 264, 267 concept of “perpetual instability” 266 geophysical context 266 inertial instability and location of ITCZ 275 ameliorating secondary circulations 284–288 inertial oscillator 288 latitude of the ITCZ: theoretical determination 277 necessary and sufficient conditions 268, 275 physical bases “contra-signed” absolute vorticity: 𝜂/f < 0 263 cross-equatorial pressure gradient (CEPG) 263 displaced zero absolute vorticity (h = 0 ) across equator 263 necessary condition static stability 275–276 sufficient conditions 265, 266 schematic depiction of centrifugal instability 264 of inertial instability 266 signatures 264 symmetric and inertial instability 263, 265, 267 vertical sections: divergence, vertical velocity and divergent wind 272–273 why is the western Indian Ocean inertially stable? 275 Intertropical Convergence Zone (ITCZ) climatology Austral summer 10–11 boreal summer 10–11 seasonal variability 10–11 wind fields 12–13 identification 9, 10 theories regarding location of ITCZ 260–262 collocation of MSLP minimum and convection 258 collocation of SST and maximum convection 260 problems over land 11 dynamic-thermodynamic optimization 261–262 Charney 261

Mitchell/Wallace: upwelling-convective feedback 262 Waliser and Somerville 262 locus of near-equatorial disturbances 261 zonally symmetric arguments 261 transients of the ITCZ 281–289 composite reconstructions 284–288 divergence and meridional wind anomaly 286 stream function and relative humidity 285 stream function heating and heating rates 287 diagnostics of ITCZ transients 288–289 inertial and wave periods 281 transients and inertial (symmetric) instability 280, 281 variability east versus west Pacific Ocean 282 filtered fields 4-8 day, 283 Intra seasonal variability (ISV) 1, 3, 345–359 angular momentum changes through cycle 347 zonally averaged surface pressure 347 composite description contemporary 191 original 30 differences between theoretical and observed phase speeds 189–190 formation regions of ISV 347 influence region of ISV 346 intraseasonal variability in TOGA COARE data 351–354 ISV components Great Cloud Bands (GCBs) 291 ISV mechanisms evolution of CAPE during ISV 355 local instabilities 351–358 convective phase 355 destabilization phase 355 restoration phase 355 longitudinal versus meridional propagation speeds 346

SST, OLR, surface pressure and winds associated with forms of ISV 349–350 zonal mass oscillation and the MJO 346–347 stretching deformation distribution in ISV 201 substructure of MJO eastward sub-elements 347–348 westward inertia-gravity waves 347–348 variance spectra 346 variability of form of ISV 348–351 canonical MJO 349 eastward decaying mode (ED) 349 eastward intensifying (EI) 351 equatorial wave feedbacks 356 impact of IOD 358 Isentropic potential vorticity substance (qs ) definition 231 definition of qs in a shallow fluid 231 isentropic mass density definition 231 qs advection: schematic diagram of 232 relationship of isentropic absolute vorticity to qs 231 seasonal distributions of qs 232–233 of cross-latitude advection of qs 232–233

k Kuroshio Current

23, 56, 67

l Laplace shallow fluid system 115 Laplace tidal equations 115 equatorial β-plane approximation 115 error analysis with approximation 116 linearization with constant basic state 117 derivation of wave equation 455–456 Doppler and non-Doppler effects 117 refractive index definition 118 Schrodinger’s equation in a complex basic state 118, 124

497

498

Index

Laplace tidal equations (contd.) total depth of the shallow fluid 116

m Matsuno’s wave equation 125, 456 Monsoon circulations anomalously strong and weak monsoons 340, 370, 371 anomalous wind fields 370–372 definition 370 impact on Indian Ocean SST 372–373 comparisons of precipitation monsoon systems 376–378 anomalous location of Asian monsoon precipitation 377–378 wind field differences: annual cycle 373–374 early descriptions Halley’s anti-trades 365 Halley’s surface wind climatology 364–365 early theories on monsoons Halley’s differential buoyancy 365 late 19th century recovery 367–370 mid-century correlation decrease 369–370 Normand’s hypothesis 367–368 monsoon leading/lagging SOI? 367–368 regression analysis 370–371 Walker’s surmise SO and Indian monsoon 366–367 forcing of the Asian summer monsoon elevated heating hypothesis 382–385 Flohn 382 Molnar et al. hypothesis 384 evidence advent of springtime precipitation of HTP 382, 383 anomalous geopotential distributions 379–383 anomalous temperature above HTP (warm dome) 378–379 humidity maximum above HTP (World Water Tower) 379 surface flux differentials surface 380–381

surface (500 hPa) streamlines HTP 382, 383 low-level easterly jet and intensity of the West African monsoon 290 macroscale structure of monsoon cross-sections PV 75∘ -85∘ , 374–375 evolution of summer monsoon April-June 375–376 isentropic PV characteristics 374–375 monsoon gyre (upper tropospheric) 374–376 and elevated heating 382, 385 extension across Africa 374 development 376 instability 400 monsoon gyre (surface) 13, 20, 21 monsoon interannual variability precipitation India 363 North Australia 363 seasonal structure: 925 hPa wind and precipitation African monsoon 22–23 Asian-Australian 21 Americas 22 West Africa/Sahel 363 quasi-biweekly PV variability 401 breaking Rossby waves 401 PV incursions around monsoon gyre 400–401 relationship of number of MISOs to strength of monsoon 393 SOI/Indian precipitation relationships 369–370 cross-wavelet 369–370 wavelet 369–370 summer monsoon intraseasonal variability (MISO) 391–401 “active” and “break” periods of the monsoon 64, 391, 392–393 composite structure of MISO 394–396 discovery of the MISO 390–391 induced bi-weekly precipitation variability 401 over Central India 362, 397–400 precipitation and phase of MISO 394–396

temperature gradient threshold for subtropical circulation 382–385 theories of the MISO 396–400 circumnavigating MJO 396 ENSO/IOD influence 396 external forcing 396–397 internal monsoon mode 397–400 interactive ground hydrology: land WISHE 398–399 modulations of Mascarene High: Rodwell’s theory 396–397 self-induction/self-regulation 246, 397 Rossby waves in strong shear (Wang) 399 time scales of monsoon variability (overview) 361–364 upper tropospheric easterly jet 386–387 influence on South Asian monsoon precipitation 387 West African monsoon 383–388 upper tropospheric monsoon gyre (AMG) 374 forcing 385 instability of 376, 400 location and scale 375, 376, 405 loci for disturbance propagation 400 seasonal evolution 376, 382 winter monsoon 388–391 cold surges and equatorial westerly bursts 404 cold surges and extratropical waves 390, 403 orographic influence: edge waves 391 oscillations of the winter monsoon 390–391 Siberian Cold Anticyclone 388–389 limitations on central pressure 389–391 winter wavelet/ analyses SOI, Indian precipitation 369–370

o Ocean buoyancy 57 annual fresh water flux 59, 60 annual surface heat flux 59, 60 approximate expression 61

Index

compensation with atmospheric buoyancy 68 concept of buoyancy 50 definition 58 differential buoyancy 50 distribution by component 60–61 gradients of buoyancy 45–46 total buoyancy 61 Ocean currents Circum-Antarctic Current (West Wind Drift) 24 definition of “counter-currents” 23, 34, 101 Ekman 24, 96 equatorial counter-current (ECC) or Cromwell 24, 35 large scale characteristics of ocean 23 major ocean currents 23–24 warm and cold surface currents 23 seasonal currents in the Indian Ocean 23 subtropical gyre currents 23, 24, 75 Sverdrup 94 von Rumford’s counter current 49 Ocean currents and transport counter-currents definition 23, 101–103 Ekman transports 96–98 Ekman pumping and suction 99 formulation 96–97 schematic of Ekman transport 97 why is Ekman transport orthogonal to the surface wind? 97–98 equatorial undercurrent (EUC or Cromwell) 24, 36 formulation 102 reversed Indian Ocean undercurrent 103 role of lateral boundaries 102 induced geostrophic currents 98 alteration of mass field to Ekman transports 99 formulation 98–99 regions of induced upwelling/downwelling 100 schematic of edge upwelling/downwelling 101

schematic of induced geostrophic currents 102 surface wind stress distribution 100 ocean geostrophic adjustment 93 formulation 93 Sverdrup wind-driven transports 95–96 formulation 95–96 Sverdrup balance 96 wind driven currents North and South equatorial currents (NEC, SEC) 103 subtropical gyres 23 Ocean dynamics fundamentals 90–101 Boussinesq approximation 92, 105 equations of ocean motion 91–92 incorporation of surface stress forcing 82 ocean Rossby radius of deformation 88 reduced gravity approximation 94, 119, 146, 154 implications convectively couple waves 195, 196 ocean gravity waves 153, 154 on ocean wave equation 119, 123 reduced surface gravity phase speed 95 shear and inertial instability 274 tsunami versus surface gravity wave 94, 105 scaling oceanic tropical motions 92 schematic of ocean structure 93 Ocean properties by water mass 46–50 coefficients thermal expansion 47 haline contraction 47 mean temperature salinity by surface area 47 salinity by volume 47 ocean cabbeling: implications of mixing masses 47 surface salinity and SST distributions WEPOCS 62 temperature–salinity as function of ocean volume 48 temperature–salinity as function of surface area 48

temperature–salinity characteristics of water masses 47 temperature–salinity profiles 47 Ocean zones abyssal layer 49 barrier layer 50, 59, 72 mixed layer 50 surface layer 49 thermocline 50

p Predictability and system complexity 446 Hofstadter’s degrees of system complexity 446 complex 446 simple 446 tangled 446 See also ENSO predictability Potential vorticity 136, 149 Ertel’s PV 214, 230, 375 isentropic potential vorticity 214 “PV thinking” and “rules of thumb” Hoskins 215, 221 quasi-geostrophic potential vorticity 205 Potential vorticity equation derivation with heating and dissipation 279, 465 Potential vorticity equation on extratropical β-plane 205, 463

r Radiation net emitted radiation TOA 52 net radiation TOA 52 net solar radiation TOA 52 Reduced β-effect and equatorial trapping 146, 149 Reversible and irreversible processes constant entropy with reversible process 243 adiabatic expansion, compression 78 propagating Rossby wave 231 definitions 243 increase of entropy with irreversible process 243 breaking Rossby waves: irreversible deformation 231 constant entropy with reversible process 243 Rossby number 81

499

500

Index

Rossby number (contd.) definition 81 in extratropics 83 magnitude In tropics 83–84 Rossby waves (extratropical) 77, 79, 103 poleward transport heat and momentum 226–227 propagation into tropics 86 impact of equatorial easterlies 219 relationship to meridional circulation 227–229 Rossby waves breaking basic state deformation and breaking 230 breaking waves in westerly duct 219–221 characteristics of breaking Rossby waves 219 definition of wave breaking 231 equatorward potential vorticity transport 242 impacts East Asia 400–401 Indian rainfall 400–401 North America 219

s Schrödinger’s equation 85, 113, 118, 123–125 Self-induction 246, 397 Bryson-Baerreis/Charney desert hypotheses 245 definition 256 versus induction 256 monsoon variability and self-induction 387, 397 Wang’s theory monsoon intraseasonal variability 397 within convective systems 357 intraseasonal variability 357 monsoon intraseasonal mode 38 Shallow water nonlinear model 142, 461 generation of nonlinear basic states 176 Southern oscillation 305–308 compilation of early observations 305, 306 initial description 31 locations used to calculate indices 309

SOI and “strategic points of world weather” 366 SOI defined 306–307 annual cycle 306, 326 Troup, Bjerknes: The SO as a coupled phenomenon 307, 319 southern oscillation, El Niño/La Niña (ENSO) 95, 305–309 Southern Oscillation Index (SOI) 31 time series and spectra 31 Walker’s global MSLP oscillation 305 Walker’s surmise: monsoon/SOI relationship 366 SST regulation theories adaptive iris feedback 433 cloud-radiation feedback (thermostat) 433 ocean feedbacks 433 surface energy balance regulation 433 See also dynamic warm pool (DWP); ocean warm pool (OWP) Steady state numerical primitive equation model 207, 208, 467 Stokes’ drift 107 Stokes’ law or theorem 451 Synergies creation and breaking of Rossby waves 242 creation of a westerly duct 241–243 creation of extratropical jet streams 241–242 divergent circulations and Rossby wave dynamics 239, 241–242 divergent circulations and the poleward transport of qs 241–242 irreversible distortion by basic state 242 recursive Rossby wave breaking and equatorward transport of qs 242

t Thermal wind definition 449–450 with elevated heating 382, 384 imposed constraints 227, 228, 230 ocean “thermal wind” 152

and vertical shear 26, 449 Tropical upper-tropospheric trough (TUTT) 1, 13, 14, 374 and eddy kinetic maxima 209 and extratropical-tropical interaction 221, 223, 224 seasonal location 224

w Walker circulation early depictions 302 longitudinal heating gradients along equator 19, 67 modification by an interactive ocean 304 numerical simulation 303–304 equatorial forcing 304 model description 467 comparison with Gill-model 467 subtropical forcing 303 Warm pool (OWP) areal changes in warm pool area last century 429–432 by basin 430 globally 430 mid-20th century constancy 430–431 atmospheric columnar heating 56–57 characteristics of the tropical OWP 49 columnar heating over the warm pool 56 convergence of atmospheric moisture flux 13, 15 expeditions examining warm pool coupling EMEX 3, 267, 277 GATE 2, 3, 29, 105, 188–190, 202 JASMINE 63–68, 187, 189, 249, 301, 353, 407 TOGA COARE 3, 63–64, 67–68, 74, 189, 197, 301, 347, 355, 359, 381, 402–408 WEPOCS I., II, III 49, 61, 351 geographical distribution of OWP 9 evolution during last century 430 mid-century plateau 430–431 Indo-Pacific warm pool 7, 8 maintenance 73 precipitation freshening 49, 50

Index

radiational heating 49, 50 OWP during last glacial maximum OWP 432 and precipitation 10 and the stability of the climate system 72 SST-convection relationship in OWP 432 traditional definition (>28∘ C) 8, 429 Warm pool dynamic DWP 434, 436–438 constant area with time 436 definition 434 evolution of threshold temperature 434, 437, 439 evolution in models 435, 436 numerical substantiation 437–438 physical explanation for area constancy 436–438 relationship to column integrated heating 434 threshold temperature definition 59, 433 Water distribution on planet fresh water flux distribution (E-P) 60 global water cycle 39 climate degrees of freedom 42, 74 reservoirs 38 residence times 28, 39 inventory of water forms 38–39 land versus ocean precipitation 39, 40 precipitation rate by season 6, 39 precipitation volume by season 6, 39

temperature-water vapor by latitude 5 temperature-water vapor by longitude 16 trade winds and evaporation 13 vapor and temperature 44 along equator 16 with latitude 5 water ice distribution 43 water vapor and specific humidity 25 Wave kinematics 107–113 definition of kinematics 107 Dirac delta function forcing 107 dispersive and non-dispersive waves 111 wave action flux convergence and divergence 111 dispersive versus non-dispersive waves 111 Einstein summation convention 110, 113 kinematics of acoustic waves, dissipation and fog horns 114 wave action flux definition 199 waves in an (x, y, z) variable basic state 109–112 frequency constancy and wavenumber 111 and wave action flux 111 Doppler shifting 113 group and phase dependencies 110 non-Doppler effects 110 waves in a quiescent basic state 107–109

golf ball in pond example 107–109 group speed: propagation of a wave packet 107, 109 phase speed: propagation of a wave crest 107, 109 phase versus group speed 107, 109 Stokes’ drift and observations for a pier 107, 108, 113 mass/energy propagation 107 dispersion of energy 111 WKBJ approximation 109, 113, 213, 221, 461 Westerly duct (WD) 1, 13, 14, 86 association with Great Cloud Bands 291, 331 definition 184 extratropical influence on ISV 357 and biweekly variability 400 impact on wave scale 177 locus for equatorward PVS flux 233, 243 recursive Rossby waves breaking in WD 219–221, 231, 232, 234 seasonal location 224, 232, 374 wave propagation through WD diagnostics 215–219, 244 numerical experiments 210–211, 238 Westerly wind bursts 361, 404 precursors 404 WISHE land and the monsoon 198 WISHE ocean (Wave–Induced Sea–Air Heat Exchange) 197–198, 357

501