Atmospheric Dynamics [1 ed.] 9783662639405, 9783662639412

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Ulrich Achatz

Atmospheric Dynamics

Atmospheric Dynamics

Ulrich Achatz

Atmospheric Dynamics

Ulrich Achatz Institut für Atmosphäre und Umwelt Goethe Universität Frankfurt am Main, Germany

ISBN 978-3-662-63940-5 ISBN 978-3-662-63941-2  (eBook) https://doi.org/10.1007/978-3-662-63941-2 © Springer-Verlag GmbH Germany, part of Springer Nature 2022 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Illustrations: Dr. Martin Lay Responsible Editor: Simon Rohlfs This Springer Spektrum imprint is published by the registered company Springer-Verlag GmbH, DE part of Springer Nature. The registered company address is: Heidelberger Platz 3, 14197 Berlin, Germany

Introduction

Dynamics is still one of the essential foundations of the atmospheric sciences. It provides basic understanding of motion and thermodynamic development of the atmosphere. Whoever is involved in developing numerical weather prediction or climate modeling depends on solid expertise in this field. This book focusses exclusively on dry dynamics. Clouds and precipitation are not covered and fill their own volumes. The same with regard to radiative processes, and exchange with soil and ocean. Questions asked are, e.g., which fundamental laws and equations control the developments of winds, temperature, and pressure distribution, and what is the reason for their characteristic spatial structure. Basic equilibria are discussed that are quite prominent on the weather map. The nature of vortical flows is discussed that are omnipresent. Wave dynamics is introduced and developed as a fundamental cornerstone. Quasigeostrophic theory helps us understand the nature and origin of extratropical weather. An excursion into the dynamics of boundary layers is offered, and finally several chapters on the interactions between waves and mean flows, including the mean circulation. The text is an outgrowth of lectures that I have been giving for a while at the Goethe Universität in Frankfurt. Chapters 1–7 (mandatory) and 8–9 (elective) cover material from the BSc Meteorology program, while Chap. 10 on gravity waves is a course given in the MSc Meteorology program. The development of an intuitive understanding is as important here as the application and training of the mathematical tools that form our language. Some appendices describe some details in that regard, and I have tried as much as possible not to jump to boldly from equation to equation. Hopefully readers will find the approach useful. This book has developed slowly from lecture notes, and the comments from students and colleagues have helped a lot in compiling it, improving it, and getting rid of all kind of errors. I would like to mention especially Gergely Bölöni, Sebastian Borchert, Stamen Dolaptchiev, Markus Ernst, Elena Gagarina, Mark Fruman, Magnus Heinz, Young-Ha Kim, Ulrike Löbl, Jewgenija Muraschko, Martin Pieroth, Kristin Raykova, Bruno

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Introduction

Ribstein, Georg Sebastian Völker, and Yannik Wilhelm. Thanks to all and especially also to Young-Ha Kim for his support with various figures from ERA5 data. Much appreciated are also the comments of Erich Becker, Oliver Bühler, and Volkmar Wirth on the final book draft. No doubt there will still be typing errors, and the presentation could be improved further. Corresponding comments are most welcome.

Contents

1

The Basic Equations of Atmospheric Motion. . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Time Derivatives in Fluids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 The Fluid Description According to Euler and Lagrange . . . . 1 1.1.2 The Material Derivative of a Fluid Element. . . . . . . . . . . . . . . 3 1.1.3 The Material Derivative of Volume Integrals. . . . . . . . . . . . . . 5 1.1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2 The Equation of Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1 A Eulerian Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.2 A Lagrangian Derivation Using the Material Derivative. . . . . 9 1.2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 The Momentum Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.1 The Volume Forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.2 Surface Forces (1): The Pressure Gradient Force . . . . . . . . . . 11 1.3.3 Surface Forces (2): Friction. . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3.4 The Total Momentum Equation. . . . . . . . . . . . . . . . . . . . . . . . 18 1.3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.4 The Equations of Motion in a Rotating Frame of Reference. . . . . . . . . . 19 1.4.1 The Time Derivative in a Rotating Frame of Reference . . . . . 19 1.4.2 The Momentum Equation in the Rotating Frame of Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.4.3 The Equation of Continuity in the Rotating Frame of Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.5 The Equations of Motion on the Sphere . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.5.1 Velocity and Material Derivative in Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.5.2 The Transformed Equations of Motion . . . . . . . . . . . . . . . . . . 26 1.5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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1.6

1.7

Synoptic Scale Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.6.1 The Geostrophic Equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.6.2 The Hydrostatic Equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Recommendations for Further Reading. . . . . . . . . . . . . . . . . . . . . . . . . . 34

2

Elementary Thermodynamics and Energetics of Dry Air. . . . . . . . . . . . . . . 35 2.1 Fundamentals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.1.1 Thermodynamic Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.1.2 Thermodynamic State and Thermodynamic Equilibrium. . . . 36 2.1.3 Temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.1.4 Equations of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.1.5 Energy Change of a Thermodynamic System. . . . . . . . . . . . . 38 2.1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.2 The Fundamental Laws of Thermodynamics. . . . . . . . . . . . . . . . . . . . . . 41 2.2.1 The First Law of Thermodynamics and Internal Energy. . . . . 42 2.2.2 The Heat Capacities of an Ideal Gas . . . . . . . . . . . . . . . . . . . . 43 2.2.3 Adiabatic and Isothermal Changes of State of an Ideal Gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.2.4 The Second Law of Thermodynamics. . . . . . . . . . . . . . . . . . . 46 2.2.5 The Carnot Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.2.6 Entropy as State Variable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.2.7 Entropy and Potential Temperature of Dry Air . . . . . . . . . . . . 57 2.2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.3 The Prognostic Equations for Temperature and Entropy in Dry Air. . . . 60 2.3.1 Prediction of Temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.3.2 The Prediction of Entropy and Potential Temperature. . . . . . . 65 2.3.3 The Equations in a Rotating Frame of Reference . . . . . . . . . . 66 2.3.4 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.4 Potential Temperature and Static Stability. . . . . . . . . . . . . . . . . . . . . . . . 67 2.4.1 Stable and Unstable Stratification . . . . . . . . . . . . . . . . . . . . . . 67 2.4.2 Buoyancy Oscillations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.5 Recommendations for Further Reading. . . . . . . . . . . . . . . . . . . . . . . . . . 71

3

Elementary Properties and Applications of the Basic Equations. . . . . . . . . 73 3.1 Summary of the Basic Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.2 The Importance of the Basic Equations for Weather Prediction. . . . . . . 74 3.3 Conservation Laws. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.3.1 Conservation of Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.3.2 Conservation of Angular Momentum. . . . . . . . . . . . . . . . . . . . 77 3.3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

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3.4 3.5

The Primitive Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 The Primitive Equations in Pressure Coordinates. . . . . . . . . . . . . . . . . . 82 3.5.1 Arbitrary Vertical Coordinates. . . . . . . . . . . . . . . . . . . . . . . . . 82 3.5.2 Pressure Coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Balanced Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.6.1 The Natural Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.6.2 Geostrophic Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.6.3 Inertial Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.6.4 Cyclostrophic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.6.5 The Gradient Wind. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Thermal Wind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Recommendations for Further Reading. . . . . . . . . . . . . . . . . . . . . . . . . . 97

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3.7 3.8 4

Vortex Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.1 Vorticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.1.1 Relative, Absolute, and Planetary Vorticity. . . . . . . . . . . . . . . 99 4.1.2 Vortex Lines, Vortex Tubes, and Vortex Flux. . . . . . . . . . . . . . 100 4.1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.2 Circulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.2.1 Relative and Absolute Circulation. . . . . . . . . . . . . . . . . . . . . . 103 4.2.2 The General Circulation Theorem. . . . . . . . . . . . . . . . . . . . . . 104 4.2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.3 The Kelvin Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.4 The Vorticity Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.4.1 The Derivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.4.2 Vortex-Tube Stretching and Vortex Tilting. . . . . . . . . . . . . . . . 111 4.4.3 Summary of the Impacts on Relative Vorticity . . . . . . . . . . . . 113 4.4.4 The Frozen-In Property of Absolute Vorticity. . . . . . . . . . . . . 113 4.4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.5 Potential Vorticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.5.1 An Algebraic Derivation of the Prognostic Equation for Potential Vorticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.5.2 A Derivation of Potential-Vorticity Conservation from the General Circulation Theorem . . . . . . . . . . . . . . . . . . 117 4.5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.6 Vortex Dynamics and the Primitive Equations . . . . . . . . . . . . . . . . . . . . 120 4.6.1 The Primitive Equations in Isentropic Coordinates. . . . . . . . . 120 4.6.2 The Primitive Vorticity Equation in Isentropic Coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.6.3 The Potential Vorticity of the Primitive Equations. . . . . . . . . . 125

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4.6.4 Flow over a Mountain Ridge. . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Recommendations for Further Reading. . . . . . . . . . . . . . . . . . . . . . . . . . 129

5

The Dynamics of the Shallow-Water Equations. . . . . . . . . . . . . . . . . . . . . . . 131 5.1 Derivation of the Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.1.1 The Momentum Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 5.1.2 The Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.2 Conservation Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.2.1 Energy Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.2.2 Potential Vorticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.3 Quasigeostrophic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.3.1 The Tangential β-Plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.3.2 Scaling the Shallow-Water Equations on the β-Plane. . . . . . . 140 5.3.3 The Quasigeostrophic Approximation: Derivation by Scale Asymptotics. . . . . . . . . . . . . . . . . . . . . . . 144 5.3.4 The Quasigeostrophic Approximation: Derivation from the Conservation of Shallow-Water Potential Vorticity. . . . . . 148 5.3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.4 Wave Solutions of the Shallow-Water Equations. . . . . . . . . . . . . . . . . . . 150 5.4.1 Perturbation Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.4.2 Waves on the f -Plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.4.3 Waves on the β Plane: Quasigeostrophic Rossby Waves. . . . . 160 5.4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 5.5 Geostrophic Adjustment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 5.5.1 The General Solution of the Linear Shallow-Water Equations on an f Plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 5.5.2 The Adjustment Process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 5.5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 5.6 Recommendations for Further Reading. . . . . . . . . . . . . . . . . . . . . . . . . . 177

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Quasigeostrophic Dynamics of the Stratified Atmosphere. . . . . . . . . . . . . . 179 6.1 Quasigeostrophic Theory and Its Potential Vorticity. . . . . . . . . . . . . . . . 179 6.1.1 Analysis of Momentum and Continuity Equation. . . . . . . . . . 179 6.1.2 Analysis of the Entropy Equation . . . . . . . . . . . . . . . . . . . . . . 188 6.1.3 Quasigeostrophic Potential Vorticity in the Stratified Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 6.1.4 The Relationship with General Potential Vorticity. . . . . . . . . . 195 6.1.5 Quasigeostrophic Theory in Pressure Coordinates . . . . . . . . . 198 6.1.6 A Quasigeostrophic Two-Layer Model. . . . . . . . . . . . . . . . . . 202 6.1.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

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6.2

Quasigeostrophic Energetics of the Stratified Atmosphere. . . . . . . . . . . 205 6.2.1 The Continuously Stratified Atmosphere. . . . . . . . . . . . . . . . . 206 6.2.2 The Two-Layer Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 6.2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Rossby Waves in the Stratified Atmosphere . . . . . . . . . . . . . . . . . . . . . . 214 6.3.1 Rossby Waves in the Two-Layer Model. . . . . . . . . . . . . . . . . . 214 6.3.2 Rossby Waves in an Isothermal Continuously Stratified Atmosphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 6.3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 Baroclinic Instability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 6.4.1 Baroclinic Instability in the Two-Layer Model. . . . . . . . . . . . 224 6.4.2 Baroclinic Instability in a Continuously Stratified Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 6.4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 Recommendations for Further Reading. . . . . . . . . . . . . . . . . . . . . . . . . . 248

6.3

6.4

6.5 7

The Planetary Boundary Layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 7.1 Anelastics and the Boussinesq Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . 250 7.1.1 The Anelastic Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 7.1.2 The Boussinesq Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 7.1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 7.2 Instabilities in the Boundary Layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 7.2.1 The Taylor–Goldstein Equation. . . . . . . . . . . . . . . . . . . . . . . . 256 7.2.2 Neutral Stratification (N 2 = 0) . . . . . . . . . . . . . . . . . . . . . . . . 258 7.2.3 No Shear (du/dz = 0) and Constant Stratification N 2. . . . . . . 264 7.2.4 The General Case: The Richardson Criterion of Howard and Miles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 7.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 7.3 The Averaged Equations of Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 7.3.1 Turbulence and Mean Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . 269 7.3.2 The Reynolds Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 7.3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 7.4 Gradient Ansatz and Mixing Length. . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 7.5 The Turbulent Kinetic Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 7.5.1 The Prognostic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 7.5.2 Sources and Sinks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 7.5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 7.6 The Prandtl Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 7.6.1 The Momentum Flux. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 7.6.2 The Wind Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 7.6.3 The Influence of Stratification. . . . . . . . . . . . . . . . . . . . . . . . . 283 7.6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

xii

Contents

7.7

7.8

The Ekman Layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 7.7.1 The Ekman-Spiral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 7.7.2 Ekman Pumping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 7.7.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 Recommendations for Further Reading. . . . . . . . . . . . . . . . . . . . . . . . . . 299

8

The Interaction Between Rossby Waves and the Mean Flow. . . . . . . . . . . . 301 8.1 Basics of Quasigeostrophic Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 8.1.1 The Governing Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 8.1.2 Conservation Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 8.1.3 The Quasigeostrophic Enstrophy Equation Within Linear Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 8.1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 8.2 Rossby-Wave Propagation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 8.2.1 Wave Propagation Within WKB Theory . . . . . . . . . . . . . . . . . 307 8.2.2 Rossby-Wave Propagation into the Stratosphere. . . . . . . . . . . 313 8.2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 8.3 The Eliassen–Palm Flux. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 8.3.1 Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 8.3.2 The Eliassen–Palm Relationship. . . . . . . . . . . . . . . . . . . . . . . 319 8.3.3 Wave Action and Eliassen–Palm Flux Within WKB Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 8.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 8.4 The Transformed Eulerian Mean (TEM). . . . . . . . . . . . . . . . . . . . . . . . . 324 8.4.1 The TEM in the Context of Quasigeostrophy . . . . . . . . . . . . . 325 8.4.2 The Mass-Weighted Circulation in Isentropic Coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 8.4.3 The Relation Between the Residual Circulation and the Mass-Weighted Circulation. . . . . . . . . . . . . . . . . . . . . 334 8.4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 8.5 The Non-acceleration Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 8.6 Recommendations for Further Reading. . . . . . . . . . . . . . . . . . . . . . . . . . 342

9

The Meridional Circulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 9.1 Some Essentials of the Empirical Basis. . . . . . . . . . . . . . . . . . . . . . . . . . 343 9.2 The Hadley Circulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 9.2.1 The Basic Equations of a Model Without Wave Driving. . . . . 344 9.2.2 A Solution Without Meridional Circulation. . . . . . . . . . . . . . . 348 9.2.3 Hide’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 9.2.4 A Simplified Description of the Hadley Cell. . . . . . . . . . . . . . 353 9.2.5 The Summer–Winter Asymmetry . . . . . . . . . . . . . . . . . . . . . . 364 9.2.6 The Wave-Driven Hadley Circulation . . . . . . . . . . . . . . . . . . . 367 9.2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376

Contents

xiii

9.3

The Circulation in the Midlatitudes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 9.3.1 The Phenomenology of the Ferrel Cell . . . . . . . . . . . . . . . . . . 377 9.3.2 Eddy Fluxes and Barotropic Jet Stream. . . . . . . . . . . . . . . . . . 380 9.3.3 A Two-Layer Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 9.3.4 The Continuously Stratified Atmosphere. . . . . . . . . . . . . . . . . 400 9.3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 Recommendations for Further Reading. . . . . . . . . . . . . . . . . . . . . . . . . . 405

9.4

10 Gravity Waves and Their Impact on the Atmospheric Flow. . . . . . . . . . . . . 407 10.1 Some Empirical Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 10.2 The Fundamental Wave Modes of an Atmosphere at Rest . . . . . . . . . . . 409 10.2.1 Equations of Motion and Energetics . . . . . . . . . . . . . . . . . . . . 410 10.2.2 Free Waves on the f -Plane in an Isothermal Atmosphere. . . . 414 10.2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 10.3.1 A Reformulation of the Dynamical Equations. . . . . . . . . . . . . 431 10.3.2 Scaling for Synoptic-Scale Flow and for Inertia-Gravity Waves . . . . . . . . . . . . . . . . . . . . . . . . . 432 10.3.3 Non-dimensional Equations and WKB Ansatz . . . . . . . . . . . . 442 10.3.4 Leading-Order Results: Equilibria, Dispersion and Polarization Relations, Eikonal Equations. . . . . . . . . . . . . . . . 447 10.3.5 The Next Order of the Equations. . . . . . . . . . . . . . . . . . . . . . . 460 10.3.6 Wave Action. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 10.3.7 Wave Impact on the Synoptic-Scale Flow. . . . . . . . . . . . . . . . 476 10.3.8 Generalization to Gravity-Wave Spectra: Phase-Space Wave-Action Density . . . . . . . . . . . . . . . . . . . . . 480 10.3.9 Conservation Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 10.3.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 10.4 Critical Levels and Reflecting Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 10.4.1 Critical Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 10.4.2 Reflecting Levels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 10.4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 10.5 The Middle-Atmosphere Gravity-Wave Impact . . . . . . . . . . . . . . . . . . . 496 10.5.1 Extension of the TEM by Gravity-Wave Effects. . . . . . . . . . . 497 10.5.2 The Gravity-Wave Effect on the Residual Circulation and on the Zonal-Mean Flow. . . . . . . . . . . . . . . . . . . . . . . . . . 500 10.5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 10.6 References and Recommendations for Further Reading. . . . . . . . . . . . . 504

xiv

Contents

11 Appendices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 11.1 Appendix A: Useful Elements of Vector Analysis. . . . . . . . . . . . . . . . . . 507 11.1.1 The Gradient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 11.1.2 The Divergence and the Integral Theorem from Gauss. . . . . . 507 11.1.3 The Curl and the Integral Theorem from Stokes. . . . . . . . . . . 508 11.1.4 Some Identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 11.1.5 Recommendations for Further Reading. . . . . . . . . . . . . . . . . . 509 11.2 Appendix B: Rotations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 11.2.1 Recommendations for Further Reading. . . . . . . . . . . . . . . . . . 512 11.3 Appendix C: Isotropic Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512 11.3.1 Isotropic Tensors of Rank One. . . . . . . . . . . . . . . . . . . . . . . . . 513 11.3.2 Isotropic Tensors of Rank Two. . . . . . . . . . . . . . . . . . . . . . . . . 513 11.3.3 Isotropic Tensors of Rank Three . . . . . . . . . . . . . . . . . . . . . . . 514 11.3.4 Isotropic Tensors of Rank Four. . . . . . . . . . . . . . . . . . . . . . . . 515 11.3.5 Recommendations for Further Reading. . . . . . . . . . . . . . . . . . 517 11.4 Appendix D: Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 11.4.1 The Local Basis Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 11.4.2 The Gradient in Spherical Coordinates . . . . . . . . . . . . . . . . . . 518 11.4.3 The Divergence in Spherical Coordinates. . . . . . . . . . . . . . . . 519 11.4.4 The Curl in Spherical Coordinates. . . . . . . . . . . . . . . . . . . . . . 521 11.4.5 Recommendations for Further Reading. . . . . . . . . . . . . . . . . . 523 11.5 Appendix E: Fourier Integrals and Fourier Series. . . . . . . . . . . . . . . . . . 523 11.5.1 Fourier Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 11.5.2 Fourier Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 11.5.3 Recommendations for Further Reading. . . . . . . . . . . . . . . . . . 527 11.6 Appendix F: Zonally Symmetric Rossby Waves in the Quasigeostrophic Two-Layer Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 11.7 Appendix G: Explicit Solution of the Initial-Value Problem of Baroclinic Instability in a Quasigeostrophic Two-Layer Model. . . . . . . 528 11.8 Appendix H: Polarization Relations of the Geostrophic Mode and all f -Plane Modes Without Buoyancy Oscillations . . . . . . . . . . . . . 529 11.9 Appendix I: The Higher Harmonics of a Gravity-Wave Field in WKB Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 11.9.1 Leading-Order Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 11.9.2 Next-Order Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536 11.9.3 Recommendations for Further Reading. . . . . . . . . . . . . . . . . . 539 Literature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547

1

The Basic Equations of Atmospheric Motion

This chapter focusses on the derivation of the basic equations allowing the prediction of wind and density from corresponding initial conditions. They also require knowledge of the evolving pressure distribution. The prediction of the latter, using a prognostic equation for the temperature and the thermodynamic equation of state, will be discussed later.

1.1

Time Derivatives in Fluids

1.1.1

The Fluid Description According to Euler and Lagrange

Basis of the formulation of the prognostic equations in meteorology is two complementary perspectives of the development of atmospheric motion. In both, a volume of air under consideration is subdivided into fluid elements. These are infinitesimally small and directly tied to the molecular particles they contain. Each fluid element follows the macroscopic motion of its particles (Fig. 1.1). Kinetic particle motion is ignored. Correspondingly, it is assumed that the particle mean free path is small in comparison with all considered length scales. This even holds when differential limits are taken. Effects of kinetic motion which cannot be neglected are described via macroscopic friction and diffusion. As a consequence of these considerations, fluid elements have the following properties: • They are infinitesimally small. • The mass of a fluid element is conserved. Moreover, each fluid element carries quantitative properties, depending on time t, such as • position x(t), • velocity v(t), © Springer-Verlag GmbH Germany, part of Springer Nature 2022 U. Achatz, Atmospheric Dynamics, https://doi.org/10.1007/978-3-662-63941-2_1

1

2

1 The Basic Equations of Atmospheric Motion

Fig. 1.1 Subdivision of a volume of air into infinitesimally small fluid elements

• density ρ(t), • temperature T (t), and • other properties, such as moisture content, tracer concentrations, etc.

The Lagrangian Perspective In the perspective named after Lagrange, all fluid elements are followed, with their evolving properties, in their spatial movement. Due to the large, in principle infinite, number of fluid elements this is quite an elaborate approach. In approximation, it is, however, used in various model descriptions of the atmosphere (Fig. 1.2). The pure concept, however, is very important as the basis of the formulation of the basic equations of motion, since it is most closely related to the fundamental physical laws. The latter describe the very behavior of fluid elements, as the smallest conceivable material units, under the impact of external forces and fluxes. The Eulerian Perspective Most of the time, an observer is interested in the development of wind, temperature, etc., at a fixed location. This is the perspective named after Euler. The fluid properties are thus not associated with specific fluid elements. They are rather defined as fields depending on space and time. Examples are

1.1 Time Derivatives in Fluids

3 θ (K) 491 487 483 479 475 471 467 463 459 455

Fig. 1.2 Illustration of the results of a transport model which uses a Lagrangian-like fluid description. Note the finite fluid elements which are followed by the model, with their (potential) temperature, in time. Figure from McKenna et al. (2002)

• velocity v(x, t), • density ρ(x, t), • temperature T (x, t) etc. This is also the perspective we are acquainted with from the daily weather map (Fig. 1.3).

1.1.2

The Material Derivative of a Fluid Element

The Material Derivative of a Scalar The basic laws of physics describe the behavior of single fluid elements. For a Eulerian description of the time and spatial development of a fluid, we therefore need a mapping from the Lagrangian picture onto the Eulerian one. This is provided by the material derivative. The corresponding operator solves the following problem: Given a field (x, t) according to Euler what is, at location x0 and time t the time derivative of  carried by the very fluid element which is at time t at the location x(t) = x0 ? For this purpose we first determine ˜ the infinitesimal change of  at the fluid-element location, i.e., of (t) =  [x(t), t]. The required change results from the explicit time development of the field at the location x0 and the movement of the fluid element itself, i.e.,

4

1 The Basic Equations of Atmospheric Motion

Fig. 1.3 Spatial dependence of the temperature over Europe (on the 850-mb pressure level) at a fixed time, as an example of a field quantity in a Eulerian picture. (Source: Deutscher Wetterdienst)

˜ = δ

∂ ∂ ∂ ∂ (x0 , t) δt + (x0 , t) δx + (x0 , t) δ y + (x0 , t) δz ∂t ∂x ∂y ∂z

(1.1)

Division by the time increment δt yields in the limit δt → 0 ˜ d ∂ dx = + · ∇ dt ∂t dt

(1.2)

The movement of the fluid element is given by the local velocity field, i.e., dx = v(x0 , t) dt

(1.3)

so that one obtains the material derivative of  as D ∂ = + v · ∇ Dt ∂t

(1.4)

Other names for the material derivative are the advective or the Lagrangian derivative. The Material Derivative of a Vector Field The material derivative of the Cartesian components of a vector field b = bx ex +b y e y +bz ez is defined in complete analogy to that of a scalar field. Thus, one has, e.g.,

1.1 Time Derivatives in Fluids

so that in summary

1.1.3

5

Dbx ∂bx = + (v · ∇)bx Dt ∂t

(1.5)

Db ∂b = + (v · ∇)b Dt ∂t

(1.6)

The Material Derivative of Volume Integrals

The Material Derivative of a Volume In various applications we will have to consider a finite material volume, always consisting of the same fluid elements. The shape of this material volume changes, as well as the volume content, as the fluid elements move in space (Fig. 1.4). What is the time dependence of the volume content? A change of volume is brought about by the dislocation of the fluid elements at the surface. For a calculation of this effect we decompose the total material surface, bounding the material volume, into infinitesimally small surface elements, each with an area d S. We further decompose the velocity through each surface element into v = vn +vt . Here vn is the normal component and vt the residual tangential component (Fig. 1.5). The latter causes a shift of the surface element along the material surface. It therefore does not lead to a volume change. The normal component, however, does cause a volume change

Fig. 1.4 A two-dimensional illustration of a material volume. It changes its shape and its volume content as the constituting fluid elements move in space

6

1 The Basic Equations of Atmospheric Motion

Fig. 1.5 Illustration of the volume change of a material volume, resulting from the normal movement of an infinitesimally small surface element. The velocity vector at the surface element is decomposed into its tangential and normal component, respectively. Only the normal component contributes to the volume change

v

surface element dS(t + δt)

vn

δr = vnδt dS(t)

δ d V = d Sδr

(1.7)

where δr = vn δt is the distance in normal direction over which the surface element is moved within the time δt. It is proportional to the velocity vn = n · v, obtained by projection onto the outwardly directed normal vector n. This velocity is positive in the case of outwardly directed motion, otherwise negative. With dS = nd S one obtains δ d V = dS · vδt

(1.8)

The total volume change is obtained by summation, i.e., integration, over all surface elements  δV = v · dSδt (1.9) S

Division by the time increment yields, in the limit δt → 0, the material derivative of the volume   D d V = v · dS (1.10) Dt V S Applying the integral theorem from Gauss (appendix A.2) we finally obtain D Dt



 dV = V

∇ · vd V V

(1.11)

1.1 Time Derivatives in Fluids

7

The limit V → V → 0 yields the change of volume of a single fluid element D V = V ∇ · v Dt

(1.12)

Thus, the rate of change of the volume of a single fluid element is: lim

1 DV =∇ ·v Dt

(1.13)

V →0 V

The Material Derivative of the Integral of a Volume-Specific Quantity For the material derivative of the product of an arbitrary scalar ξ , e.g. a volume specific quantity such as the specific moisture, with the volume of a fluid element one obtains via (1.12)   DV Dξ Dξ D (ξ V ) = ξ + V = V ξ ∇ · v + (1.14) Dt Dt Dt Dt The integral over a finite material volume yields D Dt





 dV ξ = V

V

Dξ dV ξ∇ · v + Dt

 (1.15)

For comparison: The Eulerian derivative would be the derivative for a spatially fixed volume with different fluid elements contributing as time evolves, i.e.,   ∂ξ d dV ξ = dV (1.16) dt ∂t The Material Derivative of the Integral of a Mass-Specific Quantity Consider finally the product of some other scalar quantity q, e.g., a mass-specific quantity such as the relative moisture, with density and volume of a fluid element. One has Dq D D (V ρq) = ρV +q (ρV ) Dt Dt Dt

(1.17)

However, M = ρV is the constant mass of the fluid element, so that the second term on the right-hand side does not contribute. Hence, via integration, D Dt



 d V ρq = V

dV ρ V

Dq Dt

The Eulerian time derivative would be, for comparison,   ∂ d d V ρq = d V (ρq) dt V ∂t V

(1.18)

(1.19)

8

1.1.4

1 The Basic Equations of Atmospheric Motion

Summary

• In the theory of fluids (including gases) there are two complementary perspectives: In the Lagrangian picture one considers separate fluid elements. They are characterized by their location, velocity, conserved mass, and their thermodynamic properties. The Eulerian picture assumes the perspective of a spatially stationary observer of fluid elements passing by so that the local wind velocities, density, pressure, and temperature change in time. • The material derivative provides the link between the two perspectives. It is used for calculating, from the spatially and time-dependent fields of the Euler picture, the temporal change of all dynamic properties as experienced by the very fluid element which at a given time passes a given location. • A material volume is a time-dependent volume always containing the same fluid elements. The temporal change of the integral properties of such a volume, also called material derivative, can be obtained from the local material derivative defined above.

1.2

The Equation of Continuity

The first central equation of motion to be derived here is the equation of continuity. This prognostic equation for the density is a direct consequence of the conservation of the mass of each fluid element. For further illustration of the two fluid-dynamical perspectives with their respective strengths and weaknesses two derivations of the equation of continuity will be given, one within the Eulerian picture, and one in the Lagrangian picture.

1.2.1

A Eulerian Derivation

In the Eulerian perspective one considers a spatially fixed volume in its balance between mass gain or loss and density change: For a determination of the change of mass within the volume one has to know the mass of all fluid elements passing within the incremental time δt through the surface of the volume. The volume of the fluid element leaving (in the case of vn > 0) the volume through the infinitesimally small sub-surface d S is (Fig. 1.6), in analogy to (1.8), δ d V = δr d S = v · dS δt (1.20) The corresponding mass loss is δ d M = −ρ δ d V = −ρv · dS δt

(1.21)

which is, per time, the negative scalar product of the mass flux ρv with the surface vector dS. In the case of flow into the volume (vn < 0) one obtains the same relationship. Integration over the total surface yields the total mass change

1.2 The Equation of Continuity

9

Fig. 1.6 Illustration of the volume of a fluid element passing within the infinitesimally small time δt through the surface d S of a spatially fixed volume

vt

v

vn dS

dS



δr = vnδt



δM = −

ρv · dS δt = − S

d V ∇ · (ρv) δt

(1.22)

V

On the other hand we have also directly d δ M = δt dt Thus, one obtains



 dV V



 d V ρ = δt

dV V

∂ρ ∂t

 ∂ρ + ∇ · (ρv) = 0 ∂t

(1.23)

(1.24)

This holds, however, for arbitrary volumina, i.e., also an infinitesimally small control volume. Thus, one has ∂ρ + ∇ · (ρv) = 0 (1.25) ∂t This is the equation of continuity.

1.2.2

A Lagrangian Derivation Using the Material Derivative

On the basis of the preparations above the derivation of the equation of continuity in the Lagrangian picture is considerably more elegant. With ξ = ρ in (1.15) one obtains for a material volume     D Dρ dV ρ = d V ρ∇ · v + (1.26) Dt V Dt V

10

1 The Basic Equations of Atmospheric Motion

The left-hand side, however, vanishes due to mass conservation. Thus one has    Dρ d V ρ∇ · v + =0 Dt V

(1.27)

Again this holds for arbitrary control volumina so that Dρ + ρ∇ · v = 0 Dt

(1.28)

One can easily show that this is equivalent to (1.25).

1.2.3

Summary

The continuity equation follows from the fundamental principle of mass conservation. It can be derived in two different manners: • In the Eulerian picture one considers a spatially fixed volume. Mass conservation demands that the temporal change of the mass contained by the volume is balanced by the mass flux through the boundaries of the volume. • In the Lagrangian picture one considers a material volume. Since its mass is conserved, the continuity equation directly follows from the vanishing corresponding material derivative. The continuity equation is a fundamental building stone in the prediction of the development of the fluid flow.

1.3

The Momentum Equation

The second essential equation for the prediction of the atmosphere is the momentum equation. Given the distribution of pressure, density, and winds, it determines the time derivatives of the wind fields, i.e., the velocity fields. For this we must consider the acceleration of all single fluid elements. For each fluid element it is obtained from the sum of forces acting on it. In this we must discriminate between volume forces, such as gravity or apparent forces due to rotation, and surface forces. To the latter belong the pressure gradient force and friction. In the following we will give a mathematical description of the effect of these forces.

1.3.1

The Volume Forces

Volume forces influence the motion of a fluid element in a volume immanent manner; i.e. they do not act via momentum fluxes through the surface. The material derivative of the

1.3 The Momentum Equation

11

 momentum V d V ρv of a material volume is given by the volume integral of a corresponding force density. One such example is f = Mg/V = ρg in the case of gravity, where g is the local gravitational acceleration. In other words: The material derivative of momentum of a single fluid element is, in the absence of other forces, D (Mv) = fV Dt

(1.29)

where f is force per volume. Because M = ρV , this becomes D (V ρv) = fV Dt or, integrated over all fluid elements of a material volume,   D d V ρv = fd V Dt V V Using (1.18) one obtains

 V

Dv dV ρ = Dt

(1.30)

(1.31)

 fd V

(1.32)

V

The material volume under consideration, however, is arbitrary, so that ρ

1.3.2

Dv =f Dt

(1.33)

Surface Forces (1): The Pressure Gradient Force

In contrast to the volume forces, surface forces act on the surface of a material volume. They are typically the result of the kinetic motion of the molecular components of the fluid elements. A prime example for this is the force due to external pressure. The thermal motion of the air molecules leads to collisions between molecules of neighboring fluid elements, and thus to a corresponding momentum transfer. The component of the resulting force normal to the respective contact surface is the pressure force. Consider the partial force on a material volume resulting, as in Fig. 1.7, from the action of pressure on the surface element d S. It is df = − pdS The total force is obtained by integrating over the total surface, i.e.,   D d V ρv = − pdS Dt V S

(1.34)

(1.35)

12

1 The Basic Equations of Atmospheric Motion

Fig. 1.7 Partial force df exerted on a material volume by external pressure on the surface element d S

material volume

dS df

Now consider the change of momentum in x-direction. It is given by    D d V ρvx = − pd Sx = − ( p, 0, 0) · dS Dt V S S

dS

(1.36)

Applying the integral theorem from Gauss yields D Dt



 d V ρvx = − V

V

⎛ ⎞  p ∂p ⎝ ⎠ ∇ · 0 dV = − dV V ∂x 0

Analogous calculations for the other spatial directions yield together   D d V ρv = − ∇ pd V Dt V V

(1.37)

(1.38)

Application of (1.18) and consideration of the fact that the result holds for arbitrary control volumina finally leads to Dv ρ = −∇ p (1.39) Dt The force on a fluid element thus can be determined from the gradient of the pressure field.

1.3 The Momentum Equation

1.3.3

13

Surface Forces (2): Friction

The conceptually more difficult, however in many regards equally important, kind of surface forces is friction. For an illustration Fig. 1.8 shows two neighboring fluid elements, both moving in x-direction, but at different velocities. Due to their kinetic motion, molecular particles cross the contact surface between the two fluid elements. Thus typically, by collisions with molecular particles of the other fluid element, they transfer momentum from one fluid element to the other. The resulting frictional force Fx in x-direction is, via the number of crossing particles, proportional to the contact surface A. Via the momentum exchanged thereby on average, it is also proportional to the velocity difference between the two fluid elements, and thus also to the local velocity gradient ∂vx /∂z normal to the contact surface, which here is in z-direction. We thus write Fx = Aη

∂vx ∂z

(1.40)

The coefficient of proportionality η is the dynamic viscosity. The stress η∂vx /∂z indicates the momentum flux, here of momentum in x-direction, through the contact surface, here in z-direction. Guided by the example above we now consider the case of motion in arbitrary directions, and of arbitrary contact surfaces. The most general ansatz for a surface force, including the pressure force, resulting in total from momentum fluxes, is

z

vx1 A vx2

x Fig. 1.8 Relative motion (vx1  = vx2 ) of two neighboring fluid elements leads to a frictional force

14

1 The Basic Equations of Atmospheric Motion

D Dt



 d V ρvi = − V

i · dS

(1.41)

S

Here the index i denotes one of three spatial directions. i is the vectorial flux of the ith momentum component in all three spatial directions. Using the theorem from Gauss and the differentiation rule (1.18) leads to   Dvi dV ρ d V ∇ · i (1.42) =− Dt V V We now introduce the stress tensor i j = − i j , which yields ρ

∂ i j Dvi = Dt ∂x j

(1.43)

since the momentum balance must hold for arbitrary control volumina. Here, as in the rest of the text, we use the summation convention due to Einstein, according to which a summation is done over all indices occurring twice. The pressure contribution to the stress tensor can be separated off via (1.44) i j = −δi j p + σi j so that ρ

∂σi j Dvi ∂p + =− Dt ∂ xi ∂x j

(1.45)

Here σ is the viscous stress tensor. So far the formulation is very general. What can σi j actually look like? First of all it is clear that the viscous stress tensor must be a function of the gradients of the velocity field, which vanishes in the case of absence of any gradient. Based on this we use the ansatz that the viscous stress tensor shall be a linear function of the velocity gradients, i.e., σi j = si jkl

∂vk ∂ xl

(1.46)

Here the si jkl are coefficients. In their specification symmetry considerations will be helpful. Let us first assume that the kinetic motion of and the collisions between the air molecules are not influenced significantly by gravity, and that in all other regards the frictional process is not characterized by any distinguished spatial direction, but solely by the local velocity gradients. Accordingly the functional relationship between the viscous stress tensor and the velocity gradients shall be invariant under rotations. In other words, if σij are the components of the viscous stress tensor in an arbitrarily rotated coordinate system we still demand that σij = si jkl

∂vk ∂ xl

(1.47)

1.3 The Momentum Equation

15

Here xi and vi are the components of the vectors of position and velocity in the rotated coordinate system. All aspects of the mathematics of rotations relevant for us are summarized in the appendix B. The mapping between the two coordinate systems before and after the rotation is performed by xi = Ri j x j

(1.48)

vi = Ri j v j

(1.49)

where the rotational matrix R is orthogonal, i.e., (R −1 )i j = R ji . Obviously the material derivative does not change its form under rotations, ∂ ∂ ∂ ∂ D = = + vi + vi  Dt ∂t ∂ xi ∂t ∂ xi

(1.50)

Via (1.48) and (1.49) one can also convince oneself directly that ∂x j ∂ ∂ ∂ = = (R −1 ) ji   ∂ xi ∂ xi ∂ x j ∂x j

(1.51)

and therefore, with a corresponding renaming of the indices, vi

∂ ∂ ∂ = Rik (R −1 )i j vk = vi ∂ xi ∂x j ∂ xi

(1.52)

In order to exploit (1.47) we need σij . One has   ∂σij Dvi ∂σ jk ∂p Dvi ∂p =ρ + = Ri j ρ = Ri j − −  + ∂ xi ∂ x j Dt Dt ∂x j ∂ xk and, in parallel to (1.51),

∂ x j ∂ ∂ ∂ = = R ji   ∂ xi ∂ xi ∂ x j ∂x j

(1.53)

(1.54)

Together with the orthogonality of the rotational matrix this leads in (1.53) to ∂σij ∂σ jk ∂p ∂σmn ∂p ∂p −  + = −  + Ri j Rlk = −  + Rim R jn ∂ xi ∂ x j ∂ xi ∂ xl ∂ xi ∂ x j

(1.55)

Hence, the viscous stress tensor is transformed under rotations according to σij = Rim R jn σmn

(1.56)

16

1 The Basic Equations of Atmospheric Motion

This means together with (1.46) and (1.47) that si jkl

∂vk ∂v p = Rim R jn smnpq  ∂ xl ∂ xq

(1.57)

In order to be able to derive from this constraints on the coefficients in s, we also need on the right-hand side the derivatives of the rotated velocities in the rotated coordinate system. Due to the orthogonality of the rotational matrix one has v p = Rkp vk

(1.58)

Moreover we once again use (1.54) and obtain finally si jkl

∂vk ∂vk = R R R R s im jn kp lq mnpq ∂ xl ∂ xl

(1.59)

si jkl = Rim R jn Rkp Rlq smnpq

(1.60)

This means that i.e., s is an isotropic tensor of rank four, that transforms under rotations into itself. Appendix C shows, however, that the most general form of such a tensor is given by si jkl = αδi j δkl + β(δik δ jl + δil δ jk ) + γ (δik δ jl − δil δ jk ) with free parameters α, β und γ . We insert this into (1.46) and get     ∂v j ∂v j ∂vi ∂vk ∂vi +γ +β + − σi j = αδi j ∂ xk ∂x j ∂ xi ∂x j ∂ xi

(1.61)

(1.62)

Merely by symmetry considerations it has been possible to reduce the number of free parameters in σ from 81 to three! A further argument eliminates γ : Consider a rigid rotation as in Fig. 1.9, where v(x) = ω × x

(1.63)

vi = i jk ω j xk

(1.64)

or with constant angular velocity ω. i jk is known as Levi-Civita symbol, or the total antisymmetric tensor of third rank, with the properties ⎧ ⎨ 0 if two indices are identical (1.65) i jk = 1 if i jk can be obtained by cyclic permutation from 123 ⎩ −1 else It is obvious that such a movement must not lead to frictional forces. Otherwise the conservation of angular momentum would be violated. Nonetheless, it is accompanied by

1.3 The Momentum Equation

17

Fig.1.9 Velocity field of a rigid rotation, accompanied by velocity gradients, however not by frictional forces

non-vanishing velocity gradients. Insertion of (1.63) into (1.62) must therefore lead to a vanishing stress tensor. Because of ∂ ∂vk = (klm ωl xm ) = klk ωl = 0 ∂ xk ∂ xk

(1.66)

we thus have       ∂ ∂ ∂  ∂  0=β  jkl ωk xl + γ  jkl ωk xl (ikl ωk xl ) + (ikl ωk xl ) − ∂x j ∂ xi ∂x j ∂ xi     (1.67) = β ik j ωk +  jki ωk + γ ik j ωk −  jki ωk = 2γ ik j ωk This only holds for all possible ω if γ = 0. At last we introduce the customary notation 2 α=ζ− η 3 β=η so that the final result is the most general form of the viscous stress tensor,

(1.68) (1.69)

18

1 The Basic Equations of Atmospheric Motion



∂v j ∂vi 2 ∂vk σi j = η + − δi j ∂x j ∂ xi 3 ∂ xk

 + ζ δi j

∂vk ∂ xk

(1.70)

η is usually known as dynamic shear viscosity, while ζ is the volume viscosity.

1.3.4

The Total Momentum Equation

Summarizing all possible forces, i.e., volume and surface forces, we obtain the total momentum equation Dv ρ = f −∇p +∇ ·σ (1.71) Dt where (∇ · σ )i =

∂σi j ∂x j

(1.72)

Often we will also write f = ρg with the gravitational acceleration g, which again can be expressed as g = −∇, where  is the geopotential.

1.3.5

Summary

Newton’s second law finds its expression in the momentum equation. One first considers in the Lagrangian picture the acceleration of a material volume by the forces acting on it. Then one obtains via the material derivative a prognostic equation for the wind fields. The forces can be subdivided into two classes: • Volume forces directly act on each of the fluid elements. The force acting on the material volume is the sum of these forces, i.e., can be obtained from a volume integral. In the context here, with electrically neutral air, the acting volume force is the gravitational force. • Surface forces only act on the surface of a material volume. They are due to kinetic molecular motions, which do not find an explicit description in the macroscopic perspective of fluid mechanics. – The pressure force results from collisions normal to the surface. In the momentum equation this leads to an acceleration opposite to the local pressure gradient. – Friction results from collisions with a component tangential to the surface. The empirically well-founded ansatz can be used that the viscous part of the stress tensor, corresponding to the negative of the momentum flux tensor, depends linearly on the local velocity gradients. One further postulates that friction does not have any distinguished spatial direction, so that the corresponding process is invariant under rotations. Angular-momentum conservation, finally, demands that friction does not affect rigid

1.4 The Equations of Motion in a Rotating Frame of Reference

19

rotation. These two postulates help in simplifying the viscous stress tensor so that only two remaining physical parameters turn out to be necessary for calculating it from the velocity gradients: shear viscosity and volume viscosity.

1.4

The Equations of Motion in a Rotating Frame of Reference

So far the equations of continuity and momentum have been derived for a non-accelerated, i.e., an inertial frame of reference. The observer on earth, however, has a geostationary perspective, i.e., she is in a rotating frame of reference. She therefore observes two additional apparent forces; i.e., the centrifugal force and the Coriolis force. The modification of the equations of motion under such conditions, and thus also the derivation of the apparent forces is addressed in this section.

1.4.1

The Time Derivative in a Rotating Frame of Reference

The rotation of an arbitrary vector A with the angular velocity  satisfies dA =×A dt

(1.73)

For the basis vectors ei of a frame of reference rotating with the earth one therefore has Dei =  × ei Dt

(1.74)

where here and henceforth  is the angular velocity of the rotation of earth. Now consider an arbitrary time dependent vector b = bi ei , such as position or velocity (Fig. 1.10). How does its temporal development differ between the inertial and the rotating frame of reference? The rotating, i.e., geostationary, observer sees a time derivative   Db Dbi = (1.75) ei Dt R Dt since the axes do not move in his frame of reference. In the non-rotating frame of reference, however, one has     Db Dei Dbi Db = + bi  × ei (1.76) ei + bi = Dt I Dt Dt Dt R Thus



Db Dt



 = I

Db Dt

 +×b R

(1.77)

20

1 The Basic Equations of Atmospheric Motion

Ω

Fig. 1.10 Vector b and the rotating axes (basis vectors) ei

e3

b

e1 1.4.2

e2

The Momentum Equation in the Rotating Frame of Reference

Via (1.77) we now transform the velocities and accelerations from the inertial frame of reference into the rotating frame of reference. Let x be the position of a fluid element. Its time derivative is     Dx Dx = +×x (1.78) Dt I Dt R and thus vI = vR +  × x

(1.79)

where v I = (Dx/Dt) I is the velocity in the inertial frame of reference, and v R = (Dx/Dt) R the velocity in the rotating frame of reference. A bit more work is necessary to express the inertial-frame acceleration in terms of quantities in the rotating reference frame. Due to (1.77) one has     Dv I Dv I = +  × vI (1.80) Dt I Dt R From (1.79) we obtain       Dv I Dv R D = + × x +  × vR Dt R Dt R Dt R

(1.81)

1.4 The Equations of Motion in a Rotating Frame of Reference

Fluctuations of the earth’s rotation are not considered. Thus one gets with (1.77)     D D = +× 0= Dt I Dt R 

and therefore

Hence (1.81) becomes



Dv I Dt

D Dt



 =

21

(1.82)

 =0

(1.83)

R

Dv R Dt

 +  × vR

(1.84)

 × v I =  × v R +  × ( × x)

(1.85)

R

R

Due to (1.79) one has in addition

Insertion of (1.84) and (1.85) in (1.80) finally yields 

Dv I Dt



 = I

Dv R Dt

 + 2 × v R +  × ( × x)

(1.86)

R

On the right-hand side of the equation we find, consecutively, the acceleration in the rotating frame of reference, the negative of the Coriolis acceleration −2v R × , and the negative of the centrifugal acceleration − × ( × x). We obtain the momentum equation in the rotating frame of reference by insertion of (Dv I /Dt) I from (1.86), for Dv/Dt, into (1.71). Since in the following all considerations will be in the rotating frame of reference we drop the index R and obtain   Dv ρ (1.87) + 2 × v = −∇ p + ρg + ∇ · σ Dt Here

g = g −  × ( × x)

(1.88)

is an effective gravitational acceleration, modified by the effect of the centrifugal acceleration. For most purposes one can assume, at very good accuracy, that g ≈ g. Due to its importance we finally give a short interpretation of the Coriolis acceleration. It expresses the conservation of angular momentum and can best be understood near one of the poles, such as the north pole in Fig. 1.11. A fluid element moving away from the rotational axis will be deflected, due to its angular momentum being smaller than that of the environment, against the rotation. Movement toward the rotational axis leads in analogy to a deflection into the rotation, since now the angular momentum is larger than in the environment. Similar considerations apply for arbitrary latitudes.

22

1 The Basic Equations of Atmospheric Motion

Fig. 1.11 Coriolis acceleration, expressing the conservation of angular momentum, in the case of motion near the north pole

NP

1.4.3

The Equation of Continuity in the Rotating Frame of Reference

The transformation of the equation of continuity from the inertial frame of reference into the rotating frame of reference is much simpler than in the case of the momentum equation. First of all, the density is invariant under transformations, so that     Dρ Dρ = (1.89) Dt I Dt R Furthermore (1.79) leads to ∇ · v I = ∇ · v R + ∇ · ( × x) = ∇ · v R since ∇ · ( × x) =

∂ i jk  j xk = i ji  j = 0 ∂ xi

(1.90)

(1.91)

Inserting (Dρ/Dt) I from (1.89) and ∇ · v I from (1.90) for Dρ/Dt and ∇ · v into (1.28) yields Dρ + ρ∇ · v = 0 (1.92) Dt where again the index R has been dropped. The equation of continuity in the rotating frame of reference has the same form as in the inertial frame of reference. This certainly also holds in the version (1.25).

1.5 The Equations of Motion on the Sphere

1.4.4

23

Summary

While the momentum equation from the previous section only holds in a non-accelerated inertial frame, in the rotating frame on the earth, it is to be supplemented by two apparent forces: • The centrifugal force can be handled as modification of the gravitational force. • The Coriolis force directly depends on velocity, and thus must be included explicitly. The continuity equation is not affected by rotation.

1.5

The Equations of Motion on the Sphere

The (approximate) sphericity of the earth suggests the use of an adapted coordinate system. Spherical coordinates, as illustrated in Fig. 1.12, seem to be a very good choice. The geographic longitude λ is within 0 ≤ λ ≤ 2π . The geographic latitude is φ with − π2 ≤ φ < π2 . These two are supplemented by the radial distance r from the center of earth. The positional vector in Cartesian coordinates is expressed by spherical coordinates as z

NP

eϕ

eλ

r x ϕ λ

x

Fig. 1.12 Spherical coordinates

er

y

24

1 The Basic Equations of Atmospheric Motion

⎛ ⎞ ⎛ ⎞ x r cos φ cos λ ⎝ y ⎠ = ⎝ r cos φ sin λ ⎠ z r sin φ

(1.93)

The local orthonormal unit vectors eλ , eφ , and er , each in the direction of the change of one of the spherical coordinates, denoted in the index, as well as the representation of the most important spatial differential operators in spherical coordinates, are derived in appendix D. Using these results we will now first express velocity and material derivative in spherical coordinates and then express the equations of motion accordingly.

1.5.1

Velocity and Material Derivative in Spherical Coordinates

The Velocity We first represent velocity on the basis of the local unit vectors. One has v=

∂x Dλ ∂x Dφ ∂x Dr Dx = + + Dt ∂λ Dt ∂φ Dt ∂r Dt

(1.94)

Via (11.63) one obtains v = h λ eλ

Dλ Dφ Dr + h φ eφ + h r er Dt Dt Dt

(1.95)

where h λ , h φ , and h r are metric factors. Inserting these from (11.67)–(11.69) finally leads to   Dλ Dφ Dr v = ueλ + veφ + wer ,r , (1.96) (u, v, w) = r cos φ Dt Dt Dt The wind components are the zonal wind u in west–east direction, the meridional wind v in south–north direction, and the vertical wind which is directed radially outward. The Material Derivatives In determining the material derivative of a vector field b = bλ eλ + bφ eφ + br er

(1.97)

one must take into account that the basis vectors eλ , eφ , and er are spatially dependent, and hence vary along the trajectory of a fluid element. This leads to Deφ Dbφ Deλ Der Dbλ Dbr Db = e λ + bλ + e φ + bφ + er + br Dt Dt Dt Dt Dt Dt Dt

(1.98)

1.5 The Equations of Motion on the Sphere

Therein

⎛ ⎞ ⎛ ⎞ − sin λ − cos λ Deλ D ⎝ Dλ = cos λ ⎠ = ⎝ − sin λ ⎠ Dt Dt Dt 0 0

25

(1.99)

Via projection onto the local unit vectors one obtains ⎛

⎞ − cos λ ⎝ − sin λ ⎠ = sin φ eφ − cos φ er 0

(1.100)

Furthermore one has, due to (1.96), Dλ u = Dt r cos φ

(1.101)

Inserting (1.100) and (1.101) into (1.99) finally yields u u Deλ = tan φ eφ − er Dt r r Together with analogous calculations for eφ and er one obtains in total u u Deλ = tan φ eφ − er Dt r r Deφ v u = − tan φ eλ − er Dt r r v u Der = eλ + eφ Dt r r

(1.102) (1.103) (1.104)

Inserting these results into (1.98) finally yields Db = Dt

   Dbφ u v Dbλ u u − tan φ bφ + br eλ + + tan φ bλ + br eφ Dt r r Dt r r   Dbr v u + − bλ − bφ er Dt r r



(1.105)

Here the material derivatives of bλ , bφ , and br are, as in the case of an arbitrary scalar, ∂ Dλ ∂ Dφ ∂ Dr ∂ D = + + + Dt ∂t Dt ∂λ Dt ∂φ Dt ∂r

(1.106)

which, due to (1.96), is equivalent to D ∂ u ∂ v ∂ ∂ = + + +w Dt ∂t r cos φ ∂λ r ∂φ ∂r

(1.107)

26

1.5.2

1 The Basic Equations of Atmospheric Motion

The Transformed Equations of Motion

With the preparations from the previous section and from appendix D we are now able to cast the equations of motion discussed so far into a spherical-coordinate representation. The Equation of Continuity For the transformation of the equation of continuity we start from (1.25). With (1.96) and the representation of the divergence in spherical coordinates via (11.86) we obtain ∂ρ 1 ∂ 1 ∂ 1 ∂ + (ρu) + (cos φ ρv) + 2 (r 2 ρw) = 0 ∂t r cos φ ∂λ r cos φ ∂φ r ∂r

(1.108)

The alternative formulation (1.28) written in spherical coordinates is   Dρ 1 1 ∂ 2 1 ∂u ∂ +ρ + (cos φ v) + 2 (r w) = 0 Dt r cos φ ∂λ r cos φ ∂φ r ∂r

(1.109)

The Momentum Equation In the discussion of the momentum equation we here consider only a simplified version of (1.87), without centrifugal force and friction: 1 Dv + 2 × v = ∇ p − ger Dt ρ

(1.110)

For the transformation we also need the representation of the Coriolis acceleration in spherical coordinates. We observe that, as visible in Fig. 1.13,  =  sin φ er +  cos φ eφ

(1.111)

One has moreover  × v = ( sin φ er +  cos φ eφ ) × (ueλ + veφ + wer ) = (w cos φ − v sin φ)eλ + u sin φ eφ − u cos φ er

(1.112)

since eφ × er = eλ er × eλ = eφ eφ × eλ = −er

(1.113)

1.5 The Equations of Motion on the Sphere Fig. 1.13 Decomposition of the angular velocity of the earth into components parallel to the local units vectors corresponding to spherical coordinates

27

Ω Ωϕ || eϕ

Ωr || er 2

–ϕ ϕ

One also obtains, inserting b = v with (1.96) into (1.105),     Dv Du uv uw u2 vw Dv eλ + eφ = − tan φ + + tan φ + Dt Dt r r Dt r r   Dw u 2 + v 2 + − er Dt r

(1.114)

while, according to (11.76), ∇p =

1∂p ∂p 1 ∂p eλ + eφ + er r cos φ ∂λ r ∂φ ∂r

(1.115)

Inserting (1.114), (1.112), and (1.115) into (1.110) and sorting for the parts along eλ , eφ , and er , finally yields Du uv uw − tan φ + + 2(w cos φ − v sin φ) = − Dt r r Dv u2 vw + tan φ + + 2u sin φ = − Dt r r Dw u 2 + v 2 − − 2u cos φ = − Dt r

1 ∂p ρr cos φ ∂λ 1 ∂p ρr ∂φ 1 ∂ p ∂ − ρ ∂r ∂r

(1.116) (1.117) (1.118)

These are, consecutively, the zonal, meridional, and vertical components of the momentum equation.

28

1.5.3

1 The Basic Equations of Atmospheric Motion

Summary

The geometry on earth suggests that a description in spherical coordinates is appropriate: • As spatial coordinates it uses geographic longitude, geographic latitude, and the radial distance from the center of the earth. The corresponding velocity components are the zonal, meridional, and vertical wind. • The material derivative in this coordinate system is obtained via (1.107). • The continuity equation in spherical coordinates is given by (1.108). • The three components of the momentum equation are given by (116)–(118).

1.6

Synoptic Scale Analysis

The components of the momentum equation as derived above are highly complex. Indeed they admit an enormous variety of dynamic possibilities. A comparison of the order of magnitude of the various contributing terms shows, however, that a few dominate the others. This leads to very characteristic dynamic equilibria. For an estimate of the orders of magnitude we use the following observations for mid-latitudes: • The horizontal scale of typical weather phenomena can be estimated as the radius of typical pressure anomalies (highs or lows). A reasonable value is L = 106 m. • The vertical scale of typical weather phenomena is about H = 104 m. • The order of magnitude of typical horizontal wind components u and v is about U = 10m/s. • The typically observed vertical winds w are considerably weaker. A good estimate for synoptic weather systems is W = 10−2 m/s. • The corresponding pressure fluctuations, normalized by density, are of order δ P/ρ. The corresponding numerical value will be derived below. • The time scale of typical weather variations can be estimated by the advective time scale. This is the time within which a typical weather system is passing the observer. The corresponding estimate is T = L/U = 105 s, corresponding to about a day. • The radial distance r from the center of earth is approximately the radius of earth, which we here approximate by a = 107 m. • The Coriolis terms in mid-latitudes (at φ ≈ 45◦ ) are 2 sin φ ≈ 2 cos φ ≈ f 0 = 10−4 s−1 .

1.6

Synoptic Scale Analysis

1.6.1

29

The Geostrophic Equilibrium

Consider first the zonal component of the momentum equation and therein again the material derivative Du ∂u u ∂u v ∂u ∂u = + + +w Dt ∂t r cos φ ∂λ r ∂φ ∂r The second and third term each contain horizontal derivatives so that one can estimate: Term

Order of magnitude

Value (m/s2 )

∂u ∂t

U U2 = T L

10−4

u ∂u r cos φ ∂λ

U

U U2 = L L

10−4

v ∂u r ∂φ

U

U U2 = L L

10−4

∂u ∂r

W

U WU = H H

10−5

w

To leading order the material derivative of the zonal wind is of the magnitude U 2 /L = 10−4 m/s2 . With this we obtain in summary the orders of magnitude of all terms in the zonal component of the momentum equation: Term

Order of magnitude

Du Dt

U2

uv tan φ r

U2 a

uw r 2v sin φ 2w cos φ

UW a f0U f0 W

Value (m/s2 ) 10−4

L

10−5 10−8 10−3 10−6

One sees that the Coriolis acceleration due to the meridional wind v is by far the largest. The ratio to the next smaller term is the inverse of the Rossby number Ro =

U f0 L

(1.119)

which, according to our estimates, has about the value 0.1. If, on the one hand, the horizontal Coriolis acceleration also dominated over the pressure-gradient acceleration, the horizontal

30

1 The Basic Equations of Atmospheric Motion

momentum equation could be approximated by 2v sin φ ≈ 0, with the absurd result that v ≈ 0. If, on the other hand, the pressure-gradient acceleration were even stronger, then one would obtain ∂ p/∂λ = 0. This would not make any sense either, since it would imply that the pressure does not change along latitude circles. Hence the only reasonable solution is that, to leading order, the zonal component of the momentum equation can be approximated as equilibrium between Coriolis force and pressure gradient force − fv = −

∂p 1 ρr cos φ ∂λ

(1.120)

where the local Coriolis frequency is defined as f = 2 sin φ

(1.121)

From this equilibrium results the order of magnitude of the density-normalized pressure fluctuations δP (1.122) = O( f 0 U L) ρ The order-of-magnitude estimate of the meridional component of the momentum equation yields in an analogous manner 1 ∂p fu = − (1.123) ρr ∂φ thus also here an equilibrium between Coriolis force and pressure gradient force. This is the geostrophic equilibrium. It enables an estimate of the horizontal wind u = ueλ + veφ from the pressure gradient. One obtains u ≈ ug =

1 er × ∇ p fρ

(1.124)

or component-wise 1 ∂p f ρr ∂φ 1 ∂p v ≈vg = f ρr cos φ ∂λ

u ≈u g = −

(1.125) (1.126)

The horizontal wind ug calculated in this way is the geostrophic wind. On the northern hemisphere ( f > 0) it is always directed such that the circulation about a pressure low is anti-clockwise, while that about a pressure high is clockwise (Fig. 1.14). A weather map showing this can be seen in Fig. 1.15.

1.6

Synoptic Scale Analysis

31 ug

ug

ug

ug

T

ug

ug

H

ug

ug

Fig.1.14 Schematic illustration of the geostrophic circulation about pressure lows and pressure highs

10

15

20

wind speed at 925 hPa (m/s) 25 30 35 40

45

50

55

Fig. 1.15 Map of instantaneous geopotential height (contour lines) and horizontal wind (arrows and color shading) on the 950-mb level. As discussed in chapter 3, on pressure levels the geopotential plays the same role as pressure on levels at constant height. One sees quite clearly the wind tangential to isolines of geopotential, as explained by geostrophic equilibrium. Copyright ©2021 European Centre for Medium-Range Weather Forecasts (ECMWF). Source www.ecmwf.int. This data is published under a Creative Commons Attribution 4.0 International (CC BY 4.0). https://creativecommons.org/ licenses/by/4.0/. ECMWF does not accept any liability whatsoever for any error or omission in the data, their availability, or for any loss or damage arising from their use

1.6.2

The Hydrostatic Equilibrium

The primary derivation Let us now look at the vertical component of the momentum equation. In analogy to the horizontal component one finds that the material derivative of the vertical wind is to leading order of magnitude U W /L. Thus we obtain for all terms except the pressure gradient term the estimates:

32

1 The Basic Equations of Atmospheric Motion

Value (m/s2 )

Term

Order of magnitude

Dw Dt

U

u 2 + v2 r

U2 a

10−5

2u cos φ

f0U

10−3

∂ ∂r

g

10

W L

10−7

The gravity term is by far the largest. The only term which can (and must) balance it is the pressure-gradient term. This is the hydrostatic equilibrium 1 ∂p ≈ −g ρ ∂r

(1.127)

Hydrostatics of the Longitude- and Latitude-Dependent Pressure Components The estimate above does not, however, show that also that part of the pressure is in hydrostatic equilibrium which depends on the horizontal coordinates. Nonetheless this would be necessary if one wanted to use the hydrostatic approximation in determining the pressure entering the horizontal components of the momentum equation, and thus also the geostrophic wind. Indeed one can show that the horizontally dependent pressure is also in hydrostatic equilibrium. For this purpose we split pressure and density in a (dominant) horizontal mean and the deviation therefrom: p = p(r ) + δ p(λ, φ, r )

(1.128)

ρ = ρ(r ) + δρ(λ, φ, r )

(1.129)

where according to observations, |δρ| ρ |δ p| p Thus it is clear that p and ρ are to a good approximation in hydrostatic equilibrium: 1 dp ≈ −g ρ dr

(1.130)

Insertion of (1.128) and (1.129) into the sum of the both terms on the right-hand side of the vertical momentum equation yields via Taylor expansion, up to the first term, in the small quantities

1.6

Synoptic Scale Analysis

−

33

1 ∂p 1 ∂ −g =− ( p + δ p) − g ρ ∂r ρ + δρ ∂r 1 d p δρ 1 d p 1 ∂δ p ≈− + − −g ρ dr ρ ρ dr ρ ∂r

(1.131)

Inserting the vertical derivative of the horizontally averaged pressure according to (1.130) finally yields 1 ∂p δρ 1 ∂δ p − −g≈− g− (1.132) ρ ∂r ρ ρ ∂r Herein the order of magnitude of the pressure-gradient term is, via (1.122),   L 1 ∂δ p = 10−1 m/s2 = O f0U ρ ∂r H

(1.133)

This is still larger by a factor L

1 (1.134) H than the next largest term on the left-hand side of the vertical momentum equation, the Coriolis term. Hence, even after subtraction of the horizontal means of density and pressure there must still be an equilibrium between pressure-gradient term and gravitational acceleration. To a good approximation one has 0≈− Adding (1.130) yields

1 ∂δ p δρ g− ρ ρ ∂r

ρ 1 ∂p 1 ∂p 0≈− g− ≈− −g ρ ρ ∂r ρ ∂r

(1.135)

(1.136)

which is again the hydrostatic equilibrium between total pressure and total density, now however with the certainty that it also holds under consideration of those parts of the pressure field that depends on the horizontal coordinates. A side result is the order of magnitude of the relative density fluctuations: (1.135) can only be satisfied if   f0U L δρ (1.137) =O ρ g H Hydrostatics and Shallowness For dynamic phenomena in the atmosphere with scales as assumed here the hydrostatic equilibrium seems to be a very good approximation. One should, however, be aware that (1.134) had to be used. Hence, processes with small horizontal scales are not in hydrostatic equilibrium. In calculations of phenomena with small horizontal scales the assumption of hydrostatics should correspondingly be dropped. Therefore the high-resolution models used nowadays by the weather services are formulated as non-hydrostatic models, while global climate models with coarser horizontal resolution can use the hydrostatic assumption.

34

1.6.3

1 The Basic Equations of Atmospheric Motion

Summary

An order-of-magnitude estimate of the momentum equation for processes with typical synoptic scales of daily weather yields the dominance of two essential equilibria: • In the two horizontal components the Coriolis force and the pressure-gradient force dominate at small Rossby numbers. One obtains the geostrophic equilibrium. This allows to determine the horizontal wind from the horizontal pressure gradient. • For shallow processes the vertical momentum equation is dominated by the gravitational force and the pressure gradient force. One obtains the hydrostatic equilibrium. Local density yields the local vertical pressure gradient.

1.7

Recommendations for Further Reading

Introductions into the momentum and continuity equation can be found in all books on hydrodynamics. A classic among many others is the book from Landau and Lifschitz (1987). The discussion of viscosity I have drawn most from is to be found in the text from Greiner and Stock (1991). Excellent texts on the implications of the inclusion of rotation and gravity, defining so-called geophysical fluid dynamics that is the foundation both of atmosphere and ocean dynamics, are the texts from Holton and Hakim (2013), Pedlosky (1987), and Vallis (2006).

2

Elementary Thermodynamics and Energetics of Dry Air

In the previous chapters we have derived the the equations for the prediction of wind (momentum equation) and density (continuity equation). A field quantity needed by these equations but not predicted by them is pressure. The prediction of pressure is one of the goals of this chapter. For this purpose a predictive equation for temperature is derived. Using the equation of state for air, pressure can be obtained from density and temperature. Needless to say that the prediction of temperature itself is an important outcome of this procedure as such. Further important concepts to be met on the road there are heat and entropy. In all considerations we will remain within the precincts of thermodynamics. We will only look at macroscopic properties of macroscopic systems containing a large number of molecules or atoms. A fundamental explanation of these properties and related concepts, such as temperature, pressure or heat, from the microscopic perspective is given elsewhere by statistical physics, via mechanics and quantum mechanics.

2.1

Fundamentals

2.1.1

Thermodynamic Systems

A thermodynamic system is a macroscopic system with very many elementary constituents, e.g., atoms or molecules. Specific examples in dynamic meteorology are a fluid element or a material volume. The properties of a thermodynamic system are influenced to a large degree by its possibilities for exchange with the environment. • Heat transfer is the exchange of thermal energy with the environment. Thermal energy is the energy corresponding to the disordered motion, vibrations, and rotations of molecular components of a system, which again is closely linked to the temperature. Heat transfer typically leads to an equilibration between the temperatures of the thermodynamic system © Springer-Verlag GmbH Germany, part of Springer Nature 2022 U. Achatz, Atmospheric Dynamics, https://doi.org/10.1007/978-3-662-63941-2_2

35

36

2 Elementary Thermodynamics and Energetics of Dry Air

piston

gas

Fig. 2.1 Gas in a container with a movable lid. By moving the lid inwards, and thus compressing the gas, work is done at the gas.

and its environment. An environment being large enough for its temperature T to remain about constant is a heat reservoir HR(T ). • Energy exchange via work is possible if work is done at the thermodynamic system or if it does work at the environment. A typical example is gas in a container with a movable lid (Fig. 2.1). Expansion of the gas is only possible if work is done at the lid, accompanied by a loss of energy to the environment. In a compression of the gas, on the other hand, the lid does work at the gas so that it gains energy. • A further option is the exchange of matter with the environment. With regard to the possibilities of exchange one distinguishes between important classes of thermodynamic systems: • An isolated system does not exchange matter or energy with its environment. Examples are a gas in a rigid, thermally isolated, container, or the universe. • A closed system can exchange energy with the environment, but no matter. This is the standard case considered here: A fluid element has fixed mass. As soon, however, as diffusion is possible or as it contains trace gases which can be exchanged with the environment, it is an open system. • An open system can exchange both energy and matter with the environment.

2.1.2

Thermodynamic State and Thermodynamic Equilibrium

A thermodynamic system is described by macroscopic variables, the state variables, such as densityρ, pressure p, and temperature T . The state of a thermodynamic system is given

2.1

Fundamentals

37

Fig. 2.2 Gas is initially in a chamber which is separated from another chamber by a wall with a valve. If the valve is opened, part of the gas evades into the neighboring chamber. The reverse process is never observed. The process is thus irreversible.

by the values of a complete set of state variables, e.g., p and ρ in the case of dry air. These span the state space. A thermodynamic system not changing in time is in equilibrium. Experience tells us that isolated systems often tend toward an equilibrium state. The corresponding time scale is the relaxation time. In general there are the following important possibilities for changes of state: • A quasistatic change of state is so slow that the thermodynamic system can always be seen as being in an equilibrium state. The time scales are in such a case considerably longer than the relaxation time. In fact this is an assumption which will always be made here. Changes of state which are not quasistatic are considered in non-equilibrium thermodynamics, which is much more complex than the equilibrium thermodynamics discussed here. • A reversible change of state is possible in both directions. A film record of such a change of state, played in the reverse way, will appear to the observer as an equally realistic process. • An irreversible change of state is not reversible. A good example is the free evasion of a gas from a chamber (Fig. 2.2). • A cyclic process is a change of state in which all state variables have the same values in the beginning and at the end.

2.1.3

Temperature

Temperature will here simply be postulated as macroscopic quantity. In the end it is an expression of that energy of the air molecules which cannot be associated with the systematic movement of a fluid element. Corresponding contributions are the energy of the disordered kinetic movement of the molecules, as well as their rotational and vibrational energy. A fundamental explanation of temperature on the basis of these contributions is only possible within statistical physics.

38

2.1.4

2 Elementary Thermodynamics and Energetics of Dry Air

Equations of State

If Z 1 , Z 2 , . . . , Z n are the state variables of a thermodynamic system the equation of state is a relationship (2.1) f (Z 1 , Z 2 , . . . , Z n ) = 0 which can be solved for each of the Z i . The equation of state of air which will be used here rests on the assumption that air is an ideal gas. This holds if • the air molecules have a volume which is negligible in comparison with the volume of the fluid element, and • there are no interactions between the molecules other than elastic collisions. Especially near phase transitions these assumptions are not valid anymore. As long as we stay far enough away from this regime a description of air as ideal gas is well justified. The equation of state of a fluid element seen as ideal gas is pV = N k B T

(2.2)

Here V is the volume of the fluid element, N the number of its molecules, and k B = 1.3805 · 10−23 J /K the Boltzmann constant. Division by the mass M = N m, with the molecular mass m, yields pα = RT (2.3) where α = 1/ρ is the specific volume, and R=

N kB kB = M m

(2.4)

the gas constant. For dry air R = 287J /kgK .

2.1.5

Energy Change of a Thermodynamic System

As mentioned, one possibility for exchange with the environment is the exchange of energy. This can happen via exchange of heat or by work. We use the following notational convention for such processes: • Heat transfer: δQ > 0 δQ < 0

↔ ↔

the thermodynamic system receives the heat δ Q the thermodynamic system loses the heat −δ Q

2.1

Fundamentals

39

f

Fig. 2.3 Material volume with a surface element dS and the oppositely directed force df = − pdS, caused by the external or internal pressure. A shift by δr causes a volume change dδV = dS · δr.

• Work: δW > 0

↔

δW < 0

↔

the work δW is done at the thermodynamic system so that its energy increases the thermodynamic system does the work −δW , thus losing energy.

The only work possible in the case of dry air at rest and without friction is the volume work done in connection with an expansion or a compression. For its determination we consider a material volume, and on its surface an infinitesimally small surface element d S (Fig. 2.3). In the case of an outward movement of the surface element (expansion) work must be done against the external pressure, denoted by dδW < 0. In the opposite case of an inward movement (compression) the environment does work dδW > 0 against the internal pressure.1 The relevant pressure force acting onto the surface is df = − pdS. If the distance covered by the surface element is δr, one obtains dδW = df · δr = − pdS · δr = − pdδV

(2.5)

1 We assume that everything is continuous, i.e., at the surface external and internal pressure are

identical.

40

2 Elementary Thermodynamics and Energetics of Dry Air

where dδV = dS · δr is the change of volume (of the material volume) caused by the shift of the surface element. Summation, i.e., integration over all surface elements of the material volume finally yields δW = − pd V (2.6) The corresponding mass-specific formulation is obtained via division by the total mass M so that δw = − pdα (2.7) where δw = δW /M and α = V /M = 1/ρ. Here and henceforth we use the convention that changes of state variables Z i are denoted by d Z i , while for incremental quantities of any other property G we write δG. In a cyclic process the state variables eventually assume the same values as in the beginning, i.e.,  (2.8) d Zi = 0 Such an identity, however, does not hold for volume work. In Fig. 2.4 we show a cyclic change of state of a gas. The work balance is

p

p1 –δW

p2

V1

V2

V

Fig. 2.4 Illustration of a possible cyclic change of state of a gas in state space by a p-V -diagram. The volume work is the negative of the area enclosed by the trajectory.

2.2 The Fundamental Laws of Thermodynamics



 δW = −

V2 V1

41

 p(T , V )d V −

V1

p(V , T )d V  = 0

(2.9)

V2

Since the transitions between the states ( p1 , V1 ) and ( p2 , V2 ) are conceivable at various temperatures, enabling various trajectories, the work as the negative of the enclosed area is not zero. Thus work cannot be a state variable.

2.1.6

Summary

The equations of motion derived above do not offer a closed description of the dynamics of the dry atmosphere. Pressure is still undetermined. For this purpose we need a prognostic equation for temperature which is the main goal of this chapter. • We consider a material volume as thermodynamic system with a thermal energy, which corresponds to the disordered motion of its molecular constituents, and their vibrations and rotations. As such it interacts with its environment via heat transfer and active or passive work. Exchange of matter is only possible if it contains tracer constituents, e.g., water vapor. This, however, will not be considered here. Correspondingly one discriminates between isolated, closed, and open systems. In the present context material volumes are either isolated (no exchange of heat or matter) or closed (heat transfer, but no exchange of matter). • A thermodynamic system is described by state variables. The most important are pressure, density, and temperature, of which only two are necessary for a complete description since they are connected by the equation of state. An important class of change of state are quasistatic changes of state, in which a system is always in equilibrium. As important is the concept of a reversible change of state, which are also conceivable with time reversed. Cyclic changes of state have identical initial and final states. • As long as a thermodynamic system is at rest, and not affected by external forces, e.g., friction and gravitation, its thermal energy can be modified by only two exchange processes: Heat transfer and volume work.

2.2

The Fundamental Laws of Thermodynamics

A central element of thermodynamics is its fundamental laws. The most important of them will be covered here. The first law of thermodynamics states the conservation of the energy of the total system comprised by the thermodynamic system and its environment. The second law expresses the experience that nature does not order itself. In the course of the discussion of these two central laws two further state variables will be introduced: internal energy and entropy.

42

2.2.1

2 Elementary Thermodynamics and Energetics of Dry Air

The First Law of Thermodynamics and Internal Energy

The Internal Energy The internal energy U of a thermodynamic system denotes its total energy content, after subtraction of the kinetic energy residing in the mean motion of the system and after subtraction of any potential energy in an external force field. The internal energy of an ideal gas contains the energy in the unordered kinetic motion of the atoms or molecules, as well as the energy in the oscillations and rotations of the molecular components. We assume here, as always, that the atomic or molecular components do not have any electric or magnetic moments whose orientation in an ambient electromagnetic field also would be associated with a specific energy content. It is clear that each system has a unique internal energy. It is thus a state variable so that  dU = 0 (2.10) A result of statistical physics is that the internal energy of an ideal gas depends exclusively on its temperature, thus U = U (T ) (2.11) For a gas of mono-atomic constituents one has 3 N kB T 2 3 u = RT 2

U =

(2.12) (2.13)

Here u = U /M is the mass-specific internal energy. The internal energy of a gas of diatomic molecules, such as dry air, is 5 U = N kB T 2 5 u = RT 2

(2.14) (2.15)

The difference to the mono-atomic gas results from the fact that the molecules can rotate about two different axes, with a corresponding rotational energy. The internal energy per molecule is k B T /2 per degree of freedom, where the motion in each spatial direction (three in total) and the rotation about each axis (two in total) counts as one degree of freedom each. The First Law of Thermodynamics The first law of thermodynamics states that the total energy of a thermodynamic system and its environment is conserved. Thus the internal energy of a thermodynamic system at rest, i.e., neither with kinetic energy in any mean motion nor being displaced in an external force field, can only be changed by an exchange of a corresponding energy amount with

2.2 The Fundamental Laws of Thermodynamics

43

the environment. The energy exchange of a closed thermodynamic system at rest is only possible via exchange of heat with the environment, or by work being done at or by it. The first law for a closed thermodynamic system at rest thus is dU = δ Q + δW

(2.16)

The mass-specific variant is obtained by division by the total mass: du = δq + δw

(2.17)

Here q = Q/M is the mass-specific variant of heat.

2.2.2

The Heat Capacities of an Ideal Gas

From the first law of thermodynamics one can determine the characteristic heat capacities of air. In general heat capacities denote by which temperature change dT a thermodynamic system responds to a differential heat exchange δ Q. The heat capacity under a fixed set x of state variables is   δQ Cx = (2.18) dT x and the corresponding specific heat capacity  cx =

δq dT

 (2.19) x

From the first law follows δ Q = dU − δW = dU + pd V

(2.20)

with the mass-specific analogue δq = du + pdα

(2.21)

Inserting (2.20) into (2.18) yields for an ideal gas     ∂U ∂V Cx = +p ∂T x ∂T x

(2.22)

In analogy (2.21) and (2.19) yield  cx =

∂u ∂T



 +p x

∂α ∂T

 (2.23) x

44

2 Elementary Thermodynamics and Energetics of Dry Air

The heat capacity at constant volume is the one for x = V . Since trivially (∂ V /∂ T )V = (∂α/∂ T )V = 0 one has   ∂U (2.24) CV = ∂T V 

and cV =

∂u ∂T

 (2.25) V

so that be means of (2.14) and (2.15) 5 CV = N k B 2 5 cV = R 2

(2.26) (2.27)

This also implies that U = CV T

(2.28)

u = cV T

(2.29)

The heat capacity at constant pressure is obtained in the case x = p. From the equation of state one obtains V = N k B T / p and α = RT / p so that C p = CV + N k B

(2.30)

c p = cV + R

(2.31)

and with (2.26) and (2.27) 7 C p = N kB 2 7 cp = R 2

2.2.3

(2.32) (2.33)

Adiabatic and Isothermal Changes of State of an Ideal Gas

Among others, the first law admits two special types of change of state, both implying the exertion of work. In an adiabatic change of state no heat is exchanged with the environment, i.e., δQ = 0 (2.34)

2.2 The Fundamental Laws of Thermodynamics

45

The trajectory in state space corresponding to such a change of state can be obtained in the case of an ideal gas via the first law from dU dTad = − pd Vad dT

(2.35)

Here, due to the equation of state and (2.30), p=

C p − CV N kB T = T V V

(2.36)

Furthermore one has because of (2.28) dU /dT = C V so that C p − C V d Vad dTad =− T CV V

(2.37)

(d ln T )ad = −(γ − 1)(d ln V )ad

(2.38)

or with γ =

Cp cp = CV cV

(2.39)

For dry air one obtains from (2.26), (2.27), (2.32), and (2.33) γ = 7/5. Hence one finds (d ln T V γ −1 )ad = 0

(2.40)

Thus in the case of an adiabatic change of state   d T V γ −1 = 0   d pV γ = 0   d T γ p 1−γ = 0

(2.41) (2.42) (2.43)

Here the two last identities follow from the first via the equation of state. A curve in state space satisfying (2.41) – (2.43) is called an adiabat. Division of the two first relationships by M γ −1 or M γ , respectively, yields the mass-specific analogues   d T α γ −1 = 0   d pα γ = 0   d T γ p 1−γ = 0

(2.44) (2.45) (2.46)

An isothermal change of state proceeds at constant temperature, i.e., dT = 0

(2.47)

46

2 Elementary Thermodynamics and Energetics of Dry Air

Fig. 2.5 Adiabat and an isotherme in the p-V diagram.

p adiabat p

V–γ

isotherme p

V –1

V In the case of an ideal gas this means that the internal energy does not change, i.e. dU = 0

(2.48)

From the equation of state follows, due to dT = 0, d ( pV ) = 0

(2.49)

d ( pα) = 0

(2.50)

Corresponding curves in state space are called isothermes. An adiabat and an isotherme in the p-V diagram are shown in Fig. 2.5. The adiabat is steeper because γ > 1.

2.2.4

The Second Law of Thermodynamics

The second law of thermodynamics expresses the experience that certain thermodynamic processes are never observed although they are not in contradiction to the first law of thermodynamics. This is related to the fact that the internal energy of a thermodynamic system is an expression of unordered motion and that thermodynamic systems do not order themselves. • Even though conceivable from the energy perspective, it is never observed that all molecular components of a fluid element transfer the energy from their kinetic motion so into ordered motion that the fluid element cools down and simultaneously begins to move (together with its environment so that momentum conservation is respected). • Via friction it is possible that the kinetic energy of a fluid element is transferred into heat, i.e., the internal energy of the fluid element and its environment is increased, and

2.2 The Fundamental Laws of Thermodynamics

47

correspondingly the energy of the ordered motion of the fluid element is transferred into energy in the unordered motion. The reverse process, however, is never observed. Similar considerations lead to the conclusion that it is impossible to construct a perpetuum mobile of the second kind. This would be a cyclic engine doing nothing else than exerting work while an amount of heat Q is extracted from a single heat reservoir. Such an engine would not get warm, in contrast to our experience with typical engines. This leads us to the formulation of the second law according to Kelvin: Second law according to Kelvin: There is no perpetuum mobile of the second kind

In addition, let us consider two heat reservoirs at different temperatures. A process, in which the cooler of the two loses heat to the warmer one is allowed by the first law. Nonetheless, it is never observed. Nature tends to an equilibration of the distribution of its disorder. This leads to the formulation of the second law according to Clausius: Second law according to Clausius: There is no cyclic engine which does nothing else

than extracting heat from a cooler heat reservoir and transferring it to a warmer heat reservoir.

As will be shown below the two formulations of the second law are equivalent. First, however, let us define a heat engine. This is an engine performing a cyclic process at a thermodynamic work substance (e.g., an ideal gas) between two heat reservoirs HR(T1 ) and HR(T2 ) with T1 > T2 in which the following happens: • Heat is taken from the heat reservoir HR(T1 ), i.e., the corresponding heat exchange is Q 1 > 0. • The work substance does work. The work balance is thus W < 0. • The heat reservoir HR(T2 ) receives heat, i.e., the corresponding heat exchange is Q 2 < 0. Since we consider a cyclic process we have U = 0. With the help of the first law this means that 0 = W + Q 1 + Q 2 . Therefore we have necessarily |Q 2 | < Q 1 . The efficiency of the heat engine is W η=− (2.51) Q 1

48

2 Elementary Thermodynamics and Energetics of Dry Air

It tells us how much work is done per unit of energy put into the engine. The reverse of a heat engine, with all signs taken to the negative is a heat pump, as at work in refrigerators. The Equivalence Between the Two Formulations of the Second Law According to Clausius and Kelvin In the following we will show that the two formulations of the second law according to Kelvin and Clausius are equivalent. For this purpose we will assume that one of the two formulations is wrong and deduce from this a contradiction to the other formulation. Thus either both formulations must be correct or both must be wrong. Assumption: The formulation according to Clausius is wrong, but the one according to Kelvin correct In this case one could imagine a perfect heat pump, denoted by (a) which does the following: • The warmer heat reservoir HR(T1 ) receives the heat Q a1 < 0. • No work is done, i.e., W a = 0. • The heat Q a2 > 0 is extracted from the cooler heat reservoir HR (T2 ). Due to the first law Q a2 = −Q a1 . Nothing else happens. Now we couple this heat pump to a second heat engine, denoted by (b), which does the following: • Exactly the heat Q b1 = −Q a1 is extracted from the warmer heat reservoir HR(T1 ) that has been transferred to it by the perfect heat pump (a). • The work W b < 0 is done. • The cooler heat reservoir HR(T2 ) receives the heat Q b2 < 0. Due to the first law Q b2 = −Q b1 − W b . Consider now the net effect of the coupled engine: • The total heat exchange with the warmest heat reservoir HR(T1 ) is Q 1 = Q a1 + Q b1 = 0 The heat reservoir thus neither receives heat nor loses any. • The work done is W = W a + W b = W b < 0 • The total heat exchange with the cooler heat reservoir HR(T2 ) is Q 2 = Q a2 + Q b2 = −Q a1 − Q b1 − W b = −W b > 0 The heat reservoir thus loses heat.

2.2 The Fundamental Laws of Thermodynamics

49

In total the coupled engine does nothing else but extracting heat from HR(T2 ) and using the corresponding energy for doing work. However, this is a contradiction to the formulation according to Kelvin. Thus the formulation according to Clausius must be correct if the one according to Kelvin is correct. Assumption: The formulation according to Kelvin is wrong, but the one according to Clausius correct In this case there would be a heat engine, denoted by (a), doing the following: • The warmer heat reservoir HR(T1 ) does not experience any heat transfer. Thus Q a1 = 0. • Work is done, i.e., W a < 0. • The heat Q a2 > 0 is extracted from the cooler heat reservoir HR(T2 ). Due to the first law Q a2 = −W a . Nothing else happens. Now we couple this perpetuum mobile of the second kind to an engine, denoted by (b), which transforms the work energy obtained by engine (a) into heat which is transferred to the warmer heat reservoir. It hence does the following: • The warmer heat reservoir HR(T1 ) receives the heat Q b1 = W a . • The work W b = −W a < 0 is done. • The cooler heat reservoir HR(T2 ) does not experience any heat transfer, i.e., Q b2 = 0. Consider now the net effect of this coupled engine: • The total heat exchange with the warmer heat engine HR(T1 ) is Q 1 = Q a1 + Q b1 = W a < 0 The heat reservoir thus receives heat. • The work done is W = W a + W b = 0 Thus no work is done. • The total heat exchange with the cooler heat reservoir HR(T2 ) is Q 2 = Q a2 + Q b2 = −W a > 0 Thus heat is extracted from the heat reservoir.

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2 Elementary Thermodynamics and Energetics of Dry Air

In total the coupled engine does nothing else but extracting heat from HR(T2 ) and transferring it to HR(T1 ). However, this is a contradiction to the formulation according to Clausius. Thus the formulation according to Kelvin must be correct if the one according to Clausius is correct. The equivalence between the two formulations has thus been proven.

2.2.5

The Carnot Process

For the use of the second law in the introduction of the entropy a special type of heat engine is especially useful. This is the Carnot engine which performs a reversible cyclic process consisting of two isothermes, with the two heat reservoirs at the temperatures T1 and T2 , and two adiabats. This Carnot process is illustrated in Fig. 2.6 in the p-V diagram. It performs the following work steps: a → b: From a to b the work substance is compressed adiabatically. The corresponding temperature change is T = T1 − T2 > 0. b → c: This is followed by an isothermal expansion from b to c. During this work step the heat Q 1 is received from HR(T1 ). c → d: From c to d follows an adiabatic expansion. The temperature change is now T = T2 − T1 < 0. d → a: The cycle is completed by an isothermal compression from d to a. This is accompanied by a transfer of heat Q 2 to HR(T2 ).

p T1

ΔQ1 > 0

b

c

T1

T2 d

a

C

ΔQ2 < 0 V b Va

Vc

Vd

ΔQ1

V

T2

ΔW

ΔQ2

Fig. 2.6 Carnot process in the p-V diagram (left) with two isothermal and two adiabatic changes of state. To the right is the corresponding symbolic representation. Further details are given in the main text.

2.2 The Fundamental Laws of Thermodynamics

Due to the first law one has

51

 dU = Q 1 + Q 2 + W

0=

(2.52)

so that W = −Q 1 − Q 2

(2.53)

Inserting this into (2.51) yields the efficiency η =1+

Q 2 Q 1

(2.54)

One has Q 1 > 0 and Q 2 < 0. Thus η < 1. Moreover one concludes from W < 0 and (2.51) that |Q 2 | < Q 1 and thus 0 < η < 1. Certainly there also is the reverse of a Carnot engine, a corresponding heat pump. Its symbolic representation is given in Fig. 2.7. Finally we determine the efficiency of a Carnot engine for the case that the work substance is an inviscid ideal gas at rest. First, we determine the total work W = Wab + Wbc + Wcd + Wda

(2.55)

For the separate parts one obtains the following results: a → b: Along the adiabats one has Q ab = 0. With the first law follows Wab = Uab and thus for an ideal gas (2.56) Wab = C V (T1 − T2 ) b → c: On the isothermes the temperature is constant. By the equation of state of an ideal gas the volume work is

Fig. 2.7 Symbolic representation of a reverse Carnot engine (heat pump).

T1 ΔQ1 ΔW

C ΔQ2 T2

52

2 Elementary Thermodynamics and Energetics of Dry Air

 Wbc = −

Vc



Vc

pd V = −N k B T1

Vb

Vb

dV = −N k B T1 ln V



Vc Vb

 (2.57)

c → d: Along these adiabats one also has Wcd = C V (T2 − T1 )

(2.58)

d → a: In analogy to the other isotherme one has  Wda = −N k B T2 ln

Va Vd

 (2.59)

Moreover, due to (2.41) one has γ −1

T2 Va

γ −1 T1 Vc

so that

γ −1

= T1 Vb =

γ −1 T2 Vd

Vb Va = Vd Vc

Using (2.56) – (2.59) and (2.62) in (2.55) yields together     Vc Va − N k B T2 ln W = −N k B T1 ln Vb Vb   Vc = −N k B (T1 − T2 ) ln Ti is the same for all engines. It is important that each engine is constructed such that the heat exchange δ Q i is processed directly, i.e., for the Carnot engine the heat exchange is δ Q ci = −δ Q i

(2.68)

Symbolically this is represented again in Fig. 2.9. Since δ Q ci can be both positive or negative each machine can be both a heat engine or a heat pump. For each of the Carnot engines one has moreover, with (2.66), (0) δ Q ci δ Q ci + =0 (2.69) T0 Ti From this and (2.68) one obtains (0)

δ Q ci = −

T0 T0 δ Q ci = δ Q i Ti Ti

(2.70)

The efficiency of the Carnot engine is with (2.65)

δQ1

δQn

δQ2 T1

T2

...

Tn

Fig. 2.8 Symbolic representation of the subdivision of a quasistatic cyclic process in infinitesimally small parts i at a temperature Ti each, with the corresponding heat exchange δ Q i .

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2 Elementary Thermodynamics and Energetics of Dry Air

δQ1

T1

δQc1 C T0

δQ2

T2

δQc2 δWc1

δQn

δWc2

C

...

C

(0) δQc2

(0) δQc1

Tn

δQcn

δWcn

(0) δQcn

Fig. 2.9 Symbolic representation of the subdivision of an arbitrary quasistatic cyclic process in infinitesimally small parts, each at a temperature Ti with corresponding heat exchange δ Q i , and each coupled to a Carnot engine processing δ Q i directly. The warmer heat reservoir of the Carnot engines, at the temperature T0 > Ti , is everywhere the same.

ηci = 1 −

Ti T0

(2.71)

The work done in the Carnot process is with (2.70)     Ti T0 T0 (0) δ Qi δWci = −ηci δ Q ci = − 1 − δ Qi = 1 − T0 Ti Ti

(2.72)

so that the total work, done by all Carnot engines, is    T0 1− δ Qi δWci = Wc = Ti i

(2.73)

i

Now consider the work and heat balance of the total system, composed from the cyclic process and the Carnot engines. The heat exchange with the heat reservoir HR(T0 ) is because of (2.70) Q (0) =



(0)

δ Q ci = T0

i

 δ Qi

(2.74)

Ti

i

The work balance is, using (2.67)and (2.73), W = W K + Wc = −

 i

δ Qi +

 i

1−

T0 Ti

 δ Q i = −T0

 δ Qi i

Ti

(2.75)

2.2 The Fundamental Laws of Thermodynamics

55

Nothing else has happened! Following the second law one thus has W ≥ 0, i.e.,  δ Qi i

Ti

≤0

(2.76)

If K is reversible the direction of all energy fluxes can be reversed. Thus one then has  δ Qi i

Ti

=0

(2.77)

The infinitesimal limit finally yields the Clausius inequality 

δQ ≤0 T

(2.78)

in which the equality only holds for a reversible cyclic process. These findings enable the introduction of a new state variable, the entropy. Let A be a considered state, and A0 a reference state. The entropy S(A) is then defined as 

δ Q r ev T

A

S(A) = S(A0 ) +

A0

(2.79)

where the change of state along which the integration is done shall, as denoted, be reversible. Due to the Clausius equality for reversible processes this is a well-posed definition. In order to understand this we consider two reversible changes of state between A and A0 , as in Fig. 2.10. Since they are reversible, the direction of one of them can be changed. This yields a reversible cyclic process. The Clausius equality (2.78) therefore yields  0=

A A0

δ Q r1ev − T



A A0

δ Q r2ev T

(2.80)

The choice of the specific reversible change of state for the calculation of S(A) from S(A0 ) via (2.79) thus has no influence on the result. Because of the freedom of the (once to be fixed) reference state, entropy is no absolute quantity. Usually it is defined differentially: dS =

δ Q r ev T

(2.81)

Moreover one can show that an arbitrary, not necessarily reversible, change of state Z between the states A1 and A2 satisfies the inequality  S(A2 ) − S(A1 ) ≥

A2 A1

δQ T

(2.82)

56 Fig. 2.10 Two different reversible changes of state for the calculation of the entropy difference between a state A and a reference state A0 .

2 Elementary Thermodynamics and Energetics of Dry Air

p A

δQ1rev

δQ2rev

A0

V Fig. 2.11 Cyclic process, consisting of a not necessarily reversible change of state Z between the state A1 and A2 , and a reversible change of state R, transforming A2 again into A1 .

Z

A2

A1 reversible process (R) For this we consider a cyclic process, consisting of Z and a reversible change of state R, the latter transforming A2 again into A1 (Fig. 2.11). From the Clausius inequality follows 

A1 A2

δQ + T



A2 A1

δQ ≤0 T

(2.83)

The first integral is the entropy difference S(A1 ) − S(A2 ). Thus follows (2.82). The differential formulation of this is the mathematical formulation of the second law: dS ≥

δQ T

(2.84)

2.2 The Fundamental Laws of Thermodynamics

57

Equality only holds for reversible processes. For isolated systems follows from this directly: dS ≥ 0

(2.85)

This means that in isolated systems • entropy is always increased by irreversible processes, and • entropy is maximal in an equilibrium state. Combining the second law (2.84) with the first (2.16) for closed systems finally yields the fundamental relationship of the thermodynamics of closed systems T d S ≥ dU − δW

2.2.7

(2.86)

Entropy and Potential Temperature of Dry Air

Adiabatic changes of state as in Fig. 2.1 can always be done reversibly, provided they are sufficiently slow. Since along an adiabat δ Q = 0 adiabats are curves of constant entropy. For calculating the entropy difference between two states one can thus look for the adiabats on which the two states are. Provided that one can determine the entropy difference between the two adiabats one knows the entropy difference between the two states. This can, e.g., be done by considering an isothermal change of state between the two adiabats (Fig. 2.12). Isothermal changes of state are as well always possible in a reversible manner, so that with the first law dS =

p δQ = dV T T

(2.87)

Using the equation of state one obtains from this dV V

d S = N kB

(2.88)

Since on the isothermes d( pV ) = 0 one also has d S = −N k B

dp p

(2.89)

For the specific entropy s = S/M one obtains ds = −R

dp p

(2.90)

Integration of (2.89) or (2.90) yields the desired entropy difference. What remains to be determined, however, is the pressure at the integration limits.

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2 Elementary Thermodynamics and Energetics of Dry Air

A1 p T

A2

S2 S1

V Fig. 2.12 Isothermal change of state between two adiabats, used for the determination of the entropy difference between two states A1 and A2 on the adiabats.

One is relieved from searching the adiabats by using the potential temperature. The definition of this state variable uses the fact that, due to (2.43), along an adiabat the product T p (1−γ )/γ = T p −R/c p is constant. The potential temperature  θ=T

p00 p

 R/c p (2.91)

thus also is everywhere the same on the adiabat. Here p00 is an arbitrary reference pressure. The potential temperature can be interpreted easily: On an adiabat one has  T ( p) = θ

p p00

 R/c p (2.92)

Therefore θ is the temperature assumed by an element of an ideal gas in an adiabatic change of state where its pressure p is transformed to p00 . Here it is also important that θ and s are constant on an adiabat. One might thus guess that θ = θ (s)

(2.93)

Indeed: The change of potential temperature under an isothermal change of state between two adiabats is by definition

2.2 The Fundamental Laws of Thermodynamics

R dθ = T cp



p00 p

 R −1   cp p00 R dp − 2 dp = − θ p cp p

59

(2.94)

A comparison with (2.90) yields dθ θ dθ dS = Cp θ ds = c p

(2.95)

where the second relationship follows from the first via multiplication by the mass M. Thus there is a monotonic, and thus invertible, relationship between potential temperature and entropy. The entropy difference between two states can easily be determined from this as   θ2 (2.96) s2 − s1 = c p ln θ1 Potential temperature can thus also be seen as entropy of the meteorologist.

2.2.8

Summary

A central element of thermodynamics are its two fundamental laws. The first law states that the sum of the energy of a thermodynamic system and the energy of its environment is conserved. The second law expresses the experience that nature does not order itself. In this framework we have also introduced two further state variables: Internal energy and entropy. • The internal energy of a thermodynamic system is its total energy content, up to the kinetic energy residing in its mean motion and its potential energy in the gravitational field. The internal energy of an ideal gas thus is constituted by the energy of the disordered kinetic motion of its molecular constituents, and the energy of their vibrations and rotations. It is a state variable and only depends on temperature. • The first law of thermodynamics states that the internal energy of a thermodynamic system can only change by exchange with the environment. For a closed system this means that either heat must be exchanged or work must be done. • Via the first law, the equation of state, and the temperature dependence of the internal energy one can determine the heat capacities at constant volume or fixed pressure. By the same one can also derive the state-space curves for isothermal or adiabatic changes of state. • The second law expresses the experience that thermodynamic systems do not order themselves. The two formulations by Clausius and Kelvin are equivalent. • The properties of the Carnot cycle are an important ingredient in the derivation of the Clausius inequality, by which in an arbitrary cyclic process the heat gain, weighted

60

2 Elementary Thermodynamics and Energetics of Dry Air

inversely by the temperature, must always be negative. Only in the case of reversible processes the corresponding integral vanishes. • This allows the introduction of entropy as further state variable. Its change is defined as the integrand of the integral above, under the condition that the process is reversible. The famous statement that entropy always increases only holds for isolated systems as e.g., the universe. In dry air entropy is a direct function of potential temperature. Entropy changes can be determined from potential-temperature changes.

2.3

The Prognostic Equations for Temperature and Entropy in Dry Air

After the extensive preparations in this chapter we now want to develop the prognostic equation of thermodynamics by which the momentum equation and the continuity equation must be supplemented so that one has, together with the equation of state, a closed system of equations. We begin with the prediction of temperature and after this address the prediction of entropy.

2.3.1

Prediction of Temperature

In the discussion so far the motion of a material volume of a fluid element has been neglected. We will now drop this assumption. The total energy of a material volume, up to the potential energy connected with the volume forces, thus is E = K +U

(2.97)

Here the kinetic energy of a fluid element is dK = dM

|v|2 2

(2.98)

where d M = ρd V is its mass. Summation (integration) over all fluid elements yields  K =

dV ρ

|v|2 2

(2.99)

The internal energy of a fluid element is dU = d M u = d V ρcV T and hence after integration

(2.100)

 U=

d V ρcV T

(2.101)

2.3 The Prognostic Equations for Temperature and Entropy in Dry Air

One therefore has



 E=

dV ρ

|v|2 + cV T 2

61

(2.102)

The formulation of the first law under consideration of the kinetic energy now states that the total energy of a material volume can only change via heat exchange or work, including the work done by the volume forces: δW δQ DE = + Dt dt dt

(2.103)

Consider first the heat balance. A possibility of transferring heat from or to a material volume is cooling or heating it internally. In the atmosphere this can be done via radiation (solar or terrestrial) or latent heat. The latter is released if water vapor condenses. It is extracted by moisture from air when it evaporates. The corresponding heat per mass is given by the specific heat q. A second possibility for heat transfer is heat conduction. Typically local temperature differences lead to a temperature equilibration which, as in the case of friction, is done by collisions between the molecular components of air. This is described by a heat flux Fq . Fig. 2.13 shows a typical situation. It makes sense to assume that heat flux and temperature gradients have opposite directions, i.e., Fq = −κ∇T

Fig. 2.13 Molecular heat flux Fq which is opposite to the local temperature gradient.

(2.104)

T3 > T1

T3 T2 T1

Fq dS

fluid element

62

2 Elementary Thermodynamics and Energetics of Dry Air

where the coefficient of proportionality κ is the molecular heat conductivity. Actually this ansatz works very well. Obviously heat conduction is as well an irreversible process as friction. In summary one has, using the integral theorem from Gauss,     δQ (2.105) = d V ρq − dS · Fq = d V ρq − d V ∇ · Fq dt The minus sign of the second part results from the fact that an outwardly directed heat flux leads to a loss of heat. Now consider the work δW /dt done per time at or by the material volume. The possibly contributing forces are the volume and surface forces. The work done within the time dt via the volume forces at or by a contributing fluid element results from a shift by the distance dr in a force field which is given, per volume, by f. For the work one obtains   1 δW = d V f · dr = d V f · v (2.106) d dt V dt Summation, i.e., integration, over all fluid elements yields    δW = dV f · v dt V

(2.107)

The work per time by the surface forces, again at a single fluid element, is in analogy       δW 1 d = − pd Si + σi j d S j · dri dt S dt     = − pd Si + σi j d S j vi (2.108) In the summation over all fluid elements one can make use of the fact that the surface normals from contact surfaces between neighboring fluid elements are oppositely directed, i.e., the respective integral contributions cancel each other (Fig. 2.14) so that only an integral over the outer surface remains. This is     δW = − vi pd Si + vi σi j d S j (2.109) dt S The sum of (2.107) and (2.109) yields    δW = d V v · f − pv · dS + vi σi j d S j dt

(2.110)

2.3 The Prognostic Equations for Temperature and Entropy in Dry Air

63

dS dS

Fig. 2.14 In the sum of the surface integrals over all fluid elements the contributions from the contact surfaces between neighboring fluid elements cancel each other since the corresponding surface normals are oppositely directed.

Summarizing (2.102), (2.105), (2.107), and (2.110), the first law becomes

    |v|2 D dV ρ d V v · f − pv · dS + vi σi j d S j + cV T = Dt 2   + d V ρq − d V ∇ · Fq The left-hand side of this equation can also be written



 |v|2 D |v|2 DT D dV ρ + cV T = d V ρ + cV Dt 2 Dt 2 Dt

(2.111)

(2.112)

Therein, the material derivative of the kinetic energy is, using the momentum equation (1.71), 

D |v|2 = dV ρ Dt 2



Dv d V ρv · = Dt

 d V v · [−∇ p + f + ∇ · σ ]

(2.113)

The contribution of the pressure gradient is, using the integral theorem from Gauss,    − d V v · ∇ p = − d V ∇ · ( pv) + d V p∇ · v   = − dS · pv + d V p∇ · v (2.114)

64

2 Elementary Thermodynamics and Energetics of Dry Air

The friction contribution is written, once again using the same integral theorem,   ∂σi j d V v · (∇ · σ ) = d V vi ∂x j   ∂vi ∂ = dV (vi σi j ) − d V σi j ∂x j ∂x j   ∂vi = d S j vi σi j − d V σi j ∂x j

(2.115)

(2.112) – (2.115) yield together

    |v|2 D dV ρ + cV T = − pv · dS + d V p∇ · v + d V v · f Dt 2   ∂vi + vi σi j d S j − d V σi j ∂x j  DT + d V ρcV (2.116) Dt This we insert into the energy equation (2.111) and obtain    DT ∂vi d V ρcV − ρq + ∇ · Fq = 0 + p∇ · v − σi j Dt ∂x j

(2.117)

holding, however, for arbitrary control volumina. Therefore the integrand must vanish so that DT ∂vi + ρq − ∇ · Fq (2.118) + p∇ · v = σi j ρcV Dt ∂x j We proceed by obtaining from the equation of continuity (1.28) ∇ ·v =−

1 Dρ 1 Dα = ρ Dt α Dt

(2.119)

We thus finally obtain the thermodynamic energy equation cV

DT Dα 1 +p = + q − ∇ · Fq Dt Dt ρ

(2.120)

where we define the frictional heating as =

∂vi 1 σi j ρ ∂x j

(2.121)

In (2.120) the terms on the left-hand side denote, from left to right, the rate of change of the internal energy and the negative volume work. On the right-hand side one has frictional

2.3 The Prognostic Equations for Temperature and Entropy in Dry Air

65

heating, volume heating, and heating due to conduction. Frictional heating also is, using (1.70) and some further algebra, ⎡ ⎤  2 2   ∂v j ∂vi 2 ∂vk ∂vi ⎦ 1 η + − δi j +ζ = ⎣ (2.122) ρ 2 ∂x j ∂ xi 3 ∂ xk ∂ xi i, j

The viscosities are positive, and thus also the frictional heating. Finally we also note that due to the equation of state of an ideal gas (2.3) p

DT Dp Dα =R −α Dt Dt Dt

(2.123)

so that (2.120) can also be written as cp

DT Dp 1 −α = + q − ∇ · Fq Dt Dt ρ

(2.124)

This is also called the enthalpy equation, since c p T is the specific enthalpy of an ideal gas.

2.3.2

The Prediction of Entropy and Potential Temperature

On the basis of the enthalpy equation one can easily derive a prognostic equation for potential temperature. From the definition (2.91) of potential temperature and the equation of state (2.3) follows     θ Dθ DT 1 DT R 1 Dp Dp = cp (2.125) =θ − −α Dt T Dt c p p Dt cpT Dt Dt Thus

Dθ θ = Dt cpT

  1 + q − ∇ · Fq ρ

(2.126)

Due to (2.95) one also has c p Dθ Ds = Dt θ Dt so that the prognostic equation for entropy becomes Ds 1 = Dt T



1 + q − ∇ · Fq ρ

(2.127)

 (2.128)

66

2 Elementary Thermodynamics and Energetics of Dry Air

Potential temperature and specific entropy are thus conserved if • there is no friction, • there is no volume heating, and • there is no heat conduction. Frictional heating is always positive i.e., it always produces entropy.

2.3.3

The Equations in a Rotating Frame of Reference

The transformation into the rotating frame of reference of earth is trivial. The thermodynamic state variables ( p, ρ, T , S, θ ) are independent of the frame of reference. Therefore also the prognostic equations do not change in a rotating frame of reference.

2.3.4

Spherical Coordinates

The representation of the equations in spherical coordinates is also simple. As already in the equation of continuity one must use D ∂ u v ∂ ∂ ∂ = + + +w Dt ∂t r cos φ ∂λ r ∂φ ∂r

(2.129)

Nothing else changes.

2.3.5

Summary

A prognostic temperature equation is obtained by consideration of the energy balance of an arbitrary material volume: • The generalized first law demands that the sum of kinetic and internal energy can only change by heat exchange with the environment or work. • Possible contributors to the heat balance are external heat sources or sinks, as, e.g., by radiation processes, and heat conduction. • Total work is made up by volume work and frictional work. • Evaluation of the first law together with the momentum equation leads to the desired relationship which can be formulated either as thermodynamic energy equation or as enthalpy equation.

2.4

Potential Temperature and Static Stability

67

• From this we can directly derive a prognostic equation for potential temperature or entropy. The non-conservative terms appearing there are the external heat sources or sinks, heat conduction and frictional heating in the form of the dissipation rate.

2.4

Potential Temperature and Static Stability

Next to its relation to entropy, potential temperature is also decisive in the diagnostics of the stability of atmospheric stratification. This is discussed here together with the corresponding buoyancy oscillations.

2.4.1

Stable and Unstable Stratification

Consider a fluid element in an atmosphere at rest, with a thermodynamic state, defined, e.g., by θ (r ), ρ(r ), T (r ), p(r ), which only depends on the radial distance. Consider a thought experiment in which the fluid element is displaced slightly upwards (Fig. 2.15). The question we are interested in is how it responds to this displacement. Does it experience a force in its new position which drives it further upwards or is it driven back into its original position? In the first case we conclude that the atmosphere is unstable since slight displacements can lead to irreversible changes via convective motions. In the other case the atmosphere is stable with respect to vertical displacements: Slight changes are reversed immediately. Here we discuss this at the example of an upward displacement. This is, however, no limitation. The opposite example yields the same results. In the analysis we use two assumptions: • The pressure in a fluid element adjusts immediately to that of the environment. One can show that this means that the removal of pressure gradients via sound waves is so fast that it needs not be described explicitly.

Fig. 2.15 Fluid element is displaced slightly in the vertical. Does it react to this by a movement farther away from its original position (unstable stratification), or does it return there (stable stratification)?

r + dr

r

68

2 Elementary Thermodynamics and Energetics of Dry Air

• The fluid element shows an isentropic behavior; i.e., its potential temperature does not change. This means that there is neither an impact from friction, nor from volume heating, nor from heat exchange with the environment via conduction. A more general treatment shows that these assumptions do not imply serious limitations to the conclusions derived here. We now consider the density of a fluid element after a displacement from r to r + dr , with dr > 0. By the equation of state, it is ρ = p/RT , where by the first assumption p = p(r + dr )

(2.130)

and by the second, with (2.91),  T = θ (r )

p(r + dr ) p00

R/c p (2.131)

Thus the density after the displacement is ρ=

p(r + dr ) Rθ (r )



p00 p(r + dr )

R/c p (2.132)

This is to be compared to the environmental density at r + dr . The potential temperature of the environment is θ (r + dr ), and thus ρ(r + dr ) =

p(r + dr ) Rθ (r + dr )



p00 p(r + dr )

R/c p (2.133)

The configuration is stable if the fluid element is heavier than its environment; i.e., its density is larger than that of the environment. In the opposite case one has an instability. The stable (unstable) case implies with (2.132) and (2.133) stable ⇔ θ (r ) < θ (r + dr ) unstable ⇔ θ (r ) > θ (r + dr ) We thus obtain the following result: The atmosphere is stably stratified if dθ /dr > 0. In the case dθ /dr < 0 one has an unstable stratification.

2.4

Potential Temperature and Static Stability

69

In the case of hydrostatic stratification this statement can be reformulated in terms of the vertical temperature gradient. In general one can conclude from the definition of potential temperature (2.91) and the equation of state of an ideal gas (2.3)     1 1 dp p00 R/c p dT dθ (2.134) = − dr p¯ dr c p ρ dr In the hydrostatic case one has 1 dp = −g ρ dr and thus dθ = dr



p00 p¯

where =−

 R/c p 

dT − dr

(2.135) 

g ≈ −9.74K/km cp

(2.136)

(2.137)

is the adiabatic lapse rate, corresponding to dθ /dr = 0. We thus note: An atmosphere with hydrostatic stratification is stably stratified if dT /dr > . In the case dT /dr <  one has an unstable stratification.

2.4.2

Buoyancy Oscillations

The temporal behavior of a fluid element after an infinitesimally small displacement can be given further consideration. Its vertical acceleration is given by the momentum equation (1.118) which is in the absence of horizontal motion Dw 1 ∂p =− −g Dt ρ ∂r

(2.138)

where we assume to be radially at r +dr . The density is given by (2.132), where the pressure is the same as the environmental pressure. The density can also be written as ρ = ρ(r + dr ) + δρ

(2.139)

with |δρ| ρ(r + dr ) so that in the case of hydrostatic stratification Dw 1 ∂p 1 ∂p δρ ∂ p δρ =− −g≈− + 2 − g = −g Dt ρ(r + dr ) + δρ ∂r ρ ∂r ρ ρ ∂r On the other hand one has, due to (2.132) and (2.133)

(2.140)

70

2 Elementary Thermodynamics and Energetics of Dry Air



    p00 R/c p 1 1 p − Rρ p θ (r ) θ(r + dr ) r +dr 

   p00 R/c p 1 dθ 1 dθ dr = ≈ T 2 p θ dr θ dr

δρ = ρ

r +dr

dr

(2.141)

r +dr

where we have used (2.91) again. Since by definition w = Dr /Dt = Ddr /Dt, (2.140) and (2.141) yield together D 2 dr = −N 2 dr Dt 2

(2.142)

where N2 =

g dθ θ dr

(2.143)

is the squared Brunt-Väisälä frequency. It is obvious that in the stably stratified case N 2 > 0 so that a fluid element oscillates with the Brunt-Väisälä frequency about its equilibrium position, i.e., w (0) sin (N t) (2.144) dr (t) = dr (0) cos (N t) + N In the unstably stratified case N 2 < 0 and the fluid element moves away from its initial position with exponentially increasing distance: dr (t) = dr (0) cosh (|N | t) +

w (0) sinh (|N | t) |N |

(2.145)

Climatological results for the stratification and Brunt-Väisälä frequency of the zonally averaged atmosphere (i.e., averaged with respect to geographic longitude) are shown in Fig. 2.16. As was to be expected, the climatological stratification of the atmosphere is everywhere stable.

2.4.3

Summary

The vertical profile of potential temperature plays an important role in the stability of the atmosphere. • For a corresponding analysis one considers the response of a fluid element to an infinitesimally small vertical perturbation of its position. It is assumed that its pressure adjusts instantaneously to the environmental pressure, a process which happens in reality by fast acoustic waves, and that there is no heat exchange with the environment; i.e., the considered time scales are shorter than time scales characteristic for heat conduction, friction and heating.

2.5

Recommendations for Further Reading

18

71

1

18

z (km)

2

14

3

14

4

10

5

10

6

6

7

6 –50

0 latitude

50

–50

0 latitude

50

Fig. 2.16 Potential temperature (contours in K) and the corresponding squared Brunt-Väisälä frequency (color in 10−4 s−2 ) for January (left panel) and July (right), according to Birner et al. (2006).

• Stability is given if the fluid element returns after its displacement to its initial position. One finds that the atmosphere is stable if its vertical potential-temperature gradient is positive. In the opposite case it is unstable and convective motions set in. • In a hydrostatically stratified atmosphere stability means that the vertical temperature decrease must not be steeper than the adiabatic lapse rate. • In the stable case the fluid-element displacement leads to an oscillation with the BruntVäisälä frequency whose square is proportional to the vertical potential-temperature gradient.

2.5

Recommendations for Further Reading

The thermodynamics covered here is rather elementary, and would not suffice as basis for understanding radiative or cloud processes. Standard textbooks on theoretical physics, including statistical physics, offer a broader perspective, e.g., Huang (1963) and Nolting (2012, 2014). The connection to hydrodynamics is discussed, e.g., by Landau and Lifschitz (1987), and a glance into the book by Chandrasekhar (1981) is worthwhile as well. Again for all atmospheric aspects the reader is referred to the books from Holton and Hakim (2013), Pedlosky (1987), and Vallis (2006).

3

Elementary Properties and Applications of the Basic Equations

After the derivation of the basic equations of atmospheric dynamics (for dry air) in the two previous chapters, we here first discuss elementary properties, among these energy and angular-momentum conservation. Then the reasonable approximation of the primitive equations is introduced. Within this framework we finally consider simple balanced-flow configurations.

3.1

Summary of the Basic Equations

We first summarize the basic equations again. These are the vectorial momentum equation Dv 1 1 + 2 × v = − ∇ p − ∇ + ∇ · σ Dt ρ ρ

(3.1)

the equation of continuity Dρ + ρ∇ · v = 0 Dt

(3.2)

the thermodynamic equation in one of its three versions (internal-energy equation, enthalpy equation, or entropy equation) DT Dα 1 +p =  + q − ∇ · Fq Dt Dt ρ DT Dp 1 cp −α =  + q − ∇ · Fq Dt Dt ρ   1 Dθ θ  + q − ∇ · Fq = Dt cpT ρ

cV

© Springer-Verlag GmbH Germany, part of Springer Nature 2022 U. Achatz, Atmospheric Dynamics, https://doi.org/10.1007/978-3-662-63941-2_3

(3.3) (3.4) (3.5)

73

74

3 Elementary Properties and Applications of the Basic Equations

and the equation of state p = ρ RT

(3.6)

These equations are often called the Navier–Stokes equations. In the absence of friction and heat conduction they are termed the Euler equations.

3.2

The Importance of the Basic Equations for Weather Prediction

The basic equations described above are the primary tool for estimating the development of the dry atmosphere from a given initial state. In other words, they are the basis of all numerical weather prediction. The central question of numerical weather prediction is: If the state of the atmosphere at time t = t0 is known in terms of the spatial distributions of wind, density and temperature, how does it evolve in time? This question may be addressed in principle, neglecting all numerical issues, by dividing the forecast time interval into sufficiently small subintervals of duration t and approximating the state at later times using ⎛ ⎞ ⎛ ⎞ ⎞ v v v ⎝ ρ ⎠ (x, t0 + t) = ⎝ ρ ⎠ (x, t0 ) + t ∂ ⎝ ρ ⎠ (x, t0 ) ∂t T T T ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ v v v ∂ ⎝ ρ ⎠ (x, t0 + 2t) = ⎝ ρ ⎠ (x, t0 + t) + t ⎝ ρ ⎠ (x, t0 + t) ∂t T T T ... = ... ⎛

(3.7)

(3.8)

where the Eulerian time derivatives (∂v/∂t, ∂ρ/∂t, ∂ T /∂t) (x, t0 + nt) are calculated from the basic equations via the corresponding material derivatives. For example, from the momentum equation (3.1) and ∂v/∂t = Dv/Dt − (v · ∇) v, one obtains 1 1 ∂v = − ∇ p − ∇ + ∇ · σ − 2 × v − (v · ∇) v. ∂t ρ ρ

(3.9)

Pressure is calculated from density and temperature using the equation of state (2.3). Likewise the Eulerian time derivatives of density and temperature are obtained, respectively, from the continuity equation and either the thermodynamic energy equation or the enthalpy equation. One should be aware that the exact numerical procedure for the temporal integration is subject of an own field of research, but without the basic equations derived so far weather prediction would be impossible.

3.3

Conservation Laws

3.3

75

Conservation Laws

Central to the study of atmospheric dynamics is the identification of various conservation laws. Two examples are the continuity equation, which we have already seen and which expresses the conservation of mass, and the conservation of the potential vorticity, which will be introduced in the following chapter on vortex dynamics. The prognostic thermodynamic equations are closely related to the conservation of total energy, consisting of kinetic energy, potential energy and thermal internal energy. A further central property is the conservation of angular momentum. Here the last two conservation properties are to be discussed in more detail.

3.3.1

Conservation of Energy

For a re-consideration of the conservation of energy, already assumed in the derivation of the thermodynamic energy equation, we first take the scalar product of the momentum equation with ρv:   Dv ρv · + 2 × v = −v · ∇ p − ρv · ∇ + v · ∇σ . (3.10) Dt Using D v·v D |v|2 Dv = = Dt Dt 2 Dt 2 v · ( × v) = 0 v·

v · ∇ p = ∇ · ( pv) − p∇ · v

(3.11) (3.12) (3.13)

and (2.122) in the form v · ∇σ = vi

∂σi j ∂  vi σi j − ρε, = ∂x j ∂x j

(3.14)

the kinetic energy equation can be derived: ρ D 2 |v| = −∇ · ( pv) + p∇ · v − ρv · ∇ + ∇ · (vσ ) − ρε. 2 Dt

(3.15)

Multiplication of the thermodynamic energy equation (3.3) by the density ρ gives ρ

Di Dα = −ρ p + ρε + ρq − ∇ · Fq . Dt Dt

(3.16)

Herein one has, due to the continuity equation (3.2), Dα D 1 1 Dρ 1 = =− 2 = ∇ ·v Dt Dt ρ ρ Dt ρ

(3.17)

76

3 Elementary Properties and Applications of the Basic Equations

so that the internal-energy equation is obtained: ρ

Di = − p∇ · v + ρε + ρq − ∇ · Fq . Dt

(3.18)

Since the geopotential is assumed to be time-independent, the potential energy equation is ρ

D = ρv · ∇. Dt

(3.19)

Adding (3.15), (3.18), and (3.19) yields

or

ρ

De + ∇ · ( pv) − ∇ · (vσ ) + ∇ · Fq = ρq, Dt

(3.20)



  ∂ + v · ∇ e + ∇ · pv + Fq − vσ = ρq, ∂t

(3.21)

ρ where

v·v +i + (3.22) 2 is the total energy per unit mass. Multiplying e by the continuity equation (3.2), giving 

∂ρ + ∇ · (ρv) = 0, (3.23) e ∂t e=

and adding (3.21) and (3.23) finally yields the desired conservation of energy  

 ∂ p + Fq − vσ = ρq. (ρe) + ∇ · ρv e + ∂t ρ

(3.24)

To show that this implies that total energy is conserved, we integrate over the total volume of the atmosphere and use Gauss’ theorem, giving d dt



d V ρe +

V

S



  p + Fq − vσ = dS · ρv e + d V ρq. ρ V

(3.25)

Now, if • at all boundaries the normal component of the combined energy and pressure flux vanishes, i.e., dS · v (e + p/ρ) = 0, trivially satisfied at the ground, • the normal component of the viscous energy flux vσ vanishes at all solid boundaries, • there is no heat flux through the boundaries, i.e., dS · Fq = 0, typically trivially satisfied at the boundary to space, and • there is no volume heating, i.e., q = 0 everywhere,

3.3

Conservation Laws

77

then the total energy is conserved,1 i.e., d dt

d V ρe = 0.

(3.26)

V

Finally, note that the volume-specific total energy (the total energy per unit volume) ρe comprises the volume-specific kinetic energy ρ|v|2 /2, the volume-specific internal energy ρi and the volume-specific potential energy ρ. The reader might notice that in the derivation of the thermodynamic prognostic equations only the sum of kinetic and internal energy has been considered. This is, however, no contradiction since in the energy balance there also the work has been included which corresponds to motion in the gravitational field.

3.3.2

Conservation of Angular Momentum

Angular-momentum conservation is a central property of mechanics. The total angular momentum of the atmosphere is conserved if the effect of frictional forces is neglected and if there is no angular-momentum exchange between the atmosphere and the land surface through dynamic pressure (e.g., during flow over mountains), and between the atmosphere and the ocean through water waves. This ideal case is treated in this section. Furthermore the discussion is restricted to the angular-momentum component parallel to the axis of the earth. Calculation of the angular momentum requires, first of all, the velocity of a fluid element relative to a non-rotating inertial system, which we represent in spherical coordinates. As discussed above, this velocity is (3.27) vI = v +  × r where r = r er

(3.28)

is the position of the fluid element. Figure 1.13 illustrates the representation of the earth’s angular velocity  in spherical coordinates,  =  cos φ eφ +  sin φ er ,

(3.29)

  × r =  cos φ eφ +  sin φ er × r er = r cos φ eλ .

(3.30)

so that, using eφ × er = eλ ,

Inserting the result into (3.27) yields the desired form v I = (u + r cos φ) eλ + veφ + wer .

1 In the climatological mean an equilibrium is established between the three latter processes.

(3.31)

78

3 Elementary Properties and Applications of the Basic Equations

The angular momentum per unit mass is   M = r × v I = r er × (u + r cos φ) eλ + veφ + wer = r (u + r cos φ) eφ − r veλ (3.32) where we have used eλ × er = −eφ and eφ × er = eλ . The axial component is m = ez · M.

(3.33)

ez · eφ = cos φ,

(3.34)

m = r cos φ (u + r cos φ) .

(3.35)

It is obvious that and thus that The axial component of the angular momentum of a fluid element is d V ρm. Conservation of angular momentum means the sum of the angular momenta of all fluid elements is conserved, i.e., d d V ρm = 0. (3.36) dt V This is in fact the case: Using (1.107) and (3.35), one obtains   Dm ∂ u ∂ v ∂ ∂ r cos φ (u + r cos φ) = + + +w Dt ∂t r cos φ ∂λ r ∂φ ∂r = (−v sin φ + w cos φ) (u + r cos φ)   Du + r cos φ − v sin φ + w cos φ . Dt

(3.37)

Inserting Du/Dt from the zonal momentum equation of (1.116) gives the material derivative

which is equivalent to

1 ∂p Dm =− , Dt ρ ∂λ

(3.38)

∂m 1 ∂p + v · ∇m = − . ∂t ρ ∂λ

(3.39)

Using the continuity equation (3.2), this gives   ∂m ∂ρ 1 ∂p ∂ +m =ρ − − v · ∇m − m∇ · (ρv) (ρm) = ρ ∂t ∂t ∂t ρ ∂λ ∂p =− − ∇ · (ρmv) ∂λ so that

∂ ∂p (ρm) + ∇ · (ρmv) = − . ∂t ∂λ

(3.40)

(3.41)

3.3

Conservation Laws

79

Integration over the total volume of the atmosphere, assuming that the lower boundary is at r = a, i.e., neglecting the effects of mountains and water waves, gives the result:

∂p d d V ρm = − dS · ρmv − dV dt V ∂λ S V π/2

∞ 2π ∂p 2 = − dS · ρmv − drr dφ dλ (3.42) ∂λ a −π/2 0 S Thus if • • • •

there is no friction, there is no angular-momentum flux through the earth’s surface, i.e., dS · v = 0, the density of the air vanishes in outer space, i.e., ρ = 0 as r → ∞, and the surface of the earth is at r = a; i.e., there are no mountains,

then (3.36) holds because of the periodicity of pressure along a latitude circle. In reality there is angular-momentum exchange with the solid earth by viscous processes and orography so that the angular momentum of the atmosphere experiences weak fluctuations.

3.3.3

Summary

At the center of atmospheric dynamics there are various conservation properties. • The continuity equation expresses mass conservation. • The thermodynamic prognostic equations are an expression of the conservation of total energy of the atmosphere, constituted by kinetic, internal, and potential energy. Total energy is conserved if – there is no energy and pressure flux to space (and also not through the ground), – the viscous energy flux through the ground vanishes, as well as that to space, – there is no heat flux through the ground and to space, and – there is no volume heating. An equilibrium between the three latter processes is established in the climatological mean. • In the conservative case total angular momentum of the atmosphere is conserved as well. For its axial component this requires that – there is neither viscous angular-momentum flux through the ground – nor orographic angular-momentum exchange with the solid earth. In reality these two processes are active, so that total atmospheric angular momentum exhibits weak fluctuations about its climatological mean. • Potential-vorticity conservation will be introduced in the following chapter on vortex dynamics.

80

3.4

3 Elementary Properties and Applications of the Basic Equations

The Primitive Equations

In Sect. 1.6.2 it was shown that dynamical structures with considerably larger horizontal than vertical scale are in hydrostatic equilibrium. Nowadays the weather services use numerical models which also resolve very small horizontal scales. Even if in such models an assumption of hydrostatic equilibrium is not in place anymore, the correspondingly simplified primitive equations remain very useful. They are applied as well in climate modeling as in conceptual analyses of mechanisms in atmospheric dynamics. They can be derived from the general equations of motion via three approximations: Hydrostatics: The vertical momentum equation becomes 1/ρ ∂ p/∂r = −g. Shallow atmosphere: Set r = a +z where z is the vertical distance from the earth’s surface at sea level, and furthermore replace, except in ∂/∂r = ∂/∂z, everywhere r by a. This corresponds to a neglect of the deviations of the radial distance from the center of the earth from the earth’s radius. As long as |z|  a, this should be justified. It turns out, however, that by such an approximation and by the assumption of hydrostatic equilibrium the conservation of energy and angular momentum is violated (not shown). A corresponding remedy is brought about by the third approximation. Traditional approximation: In the zonal momentum equation (1.116) the product between vertical wind and the earth’s angular velocity is neglected. This corresponds to a neglect of the part of the angular velocity which is tangential to the local surface of the earth. In the horizontal momentum equations (1.116) and (1.117) the terms uw/r and vw/r are neglected as well. For reasons of simplicity we here also neglect friction and thermal conduction. This, however, is not really necessary. With the approximations above the momentum equation becomes Du uv 1 ∂p − fv− tan φ = − Dt a aρ cos φ ∂λ Dv u2 1 ∂p + fu + tan φ = − Dt a aρ ∂φ 1 ∂p −g 0=− ρ ∂z

(3.43) (3.44) (3.45)

Here the Coriolis parameter is f = 2 sin φ

(3.46)

3.4 The Primitive Equations

81

The material derivative of a vector component or a scalar is in the primitive approximation D ∂ u ∂ v ∂ ∂ = + + +w Dt ∂t a cos φ ∂λ a ∂φ ∂z

(3.47)

The horizontal momentum equations for u = ueλ + veφ can also be written: Du 1 + f × u = − ∇z p Dt ρ

(3.48)

Here f = f er 1 ∂p 1 ∂p ∇z p = eλ + eφ a cos φ ∂λ a ∂φ and we define Du = Dt



   Dv Du uv u2 − tan φ eλ + + tan φ eφ Dt a Dt a

(3.49) (3.50)

(3.51)

The equation of continuity in its two formulations ∂ρ + ∇ · (ρv) = 0 ∂t

Dρ + ρ∇ · v = 0 Dt

(3.52)

is evaluated with the help of the general divergence (for an arbitrary vector b) ∇ ·b=

∂br 1 ∂bλ 1 ∂  + bφ cos φ + a cos φ ∂λ a cos φ ∂φ ∂z

(3.53)

Finally, thermodynamics is represented by one of the three equations DT Dα +p =q Dt Dt DT Dp cp −α =q Dt Dt Dθ θq = Dt c p T

cV

(3.54) (3.55) (3.56)

and the equation of state remains p = ρ RT

(3.57)

82

3.5

3 Elementary Properties and Applications of the Basic Equations

The Primitive Equations in Pressure Coordinates

Hydrostatic equilibrium implies a monotonic dependence of pressure p on altitude z. Thus z can be replaced by p as vertical coordinate. This leads to simplifications in the equations which are useful in several regards. Since related coordinate transformations are also useful in other contexts, we first discuss the transformation onto an arbitrary new vertical coordinate.

3.5.1

Arbitrary Vertical Coordinates

Let ζ (e.g., p) be an arbitrary new vertical coordinate so that reversible relations ζ = ζ (λ, φ, z, t)

(3.58)

z = z(λ, φ, ζ, t)

(3.59)

exist. Then one has for any variable  (e.g., u, v, w, ρ, T etc.) (λ, φ, ζ, t) = [λ, φ, z(λ, , ζ, t), t]

(3.60)

Moreover, vertical derivatives can be calculated from each other via ∂ ∂ ∂ζ = ∂z ∂ζ ∂z ∂ ∂ ∂z = ∂ζ ∂z ∂ζ On also has

 

∂ ∂λ ∂ ∂φ

 

 ζ

ζ

=  =

∂ ∂λ ∂ ∂φ

 z

∂ ∂z

z

∂ + ∂z

+ 

(3.61) (3.62)

 

∂z ∂λ ∂z ∂φ

 (3.63) ζ



(3.64) ζ

so that a transformation of the horizontal gradients can be obtained by ∇z  = ∇ζ  −

∂ ∇ζ z ∂z

(3.65)

In analogy one obtains, by exchanging z and ζ in (3.63) and (3.64), ∇z  = ∇ζ  +

∂ ∇z ζ ∂ζ

(3.66)

3.5 The Primitive Equations in Pressure Coordinates

83

and for the time derivative one gets 

∂ ∂t



 = z

∂ ∂t

 ζ

+

∂ ∂ζ



∂ζ ∂t

 (3.67) z

Thus the material derivative becomes   D ∂ ∂ + u · ∇z  + w = Dt ∂t z ∂z 

    ∂ζ ∂ ∂ ∂ζ ∂ = + u · ∇ζ  + + u · ∇z ζ + w ∂t ζ ∂ζ ∂t z ∂z ∂ζ so that D = Dt Here ζ˙ =



∂ ∂t

Dζ = Dt



 ζ

∂ζ ∂t

+ u · ∇ζ  + ζ˙

∂ ∂ζ

 + u · ∇z ζ + w z

∂ζ ∂z

(3.68)

(3.69)

(3.70)

is the ζ -velocity which replaces w as vertical velocity. The Horizontal Momentum Equation By setting  = p in (3.65) we obtain, using hydrostatic equilibrium, ∇z p = ∇ζ p −

∂p ∇ζ z = ∇ζ p + ρ∇ζ  ∂z

(3.71)

where  = zg is the geopotential. Thus the horizontal momentum equation (3.48) becomes, remembering the definition (3.51), Du 1 + f × u = −∇ζ  − ∇ζ p Dt ρ

(3.72)

The Hydrostatic Equilibrium With  = p one obtains from (3.62) ∂ p ∂z ∂ ∂p = = −ρ ∂ζ ∂z ∂ζ ∂ζ

(3.73)

where again the hydrostatic equilibrium has been used. Thus ∂ ∂p = −α ∂ζ ∂ζ

(3.74)

84

3 Elementary Properties and Applications of the Basic Equations

The Equation of Continuity For a reformulation of the equation of continuity (3.52) we first write it as ∂w 1 Dρ + ∇z · u + =0 ρ Dt ∂z

(3.75)

Here, by definition, ∇z · u =

1 a cos φ



∂u ∂λ

 + z

1 ∂ (cos φv)z a cos φ ∂φ

(3.76)

which yields, employing (3.63) for  = u and (3.64) for  = cos φv,          ∂u ∂ 1 1 ∂u ∂z ∂v ∂z + − (cos φv)ζ − cos φ ∇z · u = a cos φ ∂λ ζ ∂z ∂λ ζ a cos φ ∂φ ∂z ∂φ ζ = ∇ζ · u −

∂u ∂ζ ∂u · ∇ζ z = ∇ζ · u − · ∇ζ z ∂z ∂z ∂ζ

Thus one has ∇z · u + Moreover

(3.77)

∂w ∂ζ ∂u ∂ζ ∂w = ∇ζ · u − · ∇ζ z + ∂z ∂z ∂ζ ∂z ∂ζ

Dz w= = Dt



∂z ∂t

 ζ

+ u · ∇ζ z + ζ˙

∂z ∂ζ

(3.78)

(3.79)

so that  ∂z ∂ ζ˙ ∂z ∂z ∂u ∂2z + u · ∇ζ + · ∇ζ z + + ζ˙ 2 ∂ζ ∂ζ ∂ζ ∂ζ ∂ζ ∂ζ   ∂u D ∂z ∂ ζ˙ ∂z + = · ∇ζ z + Dt ∂ζ ∂ζ ∂ζ ∂ζ

∂ ∂w = ∂ζ ∂t



(3.80)

This in (3.78) yields ∂w ∂ζ D = ∇ζ · u + ∇z · u + ∂z ∂z Dt



∂z ∂ζ

 +

∂ ζ˙ ∂ζ

(3.81)

since (∂ζ /∂z)(∂z/∂ζ ) = 1. Since for the same reasons, however, also     1 ∂ζ D ∂z D ∂z = ∂z Dt ∂ζ ∂z/∂ζ Dt ∂ζ

(3.82)

(3.81) in (3.75) yields 0=

1 Dρ ∂ ζ˙ 1 D ∂z 1 D + + ∇ζ · u + = ρ Dt ∂z/∂ζ Dt ∂ζ ∂ζ ρ∂z/∂ζ Dt

 ρ

∂z ∂ζ

 + ∇ζ · u +

∂ ζ˙ (3.83) ∂ζ

3.5 The Primitive Equations in Pressure Coordinates

85

But due to hydrostatics one also has ∂z 1 ∂p =− ∂ζ g ∂ζ

(3.84)

∂ ζ˙ 1 D ∂p + ∇ζ · u + =0 ∂ p/∂ζ Dt ∂ζ ∂ζ

(3.85)

ρ so that one finally obtains

Thermodynamics The thermodynamic equations keep their form (3.54)–(3.56). One only must keep in mind to calculate the material derivatives via (3.69).

3.5.2

Pressure Coordinates

For a transformation to pressure coordinates we now set everywhere ζ = p. Thus one obtains, due to ∇ p p = 0, from (3.72) the horizontal momentum equation Du + f × u = −∇ p  Dt

(3.86)

Reminding the reader of the definition (3.51), this is componentwise Du uv 1 ∂ − fv− tan φ = − Dt a a cos φ ∂λ 2 Dv u 1 ∂ + fu + tan φ = − Dt a a ∂φ where

D ∂ u ∂ v ∂ ∂ = + + +ω Dt ∂t a cos φ ∂λ a ∂φ ∂p

(3.87) (3.88)

(3.89)

with the pressure velocity ω = p˙

(3.90)

∂ = −α ∂p

(3.91)

The hydrostatic equilibrium is with (3.74)

86

3 Elementary Properties and Applications of the Basic Equations

The equation of continuity becomes via (3.85) ∇p · u +

∂ω =0 ∂p

(3.92)

Here we see one advantage of the pressure coordinate. The equation of continuity does not contain any time derivatives any more. The price paid for this is a time dependent lower boundary, since the surface pressure depends on time. In many applications, however, the lower boundary is not needed or the boundary condition can be approximated by p = p0 with constant surface pressure p0 . The thermodynamic equations can be written down directly. With Dp/Dt = p˙ = ω and the equation of state the enthalpy equation becomes cp

DT RT − ω=q Dt p

(3.93)

Alternatively the entropy equation becomes Dθ 1 = Dt cp



p00 p

 R/c p q

(3.94)

where the factor θ/(c p T ) has been rewritten via the definition of the potential temperature. One sees that it only depends on pressure, i.e., the vertical coordinate.

3.5.3

Summary

It is often useful in meteorology to replace geometric altitude by an alternative vertical coordinate. • In the general case the horizontal momentum equations take the form (3.72), hydrostatics becomes (3.74), and the continuity equation is (3.85). The thermodynamic energy equation keeps its old form, but note that the material derivative must be calculated by (3.69). • An important example is the pressure coordinate. Hydrostatics implies that pressure monotonically decreases with growing altitude so that it can alternatively be used as pressure coordinate. Horizontal momentum equation, material derivative, hydrostatics, and continuity equation respectively take the forms (3.87)–(3.89), (3.91), and (3.92). Advantages of the pressure coordinate, as compared to geometric altitude, are that the pressure-gradient term simplifies, and that the continuity equation becomes a simple diagnostic non-divergence condition for the generalized wind. • One should also note that in the new formulation u still is the original horizontal wind tangential to surfaces of constant geometric altitude, and not tangential, e.g., to pressure surfaces.

3.6

Balanced Flows

3.6

87

Balanced Flows

In a first application of the primitive equations in pressure coordinates we want to examine the horizontal momentum equations for purely horizontal flows where the fluid elements move with constant speed on trajectories of fixed curvature. For this purpose we introduce horizontal coordinates which are better adapted to the problem.

3.6.1

The Natural Coordinates

First define a unit vector t which is tangential to the instantaneous horizontal velocity u of a fluid element, with absolute value V , so that u = Vt

(3.95)

In addition, define a second horizontal unit vector (the normal vector) n which is orthogonal to t and rotated with regard to this anticlockwise by π/2 (Fig. 3.1). Thus the horizontal acceleration of the fluid element is Du DV Dt =t +V Dt Dt Dt

(3.96)

dt t + dt

t

R dψ ds

n

t

Fig. 3.1 Illustration of the natural coordinates for an anticlockwise rotation of the velocity vector. The angle increment dψ and the radius of curvature R are both positive. One has |t| = |t + dt|, and in the limit case dψ → 0 also dt ⊥ t

88

3 Elementary Properties and Applications of the Basic Equations

For a calculation of Dt/Dt first consider the case that the the fluid-element trajectory turns anticlockwise (also Fig. 3.1). For small time increments the trajectory can be seen as segment of a circle with radius of curvature R > 0 which is positive by definition. The distance covered in this time is ds. The change of t is dt. The (positive) angle for the segment is thus dψ =

|dt| ds = = |dt| R |t|

(3.97)

In the case of a clockwise rotation one defines R < 0 and one obviously has dψ < 0 so that (Fig. 3.2) ds |dt| dψ = =− = −|dt| (3.98) R |t| But |t| = |t + dt| = 1 so that for dt → 0 1 = |t + dt|2 = |t|2 + 2t · dt + |dt|2 → 1 + 2t · dt

(3.99)

Thus dt is orthogonal to t and thus for anti-clockwise rotation parallel to n so that dt = |dt|n. In the case of clockwise rotation it is anti-parallel so that dt = −|dt|n. Both cases thus yield via (3.97) or (3.98) ds dt = n (3.100) R One obtains Dt Ds V Dt = = n (3.101) Dt Ds Dt R

Fig. 3.2 Illustration of the natural coordinates for a clockwise rotation of the velocity vector. The angle increment dψ and the radius of curvature R are both negative. One has |t| = |t + dt|, and in the limit case dψ → 0 also dt ⊥ t

dt t

dψ

t + dt dψ < 0 R 0 one has on the northern hemisphere ∂/∂n < 0, i.e., the geopotential minimum is to the left and the maximum to the right of the moving fluid element (Fig. 3.3). The flow is therefore anti-clockwise if about a geopotential minimum, and clockwise if about a maximum. Since, due to (3.71), geopotential gradients are parallel to pressure gradients, i.e., (3.109) ∇z p = ρ∇ p  this also holds for movement relative to pressure extrema. On the southern hemisphere the sense of rotation is reversed since there f < 0.

3.6.3

Inertial Flow

In the case of a constant geopotential, i.e., ∂/∂n = 0, one has V2 + fV =0 R

(3.110)

This is an equilibrium between the centrifugal force and the Coriolis force. The radius of curvature is

t

L

H

n

Fig. 3.3 Geostrophic equilibrium in the northern hemisphere: The geopotential gradient is directed against the normal direction so that the geopotential or pressure minimum are to the left and the maximum to the right

3.6

Balanced Flows

91

R=−

V f

(3.111)

Therefore on the northern hemisphere the sense of rotation is clockwise, on the southern hemisphere anticlockwise. The period of the circular movement is P=

|2π R| 2π = V |f|

(3.112)

In the atmosphere such an inertial flow is typically not found since the geopotential gradient always has an impact. In the ocean, however, it is an observed phenomenon.

3.6.4

Cyclostrophic Flow

The case complementary to geostrophic flow is cyclostrophic flow with very small radii of curvature (R → 0). The resulting flow has a large Rossby number Ro =

V →∞ | f R|

(3.113)

so that the Coriolis acceleration can be neglected: V2 ∂ =− R ∂n

(3.114)

One obtains an equilibrium between centrifugal force and pressure-gradient force so that  ∂ V = −R (3.115) ∂n The sign of R must be opposite to that of ∂/∂n. This means that the circulation is always, irrespectively whether it is clockwise or anticlockwise, about a geopotential minimum, i.e., a pressure low (Fig. 3.4). This is the dynamic situation in dust devils, water spouts, or tornadoes. R>0

R 0, which however is necessary by the definition of V . In the following we will separate the physical solutions from the unphysical ones. The discussion here is limited to northern-hemispheric flows where f > 0. A corresponding generalization to the southern hemisphere is left to the reader as an exercise. i) R > 0 and ∂/∂n > 0 In this case one has for real solutions  f 2 R2 ∂ fR −R < 4 ∂n 2

(3.117)

Thus one obtains for both solutions (3.116) V < 0. This case yields no physical solutions. ii) R > 0 and ∂/∂n < 0 Now



f 2 R2 ∂ fR −R > 4 ∂n 2 

and thus V =−

fR + 2

f 2 R2 ∂ −R 4 ∂n

(3.118)

(3.119)

is a physical solution. This is an anticlockwise rotating cyclonic flow. From the sign of the geopotential gradient one recognizes that it has as center a pressure low. This is a regular low which is for large R approximately in geostrophic equilibrium (Fig. 3.5). It is also seen easily that the second term dominates in the square root in the case of small curvature radii so that one has cyclostropic flow about a pressure low. Furthermore it is clear that the solution with the negative root leads to V < 0. This is an unphysical solution. iii) R < 0 and ∂/∂n > 0 Now

 fR + V =− 2

f 2 R2 ∂ −R 4 ∂n

(3.120)

3.6

Balanced Flows

93

t L n

Fig. 3.5 Regular low

is always a physical solution with V > 0. Its sense of rotation is clockwise. From the sign of the geopotential gradient we see that its center is a pressure low. Obviously it cannot be geostrophic. This is an antibaric flow or an anomalous low (Fig. 3.6). In the limit R → −∞ this is an inertial flow, while in the case of small curvature radii we again have cyclostropic flow about a pressure low. Furthermore  f 2 R2 ∂ fR −R >− (3.121) 4 ∂n 2 and thus the solution with the negative root leads to V < 0. It is unphysical. iv) R < 0 and ∂/∂n < 0 Here, if real, the square root is 

f 2 R2 ∂ fR −R | f R|/2, and thus the Rossby number is Ro > 1/2. The flow cannot have a geostrophic equilibrium. This is an anomalous high (Fig. 3.7). The other solution is  f 2 R2 fR ∂ V =− − −R (3.124) 2 4 ∂n It is also anti-cyclonic. Its center is also a pressure high. Now, however, one always has V < | f R|/2, and thus a Rossby number Ro < 1/2. This is the gradient-wind solution for the flow about a pressure high with approximate geostrophic equilibrium. This is a regular pressure high, with its structure given by Fig. 3.7 as well.

3.6.6

Summary

In analyzing within the framework of a pressure-coordinate formulation cyclic flows where fluid elements move on circular trajectories it is useful to introduce a natural curved coordinate system where one of the two coordinates measures the distance along the circular motion, while the other points, as seen from the respective fluid element, to the left in normal direction. Within this formulation one recognizes that in normal direction an equilibrium between Coriolis force, centrifugal (or inertial) force, and pressure gradient force must hold, if the motion is along geopotential isolines. One obtains the following possible equilibria:

3.7 Thermal Wind

95

t H

n

Fig. 3.7 A regular or an anomalous high

• At small Rossby numbers one has an equilibrium between pressure gradient force and Coriolis force, leading to geostrophic flow as typically observed on synoptic scales. • If pressure gradients are negligible one gets inertial flow, resulting from the equilibrium between inertial force and Coriolis force. The corresponding circular period is the inertial period. • At large Rossby numbers the equilibrium is between pressure gradient force and centrifugal force. This leads to cyclostrophic flow as e.g., in a tornado. • In the general case, when no term can be neglected, one obtains the gradient wind. It enables us to estimate quantitatively the deviations from the special cases listed above.

3.7

Thermal Wind

The combination of geostrophy and hydrostatics yields an important relationship between vertical wind gradient and horizontal temperature gradient which is useful in many applications. Consider the pressure-coordinate formulation of the geostrophic equilibrium in (3.72) between Coriolis force and pressure gradient force f er × u = −∇ p 

(3.125)

The solution is the geostrophic wind ug = with its components

1 er × ∇ p  f

(3.126)

96

3 Elementary Properties and Applications of the Basic Equations

1 1 ∂ f a ∂φ 1 1 ∂ vg = f a cos φ ∂λ

ug = −

The pressure derivative is

∂ug ∂ 1 = er × ∇ p ∂p f ∂p

(3.127) (3.128)

(3.129)

The hydrostatic equilibrium (3.91) can be written, via the equation of state, ∂ RT =− ∂p p

(3.130)

so that (3.129) becomes the thermal-wind relation ∂ug R = − er × ∇ p T ∂p fp

(3.131)

Component-wise it is ∂u g R 1 ∂T = ∂p f p a ∂φ ∂vg R 1 ∂T =− ∂p f p a cos φ ∂λ

(3.132) (3.133)

A vertical gradient of the zonal wind thus is connected with a meridional gradient of temperature, a vertical gradient of the meridional wind with a zonal temperature gradient. The thermal-wind relation is clearly visible in the climatology of wind and temperature. Taking the zonal mean of the zonal thermal wind yields ∂u g  R 1 ∂T  = ∂p f p a ∂φ

(3.134)

Here we define for some arbitrary   =

1 2π



2π

dλ

(3.135)

0

Since the tropics in the troposphere are warm, and the polar regions are cold, one obtains on both hemispheres 1 ∂T  0

(5.298)

Finally we turn to the requirement that η and dη /d x be continuous at x = 0. From (5.298) follows ⎧ a+ x/L d ⎪ e at x < 0 ⎨  dη Ld (5.299) = a ⎪ dx ⎩ − − e−x/L d at x > 0 Ld Thus #

η0 + a+ at x < 0 −η0 + a− at x > 0 ⎧ a+ ⎪ at x < 0 ⎨  dη Ld lim = ⎪ − a− at x > 0 x→0 d x ⎩ Ld lim η =

x→0

(5.300)

(5.301)

Therefore continuity of dη /d x implies a+ = −a−

(5.302)

η0 + a+ = −η0 + a−

(5.303)

a+ = −η0

(5.304)

a − = η0

(5.305)

 # 1 − e x/L d at x < 0  η (x, t → ∞) = η0 −x/L d e − 1 at x > 0

(5.306)

Continuity of η leads to Together with (5.302) this yields

so that finally

5.5

Geostrophic Adjustment

175

η v η0

x

–η0

Fig. 5.12 Distribution of η and v  , resulting after the geostrophic adjustment of the pressure jump from Fig. 5.11

150

t=0 t = 40 t = 80 t = 120 t = 160 t = 200

100 η’ [m]

50 0 −50 −100 −150 −100

−50

0

50

100

x/1000km Fig. 5.13 Temporal development (time in units of 1000 s) of the geostrophic adjustment of a pressure jump. Note the gravity-wave fronts moving outwards while a geostrophically balanced state remains in the center

176

5 The Dynamics of the Shallow-Water Equations

The geostrophic wind resulting from this with (5.287) and (5.288) is purely meridional, i.e., u  = 0 and # x/L η0 g e d at x < 0 (5.307) v  (x, t → ∞) = − −x/L d at x > 0 f0 L d e One obtains a meridional jet with maximal intensity at x = 0. The distribution of η and v  is shown in Fig. 5.12, while Fig. 5.13 illustrates the temporal development of the adjustment process. For the qualitative illustration of the radiation of gravity waves from strong pressure gradients we show in Fig. 5.14 a snapshot of geopotential and horizontal divergence on the 200mbar surface over Europe.

Fig. 5.14 For the qualitative demonstration of the relevance of the geostrophic adjustment process, a snapshot of horizontal divergence (red and blue) on the 80mbar pressure level and geopotential (black) at 300mb over North America and the North Atlantic. Shading indicates the wind strength. In pressure coordinates the geopotential plays the same role as elsewhere pressure as streamfunction in geostrophic scaling. Since the geostrophic flow has zero horizontal divergence the latter can be used very well as indicator of gravity-wave activity. Note the increased gravity-wave intensity in the vicinity of strong gradients of the geopotential, where the dynamics tends to deviate from geostrophic scaling. Figure from Wu and Zhang (2004)

5.6

Recommendations for Further Reading

5.5.3

177

Summary

An initially non-geostrophic state has contributions both from Rossby waves and from gravity waves. Since the latter have a much larger group velocity they are radiated away so that finally only the geostrophically balanced Rossby-wave part remains. • For an analysis of this process we consider the limit of linear dynamics on the f -plane. • In a general solution of the initial-value problem one can show that at any time the state is constituted by a steady geostrophic part and propagating inertia-gravity waves. The corresponding contributions are obtained via projection of the initial state onto these eigenmodes. • The finally remaining part can be obtained from potential-vorticity conservation. The potential vorticity of the final state yields the streamfunction or surface elevation by quasigeostrophic theory.

5.6

Recommendations for Further Reading

Excellent texts on shallow-water dynamics are the books by Pedlosky (1987), Salmon (1998), Vallis (2006), and Zeitlin (2018).

6

Quasigeostrophic Dynamics of the Stratified Atmosphere

Strictly spoken the shallow-water equations only hold under conditions not valid for the atmosphere. Both the assumption of a constant density and the one of vanishing vertical gradients in the horizontal wind are not realistic. These will now be dropped. We will not, however, treat the atmosphere in full generality but rather focus on the synoptic scales and derive the corresponding quasigeostrophic theory. This will enable us to describe not only the vertical structure and vertical propagation of Rossby waves but also the generation of synoptic-scale extratropical weather by baroclinic instability.

6.1

Quasigeostrophic Theory and Its Potential Vorticity

6.1.1

Analysis of Momentum and Continuity Equation

Scale Analysis Much in the derivation of the quasigeostrophic theory of the baroclinic atmosphere resembles the corresponding theory for the shallow-water equations. In addition to there we here split the thermodynamic fields into a part from a hydrostatic reference atmosphere at rest, with only vertical spatial dependence, and the deviations therefrom. We thus write, with z = r −a, ρ = ρ (z) + ρ˜ (λ, φ, z, t)

(6.1)

p = p (z) + p˜ (λ, φ, z, t)

(6.2)

and where the reference-atmosphere part satisfies dp = −gρ dz © Springer-Verlag GmbH Germany, part of Springer Nature 2022 U. Achatz, Atmospheric Dynamics, https://doi.org/10.1007/978-3-662-63941-2_6

(6.3)

179

180

6 Quasigeostrophic Dynamics of the Stratified Atmosphere

We also demand that, in agreement with observations, the density deviations from the density of the reference atmosphere are small: |ρ| ˜ ρ

(6.4)

For reasons which will become clear below we choose an altitude-dependent scaling for the pressure and density fluctuations: p˜ = P (z) pˆ

(6.5)

ρ˜ = R (z) ρˆ

(6.6)

Following (6.4) we have R  ρ. As already in quasigeostrophic shallow-water theory we introduce a horizontal length scale L = 103 km and a horizontal-wind scale U = 10 m/s so that, with some reference longitude and latitude λ0 and φ0 , respectively, and consistent with the estimates (5.38) and (5.39)       L λˆ λ λ0 (6.7) + = φ0 φ a φˆ and u = U uˆ

(6.8)

The time scale is again L tˆ (6.9) U In addition we introduce a vertical length scale H = 10 km and a scale W for the vertical wind so that t=

z = H zˆ

(6.10)

w = W wˆ

(6.11)

The vertical length scale approximately corresponds to the height of typical synoptic-scale weather structures, but also to the vertical extent of the troposphere and its hydrostatic scale height. The vertical-wind scale can be related to the horizontal-wind scale via the continuity equation Dρ + ρ∇ · v = 0 (6.12) Dt Due to (6.4) the latter is approximately w

dρ + ρ∇ · v = 0 dz

(6.13)

6.1

Quasigeostrophic Theory and Its Potential Vorticity

181

or, in local Cartesian coordinates, ∇ ·u+

1 ∂ (ρw) = 0 ρ ∂z

(6.14)

The non-dimensionalization of this equation yields W 1 ∂   U ˆ ∇ · uˆ + ρ wˆ = 0 L H ρ ∂ zˆ

(6.15)

For an equilibration between the two terms one needs H W = U (6.16) L Therefore the vertical winds must be weaker than the horizontal winds by at least two orders of magnitude. Below we will see that this is actually only an estimate of an upper bound. No we turn to the two horizontal-momentum equations. Because of r = a + z and (6.10) one has   H r = a 1 + zˆ (6.17) a where H /a ≈ 10−3  1. Inserting this together with (6.7–6.11) and (6.16) into the material derivative (1.107) yields D U D˜ (6.18) = Dt L D tˆ with the non-dimensional material derivative D˜ ∂ uˆ ∂ vˆ ∂ ∂  = + + + wˆ H H ˆ ˆ ∂ zˆ D tˆ ∂ tˆ ∂ λ 1 + zˆ ∂ φ 1 + zˆ cos φ a a Moreover, one has

(6.19)

2 sin φ = f = f 0 fˆ

(6.20)

f 0 = 2 sin φ0 sin φ fˆ = sin φ0

(6.21)

with

and 2 cos φ = aβ

cos φ cos φ0

(6.22)

(6.23)

with 2 cos φ0 f0 = Roβˆ a L 1 L βˆ = cot φ0 = O(1) Ro a

β=

(6.24) (6.25)

182

6 Quasigeostrophic Dynamics of the Stratified Atmosphere

Non-dimensionalization of the horizontal-momentum equations in (1.105) via (6.1), (6.2), (6.5–6.11), and (6.16) yields, also using (6.17) and (6.18), L D˜ uˆ − Ro a D tˆ

H L H a uˆ vˆ uˆ wˆ cos φ tan φ + − fˆvˆ + wˆ Ro Roβˆ H H L a L L cos φ0 1 + zˆ 1 + zˆ a a P 1 1 1 ∂ pˆ    (6.26) =− H R ρ L f0U cos φ ∂ λˆ 1 + ρˆ 1 + zˆ ρ a 2 ˜ L H L D vˆ uˆ uˆ wˆ + Ro tan φ + + fˆuˆ Ro Ro H H a L a D tˆ 1 + zˆ 1 + zˆ a a P 1 1 ∂ pˆ    (6.27) =− H R ρ L f0U ∂ φˆ 1 + ρˆ 1 + zˆ ρ a Ro

Since the Rossby number is with the chosen scaling Ro = O(10−1 ) the Coriolis term is the only term on the left-hand side of these equations which is not small. It can only be balanced by the pressure-gradient term on the right-hand side if

P = ρ L f0U

(6.28)

Thus we indeed obtain an altitude-dependent pressure scale so that p = p(z) + ρ L f 0 U pˆ

(6.29)

In the treatment of the vertical-momentum equation in (1.105) we first rewrite the right-hand side via (6.1) and (6.2):    d p ∂ p˜ Dw u 2 + v 2 1 +g (6.30) − − 2 cos φu = − + Dt r ρ + ρ˜ dz ∂z We further use the hydrostatic equilibrium (6.3) of the reference atmosphere and obtain  Dw u 2 + v 2 g ρ˜ 1 ∂ p˜ (6.31) − − 2 cos φu = − + Dt r ρ + ρ˜ ρ + ρ˜ ∂z Now we proceed as in the non-dimensionalization of the horizontal-momentum equations, also using (6.6), and obtain

6.1

Quasigeostrophic Theory and Its Potential Vorticity

183

L a H D˜ wˆ uˆ 2 + vˆ 2 cos φ − Ro − Roβˆ uˆ Ro H L a L cos φ0 D tˆ 1 + zˆ a ⎡ =−

L ⎢ ⎢ H⎣

⎤

⎥ ρˆ Rg H 1 1 ∂ ˆ ⎥ + (ρ p) ⎦ R ρ f0U L R ρ ∂ zˆ 1 + ρˆ 1 + ρˆ ρ ρ

(6.32)

Among the terms on the left-hand side the last is the largest. It is of order O(1) and thus still small compared to the factor H /L  1 on the right-hand side. We conclude that at least to leading order the terms in the bracket on the right-hand side must cancel each other, which again only is possible if L f0U ρ L2 R= (6.33) = ρ Ro 2 gH Ld One thus also sees that R  ρ, consistent with the basic assumptions. Moreover one obtains   L2 (6.34) ρ = ρ 1 + ρ Ro 2 ρˆ Ld With this choice the vertical-momentum equation becomes L a H D˜ wˆ uˆ 2 + vˆ 2 cos φ − Ro − Roβˆ uˆ Ro H L a L cos φ0 D tˆ 1 + zˆ a ⎡ =−

L H

⎢ ⎢ ⎢ ⎣

ρˆ L2 1 + Ro 2 ρˆ Ld

+

1 L2 1 + Ro 2 ρˆ Ld

⎤

⎥ 1 ∂ ⎥ ˆ ⎥ (ρ p) ⎦ ρ ∂ zˆ

(6.35)

The non-dimensionalization of the continuity equation works the same way. We first nondimensionalize the divergence in (1.109) as ⎫ ⎧ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪  ⎬ U ⎨ˆ 1 H 2 ∂ ∇ ·v = 1 + ∇ · uˆ +  w ˆ z ˆ  ⎪ ⎪ L ⎪ a H 2 ∂ zˆ ⎪ ⎪ ⎪ ⎭ ⎩ 1 + zˆ a    U ˆ ∂ wˆ H = ∇ · uˆ + (6.36) +O L ∂ zˆ a

184

6 Quasigeostrophic Dynamics of the Stratified Atmosphere

with the non-dimensional horizontal divergence   1 1 ∂  1 ∂ uˆ ˆ ∇ · uˆ = + cos φ vˆ H cos φ ∂ λˆ cos φ ∂ φˆ 1 + zˆ a

(6.37)

This and (6.34) lead, in a non-dimensionalization closely analogous to what has been demonstrated above, to     L2 H L 2 D˜ ρˆ 1 ∂ ˆ + 1 + Ro 2 ρˆ ∇ · uˆ + Ro 2 ˆ +O (ρ w) =0 (6.38) ˆ ρ ∂ z ˆ a D t Ld Ld

Local Geometry and Characterization in Terms of Powers of the Rossby Number From the definition (5.85) of the external Rossby deformation radius follows, with H = 10 km and f 0 = 10−4 s−1 , that (6.39) L d ≈ 3000 km But L = 1000 km, so that we can use the central assumption L2 = O(Ro) L 2d

(6.40)

where at the given scaling Ro = O(10−1 ). One should note that this differs from the assumption L 2 /L 2d = O(1) used in the derivation of shallow-water quasigeostrophic theory. With our choice of length scales one obtains moreover H = O(Ro2 ) L

L = O(Ro) a

H = O(Ro3 ) a

(6.41)

Since L/a = O(Ro)  1, we expand the various trigonometric functions of the geographic latitude about the reference latitude and thus arrive at the local geometry of the β-plane. Thus, using (6.7),     L 2 1 L ∂ 1 ∂ = 1 + tan φ0 φˆ + O cos φ ∂ λˆ a a cos φ0 ∂ λˆ     L ∂ = 1 + tan φ0 yˆ (6.42) + O Ro2 a ∂ xˆ where



   x, ˆ yˆ = cos φ0 λˆ , φˆ

(6.43)

are the non-dimensional horizontal coordinates of the β-plane tangential at (λ0 , φ0 ), as can be read via (5.40), (5.41), and (x, y) = L(x, ˆ yˆ ) from (6.7). Correspondingly one also has

6.1

Quasigeostrophic Theory and Its Potential Vorticity

∂ ∂ = ˆ ∂ yˆ ∂φ so that the material derivative can be rewritten, also using     H 1 = 1 + O Ro3 =1+O H a 1 + zˆ a as

185

(6.44)

(6.45)

  Dˆ L/a D˜ ∂ = + Ro tan φ0 + O Ro2 ˆ ˆ Ro ∂ x ˆ Dt Dt

(6.46)

∂ ∂ Dˆ ∂ ∂ + vˆ + wˆ = + uˆ ∂ xˆ ∂ yˆ ∂ zˆ D tˆ ∂ tˆ

(6.47)

with the definition

Beyond this we have tan φ = O (1) cos φ = O (1) cos φ0

(6.48) (6.49)

and, with (6.22) and fˆ0 = 1,    L 2 L fˆ = fˆ0 + cot φ0 φˆ + O a a  2 = fˆ0 + Roβˆ yˆ + O Ro and

    1 L2 1 = 1 + O Ro 2 = 1 + O Ro2 = 2 R L Ld 1 + ρˆ 1 + Ro 2 ρˆ ρ Ld

(6.50)

(6.51)

Using all these estimates we now rewrite the horizontal-momentum equations (6.26) and (6.27) so that only the larger terms are expressed explicitly that will be needed in the further treatment below. One obtains        Dˆ uˆ + O (Ro) − fˆ0 + Roβˆ yˆ + O Ro2 vˆ + O Ro2 Ro D tˆ     L ∂ pˆ = − 1 + tan φ0 yˆ (6.52) + O Ro2 a ∂ xˆ

186

6 Quasigeostrophic Dynamics of the Stratified Atmosphere



      Dˆ vˆ + O (Ro) + fˆ0 + Roβˆ yˆ + O Ro2 uˆ + O Ro2 Ro ˆ Dt =−

  ∂ pˆ + O Ro2 ∂ yˆ

(6.53)

Likewise the vertical-momentum equation (6.35) becomes   2   1 ∂   L L ρˆ + O (1) = − ρ pˆ + O Ro = O Ro−2 H ρ ∂ zˆ H

(6.54)

For the continuity equation (6.38) we first reformulate ∇˜ · uˆ in (6.37). Expansion about the reference latitude φ0 yields, with φˆ = yˆ ,    ∂ vˆ L 1 ∂  cos φ vˆ = − tan φ0 vˆ + O Ro2 cos φ ∂ φˆ ∂ yˆ a This together with (6.42) leads to         ∂ uˆ L ∇˜ · uˆ = 1 + O Ro3 ∇ˆ · uˆ + tan φ0 yˆ − vˆ + O Ro2 a ∂ xˆ where

∂ uˆ ∂ vˆ ∇ˆ · uˆ = + ∂ xˆ ∂ yˆ

(6.55)

(6.56)

(6.57)

Hence (6.38) becomes   0 = O Ro2 

L2 + 1 + Ro 2 Ld

+

!



  1 + O Ro3

  1 ∂   ρ wˆ + O Ro3 ρ ∂ zˆ

"

     ∂ uˆ L ∇ˆ · uˆ + tan φ0 yˆ − vˆ + O Ro2 a ∂ xˆ (6.58)

Scale-Asymptotic Treatment As in the shallow-water case we now expand all fields in the Rossby number: ⎛ ⎞ ⎛ ⎞ vˆ vˆ i ∞ ' ⎝ ρˆ ⎠ = Roi ⎝ ρˆi ⎠ i =0 pˆ i pˆ

(6.59)

This is inserted into (6.52–6.54) and (6.58), and then all is sorted in terms of powers of Ro.

6.1

Quasigeostrophic Theory and Its Potential Vorticity

187

The leading order of the horizontal-momentum equations is O(1). It yields vˆ0 =

1 ∂ pˆ 0 fˆ0 ∂ xˆ

uˆ 0 = −

1 ∂ pˆ 0 fˆ0 ∂ yˆ

(6.60) (6.61)

This is the geostrophic equilibrium of the horizontal wind to leading order. A consequence is that the latter has no divergence: (6.62) ∇ˆ · uˆ 0 = 0 The leading order of the vertical-momentum equation is O(Ro−2 ). One obtains ρˆ0 +

1 ∂ (ρ pˆ 0 ) = 0 ρ ∂ zˆ

(6.63)

This means that the leading-order pressure and density fluctuations are in hydrostatic equilibrium. The leading order of the continuity equation is O(1), yielding 1 ∂ ∇ˆ · uˆ 0 + (ρ wˆ 0 ) = 0 ρ ∂ zˆ

(6.64)

which is, because of (6.62), equivalent to 1 ∂ (ρ wˆ 0 ) = 0 ρ ∂ zˆ

(6.65)

Since ρ → 0 for z → ∞, a diverging vertical wind at infinity can only be avoided by having everywhere wˆ 0 = 0 (6.66) Hence, the vertical wind is not only weaker than the horizontal wind by a factor H /L but even by a factor Ro H /L. Now turning to the next order O(Ro) of the horizontal-momentum equations, we obtain D0 ∂ pˆ 1 ∂ pˆ 0 L/a uˆ 0 − fˆ0 vˆ1 − βˆ yˆ vˆ0 = − − tan φ0 yˆ ∂ xˆ Ro ∂ xˆ D tˆ ∂ p ˆ D0 1 vˆ0 + fˆ0 uˆ 1 + βˆ yˆ uˆ 0 = − ∂ yˆ D tˆ where

∂ ∂ ∂ D0 = + uˆ 0 + vˆ0 ∂ xˆ ∂ yˆ D tˆ ∂ tˆ

(6.67) (6.68)

(6.69)

is the non-dimensional form of the quasigeostrophic material derivative. Via ∂ (6.68) /∂ x −∂ (6.67) ∂ y we obtain the equation

188

6 Quasigeostrophic Dynamics of the Stratified Atmosphere

   ∂ uˆ 0 L/a D0 ζˆ0 ∂ + vˆ0 βˆ yˆ = − fˆ0 ∇ˆ · uˆ 1 + tan φ0 yˆ − vˆ0 ∂ yˆ Ro ∂ xˆ D tˆ for the non-dimensional quasigeostrophic vorticity   ∂ vˆ0 ∂ 2 pˆ 0 ∂ uˆ 0 1 ∂ 2 pˆ 0 ζˆ0 = + − = ∂ xˆ ∂ yˆ ∂ yˆ 2 fˆ0 ∂ xˆ 2

(6.70)

(6.71)

where again (6.62) has been used, and, also resulting therefrom, ∂ uˆ 0 ˆ · ∇ vˆ0 = 0 ∂ xˆ ∂ uˆ 0 ˆ · ∇ uˆ 0 = 0 ∂ yˆ On the other hand, the O(Ro) of the continuity equation yields   ∂ uˆ 0 1 ∂ L/a 0 = ∇ˆ · uˆ 1 + tan φ0 yˆ − vˆ0 + (ρ wˆ 1 ) Ro ∂ xˆ ρ ∂ zˆ

(6.72) (6.73)

(6.74)

so that we obtain the quasigeostrophic vorticity equation   D0  1 ∂  ρ wˆ 1 ζˆ0 + βˆ yˆ = ρ ∂ zˆ D tˆ

(6.75)

This equation is not closed. All terms on the left-hand side can be calculated from pˆ 0 and its derivatives. A connection to wˆ 1 , however, is not yet discernible. For further progress we now turn to the thermodynamics, in the form of the entropy equation.

6.1.2

Analysis of the Entropy Equation

We first consider potential temperature. By way of (6.28), (5.70), and (5.85) the pressure can be written as   gHρ L2 (6.76) Ro 2 pˆ p = p 1+ p Ld Since the reference atmosphere is hydrostatic and H approximately corresponds to the hydrostatic scale height, one has g H ρ/ p = O(1) so that the second term in (6.76) is small. Expressing the temperature in the definition (2.91) of the potential temperature via the equation of state (2.3) in terms of pressure and density we obtain p00 θ= Rρ



p00 p

 R/c p −1 (6.77)

6.1

Quasigeostrophic Theory and Its Potential Vorticity

189

which yields, with the help of (6.1), (6.6), (6.33), and (6.76) 1−R/c p ρ L2 1 + g H Ro 2 pˆ p Ld   θ =θ L2 1 + Ro 2 ρˆ Ld 

where p00 θ= Rρ



p00 p

(6.78)

 R/c p −1 (6.79)

is the reference-atmosphere potential temperature. Due to (6.40) this leads to     L2 R gHρ L2 θ = 1 − Ro 2 ρˆ + 1 − Ro 2 pˆ + O Ro4 cp p Ld Ld θ

(6.80)

The order of magnitude of the second but last term in (6.80) needs closer consideration. First, due to the hydrostatic equilibrium (6.3) of the reference atmosphere, and (6.10), one has 1 dp (6.81) = −gρ H d zˆ so that     R gHρ 1 dp R 1− −1 = (6.82) cp p cp p d zˆ Moreover from (6.79) follows 1 dθ 1 dρ =− − ρ d zˆ θ d zˆ so that (6.82) becomes





1 dp p d zˆ

(6.83)

  1 dθ 1 dρ R ρg H =− − 1− cp p d z ˆ ρ d zˆ θ

(6.84)

R −1 cp

Since H corresponds to the atmospheric scale height one has 1 dρ = O (1) ρ d zˆ

(6.85)

On the other hand, due to (2.143) and (6.10) 1 dθ H N2 = g θ d zˆ

(6.86)

190

6 Quasigeostrophic Dynamics of the Stratified Atmosphere

holds. In the troposphere typically N 2 = O(10−4 s −2 ) so that   4 10 · 10−4 H N2 = O (Ro) =O g 10

(6.87)

Thus one can write

where

1 dθ = Ro Nˆ 2 θ d zˆ

(6.88)

H N2 = O (1) Nˆ 2 = g Ro

(6.89)

Using (6.84) and (6.88) one can finally rewrite (6.80) as 

L2 θ = θ 1 − Ro 2 Ld We therefore write

    3 pˆ dρ ρˆ + + O Ro ρ d zˆ 

L2 θ = θ 1 + Ro 2 θˆ Ld and expand θˆ in terms of Ro: θˆ =

∞ '

(6.90)



Roi θˆi

(6.91)

(6.92)

i=0

The comparison with (6.90) yields θˆ0 = −ρˆ0 −

pˆ 0 dρ ρ d zˆ

(6.93)

which gives together with (6.63) θˆ0 =

∂ pˆ 0 ∂ zˆ

(6.94)

To leading order the deviation of potential temperature from its reference-atmosphere value is thus determined by pˆ 0 . A consequence of this is the thermal-wind relation, since the vertical derivatives of (6.60) and (6.61) yield, using (6.94) and fˆ0 = 1, ∂ uˆ 0 ∂ θˆ0 =− ∂ zˆ ∂ yˆ ∂ vˆ0 ∂ θˆ0 = ∂ zˆ ∂ xˆ

(6.95) (6.96)

As a consequence of hydrostatic and geostrophic equilibrium, the horizontal potentialtemperature gradients are thus equivalent to vertical gradients of the horizontal wind.

6.1

Quasigeostrophic Theory and Its Potential Vorticity

191

Now consider the entropy equation (2.126) without friction and heat conduction. In analogous manner to the treatment of the momentum and continuity equations, also using (6.91), we first non-dimensionalize the left-hand side. The result is    U D˜ L2 qθ ˆ θ 1 + Ro 2 θ (6.97) = ˆ L Dt cpT Ld Since θ is only altitude-dependent one obtains, also with the help of (6.88),   Ro Nˆ 2 L2 q L 2d θ D˜ θˆ + 1 + Ro 2 θˆ w ˆ = Ro 2 cpT U L θ D tˆ Ld L 2 /L d

(6.98)

Due to (6.46) and (6.47) the material derivative is given to leading order by the quasigeostrophic material derivative (6.69). Moreover, the vertical wind becomes to leading order Rowˆ 1 . Further resorting to (6.40) one sees that the leading order of the left-hand side of this equation is O(Ro) so that we write for consistency q L 2d θ = Ro Qˆ cpT U L θ

(6.99)

The leading order O(Ro) of the total equation is thus D0 θˆ0 + S wˆ 1 = Qˆ D tˆ

(6.100)

where

L 2d 2 (6.101) Nˆ = O (1) L2 is a stability parameter. Now we have succeeded since (6.100) can be solved for wˆ 1 , yielding S = Ro

1 wˆ 1 = S

6.1.3



D0 θˆ0 Qˆ − D tˆ

 (6.102)

Quasigeostrophic Potential Vorticity in the Stratified Atmosphere

The vertical wind from the estimate (6.102) above is now used in the vorticity equation (6.75). One has     1 ∂ ρ D0 θˆ0 1 ∂ Qˆ 1 ∂ − (6.103) ρ (ρ wˆ 1 ) = ρ ∂ zˆ ρ ∂ zˆ S ρ ∂ zˆ S D tˆ Since ρ/S only depends on zˆ , the second term is, with the definition (6.69) of the quasigeostrophic material derivative,

192

6 Quasigeostrophic Dynamics of the Stratified Atmosphere

1 ∂ ρ ∂ zˆ



ρ D0 θˆ0 S D tˆ



   1 ∂ D0 ρ ˆ θ0 = ρ ∂ zˆ D tˆ S      1 ∂ uˆ 0 ∂ θˆ0 ∂ vˆ0 ∂ θˆ0 D0 1 ∂ ρ θˆ0 + + = S ∂ zˆ ∂ xˆ ∂ zˆ ∂ yˆ D tˆ ρ ∂ zˆ S    D0 1 ∂ ρ θˆ0 = D tˆ ρ ∂ zˆ S

(6.104)

In the last step the thermal-wind relations (6.95) and (6.96) have been used. Thus one has       1 ∂ ρ ˆ 1 ∂  D0 1 ∂ ρ θˆ0 (6.105) ρ wˆ 1 = Q − ρ ∂ zˆ ρ ∂ zˆ S D tˆ ρ ∂ zˆ S Inserting this into (6.75) finally yields the desired non-dimensional conservation equation      1 ∂ ρ ˆ 1 ∂ ρ D0 ˆθ0 ˆζ0 + βˆ yˆ + = Q (6.106) ρ ∂ zˆ S ρ ∂ zˆ S D tˆ Now also defining the non-dimensional streamfunction ψˆ = pˆ 0

(6.107)

∂ ψˆ ∂ zˆ

(6.108)

so that due to (6.94) θˆ0 =

and, because of the geostrophy (6.60) and (6.61) of the horizontal wind and fˆ0 = 1, one also has ∂ ψˆ ∂ yˆ ˆ ∂ψ

uˆ 0 = −

(6.109)

vˆ0 =

(6.110)

∂ xˆ

the conservation equation becomes 

∂ ∂ ψˆ ∂ ∂ ψˆ ∂ − + ˆ ∂ yˆ ∂ xˆ ∂ xˆ ∂ yˆ ∂t



∂ 2 ψˆ ∂ 2 ψˆ 1 ∂ + + βˆ yˆ + 2 2 ∂ xˆ ∂ yˆ ρ ∂ zˆ



ρ ∂ ψˆ S ∂ zˆ



1 ∂ = ρ ∂ zˆ



ρ ˆ Q S



(6.111) For practical use we now re-introduce the dimensions. First we define for the streamfunction ψ = U L ψˆ

(6.112)

6.1

Quasigeostrophic Theory and Its Potential Vorticity

so that the geostrophic wind

 ug =

ug vg



 =U

193



uˆ 0 vˆ0

(6.113)

can be calculated from this, using (6.109) and (6.110), as ∂ψ ∂y ∂ψ vg = ∂x

ug = −

(6.114) (6.115)

Moreover one has on the β-plane     x xˆ =L y yˆ

(6.116)

Via the definition (5.85) of the external Rossby deformation radius and (6.89) one also finds that L2 (6.117) S = di2 L where HN L di = (6.118) f0 is the internal Rossby deformation radius. Finally also using (6.9) for the redimensionalization of time, (6.10) for that of zˆ , and taking (6.24) and (6.25) into consideration, one finally obtains the conservation equation Dg π 1 ∂ = Dt ρ ∂z



f0 g q ρ 2 N cpT

 (6.119)

for the quasigeotrophic potential vorticity π= where

∇h2 ψ

1 ∂ + f0 + β y + ρ ∂z



f 2 ∂ψ ρ 02 N ∂z



Dg ∂ ∂ ∂ ∂ ∂ψ ∂ ∂ψ ∂ = + ug + vg = − + Dt ∂t ∂x ∂y ∂t ∂ y ∂x ∂x ∂ y

(6.120)

(6.121)

is the quasigeostrophic material derivative. Here we have used the approximation, at good accuracy, that θ/θ T = T . Furthermore one has   2 ∂ ∂2 2 (6.122) + 2 ψ ∇h ψ = ∂x2 ∂y

194

6 Quasigeostrophic Dynamics of the Stratified Atmosphere

Without heating (and friction and heat conduction) the quasigeostrophic potential vorticity π is thus conserved. Most importantly, the conservation equation is a prognostic equation for the streamfunction from which all other fields can be determined. The horizontal wind follows from geostrophy. The vertical wind is via (6.94), (6.102), and (6.107)     W ˆ W ˆ D0 θˆ0 D0 ∂ ψˆ (6.123) = Ro Q− Q− w = W Ro wˆ 1 = Ro S S D tˆ D tˆ ∂ zˆ With the help of (6.16), (5.85), (6.99), (6.117), (6.118) and all the redimensionalization steps having led to (6.119) one obtains from this w=

g q f 0 Dg ∂ψ − 2 N 2 cpT N Dt ∂z

(6.124)

Pressure is obtained via (6.29), (6.107), and (6.112), yielding p = p + f 0 ρψ Finally potential temperature is, via (6.91), (5.85), (6.107), (6.112), and (6.10)   f 0 ∂ψ θ =θ 1+ g ∂z while it is left as an exercise to the interested reader to show that  f0 1 ∂ ρ =ρ 1− (ρψ) g ρ ∂z

(6.125)

(6.126)

(6.127)

In a summary all fields, obtained from the streamfunction, are ∂ψ ∂y ∂ψ v= ∂x g q f 0 Dg ∂ψ w= 2 − 2 N cpT N Dt ∂z u=−

p = p + f 0 ρψ  f0 1 ∂ ρ =ρ 1− (ρψ) g ρ ∂z   f 0 ∂ψ θ =θ 1+ g ∂z

(6.128) (6.129) (6.130) (6.131) (6.132) (6.133)

In an analogous manner one finally also finds that the dimensional form of the thermal-wind relations (6.95) and (6.96) is

6.1

Quasigeostrophic Theory and Its Potential Vorticity

  ∂u g ∂ θ =− ∂z f0 ∂ y θ   ∂v g ∂ θ = ∂z f0 ∂ x θ

195

(6.134) (6.135)

where θ  = θ − θ is the deviation of potential temperature from that of the reference atmosphere.

6.1.4

The Relationship with General Potential Vorticity

Closer inspection shows that the quasigeostrophic potential vorticity is not simply an approximation of Ertel’s potential vorticity =

ωa · ∇θ ρ

(6.136)

in the limit of synoptic scaling. Rather its conservation (in the absence of heating, friction, and heat conduction) follows from scale-asymptotic analyses of the conservation equations for general potential vorticity and potential temperature. Beyond this also the result from the continuity equation finds application that to leading order the quasigeostrophic flow is horizontal, which again is a consequence of the vanishing of its horizontal divergence. This shall be demonstrated here. First we decompose the absolute vorticity ωa = ω + 2

(6.137)

via (3.29) and (4.97–4.99) into its spherical-coordinate components so that =

ωaφ 1 ∂θ ωaλ 1 ∂θ ωar ∂θ + + ρ r cos φ ∂λ ρ r ∂φ ρ ∂r

(6.138)

where 1 ∂w 1 ∂ − (r v) r ∂φ r ∂r 1 ∂ 1 ∂w = 2 cos φ + (r u) − r ∂r r cos φ ∂λ 1 ∂v 1 ∂ = 2 sin φ + − (cos φ u) r cos φ ∂λ r cos φ ∂φ

ωaλ =

(6.139)

ωaφ

(6.140)

ωar

(6.141)

By way of the scaling steps and results from Sects. 6.1.1 and 6.1.2 we now analyze the three contributing terms. They are

196

6 Quasigeostrophic Dynamics of the Stratified Atmosphere

ωaλ 1 ∂θ = ρ r cos φ ∂λ

Ro2

L 2 θ f0 L 2d ρ H



ˆ λˆ ∂ θ/∂  H 1 + zˆ cos φ a

ρˆ L 2d ⎧ ⎫ ⎪   ⎪ ⎨ H 2 ∂ w/∂ ⎬ H 1 ∂ ˆ φˆ 1 + − z ˆ v ˆ × 2 H ∂ zˆ ⎪ ⎪ a ⎩ L 1 + H zˆ ⎭ 1 + zˆ a a     L 2 θ f0 θ f0 = O Ro3 = O Ro2 2 ρ H Ld ρ H

ωaφ 1 ∂θ = ρ r ∂φ

Ro2

L 2 θ f0 L 2d ρ H

1 + Ro ⎧ ⎪ ⎪ ⎨

1 + Ro

L2

L2 ρˆ L 2d



∂ θˆ /∂ φˆ  H 1 + zˆ a 

 H H2 1 + zˆ uˆ − 2  a L

1 ∂ Ha βˆ + 2 H ⎪ L ∂ zˆ ⎪ ⎩ 1 + zˆ a     2 θ f L θ f 0 0 = O Ro2 2 = O Ro3 ρ H Ld ρ H ×

(6.142)

⎫ ⎪ ⎪ ⎬

∂ w/∂ ˆ λˆ  ⎪ H ⎭ 1 + zˆ cos φ ⎪ a

θ f0    L2 L 2 ∂ θˆ ρ H 2 ˆ 1 + Ro 2 ρˆ Ro N + Ro 2 L2 Ld L d ∂ zˆ 1 + Ro 2 ρˆ Ld ⎧ ⎫ ⎪ ⎪  ⎨  ⎬ Ro 1 ∂  1 ∂ vˆ × fˆ + − cos φ uˆ H ⎪ ⎪ cos φ ∂ λˆ cos φ ∂ φˆ ⎩ ⎭ 1 + zˆ a   θ f0  = 1 + O Ro2 ρ H     ! L 2 /L 2d ∂ θˆ ∂ v ˆ ∂ u ˆ 2 2 2 + × Ro fˆ0 Nˆ + Ro Nˆ βˆ yˆ + − fˆ0 ∂ xˆ ∂ yˆ Ro ∂ zˆ "   +O Ro3

(6.143)

ωar ∂θ = ρ ∂r

(6.144)

6.1

Quasigeostrophic Theory and Its Potential Vorticity

197

so that potential vorticity takes the asymptotic form    " !    L 2 /L 2d ∂ θˆ ∂ v ˆ θ f0 ∂ u ˆ 2 2 2 + O Ro3 Ro fˆ0 Nˆ + Ro Nˆ βˆ yˆ + − + fˆ0 = ρ H ∂ xˆ ∂ yˆ Ro ∂ zˆ (6.145) In the application of the material derivative we use (6.46) and wˆ 0 = 0 so that !   "  2 L/a ∂ ∂ ∂ U D0 D + O Ro + Ro = tan φ0 yˆ uˆ 0 + uˆ 1 + vˆ1 + wˆ 1 Dt L D tˆ Ro ∂ xˆ ∂ yˆ ∂ zˆ (6.146) and hence D Dt U θ f0 ˆ 2 = N Lρ H      ! ∂ vˆ0 ρ wˆ 1 d θ ∂ uˆ 0 fˆ0 ∂ θˆ0 2 D0 2 + × Ro − + βˆ yˆ + fˆ0 Nˆ ∂y S ∂ zˆ ρ D tˆ ∂ xˆ θ Nˆ 2 d zˆ "   +O Ro3 (6.147)

0=

Herein one has, due to (6.88) and (6.101),        w ˆ ρ wˆ 1 d θ ρ θ wˆ 1 d S d 1 2 ˆ ˆ ˆ ˆ + O (Ro) = ρ f0 N f0 S = f0 ρ ρ S d zˆ ρ θ Nˆ 2 d zˆ θ S d zˆ

(6.148)

so that to leading order     ρ d S fˆ0 ∂ θˆ0 D0 ˆ ˆ ˆ + f 0 wˆ 1 ζ0 + β yˆ + 0= S ∂ zˆ S d zˆ ρ D tˆ

(6.149)

Now we use conservation of potential temperature in the adiabatic case Qˆ = 0. Inserting the corresponding result (6.102) for wˆ 1 leads to the conservation equation    1 ∂ ρ D0 θˆ0 ζˆ0 + βˆ yˆ + (6.150) 0= ρ ∂ zˆ S D tˆ which agrees with (6.106) in the case Qˆ = 0.

198

6.1.5

6 Quasigeostrophic Dynamics of the Stratified Atmosphere

Quasigeostrophic Theory in Pressure Coordinates

Beginning with the primitive equations we can use an analogous scale asymptotics as above for deriving also in pressure coordinates a potential vorticity and its conservation equation. Instead of this rather formal procedure, however, we choose here a heuristic approach. This somewhat better illuminates how the various basic assumptions of quasigeostrophic theory act together. For simplicity we right away begin with the primitive equations on the β-plane. The two momentum equations are ∂ Du − ( f 0 + β y)v = − Dt ∂x Dv ∂ + ( f 0 + β y)u = − Dt ∂y

(6.151) (6.152)

To leading order the Coriolis effect without β-term dominates in these together with the pressure-gradient acceleration so that the horizontal wind is approximately in geostrophic equilibrium: 1 ∂g f0 ∂ y 1 ∂g v ≈ vg = f0 ∂ x

u ≈ ug = −

(6.153) (6.154)

Here we have decomposed the geopotential  = ( p) + g + a into the referenceatmosphere part, and a fluctuating remainder that is dominated by a part g participating in the geostrophic equilibrium, and has an ageostrophic remainder a . The geostrophic horizontal wind thus also is non-divergent so that ∇ · ug = 0

(6.155)

We now decompose the wind into its dominant geostrophic part and the ageostrophic rest: ⎛

⎛ ⎞ ⎞ ⎛ ⎞ u u u ⎝v ⎠=⎝v ⎠ +⎝v ⎠ w w a w g

(6.156)

The continuity Eq. (3.92) is to leading order: ∇ · ug +

∂ωg =0 ∂p

(6.157)

Due to the non-divergence of the geostrophic wind one thus has ∂ωg =0 ∂p

(6.158)

6.1

Quasigeostrophic Theory and Its Potential Vorticity

199

The upper boundary condition is ω ( p → 0) = 0, or to leading order ωg ( p → 0) = 0, yielding (6.159) ωg = 0 With these estimates the material derivative becomes to leading order Dg ∂ ∂ ∂ D = +u·∇ +ω ≈ = + ug · ∇ Dt ∂t ∂p Dt ∂t

(6.160)

Now we turn to the momentum equations (6.151) and (6.152), use there the horizontal-wind decomposition (6.156) and the approximation (6.160) of the material derivative. Further taking the geostrophic equilibrium (6.153–6.154) into account and neglecting β yua in comparison to β yug we obtain Dg u g ∂a − f 0 va − β yvg = − Dt ∂x Dg vg ∂a + f 0 u a + β yu g = − Dt ∂y

(6.161) (6.162)

∂ (6.162)/∂ x − ∂ (6.161)/∂ y yields the equation Dg (ζg + f ) = − f 0 ∇ · ua Dt

(6.163)

for the quasigeostrophic vorticity ζg =

∂vg ∂u g 1 − = ∇h2  ∂x ∂y f0

(6.164)

The divergence of the ageostrophic wind can be obtained from the continuity equation. Due to the non-divergence of the geostrophic wind and the vanishing of the geostrophic pressure velocity the latter is ∂ωa ∇ · ua + =0 (6.165) ∂p so that the vorticity equation becomes Dg ∂ωa (ζg + f ) = f 0 Dt ∂p

(6.166)

In complete analogy to the procedure above in Sect. 6.1.2 we now use the entropy equation (here without heating, friction, heat conduction) for an estimate of the contribution from vortex-tube stretching on the right-hand side of the vorticity equation. As there we split potential temperature into the contribution from the reference atmosphere and the rest, i.e., θ = θ ( p) + θ 

(6.167)

200

6 Quasigeostrophic Dynamics of the Stratified Atmosphere

and thus approximate the entropy equation as Dg θ  dθ + ωa =0 Dt dp

(6.168)

Dg θ  dθ/d p Dt

(6.169)

This way we have ωa = −

1

Due to the hydrostatic equilibrium (3.91), the equation of state (2.3) and the definition (2.91) of potential temperature one has 

R

∂ ∂p

(6.170)

 =  ( p) + g + a

(6.171)

p θ =− R

p00 p

cp

Also the geopotential is split as

so that one finds p θ =− R θ ≈ −

p R

 

p00 p p00 p

R

cp

R

cp

∂ ∂p

(6.172)

∂g ∂p

(6.173)

Since, however, ( p00 / p) R/c p = θ /T , where T ( p) is the reference-atmosphere temperature, one obtains after again using the equation of state θ  = −ρθ

∂g ∂p

(6.174)

Here ρ( p) is the reference-atmosphere density. Finally, the pressure velocity (6.169) becomes   Dg ρθ ∂g ω = ωa = (6.175) Dt dθ /d p ∂ p This we now insert into the vorticity equation (6.166), leading us to the conservation equation for quasigeostrophic potential vorticity. First one obtains   Dg ρθ ∂g ∂ Dg f0 (6.176) (ζg + f ) = Dt ∂ p Dt dθ /d p ∂ p

6.1

Quasigeostrophic Theory and Its Potential Vorticity

201

The pressure derivative of the geostrophic wind (6.153–6.154) leads us to the relationships ∂u g 1 ∂ 2 g =− ∂p f 0 ∂ p∂ y ∂vg 1 ∂ 2 g = ∂p f 0 ∂ p∂ x

(6.177) (6.178)

by the help of which (6.176) is simplified to become   Dg ∂ Dg ρθ ∂g f0 (ζg + f ) = Dt Dt ∂ p dθ /d p ∂ p Now we define the streamfunction

(6.179)

g f0

(6.180)

1 dθ ρθ d p

(6.181)

ψ= and the stability parameter σ =−

and finally obtain the desired conservation equation Dg π =0 Dt with

π=

∇h2 ψ

∂ + f + ∂p



Dg ∂ ∂ψ ∂ ∂ψ ∂ = − + Dt ∂t ∂ y ∂x ∂x ∂ y

f 02 ∂ψ σ ∂p

 (6.182)

(6.183)

Note again that it holds in the absence of friction, heating, and heat conduction. A corresponding extension, however, is possible. A characteristic value for the stability parameter in midlatitudes is σ = 2·10−6 m2 /Pa2 s2 . Note also that (6.177) and (6.178) can be rewritten as the thermal-wind relationships   ∂u g 1 ∂ θ = ∂p f0 ρ ∂ y θ   ∂vg 1 ∂ θ =− ∂p f0 ρ ∂ x θ

(6.184) (6.185)

202

6.1.6

6 Quasigeostrophic Dynamics of the Stratified Atmosphere

A Quasigeostrophic Two-Layer Model

The vertical structure of important synoptic-scale processes is simple enough so that it may often suffice to consider the atmosphere in the approximation of two layers. The corresponding equations will be derived here, where we limit ourselves directly to the geometry of the β-plane. Starting point is the vorticity equation (6.166) which we here write  Dg  ∂ω ζg + f = f 0 Dt ∂p

(6.186)

and the entropy equation (6.168) which we express via (6.174) and (6.180) in the form Dg ∂ψ σ + ω=0 Dt ∂ p f0

(6.187)

For a representation of the dynamics we now pick two pressure layers as in Fig. 6.1. Streamfunction and horizontal winds are defined on the two main levels 1 and 2, while the pressure velocity is defined on the side and intermediate levels at the top (t) and bottom (b) boundary and between the two layers (m). On the latter we also define potential temperature. For a discretization we now approximate the vertical derivatives by finite differences. Thus one approximates (6.186) on the upper level at p = p1 as  Dg  2 ωm − ωt ∇ h ψ1 + f = f 0 Dt pm − pt

(6.188)

where the quasigeostrophic material derivative of an arbitrary field A in this layer is defined as Dg A ∂A (6.189) = + J (ψ1 , A) Dt ∂t The Jacobi operator applied to arbitrary fields B and C is

pt = 0 p1 = 250 mb pm = 500 mb p2 = 750 mb pb = 1000 mb Fig. 6.1 The vertical discretization of a two-layer model

6.1

Quasigeostrophic Theory and Its Potential Vorticity

J (B, C) =

∂ B ∂C ∂ B ∂C − ∂x ∂ y ∂ y ∂x

203

(6.190)

It has the following useful properties: J (A + B, C) = J (A, C) + J (B, C) J (α A, β B) = αβ J (A, B)

(6.191) (6.192)

J (A, B) = −J (B, A)

(6.193)

J (A, A) = 0

(6.194)

where α and β are constant factors. Furthermore as before the upper boundary condition for the pressure velocity is ωt = 0, and we define pm − p t = p b − p m =  p

(6.195)

so that (6.188) can also be written as   ∂∇h2 ψ1 f0 + J ψ1 , ∇h2 ψ1 + f = ωm ∂t p

(6.196)

The procedure for the lower level at p = p2 is analogous. There we also neglect the effects of friction and orography so that ωb = 0, thus yielding   ∂∇h2 ψ2 f0 + J ψ2 , ∇h2 ψ2 + f = − ωm ∂t p

(6.197)

Finally we discretize for the elimination of the pressure velocity ωm also the quasigeostrophic entropy equation (6.187) on the intermediate level at p = pm . For this purpose we approximate the geostrophic horizontal winds and the streamfunction there by the corresponding arithmetic mean between the two full levels. One obtains     ψ1 + ψ2 ψ2 − ψ1 σ ∂ ψ2 − ψ1 +J + ωm = 0 (6.198) , ∂t p 2 p f0 Now, however,

ψ2 − ψ1 ψ2 − ψ1 ψ1 + ψ2 = ψ1 + = ψ2 − 2 2 2 so that we can derive via (6.191–6.194) from (6.198) the two identities   f0 ∂ f0 ωm = − (ψ2 − ψ1 ) − J ψ1 , (ψ2 − ψ1 ) ∂t σ  p σp   f0 ∂ f0 ωm = − (ψ2 − ψ1 ) − J ψ2 , (ψ2 − ψ1 ) ∂t σ  p σp

(6.199)

(6.200) (6.201)

204

6 Quasigeostrophic Dynamics of the Stratified Atmosphere

Inserting (6.200) into (6.196), and (6.201) into (6.197), finally yield the two conservation equations 0=

∂πi + J (ψi , πi ) ∂t

π1,2 = ∇h2 ψ1 + f ± F(ψ2 − ψ1 )

where we have F=

f 02 σ  p2

(6.202)

(6.203)

π1 and π2 are, respectively, the potential vorticity in the upper and lower layer. For later reference we also note that the contribution ±F (ψ1 − ψ2 ) results from the elimination of ωm , and thus effectively from the vortex-tube stretching.

6.1.7

Summary

In the stratified atmosphere with variable density and altitude-dependent horizontal winds a comparatively closed treatment of dynamic phenomena on the synoptic scale can be achieved within the framework of quasigeostrophic theory. • For this purpose pressure and density are decomposed into a hydrostatic reference part and small deviations. • An analysis of the continuity equation shows that the ratio between the vertical and horizontal-wind scales cannot be larger than the ratio between the vertical and horizontal length scales. • The scale of the pressure fluctuations follows, under the assumption of small Rossby numbers, directly from the horizontal-momentum equation, where the pressure gradient must be balanced by the Coriolis force. • Since the horizontal scale is smaller than the earth’s radius by a factor of the order of the Rossby number one can use again the approximation of the tangential β-plane. • The ratios between vertical scale and horizontal scale or earth’s radius are also small, more precisely they are of order Ro2 and Ro3 . Finally we also assume that the squares of the horizontal scale and the external Rossby deformation radius have a ratio of O(Ro). This last assumption differs from quasigeostrophic shallow-water theory. • The scale of the density fluctuations follows under the same assumptions from the vertical momentum equation, where the vertical pressure gradient must balance gravity. • A Rossby-number expansion of all dynamic fields yields the following leading-order results: – The horizontal wind is in geostrophic balance. The pressure fluctuations act as streamfunction. – The fluctuations of pressure and density are in hydrostatic equilibrium.

6.2

•

•

• •

• •

Quasigeostrophic Energetics

205

– The vertical wind vanishes to leading order. This means that the scale estimate above for the vertical wind must be corrected by a Rossby-number factor. The resulting quasigeostrophic vorticity equation contains vortex-tube stretching. The vertical wind there must be determined from the entropy equation. The analysis of the latter and of potential temperature yields: – The vertical gradient of reference-potential temperature is sufficiently weak so that the potential-temperature fluctuations are of the order Ro2 . – Therefore they can be determined directly from the vertical gradient of the pressure fluctuations, which also leads to thermal-wind balance. – The vertical wind can be determined via the entropy equation from stability, heat sources and the geostrophic material derivative of the potential-temperature fluctuations. Inserting this vertical wind into the vorticity equation yields the conservation equation for quasigeostrophic potential vorticity. The latter can be determined, as all dynamic fields, from the streamfunction, i.e., the pressure fluctuations! Stability can be written as S = L 2di /L 2 , where L di is the important internal Rossby deformation radius. Similar to shallow-water theory the quasigeostrophic potential-vorticity-conservation equation can as well be derived directly from its general analogue, however under additional application of entropy equation. Quasigeostrophic potential vorticity is not simply an approximation of general potential vorticity under synoptic scaling. A heuristic derivation of quasigeostrophic theory in pressure coordinates illustrates the main steps further. Moreover, the formulation in pressure coordinates forms the starting point for the derivation of a quasigeostrophic two-layer model, reducing the dynamics in the vertical, via discretization, onto two pressure layers.

6.2

Quasigeostrophic Energetics of the Stratified Atmosphere

Energy conservation is a fundamental property both of the general equations of motion and of the primitive equations. Obviously, in the absence of friction, heating and heat conduction quasigeostrophic dynamics should have a corresponding property as well, and this section demonstrates that this is indeed the case. It also shows that energy can be exchanged between kinetic and available potential energy, the latter to be defined below, and how this can happen. For this we first consider the dynamics of the continuously stratified atmosphere and then that of the two-layer model. In both cases we neglect effects of friction, heating, and heat conduction. For simplicity we use the boundary conditions of the β-channel. The results also hold, however, in the case of periodic boundary conditions in both horizontal directions, or in the case of solid-wall boundary conditions in all horizontal directions.

206

6.2.1

6 Quasigeostrophic Dynamics of the Stratified Atmosphere

The Continuously Stratified Atmosphere

A common variant of β-plane dynamics, for the approximation of extratropical processes, uses the boundary conditions of a zonal channel (β-channel), spanning the globe parallel to a latitude circle. The model volume thus is 0 ≤ x ≤ Lx 0 ≤ y ≤ Ly 0≤ z 0. Thus kinetic energy is produced at the cost of available potential energy if either cold air sinks or warm air rises.

6.2.2

The Two-Layer Model

The quasigeostrophic two-layer model satisfies energy conservation in a manner very similar to that of the continuously stratified atmosphere, shown above. Beyond this one can also recognize in its context the transformation of baroclinic kinetic energy into barotropic kinetic energy, a process of relevance in the late development of extratropical weather systems. Again we assume in the horizontal the boundary conditions of the β-channel. Moreover, already in the derivation of the equations of the two-layer model we have assumed that ωb = ωt = 0.

The Conservation Law For a derivation of energy conservation we respectively multiply the two equations in (6.202) by the negative of the corresponding streamfunction, take the sum, and integrate the result over the total area of the β-channel, i.e., we form −

2 ) '

)

Ly 0

i=1

Lx

dy

 d x ψi

0

∂πi + J (ψi , πi ) = 0 ∂t

(6.230)

By steps completely analogous to the ones in the previous chapter we obtain the conservation equation dE =0 (6.231) dt for the total energy E=K+A (6.232) consisting of the kinetic energy 1 K = 2

)

)

Ly

dy 0

Lx

d x (∇h ψ1 · ∇h ψ1 + ∇h ψ2 · ∇h ψ2 )

(6.233)

0

with the horizontal streamfunction gradient ∇h ψi =

∂ψi ∂ψi ex + ey ∂x ∂y

(6.234)

212

6 Quasigeostrophic Dynamics of the Stratified Atmosphere

and of the available potential energy A=

1 2

)

)

Ly

Lx

dy 0

dx 0

Here we have κ=

κ2 (ψ1 − ψ2 )2 2

√ 2F

(6.235)

(6.236)

It has the dimension of a wave-number, and the corresponding wavelength is in midlatitudes of the order 2π/κ ≈ 3000km. Introducing the barotropic streamfunction ψ=

ψ1 + ψ 2 2

(6.237)

τ=

ψ1 − ψ 2 2

(6.238)

ψ1,2 = ψ ± τ

(6.239)

and the baroclinic streamfunction

so that one finally obtains dE =0 dt

E=K+A

(6.240)

with )

)

Ly

K =

Lx

dy )

0

)

Ly

A=

Lx

dy 0

d x (∇h ψ · ∇h ψ + ∇h τ · ∇h τ )

(6.241)

d x κ 2τ 2

(6.242)

0

0

The Exchange Rates The kinetic energy has barotropic and baroclinic parts. In the following we examine the exchange between the two, and between kinetic and available potential energy. For this we somewhat rewrite the basic equations. The mean [(6.196) + (6.197)]/2 of the two vorticity equations yields ∂ 2 (6.243) ∇ ψ + J (ψ, ∇h2 ψ + f ) = −J (τ, ∇h2 τ ) ∂t h while one obtains from the difference [(6.196) − (6.197)]/2 f0 ∂ 2 ∇h τ + J (τ, f ) = −J (τ, ∇h2 ψ) − J (ψ, ∇h2 τ ) + ωm ∂t p

(6.244)

6.2

Quasigeostrophic Energetics

213

Moreover, the entropy equation (6.198) can be written as ∂τ σp ωm + J (ψ, τ ) = ∂t 2 f0

(6.245)

We then obtain the desired exchange rates as follows: Multiplication of (6.243) by −2ψ and integration over the area of the β-channel yields )

d dt

)

Ly

Lx

dy 0

)

)

Ly

d x ∇h ψ · ∇h ψ = 2

Lx

dy

0

0

0

d x ψ J (τ, ∇h2 τ )

(6.246)

while multiplication of (6.244) by −2τ and the corresponding integration gives

d dt

)

)

Ly

Lx

dy 0

0

)

)

Ly

= −2

Lx

dy 0

0

Here we have used ) ) Ly dy 0

d x ∇h τ · ∇h τ d x ψ J (τ, ∇h2 τ ) −

)

Lx

d x τ J (ψ, ∇h2 τ ) = −

0

2 f0 p

)

Lx

d x τ ωm

(6.247)

d x ψ J (τ, ∇h2 τ )

(6.248)

dy 0

)

Ly

)

Ly

0

Lx

dy 0

0

which can easily be verified using the boundary conditions. Finally one obtains, integrating −2κ (6.245), d dt

)

)

Ly

Lx

dy 0

0

2 f0 dx κ τ = p

)

)

Ly

2 2

Lx

dy 0

d x τ ωm

(6.249)

0

In total one thus finds that ) K Cψτ

=2

)

Ly

Lx

dy 0

0

d x ψ J (τ, ∇h2 τ )

(6.250)

describes the transformation of baroclinic kinetic energy into barotropic kinetic energy, while ) ) Lx 2 f0 L y dy d x τ ωm (6.251) C AK = − p 0 0 describes the transformation of available potential energy into baroclinic kinetic energy. The last process we have already met in the continuously stratified case. Since, due to (6.174), (6.180), and (6.238) the potential temperature in the intermediate layer at p = pm is   θm = 2 f 0 ρθ p = p τ m

(6.252)

one finds that C AK > 0 if θm ωm < 0, thus again if colds air sinks or warm air rises.

214

6.2.3

6 Quasigeostrophic Dynamics of the Stratified Atmosphere

Summary

A fundamental property of quasigeostropic theory is that, beyond the material conservation of its potential vorticity, it also conserves its form of total energy. • In the absence of friction, heat conduction, and heat sources or sinks the sum of kinetic and available potential energy is conserved. The latter is contained in potential-temperature fluctuations. • Available potential energy can be transformed into kinetic energy if warm air rises and cold air sinks. • In addition, the two-layer model demonstrates the process of exchange between baroclinic kinetic energy and barotropic kinetic energy. • These theorems have been shown for β-channel geometry. They hold, however, also for other geometries.

6.3

Rossby Waves in the Stratified Atmosphere

Just as the linearized quasigeostrophic dynamics of the shallow-water equations yields free wave solutions, the stratified atmosphere does so too. As, e.g., in Fig. 6.2 such wave structures are always prominent in atmospheric data. These Rossby waves shall here be discussed in the two-layer-model approximation, followed by a treatment of the continuously stratified case. Effects of friction and orography are neglected. As boundary conditions we use those of the β-channel.

6.3.1

Rossby Waves in the Two-Layer Model

As easily verified, the two-layer-model equations (6.202) are satisfied by an altitudeindependent zonal flow (6.253) ψ1 = ψ2 = −U y The perturbation ansatz

ψi = −U y + ψi

(6.254)

yields, neglecting all nonlinear terms in the infinitesimally small ψi ,  ∂ψ  ∂ ∂  2  ∇h ψ1 + F(ψ2 − ψ1 ) + β 1 = 0 +U ∂t ∂x ∂x    ∂ψ ∂ ∂  2  ∇h ψ2 + F(ψ1 − ψ2 ) + β 2 = 0 +U ∂t ∂x ∂x 

(6.255) (6.256)

6.3

Rossby Waves in the Stratified Atmosphere

215

temperature at 1000 hPa (C) –72 –67 –64 –60 –56 –52 –48 –44 –40 –36 –32 –28 –24 –20 –16 –12 –8 –4

0

4

8

12 16 20 24 28 32 36 40 44 48 50

Fig. 6.2 Snapshot of geopotential (contours) and temperature (color shading) at 1000 mb. Note the clear wave structures in the geopotential. Copyright ©2021 European Center for Medium-Range Weather Forecasts (ECMWF). Source www.ecmwf.int. This data is published under a Creative Commons Attribution 4.0 International (CC BY 4.0). https://creativecommons.org/licenses/by/4.0/. ECMWF does not accept any liability whatsoever for any error or omission in the data, their availability, or for any loss or damage arising from their use

Again we decompose into 1  (ψ + ψ2 ) 2 1 1 τ  = (ψ1 − ψ2 ) 2

ψ =

so that

 ψ1,2 = ψ ± τ 

and form [(6.255) + (6.256)]/2, with the result   ∂ ∂ψ  ∂ +U ∇h2 ψ  + β =0 ∂t ∂x ∂x

(6.257) (6.258)

(6.259)

(6.260)

This is the equation for the barotropic mode. The one for the baroclinic mode is obtained from [(6.255) − (6.256)]/2 as    ∂τ  ∂ ∂  2  ∇h τ − κ 2 τ  + β +U =0 (6.261) ∂t ∂x ∂x Obviously in the linear limit the two modes are completely decoupled.

216

6 Quasigeostrophic Dynamics of the Stratified Atmosphere

Since the coefficients of their respective prognostic equations do not have any spatial or temporal dependence, a solution via Fourier ansatz seems possible. As sketched in Appendix 11.5.2, both of the two streamfunctions can be expressed, due to their periodicity in x, as Fourier series, ∞ ' 2π ψin (y, t) eikn x ; kn = n (6.262) ψi (x, y, t) = Lx n=−∞ The meridional boundary conditions are vi =

∂ψi =0 ∂x



One thus has for all n



ikn ψin = 0 Hence for all n = 0 ψin = 0



y = 0, L y

y = 0, L y

y = 0, L y





(6.263)



(6.264)

(n = 0)

(6.265)

π Ly

(6.266)

Following Appendix 11.5.2 one can then write ψin (y, t) =

∞ '

ψinm (t) sin(lm y) lm = m

m =1

(n = 0)

As shown in appendix F one can derive for the zonally symmetric part n = 0 via (6.211) ψi0 (y, t) = Di0 (y) +

∞ '

ψi0m (t) cos (lm y)

(6.267)

m =1

where Di0 is a quadratic polynomial in y. Hence ψi (x, y, t) = Di0 (y) ∞ ∞ ' ' ψinm (t) [δn0 cos(lm y) + (1 − δn0 ) sin(lm y)] eikn x (6.268) + n = −∞ m = 1

so that ψ  (x, y, t) = Dψ0 (y) +

∞ ∞ ' '

ψ nm (t) [δn0 cos(lm y) + (1 − δn0 ) sin(lm y)] eikn x (6.269)

n = −∞ m = 1

τ  (x, y, t) = Dτ0 (y) ∞ ∞ ' ' + τ nm (t) [δn0 cos(lm y) + (1 − δn0 ) sin(lm y)] eikn x (6.270) n = −∞ m = 1

6.3

Rossby Waves in the Stratified Atmosphere

217

where   ψ nm = ψ1nm + ψ2nm /2   τ nm = ψ1nm − ψ2nm /2   Dψ0 = D10 + D20 /2   Dτ0 = D10 − D20 /2

(6.271) (6.272) (6.273) (6.274)

Finally we express all ψ nm and τ nm as Fourier integrals in time so that ψ  (x, y, t) = Dψ0 (y) ∞ )∞ ∞ ' '

+

dω ψ nmω [δn0 cos(lm y) + (1 − δn0 ) sin(lm y)] ei(kn x−ωt)

n = −∞ m = 1−∞

(6.275) 

τ (x, y, t) =

Dτ0 (y)

∞ )∞ ∞ ' '

+

dω τ nmω [δn0 cos(lm y) + (1 − δn0 ) sin(lm y)] ei(kn x−ωt)

n = −∞ m = 1−∞

(6.276) We first discuss the barotropic mode. (6.275) inserted into (6.260) yields 0=

∞ )∞ ∞ ' '

dω ei(kn x−ωt) ψ nmω

n = −∞ m = 1−∞

  *  + × δn0 ω cos(lm y) − (1 − δn0 ) (ω − kn U ) kn2 + ln2 + kn β sin(lm y) (6.277) Therefore, with k0 = 0, nontrivial solutions ψ nmω = 0 require fulfillment of the dispersion relation βkn ω = ωψ (kn , lm ) = kn U − 2 (6.278) kn + lm 2 for barotropic Rossby waves. Hence one has  ψ nmω =  nm δ ω − ωψ (kn , lm )

(6.279)

with (nearly) free complex  nm . Inserting these results into (6.275) leads to  ψ  (x, y, t) = D10 (y) + D20 (y) /2 +

∞ ∞ ' '

 nm ei[kn x−ωψ (kn ,lm )t] [δn0 cos(lm y) + (1 − δn0 ) sin(lm y)]

n = −∞ m = 1

(6.280)

218

6 Quasigeostrophic Dynamics of the Stratified Atmosphere

Since ψ  is real, and ωψ (−kn , lm ) = −ωψ (kn , lm ), one must have  nm∗ =  −nm and hence with the decomposition  nm = | nm |eiαnm into amplitude and phase, and  0m ∈ R, 

ψ (x, y, t) =



D10 (y) +

D20 (y)

/2 +

∞ '

 0m cos(lm y)

m =1

+2

∞ ' ∞ '

| nm | sin(lm y) cos[kn x − ωψ (kn , lm )t + αnm ] (6.281)

n=1 m =1

Herein the zonally symmetric part is steady, while the longitude dependent part consists of Rossby waves, each propagating with a phase velocity cnm =

ωψ (kn , lm ) β =U− 2 kn kn + lm2

(6.282)

in zonal direction. With respect to the basic flow they move westwards. Their spatial structure is of interest as well. At the specific time t when ωψ (kn , lm )t − αnm = π/2 it is of the form sin(lm y) sin(kn x), illustrated for a few examples in Fig. 6.3. One recognizes the typical sequences of pressure highs and lows characteristic for synoptic-scale weather systems in midlatitudes.

n=1 m=1 y/L y

1 0.5 0

0

0.2

0.4

0.6

0.8

1

1 0.5 0 −0.5 −1

n=2 m=1 y/L y

1 0.5 0

0

0.2

0.6

0.4

0.8

1

1 0.5 0 −0.5 −1

n=1 m=2 y/L y

1 0.5 0

0

0.2

0.4

x/L x

0.6

0.8

1

1 0.5 0 −0.5 −1

Fig. 6.3 Horizontal structure of various barotropic Rossby waves, with zonal wavenumber kn and meridional wavenumber lm , at time t = (π/2 + αnm )/ωψ (kn , lm )

6.3

Rossby Waves in the Stratified Atmosphere

219

The dynamics at the base of the westward propagation can be understood be noting that, due to the vanishing baroclinic streamfunction τ  = 0 the streamfunction in both layers is identical to the barotropic streamfunction, i.e., ψ1 = ψ2 = ψ 

(6.283)

Hence (6.260) corresponds layer-wise to   ∂ψ  ∂ ∂ ∇h2 ψi + β i = 0 +U ∂t ∂x ∂x

(i = 1, 2)

(6.284)

which is each the linearization of D (ζi + f ) = 0 Dt

(6.285)

Therefore, barotropic Rossby waves conserve their absolute vorticity, which leads to westward propagation by the same mechanism as already discussed for short-wave shallow-water Rossby waves. The calculations for the baroclinic mode are completely analogous. Use of (6.276) in (6.261) leads to the dispersion relation ω = ωτ (kn , lm ) = kn U −

kn

2

βkn + lm 2 + κ 2

(6.286)

of baroclinic Rossby waves. These as well have a westward directed phase velocity with respect to the basic flow. Their structure is given by ∞ '  T 0m cos(lm y) τ  (x, y, t) = D10 (y) − D20 (y) /2 + m =1

+2

∞ '

∞ '

|T nm | sin(lm y) cos[kn x − ωτ (kn , lm )t + βnm ] (6.287)

n=1 m =1

with real T 0m and otherwise free T nm with corresponding phase βnm . Their barotropic streamfunction is ψ  = 0 so that the streamfunctions in the two layers are in opposite phase, i.e., (6.288) ψ1 = −ψ2 = τ  In the case of short baroclinic Rossby waves, for K nm 2 = kn 2 +lm 2 >> κ 2 , one can neglect in (6.261) κ 2 τ  in comparison with ∇h2 τ  so that approximately 

∂ ∂ +U ∂t ∂x



∇h2 τ  + β

∂τ  =0 ∂x

(6.289)

220

6 Quasigeostrophic Dynamics of the Stratified Atmosphere

Hence (6.284) holds layer-wise, so that the westward propagation is caused again by the conservation of absolute vorticity. In the case of long baroclinic Rossby waves with K nm 2  κ 2 the corresponding approximation is   ∂ ∂τ  ∂ κ 2τ  + β − +U =0 (6.290) ∂t ∂x ∂x Due to (6.288) the corresponding prognostic equations in the two layers are     ∂ψ  ∂ ∂ F ψ2 − ψ1 + β 1 = 0 +U ∂t ∂x ∂x       ∂ψ ∂ ∂ F ψ1 − ψ2 + β 2 = 0 +U ∂t ∂x ∂x

(6.291) (6.292)

This is the linear approximation of D Dt D Dt

    F ψ2 − ψ1 + f = 0

(6.293)

    F ψ1 − ψ2 + f = 0

(6.294)

These waves are therefore characterized by a balance between vortex-tube stretching and planetary-vorticity advection. For an illustration we consider the situation in Fig. 6.4 where

H

ϕ1' ϑ' = –ρsϑs ϕ2'

f ↓ → ψ1 – ψ2 ↑

∂ϕ' ϕ' – ϕ' ≈ ρsϑs 1 2 > 0 ∂p Δp

L

f ↑ → ψ 1 – ψ2 ↓

westward propagation of the anomaly Fig. 6.4 Dynamics of a positive potential-temperature anomaly in a long-wave baroclinic Rossby wave

6.3

Rossby Waves in the Stratified Atmosphere

221

we see a positive anomaly of the shear streamfunction, equivalent to a positive potentialtemperature anomaly. The upper-layer anomaly of geopotential and streamfunction is also positive (high-pressure anomaly), while the lower-layer anomaly is negative (low-pressure anomaly). Correspondingly the upper-layer geostrophic wind on the western flank is directed northward, while it is southward on the eastern flank. This corresponds to an increase (decrease) of planetary vorticity on the western (eastern) flank. Therefore the geopotential anomaly must increase (decrease) on the western (eastern) flank. In the lower layer conditions are opposite. The potential-temperature anomaly thus moves westwards, just as described by the dispersion relation.

6.3.2

Rossby Waves in an Isothermal Continuously Stratified Atmosphere

A continuously stratified case where Rossby waves can be treated comparatively easily is the one of a reference atmosphere with constant temperature T . It has a constant scale height H = RT /g so that pressure and density have the exponential profiles p(z) = p0 e−z/H p ρ(z) = RT

(6.295) (6.296)

with fixed reference surface pressure p0 . Its potential temperature then is  θ (z) = T

p00 p0

 R/c p

R z H

e cp

(6.297)

so that N2 =

g dθ R g = dz c θ p H

(6.298)

also is a constant. It is easy to convince oneself that the quasigeostrophic basic equation (6.119) is satisfied in the absence of heating by the constant zonal flow

The perturbation ansatz

ψ = −U y

(6.299)

ψ = −U y + ψ 

(6.300)

ψ ,

yields, neglecting all nonlinear terms in     f 02 ∂ψ  ∂ψ  1 ∂ ∂ ∂ 2  ∇h ψ + +β ρ 2 +U 0= ∂t ∂x ρ ∂z N ∂z ∂x

(6.301)

222

6 Quasigeostrophic Dynamics of the Stratified Atmosphere

Here the altitude dependence of the coefficients can be removed by the substitution ψ  = e z/2H ψr with the result  0=

∂ ∂ +U ∂t ∂x

 ∇h2 ψr



f2 + 02 N

(6.302)

∂ 2 ψr ψr − ∂z 2 4 H2

 +β

∂ψr ∂x

(6.303)

The rest is routine. Due to the β-channel boundary conditions ψr (x, y, z, t) =D 0 (y, z) ∞ )∞ ∞ ' '

+

)∞ dm

n = −∞ p = 1−∞

 dω ψ npmω δn0 cos(l p y) + (1 − δn0 ) sin(l p y)

−∞

× ei(kn x+mz−ωt) (6.304) corresponding to ψ  (x, y, z, t) = e z/2H D 0 (y, z) ∞ )∞ ∞ ' '

+

)∞ dm

n = −∞ p = 1−∞

 dω ψ npmω δn0 cos(l p y) + (1 − δn0 ) sin(l p y)

−∞

× e z/2H +i(kn x+mz−ωt) (6.305) As dispersion relation for non-trivial

ψ npmω

we obtain

ω = kn U − K + 2

βkn 2 f0 2 m N2

1 + 4L 2di

(6.306)

where K 2 = kn2 + l 2p is the squared total horizontal wavenumber. Clearly, κ 2 in the twolayer model here corresponds to m 2 f 02 /N 2 + 1/4L 2di . The special treatment of the zonally symmetric case n = 0 is analogous to its treatment in the two-layer model (appendix F).

6.3.3

Summary

As in the shallow-water equations the synoptic-scale variability of the stratified atmosphere is carried by Rossby waves.

6.4

Baroclinic Instability

223

• They can be determined as solutions of the linear equations obtained by expanding the dynamics about a state with constant zonal flow. • In the two-layer model one finds a barotropic and a baroclinic mode. – The dynamics of the barotropic mode corresponds to that of short-wave Rossby waves in the shallow-water equations. The advection of planetary vorticity is balanced by relative-vorticity advection, so that absolute vorticity is conserved. – In the baroclinic mode the two streamfunctions are opposite in phase so that they incorporate potential-temperature fluctuations. In the case of short wavelengths the dynamics on each layer is governed again by the conservation of absolute vorticity. At short wavelengths the planetary-vorticity advection is balanced by vortex-tube stretching. • In the continuously stratified atmosphere the isothermal case can be solved analytically. Instead of just two modes, as in the two-layer case, one obtains separate solutions for every vertical wavelength. • In general the β-channel boundary conditions lead to a horizontal Rossby-wave structure characterized by sequences of cyclones and anti-cyclones, as is characteristic for midlatitude synoptic-scale weather systems.

6.4

Baroclinic Instability

The daily extratropical synoptic-scale weather is essentially carried by baroclinic waves, as also discernible in the low-level geopotential heights in Fig. 6.5. The basic mechanism in the generation of these waves is the baroclinic instability of the zonal-mean atmosphere. Differential solar heating of the atmosphere produces warm tropics and cold polar regions. The corresponding potential-temperature distribution has meridional gradients ∂θ/∂ y on the northern (southern) hemisphere which are negative (positive). Due to the thermal-wind relation this implies ∂u/∂z > 0 which finds its expression in pronounced jet streams in midlatitudes (Fig. 3.8). These gradients are baroclinically unstable. The atmosphere reacts by the generation of baroclinic waves which transport heat from the tropics into the polar regions, thus working against the origin of the instability. The latter, the primary generator of synoptic weather in midlatitudes, shall be discussed here. We first consider the process in the two-layer-model approximation, followed by a discussion of the continuously stratified case. Without restriction of generality we limit ourselves to a discussion of the dynamics on the northern hemisphere.

224

6 Quasigeostrophic Dynamics of the Stratified Atmosphere

Fig. 6.5 950 mb geopotential height as predicted by DWD for September 30 in 2020. Note the chains of lows and highs discernible in the extratropics, to a large part due to baroclinic instability. Source: Deutscher Wetterdienst

6.4.1

Baroclinic Instability in the Two-Layer Model

The Linear Equations As starting point we take the inviscid equations of the quasigeostrophic two-layer model in a β channel without orography. The first of these is the prognostic equation (6.243) for the barotropic streamfunction which we repeat here: ∂ 2 ∇ ψ + J (ψ, ∇h2 ψ + f ) = −J (τ, ∇h2 τ ) ∂t h

(6.307)

The second equation we obtain by elimination of ωm from (6.244) and (6.245):  ∂  2 ∇ τ − κ 2 τ + J (τ, f ) = −J (τ, ∇h2 ψ) − J (ψ, ∇h2 τ − κ 2 τ ) ∂t h

(6.308)

It is easy to convince oneself that these equations are solved by ψ = −U y

(6.309)

τ = −U y

(6.310)

ψ1,2 = − (U ± U ) y

(6.311)

u 1,2 = U ± U

(6.312)

v1,2 = 0

(6.313)

so that

6.4

Baroclinic Instability

225

Here U is the barotropic part of the zonal wind velocity, and U the baroclinic part. The latter corresponds to a meridional potential-temperature gradient so that potential temperature is decreasing from south to north. One also finds, e.g., by inserting into (6.245), that the solution does not entail any vertical flow: (6.314) ωm = 0 We now examine the dynamics of infinitesimally small perturbations of this solution. We thus set       ψ −U y ψ (6.315) = + τ −U y τ with infinitesimally small ψ  and τ  . Inserting into (6.307) and (6.308) yields, neglecting all terms nonlinear in the perturbation fields,  ∂ ∂ψ  ∂ ∂ ∇h2 ψ  + β +U = − U ∇h2 τ  ∂t ∂x ∂x ∂x     ∂ ∂τ ∂ ∂  2  (∇h2 τ  − κ 2 τ  ) + β +U = − U ∇h ψ + κ 2 ψ  ∂t ∂x ∂x ∂x 

(6.316) (6.317)

In the following these equations shall be solved for arbitrary initial fields of ψ  and τ  .

The Solution of the Initial-Value Problem As shown in Sect. 6.3.1, due to the β-channel boundary conditions the barotropic and baroclinic streamfunction can be decomposed according to (6.269) and (6.270). Inserting into (6.316) and (6.317) eliminates Dψ0 and Dτ0 , since they do not depend on t and x. One obtains

0=

0=

∞ '

∞ ! '

dψ nm dt n = −∞ m =1   d 2 ψ nm + ikn U K nm + (1 − δn0 ) sin(lm y) dt " 2 τ nm −ikn βψ nm + ikn U K nm ∞ ' n = −∞

eikn x

eikn x

∞ ! ' m =1

2 δn0 cos(lm y)lm

(6.318)

 nm  2 + κ 2 dτ δn0 cos(lm y) lm dt

   d 2 + κ 2 τ nm − ik βτ nm K nm + (1 − δn0 ) sin(lm y) + ikn U n dt "    2 − κ 2 ψ nm +ikn U K nm

(6.319)

226

6 Quasigeostrophic Dynamics of the Stratified Atmosphere

yielding for the zonally symmetric parts with n = 0 and any m dτ 0m dψ 0m = =0 dt dt

(6.320)

i.e., the zonally symmetric part of the infinitesimally small perturbation does not develop in time. This is not the case for the longitude-dependent part, since in the subspaces for each n = 0 and m   ∂ 2 2 i − kn U K nm ψ nm + kn βψ nm = kn U K nm τ nm (6.321) ∂t   ∂ 2 2 i − kn U (K nm + κ 2 )τ nm + kn βτ nm = kn U (K nm − κ 2 )ψ nm (6.322) ∂t 2 = k 2 + l 2 is the squared total horizontal wave number. Now we must hold. Here again K nm n m form twice the imaginary part of ψ nm∗ (6.321) + τ nm∗ (6.322). The result is1

  2 ∂  2 + κ 2 |τ nm |2 = 2kn U κ 2 Im(ψ nm∗ τ nm ) K nm |ψ nm |2 + K nm ∂t

(6.323)

Hence, in the absence of any velocity shear U , the pseudoenergy   2 2 E  = K nm |ψ nm |2 + K nm + κ 2 |τ nm |2 is conserved within linear dynamics. Thus motivated we define the vector   K nm ψ nm nm  (t) = , 2 + κ 2 τ nm K nm with pseudoenergy norm

||2 = E 

(6.324)

(6.325)

(6.326)

The transformed Eqs. (6.321) and (6.322) can then be written in the compact form   ∂ i − kn U  nm = Hnm  nm ∂t with

 Hnm =

1 The asterisk denotes complex conjugation.

ωψ α α − γ ωτ

(6.327)

 (6.328)

6.4

Baroclinic Instability

227

Here βkn 2 K nm βkn ωτ = − 2 K nm + κ 2

ωψ = −

(6.329) (6.330)

are the intrinisc frequencies of barotropic and baroclinic Rossby waves in a reference frame moving zonally at velocity U , and kn U K nm α= , 2 + κ2 K nm

(6.331)

κ2 α 2 K nm

(6.332)

γ =

are contributions to Hnm , nonzero only if a zonal-wind shear U = 0 exists. Fourier transformation of (6.327) in time, so that ) ∞  nm (t) = dω nmω e−iωt −∞

(6.333)

yields the eigenvalue equation ω ˆ nmω = Hnm  nmω

(6.334)

ωˆ = ω − kn U

(6.335)

where is the intrinsic frequency observed in a reference frame moving at velocity U in zonal direction. Non-trivial solutions  nmω must be eigenvectors of Hnm . The two eigenvalues ωˆ 1,2 are determined via   (6.336) det Hnm − ωˆ i I = 0 They hence solve (ωˆ i − ωψ )(ωˆ i − ωτ ) = α (α − γ )

(6.337)

Since the coefficients of Hnm are all real, the two eigenvalues are either real or a complexconjugate pair, i.e., ωˆ 1 = ωˆ 2∗ . Up to a normalization factor the corresponding eigenvectors  nm 1,2 are determined by ωˆ i  inm = Hnm  inm

(6.338)

The general solution of (6.327) is therefore  nm (t) =

2 ' j =1

nm −iω j t  nm j Aj e

(6.339)

228

6 Quasigeostrophic Dynamics of the Stratified Atmosphere

or  nm (t) =

2 '

nm −i (kn U +ωˆ j )t  nm j Aj e

(6.340)

j=1

For the determination of the Anm j from the initial state we additionally consider the adjoint problem t inm (6.341) αˆ i inm = Hnm As one can easily convince oneself, the eigenvalues are the same as above. Because they are either real-valued or a complex-conjugate pair, we can order them as αˆ i = ωˆ i∗

(6.342)

Since  †   †  nm †  t nm † nm ωˆ j inm  nm Hnm  nm  j = ωˆ i inm  nm j = i j = Hnm i j one has



ωˆ j − ωˆ i

and therefore for i = j



  nm † nm i j = 0

inm

†

 nm j =0

(6.343)

(6.344) (6.345)

Eigenvectors to different eigenvalues are orthogonal to each other. Without loss of generality we choose their normalization factors so that 

inm

†

 nm j = δi j

(6.346)

The initial perturbation  nm (0) =

2 '

nm  nm j Aj

(6.347)

j =1

can be projected directly onto the eigenvectors, with the result †  nm Anm  nm (0) j = j

(6.348)

The solution in physical space is reconstructed by determining, according to (6.325), for the nm j-th eigenvector ψ nm j and τ j so that   nm j

=

,

K nm ψ nm j 2 + κ 2 τ nm K nm j

 (6.349)

6.4

Baroclinic Instability

229

where the index j does not indicate a layer! Then the general solution is 

ψ τ



 (x, y, t) =

Dψ0 Dτ0

 (y)

 0m  ∞  ' ψ + δn0 cos (lm y) e τ 0m n = −∞ m =1   2 nm ' ψ nm −i (kn U +ωˆ j )t j + (1 − δn0 ) sin(lm y) e Aj τ nm j ∞ '

ikn x

(6.350)

j=1

where we also express the fact that only real fields can result from real initial conditions. This solution is brought into a more explicit form in appendix G.

Baroclinic Waves and Their Structure In the following we consider three cases. For transparency of notation we often suppress the indices n and m. No Zonal-Wind Shear (U = 0) In the absence of zonal-wind shear one has α (α − γ ) = 0. With this one obtains as solutions a superposition of the well-known free Rossby waves with (6.351) ωˆ 1,2 = ωψ,τ Zonal-Wind Shear, but no β-effect (U = 0, β = 0) In this case we have ωψ = ωτ = 0

(6.352)

ωˆ i2 = α(α − γ )

(6.353)

The intrinsic frequencies thus satisfy

or ωˆ i2 = k 2 U 2

2 − κ2 K nm 2 + κ2 K nm

(6.354)

2 < κ 2 . For these we get The interesting case is the one of long waves with K nm

ωˆ 1,2 = ±i with a growth rate

 = kU

2 κ 2 − K nm 2 κ 2 + K nm

(6.355)

(6.356)

230

6 Quasigeostrophic Dynamics of the Stratified Atmosphere

The time dependence of the perturbation is   2 nm ' nm ψ j e−i (kn U +ωˆ j )t Aj τ nm j j =1  nm   nm  ψ1 −ikn U t t nm ψ2 e e−ikn U t e−t = Anm e + A 1 2 τ1nm τ2nm

(6.357)

The first part thus grows exponentially! This is the baroclinic instability.2 From a virtually arbitrary initial perturbation, with a nonzero projection onto such a baroclinic wave, the latter will grow and progressively dominate. This growth can only be halted by nonlinear dynamics. Since an initial perturbation typically has contributions from different n and m, one usually observes a superposition of growing waves, which again will finally be dominated by the fastest growing mode. We note the following: • Without β-effect one finds an instability for all U . • This necessitates K < κ. The corresponding wavelength is in midlatitudes λ=

2π 2π > ≈ 3000 km K κ

(6.358)

Baroclinic instability is characterized by relatively large wavelengths. The extension of the corresponding pressure anomalies is approximately λ/2. This is consistent with the synoptic-scale estimate L = 1000 km. • The growth rate is largest at ∂ ∂ = =0 (6.359) ∂k ∂l √  2 = 2 − 1 κ 2 , and thus This leads to l = 0 and K nm . (k, l)max =

√ 2 − 1κ, 0

 (6.360)

However, since the smallest possible meridional wavenumber component is l = π/L y , the more precise result is ⎧ / ⎫ ⎤ 0⎡/  0  ⎪ ⎪ ⎨ 0 0 0 π2 π ⎬ π2 (6.361) (k, l)max = κ 1⎣12 1 + 2 2 − 1⎦ − 2 2 , ⎪ κ Ly κ Ly Ly ⎪ ⎩ ⎭ For L y  π/κ ≈ 1500 km, however, the difference is negligible. The zonal wavelength for maximum growth then is in midlatitudes 2 A critical reader might remark that complex frequencies are not really admitted by Fourier trans-

forms. The mathematically cleanest way would be applying Laplace transforms, but the final results are basically the same.

6.4

Baroclinic Instability

231

λ=

2π 2π ≈ ,√ ≈ 5000 km kmax 2 − 1κ

(6.362)

For a determination of the structure of a baroclinically unstable wave we Fourier transform (6.321) in time, obtaining 2 2 ωK ˆ nm ψ nmω + kn βψ nmω = kn U K nm τ nmω

With β = 0, (6.355), and (6.356) this leads for the growing baroclinic wave to 2 π  κ 2 − K nm nm nm τ1 = i ei 2 ψ1nm ψ1 = 2 2 kn U κ + K nm Thus the corresponding streamfunctions in the two layers are   2 κ 2 − K nm nm nm nm nm 1±i (ψ1,2 )1 = ψ1 ± τ1 = ψ1 2 κ 2 + K nm √ 2κ e±i ψ1nm = , 2 2 κ + K nm -

where  = arctan

2 κ 2 − K nm 2 κ 2 + K nm

(6.363)

(6.364)

(6.365)

(6.366)

is half the phase difference between the upper and the lower layer. In the subspace of the nm iα wavenumber combination with largest growth we may now decompose Anm 1 ψ1 = A ψ e , so that the barotropic streamfunction of the growing part is, due to (6.350),   nm i(kn x−kn U t) t ψ  (x, y, t) =  Anm e 1 ψ1 sin(lm y) e = Aψ sin(lm y) cos (kn x − kn U t + α) et and thus the baroclinic streamfunction, via (6.364),   nm i(kn x−kn U t) t τ  (x, y, t) =  Anm τ sin(l y) e e m 1 1 2 κ 2 − K nm π = Aψ sin(lm y) cos(kn x − kn U t + α + )et 2 2 κ + K nm 2

(6.367)

(6.368)

It thus leads the barotropic streamfunction by a phase difference π/2, correspondingly in x by x = λ/4, where λ = 2π/kn is the zonal wavelength of the wave. The streamfunctions in the two layers thus are

232

6 Quasigeostrophic Dynamics of the Stratified Atmosphere  ψ1,2 (x, y, t) = ψ ± τ √

= ,

κ2

2κ

2 + K nm

Aψ sin(lm y) cos(kn x − kn U t + α ± )et

(6.369)

The upper-layer streamfunction leads the barotropic streamfunction in phase by , or in x by x = λ/2π . By the same difference the lower-layer streamfunction follows the barotropic streamfunction. The resulting westward tilt of the phase with increasing altitude is sketched in Fig. 6.6. Note that the middle-layer potential temperature θm is given, due to (6.252), up to a constant factor by the baroclinic streamfunction. The meridional-wind velocity there is v  = ∂ψ  /∂ x, leading ψ  in phase by π/2. Thus v  is in phase with θ  . The westward tilt thus implies that warm air is transported northwards, and cold air southwards. The baroclinic wave thus operates against the cause of its instability. Zonal-Wind Shear and β-Effect (U = 0, β = 0) In the general case the solution of (6.337) is   ωψ + ω τ ωψ − ωτ 2 + α(α − γ ) (6.370) ± ωˆ 1,2 = 2 2 The argument of the square root must be negative for an instability. This implies

ridge

ridge trough

p = p1

ψ1

ψ

p = pm

τ ψ2

p = p2

0

λ 4

λ 2

3λ 4

λ x – Ut

Fig. 6.6 Longitude–altitude structure of a growing baroclinic wave. Note the westward tilt of the phase. The potential temperature in the intermediate layer, proportional to τ , is in phase with the meridional wind v = ∂ψ/∂ x at the same altitude

6.4

Baroclinic Instability

233

 α (α − γ ) < 0 and

ωψ − ωτ 2

2 < α (γ − α)

leading to 2 < κ2 K nm

and

4 4 β 2 κ 4 < 4U 2 K nm (κ 4 − K nm )

(6.371)

The β-effect thus stabilizes the flow. At a given total K nm an instability is only possible if U 2 > G (K nm ) =

β2κ 4 4 (κ 4 − K 4 ) 4K nm nm

(6.372)

This is also sketched in Fig. 6.7. No instability is possible if U 2 is below the minimum of G. The latter is κ2 β2 2 =√ min G (K ) = 4 at K nm κ 2 Thus the flow is only unstable if U 2 >

β2 κ4

(6.373)

The following is also important:

∆U2

unstable region G(K) =

β2 κ4 – K4)

4K4(κ4

β2/κ4

κ2/√2

κ2

K2

Fig. 6.7 For a quasigeostrophic two-layer model, the baroclinically unstable region in a K 2 − U 2 diagram

234

6 Quasigeostrophic Dynamics of the Stratified Atmosphere

• In the unstable regime the growth rate =



α(γ − α) −

ω ψ − ωτ 2

2 (6.374)

is again largest at l = π/L y . The zonal wavenumber with largest instability is close to kmax ≈

κ 21/4

(6.375)

corresponding to a wavelength λ = 21/4 2π/κ ≈ 3000 km. • The latitude dependence of the potential for baroclinic instability is also interesting. The definitions of κ and β imply β σ  p 2 cos φ0 = (6.376) κ2 4a sin2 φ0 The minimum zonal-wind shear which must be exceeded for an instability also depends on the reference latitude φ0 . On the synoptic scale, which here is the focus, the tropics are much more stable than the midlatitudes (Fig. 6.8). At φ0 = 45◦ one finds that U > 3 m/s must be satisfied, corresponding to a zonal-wind shear between the two layers of 6 m/s.

Fig. 6.8 Latitude dependence of the minimum zonal-wind shear necessary for a baroclinic instability. The tropics are much more stable than the midlatitudes

β/κ2

tropics more stable!

equator

ϕ0

6.4

Baroclinic Instability

235

Mechanisms and Energetics For the further analysis of the mechanisms and energetics of the baroclinic instability we linearize (6.243), (6.244), and (6.245) directly about ⎛

⎞ ⎛ ⎞ ψ −U y ⎝ τ ⎠ = ⎝ −U y ⎠ ωm 0

(6.377)

The result is 

 ∂ψ  ∂ ∂ ∂ +U ∇h2 ψ  + β = −U ∇h2 τ  ∂t ∂x ∂x ∂x    ∂τ f0  ∂ ∂ ∂ ∇h2 τ  + β +U = −U ∇h2 ψ  + ω ∂t ∂x ∂x ∂x p m   ∂ ∂ψ  ∂ σp  τ  − U ω +U = ∂t ∂x ∂x 2 f0 m

(6.378) (6.379) (6.380)

2  Together with the boundary conditions of the β-channel the integral −2 d 2 x ψ  (6.378) + τ  (6.379) ] is )L y

dK 2 f0 =− dt p K =

)L y

while 2κ 2

2

0

 d x τ  ωm

(6.381)

0

)L x dy

0

)L x dy

  d x ∇h ψ  · ∇h ψ  + ∇h τ  · ∇h τ 

(6.382)

0

d 2 xτ  (6.380) yields

d A 2 f0 = dt p A =

)L y

)L x dy

0

)L y

0

)L x dy

0

dx

 τ  ωm

)L y + 2U κ

2

)L x dy

0

d x κ 2 τ 2

0

dx τ

∂ψ  ∂x

(6.383)

(6.384)

0

A perturbation grows if d (K + A) > 0 dt

(6.385)

236

6 Quasigeostrophic Dynamics of the Stratified Atmosphere

which is equivalent to )L y 2U κ

2

)L x dy

0

0

dx τ

∂ψ  >0 ∂x

(6.386)

In the integral one must also have predominantly τ  ∂ψ  /∂ x > 0, which is equivalent to f 0 θm v  > 0 so that warm air is transported to the pole and cold air to the equator. As a consequence, available potential energy A¯ of the basic flow is reduced and exchanged into available potential energy A of the perturbations. As we have already convinced us above, this is the case for growing baroclinic waves. Vice versa an oppositely directed mean transport so that f 0 θm v  < 0 implies that the perturbation is damped. The other partial process we have already met in the discussion of the general energetics of the two-layer model: If )L y )L x 2 f0  dy d x τ  ωm >0 (6.387) − p 0

0

A

available potential energy of the perturbation is transformed into kinetic energy K  of the perturbation. Since up to a factor τ  is equivalent to θm , the condition for this is that warm air rises and cold air sinks. Also this can be checked directly for growing baroclinic waves. For simplicity we limit ourselves for this to the case β = 0: First, the Fourier transform of (6.380) yields 2i f 0 nmω ωm =− (6.388) (ωτ ˆ nmω + kn U ψ nmω ) σp Together with ωˆ = ωˆ 1 = i, (6.356), and (6.364) one obtains from this for the growing baroclinic wave 4 f0 K2 (ωm )nm (6.389)  2 nm 2 τ1nm 1 =− σ  p κ − K nm  and τ  have opposite phases, as we could show. Figures 6.9 One thus sees that in this wave ωm and 6.10 summarize what we have learned.

6.4.2

Baroclinic Instability in a Continuously Stratified Atmosphere

The Linear Equations Starting point for the analysis of a continuously stratified atmosphere is the quasigeostrophic potential-vorticity conservation Eq. (6.119) without heating. As boundary conditions we use in the horizontal those of the β-channel, i.e., (6.204–6.206). Orography and heating are neglected, so that the vertical boundary condition at the ground is given by (6.207). As upper boundary condition we can use (6.208). In the case of an approximation where a solid upper boundary of the atmosphere is assumed at altitude H , we can use alternatively, in analogy with (6.207),

6.4

Baroclinic Instability

237

ridge

trough

ridge ωm ' 0 τ' > 0 warm

ϑ' < 0 τ' < 0 cold

v' < 0

v' > 0

Fig. 6.9 Longitude–altitude section of the exchange processes determining the energetics of a growing baroclinic wave in the northern hemisphere: The meridional heat transport conveys warm (cold) air to the north (south). The vertical transport leads to upward (downward) motion of warm (cold) air masses

A

available potential energy of the basic flow

v' ϑ' > 0

A'

ω' ϑ' < 0

K'

wave

Fig. 6.10 Schematic representation of the energy-exchange processes in a baroclinic instability. The meridional transport transforms available potential energy in the basic flow into available potential energy of the growing wave, while vertical heat transport transforms the latter into kinetic energy of the wave

238

6 Quasigeostrophic Dynamics of the Stratified Atmosphere

( Dg ∂ψ (( =0 Dt ∂z (z = H

(6.390)

It is easy to convince oneself that these equations are solved by a zonally symmetric and steady streamfunction ψ = ψ¯ (y, z) (6.391) with corresponding horizontal wind fields u¯ = − v¯ = 0

∂ ψ¯ ∂y

(6.392) (6.393)

At a sufficiently strong vertical gradient of u¯ we again expect a baroclinic instability. With the aim of a corresponding analysis we linearize the equations about this basic flow, i.e., we use the perturbation ansatz ψ = ψ¯ (y, z) + ψ  (x, y, z, t)

(6.394)

with infinitesimally small ψ  , insert this into the equations and then neglect all contributions which are nonlinear in ψ  . Inserting this into (6.119) thus leads to 

where

∂ ∂ + u¯ ∂t ∂x



π +

1 ∂ π  = ∇h2 ψ  + ρ ∂z

∂ψ  ∂ π¯ =0 ∂x ∂ y 

ρ f 02 ∂ψ  N 2 ∂z

(6.395)

 (6.396)

is the quasigeostrophic potential vorticity of the perturbation, and   ∂ 2 ψ¯ 1 ∂ ρ f 02 ∂ ψ¯ π¯ = + f + ∂ y2 ρ ∂z N 2 ∂z

(6.397)

that of the basic flow, with meridional gradient ∂ π¯ 1 ∂ ∂ 2 u¯ =− 2 +β − ∂y ∂y ρ ∂z



ρ f 02 ∂ u¯ N 2 ∂z

 (6.398)

The linearization of the meridional boundary conditions (6.206) leads to ∂ψ  =0 ∂x

(y = 0, L y )

(6.399)

6.4

Baroclinic Instability

239

while the vertical boundary condition (6.207) yields   ∂ ∂ ∂ψ  ∂ψ  ∂ ∂ ψ¯ 0= + u¯ + ∂t ∂ x ∂z ∂ x ∂ y ∂z or

 0=

∂ ∂ + u¯ ∂t ∂x



(z = 0)

∂ψ  ∂ψ  ∂ u¯ − ∂z ∂ x ∂z

(z = 0)

(6.400)

(6.401)

If needed such a boundary condition can also be applied at a solid upper boundary at z = H . As in the case of the two-layer model we can assume, without restriction of generality, that the perturbation can be represented as ') ∞  (6.402) dωei(kx−ωt) ψˆ (k, y, z, ω) ψ = k

−∞

where the contributing zonal wavenumbers are k=n

2π Lx

(n  Z)

(6.403)

Inserting this into (6.395) yields 

1 ∂ ∂ 2 ψˆ ¯ −k ψˆ + + (ω − k u) 2 ∂y ρ ∂z



2

f 2 ∂ ψˆ ρ 02 N ∂z

 − k ψˆ

∂ π¯ =0 ∂y

(6.404)

while the boundary conditions (6.399) and (6.401) lead to ψˆ = 0 and ¯ (ω − k u)



y = 0, L y k = 0

∂ ψˆ ∂ u¯ + k ψˆ =0 ∂z ∂z



(z = 0)

(6.405)

(6.406)

Similar to the two-layer model we do not expect any wave growth in the zonally symmetric case k = 0 so that we limit ourselves in the following to longitude dependent perturbations with k = 0.

The Rayleigh Theorem A closed analytical treatment of the linear equations is only possible in special cases. Beyond these, however, there is a general theorem that tells us under which conditions a zonally symmetric flow can become unstable within the framework of quasigeostrophic theory at all. For this we assume à priori that ω = ωr + i

( > 0)

and examine under which cases this does not lead to a contradiction.

(6.407)

240

6 Quasigeostrophic Dynamics of the Stratified Atmosphere

We first take the real and imaginary part of (6.404)/(ω − k u): ¯     f 02 ∂ ψˆ r 1 ∂ ∂ π¯ ∂ π¯ ∂ 2 ψˆ r 2 − k ψˆ i = 0 + ρ + kδ ψˆ r − kδi r ∂ y2 ρ ∂z N 2 ∂z ∂y ∂y     f 02 ∂ ψˆ i ∂ 2 ψˆ i 1 ∂ ∂ π¯ ∂ π¯ 2 − k + kδr ψˆ i + kδi ψˆ r = 0 + ρ 2 2 ∂y ρ ∂z N ∂z ∂y ∂y

(6.408) (6.409)

where the Fourier transform of the streamfunction has been decomposed into ψˆ = ψˆ r + i ψˆ i

(6.410)

and we have δr = δi =

ωr − k u¯

(6.411)

¯ 2 + 2 (ωr − k u) 

(6.412)

¯ 2 + 2 (ωr − k u)

Moreover, real and imaginary part of (6.406)/(ω − k u) ¯ are  ∂ ψˆ r ∂ u¯  δr ψˆ r + δi ψˆ i = 0 +k ∂z ∂z  ∂ ψˆ i ∂ u¯  δr ψˆ i − δi ψˆ r = 0 +k ∂z ∂z

(6.413) (6.414)

Now we form ψˆ i (6.408) −ψˆ r (6.409) and multiply this by ρ, with the result      f 02 ∂ ψˆ r ∂ ψˆ i ∂ ψˆ r ∂ ψˆ i ∂ π¯ ∂ ∂ ψˆ i ρ ρ 2 ψˆ i − ψˆ r + − ψˆ r − ρkδi ∂y ∂y ∂y ∂z N ∂z ∂z ∂y

( (2 ( ˆ( (ψ ( = 0 (6.415)

This we integrate in y and z and obtain )∞ 0



∂ ψˆ r ∂ ψˆ i dzρ ψˆ i − ψˆ r ∂y ∂y

)∞ =

)L y dz

0

dyρkδi 0

∂ π¯ ∂y

L y

( (2 ( ˆ( (ψ (

)L y +

0

0



f2 dy ρ 02 N

 ψˆ i

∂ ψˆ r ∂ ψˆ i − ψˆ r ∂z ∂z

∞ 0

(6.416)

6.4

Baroclinic Instability

241

Application of the boundary conditions (6.405) and (6.208) yields )∞

)L y dz

0

0

∂ π¯ dyρkδi ∂y

L    ( (2 ) y f 02 ∂ ψˆ r ∂ ψˆ i ( ˆ( ˆ ˆ − ψr (ψ ( + dy ρ 2 ψi N ∂z ∂z 0

=0

(6.417)

z=0

From (6.413) and (6.414) we furthermore obtain ψˆ i

∂ ψˆ r ∂ ψˆ i ∂ u¯ (( ((2 − ψˆ r = −k δi (ψˆ ( ∂z ∂z ∂z

(6.418)

so that (6.417) becomes, using (6.412), ⎛ ⎡ ( (2 ( (2 ⎞ Ly ( ˆ( ( ˆ( )∞ )L y ) 2 ψ ψ ∂ π¯ ( ( ⎜ f 0 ∂ u¯ ( ( ⎢ ⎟ − dy ρ k ⎣ dzρ dy ⎝ ⎠ ∂ y |ω − k u| N 2 ∂z |ω − k u| ¯2 ¯2 0

0

0

⎤ ⎥ ⎦=0

(6.419)

z=0

In the case of an instability, however, one has  > 0 so that we obtain as condition for this )∞

)L y dzρ

0

dy 0

( (2 ( ˆ( (ψ (

∂ π¯ − ∂ y |ω − k u| ¯2

)L y 0

⎛

( (2 ⎞ ( ˆ( 2 ψ f ⎜ 0 ∂ u¯ ( ( ⎟ dy ⎝ρ 2 ⎠ N ∂z |ω − k u| ¯2

=0

(6.420)

z=0

In the application of this three interesting cases must be considered: No Wind Shear at the Ground ( ∂ u/∂z| ¯ z=0 = 0): In this case the second integral vanishes. For the other to also do so, ∂ π/∂ ¯ y must change sign in the model volume. This is the Rayleigh condition, also to be used as condition for a barotropic instability. Positive Meridional Gradient of Basic-Flow Potential Vorticity (∂ π/∂ ¯ y ≥ 0): This is the typical case, since typically planetary vorticity dominates so that ∂ π¯ ≈β>0 ∂y Clearly, in this case we must have at least locally ( ∂ u¯ (( >0 ∂z (z=0 This is the scenario of a baroclinic instability.

(6.421)

(6.422)

242

6 Quasigeostrophic Dynamics of the Stratified Atmosphere

Negative Wind Shear Everywhere at the Ground ( ∂ u/∂z| ¯ z=0 < 0): This requires that at least locally ∂ π¯ 0 is also positive. Whenever they are absorbed in the stratosphere, e.g., by wave breaking near critical layers, one gets ∂ Fz /∂z < 0. Likewise the absorption of waves also propagating horizontally leads in general to a convergence of the Eliassen–Palm flux ∇ · F < 0. Hence f 0 v∗ > 0 so that the residual meridional velocity in the northern (southern) hemisphere is positive (negative); i.e., it is directed from the equator to the poles.

4 In the mesosphere higher up small-scale gravity waves play the dominant role. In the present

formulation their influence can be captured via corresponding forcing F = 0 and heating Q = 0.

342

8.6

8 The Interaction Between Rossby Waves and the Mean Flow

Recommendations for Further Reading

Good textbook coverage of wave–mean flow interactions can be found in the books of Holton and Hakim (2013), Pedlosky (1987), and Vallis (2006), but especially worthwhile are the texts from Andrews et al. (1987) and Bühler (2009). The latter gives an excellent account of the generalized Lagrangian mean theory that has been introduced by Andrews and McIntyre (1978a, b), and that forms an important part of the basis of our understanding of the interactions between waves and mean flows, including the TEM and the non-acceleration theorem. Fundamental publications on meridional and vertical Rossby-wave propagation are Charney and Drazin (1961), Hoskins and Karoly (1981), and Karoly and Hoskins (1982). The Eliassen–Palm flux has been introduced by Eliassen and Palm (1961).

9

The Meridional Circulation

The knowledge gathered so far, on the dynamics of the atmosphere in general and on wavemean-flow interaction in particular, will now be applied in a discussion of the mechanism of the general circulation of the atmosphere. The goal is to achieve a better understanding, beyond diagnostics, why circulation, zonal mean of wind and temperature, and waves arrange themselves as observed. Important open questions still remain, but the theory of atmospheric dynamics can explain a lot. We will here first sketch a few essentials of the empirical findings, then discuss the circulation in the tropics and finally we will consider the midlatitudes.

9.1

Some Essentials of the Empirical Basis

We first consider the radiation budget. Figure 9.1 shows the latitude-dependent power density of the zonal mean of the incoming solar radiation at the top of the atmosphere and the zonal mean of the outgoing infrared radiation. The two profiles do not match exactly. According to the Stefan–Boltzmann law, the outgoing atmospheric radiation L corresponds to a radiative temperature T so that L = σ T 4 . Hence, the atmospheric radiative temperature is in polar regions warmer and in the tropics colder than expected from simple radiative equilibrium. Obviously there must be transport processes that transport thermal energy from the tropics to the polar regions. In addition to the oceans, the atmosphere has a considerable share in this effect. Here the direct atmospheric latitude–altitude–circulation contributes as well as synoptic-scale waves due to baroclinic instability. The resulting latitude–altitude distribution of the zonally averaged potential temperature of the troposphere, together with the zonal-mean zonal wind, is shown in Fig. 9.2. Besides the general decrease in temperature from the equator to the poles one notes a strongly baroclinic zone in the subtropics, together with the corresponding thermal westerlies. The jet stream, however, extends further into the midlatitudes, where it has a more barotropic structure. Correspondingly the mid latitude surface winds are westerlies as well, while in © Springer-Verlag GmbH Germany, part of Springer Nature 2022 U. Achatz, Atmospheric Dynamics, https://doi.org/10.1007/978-3-662-63941-2_9

343

344

9 The Meridional Circulation 350 absorbed shortwave

300 250

outgoing longwave

200 Irradiance (W m–2)

Fig. 9.1 Schematic representation of the latitudinal dependency of the zonal mean of the incoming solar radiation at the top of the atmosphere, the emitted infrared radiation, and the net radiation (incoming–outgoing) resulting from the two

150 100 net radiation 50 0 –50 –100 –150 0

30°N

60°N

90°N

latitude

the tropics and in the polar latitudes easterlies prevail. We also recall Fig. 8.7, showing the Eulerian-mean meridional circulation for northern-hemisphere winter. One can see the two Hadley cells in the tropics, with a much stronger cell on the winter side. This circulation is direct, i.e., in concordance with the thermal structure so that warm air masses rise and cold air masses sink. The Hadley circulation is flanked by the Ferrel cells that are thermally indirect. Obviously wave driving must be of significance here. In polar latitudes one can identify again weak direct circulation cells.

9.2

The Hadley Circulation

With regard to the circulation in the tropics we begin by discussing the zonally symmetric dynamics without waves and with symmetry between the northern and southern hemispheres. This is based on the work of Schneider (1977) as well as Held and Hou (1980). Next we will address the summer–winter–asymmetry and finally also consider the influence of waves.

9.2.1

The Basic Equations of a Model Without Wave Driving

We consider the primitive equations on a sphere, where we assume that there are no waves so that everything is zonally symmetric. Additionally we only look for steady solutions. As important dynamical aspects, we allow for turbulent friction and diffusion in the planetary boundary layer, and differential heating between the equator and the poles is included as well. Therefore we have

9.2 The Hadley Circulation

345

pressure (hPa)

200 400 600 800 1000 90°S

60°S

30°S

60°S

30°S

0

30°N

60°N

90°N

0 30°N latitude

60°N

90°N

pressure (hPa)

200 400 600 800 1000 90°S

Fig. 9.2 The latitude–altitude distribution of the zonal mean of the zonal wind (contour interval 5 m/s, areas with >20 m/s are shaded) and the potential temperature (contour interval 10 K) in the troposphere in northern-hemisphere winter (top picture) and summer (bottom) in the ERA5 data (Hersbach et al., 2020)

346

9 The Meridional Circulation

uv ∂ v · ∇u − f v − tan φ = a ∂z v · ∇v + f u +



∂u K ∂z



  u2 1 ∂p ∂ ∂v tan φ = − + K a aρ ∂φ ∂z ∂z 1 ∂p 0=− −g ρ ∂z ∇ · (ρv) = 0   ∂θ θ − θE ∂ K − v · ∇θ = ∂z ∂z τ

(9.1) (9.2) (9.3) (9.4) (9.5)

Due to the zonal symmetry we have v·∇ =

∂ v ∂ +w a ∂φ ∂z

(9.6)

and for every scalar field α ∇ · (αv) =

∂ ∂ 1 (cos φ vα) + (wα) a cos φ ∂φ ∂z

(9.7)

Furthermore we have approximated the vertical turbulent fluxes by a simple flux-gradient relationship, with identical viscosity and diffusion coefficient K , and the heating by a relaxation ansatz θ − θE Q=− (9.8) τ where θ E (φ, z) is the potential temperature of the radiative equilibrium and τ a relaxation time within which the atmosphere would adjust, in the absence of dynamics, to the radiative equilibrium. One can easily see that Q cools (heats) the atmosphere where the potential temperature is above (below) θ E . For reasons of simplicity we also use the Boussinesq approximation, i.e., we assume ρ = ρ0 + ρ˜ p = p + p˜ θ = θ + θ˜

|ρ| ˜  ρ0 | p| ˜ p ˜ |θ |  θ0

(9.9) (9.10) (9.11)

where θ (z) = θ0 + δθ (z) with |δθ |  θ0 . ρ0 and θ0 are constants, and ρ0 and p(z) are in hydrostatic balance so that dp (9.12) = −ρ0 g dz

9.2 The Hadley Circulation

347

As already discussed with regard to boundary layer theory, (9.2) and (9.3) then become   ∂v 1 ∂P ∂ u2 K (9.13) tan φ = − + v · ∇v + f u + a a ∂φ ∂z ∂z ∂P θ − θ0 0=− (9.14) +g ∂z θ0 with P=

p˜ ρ0

(9.15)

The continuity Eq. (9.4) simplifies to ∇ ·v =0

(9.16)

Therefore all advection terms can be rewritten as flux terms, e.g., v · ∇u = ∇ · (uv)

(9.17)

In summary one obtains uv ∂ tan φ = a ∂z

∇ · (vu) − f v − ∇ · (vv) + f u +

 K

∂u ∂z



  ∂v u2 1 ∂P ∂ K tan φ = − + a a ∂φ ∂z ∂z ∂P θ − θ0 0=− +g ∂z θ0 ∇ ·v =0   ∂ ∂θ θ − θE ∇ · (vθ ) = K − ∂z ∂z τ

(9.18) (9.19) (9.20) (9.21) (9.22)

In order to solve the equations we need boundary conditions. At the upper boundary (z = H ) we assume that there is no vertical motion, i.e., z=H:

w=0

(9.23)

and therefore all vertical turbulent fluxes, for example u  w , vanish as well. Within the flux-gradient approximation these are proportional to the vertical gradient of the mean fields, e.g., ∂u u  w  = −K (9.24) ∂z Hence one has ∂v ∂θ ∂u = = =0 (9.25) z=H: ∂z ∂z ∂z

348

9 The Meridional Circulation

At the lower boundary (z = 0) we similarly assume z=0:

w=0

(9.26)

and we also neglect the turbulent heat flux through the ground, i.e., z=0:

∂θ =0 ∂z

(9.27)

However, ignoring the turbulent momentum flux is not possible. The turbulence of the boundary layer communicates the effect of the molecular viscosity so that momentum is transferred from the solid earth to the atmosphere. In doing so it tends to decelerate the laminar flow. Hence we can simply assume u  w  = −K

∂u = −Cu ∂z

(9.28)

with C > 0 a constant friction coefficient. Hence one uses z=0:

9.2.2

K

∂u = Cu ∂z

K

∂v = Cv ∂z

(9.29)

A Solution Without Meridional Circulation

Without turbulent fluxes, i.e., when K =0

(9.30)

there is a simple solution without meridional circulation, v=w=0

(9.31)

u(z = 0) = 0

(9.32)

and without surface winds, that is in perfect radiative balance, i.e., θ = θE

(9.33)

This solution trivially satisfies the zonal-momentum equation, the continuity equation and the entropy equation. The meridional and the vertical-momentum equations become fu +

1 ∂P u2 tan φ = − a a ∂φ θ E − θ0 ∂P +g 0=− ∂z θ0

(9.34) (9.35)

9.2 The Hadley Circulation

349

i.e., it is hydrostatic and in a generalized geostrophic balance. The vertical derivative of (9.34) yields, with the aid of (9.35),     u2 θ E − θ0 g ∂θ E 1 ∂ ∂ fu + g =− tan φ = − (9.36) ∂z a a ∂φ θ0 aθ0 ∂φ which is a generalization of the thermal-wind relation. Because there are no surface winds, vertical integration of (9.36) yields fu +

g u2 tan φ = − a aθ0

z

dz 

0

∂θ E ∂φ

This quadratic equation in u has only one solution in agreement with (9.32): ⎡⎛ ⎤ ⎞1/2 z g ∂θ E ⎠ ⎢ ⎥ u = a cos φ ⎣⎝1 − 2 2 dz  − 1⎦

a sin φ cos φ ∂φ

(9.37)

(9.38)

0

The potential temperature of the radiative equilibrium decreases from the equator to the poles, and therefore 1 ∂θ E 0 =− 1− 2 2 ∂z 2 a sin φ ∂φ

a sin φ cos φ ∂φ 0

(9.42)

350

9.2.3

9 The Meridional Circulation

Hide’s Theorem

One could expect that it is possible to obtain a solution for small K by using the one above as a starting point and expanding around it. Surprisingly, however, solutions for K  = 0 do not converge to superrotation without meridional circulation as K → 0. This results from the conservation of angular-momentum. The corresponding equation can be obtained by first rewriting the zonal-momentum Eq. (9.18), using the fact that the wind field has no divergence,   uv ∂ ∂u v · ∇u − 2 sin φv − tan φ = K (9.43) a ∂z ∂z Multiplying this with a cos φ and using (9.6) yields ∂m ∂ v ∂m +w = a ∂φ ∂z ∂z



∂m K ∂z

 (9.44)

with m = a 2 cos2 φ + ua cos φ

(9.45)

the mass-specific density of the axial angular-momentum component. Using (9.6) and again the non-divergence of the wind field leads the following conservation equation   ∂ ∂m 0 = −∇ · (mv) + K (9.46) ∂z ∂z Using this one can show that the maximum of m must be located at the lower boundary of the atmosphere, and that it must be in a region with easterlies u < 0. Hence the global maximum of m must satisfy (9.45) (9.47) m 0 < a 2 so that u=

m − a 2 cos2 φ m 0 − a 2 cos2 φ sin2 φ < < a a cos φ a cos φ cos φ

(9.48)

This requires easterlies at the equator, in contrast to superrotation. Moreover, at the ground close to the maximum of m one must have ∂u/∂z < 0 which cannot approach (9.42) in a continuous manner as K → 0. We first show that the maximum cannot be located in the interior of the atmosphere, by demonstrating that this would lead to a contradiction. If there were a maximum m 0 in the inner zone of the atmosphere, there would also be, in sufficient vicinity, a closed contour C, around the location of the maximum, where the angular-momentum density has a constant value m = m C < m 0 . This is shown in Fig. 9.3. Integrating (9.46) over the enclosed area SC yields     ∂m ∂ K (9.49) 0 = − d S∇ · (mv) + d S ∂z ∂z SC

SC

9.2 The Hadley Circulation

351

Fig. 9.3 A closed contour C with a constant angular-momentum density m C , that includes a maximum m 0 . The vector ds is parallel to the outward directed normal vector, and its magnitude agrees with the length of the curve element ds

ds

dS m0

m-contour around m0 with mc < m0 C

Applying Gauss’ theorem twice, and using the non-divergence of the flow field, yields for the first integral     d S∇ · (mv) = ds · mv = m C ds · v = m C d S∇ · v = 0 (9.50) SC

C

C

SC

With the help of Fig. 9.4 one can see that the second integral is  SC

∂ dS ∂z



∂m K ∂z



 = SC

Fig. 9.4 A helpful segmentation of the contour C in Fig. 9.3, into parts that can be written as z i (φ)

∂ dzdφ a ∂z



∂m K ∂z



φ2 = φ1



∂m dφ a K ∂z

z 2 (φ) z 1 (φ)

0 are only possible by numerical methods. Assumption i) yields the zonal wind in the upper branch of the Hadley cell, because by inserting (9.45) into (9.65) one obtains z=H:

a 2 cos2 φ + ua cos φ = a 2 + a u|φ=0

(9.70)

Because air masses originate at the equator from the rising branch of the Hadley cell, one can use assumption iii) to neglect u|φ=0 , leading to z=H:

u = u M = a

sin2 φ cos φ

(9.71)

Due to the conservation of angular-momentum, the zonal wind increases strongly from the equator to the midlatitudes. As will be explained below, one obtains outside the Hadley cell the above discussed radiative-equilibrium solution so that a jet stream results, as sketched in Fig. 9.8. Figure 9.9 shows the analytical jet stream solution together with numerical results for different turbulent viscosities. The corresponding latitude–altitude dependency is shown in Fig. 9.10. Fig. 9.8 Latitude dependence of the zonal wind in the upper troposphere according to the simplified model by Held and Hou (1980)

u(H)

uE

ϕH

ϕ

356

9 The Meridional Circulation

80

60 m s–1

theory K= 1 K=5 K = 25

40

20

0

15°

30°

45° latitude

60°

75°

90°

Fig. 9.9 Latitude dependence of the zonal wind in the upper troposphere, in the simplified model by Held and Hou (1980), together with numerical results for different turbulent viscosities (K ), reprinted from Held and Hou (1980)

Due to assumption ii) one has the generalized geostrophic balance (9.66). The difference between the equilibria at the upper and lower boundaries, respectively, is f [u(H ) − u(0)] +

 1 ∂ tan φ  2 u (H ) − u 2 (0) = − [P(H ) − P(0)] a a ∂φ

(9.72)

The corresponding right-hand side is obtained by vertical integration of the hydrostatic balance θ − θ0 ∂P (9.73) =g ∂z θ0 leading to

{θ } − θ0 1 [P(H ) − P(0)] = g H θ0

where for an arbitrary field X 1 {X } = H

(9.74)

H dz X

(9.75)

0

denotes the vertical mean. Further applying the approximation (9.67) of weak surface winds and the latitude independence of θ0 gives z=H:

fu +

g H ∂ {θ } tan φ 2 u =− a aθ0 ∂φ

(9.76)

9.2 The Hadley Circulation

357

K = 25 m2 s–1

H

height

5674

½H

0 0°

15°

30°

45°

0°

15°

30°

45°

60°

75°

90°

45°

0°

15°

30°

45°

60°

75°

90°

45°

0°

15°

30°

45° latitude

60°

75°

90°

K = 10 m2 s–1

H

height

2680

½H

0 0°

15°

30°

K = 5 m2 s–1

H

height

1554

½H

0 0°

30° 15° latitude

Fig. 9.10 Numerical results for the latitude–altitude dependence of the mass streamfunction (left) and the zonal wind (right) in the simplified model of Held and Hou (1980)

Inserting (9.71) and f = 2 sin φ yields

2 a ∂ sin4 φ g H ∂ {θ } =− 2 ∂φ cos2 φ aθ0 ∂φ

(9.77)

Integrating this relation over the latitude finally results in {θ } (φ) − {θ } (0)

2 a 2 sin4 φ =− θ0 2g H cos2 φ

(9.78)

358

9 The Meridional Circulation

Up to this point the latitude φ H of the Hadley cell as well as the mean potential temperature {θ } (0)/θ0 at the equator remain undetermined. Moreover, no proof has been given yet that outside the Hadley cell one has radiative balance so that the zonal wind is given by the corresponding thermal wind (9.38). The latter can be deduced from (9.68), because outside of the Hadley cells v = w = 0 so that θ = θE

(9.79)

Combined with (9.76) this yields the desired result. The latitude φ H can be found by assuming a meaningful profile for θ E as well as continuity in the potential temperature so that φ = φH :

{θ } = {θ E }

(9.80)

Because by definition the normal velocity component vanishes at the boundaries of the Hadley cell, see Fig. 9.11, the integration of the entropy Eq. (9.68) over the total cross section of the cell yields, using Gauss’ theorem,  1 d S(θ − θ E ) (9.81) 0=− τ SH

This leads to

φ H

φ H dφ a {θ } =

0

dφ a {θ E }

(9.82)

0

The area under the curve of the vertically averaged potential temperature is the same as in radiative balance, as shown in Fig. 9.12. At this point an explicit spatial dependency of the radiative-equilibrium potential temperature is needed. A good compromise between realism and simplicity is

Fig. 9.11 Velocity field within a Hadley cell, with vanishing normal component at its boundaries

n z=H n

z=0 ϕ=0

n

n

ϕ = ϕH

9.2 The Hadley Circulation

359

vertically averaged potential temperature

{θ}

{θE}

ϕH

equator

pole

latitude

Fig.9.12 Area under the curve of latitude dependence of the vertically averaged potential temperature is the same as in radiative balance. Therefore the shaded areas agree as well. Reprinted from (Held and Hou, 1980)

θE 2 = 1 −  H P2 (sin φ) + v θ0 3



z 1 − H 2

 (9.83)

Here  H is the fraction by which the potential temperature differs between equator and pole, whereas v describes the fractional potential-temperature change between ground and top of the troposphere. Finally, 3 1 P2 (x) = x 2 − (9.84) 2 2 is a Legendre polynomial. Because one can assume in the tropics |φ|  1

(9.85)

(9.83) can be approximated well by θE H =1+ −  H φ 2 + v θ0 3 whence



z 1 − H 2



{θ E } H =1+ − H φ2 θ0 3

(9.86)

(9.87)

Likewise applying the low-latitude approximation to (9.78) yields {θ } {θ } (0) 2 a 2 4 = − φ θ0 θ0 2g H

(9.88)

360

9 The Meridional Circulation

Inserting both approximations into (9.80) and (9.82) leads to {θ E } (0) {θ } (0) 2 a 2 4 − −  H φ 2H φH = θ0 2g H θ0 {θ E } (0)  H 2 {θ } (0)

2 a 2 4 − − φH = φ θ0 10g H θ0 3 H where

(9.89) (9.90)

{θ E } (0) H =1+ θ0 3

(9.91)

2 2 a 2 4 2 φ H =  H φ 2H 5 gH 3

(9.92)

The difference (9.90)–(9.89) gives

or 

5 R 3

(9.93)

g H H

2 a 2

(9.94)

φH = with R=

The horizontal extent of the Hadley circulation increases with the baroclinicity of the atmosphere, while it decreases with the radius and the rotation frequency of the planet. For the earth a reasonable value for the baroclinicity is  H = 1/6, which yields φ H ≈ 20◦ . Inserting (9.93) into (9.89) finally gives {θ E } (0) {θ } (0) 5 = − R H θ0 θ0 18

(9.95)

The difference between the two vertical-mean potential temperatures at the equator increases with the horizontal extent of the Hadley cell, because the strength of the circulation increases, and hence also the heat transport. The latitude dependence of the vertical-mean heat flux can be obtained by reconsidering the entropy equation ∂ ∂ θ − θE 1 (cos φvθ ) + (wθ ) = − a cos φ ∂φ ∂z τ

(9.96)

Integrating this equation in the vertical and using the boundary conditions (9.23) and (9.26) leads to

9.2 The Hadley Circulation

1 H

361

H dz 0

{θ } − {θ E } 1 ∂ (cos φvθ ) = − a cos φ ∂φ τ

(9.97)

Here (9.88), (9.87), and (9.95) yield in the low-latitude approximation {θ } (0) − {θ E } (0) 2 a 2 4 {θ } − {θ E } 5

2 a 2 4 = − φ +  H φ 2 = − R H − φ + H φ2 θ0 θ0 2g H 18 2g H (9.98) so that ⎞ ⎛   H 1 ∂ ⎝1 θ0 5

2 a 2 4 (9.99) dz vθ ⎠ = R H + φ − H φ2 a ∂φ H τ 18 2g H 0

where we have used the low-latitude approximation cos φ ≈ 1. At the equator we have v = 0 so that  H  1 dz vθ  =0 (9.100) H 0

φ=0

Hence, integrating in latitude from 0 to φ results in 1 1 aH

H 0

θ0 dz vθ = τ



5

2 a 2 5  H 3 R H φ + φ − φ 18 10g H 3

 (9.101)

which can easily be reformulated as 1 θ0

H 0

5 dz vθ = 18

  1/2      5 H a H 3/2 φ φ 3 φ 5 −2 + R 3 τ φH φH φH

(9.102)

This result, together with numerical results for K > 0, is shown in Fig. 9.13. To estimate the surface winds we need two additional basic assumptions. We first observe, however, that the vertical integral of the continuity equation 1 ∂ ∂w (cos φv) + =0 a cos φ ∂φ ∂z with the boundary conditions (9.23) and (9.26) results in    1 ∂ cos φ dz v = 0 a cos φ ∂φ

(9.103)

(9.104)

362

9 The Meridional Circulation 1.4

1 υθ dz (10–2 m s–1 Hθ0 ∫

1.2 25 m2 s–1 10 5

1.0 0.8

2.5 1 0.5

0.6 0.4 0.2 0 0

theory (K → 0)

6

12

18

24

30 36 42 latitude

48

54

60

66

72

Fig. 9.13 Latitude dependence of the vertical-mean meridional heat transport. Shown are the analytical result for K = 0 and numerical results for K > 0, according to Held and Hou (1980)

But according to (9.69) there is no mass flux across the equator, i.e., H φ=0:

dz v = 0

(9.105)

0

so that one has in general

H dz v = 0

(9.106)

0

Hence the poleward mass flux in the upper branch of a Hadley cell is balanced exactly by the equatorward flux in the lower branch. We now specify the two additional assumptions needed for the determination of the surface winds: vi) As sketched in Fig. 9.14 the meridional flux is confined to thin layers near the top and the bottom boundaries. Following from the discussion above the two fluxes are of opposite sign and equal size: V |z=H = v(H )z(H ) = − V |z=0 = −v(0)z(0)

(9.107)

vii) Furthermore we assume that the static stability is not altered significantly by the circulation, i.e., θ |z=H − θ |z=0 = v θ0

(9.108)

9.2 The Hadley Circulation

363

∫ dz v = V ≈ v(H) Δ z(H)

v≈0

ϕ=0

ϕH

V = v(0) Δ z(0)

Fig. 9.14 Meridional flux in the Hadley cell is confined to thin layers near the top and bottom boundaries

These additional assumptions lead to the following estimate for the vertical-mean heat flux: 1 θ0

H dzvθ ≈ 0

 1  (V θ )|z=H + (V θ )|z=0 = V |z=H v θ0

(9.109)

Likewise one obtains for the meridional momentum flux H dz uv ≈ (V u)|z=H + (V u)|z=0 ≈ V |z=H u M

(9.110)

0

where the assumption (9.67) has been used in the last step, together with the result (9.71) for the latitude dependence of the zonal wind in the upper troposphere. Combined with (9.109) one finds the useful result H H uM 1 dz uv = dz vθ (9.111) v θ0 0

0

Again using the boundary conditions (9.23) and (9.26) we now take the vertical mean of the angular-momentum conservation (9.46) and obtain ⎛ ⎞   H 1 ∂ ⎝ ∂m H cos φ dz vm ⎠ = K a cos φ ∂φ ∂z 0 0

(9.112)

364

9 The Meridional Circulation

Here we can simplify the vertical mean of the meridional angular-momentum flux on the left-hand side by inserting (9.45) and using the fact that the meridional mass flux vanishes (9.106). Hence H H dz vm = a cos φ dz uv (9.113) 0

0

On the right-hand side we use the fact that, due to the boundary conditions  0 z=H ∂m ∂u K = a cos φ K = ∂z ∂z a cos φ Cu z = 0 and hence

⎛ ⎞ H ∂ ⎝ 1 a cos2 φ dz uv ⎠ = −a cos φ C u|z=0 a cos φ ∂φ

(9.114)

(9.115)

0

Therefore the surface wind is u|z=0

⎞ ⎛ H 1 ∂ ⎝ 2 =− cos φ dz uv ⎠ Ca cos2 φ ∂φ ⎛

0

1 ∂ ⎝ 2 uM =− cos φ 2 v Ca cos φ ∂φ θ0

H

⎞ dz vθ ⎠

(9.116)

0

Finally using the results (9.102) and (9.71) yields

z=0:

25 a H  H 2 u=− R 18 Cτ v



φ φH

2

10 − 3



φ φH

4

7 + 3



φ φH

6  (9.117)

In Fig. 9.15 the profile is shown together with numerical results for K > 0. The easterlies in the tropics are quite prominent.

9.2.5

The Summer–Winter Asymmetry

Held and Hou (1980) have assumed symmetry with respect to the equator. This is a reasonable assumption for spring and autumn, because in these seasons the radiative equilibrium is almost symmetric. In summer and winter, however, neither the radiative-equilibrium potential temperature nor the resulting Hadley circulation are symmetric. This situation is sketched in Fig. 9.16. On the summer and winter sides the cells are bounded by the latitudes φ S and φW . The common rising branch of both cells is located at latitude φ1 . Following Lindzen

9.2 The Hadley Circulation

365

10

5

m s–1 0 theory K=1 K=5 K = 25

–5

–10 15°

30°

45° latitude

60°

75°

90°

Fig. 9.15 Latitude dependence of the surface winds in the tropics. The analytical result for K = 0 is shown together with the numerical results for K > 0. Reprinted from Held and Hou (1980) z z=H

ϕH–

0

ϕ0

ϕ1

ϕH+

ϕ

Fig. 9.16 Schematic representation of the summer–winter asymmetry of the Hadley circulation. On the summer and winter side the cells are bounded by the latitudes φ S (here φ H + ) and φW (here φ H − ), respectively. The common rising branch of both cells is located at latitude φ1 . The latitude of the maximum of the radiative-equilibrium potential temperature is φ0 . Reprinted from Lindzen and Hou (1988)

366

9 The Meridional Circulation

and Hou (1988), the radiative-equilibrium potential temperature is approximated by   z H 1 θE 2 (9.118) =1+ −  H (sin φ − sin φ0 ) + v − θ0 3 H 2 It reaches its maximum at all altitudes at the latitude φ0 . Note that in general φ1  = φ0 . The latitude of the rising branch is to be determined from the calculations. The assumptions for these are the same as the ones used in the symmetric calculations of Held and Hou (1980), assumption v) excepted. Thus we again assume conservation of angular-momentum in the upper branch of both cells. Hence z=H:

m = m|φ=φ1

(9.119)

Because u(φ1 , H ) = 0

(9.120)

a 2 cos2 φ + ua cos φ = a 2 cos2 φ1

(9.121)

this leads to or z=H:

u = u M = a

cos2 φ1 − cos2 φ cos φ

(9.122)

One should note that the resulting winds at the equator are always easterlies. For the calculation of the distribution of the potential temperature we assume generalized geostrophic balance, hydrostatics, and we neglect the zonal surface winds in comparison with their counterparts in the upper troposphere. One obtains g H ∂ {θ } u2 tan φ = − a aθ0 ∂φ

(9.123)

 2 g H ∂ {θ }

2 a ∂ sin2 φ1 − sin2 φ =− 2 ∂φ cos2 φ aθ0 ∂φ

(9.124)

 2 {θ } (φ1 ) 2 a 2 sin2 φ1 − sin2 φ {θ } = − θ0 θ0 2g H cos2 φ

(9.125)

z=H:

fu +

Combined with (9.122), this leads to

Integration yields

Finally the integration of the entropy equation without turbulent diffusion over the total area of either of the two cells yields

9.2 The Hadley Circulation

367

φ1 dφ a({θ } − {θ E }) = 0

(9.126)

dφ a({θ } − {θ E }) = 0

(9.127)

φW

φ S φ1

where Gauss’ theorem has been applied. Furthermore, outside of the circulation cells one has the radiative-equilibrium solution. Continuity of potential temperature requires {θ } (φ S ) = {θ E } (φ S )

(9.128)

{θ } (φW ) = {θ E } (φW )

(9.129)

Hence we have the four equations (9.126)–(9.129) for the four unknowns φ S , φW , φ1 , and {θ } (φ1 ). They can only be solved numerically. Solutions for northern-hemisphere spring and summer are shown in Fig. 9.17. Note that not only there is an asymmetry in the potential temperature in the second case, but that there is also a significantly larger deviation from radiative balance on the winter side. This indicates a far stronger circulation on this side.

9.2.6

The Wave-Driven Hadley Circulation

The models considered so far have different drawbacks. The neglect of the evolution in time (beyond the seasonal cycle) as well as the neglect of the effects of turbulent viscosity cause errors. The most important correction, however, concerns the wave forcing of the tropical circulation, that we did not need to consider in the zonally symmetric models. In the discussion of this aspect we extend the equations by the wave forcing. For the sake of easier progress we then again apply appropriate approximations, so that the results to be derived complement the findings of the models of Schneider (1977), Held and Hou (1980) and Lindzen and Hou (1988) instead of generalizing them. First we recall that the product of a cos φ with the inviscid zonal-momentum equation in the Boussinesq approximation u2 1 ∂P ∂u + v · ∇u − f v + tan φ = − ∂t a a cos φ ∂λ

(9.130)

yields the conservation equation for angular-momentum ∂P ∂m + ∇ · (vm) = − ∂t ∂λ

(9.131)

368

9 The Meridional Circulation 1.000 0.996 0.992 0.988 0.984 0.980 0.976 0.972

–20°

–10°

0°

10°

–20°

–10° 0° latitude

20°

1.00 0.98 0.96 0.94 0.92 –40°

–30°

1 0°

20°

30°

Fig. 9.17 Numerical results for the vertical-mean potential temperature {θ } (solid line) in northernhemisphere spring (top picture, φ0 = 0) and in the summer (bottom picture, φ0 = 6◦ ), together with the vertical mean {θ E } (dashed) of the radiative-equilibrium potential temperature. Reprinted from Lindzen and Hou (1988)

Here we split all fields into zonal mean and waves m = m + m  v = v + v



P = P + P

(9.132) (9.133)



(9.134)

and take the zonal mean of the equation. The result is ∂m + ∇ · (vm) + ∇ · m  v  = 0 ∂t

(9.135)

Averaging the continuity equation yields ∇ · v = 0

(9.136)

9.2 The Hadley Circulation

369

so that one can write the angular-momentum equation ∂m + v · ∇m = −∇ · m  v  ∂t

(9.137)

or more in detail  ∂m v ∂m ∂ ∂m 1 ∂    + + w =− m v  cos φ − m  w  ∂t a ∂φ ∂z a cos φ ∂φ ∂z

(9.138)

For estimates of the magnitude of each of the contributing terms we assume, for reasons of simplicity, synoptic scaling as in quasigeostrophic theory. First, due to (9.45) the advection terms on the left-hand side are ∂u ∂m = a cos φ w ∂z ∂z v ∂ v ∂m = −a cos φ f v + (a cos φ u) a ∂φ a ∂φ

w

(9.139) (9.140)

with f = 2 sin φ = O( f 0 ). Here u = O(U )

(9.141)

v = O(Ro U )   H w = O Ro U L

(9.142) (9.143)

whence   U2 ∂m = O Ro a ∂z L  2 U a cos φ f v = O a L   v ∂ U2 (a cos φ u) = O Ro a a ∂φ L w

(9.144) (9.145) (9.146)

so that one can neglect vertical advection in (9.138). Different than suggested by a comparison of (9.145) and (9.146), however, we keep the total merdional advection of angularmomentum, i.e., the term in (9.146) is not neglected! This is done on the one hand because, especially in the tropics but also in the subtropics, the dominance of the Coriolis term in (9.145) is less pronounced than in midlatitudes. On the other hand the term in (9.146) is the one responsible, in the zonally symmetric case, for the impact of angular-momentum conservation on the subtropical jet stream. We finish the analysis of the angular-momentum Eq. (9.138) by considering the flux terms on the right-hand side. Due to (9.45) these are

370

9 The Meridional Circulation

  1 1 ∂  ∂  cos φ m  v   = a cos2 φ u  v   a cos φ ∂φ a cos φ ∂φ ∂ ∂ m  w  = a cos φ u  w  ∂z ∂z

(9.147) (9.148)

Herein u  = O(U ) 

v = O(U )   H  w = O Ro U L so that

 2  U 1 ∂    cos φ m v  = O a a cos φ ∂φ L   ∂ U2 m  w  = O Ro a ∂z L

(9.149) (9.150) (9.151)

(9.152) (9.153)

Hence the vertical angular-momentum flux can be neglected in comparison with the horizontal flux. In summary, a reasonable approximation of the angular-momentum equation is  ∂m v ∂m 1 ∂    + ≈− u v a cos2 φ ∂t a ∂φ a cos φ ∂φ

(9.154)

The momentum flux on the right-hand side is shown both for northern-hemispheric winter and summer in Fig. 9.18. Now consider, e.g., the upper branch of the Hadley cell in the northern hemisphere. There v > 0 (9.155) and also −

 1 ∂  a cos2 φ u  v   < 0 a cos φ ∂φ

(9.156)

Hence, in the steady case

∂m =0 ∂t angular momentum decreases with increasing latitude, i.e., ∂m 0 (9.163) ∂φ 2 so that the waves enhance the latitudinal gradient of the equilibrium potential temperature, i.e. ∂θ EW ∂θ E < 0 with rising air masses in the tropics and sinking air masses in the subtropics. This is summarized in Fig. 9.21. In the southern hemisphere one has likewise ψ < 0. • Likewise one derives from Fig. 9.18 that in the tropopshere

Fig. 9.21 Qualitative impact of the heat fluxes on the Hadley circulation. They cause relative heating in the tropics and cooling in the subtropics, so that air masses rise in the tropics and sink in the extratropics

J>0

equator

ψ>0

J 0, as summarized in Fig. 9.22. Again one obtains for the southern hemisphere ψ < 0. • One also sees that a reduction of the static stability N 2 leads to an enhancement of the meridional circulation as well.

9.2.7

Summary

To leading order the Hadley circulation in the tropics can already be understood without the impact of eddies. Those, however, represent an important correction. • A discussion of the dynamics without waves can be done using the stationary zonally symmetric primitive equations in Boussinesq approximation. Essential are radiative heating, with a potential temperature characterizing radiative equilibrium that decreases in spring and fall from the equator to the poles, and momentum exchange with the solid earth via turbulent surface friction. In the inviscid case the equations admit a geostrophic-hydrostatic equilibrium solution without meridional circulation. Even the weakest friction, however, makes this solution impossible, and a solution is to be found where a maximum of angular momentum at the ground is accompanied by easterlies in its vicinity.

9.3 The Circulation in the Midlatitudes

377

• In the case of weak turbulent viscosity an analytical solution is possible with a subtropical jet stream resulting from angular-momentum conservation in a merdidional circulation directed polewards in the upper troposphere. The latitude dependence of the corresponding potential temperature follows from the thermal wind and a continuous transition to radiative equilibrium outside of the Hadley cell. One finds that the horizontal extent of the Hadley circulation increases with the equator-pole contrast in radiative-equilibrium potential temperature, while it decreases with radius and rotation of the planet. The stronger the equator-pole contrast the stronger the circulation, and the stronger as well the reduction of potential temperature in the tropics. The corresponding heat flux is everywhere directed from the tropics to the middle latitudes. An equilibrium between angularmomentum flux and the viscous-turbulent angular-momentum sink by surface friction leads in the vertically integrated angular-momentum equation to surface easterlies near the equator. • A modification of the zonally symmetric model for summer and winter conditions leads to easterlies everywhere above the equator. The Hadley cell on the winter side is considerably stronger than the summer cell. • Eddies turn out to have a considerable influence. With the latitude dependence of their momentum fluxes in the upper troposphere they represent an angular-momentum sink so that the subtropical jet stream with eddies is weaker than without eddies. In addition, the latitude dependence of the eddy heat fluxes enhances the equator-pole contrast between heating and cooling so that the Hadley circulation is forced additionally. These impacts can be captured quite conveniently in an elliptic equation for the mass streamfunction. They are the stronger the weaker the stratification is.

9.3

The Circulation in the Midlatitudes

While the circulation in the tropics can be described to some extent without waves, the midlatitudes are not to be understood without the effect of waves at all. The dependency of the heating rates on longitude, for example because of the land–sea contrast, orographic wave generation and particularly baroclinic instability continuously excite waves in the extra tropics. Hence the dynamics of this latitude region is intrinsically turbulent. Important characteristics of the resulting circulation are the Ferrel cells and the barotropic jet stream in the midlatitudes. In the following section we want to discuss the dynamics of these phenomena.

9.3.1

The Phenomenology of the Ferrel Cell

For a phenomenological description of the Ferrel cells we can go back to the considerations of the last section. In the context of Boussinesq theory it is possible here as well to introduce

378

9 The Meridional Circulation

a meridional streamfunction ψ defining the zonal-mean meridional and vertical winds via (9.177) and (9.178). Using the effective heating rate J =−

g θ  − θ E ∂   b v  − ∂y θ0 τ

and the effective acceleration ∂ ∂ M = − u  v   + ∂y ∂z



∂u K ∂z

(9.184)

 (9.185)

supplemented here by the effect of turbulent friction in the boundary layer, this streamfunction can be determined from the elliptical Eq. (9.180). The corresponding derivation has not made use of any specific aspects of tropical dynamics, so that all can be used in midlatitudes right away. In fact the application of quasigeostrophic theory is much better justified in the present context. Different here is that we can neglect the direct heating, as we did in the tropics with regard to turbulent friction outside of the boundary layer, and the sign of the wave forcing changes as well. Thus, according to Fig. 9.18 one finds that the momentum-flux convergence in midlatitudes ∂   (9.186) u v  < 0 ∂y increases with altitude up to the tropopause so that ∂M ∂2 =− u  v   > 0 ∂z ∂z∂ y

(9.187)

As a consequence of the baroclinic instability active there the heat flux has its maximum in midlatitudes so that ∂2   v b  < 0 (9.188) ∂ y2 whence

∂J >0 ∂y

(9.189)

All of this taken together shows that the streamfunction ψ 0

(9.195)

0

the vertical integral of the meridional wind. Here, the boundary conditions are such that the turbulent momentum flux vanishes at the top of the boundary layer. At the lower boundary the turbulent momentum flux can be approximated by a drag coefficient,  0 at z = z ∂u (9.196) K = ∂z Cu at z = 0 Hence one obtains

f0 V >0 C i.e., westerlies prevail so that friction is balanced by the Coriolis effect. z = 0 : u =

(9.197)

380

9 The Meridional Circulation

tropopause ƒv ~ ∂ u' v' < 0 ∂y

w ~ – ∂ v' θ' < 0 ∂y

w ~ – ∂ v' θ' > 0 ∂y

boundary layer

ƒv ~ C u | z = 0 > 0

ground subtropics

latitude

subpolar

Fig. 9.23 Wave forcing of the Ferrel cell and the surface westerlies in midlatitudes

• On the other hand, one can integrate over the whole troposphere, yielding H − f0

H dzv = −

0

dz 0

∂   u v  − C u|z=0 ∂y

(9.198)

But according to (9.106) the total meridional mass flux vanishes, simplifying the equation to H 1 ∂ z=0: u = − dz u  v   > 0 (9.199) C ∂y 0

because the mid latitude momentum flux is convergent, as can also be read from Fig. 9.18. Following Vallis (2006), the whole is summarized in Fig. 9.23.

9.3.2

Eddy Fluxes and Barotropic Jet Stream

A distinguished characteristic of the midlatitudes is their barotropic jet stream, most evident in local cuts without zonal average. In Fig. 9.24 the baroclinic jet stream in the subtropics, with surface easterlies, is clearly distinguishable from the barotropic jet stream in midlatitudes, where westerlies extend to the ground. This wave-driven phenomenon shall be discussed

9.3 The Circulation in the Midlatitudes

381

pressurek (hPa)

200 400 600 800 1000 90°S

60°S

0

30°S

30°N

60°N

90°N

latitude Fig. 9.24 Latitude–altitude dependence of the zonal wind at 150◦ W above the Pacific in northernhemispheric spring. Data from ERA5 (Hersbach et al., 2020)

in the following. We will also see there that the configuration of the eddy fluxes causing this jet stream are a direct result of baroclinic instability and the conservation of vorticity as described by the Kelvin theorem. The Basic Mechanism Baroclinicity is essential in the dynamics of the jet stream, this however only for the explanation of the mid latitude wave source by baroclinic instability. Beyond this all can be discussed within the framework of barotropic dynamics. Take, e.g., a barotropic incompressible β-channel with periodic boundary conditions in zonal direction. Because the flow is purely horizontal one has from incompressibility

therefrom

∂v ∂u + =0 ∂x ∂y

(9.200)

∂v =0 ∂y

(9.201)

v = 0

(9.202)

and hence also because the flow at the meridional boundaries of the channel must be zonal. Moreover, there is only vertical relative vorticity ∂v ∂v ζ = − (9.203) ∂x ∂y

382

9 The Meridional Circulation

The planetary vorticity is f = f 0 + β y, and the absolute vorticity ωaz = f + ζ consists of the latter and relative vorticity. Assume moreover that the atmosphere is initially at rest so that its absolute vorticity is identical with the planetary vorticity, increasing from south to north. Now the atmosphere is put in midlatitudes into irregular motion, e.g., by a baroclinic wave source. Following the Kelvin theorem (4.33), outside of the region of the wave source all material surface elements conserve their absolute vorticity flux a = ωaz d S

(9.204)

The velocity field, however, is non-divergent, and hence due to the two-dimensional variant of (1.12) the material surface d S is conserved as well, so that the material surface elements transport their absolute vorticity. The latter is given initially by the planetary vorticity so that surface elements moving northwards carry low absolute vorticity and southwards moving surface elements transport high vorticity. Hence, in the zonal mean one obtains outside of the baroclinic source region a negative vorticity flux  vωaz  = v  ωaz  0

u' v' < 0

u

y Fig.9.25 Westerly jet stream with its easterly flanks, as it originates in midlatitudes from momentumflux convergence, that is again due to a baroclinic source there and vorticity conservation according to the Kelvin theorem

A further angle to this result is offered by the properties of Rossby waves, now without the assumption of barotropicity and incompressibility, however within the framework of linear quaisgeostrophic theory. Consider the baroclinic instability as a mid latitude source of Rossby waves. Following (8.146) these propagate with a meridional group velocity cgy =

∂π   ∂y

2lk f2 1 k 2 + l 2 + 02 m 2 + N 4L 2di

2

(9.210)

where k and l are the zonal and meridional wavenumber, respectively, and π  the zonal mean of potential vorticity. Although the latter also contains contributions from the atmospheric flow field, its meridional derivative is dominated by that of the planetary vorticity, i.e. ∂π  ≈β>0 ∂y

(9.211)

Because the waves originate from the mid latitude source region one finds that to the north of the source region:

cgy > 0 ⇒ kl > 0

(9.212)

to the south of the source region:

cgy < 0 ⇒ kl < 0

(9.213)

This has consequences for the meridional momentum flux. The contribution of each Rossby wave is, due to (8.144), |A|2 u  v   = − kl (9.214) ρ

384

9 The Meridional Circulation

Fig. 9.26 Characteristic boomerang shape of the streamlines of midlatitudes Rossby waves with a convergent momentum flux

u' v' < 0 u' v' < 0 u y

max

u' v' > 0 u' v' > 0 x

Hence to the north of the source region: to the south of the source region:

u  v   < 0  

u v  > 0

(9.215) (9.216)

whence follows (9.209). The Rossby-wave streamlines have a characteristic boomerang shape as sketched in Fig. 9.26.

A Closed Description with Wave Source and Dissipative Sink For the development of simple closed equations for the jet stream we return to barotropic dynamics. Consider a β-channel with constant density ρ0 and purely horizontal flow. The equation of continuity then leads to (9.200). The horizontal-momentum equations are in this framework ∂u ∂P + u · ∇u − f v = − + Fu − Du ∂t ∂x ∂v ∂P + v · ∇v + f u = − + Fv − Dv ∂t ∂y

(9.217) (9.218)

where P = p/ρ0 is the density-normalized pressure, F the baroclinic source, and D the viscous-turbulent term. As usual ∂(9.218)/∂ x − ∂(9.217)/∂ y with (9.200) leads to the prognostic equation ∂ζ (9.219) + u · ∇ζ + βv = Fζ − Dζ ∂t

9.3 The Circulation in the Midlatitudes

385

for relative vorticity (9.203), where Fζ =

∂ Fv ∂ Fu − ∂x ∂y

(9.220)

∂ Dv ∂ Du − ∂x ∂y

(9.221)

is the baroclinic vorticity source, and Dζ =

the viscous-turbulent sink. Due to the non-divergence (9.200) the zonal-momentum equation (9.217) can also be written ∂u ∂P + ∇ · (uu) − f v = − + Fu − Du ∂t ∂x

(9.222)

whence in the zonal mean ∂u ∂ + uv − f v = Fu  − Du  ∂t ∂y

(9.223)

Now, however, due to the non-divergence (9.200) and the impermeability of the meridional boundaries of the β-channel, the zonal-mean meridional velocity vanishes as in (9.202) so that (9.224) uv = uv + u  v   = u  v   Beyond this it is reasonable to assume Fu  = 0

(9.225)

because F represents the effect of baroclinic instability, generating essentially waves without zonal mean. Finally, we choose as most simple ansatz for the description of turbulent friction Du  = r u

(9.226)

All of this taken together we obtain from (9.223) ∂u ∂ = − u  v   − r u ∂t ∂y

(9.227)

But due to (9.208) the vorticity flux is identical with the momentum-flux convergence so that these equations can also be written, again using (9.202), ∂u = v  ζ   − r u ∂t

(9.228)

The mean-flow acceleration can be related to wave transience. The zonal mean of the vorticity equation (9.219) is ∂ζ  ∂   + v ζ  = Fζ  − Dζ  ∂t ∂y

(9.229)

386

9 The Meridional Circulation

Subtracting this from (9.219) results, under neglect of all terms nonlinear in the wave contributions, in a prognostic equation for the eddy vorticity, ∂ζ  ∂ζ  + u + γ v  = Fζ − Dζ ∂t ∂x where γ =β+

∂ζ  ∂y

(9.230)

(9.231)

is the meridional derivative of absolute vorticity. Similar to Sect. 8.3.2 we now assume that the latter varies only very slowly in time. Then, multiplication of the eddy-vorticity equation by ζ  /γ , and zonal averaging of the result, yields the equation  ∂A 1   + v  ζ   = ζ Fζ  − ζ  Dζ ∂t γ for the wave-action density



ζ 2 A= 2γ

(9.232)

 (9.233)

The sum of (9.228) and (9.232) is  ∂u ∂ A 1   + = ζ Fζ  − ζ  Dζ  − r u ∂t ∂t γ

(9.234)

This yields a barotropic variant of the non-accelaration theorem. In the time mean, here of interest, we find  1   r u = (9.235) ζ Fζ  − ζ  Dζ  γ Hence the mean zonal wind results from a balance between the baroclinic source ζ  Fζ  of wave action, largest in midlatitudes, and the viscous-turbulent sink ζ  Dζ . This balance integrates to zero: The meridional integral of (9.232) is in the time mean    1 (9.236) dy ζ  Fζ  − ζ  Dζ  0= γ because, due to the meridional boundary conditions of the β-channel   ∂ dyv  ζ   = − dy u  v   = 0 ∂y

(9.237)

It is obvious that in the middle latitudes, close to the baroclinic source, ζ  Fζ  > ζ  Dζ 

(9.238)

u > 0

(9.239)

whence

9.3 The Circulation in the Midlatitudes

387

while at the flanks of the jet stream ζ  Fζ  < ζ  Dζ 

(9.240)

u < 0

(9.241)

and hence The decisive balances are sketched in Figs. 9.27 and 9.28.

90

60 latitude

Fig. 9.27 Latitude distribution of the zonal-mean wind and the eddy velocity corresponding to the wave-action density, resulting from a barotropic model with mid latitude wave source and viscous-turbulent sink. Redrawn from Vallis (2006) with permission from Cambridge University Press

30

mean zonal wind eddy velocity 0 –10

m s–1

10

20

90

60 latitude

Fig. 9.28 As in Fig. 9.27, but now showing the wave source ζ  Fζ , the wave sink ζ  Dζ , and their net effect, agreeing with r u. Redrawn from Vallis (2006) with permission from Cambridge University Press

0

30 wave source wave sink sum 0

–4

0

4 s3 × 10–15

8

388

9.3.3

9 The Meridional Circulation

A Two-Layer Model

The discussion above of purely barotropic dynamics does not admit the explicit treatment of a baroclinic wave source. Moreover, it also does not allow a description of an altitudedependent mean circulation. The most simple framework to make this possible is a two-layer model. The Model Equations In the baroclinic perspective as well the mechanism of vorticity conservation pertains as it is described by the Kelvin theorem. Now however, as in Chap. 4.5.2, it is to be applied to isentropic material surface elements, and vorticity is to be replaced by potential vorticity. Therefore the most direct route from the barotropic to the baroclinic perspective is a model with two isentropic layers (Held, 2000). A bit easier, however, is the approach of Vallis (2006), where a model with two layers is considered that each have constant density. The resulting equations are the same as in the isentropic two-layer model. The conditions are sketched in Fig. 9.29: One has two layers with constant densities ρ1 = ρ0 − ρ˜1

ρ˜1  ρ0

(9.242)

ρ2 = ρ0 + ρ˜2

ρ˜2  ρ0

(9.243)

that do not differ much. The upper lid is rigid, which is only possible if a temporally and spatially dependent pressure pT (x, y, t) is applied there. The interface between the two layers is variable so that its vertical displacement relative to the equilibrium position is η(x, y, t). The respective layer thicknesses are h 1 (x, y, t) and h 2 (x, y, t), with corresponding equilibrium values H1 and H2 . As we are interested in synoptic-scale eddies, these are in geostrophic and hydrostatic equilibrium so that η is small in comparison with H1 and H2 . The total thickness of the model atmosphere is

p = pT ρ1 = const.

H1

h1

u1

H

η H2

h2

u2

ρ2 = const.

Fig. 9.29 Geometry of a two-layer model for the discussion of the mean circulation in midlatitudes

9.3 The Circulation in the Midlatitudes

389

H = h 1 + h 2 = H1 + H2 = const.

(9.244)

As in the derivation of the shallow-water equations one assumes that the horizontal winds u1 and u2 in the layers do not depend on altitude. The corresponding pressure results from hydrostatic equilibrium ∂ pi = −gρi ∂z

(9.245)

i.e., p1 = pT + ρ1 g(H − z)

(9.246)

p2 = pT + ρ1 g(H1 − η) + ρ2 g(H2 + η − z)

(9.247)

Hence the pressure-gradient accelerations acting in the two horizontal-momentum equations are 1 1 1 ∇ h p1 = ∇ pT ≈ ∇ pT ρ1 ρ1 ρ0 1 1 ρ2 − ρ1 1 ∇ h p2 = ∇ pT + g ∇η ≈ ∇ pT + g  ∇η ρ2 ρ2 ρ2 ρ0

(9.248) (9.249)

Here we have assumed ρi ≈ ρ0 in the denominator. Moreover, g = g

ρ2 − ρ1 ρ0

(9.250)

is the so-called reduced gravity. Including simple turbulent friction in the lower layer, the horizontal-momentum equations becomes Du1 + f ez × u1 = − ∇ Dt Du2 + f ez × u2 = − ∇ Dt

pT ρ0 pT − g  ∇η − r u2 ρ0

(9.251) (9.252)

Here r is the turbulent drag coefficient. The continuity equation in each layer is ∇ · ui +

∂wi =0 ∂z

(9.253)

Elimination of the vertical wind is achieved as in shallow-water theory: Vertical integration, e.g., of the upper layer, first yields h 1 ∇ · u1 + [w1 ] H H1 −η = 0

(9.254)

390

9 The Meridional Circulation

However  [w1 ] H H1 −η so that one obtains

=

Dz Dt

H H1 −η

=

Dh 1 Dt

Dh 1 + h 1 ∇ · u1 = 0 Dt

(9.255)

(9.256)

or correspondingly

∂h 1 + ∇ · (h 1 u1 ) = 0 ∂t The treatment of the lower layer is completely analogous, so that generally ∂h i + ∇ · (h i ui ) = 0 ∂t

i = 1, 2

(9.257)

(9.258)

For reasons that will become clearer farther below these equations shall also be extended by sources and sinks Si describing a mass exchange between the two layers that is due to sinking and rising air masses. Hence ∂h i + ∇ · (h i ui ) = Si ∂t

(9.259)

are used. Geostrophic and Thermal Wind The geostrophic equilibria between Coriolis force and pressure-gradient force are respectively pT ρ 0  pT  f 0 ez × u2 = −∇ +gη ρ0 f 0 ez × u1 = −∇

(9.260) (9.261)

where f 0 is the value of the Coriolis parameter at a mean reference latitude, so that the geostrophic winds are 1 pT ∇ f 0 ρ0   1 pT =ez × ∇ + g η f0 ρ0

u1g =ez ×

(9.262)

u2g

(9.263)

The difference between these yields the thermal-wind relation   f 0 u1g − u2g = −g  ez × ∇η

(9.264)

9.3 The Circulation in the Midlatitudes

391

which is component-wise   ∂η f 0 u 1g − u 2g = g  ∂y   ∂η f 0 v1g − v2g = −g  ∂x

(9.265) (9.266)

Hence, −η takes the role of potential temperature. An interface slope as in Fig. 9.30, i.e., with increasing altitude from the equator to the pole, corresponds to a vertically increasing zonal wind. In order to be close to real atmospheric conditions one therefore needs source terms Si transferring at the equator mass from the lower layer to the upper layer, as would result from direct heating, and transferring at the pole mass from the upper layer to the lower layer, corresponding to cooling there. The Zonal-Mean Equations in Quasigeostrophic Scaling In midlatitudes we can assume geostrophic winds ui ≈ uig

(9.267)

∇ · uig = 0

(9.268)

Since the latter are non-divergent

u1

u1 > u2 ∂η >0 ∂y

u2 equator S1 > 0 S2 < 0

pole S1 < 0 S2 > 0 radiation

Fig. 9.30 The interface of the two-layer model is tilted upwards from the tropics to the pole, corresponding to a decrease of potential temperature and a positive zonal thermal wind. This tilt is due to source terms in the continuity equations that describe the effect of solar heating. The latter leads to rising air masses in the tropics and sinking air masses in the polar regions

392

9 The Meridional Circulation

the zonal-momentum equations can be written at good accuracy ∂ pT ∂u 1 + ∇ · (u1 u 1 ) − f v1 = − ∂t ∂ x ρ0   pT ∂u 2 ∂ + g η − r u 2 + ∇ · (u2 u 2 ) − f v2 = − ∂t ∂ x ρ0

(9.269) (9.270)

The zonal mean of these is ∂u 1  ∂    + v u − f v1  = 0 ∂t ∂y 1 1 ∂u 2  ∂    + v u − f v2  = −r u 2  ∂t ∂y 2 2

(9.271) (9.272)

As in barotropic theory one has also, however, ! "  ∂ !   " ∂      vgi ≈ − ζi vi vi u i ≈ vgi u gi = − ζgi ∂y ∂y

(9.273)

f vi  = f 0 vi  + O(Ro f 0 U )

(9.274)

Moreover because the β-term is only O(Ro) in comparison with the leading term. Hence the zonalmean of the zonal-momentum equation becomes approximately  ∂u 1  − f 0 v1  = v1 ζ1 ∂t  ∂u 2  − f 0 v2  = v2 ζ2 − r u 2  ∂t

(9.275) (9.276)

Averaging the continuity equations likewise results in ∂h i  ∂ ∂ + (h i vi ) = − h i vi  + Si , ∂t ∂y ∂y

i = 1, 2

(9.277)

so that the Eulerian mean of the two-layer model is  ∂u 1  − f 0 v1  = v1 ζ1 ∂t  ∂u 2  − f 0 v2  = v2 ζ2 − r u 2  ∂t ∂h i  ∂   ∂ + h v  + Si , (h i vi ) = − ∂t ∂y ∂y i i

(9.278) (9.279) i = 1, 2

(9.280)

Herein h i  ≈ Hi because η is small, an approximation not applicable in meridional derivatives of the zonal-mean thicknesses.

9.3 The Circulation in the Midlatitudes

393

Now for the transformed Eulerian mean. First, as in shallow-water theory one can easily show that in the absence of friction (r = 0) and sinks and sources (S1 = S2 = 0)   ∂ (9.281) + ui · ∇ i = 0 ∂t where i =

ζi + f hi

(9.282)

is the ith-layer potential vorticity of the two-layer model. In synoptic scaling one has |η|  h i

(9.283)

|ζi |  f 0

(9.284)

so that i ≈ πi /Hi , where πi = ζi + f − f 0

h i − Hi Hi

(9.285)

is the corresponding quasigeostrophic potential vorticity. Hence          vh vi πi = vi ζi − f 0 i i Hi

(9.286)

However, the mass of a fluid column is proportional to its thickness so that a mass-weighted (transformed Eulerian) mean of the meridional velocity in each layer is

because

   vh vi h i  vi ∗ = = vi  + i i h i  h i 

(9.287)

h i vi  = h i vi  + h i vi 

(9.288)

Hence, and because h i  ≈ Hi , the zonal-mean momentum equations become in good approximation ∂u 1  − f 0 v1 ∗ = v1 π1  ∂t ∂u 2  − f 0 v2 ∗ = v2 π2  − r u 2  ∂t

(9.289) (9.290)

Likewise the zonal-mean continuity equations can be re written ∂h i  ∂ + (h i vi ∗ ) = Si , ∂t ∂y

i = 1, 2

(9.291)

394

9 The Meridional Circulation

so that in summary the transformed Eulerian mean (TEM) of the two-layer model is ∂u 1  − f 0 v1 ∗ =v1 π1  ∂t ∂u 2  − f 0 v2 ∗ =v2 π2  − r u 2  ∂t ∂h i  ∂ i = 1, 2 + (h i vi ∗ ) =Si , ∂t ∂y

(9.292) (9.293) (9.294)

Integral Properties From the equations follow two important properties of vertical integrals of the model. First, due to the non-divergence of the approximately geostrophic wind, the continuity equations can also be written ∂h i i = 1, 2 (9.295) + ui · ∇h i = Si , ∂t Inserting h 1 = H1 − η

(9.296)

h 2 = H2 + η

(9.297)

yields −

∂η − u1 · ∇η = S1 ∂t ∂η + u2 · ∇η = S2 ∂t

(9.298) (9.299)

The sum of these two equations is − (u1 − u2 ) · ∇η = S1 + S2

(9.300)

Due to the thermal-wind relationship (9.264), however, the left-hand side of this equation vanishes so that S1 + S2 = 0 (9.301) and there is a mass-exchange term S from which S1 and S2 can be determined via S2 = S

(9.302)

S1 = −S

(9.303)

where S(y = 0) < 0

S(y = L y ) > 0

dS ≥0 dy

(9.304)

9.3 The Circulation in the Midlatitudes

395

Inserting this into the transformed Eulerian mean of the continuity equations yields −

so that in sum

∂η ∂ + (h 1 v1 ∗ ) = −S ∂t ∂y ∂η ∂ + (h 2 v2 ∗ ) = S ∂t ∂y

∂ (h 1 v1 ∗ + h 2 v2 ∗ ) = 0 ∂y

(9.305) (9.306)

(9.307)

Once more we apply the meridional boundary conditions y = 0, L y :

vi = 0

(9.308)

so that h 1 v1 ∗ + h 2 v2 ∗ = 0

(9.309)

H1 v1 ∗ + H2 v2 ∗ = 0

(9.310)

or, due to h i ≈ Hi ,

Hence the vertical integral of the mass flux vanishes. Moreover, from the relationship (9.266) for the meridional component of the thermal wind follows at good accuracy   g  ∂ η2 η v1 − v2 = − f0 ∂ x 2

(9.311)

  η v1 − v2  = 0

(9.312)

η = −h 1 = h 2

(9.313)

v1 h 1  + v2 h 2  = 0

(9.314)

so that in the zonal mean Because due to (9.296) and (9.297)

one obtains from this

i.e., in the vertical mean the eddies do not transport any mass. Hence the approximate vertical integral of the potential-vorticity fluxes is       f0  f0  h 1 + H2 v2 ζ2 − h2 H1 v1 π1  + H2 v2 π2  = H1 v1 ζ1 − H1 H2     = H1 v1 ζ1  + H2 v2 ζ2   ∂  (9.315) H1 u 1 v1  + H2 u 2 v2  =− ∂y

396

9 The Meridional Circulation

where we have used (9.273) in the last step. Because, due to the meridional boundary conditions of the β-channel, there are no momentum fluxes, one finally obtains L y

  dy H1 v1 π1  + H2 v2 π2  = 0

(9.316)

0

Hence the vertical and meridional mean of the potential-vorticity flux vanishes. The Dynamics of the Climatological Mean Based on the relationships derived above, we now discuss the dynamics of a climatological mean, where all time derivatives disappear by time averaging and where all zonal means are replaced by zonal and temporal means. First, integration of the climatological means of (9.305) and (9.306) yield, again using the meridional boundary conditions for vi ,  y 1 v1 ∗ ≈ − S (9.317) H1 0  y 1 v2 ∗ ≈ S (9.318) H2 0 Due to (9.304), however, this means that v1 ∗ >0

(9.319)

v2 ∗ β0 ∂y ∂π2  0 ∂y

(9.331) (9.332)

Here the result for the lower layer cannot be understood without the comment that, from a calculation as in the derivation of the Rayleigh theorem in Sect. 6.4.2, left to the interested reader as an exercise, baroclinic instability is only possible if 2 # i=1



Ly

Hi

dy 0

∂πi  |ψi |2 =0 |ω − ku i |2 ∂ y

(9.333)

Here ω is the complex eigenfrequency of the baroclinic instability, ψˆ i the corresponding streamfunction amplitude in the ith layer, where the quasigeostrophic streamfunctions are approximately

398

9 The Meridional Circulation

1 pT f 0 ρ0   1 pT  +gη ψ2 = f 0 ρ0 ψi =

so that η=−

f0 (ψ1 − ψ2 ) g

(9.334) (9.335)

(9.336)

For the two-layer model to generate waves at all, (9.333) must be fulfilled, and hence the lower-layer potential-vorticity gradient must be negative at some locations. In this aspect the model differs perhaps from reality. The climatological gradient of potential vorticity from observation is shown in Fig. 9.31, and one sees that the gradient is predominantly positive at all altitudes. The essential result remains, however, so that the upper-layer Rossby waves are faster and hence are better able to spread potential-vorticity fluctuations. Therefore it is to be expected that the distribution of v1 π1  < 0 is broader than that of v2 π2  > 0. Because of (9.316), however, the areas under the two distributions agree with each other, so that u 2  > 0 in middle latitudes u 2  < 0 at the flanks of the jet stream

(9.337)

This is sketched in Fig. 9.32 and agrees with the empirical findings. The upper-layer winds result therefrom and the baroclinic shear of the zonal-mean atmosphere according to the thermal-wind equation (9.264). It is important to realize that it is the surface winds that are controlled by the eddies together with surface friction, and that the upper-tropopsheric winds simply result from those! Finally, the upper branch of the Ferrel circulation is obtained by considering the climatological and Eulerian-mean zonal-momentum equation − f 0 v1  = v1 ζ1 

(9.338)

As in barotropic dynamics, e.g., via the relationship between Rossby-wave group velocity and momentum flux, we have here as well v1 ζ1  > 0

(9.339)

v1  < 0

(9.340)

Hence In the lower layer, however, the eddy fluctuations are weak so that v2 ζ2  can be neglected in the corresponding Eulerian-mean zonal-momentum equation, whence the climatological mean of the latter is (9.341) − f 0 v2  ≈ −r u 2 

9.3 The Circulation in the Midlatitudes

399

pressure (hPa)

101

102

103 90°S

60°S

30°S

60°S

30°S

0

30°N

60°N

90°N

0 30°N latitude

60°N

90°N

pressure (hPa)

101

102

103 90°S

Fig. 9.31 Zonal-mean potential vorticity (blue contour lines) and zonal-mean potential temperature (red) in the annual mean (top panel) and in northern-hemispheric winter (bottom) from ERA5 analysis data (Hersbach et al., 2020)

requiring that in middle latitudes v2  > 0. In summary, one obtains the Ferrel cell in middle latitudes:

v1  < 0 v2  > 0

(9.342)

400

9 The Meridional Circulation

v'2 2' u2

y

v'1 1'

Fig. 9.32 Mid latitude surface winds are due to the balance between potential-vorticity fluxes in both layers, resulting in westerlies in middle latitudes, and easterlies at the corresponding flanks

9.3.4

The Continuously Stratified Atmosphere

Based on the discussions above, the representation of the mid latitude circulation in the continuously stratified atmosphere shall be done relatively quickly. A few parallels, but also differences, shall be described here in addition. The Surface Winds The connection between the surface winds and the potential-vorticity fluxes is similar to the one in the two-layer model. We recall that the TEM zonal-momentum equation is within the quasigeostrophic approximation ∂u − f 0 v∗ = v  π   + F ∂t where v∗ = v −

1 ∂ ρ ∂z



 ρ   v b  N2

is the residual meridional wind, and where turbulent friction is approximated via   ∂u 1 ∂ ρK F = ρ ∂z ∂z

(9.343)

(9.344)

(9.345)

The climatological mean yields − ρ f 0 v∗ = ρv  π   + ρF

(9.346)

9.3 The Circulation in the Midlatitudes

401

or, after inserting (9.344) ∂ − f 0 ρv + f 0 ∂z



   ρ   ∂ ∂u   v b  = ρv π  + ρK N2 ∂z ∂z

(9.347)

The vertical integral of this equation, defined for an arbitrary field X by ∞ {X } =

dz X

(9.348)

0

yields

 − f 0 {ρv} + f 0

   ∂u ∞ ρ   ∞ $   % v b  = ρv π  + ρ K N2 ∂z 0 0

(9.349)

In quasigeostrophic scaling the continuity equation is ∇ · (ρu) +

∂ (ρw) = 0 ∂z

(9.350)

leading in the zonal mean to

Vertical integration gives

∂ ∂ (ρv) + (ρw) = 0 ∂y ∂z

(9.351)

∂ {ρv} = 0 ∂y

(9.352)

due to the boundary conditions ρ −−−→ 0

(9.353)

=

(9.354)

z→∞

w|z=0

0

Since v = 0 holds approximately at the poles ones has in general {ρv} = 0 Moreover, likewise,



   ρ   ∞ ρ   v b  = − v b  N2 N2 z=0 0

For the negative turbulent momentum flux we assume  0 z→∞ ∂u ρK = ∂z r ρu|z=0 z = 0

(9.355)

(9.356)

(9.357)

402

9 The Meridional Circulation

so that one obtains in total z=0:

r u =

f0   1$   % v b  + ρv π  2 N ρ

(9.358)

In the continuously stratified atmosphere as well, the surface winds result from the vertical integral of the potential-vorticity fluxes, however substituted by a contribution from the surface buoyancy fluxes. The Potential-Vorticity Flux Because the zonal-mean meridional potential-vorticity flux agrees with the divergence of the Eliassen–Palm flux, v  π   =

1 ∇ ·F ρ

(9.359)

F = −ρu  v  e y + ρ

f0   v b ez N2

(9.360)

pressure (hPa)

consider the latter in Fig. 9.33. One sees a dominance of the vertical component in the mid latitude lower troposphere. This is the meridional heat flux resulting there from baroclinic instability. Because the Eliassen–Palm flux is also the flux of wave-action density, one sees that the latter is transported upwards and then equatorwards. The upwards increasing horizontal component is explained by the upwards increase of the meridional gradient in the zonal-mean potential vorticity (see Fig. 9.31) and hence also the meridional Rossby-wave group velocity. The signs of the Eliassen–Palm-flux divergence are such that it is negative in the upper troposphere and positive in the lower troposphere. This agrees well with the findings from the two-layer model, and it can also be identified in Fig. 9.34. One can see there as well, with an additional glance at Fig. 8.9, that the upper-troposphere wave fluxes

200

200

400

400

600

600

800

800

1000

1000 0

20°N

40°N 60°N latitude

80°N

0

20°N

40°N 60°N latitude

80°N

Fig.9.33 From ERA5 analysis data (Hersbach et al., 2020), the Eliassen–Palm flux and its divergence (left panel, negative values indicated by dashed contours) together with the zonal-mean wind (right), both for the northern hemisphere in northern-hemispheric winter

pressure (hPa)

9.3 The Circulation in the Midlatitudes

403

101

101

102

102

103 90°S

60°S

30°S

0 30°N latitude

60°N

90°N

103 90°S

60°S

30°S

0 30°N latitude

60°N

90°N

pressure (hPa)

Fig. 9.34 From ERA5 analysis data (Hersbach et al., 2020), the Eliassen–Palm-flux divergence (thin contours, negative values indicated by shading) and the zonal-mean zonal wind (fat contours) in the yearly mean (left panel) and in northern-hemispheric winter (right) 101

101

102

102

103 90°S

60°S

30° S

30°N 0 latitude

60°N

90°N

103 90°S

60°S

30°S

0 30°N latitude

60°N

90°N

Fig. 9.35 From ERA5 analysis data (Hersbach et al., 2020) for northern-hemispheric winter: The decomposition of the Eliassen–Palm-flux divergence into its horizontal part (momentum-flux convergence, left panel) and its vertical part (heat flux, right), each together with the zonal-mean zonal wind as in Fig. 9.34

balance the effect of a poleward residual circulation, and that the lower-troposphere fluxes do so as well (in parts) for the equatorward residual circulation there. The decomposition of the Eliassen–Palm flux divergence into the respective contributions from the momentum and heat fluxes, shown in Fig. 9.35, exhibits quite clearly the pattern of momentum-flux convergence in midlatitudes and divergence at the flanks of the jet stream, as could already be predicted from barotropic theory. Note again that it is not the case at all that positive Eliassen–Palmflux divergence leads to westerlies and negative Eliassen–Palm flux divergence to easterlies! This is because we are considering climatological means where all time derivatives are averaged out. A more intricate situation arises where the surface winds are controlled by a boundary-layer balance between turbulent friction and the vertical-mean Eliassen–Palm

404

9 The Meridional Circulation

flux convergence, supplemented by surface buoyancy fluxes. The upper-tropospheric winds follow from the surface winds and thermal-wind balance.

9.3.5

Summary

Different to the tropics planetary-and synoptic-scale waves in middle latitudes, generated by the land–sea contrast of atmospheric heating, by orography, and by the process of baroclinic instability, are not just one additional factor in the explanation of the mean flow. Here they are essential from the beginning. • Phenomenologically the meridional circulation in the Eulerian mean can be determined by the solution of the elliptic equation for the mass streamfunction. The vertical derivative of the momentum-flux convergence and the latitudinal derivative of the buoyancy-flux convergence are structured such that they force in midlatitudes indirect Ferrel cells. At the surface one obtains westerlies, needed for balancing in the Eulerian-mean meridional momentum equation, via friction, the poleward circulation at the ground. Understanding the wave-flux signs is, however, not possible without the following arguments. • A conspicuous mid latitude phenomenon is a jet stream with a considerably more barotropic structure than the subtropical jet stream. This barotropic jet stream is forced by waves. An important factor is the mixing of (potential) vorticity. First clues to this can already be provided by a barotropic model. The conservation of absolute vorticity due to the Kelvin theorem explains that a mid latitude wave source unavoidably leads to momentum-flux convergence there, and hence a westerly jet stream flanked by easterlies. Alternatively this can be explained by the connection between of the meridional group velocity and the momentum flux of Rossby waves radiated by the wave source. • The most simple framework for the study of non-barotropic aspects, i.e., the wave source, latitude-dependent temperature, baroclinicity of the winds, and the altitude-dependent meridional circulation is a two-layer model, where the role of vorticity is taken by potential vorticity. Essential are sinks and sources in the two continuity equations in this model capturing the rising (sinking) of air masses due to heating in the tropics (cooling at the poles). The residual circulation in the TEM of this model is the mass-weighted meridional flow. Due to mass conservation there is no vertical-mean flow. In the upper troposphere air masses move polewards while they move equatorwards in the lower troposphere. Due to the thermal wind potential-vorticity fluxes are balanced in the vertical and meridional mean. The upper-layer potential-vorticity flux must be directed polewards in order to balance in the climatological-mean zonal-momentum equation the Coriolis acceleration due to the meridional circulation. Hence the lower-layer potential-vorticity flux must be directed equatorwards. In the climatological mean of the zonal-momentum equation the

9.4

Recommendations for Further Reading

405

vertical integral of the potential-vorticity fluxes must be balanced by surface friction. This determines structure and sign of the surface winds. The different mean potentialvorticity gradients lead to differences in the group velocities of the Rossby waves so that the positive upper-layer potential-vorticity flux has a broader distribution than the oppositely directed flux in the lower layer. Hence one obtains in midlatitudes surface westerlies flanked by easterlies. It is the surface winds that are forced directly by the waves, while the upper-layer jet stream follows from these by thermal-wind balance. From the above the Ferrel circulation can be understood as well, via balancing of the Eulerian-mean momentum equation, without needing diagnostic input from analyses. • Simulations and analyses of the continuously stratified atmosphere essentially support the role of the mechanisms discussed above. Again the surface winds follow from a balance between the vertical-mean potential-vorticity fluxes and surface friction. Here, however, surface buoyancy fluxes contribute as well.

9.4

Recommendations for Further Reading

The textbooks by Andrews et al. (1987), Holton and Hakim (2013), Lindzen (1990), and Vallis (2006) can all be helpful in deepening the material in this chapter. The same holds for the original publications by Schneider (1977), Held and Hou (1980) and Lindzen and Hou (1988). Sources on the effect of Rossby waves on the Hadley circulation are Becker et al. (1997), Vallis (2006), and Walker and Schneider (2006). The discussion of the circulation in midlatitudes is based on Held (2000) and Vallis (2006). Useful sources on the energetics of the general circulation are Lorenz (1967) and the textbooks of Peixoto and Oort (1992) and Hartmann (2016).

Gravity Waves and Their Impact on the Atmospheric Flow

10

To leading order the synoptic- and planetary-scale flow of the troposphere can be understood in terms of a zonal-mean circulation interacting with Rossby waves. In the stratosphere above the tropopause and especially in the mesosphere above the stratopause at about 50 km altitude, however, mesoscale gravity waves are a factor not to be ignored. Gravity waves are emitted by various processes in the troposphere, most prominently flow over mountains, convection, and so-called spontaneous imbalance near jets and fronts. From all of these processes gravity waves radiate upwards into the middle atmosphere. Measurements and observations typically show strong gravity-wave activity in these altitudes. Their dynamics is a wide field, easily filling books on their own, that this chapter cannot cover by far. Instead, it gives a short overview on the empirical findings why it is relevant, then discusses the fundamental wave motions of an atmosphere at rest, moves on to the description of the interaction between mesoscale gravity waves and a synoptic-scale flow, and finally demonstrates how that theory can explain the observed gravity-wave effects in the middle atmosphere.

10.1

Some Empirical Facts

The impact of gravity waves is most conspicuous at the summer mesopause at about 80– 90 km altitude. To see this, let us first have a look at Fig. 10.1 that shows the radiativeequilibrium temperature the atmosphere would have in northern-hemispheric winter without waves and without a meridional circulation. One recognizes a temperature maximum in the tropical troposphere, another one around the summer stratopause, at an altitude of about 50 km, and a third one in the thermosphere above the mesopause. The summer-stratopause maximum results from the absorption of solar radiation by stratospheric ozone. Cold temperatures are seen in the winter stratosphere and mesosphere, between about 10 and 90 km altitude, especially in polar latitudes, where the solar radiative influx is minimal. The figure © Springer-Verlag GmbH Germany, part of Springer Nature 2022 U. Achatz, Atmospheric Dynamics, https://doi.org/10.1007/978-3-662-63941-2_10

407

408

10 Gravity Waves and Their Impact on the Atmospheric Flow uE (m s–1)

TE (K) 100

300

100

80

altitude (km)

280

80

260 240

60

220 200

40

60

80

40 20

60

0 –20

40

180 160

20

–40 –60

20

140

0

–80

0 60°S 30°S

0° 30°N 60°N latitude

60°S 30°S

0° 30°N 60°N latitude

Fig. 10.1 Zonal-mean atmosphere as it would result from radiative equilibrium in northernhemispheric winter. Shown are the zonal-mean temperature in K (left panel) and the zonal-mean zonal wind, in m/s, resulting therefrom via the thermal-wind relationship, with zero winds assumed at the ground (right). Reprinted from Becker (2012).

altitude (km)

also shows the zonal wind one obtains from this temperature by integrating the thermalwind relationship upward, assuming zero winds at the ground. No winds are shown near the equator, where the Coriolis parameter passes through zero, so that the thermal wind would be singular there. In the mesosphere one obtains strong westerlies on the winter side and easterlies on the summer side. Both mesospheric jets are nearly barotropic. This is to be contrasted with the zonal-mean climatology observed by satellite, as shown in Fig. 10.2. Both the hot tropical troposphere and the warm summer stratopause appear there as well. What is different, however, is a conspicuous temperature minimum at the summer mesopause and a

100

100

80

80

60

60

40

40

20

20

0

0 60°S

150

30°S

180

0 30°N latitude 210

240

270

60°N

300

60°S

30°S

–60 –40 –20

0 30°N latitude 0

20

40

60°N

60

Fig. 10.2 From URAP data (Swinbank and Ortland, 2003), the zonal-mean atmosphere as it is observed by satellite in northern-hemispheric winter. Shown are the zonal-mean temperature in K (left panel) and the zonal-mean zonal wind, in m/s (right).

10.2 The Fundamental Wave Modes of an Atmosphere at Rest 110

100

altitude (km)

Fig. 10.3 From rocket soundings, vertical profiles of temperature fluctuations in K, between 70 and 110 km altitudes. The red curves indicate an amplitude increase that would be inversely proportional to the square root of the density in an isothermal atmosphere. Courtesy by M. Rapp.

409

90

80

← ~ ez/2H 70 –30

–20

–10

0 ΔT (K)

10

20

30

temperature maximum at the winter stratopause. The corresponding zonal-mean zonal wind also shows easterlies in the summer mesosphere and westerlies in the winter mesosphere. They change their direction, however, at the mesopause. As one crosses the mesopause, the summer easterlies turn into westerlies and the winter westerlies into easterlies. These winds are in reasonable thermal-wind equilibrium with the zonal-mean temperatures. Obviously these structures cannot be explained by radiation alone. Rossby waves are also observed in the mesosphere, but the main agent for the features described above is gravity waves that typically show up as a clear signal in middle-atmosphere measurements. An example is given in Fig. 10.3, showing vertical profiles of temperature fluctuations. Quite strong wavy fluctuations are discernible. One sees an exponential increase of the amplitude of the temperature fluctuations up to about 20 K near 95 km altitude. Further up the amplitude remains close to constant. As will be explained below, this indicates breaking of the measured gravity waves due to some instability process. This instability is instrumental in bringing irreversible mean-flow impacts about.

10.2

The Fundamental Wave Modes of an Atmosphere at Rest

Before we turn to the impact of gravity waves on the large-scale flow, let us first recognize them as one of the fundamental wave modes in the atmosphere. For a better understanding of their structure it is worthwhile to begin by considering the corresponding energetics. This will be followed by a derivation of the dispersion and polarization relations of the fundamental wave modes. A preliminary discussion of the properties of gravity waves will conclude this section.

410

10 Gravity Waves and Their Impact on the Atmospheric Flow

10.2.1 Equations of Motion and Energetics Review of the Equations on an f -Plane To keep things as simple as possible, we consider the equations of motion on an f -plane. From Chaps. 1 and 2 we recall that these are the momentum equation 1 1 Dv + f ez × u = − ∇ p − gez + ∇ · σ , Dt ρ ρ

(10.1)

the entropy equation Dθ θ = Q Dt cpT

Q =+q −

1 ∇ · Fq ρ

(10.2)

the continuity equation Dρ ∂ρ + ρ∇ · v = + ∇ · (ρv) = 0 Dt ∂t

(10.3)

p = ρ RT

(10.4)

and the equation of state

Here u = uex + ve y is the horizontal wind, v = u + wez the total wind, f the constant Coriolis parameter, ρ the density, p the pressure, g the gravitational acceleration, σ the viscous stress tensor with Cartesian elements   ∂v j ∂vi 2 ∂vk ∂vk + ζ δi j + − δi j (10.5) σi j = η ∂x j ∂ xi 3 ∂ xk ∂ xk that are linear in ∇v, θ = T ( p00 / p) R/c p the potential temperature, T the temperature, =

∂vi 1 σi j ρ ∂x j

(10.6)

the rate of viscous dissipation, and Fq = −κ∇T the diffusive heat flux. From the definition of the potential temperature one can easily find that the equation of state also has the alternative form   p cV /c p p00 (10.7) ρ= Rθ p00 As discussed in Sect. 3.3.1, the equations of motion satisfy an energy-conservation law     ∂ p (ρe) + ∇ · ρv e + + Fq − v · σ = ρq ∂t ρ

e=

|v|2 + cV T + gz 2

(10.8)

10.2 The Fundamental Wave Modes of an Atmosphere at Rest

411

 so that energy E = d V ρe is conserved when through the vertical boundaries (at the ground and at infinite altitude) there are neither viscous energy flux, nor diffusive energy flux, nor mechanical energy flux, and when there is no net volume heating. Energetics of Weak Fluctuations in an Atmosphere at Rest There is also a corresponding conservation law for the energy of infinitesimally small deviations from an atmospheric state at rest. The latter is defined as a solution of the equations of motion with v=0

(10.9)

ρ = ρ(z)

(10.10)

p = p(z)

(10.11)

θ = θ (z)

(10.12)

where pressure and density must be in hydrostatic equilibrium, 0=−

1 dp −g ρ dz

(10.13)

and also volume heating must be balanced by diffusive heat fluxes, i.e., 0=q+

κ d2 T ρ dz 2

(10.14)

We now consider infinitesimally small deviations v = v very small ρ =ρ+ρ



p= p+ p θ =θ +θ





(10.15)



(10.16)



(10.17)

|ρ |  ρ |p |  p 

|θ |  θ

(10.18)

dρ dθ cV d p =− + ρ θ cp p

(10.19)

Via differentiation (10.7) becomes

yielding the linearized equation of state ρ θ cV p  =− + ρ θ cp p

(10.20)

412

10 Gravity Waves and Their Impact on the Atmospheric Flow

Similarly one obtains, using the hydrostatic equilibrium (10.13), 1 dρ 1 dθ cV 1 d p 1 dθ cV ρ =− + =− −g ρ dz c p p dz cp p θ dz θ dz

(10.21)

Linearization of the momentum equation (10.1) gives 1 ρ d p 1 ∂v + f e z × u = − ∇ p  + 2 ez + ∇ · σ  ∂t ρ¯ ρ ρ dz

(10.22)

where, with hydrostaticity (10.13), the linearized equation of state (10.20), and (10.21), the two first terms on the right-hand side can be rewritten    1 ρ d p p p dρ ρ  ez − ∇p + 2 ez = −∇ − +g ρ¯ ρ ρ ρ dz ρ 2 dz      cV p  p θ p  1 dθ cV ρ ez − = −∇ + +g ez + g ρ ρ θ dz cp p cp p θ (10.23) so that (10.22) becomes ∂v p + f ez × u = −∇ + ∂t ρ



 1 dθ p  1 + b  ez + ∇ · σ  dz ρ ρ θ

(10.24)

where b = gθ  /θ is buoyancy. Moreover, since  is nonlinear in ∇v , linearization of the entropy equation (10.2) leads to   θ κ 2  κ d2T ρ dθ ∂θ   q + ∇ T − , (10.25) +w = ∂t dz ρ ρ dz 2 ρ cpT or ∂b g Q + N 2 w = ∂t cpT

Q = q  +

κ 2  κ d2T ρ ∇ T − ρ ρ dz 2 ρ

(10.26)

where g dθ (10.27) θ dz is the squared Brunt–Väisälä frequency. Finally we also linearize the continuity equation (10.3), obtaining N 2 (z) =

∂ρ  + ∇ · (ρv ) = 0 ∂t

(10.28)

10.2 The Fundamental Wave Modes of an Atmosphere at Rest

413

Now we are ready for the derivation of an energy-conservation law for the linear dynamics of the perturbations. First, taking the scalar product of ρv with the linearized momentum equation (10.24) yields p ∂ |v |2 1 dθ   ρ = −ρv · ∇ + p w + ρb w + v · ∇ · σ ∂t 2 ρ θ dz

(10.29)

or ∂(ρeκ ) + ∇ · ( p  v − v · σ  ) = Ce + Ca − ρ  ∂t where eκ =

|v |2 2

is the mass-specific kinetic energy density,

p ρ dθ  p  ∇ · (ρv ) + ∇ · (ρθ v ) w = Ce = ρ θ dz ρθ the elastic exchange rate,

(10.30)

(10.31)

(10.32)

Ca = ρb w

(10.33)

1  ∂vi σ ρ ij ∂x j

(10.34)

the anelastic exchange rate, and  =

the quasilinear viscous dissipation rate. Moreover, multiplying the linearized entropy equation (10.26) by ρb /N 2 yields the prognostic equation ∂(ρea ) gρ b Q  = −Ca + ∂t cpT N 2

(10.35)

for the anelastic potential energy density b 2 (10.36) 2N 2 In a next step we divide the linearized continuity equation (10.28) by ρ and use the linearized equation of state (10.20), obtaining     1 1 ∂ ρ 1 ∂b ∂ cV p   + ∇ · (ρv ) = − + ∇ · (ρv ) 0= (10.37) + ∂t ρ ρ g ∂t ∂t c p p ρ ea =

414

10 Gravity Waves and Their Impact on the Atmospheric Flow

Together with the buoyancy equation (10.26) this yields a prognostic equation for the perturbation pressure ∂ ∂t



1 p cs 2 ρ



1 dθ  Q Q Ce 1 =−  + = − ∇ · (ρv ) − w + ρ p θ dz cpT cpT

(10.38)

cp RT cV

(10.39)

where cs2 =

is the squared speed of sound. Multiplying (10.38) by p  finally gives the prognostic equation ∂ p Q  (ρee ) = −Ce + ∂t cpT

(10.40)

for the elastic potential energy density ee =

1 2cs2



p ρ

2 (10.41)

Finally we take the sum of the three prognostic energy equations (10.30), (10.35), and (10.40), thereby obtaining the conservation law 1 ∂  ρe + ∇ · ( p  v − v · σ  ) = −ρ  + ∂t cpT



 ρg   Q b + p N2

(10.42)

for the total energy of the fluctuations, with mass-specific density e = ek + ea + ee

(10.43) 

In the absence of friction, diffusion, and volume heating the fluctuation energy d V ρe is conserved, provided there is no mechanical energy flux through the vertical boundaries. Even in the conservative case, however, there is an exchange between kinetic and elastic potential energy, resting in the pressure fluctuations, via Ce , and between kinetic and anelastic potential energy, contained in the buoyancy fluctuations, via Ca .

10.2.2 Free Waves on the f -Plane in an Isothermal Atmosphere The Linear Equations in Fourier Space Let us now turn to the fundamental wave modes of the linear dynamics. For this purpose we consider the linearized equation of state (10.20), momentum equation (10.24), buoyancy equation (10.26), and pressure equation (10.38) without volume heating, diffusive heat fluxes and friction. Since the reference atmosphere does not depend on x, y, and t, we perform a

10.2 The Fundamental Wave Modes of an Atmosphere at Rest

415

Fourier transformation of the equations with respect to these variables. The dependence of the volume-specific energy density ρe = ρ

|v |2 b 2 ρ + 2 +ρ 2 2 N2 2cs



p ρ

2

on the fluctuating fields suggests for their vertical dependence an ansatz 1 N b ∼ √ v ∼ √ ρ ρ

 p  ∼ cs ρ

(10.44)

We thus write, with ρ0 a constant reference density, ⎛

v˜ √ ⎜ ρ/ρ ⎛ ⎞ 0 ⎜  v ˜ ⎜ bN ⎜ i(kx+ly−ωt) ⎝ b ⎠ = dk dl dω e ⎜ √ρ/ρ ⎜  0  ⎜ p ⎝ ρ pc ˜ s ρ0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(10.45)

Inserting this into (10.24) yields u˜ v˜ cs −iω √ − f√ = −ik p˜ √ ρ/ρ0 ρ/ρ0 ρρ0 u˜ cs v˜ + f√ = −il p˜ √ −iω √ ρ/ρ0 ρ/ρ0 ρρ0   cs w˜ d N 2 cs N −iω √ =− p˜ + √ p˜ + b˜ √ √ dz ρ/ρ0 ρ/ρ0 ρρ0 g ρρ0

(10.46) (10.47) (10.48)

In the isothermal case T = const. the speed of sound in (10.39) is a constant, and the reference-atmosphere density and pressure in hydrostatic equilibrium (10.13) have exponential profiles p = p0 e−z/H p ρ= = ρ0 e−z/H RT where we have made the choice ρ0 =

p0 RT

(10.49) (10.50)

(10.51)

for ρ0 , and where H=

RT g

(10.52)

416

10 Gravity Waves and Their Impact on the Atmospheric Flow

is the constant pressure and density scale height. Moreover, the potential temperature and stratification are     p00 R/c p p00 R/c p z/Hθ θ=T = θ0 e θ0 = T (10.53) p p0 N2 =

g dθ g = Hθ θ dz

(10.54)

with the potential-temperature scale height Hθ =

cp H R

(10.55)

Under these conditions the transformed momentum equations (10.46)–(10.48) become cs ρ0 cs −iωv˜ + f u˜ = −il p˜ ρ0   cs 1 1 d p˜ −iωw˜ = − + N b˜ − Hθ 2H dz ρ0 −iωu˜ − f v˜ = −ik p˜

(10.56) (10.57) (10.58)

In the same manner, using the Fourier decomposition (10.45) in the buoyancy equation (10.26) yields − iωb˜ + N w˜ = 0 (10.59) while the pressure equation (10.38) becomes p˜ = −ik u˜ − il v˜ − − iω cs ρ0



1 1 d − + Hθ 2H dz

 w˜

(10.60)

We now see that, due to the constant reference-atmosphere temperature, and because of the transformation (10.45), all coefficients in (10.56)–(10.60) are constants. Hence, ignoring the existence of a solid boundary at the ground, a further vertical Fourier transformation ⎛ ⎞ ⎛ ⎞  vˆ v˜ ⎝ b˜ ⎠ = dm eimz ⎝ bˆ ⎠ p˜ pˆ

(10.61)

can be applied, yielding the system of equations cs ρ0 cs −iωvˆ + f uˆ = −il pˆ ρ0   1 1 cs −iωwˆ = − + N bˆ − im pˆ Hθ 2H ρ0

−iωuˆ − f vˆ = −ik pˆ

(10.62) (10.63) (10.64)

10.2 The Fundamental Wave Modes of an Atmosphere at Rest

−iωbˆ + N wˆ = 0 −iω

pˆ = −ik uˆ − il vˆ − cs ρ0



417



1 1 − + im wˆ Hθ 2H

(10.65) (10.66)

In each Fourier subspace corresponding to a specific choice of wavenumber k = kex +le y + ˆ ˆ v, ˆ w, ˆ b, mez and frequency ω, these are five linear equations for the Fourier amplitudes u, and p. ˆ The Dispersion Relations The situation is the same as in the case of shallow-water waves on the f -plane, discussed in Sect. 5.4.2: The system of equations (10.62)–(10.66) has non-trivial solutions only if the corresponding matrix is singular. The roots of its determinant yield the possible dispersion relations. Instead of using this determinant, we here take an equivalent approach where we consecutively eliminate more and more of the unknown variables so that eventually an algebraic equation for one of those is achieved that can only be solved in a non-trivial manner if dispersion relations between frequency and wavenumber are satisfied. The advantage of this is that half of the work necessary for eventually also determining the polarization relations between the various Fourier amplitudes is already done this way, and also that corresponding structural understanding can be obtained more directly. To begin with, we solve the transformed horizontal-momentum equations (10.62) and (10.63) for the horizontal-wind amplitudes. One obtains   ˆ s kω + il f pc (10.67) uˆ = 2 2 ω − f ρ0   ˆ s lω − ik f pc vˆ = (10.68) ω2 − f 2 ρ0 Inserting this into the transformed pressure equation (10.66) leads to     1 1 − + im ω2 − f 2 ics Hθ 2H pˆ = −ρ0 wˆ ω ω2 − f 2 − cs2 kh2

(10.69)

However, the transformed buoyancy equation (10.65) implies wˆ =

iω ˆ b N

(10.70)

so that one finally gets  pˆ = ρ0

cs N

   1 1 − + im ω2 − f 2 ωˆ Hθ 2H b 2 2 2 2 ω ω − f − cs k h

(10.71)

418

10 Gravity Waves and Their Impact on the Atmospheric Flow

Note that we do not set ω/ω = 1, anticipating the possibility ω = 0. Inserting (10.70) and (10.71) into (10.64) gives      1 1 1 1 − − − im + im ω2 − f 2 2 2 ωˆ ω ˆ c Hθ 2H Hθ 2H b= s b + N bˆ (10.72) N N ω ω2 − f 2 − cs2 kh2 A solution with non-vanishing buoyancy fluctuations bˆ  = 0 implies 

  2    2 1 1 2 2 2 2 2 2 2 2 ω − f2 ω − +m ω(ω − N ) ω − f − cs kh = cs Hθ 2H

(10.73)

There are two possibilities to satisfy this equation. One is ω = ωg = 0

(10.74)

This is the eigenfrequency of the balanced (geostrophic/ hydrostatic) mode. Inserting ω = 0 into the transformed momentum equations (10.46)–(10.48) one sees directly that this mode is indeed in geostrophic and hydrostatic equilibrium. The other possibility is that the quadratic equation in ω2 , 2

 1 1 (10.75) − 0 = ω4 − ω2 f 2 + cs2 (k 2 + l 2 + m 2 ) + N 2 + cs2 Hθ 2H 

 1 1 2 2 2 2 2 2 2 2 2 + N cs k h + f cs m + N + cs (10.76) − Hθ 2H be satisfied where it follows from the above that     c2 1 1 1 2 c2 1 + s2 N 2 + cs2 − = N2 + s − Hθ 2H Hθ Hθ H 4H with

1 1 − = Hθ H



so that

 N 2 + cs2

We thus obtain

R −1 cp



1 cV 1 cV g g =− =− =− 2 H cp H c p RT cs

1 1 − Hθ 2H

2 = N2 −

g c2 c2 + s2 = s2 Hθ 4H 4H

   1 0 = ω4 − ω2 f 2 + cs2 k 2 + l 2 + m 2 + 4H 2    1 + cs2 N 2 kh2 + f 2 m 2 + 4H 2

(10.77)

(10.78)

(10.79)

(10.80)

10.2 The Fundamental Wave Modes of an Atmosphere at Rest

419

which is solved by    1 1 2 2 2 2 2 f + cs k h + m + ω = 2 4H 2    2    1 1 1 2 2 2 2 2 2 2 2 2 f + cs k h + m + ± − cs N k h + f m + 4 4H 2 4H 2 (10.81) where kh2 = k 2 + l 2 is the squared horizontal wavenumber. Moreover, with the definitions (10.39) of the speed of sound, (10.52) of the isothermal atmospheric scale height, and L 2d = g H / f 2 the squared external Rossby deformation radius one has for typical referenceatmosphere temperatures of a few 100K f2 4 H2 = 1 cs2 /4H 2 c p /cV L 2d

(10.82)

so that (10.81) can be simplified to   c2 1 ω2 ≈ s kh2 + m 2 + 2 4H 2       cs4 1 2 1 2 N 2k2 + f 2 m2 + ± − c kh2 + m 2 + s h 4 4H 2 4H 2

(10.83)

However, due to (10.82) one also has cs4 4

 kh2

1 +m + 4H 2

2

2



cs2

 N 2 kh2

+ f

2

1 m + 4H 2



2

This is directly visible at zero wavenumbers and follows thence from the rise of the first term with wavenumber to the fourth power, as compared to the squared-wavenumber dependence of the second term. Hence, the square root can be expanded, and one obtains to a very good approximation from the plus sign in front of the root the dispersion relation of sound waves  ω2 = ωs2 ≈ cs2 kh2 + m 2 +

1 4H 2

 (10.84)

while the minus sign yields that of gravity waves

2 ω2 = ωgw ≈

 N 2 kh2 + f 2 m 2 + kh2 + m 2 +

1 4H 2

1 4H 2

 (10.85)

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10 Gravity Waves and Their Impact on the Atmospheric Flow

Alternatively, (10.72) is trivially satisfied if the buoyancy amplitude vanishes, bˆ = 0

(10.86)

wˆ = 0

(10.87)

In this case (10.65) implies that also so that the vertical-momentum equation (10.64) becomes   1 1 cs 0= − − im pˆ Hθ 2H ρ0

(10.88)

Concluding from this that pˆ = 0 would also lead, via the transformed horizontal-momentum equations (10.62) and (10.63), and the transformed pressure equation (10.66) to the trivial solution uˆ = vˆ = 0, i.e., no fluctuations at all! However, we also have the possibility that m=

i i − 2H Hθ

(10.89)

implying purely horizontal propagation. Inserting now wˆ = 0 along with (10.67) and (10.68) into the transformed pressure equation (10.66) yields ω

k2 pˆ pc ˆ s = 2 h 2ω cs ρ0 ω − f ρ0

(10.90)

leading us to the geostrophic mode without buoyancy fluctuations with again ω = ωg = 0

(10.91)

cs2 kh2 ω2 − f 2

(10.92)

or 1= i.e., the dispersion relation

ω2 = ω2L = f 2 + cs2 kh2

(10.93)

of a Lamb wave. Figure 10.4 summarizes all branches for the non-rotating case ( f = 0) and horizontal propagation in x-direction (l = 0). Sound waves are the highest-frequency waves, while the frequency of gravity waves is between f and N . The Polarization Relations for Gravity Waves and Sound Waves Let us now have a look at what these wave solutions look like. How is the phase- and amplitude relation between the various dynamical fields? This question is answered by the polarization relations. Every superposition of waves satisfying the various dispersion relations introduced above is a possible solution of the linear dynamics. We here address

10.2 The Fundamental Wave Modes of an Atmosphere at Rest

8

421

gw (mH = 0) gw (mH = 1) sw (mH = 0) sw (mH = 1) Lamb wave geostr. mode

7 6

ω/N

5 4 3 2 1 0 −2

−1. 5

−1

−0. 5

0 kH

0.5

1

1.5

2

Fig. 10.4 Eigenfrequencies of all free modes of a rotating ( f = 0.1N ) isothermal atmosphere at rest, propagating in the horizontal in the x-direction (l = 0).

gravity waves and sound waves. The geostrophic mode and then the two wave types without buoyancy oscillations are discussed in Appendix H. ω  = 0 implies, as previously derived, the relation (10.70) between buoyancy and vertical wind, and the relation (10.71) between buoyancy and pressure. Inserting these into (10.67) and (10.68) leads to   1 1 − + im (kω + il f ) c2 Hθ 2H bˆ (10.94) uˆ = s N ω2 − f 2 − cs2 kh2   1 1 − ik f − + im (lω ) c2 Hθ 2H bˆ (10.95) vˆ = s 2 2 N ω − f − cs2 kh2 Here the part of bˆ due to gravity waves or sound waves is only non-zero when one of the corresponding dispersion relations is satisfied, i.e., when ω agrees with one of the four options   ωs± (k) = ± ωs2 (k)

± 2 (k) ωgw (k) = ± ωgw

(10.96)

422

10 Gravity Waves and Their Impact on the Atmospheric Flow

± (k) so that Therefore for each combination of k and ω there are amplitudes Bˆ s± (k) and Bˆ gw



ˆ ω) = b(k,

α=s,gw ν=+/−

  Bˆ αν (k)δ ω − ωαν (k)

(10.97)

Hence, from (10.61), ˜ l, z, ω) = b(k,





dm eimz

α=s,gw ν=+/−

  Bˆ αν (k)δ ω − ωαν (k)

(10.98)

Inserting this into the original Fourier representation (10.45) one finally obtains, with the help of the density profile (10.50), the spatio-temporal dependence   ν  b (x, t) = d 3 k eik·x+z/2H Bαν (k)e−iωα (k)t (10.99) α=s,gw ν=+/−

of the buoyancy field of a superposition of sound waves and gravity waves, where Bαν (k) = N Bˆ αν (k)

(10.100)

It is important to realize that Bα+ and Bα− are not independent from each other. This follows from (10.101) ωαν (−k) = ωαν (k) ωα−ν = −ωαν and from the fact that the buoyancy field is real-valued. Hence, with an asterisk denoting the complex conjugate, and by the substitutions k → −k and ν → −ν     ∗ ν 3 ik·x+z/2H ν −iωαν (k)t  ∗ d ke Bα (k)e = b = b = d 3 k e−ik·x+z/2H Bαν (k)eiωα (k)t  =

α,ν

d 3 k eik·x+z/2H

α,ν

 =



d 3 k eik·x+z/2H

 α,ν

ν

Bαν ∗ (−k)eiωα (−k)t = ∗

Bα−ν (−k)e

 d 3 k eik·x+z/2H

−iωαν (k)t

 α,ν

α,ν ∗

−ν (−k)t

Bα−ν (−k)eiωα

(10.102)

Therefore both for gravity waves and sound waves the respective amplitudes Bα+ and Bα− are related by Bα−ν (k) = Bαν ∗ (−k)

(10.103)

10.2 The Fundamental Wave Modes of an Atmosphere at Rest

so that b =

 d 3 k eik·x+z/2H

=

α



 3

d k



e

ik·x+z/2H

+

−

Bα+ (k)e−iωα (k)t + Bα− (k)e−iωα (k)t

 α

423

+ Bα+ (k)e−iωα (k)t

+e

−ik·x+z/2H



 α

+ Bα+∗ (k)eiωα (k)t

(10.104) leading finally to b = 2

 d 3 k eik·x+z/2H

α=s,gw

 = 2



d 3 k eik·x+z/2H



α=s,gw

+

Bα+ (k)e−iωα (k)t −

Bα− (k)e−iωα (k)t

(10.105)

Thus, either the amplitudes Bα+ or their counterparts Bα− contain all information needed! Likewise one proceeds to obtain the representation of the wind fields. One inserts the decomposition (10.97) into the relations (10.94), (10.95), and (10.70) to calculate vˆ , gets from that via (10.61) v˜ , and finally uses the latter, together with the density profile (10.50), to obtain the wind-field representation of a superposition of sound waves and gravity waves   ν  Vαν (k)e−iωα (k)t (10.106) v = d 3 k eik·x+z/2H α=s,gw ν=+/−

where the polarization relations ωαν ν B N2 α    ν  1 1 kω + il f − + im α c2 Hθ 2H Uαν = s2 Bαν N ωα 2 − f 2 − cs2 kh2    ν  1 1 lωα − ik f − + im c2 Hθ 2H Vαν = s2 Bαν N ωα 2 − f 2 − cs2 kh2

Wαν = i

(10.107)

(10.108)

(10.109)

relate the wind-field amplitudes to the buoyancy amplitudes. Since v is real, the same arguments as above for the buoyancy lead to Vα−ν (k) = Vαν∗ (−k)

(10.110)

424

10 Gravity Waves and Their Impact on the Atmospheric Flow

which can also be verified explicitly from (10.107)–(10.109). Thus one obtains v = 2

 d 3 k eik·x+z/2H

α=s,gw

 = 2



d 3 k eik·x+z/2H



α=s,gw

+

Vα+ (k)e−iωα (k)t −

Vα− (k)e−iωα (k)t

Finally, analogous arguments for the pressure yield, using (10.71),   ν p  = d 3 k eik·x−z/2H Pαν (k)e−iωα (k)t

(10.111)

(10.112)

α=s,gw ν=+/−

where  Pαν = ρ0

cs2 N2

   1 1 − + im ωα 2 − f 2 Hθ 2H Bαν ωα2 − f 2 − cs2 kh2

(10.113)

is the polarization relation between pressure and buoyancy. Again one has Pα−ν (k) = Pαν∗ (−k)

(10.114)

whence 



p = 2

d 3 k eik·x−z/2H

α=s,gw

 = 2



d 3 k eik·x−z/2H



α=s,gw

+

Pα+ (k)e−iωα (k)t −

Pα− (k)e−iωα (k)t

(10.115)

Note that buoyancy and velocity increase with altitude ∼ e z/2H whereas the pressure decreases ∼ e−z/2H ! The corresponding increase in the temperature amplitude is visible in Fig. 10.3. As will be discussed below, the exponential increase in the buoyancy amplitude makes these waves statically unstable when they reach sufficiently high altitudes. Shear instabilities due to strong velocity gradients are possible as well. An important, actually the most frequently met, special case is that of waves with vertical wavelengths λz  4π H sufficiently short so that m2

1 4 H2

(10.116)

10.2 The Fundamental Wave Modes of an Atmosphere at Rest

425

For these the dispersion relations (10.84) and (10.85) simplify to   ωs2 = cs2 kh2 + m 2

(10.117)

N 2 kh2 + f 2 m 2 kh2 + m 2

2 ωgw =

(10.118)

A short calculation then shows that, with N 2 f 2 , and using the definitions (10.39) of the speed of sound, and (10.52) and (10.55) of the isothermal atmospheric density and potential-temperature scale heights 2 − f2 ωgw

cs2 kh2

=

N2 − f 2 cV R N2 N2 H2 = 4 2 = O(1) ≈  4 2 2 2 2 2 2 2 cs cp cs (kh + m ) cs (kh + m )

(10.119)

Hence, and because (10.116) also entails    1 1   |m|  − Hθ 2H 

(10.120)

the polarization relations (10.107)–(10.109) and (10.113) for gravity waves can be approximated well by

ν Ugw =−

ν Vgw =− ν Wgw =i

  ν + il f im kωgw N 2 kh2

  ν − ik f im lωgw

ν ωgw

N2

N 2 kh2

ν Bgw

(10.121)

ν Bgw

(10.122)

ν Bgw   2 − f2 m ωgw

ν Pgw = − iρ0

N 2 kh2

(10.123) ν Bgw

(10.124)

For a further illustration of the meaning of these complex amplitudes we finally introduce the real-valued amplitude A(k) and phase φ(k) so that A iφ (10.125) e 2 and hence the gravity-wave part of the dynamical fields according to (10.105), (10.111), and (10.115) becomes − = Bgw

426

10 Gravity Waves and Their Impact on the Atmospheric Flow 

b =

 d 3 k A(k)e z/2H cos 



  − k · x − ωgw (k)t + φ(k))

   A(k) z/2H  − ml f cos k · x − ωgw e (k)t + φ(k) 2 2 N kh    − − + mkωgw sin k · x − ωgw (k)t + φ(k)      A(k)  − (k)t + φ(k) v = d 3 k 2 2 e z/2H −mk f cos k · x − ωgw N kh    − − + mlωgw sin k · x − ωgw (k)t + φ(k)  −    ωgw − (k)t + φ(k) w = − d 3 k 2 A(k) e z/2H sin k · x − ωgw N   2 − f2     m ωgw −z/2H − p  = d 3 k ρ0 A(k) e sin k · x − ω (k)t + φ(k) gw N 2 kh2 u =

(10.126)

d 3k

(10.127)

(10.128) (10.129)

(10.130)

Again we stress that the choice for the −-branch was arbitrary. Analogous results can be − → B + and ω− → ω+ . obtained after choosing the +- branch, i.e., by replacing Bgw gw gw gw It is an instructive exercise, left to the reader, to verify that the short-wavelength gravitywave dispersion relation, along with the corresponding polarization relations (10.121)– (10.124) can also be obtained from the Boussinesq equations introduced in Sect. 7.1.2. This is because a basic assumption of Boussinesq theory is that all relevant vertical scales are smaller than the atmospheric density scale height, in agreement with the assumption (10.116) used here. Many studies on gravity-wave dynamics therefore use this more simple framework. What it cannot explain, however, is the exponential increase in gravity-wave amplitude due to the decreasing reference-atmosphere density.

Some Properties of Gravity Waves At this stage one can already discern some important properties of gravity waves. To begin with, the gravity-wave dispersion relation (10.118) can also be written 2 = N 2 cos2 θ + f 2 sin2 θ ωgw

(10.131)

where here θ is the angle between the wave vector and the horizontal (Figs. 10.5 and 10.6). Note that the frequency is only determined by this angle, but not be the total wavelength. One sees that the squared frequency obeys ! 2 N if θ → 0, π 2 ωgw → f 2 if θ → ± π2 Thus, the phase propagation of low-frequency gravity waves is close to vertical. Their vertical wavenumber is considerably larger than the horizontal wavenumber, so that their horizontal

10.2 The Fundamental Wave Modes of an Atmosphere at Rest

427

cg

ϕ z

cp

θ x Fig. 10.5 Geometry of a gravity wave in Boussinesq approximation. The wave vector is assumed to lie in the x − z-plane. It is directed at an angle θ with respect to the horizontal. Lines of constant phase are indicated by blue coloring. The group velocity is parallel to lines of constant phase. If it is directed upward, as indicated here, the phase velocity of the wave is antiparallel to the wave vector.

scale is much larger than their vertical scale. These waves are therefore hydrostatic waves. High-frequency wave gravity waves, on the contrary, have a significant phase-velocity component in the horizontal. They are non-hydrostatic. The dispersion relation is also illustrated in Fig. 10.6. The phase velocity is ⎛ ⎞ k ω ω ⎝ ⎠ k= cp = l |k|2 |k|2 m

(10.132)

while it is an easy exercise to show that the group velocity is ⎛ ⎞ km 2 N2 − f 2 ⎝ cg = ∇k ω = lm 2 ⎠ ω|k|4 −kh2 m

(10.133)

cg · c p = 0

(10.134)

and thus

428

10 Gravity Waves and Their Impact on the Atmospheric Flow

ω N cgz =

∂ω >0 ∂m

cgz < 0 f

m cgz < 0

–f

cgz > 0

–N Fig. 10.6 Gravity-wave frequency dependence (in an atmosphere at rest) on vertical wavenumber. Low vertical wavenumbers (long vertical wavelengths, non-hydrostatic case) imply high frequencies where high vertical wavenumbers (short vertical wavelengths, hydrostatic case) go along with low frequencies close to the inertial frequency.

It is a very interesting property of gravity waves that the group velocity is perpendicular to the phase velocity! The wave energy actually propagates along lines of constant phase. This is well illustrated in Fig. 10.7 which shows the radiation of gravity waves from a convective event. We also note that the two gravity-wave branches satisfy the following relationship between vertical wavenumber and vertical group velocity: ! cgz < 0 if m > 0 + (10.135) : ω = ωgw cgz > 0 if m < 0 ! cgz > 0 if m > 0 − ω = ωgw (10.136) : cgz < 0 if m < 0 One sees that signs of vertical wavenumber and vertical group velocity match in the case of the negative-frequency branch, as also visible in Fig. 10.6. For this reason it is often more convenient to choose this one to represent the gravity-wave field. It is left as an exercise to the reader to verify that the four possible configurations of phase and group velocity, determined by horizontal and vertical wavenumber components are as indicated in Fig. 10.8.

10.2 The Fundamental Wave Modes of an Atmosphere at Rest

429

90 32 16 8 4 2 1 .5 –.5 –1 –2 –4 –8 –16 –32

80 70

z (km)

60 50 40 30 20 10

a

0 90

b

32 16 8 4 2 1 .5 –.5 –1 –2 –4 –8 –16 –32

80 70

z (km)

60 50 40 30 20 10 0

c 0

d 500

1000 x (km)

1500

2000

0

500

1000 x (km)

1500

2000

Fig.10.7 Gravity-wave radiation from a convective event in the troposphere. Shown are vertical wind (color shading), the log of the potential temperature (black contours) and the cloud outline (white contour) at 2h (a), 4h (b), 6h (c), and 8h (d) after the initialization of the simulation. Note how the energy propagates parallel to lines of constant phase. Reprinted from Holton and Alexander (1999) .

10.2.3 Summary The linear dynamics of perturbations of an f -plane atmosphere at rest highlights some fundamental wave modes of atmospheric dynamics and their properties: • In the absence of heating, friction, and heat conduction not only the nonlinear dynamics conserves energy, but also its linear counterpart. Next to kinetic energy two potentialenergy reservoirs appear. Anelastic potential energy is carried by potential-temperature or buoyancy fluctuations, while elastic potential energy is due to pressure fluctuations. The

430

10 Gravity Waves and Their Impact on the Atmospheric Flow

z

cg

cg

cp

cp

g

x

θ

cp

cg

cp

cg

Fig. 10.8 Four possible configurations of the group velocity cg and phase velocity c p of plane gravity waves, with wavenumber vector inclined at an angle θ to the horizontal, obeying the dispersion relation (10.131).

volume-specific density of the kinetic and anelastic potential energy, respectively, weights the squared wind and buoyancy fluctuations with the reference-atmosphere density, while the squared pressure fluctuations contribute to the elastic energy density with an inverse density weight. • This has consequences for the wave modes occurring in the linear dynamics. Wind and buoyancy amplitudes are inversely proportional to the square root of the referenceatmosphere density. Hence they grow considerably with altitude. The opposite is the case for the pressure amplitudes that are strongest at the ground. Three wave modes arise, i.e., a geostrophic mode with zero frequency, sound waves, and gravity waves. These modes are supplemented by various modes with an imaginary vertical wavenumber that can only propagate in the horizontal. The Lamb wave is the most important of these. All modes satisfy polarization relations relating the amplitudes in the various fields with each other, so that one of them determines all others. Gravity waves are highly

10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow

431

dispersive, i.e., group and phase velocity do not agree. They are actually orthogonal to each other so that the propagation of a wave group is along its phase lines.

10.3

The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow

In order to understand how gravity waves influence the atmospheric circulation we must allow for the presence of a flow with scales larger than those of the gravity waves of interest. We will now do so, again however with a focus on the extratropics. To keep things as simple as possible we will restrict ourselves to inviscid and adiabatic dynamics on an f -plane where x and y denote the zonal and meridional direction, respectively. Mesoscale gravity waves will be considered together with a synoptic-scale flow. In the extratropics the latter is described by quasigeostrophic theory. This theory will be extended to incorporate the interaction of small-scale gravity waves with the synoptic-scale flow. For this we will use Wentzel–Kramers–Brillouin (WKB) theory and consider slowly modulated gravity waves in a slowly varying and large-scale background.

10.3.1 A Reformulation of the Dynamical Equations It has turned out convenient to slightly reformulate the dynamical equations, by introducing the Exner pressure  π=

p p00

 R/c p (10.137)

so that temperature and potential temperature are linked by T = π θ . This inserted together with p = p00 π c p /R into the equation of state p = ρ RT yields for the latter the alternative formulation ρ=

p00 cV /R π Rθ

(10.138)

whence one determines directly for the pressure-gradient acceleration 1 ∇ p = c p θ ∇π ρ

(10.139)

so that the inviscid momentum equation on an f -plane takes the form Dv + f ez × u = −c p θ ∇π − gez Dt

(10.140)

432

10 Gravity Waves and Their Impact on the Atmospheric Flow

The adiabatic entropy equation remains Dθ =0 Dt

(10.141)

and instead of the continuity equation we use the prognostic equation for the Exner pressure: From the equation of state (10.138) one obtains in the absence of heating processes 1 Dθ cV 1 Dπ cV 1 Dπ 1 Dρ =− + = ρ Dt θ Dt R π Dt R π Dt

(10.142)

Due to the continuity equation, however, the left-hand side is −∇ · v so that one obtains the prognostic equation Dπ R π∇ · v = 0 + Dt cV

(10.143)

We will use the system (10.138), (10.140), (10.141), and (10.143) to derive prognostic equations for the interaction between mesoscale gravity waves and a synoptic-scale flow.

10.3.2 Scaling for Synoptic-Scale Flow and for Inertia-Gravity Waves Quasigeostrophic theory, as derived and applied in Chap. 6, has proven a powerful tool for the analysis of synoptic-scale motions in the extratropics. In its derivation one identifies typical synoptic length and time scales, does the same for the magnitude of synoptic-scale fluctuations in some dynamic variables, e.g., horizontal wind, hence derives the magnitude of the fluctuations in the remaining dynamic variables, e.g., pressure, and then uses all these to non-dimensionalize the equations of motion, thereby obtaining estimates of the magnitude of each term there in terms of powers of the Rossby number. Expanding all fields in powers of the Rossby number as well, inserting this into the non-dimensional equations, and sorting the result in terms of powers of the Rossby number finally leads to the conservation equation for quasigeostrophic potential vorticity. In principle we will follow the same route here as well. A major difference is, however, that instead of one time scale, one horizontal length scale, one vertical length scale, one scale for the wind fluctuations, etc., we have now two in each case, one for the synoptic-scale flow, and one for the mesoscale motions. These scales will have to be defined before we can proceed. We therefore first review shortly the essentials of synoptic scaling and then move on to the identification of characteristic mesoscales to be used in the further analysis. Synoptic Scaling within Quasigeostrophic Theory As a first step we review the synoptic scaling which quasigeostrophic theory is built on. The theory assumes the synoptic-scale flow to have typical horizontal and vertical length scales L s and Hs . Velocity scales for horizontal and vertical wind are Us and Ws , satisfying

10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow

433

Hs Us Ls

(10.144)

Ls Hs = Us Ws

(10.145)

Ws = The time scale Ts is the advective time scale Ts =

The synoptic time scale is much larger than the inertia time scale so that the Rossby number is small: 1/ f Us = 1 (10.146) Ro = f Ls Ts The density scale height Hρ is of the same order as the vertical scale, i.e., −

Hs dρ Hs = O(1) = ρ dz Hρ

(10.147)

From p = ρ RT the pressure, density, and temperature scale heights are related via 1 1 dp 1 1 1 dρ 1 dT 1 =− + ≈ =− − = Hp p dz ρ dz Hρ HT Hρ T dz

(10.148)

because typically the temperature scale height is larger than the density and pressure scale height. Hence the density scale height agrees approximately with the pressure scale height that is, due to the hydrostaticity of the reference atmosphere, Hp = −

p p RT = = d p/dz gρ g

(10.149)

Therefore for both scale heights we can use the estimate, appropriate as well for the vertical length scale Hs ,   RT00 (10.150) Hρ ≈ H p = O g where T00 is an order-of-magnitude of the reference-atmosphere temperature T = O(T00 ), e.g., T00 = 300K. Hence a reasonable choice for the vertical length scale is Hs =

RT00 g

(10.151)

The potential-temperature scale height, however, is Hθ =

θ dθ /dz

=

g N2

(10.152)

and the classic derivation of quasigeostrophic theory assumes that it is larger than the density and pressure scale heights, i.e.,

434

10 Gravity Waves and Their Impact on the Atmospheric Flow

Hs = O(Ro) Hθ

(10.153)

We will keep this assumption, for simplicity, although it is less appropriate for the more strongly stratified stratosphere.1 Moreover, an important result of baroclinic-instability theory is that the internal Rossby deformation radius N Hs (10.154) L di = f is the horizontal scale of the most rapidly growing waves, i.e., one has Ls = O(1) L di and therefore also

Hs =O Ls



f N

(10.155)

 (10.156)

The latter ratio is under typical tropospheric and mesospheric conditions f = O(Ro2 ) N

(10.157)

The correspondence (10.155) between horizontal length scale and internal Rossby deformation radius also implies that the external Rossby deformation radius √ g Hs Ld = (10.158) f satisfies Ls2 =O Ld 2



L 2di L 2d



 =O

N 2 Hs2 f 2 f 2 g Hs



 =O

Hs g/N 2



 =O

Hs Hθ

 = O(Ro) (10.159)

After the considerations above we are now ready for order-of-magnitude estimates of the fluctuations in the dynamical fields. To begin with, geostrophic equilibrium implies for the Exner-pressure fluctuations π  = π − π f ez × u ≈ −c p θ∇h π  

whence 

π =O

f Us L s

(10.160)



cpθ

(10.161)

and therefore, using the estimate (10.151) of the vertical scale height, the definition (10.146) of the Rossby number, and the definition (10.158) of the external Rossby deformation radius, 1 A generalization incorporating stratospheric conditions is possible.

10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow

π =O π



f Us L s c p πθ



 =O

f Us L s



cpT



R f Us L s =O c p g Hs



 =O

R L 2s Ro c p L 2d

435

 (10.162)

so that the relative magnitude of the synoptic-scale Exner-pressure fluctuations is, with the help of the ratio (10.159),   π = O Ro2 π

(10.163)

Moreover, hydrostatic equilibrium implies 0 ≈ −c p θ

∂π −g ∂z

(10.164)

As usual we assume that the reference atmosphere is in exact hydrostatic equilibrium, i.e., g/c p dπ =− dz θ

(10.165)

Then, inserting the decompositions π = π + π  and θ = θ + θ  into (10.164) and neglecting the term nonlinear in the fluctuations, one obtains g/c p θ  ∂π  = ∂z θ θ

(10.166)

and therefrom, using ∂π  /∂z = O(π  /Hs ) and the estimate (10.150) of the vertical scale height,         cp π cp πθ π cp T π θ π θ (10.167) =O =O =O =O g/c p Hs g Hs π g Hs π R π θ Hence the relative magnitude of the synoptic-scale potential-temperature fluctuations is θ θ

  = O Ro2

(10.168)

The order of magnitude of the synoptic-scale horizontal wind is determined by solving the estimate (10.161), of the Exner-pressure fluctuations, for Us ,     c p πθ π  c p RT π  =O (10.169) Us = O f Ls π R f Ls π Here one has, due to definition (10.151) of the vertical length scale and the definition (10.158) of the external Rossby deformation radius,    RT = O g Hs = O( f L d ) (10.170)

436

so that

10 Gravity Waves and Their Impact on the Atmospheric Flow



cp Ld π  Us = O RT R Ls π

 (10.171)

whence, using the ratios (10.159) and (10.163),  Us = Ro3/2 RT00

(10.172)

turns out to be a good choice as velocity scale for the horizontal-wind fluctuations. The vertical-wind scale can be obtained from this, using (10.144), the aspect ratio (10.156) and the ratio (10.157), resulting in  Ws = Ro7/2 RT00

(10.173)

But because the definition of the Rossby number implies that L s = Us /(Ro f ), (10.172) also fixes the horizontal length scale as  L s = Ro1/2 RT00 / f

(10.174)

which, together with the aspect ratio (10.156) and the ratio (10.157), again determines the vertical length scale as  Hs = Ro5/2 RT00 / f

(10.175)

so that in this scaling the gravitational acceleration is actually not free anymore, but rather satisfies due to (10.151)  g = Ro−5/2 f RT00 (10.176) √ Certainly one could as well get from this RT00 in terms of g and f and express all scales in terms of those constants. This would not change the results below. Note that Hs is also an estimate of the scale height of reference-atmosphere density and pressure, while as expressed in (10.153), the potential-temperature scale height is longer Hρ = O(Hs )

H p = O(Hs )

Hθ = O(Hs /Ro)

(10.177)

To complete the list, we also re-express the synoptic time scale, using the definition (10.146) of the Rossby number, as Ts = Ro−1 / f

(10.178)

Thus we have expressed all scales of interest, as well as the relative strength of the Exnerpressure and potential-temperature fluctuations, in terms of the Coriolis parameter f , the √ rough estimate RT00 of the speed of sound, and the Rossby number Ro.

10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow

437

Scaling of Inertia-Gravity Waves Close to Breaking Atmospheric gravity waves of interest fall in spatial scales below the corresponding synoptic scales. As can be seen in Fig. 10.9, e.g., their spectrum is quite wide, with horizontal scale at least between a few km and a few 100 km. We have to make a choice here which part of the gravity-wave spectrum we want to consider. We assume typical horizontal and vertical √ wavenumbers k, l, and m with corresponding scales L w = 1/kh = 1/ k 2 + l 2 and Hw = 1/m. We want these scales to be shorter than the corresponding synoptic scales, by a scaleseparation parameter ε  1, so that L w = εL s

Hw = ε Hs

(10.179)

Our choice for the scale-separation parameter is ε = Ro

(10.180)

whence, from the definitions (10.174) and (10.175) of the synoptic length scales, the wave length scales are

109

total energy partition

power spectral density (m3 s–2)

108 107 106 105 104 103 102 10–7

observed total energy inertia-gravity wave component residual geostrophic component 10–6 10–5 inverse wavelength (m–1)

10–4

Fig. 10.9 From an analysis of upper-troposphere aircraft data, a decomposition of the horizontal kinetic-energy spectrum (in terms of horizontal wavenumber) into contributions due to gravity waves and the residual geostrophic flow. Figure adapted from Callies et al. (2014).

438

10 Gravity Waves and Their Impact on the Atmospheric Flow

√

RT00 f √ RT 00 L w = ε3/2 f

Hw = ε

7/2

(10.181) (10.182)

With this choice of vertical scale one is in the regime Hw  2Hs . For an estimate of the gravity-wave time scale we therefore assume, as will be verified in the derivations below, the gravity-wave dispersion relation (10.118), now however supplemented by a Doppler term due to advection by the horizontal synoptic-scale flow,2 i.e.,  N 2 kh2 + f 2 m 2 ω =k·u± (10.183) kh2 + m 2 Because the advecting wind u is due to the synoptic-scale flow, the Doppler term scales as       Us Us /L s 1/Ts =O (10.184) =O k·u =O Lw ε ε Moreover, the gravity-wave aspect ratio is Hw /L w = ε2 . Because, as expressed in (10.157), this is also the ratio f /N , the intrinsic part of the anticipated gravity-wave frequency scales as       N 2 kh2 + f 2 m 2 N 2 /L 2w + f 2 /Hw2 N 2 Hw2 = O f 1 + 2 2 = O( f ) =O 1/Hw2 f Lw kh2 + m 2 (10.185) But due to (10.178) one has f = (1/Ts )/ε so that the gravity-wave time scale Tw , with ω = O(1/Tw ), is Tw = εTs = 1/ f

(10.186)

Another decision must be made on how large the gravity-wave amplitudes are to be. As we have already seen above, gravity waves grow in their vertical propagation considerably in amplitude, so that a wide range of amplitudes can be considered. Most interesting, however, is the dynamics of large-amplitude gravity waves. Here a critical bound is given by the threshold of static instability. Consider the potential-temperature field of a monochromatic gravity wave with slowly varying amplitude θw in an atmosphere with a synoptic-scale potential temperature θs θ (x, t) = θ (z) + θs (x, t) + θw (x, t) cos(k · x − ωt)

(10.187)

2 As will be shown below, it is a peculiar property of the gravity waves in the considered scaling

regime that there is no Doppler term due to self-advection by the gravity-wave winds themselves. This holds as long as the gravity-wave field is locally monochromatic.

10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow

439

As can be inferred from the discussion in Sect. 2.4, such a potential-temperature field will be unstable, and lead to the onset of convection, whenever and wherever its vertical spatial derivative becomes negative, i.e., dθ ∂θs ∂θw ∂θ = + + (x, t) cos(k · x − ωt) − mθw sin(k · x − ωt) < 0 ∂z dz ∂z ∂z

(10.188)

Here, due to (10.168) and (10.177) ∂θs /∂z dθ /dz

 =O

θ  /Hs θ /Hθ

 = O(ε)

(10.189)

i.e., the vertical derivative of the synoptic-scale flow can be neglected. Moreover, we assume that the gravity-wave amplitude is slowly varying, so that ∂θw /∂z can be neglected in comparison with mθw . Therefore, static instability can arise when mθw is larger in magnitude than the derivative of the reference-atmosphere potential temperature. Whenever this happens, convection leads to the onset of turbulence, and the latter would have to be considered as well. A statically stable and a statically unstable case are illustrated in Fig. 10.10. We therefore consider here gravity waves which have not quite reached the threshold of

a = 1.5

2

0.25 0.2

z/λ z

1.5

0.15

1

0.1 0.05

0.5 0

0

0

0.5

1

1.5

2

a = 0.5

2

0.25 0.2

z/λ z

1.5

0.15

1

0.1 0.05

0.5 0

−0.05

0

0

0.5

1 x/λ x

1.5

2

−0.05

Fig. 10.10 The potential-temperature field of the reference atmosphere superposed by a lowamplitude gravity wave not (with horizontal and vertical wavelength λx and λz , respectively) leading to static instability (bottom panel, wave amplitude relative to the threshold of static instability a = 0.5) or a large-amplitude gravity wave bound to break due to the instability (top). Important are the regions where high potential temperature is located below low potential temperature.

440

10 Gravity Waves and Their Impact on the Atmospheric Flow

static instability, but that are nonetheless close to it. Hence, with mθw = O(θw /Hw ) and dθ /dz = O(θ/Hθ ) the relative amplitude of the gravity-wave fluctuations considered is   Hw θw =O (10.190) Hθ θ which becomes, using (10.177) and (10.179) θw θ

= O(ε2 )

(10.191)

The gravity-wave-buoyancy scale Bw can be estimated by multiplying (10.190) by g, yielding     g θw =O g (10.192) Hw = O N 2 Hw Hθ θ so that Bw = N 2 Hw

(10.193)

is the appropriate choice. This estimate will now be used in the polarization relations (10.121)–(10.124) to determine the scaling for the winds and the Exner pressure. We will basically rederive these polarization relations below so that there is no reason to worry about using results for an atmosphere at rest. Here they hold in the reference frame moving with the wind so that one uses in them instead of the frequency ω the intrinsic frequency  N 2 kh2 + f 2 m 2 (10.194) ωˆ = ω − k · u = ± kh2 + m 2 First we consider the horizontal wind in x-direction. Inserting the wave-buoyancy scale into (10.121) one obtains a wave amplitude in u,   im k ωˆ + il f uw = − Bw (10.195) N 2 kh2 Since both k ωˆ and l f are O ( f /L w ), this yields, together with (10.193) and (10.182),      1 f /L w 2 3/2 = ε (10.196) N H O ( f L ) = O RT uw = O w w 00 Hw N 2 /L 2w so that the appropriate choice for the gravity-wave horizontal-wind scale is  Uw = ε3/2 RT00

(10.197)

Comparing with (10.172) we see that this agrees with the synoptic horizontal-wind scale Us ! These waves are quite strong in amplitude. Finally, the interested reader will easily verify

10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow

441

that considering instead the horizontal wind in y-direction, i.e., (10.122), leads to the same result. Likewise, the vertical-wind scale Ww can be estimated by inserting the buoyancy scale Bw into the polarization relation (10.123), yielding a vertical-wind amplitude      f 2 i ωˆ 7/2 (10.198) N H RT00 ww = 2 Bw = O w = O ( f Hw ) = O ε 2 N N so that we choose  Ww = ε7/2 RT00

(10.199)

which agrees with the synoptic vertical-wind scale Ws ! Finally, from the definition of the Exner pressure one obtains at a good accuracy, as long as the pressure fluctuations are weak, π R p = π cp p

(10.200)

Therefore the relative amplitude of the gravity-wave Exner-pressure fluctuations can be obtained from the corresponding result for the pressure fluctuations. Estimating the latter from inserting the gravity-wave-buoyancy scale Bw into the polarization relation (10.124) one finds πw R ρ0 m(ωˆ 2 − f 2 ) Bw (10.201) = π cp p N 2 kh2 Since N 2 f 2 and L w Hw , and hence m 2 kh2 , the dispersion relation (10.118) yields ωˆ 2 − f 2 =

(N 2 − f 2 )kh2 kh2 + m 2

≈

N 2 kh2 m2

(10.202)

leading, with ρ00 and p00 local density and pressure scales so that p00 = ρ00 RT00 , to       R ρ00 1/Hw 2 R N2 2 R g/Hθ 2 πw =O (10.203) N Hw = O H H =O π c p p00 1/Hw2 c p RT00 w c p g Hs w or

    πw R Hw Hw R 3 =O ε =O π c p Hs Hθ cp

(10.204)

so that an appropriate estimate for the relative amplitude of the gravity-wave Exner-pressure fluctuations is   πw = O ε3 π The pressure fluctuations in gravity waves are extremely weak.

(10.205)

442

10 Gravity Waves and Their Impact on the Atmospheric Flow

10.3.3 Non-dimensional Equations and WKB Ansatz The estimates above will now be used to first non-dimensionalize the dynamical equations and then formulate a decomposition of the dynamical fields reflecting the simultaneous presence of reference atmosphere, synoptic-scale flow, and gravity waves. In this decomposition the wave part will be expressed by a WKB ansatz, as introduced heuristically in Sect. 8.2.1. With the preparations there we will here take a more rigorous, and therefore safer, scale-asymptotic approach. Non-dimensionalization of the Equations of Motion Other than in the derivation of quasigeostrophic theory we now choose the wave scales to non-dimensionalize the inviscid and adiabatic equations of motion in Sect. 10.3.1. In order to economize on the notation, we simply replace (u, w) → (Uw u, Ww w) (x, y, z, t) → [L w (x, y), Hw z, Tw t] (θ, π, ρ) → (T00 θ, π, ρ00 ρ) f → f f0

(10.206) (10.207) (10.208) (10.209)

while the local pressure and density scales of the reference atmosphere are linked to the temperature scale by p00 ρ00 = (10.210) RT00 The horizontal-momentum equation thus becomes c p T00 Uw2 Du θ ∇h π + f U f 0 ez × u = − L w Dt Lw

(10.211)

However, using the definitions of the gravity-wave scales above, one finds R 3 Uw2 /L w = ε c p T00 /L w cp

f 0 Uw R 3 = ε c p T00 /L w cp

(10.212)

so that one finally obtains the non-dimensional horizontal momentum equation  ε3

 cp Du + f 0 e z × u = − θ ∇h π Dt R

(10.213)

Likewise, the vertical-momentum equation becomes c p T00 ∂π Uw Ww Dw θ =− −g L w Dt Hw ∂z

(10.214)

10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow

443

where from the definitions above Uw Ww /L w R 7 = ε c p T00 /Hw cp

g =ε c p T00 /Hw

(10.215)

yielding the non-dimensional vertical-momentum equation ε7

c p ∂π Dw =− θ −ε Dt R ∂z

(10.216)

The non-dimensionalization of the entropy equation is trivial. One first gets Dθ Uw T00 =0 Lw Dt

(10.217)

Dθ =0 Dt

(10.218)

whence

Similarly one finds from the Exner-pressure equation first Uw Dπ R Uw π ∇ ·v =0 + L w Dt cV L w

(10.219)

Dπ R π∇ · v = 0 + Dt cV

(10.220)

and then

Finally, the equation of state becomes ρ00 ρ =

p00 π cV /R RT00 θ

(10.221)

which leads via (10.210) to ρ=

π

cV R

θ

(10.222)

Multiscale Asymptotics and WKB-Ansatz In the following, we consider the superposition of a reference atmosphere at rest, a synopticscale flow, and a locally monochromatic gravity-wave field. Special measures must be taken to express that they have different length and time scales. To begin with, length and time scale of the synoptic-scale flow are all longer than the corresponding wave scales by a factor 1/ε, i.e., Hs = Hw /ε Ts = Tw /ε (10.223) L s = L w /ε

444

10 Gravity Waves and Their Impact on the Atmospheric Flow

This is expressed by all synoptic-scale fields depending on the compressed coordinates X = εx, T = εt

(10.224)

i.e., the synoptic-scale fields, e.g., the horizontal wind of the synoptic-scale flow, have a spatial and time dependence of the form u(X, T ) so that changes of O(1) are required in the compressed coordinates in order to bring about changes of O(1) in u. By definition this implies that the original non-dimensional coordinates x and t must change by O(ε−1 ), as is required by the scaling (10.223) and the original non-dimensionalization using the wave scales. This is then reflected in a corresponding prefactor ε of all derivatives of any function f (X, T ) which are, by the chain rule, ∂T ∂ f ∂f ∂f = =ε ∂t ∂t ∂ T ∂T ∂f ∂ Xi ∂ f ∂f = =ε ∂ xi ∂ xi ∂ X i ∂ Xi

(10.225) (10.226)

and therefore also ∇ f = ε∇ X f

with

∇ X = ex

∂ ∂ ∂ + ey + ez ∂X ∂Y ∂Z

(10.227)

Moreover, because the non-dimensional dynamical equations contain factors with various powers of ε, all synoptic-scale fields are also expanded in terms of ε, as we are already accustomed to from the development of quasigeostrophic theory. Here not to be forgotten, however, are the results (10.163) and (10.168), telling us that the leading-order terms in the expansion of the synoptic-scale Exner pressure and potential temperature are both O(ε2 ). Next, the wave field is assumed to have the following scaling properties: • Its wavelengths and periods are characterized by the wave scales assumed above. • Wavelengths and periods vary in space and time, in response to the interaction with the synoptic-scale wind. The spatial and time scales of these variations are therefore the synoptic scales. • Likewise also the wave amplitude has a corresponding weak spatial and temporal dependence. This can be expressed by a WKB ansatz, as introduced heuristically for Rossby waves in Sect. 8.2.1. In the case of the horizontal wind in x-direction, e.g., this is achieved by   u(x, t) = U (X, T )eiφ(X,T )/ε with amplitude U and phase φ/ε. Time derivative and spatial gradient of the latter define the local frequency ω and local wavenumber k, respectively,

10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow

  ∂ φ ∂φ ω(X, T ) = − =− ∂t ε ∂T   φ = ∇X φ k(X, T ) =∇ ε

445

(10.228) (10.229)

This way one has frequencies and wavenumbers which are O(1) in the wave scales, but do vary on the synoptic scales. With these definitions of local frequency and wavenumber one therefore has for any function f (X, T ), using the rules (10.225)–(10.227),    ∂  iφ/ε  ∂f = −iω f + ε (10.230) fe eiφ/ε ∂t ∂T     ∇ f eiφ/ε = (ik f + ε∇ X f ) eiφ/ε (10.231) Again we also expand the wave amplitudes in terms of ε, keeping however the results (10.191) and (10.205) in mind, so that the leading-order terms in the expansions of the gravity-wave potential temperature and Exner pressure are O(ε2 ) and O(ε3 ), respectively. Finally turning to the reference atmosphere, we note that the potential-temperature scale height relates to the vertical wave scale via Hθ = O(Hw /ε2 )

(10.232)

so that dθ /dz = O(θ /Hθ ) = O(ε2 θ/Hw ). This weak dependence of reference-atmosphere potential temperature on the vertical direction, scaled by the gravity-wave vertical scale, can be expressed by (0) (1) θ = θ + εθ (Z ) (10.233) with θ

(0)

a constant. This ensures a sufficiently weak stratification, (1)

(1)

(1)

1 dθ ε dθ ε d Z dθ ε2 dθ = = = = O(ε2 ) θ dz θ dz θ dz d Z θ dZ

(10.234)

H p = O(Hs ) = O(Hw /ε)

(10.235)

Likewise, since the reference-atmosphere Exner pressure has a leading-order term depending on Z : π = π (0) (Z ) + επ (1) (Z )

(10.236)

446

10 Gravity Waves and Their Impact on the Atmospheric Flow

Putting all together, our flow decomposition is

v=

∞ 

( j)

ε j V0 (X, T ) +

j=0

#$

"

%

∞ 

( j)

ε j V1 (X, T )eiφ(X,T )/ε

j=0

"

#$

θ=

1  j=0

"

εjθ

( j)

(Z ) +

#$

%

reference

π=

1  j=0

"

#$

( j)

ε j 0 (X, T ) +

j=2

"

#$

%

∞ 

"

( j)

ε j 1 (X, T )eiφ(X,T )/ε

j=2

#$

wave

synoptic−scale part

ε j π ( j) (Z ) + reference

∞ 

%

wave

synoptic−scale part

%

∞ 

( j)

ε j 0 (X, T ) +

j=2

#$

"

%

(10.237)

"

∞ 

%

( j)

ε j 1 (X, T )eiφ(X,T )/ε

j=3

#$

wave

synoptic−scale part

(10.238)

(10.239)

%

Note that the expansions of the synoptic-scale and wave-scale potential temperature begin at O(ε2 ), while the Exner-pressure expansions begin at O(ε2 ) in the synoptic-scale part and at O(ε3 ) in the wave-scale part. It should also be mentioned that a more complete treatment would take higher harmonics into account as well, with double, triple, etc. wavenumbers and frequencies, caused by nonlinear self-interactions of the wave field. As shown in the Appendix I, however, the gravity-wave higher harmonics do not show up to leading order, so that we just neglect them here. Finally, inserting the expansions (10.238) and (10.239) of potential temperature and Exner pressure into the non-dimensional equation of state (10.210) yields  ρ=

π (0) + επ (1) + O(ε2 ) θ

(0)

+ εθ

(1)

cV /R (10.240)

+ O(ε2 )

Expanding this ratio in terms of ε one finally obtains for the non-dimensional density ρ=

1 

ε j ρ ( j) (Z ) + O(ε2 )

(10.241)

j=0

with ρ (0) = ρ

(1)

P¯ (0) θ

=ρ

(0)

(0)



(10.242) (1)

θ cV π (1) − (0) R π (0) θ

 (10.243)

10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow

where

cV /R P¯ (0) = π (0)

447

(10.244)

Higher orders of the expansion will not be needed in the following.

10.3.4 Leading-Order Results: Equilibria, Dispersion and Polarization Relations, Eikonal Equations The next to be done is inserting the expansions (10.237)–(239) into the non-dimensional dynamical equations (10.213), (10.216), (10.218), and (10.220), collecting the leading-order terms and extracting from these first results. Exner-Pressure Equation We first address the Exner-pressure equation (10.220). In the material time derivative Dπ/Dt = ∂π/∂t + v · ∇π the Eulerian time derivative of the Exner-pressure expansion (10.239) is, by the rules (10.225) and (10.230),   ∞ ∞ ( j) ( j)   ∂π ( j) j+1 ∂0 j j+1 ∂1 eiφ/ε = O(ε3 ) (10.245) −ε iω1 + ε ε = + ∂t ∂T ∂T j=2

j=3

while the advective part yields, inserting the expansions (10.237) and (10.239) of wind and Exner pressure, respectively, and using the rules (10.227) and (10.231), ⎛

⎞

∞ 

∞  ( j) ( j) v · ∇π = ⎝ ε j V0 + ε j V1 eiφ/ε ⎠ j=0 j=0

   ∞ ∞  1    dπ k (k) (k) (k) · ez εk ik1 + εk+1 ∇ X 1 eiφ/ε εk+1 εk+1 ∇ X 0 + + dZ 

k=0

(0)



k=2

(0)

k=3

 dπ (0)

= ε W0 + W1 eiφ/ε dZ & '   dπ (1)   dπ (0)   (0) (0) iφ/ε (1) (1) iφ/ε 2 W0 + W1 e + W0 + W1 e +ε dZ dZ + O(ε3 )

(10.246)

Likewise one finds

⎛ ⎞ 1 ∞ ∞  R ⎝ j ( j)  j ( j) R ( j) π∇ · v = ε π + ε 0 + ε j 1 eiφ/ε ⎠ cV cV j=0 j=2 j=3

∞ ∞     (k) (k) (k) k+1 k k+1 iφ/ε ε ik · V1 + ε ∇ X · V1 e ε ∇ X · V0 + k=0

k=0

448

10 Gravity Waves and Their Impact on the Atmospheric Flow

=

 R (0)  (0) π ik · V1 eiφ/ε cV !     R (0) (1) (0) +ε π (0) ∇ X · V0 + ik · V1 + ∇ X · V1 eiφ/ε cV (  (0) + π (1) ik · V1 eiφ/ε + ε2

!     R (1) (2) (1) π (0) ∇ X · V0 + ik · V1 + ∇ X · V1 eiφ/ε cV     (0) (1) (0) + π (1) ∇ X · V0 + ik · V1 + ∇ X · V1 eiφ/ε (  (2) (0) + O(ε3 ) (10.247) + 0 ik · V1 eiφ/ε

Gathering now everything in the Exner-pressure equation (10.220), and collecting terms of like power in ε, we find that the leading order O(1) is simply  R (0)  (0) π ik · V1 eiφ/ε = 0 cV

(10.248)

yielding (0)

k · V1 = 0

(10.249)

The leading-order velocity amplitude of the gravity-wave part is orthogonal to the local wave vector. This fact will be used frequently in the following. Entropy Equation Next we turn to the entropy equation (10.218). Herein, the Eulerian time derivative of the potential-temperature expansion (10.238) is   ∞ ∞ ( j) ( j)   ∂θ ( j) j+1 ∂0 j j+1 ∂1 eiφ/ε −ε iω1 + ε ε = + ∂t ∂T ∂T j=2 j=2   & (2)

' (2)   ∂1 (2) iφ/ε (3) 2 3 ∂0 iφ/ε +ε = ε −iω1 e + −iω1 + e ∂T ∂T + O(ε4 )

(10.250)

while the advective part is, after inserting the expansions (10.237) and (10.238) of wind and potential temperature, respectively,

10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow

⎛ v · ∇θ = ⎝

∞ 

( j)

ε j V0 +

j=0



∞ 

449

⎞ ( j)

ε j V1 eiφ/ε ⎠

j=0

∞ ∞     dθ (k) (k) (k) εk ik1 + εk+1 ∇ X 1 eiφ/ε εk+1 ∇ X 0 + ez + · ε2 dZ k=2 k=2 

 ! (1) (1) dθ dθ (0) (0) (0) (2) = ε 2 W0 + W1 + ik · V0 1 eiφ/ε dZ dZ   (  (0) (2) + V1 eiφ/ε · ik1 eiφ/ε (1)



!

 dθ (1)  (1) (1) W0 + W1 eiφ/ε dZ        (0) (0) (2) (3) (2) + V0 + V1 eiφ/ε · ∇ X 0 + ik1 + ∇ X 1 eiφ/ε   (   (1) (1) (2) + O(ε4 ) (10.251) + V0 + V1 eiφ/ε · ik1 eiφ/ε

+ ε3

Inspecting these expansions one finds that the leading O(ε2 ) in the entropy equation is 

 (1) (1) (0) dθ (2) (0) (2) (0) dθ iφ/ε e W0 + −iω1 + ik · V0 1 + W1 dZ dZ     (0) (2) (10.252) + V1 eiφ/ε · ik1 eiφ/ε = 0 Herein the last term on the left-hand side is due to self-advection of the wave part, i.e., advection of the wave potential temperature by the wave velocity. Here and at all other places below such leading-order self-advection terms vanish due to the orthogonality relation (10.249). Hence one gets 

 (1) (1) (0) dθ (2) (0) dθ eiφ/ε = 0 W0 (10.253) + −i ω ˆ 1 + W1 dZ dZ where

(0)

ωˆ = ω − k · V0

(10.254)

is the non-dimensional intrinsic frequency that an observer would measure in a reference frame moving with the leading-order synoptic-scale velocity. Since for any A one has (Aeiφ/ε ) = A/2 eiφ/ε + A∗ /2 e−iφ/ε , equation (10.253) is of the form ∞ 

an (X, T )ei(n/ε)φ = 0

(10.255)

n=−∞

i.e., a Fourier series in φ with period ε 2π , where in the limit ε → 0 the coefficients an are constants. Hence they must all vanish, i.e., an = 0 for all n. Applying this rule to the

450

10 Gravity Waves and Their Impact on the Atmospheric Flow

mean-flow part in (10.253), with n = 0, one sees that the first term in this equation must vanish, whence (0)

W0

=0

(10.256)

This reproduces the well-known result that the leading-order synoptic-scale wind has no vertical component. Finally, the remaining wave contributions, n = ±1, in (10.253) must then be such that the bracket in the second term of the equation vanishes as well. Dividing (0) that by θ finally leads to the linear buoyancy equation (2)

(0)

−i ωB ˆ 1 + W1 N02 = 0 where

(10.257)

( j)

( j)

B1 =

1 θ

(10.258)

(0)

is the O(ε j ) non-dimensional wave-buoyancy amplitude, and (1)

N02 =

1 dθ (0) d Z θ

(10.259)

the non-dimensional squared reference-atmosphere Brunt–Väisälä frequency. Vertical Momentum Equation Next we analyze the vertical momentum equation (10.216). Here the Eulerian time derivative of the vertical wind in the expansion (10.237) is, taking into account that according to (10.256) there is no O(1) synoptic-scale vertical wind,   ∞ ∞ ( j) ( j)   ∂ W ∂ W ∂w ( j) 0 1 ε7 ε j+8 = + eiφ/ε −ε j+7 iωW1 + ε j+8 ∂t ∂T ∂T j=1 j=0 

 (0)   ∂ W1 (0) iφ/ε (1) 7 8 iφ/ε e + O(ε9 ) + ε −iωW1 + = ε −iωW1 e ∂T (10.260) The advective term is, again inserting the wind expansion (10.237), ⎛ ⎞ ∞ ∞   ( j) ( j) ε j+7 V0 + ε j+7 V1 eiφ/ε ⎠ ε7 v · ∇w = ⎝ j=0

·

∞ 

j=0

ε

k+1

k=1

=ε

7



(0) V0

(k) ∇ X W0

∞    (k) (k) εk ikW1 + εk+1 ∇ X W1 eiφ/ε + k=0

    (0) (0) + V1 eiφ/ε · ikW1 eiφ/ε



10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow

+ ε8

!

451

     (0) (0) (1) (0) V0 + V1 eiφ/ε · ikW1 + ∇ X W1 eiφ/ε

   (  (1) (1) (0) + V0 + V1 eiφ/ε · ikW1 eiφ/ε + O(ε9 )

(10.261)

Finally, inserting the potential-temperature and Exner-pressure expansions (10.238) and (10.239), respectively, into the pressure-gradient term yields ⎛ ⎞ 1 ∞ ∞  c p  j ( j)  j ( j) c p ∂π ( j) ⎝ ε θ + ε 0 + ε j 1 eiφ/ε ⎠ =− − θ R ∂z R j=0

·

 1

∞ 

j=2

∞ 

dπ ( j) ε + dZ k=2 

ε j+1

j=0

+

j=2

ε

k

(k) im1

k=3

+ε 

( j) k+1 ∂0

∂z (k) k+1 ∂1

∂Z



 eiφ/ε

 (0) (1) c p (0) dπ (0) (1) dπ (0) dπ 2 cp θ −ε +θ =−ε θ R dZ R dZ dZ (2)

!   (1) (0) c p (0) ∂0 (1) dπ (3) (2) dπ − ε3 θ + im1 eiφ/ε + θ + 0 R ∂Z dZ dZ  ( (0) (2) dπ + 1 eiφ/ε dZ  (3) 

 ! (3) ∂1 (0) ∂0 (4) 4 cp iφ/ε e −ε θ + im1 + R ∂Z ∂Z (2)

  (1) ∂0 (3) iφ/ε +θ + im1 e ∂Z  dπ (1)   (2) (2) + 0 + 1 eiφ/ε dZ   dπ (0) (  (3) iφ/ε + (3) (10.262) + O(ε5 ) 0 + 1 e dZ Gathering (10.260)–(10.262) in the vertical-momentum equation (10.216) we find that the leading order is O(ε), yielding 0=− or

c p (0) dπ (0) θ −1 R dZ

(10.263)

452

10 Gravity Waves and Their Impact on the Atmospheric Flow

R/c p dπ (0) = − (0) dZ θ

(10.264)

At O(ε2 ) we find

c p (1) dπ (0) c p (0) dπ (1) θ − θ R dZ R dZ leading, with the help of (10.264), to 0=−

(10.265)

(1)

R/c p θ dπ (1) = (0) (0) dZ θ θ

(10.266)

This is nothing else than the hydrostatic equilibrium of the reference atmosphere. The next order O(ε3 ) is the first with nonzero wave contributions. Here the most straightforward procedure would be to accept that the material derivative does not contribute to this. Keeping in mind that it is never allowed to neglect large terms, but that it is possible to retain small terms,3 we here keep the lowest-order contributions from the material derivative. This is done in order to retain, as is to be seen below, the complete dispersion relation for gravity waves with vertical scale at the order of or smaller than the atmospheric scale height. We absorb the O(ε7 ) terms from the Eulerian time derivative (10.260) and the advection term (10.261) into the O(ε3 ) of the vertical-momentum equation. Taking also the orthogonality condition (10.249) into account, this yields    (0) (0) (0) ε4 −iωW1 + ik · V0 W1 eiφ/ε ! (2) (1) (0) c p (0) ∂0 (1) dπ (2) dπ θ =− +θ + 0 R ∂Z dZ dZ  

( (0) (0) (3) (2) dπ iφ/ε e + imθ 1 + 1 (10.267) dZ With the definitions (10.254) of the intrinsic frequency and (10.258) of the wave-buoyancy the wave part leads, taking the hydrostatic equilibrium (10.264) of the leading-order reference-atmosphere Exner pressure into account, to the linear vertical-momentum equation (0)

(2)

−ε4 i ωW ˆ 1 − B1 + im

c p (0) (3) θ 1 = 0 R

(10.268)

3 This is possible as long as it is taken into account in the analysis of the corresponding higher orders

in ε.

10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow

453

while the mean-flow part yields, employing the hydrostatic equilibria (10.264) and (10.266), ⎡

(2)

∂0 ∂Z

=

R/c p θ

(0)



⎣ B (2) − 0

θ

(1)

θ

(0)

2 ⎤ ⎦

(10.269)

which expresses the hydrostatic equilibrium of the synoptic-scale flow. Here ( j)

0

( j)

B0 =

(10.270)

(0)

θ is the non-dimensional synoptic-scale buoyancy at O(ε j ).

Horizontal Momentum Equation Finally we address the horizontal-momentum equation (10.213), where the Eulerian time derivative of the horizontal wind in the expansion (10.237) is   ∞ ∞ ( j) (0)   ( j) 3 ∂u j+4 ∂U0 j+3 j+4 ∂U1 ε ε = + eiφ/ε −ε iωU1 + ε ∂t ∂T ∂T j=0 j=0   & (0)

' (0)   ∂U1 (0) iφ/ε (1) 3 4 ∂U0 iφ/ε +ε = ε −iωU1 e + −iωU1 + e ∂T ∂T + O(ε5 )

(10.271)

The advective term is, again inserting the wind expansion (10.237), ⎛ ε3 (v · ∇)u = ⎝

∞ 

( j) ε j+3 V0 +

j=0

⎡

·⎣ 

∞ 

⎞ ( j) ε j+3 V1 eiφ/ε ⎠

j=0

∞ 

k=0





⎤

(k) (k) εk ikU1 + εk+1 ∇ X U1 eiφ/ε ⎦

k=0

(0) (0) = ε3 V0 · ikU1 eiφ/ε

+ ε4

∞  

(k) εk+1 ∇ X U0 +



    (0) (0) + V1 eiφ/ε · ikU1 eiφ/ε

!       (0) (0) (0) (1) (0) V0 + V1 eiφ/ε · ∇ X U0 + ikU1 + ∇ X U1 eiφ/ε    (  (1) (1) (1) + V0 + V1 eiφ/ε · ikU1 eiφ/ε + O(ε5 )

(10.272)

and inserting the horizontal wind from that same expansion into the Coriolis term yields ⎞ ⎛ ∞ ∞   ( j) ( j) ε j+3 U0 + ε j+3 U1 eiφ/ε ⎠ ε 3 f 0 ez × u = f 0 ez × ⎝ j=0

= ε f 0 ez × 3

+ O(ε5 )



(0) U0

j=0

     (0) (1) (1) + U1 eiφ/ε + ε4 f 0 ez × U0 + U1 eiφ/ε (10.273)

454

10 Gravity Waves and Their Impact on the Atmospheric Flow

while inserting the potential-temperature and Exner-pressure expansions (10.238) and (10.239), respectively, into the pressure-gradient term gives ⎛ ⎞ 1 ∞ ∞    cp cp ( j) ( j) ( j) ⎝ εjθ + ε j 0 + ε j 1 eiφ/ε ⎠ − θ ∇h π = − R R j=0 j=2 j=2

∞ ∞    (k) (k) (k) εk ikh 1 + εk+1 ∇ X ,h 1 eiφ/ε εk+1 ∇ X ,h 0 + k=2  3 c p (0)

k=4

  (3) θ + ikh 1 eiφ/ε R !     (0) (3) (4) (3) 4 cp ∇ X ,h 0 + ikh 1 + ∇ X ,h 1 eiφ/ε −ε θ R   (  (1) (2) (3) iφ/ε +θ ∇ X ,h 0 + ikh 1 e + O(ε5 ) (10.274) (2) ∇ X ,h 0

=−ε

Here ∇ X ,h = ex ∂/∂ X + e y ∂/∂Y denotes the horizontal part of the non-dimensional compressed-coordinate gradient, and kh the horizontal part of the wavenumber vector. Now gathering all in the horizontal-momentum equation (10.213), ones sees that the leading O(ε3 ) is      (0) (0) (0) (0) (0) −iωU1 + ik · V0 U1 eiφ/ε + (V1 eiφ/ε ) · ikU1 eiφ/ε      c p (0)  (0) (0) (2) (3) ∇ X ,h 0 + ikh 1 eiφ/ε + f 0 ez × U0 + U1 eiφ/ε = − θ R (10.275) Once more the nonlinear term vanishes due to the orthogonality relation (10.249), leaving a wave contribution (0)

(0)

−i ωU ˆ 1 + f 0 ez × U1 +

c p (0) (3) θ ikh 1 = 0 R

(10.276)

and a mean-flow part (0)

f 0 ez × U0 = −

c p (0) (2) θ ∇ X ,h 0 R

(10.277)

The latter expresses the geostrophic equilibrium of the synoptic-scale flow. Dispersion Relations and Polarization Relations So far we have rederived the hydrostatic equilibrium of the reference atmosphere, and the geostrophic and hydrostatic equilibria of the synoptic-scale flow. Beyond this we have obtained linear momentum, buoyancy, and Exner-pressure equations that are obviously related to the linear equations (10.62)–(10.66) from which one derives the dispersion relations for the various fundamental modes in an atmosphere at rest. A difference is that (10.66) allows

10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow

455

for an oscillation of the pressure, while the corresponding orthogonality relation (10.249) is without any frequency contribution. This is basically because pressure fluctuations in gravity waves are weak. We will now use our linear equations to derive the dispersion and polarization relations of our problem, and we will see that no sound waves occur, while the derived gravity-wave properties (and also those of the geostrophic mode) are very closely related to those in an atmosphere at rest. Hence, as quasigeostrophic theory is a theory for a synoptic-scale flow with gravity waves and sound waves filtered out, we here have a theory for synoptic-scale flow and gravity waves with sound waves filtered out. Dispersion Relation The procedure for the derivation of the wave properties of interest is very similar to that for the fundamental modes in an atmosphere at rest. The leading-order wave equations (10.249), (10.257)/N0 , (10.268), and (10.276) can be summarized as ⎞ ⎛ ⎞⎛ U1(0) −i ωˆ − f 0 0 0 ik ⎟ ⎜ (0) ⎜ f −i ωˆ ⎟ V1 0 0 il ⎟ ⎜ 0 ⎟⎜ ⎟ ⎜ ⎜ (0) 4 −N im ⎟ ⎜ ⎟=0 (10.278) 0 0 −i ωε ˆ W ⎜ ⎟⎜ 0 1 ⎟ ⎜ ⎟ ⎜ (2) ⎟ ⎝ 0 0 N0 −i ωˆ 0 ⎠ ⎝ B1 /N0 ⎠ c p (0) (3) ik il im 0 0 θ 1 " #$ % R M1 = M(k, ω) Non-trivial wave amplitudes require det M1 = 0, yielding -   . 0 = ωˆ ωˆ 2 ε4 kh2 + m 2 − N02 kh2 − f 02 m 2

(10.279)

ωˆ = 0

(10.280)

Thus either

which is the geostrophic-mode solution, or ωˆ 2 =

N02 kh2 + f 02 m 2 ε4 kh2 + m 2

(10.281)

which is the dispersion relation for gravity waves. In order to convince ourselves of this we redimensionalize the dispersion relation by the substitutions   (k, l, m, ω) ˆ → L w (k, l) , Hw m, Tw ωˆ     (0) (1) → θ /T00 , dθ/(εT00 ) θ , dθ

(10.282) (10.283)

Z → εz/Hw

(10.284)

f0 → f / f = 1

(10.285)

456

10 Gravity Waves and Their Impact on the Atmospheric Flow

Thereby we first obtain

(1)

N02 =

1 dθ 1 Hw dθ = (0) d Z θ ε2 dz θ

(10.286)

whence, using the definitions (10.181) and (10.182) of the wave scales, f 2 Hw 1 dθ 2 1 Hw dθ 2 2 L w kh + Hw2 m 2 k + f 2 m2 2 ε6 θ dz h 2 2 θ ε dz = ωˆ = f ε4 L 2w kh2 + Hw2 m 2 kh2 + m 2 √ ε−5/2 f RT00 dθ 2 k + f 2 m2 dz h θ = kh2 + m 2

(10.287)

Via the relation (10.176) for the gravitational acceleration this leads indeed to the dimensional dispersion relation ωˆ 2 =

N 2 kh2 + f 2 m 2 kh2 + m 2

(10.288)

with g dθ (10.289) θ dz Note that, as compared to the dispersion relations in an atmosphere at rest, the frequency is now replaced by the intrinsic frequency, which is in dimensional units N2 =

ωˆ = ω − k · u Here

(0)

u = Uw U0

(10.290)

(10.291)

is the leading-order synoptic-scale horizontal wind in dimensional units that one obtains by averaging the total horizontal wind over the phase, achieved by either averaging at fixed location over a local period or by averaging at fixed time over a local horizontal wavelength. Hence in a local reference frame moving with the synoptic-scale velocity, the dispersion relations for an atmosphere of rest are recovered. Gravity-Wave Polarization Relations The geostrophic mode is an interesting solution in itself, but we here focus exclusively on gravity waves. Their structure is given by the null vector of M1 , using ωˆ from the dispersion relation (10.281). First, the two components of the linear horizontal-momentum equation (10.276), i.e.,

10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow (0)

(0)

(0)

(0)

−i ωU ˆ 1 − f 0 V1 ˆ 1 f 0 U1 − i ωV

c p (0) (3) θ 1 R c p (0) (3) = −il θ 1 R = −ik

457

(10.292) (10.293)

yield the relations (0)

U1

(0)

V1

k ωˆ + il f 0 c p (0) (3) θ 1 ωˆ 2 − f 02 R l ωˆ − ik f 0 c p (0) (3) = 2 θ 1 ωˆ − f 02 R =

(10.294) (10.295)

between the gravity-wave horizontal-wind amplitudes and the gravity-wave pressure amplitude, in vector form kh ωˆ − i f 0 ez × kh c p (0) (3) (0) θ 1 (10.296) U1 = R ωˆ 2 − f 02 From the linear buoyancy equation (10.257) one obtains (0)

W1

=

i ωˆ (2) B N02 1

(10.297)

Using this in the linear vertical-momentum equation (10.268) leads to c p (0) (3) i ε4 ωˆ 2 − N02 (2) θ 1 = B1 R m N02

(10.298)

so that (10.296) finally becomes (0)

U1 =

 (2) i ε4 ωˆ 2 − N02  kh ωˆ − i f 0 ez × kh B1 2 2 2 m N0 ωˆ − f 0

(10.299)

The redimensionalization of these relations is achieved by the replacements (10.282)– (10.285) and     (0) (0) (2) (3) −2 −3 ˆ w , w/W ˆ ˆ (10.300) U1 , W1 , B1 , 1 → u/U w , ε θˆ /θ , ε π whence one obtains, with bˆ = g θˆ /θ  i ωˆ 2 − N 2  kh ωˆ − i f ez × kh bˆ m N ωˆ 2 − f 2 i ωˆ wˆ = 2 bˆ N i ωˆ 2 − N 2 ˆ c p θ πˆ = b m N2 uˆ =

(10.301) (10.302) (10.303)

458

10 Gravity Waves and Their Impact on the Atmospheric Flow

With the help of the dispersion relation (10.288) one can convince oneself that this is equivalent to the polarization relations (10.121)–(10.124) for gravity waves in an atmosphere at rest, provided one replaces the frequency there by the intrinsic frequency. The Eikonal Equations From (10.256) and (10.281) follows  (0)

ω(X, T ) = (X, T , X) = k · U0 (X, T ) ±

N02 (Z )kh2 + f 02 m 2 ε4 kh2 + m 2

(10.304)

Both ω and k depend on (X, T ), while in  the explicit dependence on X and T arises (0) exclusively from the corresponding dependencies of U0 and N02 . Prognostic equations for frequency and wavenumber can be obtained from the constraining dispersion relation (10.304). One first obtains ∂k ∂ ∂ω = + cg · (10.305) ∂T ∂T ∂T where the group velocity cg = ∇k 

(10.306)

is the gradient of the local frequency in wave-number space. But we also have, due to the definitions (10.228) and (10.229) of local wavenumber and frequency, ∂k ∂φ ∂ = ∇X φ = ∇X = −∇ X ω ∂T ∂T ∂T

(10.307)

so that we can obtain a prognostic equation for local frequency, 

 ∂ ∂ + cg · ∇ X ω = ∂T ∂T

(10.308)

On the other hand, (10.307) implies component-wise, using also the definitions (10.228) and (10.229) of the local wavenumber and frequency, respectively, and the dispersion relation (10.304), ∂ki ∂ ∂ω ∂k j ∂ ∂ω ∂ki ∂ω =− − =− − =− ∂T ∂ Xi ∂ Xi ∂k j ∂ X i ∂ Xi ∂k j ∂ X j

(10.309)

whence 

 ∂ + cg · ∇ X k = −∇ X  ∂T

(10.310)

The eikonal equations (10.308) and (10.310) are prognostic equations for the wave-number and frequency fields. They are, however, often written in a Lagrangian manner. For this we

10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow

459

define rays as the characteristics of these equations, to be understood as paths along which wave packets would propagate, satisfying dX = cg dT

(10.311)

Note that the rays are time-dependent since the group velocity is not constant! Along these rays frequency and wavenumber satisfy the ray equations dω ∂ = dT ∂T dk = −∇ X  dT

(10.312) (10.313)

In the special case shown here the eikonal equations are 

 (0) ∂U0 ∂ + cg · ∇ X ω = k · (10.314) ∂T ∂T     kh2 ez d N02 ∂ (0)  + cg · ∇ X k = − ∇ X U0 · k ∓    ∂T 2 dZ ε4 kh2 + m 2 N02 kh2 + f 02 m 2 (10.315)

Redimensionalizing the results above is straightforward, following the same procedures as above and replacing (Xh , Z , T ) → ε(xh /L w , z/Hw , t/Tw )

(10.316)

This leads to

ω(x, t) = (x, t, k) = k · u(x, t) ±

/

 N 2 (z)kh2 + f 2 m 2 /|k|2

cg = ∇k   kh2 /|k| ez d N 2 ∂  + cg · ∇ k = − ∇ = − (∇u) · k ∓ ∂t 2 dz N 2 kh2 + f 2 m 2   ∂u ∂ ∂ + cg · ∇ ω = =k· ∂t ∂t ∂t 

(10.317) (10.318) (10.319) (10.320)

460

10 Gravity Waves and Their Impact on the Atmospheric Flow

Rays are defined by dx = cg dt

(10.321)

and along these one has dω ∂ = dt ∂t dk = −∇ dt

(10.322) (10.323)

10.3.5 The Next Order of the Equations The leading-order parts of the dynamical equations have established well-known equilibria for the reference atmosphere and the synoptic-scale flow, and the dispersion and polarization relations for gravity waves with vertical scale of the order of or shorter than the atmospheric scale height. Since the vertical scale of the waves was assumed to be smaller, by one order in ε, than that of the synoptic-scale flow, and because the vertical scale of the latter agrees with the atmospheric scale height, it is not surprising that the gravity waves are found to locally follow the identified dynamics. Only waves with vertical scale larger than the atmospheric scale height, which however is identical to the vertical scale of the synoptic-scale flow, can deviate from this in a significant manner. Moreover, due to the chosen scales sound waves are filtered from the theory. We have also derived prognostic equations for the local gravity-wave frequency and wavenumber. They respond to gradients and time derivatives of the reference atmosphere and the synoptic-scale flow. However, what has not been touched so far is whether and how the gravity-wave amplitude responds to the synoptic-scale flow, and whether and how gravity waves can influence the latter. This can be settled by looking at the respective next order of the basic equations. In this section the next-order terms will be identified. They will be used further below for the derivation of a gravity-wave amplitude equation and for analyzing the gravity-wave impact on the synoptic-scale flow. Exner-Pressure Equation Just as the leading-order O(1) terms in the Exner-pressure equation (10.220) had been identified, we obtain by inspection of the expansions (10.245)–(10.247) the next O(ε)

10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow

 dπ (0)  (0) (0) W0 + W1 eiφ/ε dZ !     R (0) (1) (0) π (0) ∇ X · V0 + ik · V1 + ∇ X · V1 eiφ/ε + cV (  (0) + π (1) ik · V1 eiφ/ε =0

461



(10.324)

Recalling the definition (10.244) of P¯ (0) , and the orthogonality relation (10.249), the wave contributions yield   R (0) R π (0) (1) (0) (0) ¯ =0 π ik · V1 + ∇ · P V X 1 cV cV P¯ (0) whence (1)

ik · V1 = −

  1 ¯ (0) V(0) ∇ · P X 1 P¯ (0)

(10.325)

(10.326)

Since, however, P¯ (0) differs, according to the leading-order reference-atmosphere equation of state (10.242), from the leading-order reference-atmosphere density ρ (0) only by a (0) constant factor 1/θ , this can also be written (1)

  (0) (0) ∇ · ρ V X 1 (0)

1

(1)

ik · V1 = Rπ,1 ≡ −

ρ

(10.327)

Because, according to (10.256), W0(0) = 0, the mean-flow part is   R π (0) ¯ (0) U(0) = 0 ∇ · P X ,h 0 cV P¯ (0)

(10.328)

or simply (0)

(0)

∇ X · V0 = ∇ X ,h · U0 = 0

(10.329)

This is not really a new result since the horizontal synoptic-scale flow is by (10.277) in geostrophic equilibrium, and therefore non-divergent. Finally, from the next O(ε2 ) terms of the Exner-pressure equation (10.220) we will not need the wave part in the following, but the mean-flow part is of interest. Inspecting the (0) expansions (10.245)–(10.247), again using W0 = 0, but also the non-divergence (10.329) just derived, this mean-flow part turns out to be (1) dπ

W0

(0)

dZ

+

 R  (0) (1) π ∇ X · V0 = 0 cV

(10.330)

462

10 Gravity Waves and Their Impact on the Atmospheric Flow

which leads with the definition (10.244) of P¯ (0) to   (1) ∇ X · P¯ (0) V0 = 0

(10.331)

Again we use the leading-order reference-atmosphere equation of state (10.242) to simplify this to   (1) (10.332) ∇ X · ρ (0) V0 = 0 As in quasigeostrophic theory, the synoptic-scale flow turns out to have a non-divergent mass flux. Entropy equation Likewise, inspecting the expansions (10.250) and (10.251), the O(ε3 ) contributions to the entropy equation (10.218) are   (2) (2)    dθ (1)   ∂1 iφ/ε ∂0 (3) iφ/ε (1) (1) iφ/ε + W0 + W1 e + + −iω1 e e ∂T ∂T dZ        (0) (0) (2) (3) (2) + V0 + V1 eiφ/ε · ∇ X 0 + ik1 + ∇ X 1 eiφ/ε      (1) (1) (2) (10.333) + V0 + V1 eiφ/ε · ik1 eiφ/ε = 0 Herein, due to the orthogonality relation (10.249),     (0) (3) V1 eiφ/ε · ik1 eiφ/ε = 0

(10.334)

but two other nonzero products nonlinear in the wave amplitudes appear, that both arise from potential-temperature advection,     iφ/ε V1(0) eiφ/ε · ∇ X (2) e 1  1 ∗ 1 (0) ∗ 1  (0) (2) (2) ∗ (0) (2) = V1 · ∇ X 1 e−2iφ/ε + V1 · ∇ X 1 + V1 · ∇ X 1 e2iφ/ε 4 2 4 (10.335)     (1) (2) V1 eiφ/ε · ik1 eiφ/ε  1 1 (1) ∗ 1  (1) (2) ∗ (2) ∗ (1) (2) = − V1 · ik1 e−2iφ/ε − V1 · ik1 + V1 · ik1 e2iφ/ε (10.336) 4 2 4 Within each of these the middle term is a mean-flow term, while the two others contribute to the forcing of a weak second harmonic of the gravity-wave field considered here. As already mentioned before, those second harmonics can be shown to not contribute to the leading-order of the gravity-wave mean-flow interaction, so that we just ignore them here.4 (0) With these considerations the wave part of (10.333) turns out to be, after division by θ 4 The proper generalization including higher harmonics is given in the Appendix I.

10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow

463

and using the definitions (10.258) and (10.259) for the wave-buoyancy amplitude and the non-dimensional Brunt–Väisälä frequency, (3)

(1)

(1)

− i ωB ˆ 1 + N02 W1 = Rb,1   ∂ (0) (1) (2) (0) (2) B1 − V1 · ∇ X B0 ≡− + U0 · ∇ X + ik · V0 ∂T

(10.337)

while the mean-flow part is, with the help of (10.327), 

 (1) ∂ (0) (2) (1) dθ + V0 · ∇ X 0 + W0 ∂T dZ  1   ∗ 1  (0) (2) (1) (2) ∗ + ik · V1 1 = − V1 · ∇ X 1 2 2 (  1 ! 1    1  (0) (2) ∗ (2) ∗ (0) (0) = − V1 · ∇ X 1 − ∇  · ρ V X 1 1 2 2 ρ (0)

(10.338)

(0)

and thus, after division by θ , and remembering the definition (10.270) of the synoptic-scale (0) buoyancy and that W0 = 0 according to (10.256),

  ∂ ρ (0)  (0) (2) ∗  1 (0) (2) 2 (1) (10.339) + U0 · ∇ X B0 + N0 W0 = − (0) ∇ X · V1 B1 ∂T 2 ρ Now we use in a last step the polarization relation (10.297) linking the gravity-wave verticalwind and buoyancy amplitudes, showing that they have a phase difference π/2 so that  ∗ W1(0) B1(2) = 0 (10.340) This finally leads to 

∂ (0) + U0 · ∇ X ∂T



(2)

(1)

B0 + N02 W0

1  (0) (2) ∗  = −∇ X ,h · U1 B1 2

(10.341)

This equation shows how the synoptic-scale buoyancy is forced by the gravity-wavebuoyancy fluxes. To understand this we note that the leading-order gravity-wave fluctuations in horizontal wind and buoyancy (or potential temperature) are in dimensional units   u =Uw U1(0) eiφ/ε (10.342)    θ (2) (10.343) b = g =ε2 g B1 eiφ/ε θ

464

10 Gravity Waves and Their Impact on the Atmospheric Flow

whence u b = ε2 gUw



1 (0) ∗ (2) ∗ −2iφ/ε 1  (0) (2) ∗  1 (0) (2) 2iφ/ε + U1 B1 e B1 e + U1 B1 U 4 1 2 4

 (10.344)

Averaging this over the phase yields the gravity-wave horizontal buoyancy flux u b  =

1  (0) (2) ∗  u θ   = ε2 gUw U1 B1 2 θ g

Using this together with the replacements (10.283), (10.316), (10.286), and     U0(0) , W0(1) , B0(2) → u/Uw , ε−1 w/Ww , ε−2 b/g

(10.345)

(10.346)

one finds that the prognostic equation (10.341) for the synoptic-scale buoyancy takes the dimensional form 

 ∂ + u · ∇ b + N 2 w = −∇ · u b  ∂t

(10.347)

or for the synoptic-scale potential-temperature fluctuations δθ  = θ b/g 

 ∂ dθ + u · ∇ δθ  + w = −∇ · u θ   ∂t dz

(10.348)

Vertical-Momentum Equation For the terms of O(ε4 ) in the vertical-momentum equation (10.216) we inspect the expansions (10.260)–(10.262). Again we also incorporate the material derivative, now by the O(ε8 ) terms in the Eulerian time derivative (10.260) and the advection part (10.261). Gathering all together we obtain ! ε4



(0) (0)  ∂ W0 ∂ W1 (1) + −iωW1 + eiφ/ε ∂T ∂T       (0) (0) (1) (0) iφ/ε e + V0 + V1 eiφ/ε · ikW1 + ∇ X W1   (   (1) (1) (0) + V0 + V1 eiφ/ε · ikW1 eiφ/ε

 



! (3) (3)  (2)   ∂1 c p (0) ∂0 (1) ∂0 (4) (3) iφ/ε iφ/ε + im1 + + im1 e θ =− +θ e R ∂Z ∂Z ∂Z  dπ (1)   dπ (0) (    (2) (2) (3) (3) + 0 + 1 eiφ/ε (10.349) + 0 + 1 eiφ/ε dZ dZ ((0))

Again remembering the vanishing of W0 according to (10.256) and the orthogonality relation (10.249), and taking the hydrostatic equilibrium (10.264) and (10.266) into account, one sees that the wave part therein is

10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow

c p (0) (4) (1) (3) (1) − i ωε ˆ 4 W1 − B1 + im θ 1 = Rw,1 R   ∂ (0) (1) (0) ≡ −ε4 + U0 · ∇ X + ik · V0 W1 ∂T   (3) (1) cp (0) ∂1 (1) (3) (2) ∂π − θ + θ im1 + 1 R ∂Z ∂Z

465

(10.350)

The corresponding mean-flow part will not be needed below. Horizontal Momentum Equation Finally we collect all terms of O(ε4 ) in the horizontal momentum equation (10.213), using the expansions (10.271)–(10.274). This yields 

 (0) (0) ∂U1 ∂U0 (1) iφ/ε e + −iωU1 + ∂T ∂T        (0) (0) (0) (1) (0) + V0 + V1 eiφ/ε · ∇ X U0 + ikU1 + ∇ X U1 eiφ/ε         (1) (1) (1) (1) (1) + V0 + V1 eiφ/ε · ikU1 eiφ/ε + f 0 ez × V0 + V1 eiφ/ε !    c p (0)  (3) (4) (3) ∇ X ,h 0 + ikh 1 + ∇ X ,h 1 eiφ/ε θ =− R   (  (1) (2) (3) iφ/ε +θ ∇ X ,h 0 + ikh 1 e (10.351) The wave part herein is (0) c p (4) (1) (1) (1) − i ωU ˆ 1 + f 0 ez × U1 + ikh θ 1 = Ru,1 R     ∂ (0) (1) (0) (0) (0) ≡− + U0 · ∇ X + ik · V0 U1 − V1 · ∇ X U0 ∂T  c p  (0) (1) (3) (3) − θ ∇ X ,h 1 + θ ikh 1 R (0)

(10.352)

while the mean-flow part yields, again using W0 = 0 and the orthogonality relation (10.249), and using analogous considerations for the terms nonlinear in the wave amplitudes as leading from the O(ε3 ) terms (10.333) in the entropy equation to the corresponding mean flow part (10.338),

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10 Gravity Waves and Their Impact on the Atmospheric Flow



 ∂ (0) (0) (1) + U0 · ∇ X U0 + f 0 ez × U0 ∂T  c p  (0) (1) (3) (2) =− θ ∇ X ,h 0 + θ ∇ X ,h 0 R   1    1  (0) (0) ∗ (1) (0) ∗ + ik · V1 U1 − V1 · ∇ X U1 2 2

(10.353)

With the help of (10.327) this gives 

 ∂ (0) (0) (1) + U0 · ∇ X ,h U0 + f 0 ez × U0 ∂T     c p  (0) 1 (1) (3) (2) (0) (0) ∗ (0) 1 =− θ ∇ X ,h 0 + θ ∇ X ,h 0 − (0) ∇ X · ρ V1 U1 R 2 ρ

(10.354) This describes the impact of the gravity-wave momentum fluxes on the synoptic-scale horizontal flow. Adding the geostrophic equilibrium to this equation and redimensionalizing yields 

   ∂ 1 + u · ∇ u + f ez × u = −c p θ ∇h π  − ∇ · ρv u  ∂t ρ

where π  is the synoptic-scale Exner pressure, and    (0) (0) v = Uw U1 + Ww W1 ez eiφ/ε

(10.355)

(10.356)

are the gravity-wave velocity fluctuations. The synoptic-scale horizontal wind is forced by the convergence of the gravity-wave momentum flux.

10.3.6 Wave Action Now we are ready for deriving from the next-order wave equations (10.327), (10.337), (10.350), and (10.352) a prognostic equation for the wave amplitude. This equation expresses the conservation of wave action. One first derives a wave-energy theorem, then reformulates therein energy flux, shear, and buoyancy production using the dispersion and polarization relations as well as the mean-flow balance conditions, and finally combines all, using also the eikonal equations.

10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow

467

Wave-Energy Theorem The wave equations (10.327), (10.337), (10.350), and (10.352) can be summarized as (1)

(1)

M1 Z1 = R1 with



t Z(1) 1

R1(1)

t

(3)

B1 c p (0) (4) = , θ 1 N0 R   (1) (1) t (1) Rb,1 (1) = Ru,1 , Rw,1 , , Rπ,1 N0 (1) t (1) U1 , W1 ,

(10.357)  (10.358)

(10.359)

As we have seen before, M1 must be singular so that it has a non-vanishing null space, given (1) by the gravity-wave polarization relations. By the so-called Fredholm alternative, R1 may not project onto this null space. To see this we note that the latter is given, up to a constant factor, by the null vector   (2) c p (0) (3) (0) t (0) t (0) B1 (10.360) , θ 1 Z1 = U1 , W1 , N0 R as follows from the polarization relations (10.297), (10.298), and (10.299). By definition, it †

(0) † satisfies M1 Z(0) 1 = 0, thus also Z1 M1 = 0, and finally (0) †

Z1

M1 = 0

(10.361) (0) †

since M1 is anti-Hermitian, i.e., M1† = −M1 . Thus, multiplying (10.357) by Z1 (0) †

0 = Z1

(1)

R1

yields (10.362)

Herein, minus half the real part is   (2) ∗ (1) B1 Rb,1 c p (0) (3) ∗ (1) 1 (0) ∗ (1) (0) ∗ (1) + θ 1 Rπ,1 0 = − U1 · Ru,1 + W1 Rw,1 + 2 N0 N0 R   !   1 ∂ (0) ∗ (0) (1) (0) (0) (0) = U1 · + U0 · ∇ X + ik · V0 U1 + V1 · ∇ X U0 2 ∂T  c p  (0) (1) (3) (3) + θ ∇ X ,h 1 + θ ikh 1 R    ∂ (0) ∗ 4 (0) (1) (0) ε + U0 · ∇ X + ik · V0 W1 + W1 ∂T

468

10 Gravity Waves and Their Impact on the Atmospheric Flow

cp + R (2) ∗ 

 θ

(3) (0) ∂1

∂Z

+θ

(1)

(3) im1

∂ (0) (1) + U0 · ∇ X + ik · V0 ∂T (  c p (0) (3) ∗ 1 (0) (0) + θ 1 ∇ · ρ V X 1 R ρ (0)

B + 1 N0

(2) ∂π + 1

(1)



∂Z



(2)

(0)

B1 V (2) + 1 · ∇ X B0 N0 N0



(10.363)

or, after multiplication by ρ (0) , and using once more the orthogonality relation (10.249),    1  ∂ (0) (0) ∗ (0) (0) + U0 · ∇ X E w + ρ (0) U1 V1 · ·∇ X U0 0= ∂T 2   ρ (0) (2) ∗ (0) 1 1  (0) ∗ (2)  (0) ∂π (1) (2) · ∇ + B V B + W1 B1 θ X 1 0 2 2 ∂Z N02 1       c p (0) (3) c p (0) (3) ∗ 1 1 (0) ∗ (0) + + ρ (0) V1 · ∇ X θ 1 θ 1 ∇ X · ρ (0) V1 2 R 2 R (10.364) ⎛      ⎞  (0) 2  (0) 2  (2) 2 U  W  B 1 1  ⎟ 1 ⎜ 1 Ew = + ε4 + 2 ⎝ ⎠ 2 2 2 2 N0

where

ρ (0)

(10.365)

is the non-dimensional wave-energy density, and we use for brevity the notation, for any three vectors a, b, c,  ∂ci ab · ·∇c = ai b j (10.366) ∂x j i, j

Finally we recall that there is no vertical wave-buoyancy flux, as expressed by (10.340), so that we arrive at the non-dimensional wave-energy theorem 

  c p (0) (3) ∗ (0)  ∂ 1 (0) + U0 · ∇ X ,h E w + ∇ X · ρ (0) θ 1 V1 ∂T 2 R     ρ (0) (2) ∗ (0) 1 1 (0) (2) (0) (0) ∗ (0) · ∇ X ,h B0 B U1 = − ρ U1 V1 · ∇ X U0 − 2 2 N02 1

(10.367)

which has the dimensional equivalent 

∂ + u · ∇h ∂t



1 1 1 0 0 ρ 0 E w + ∇ · c p ρθ π  v = −ρ u v · ·∇u − 2 b u · ∇h b N (10.368)

with now Ew =

ρ 2



|v |2  +

b2  N2

 (10.369)

10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow

469

the dimensional wave-energy density. Both advection by the (to leading order non-divergent) mean flow and pressure or energy flux, the last term on the left-hand side, redistribute wave energy. That the last term is indeed a pressure flux can be seen by noting that due to the definition (10.137) of the Exner pressure π R p = π cp p

(10.370)

whence cp p  cp T ρ R π  = c p ρθ π  (10.371) π = R π R π On the right-hand side of the energy equations (10.367) and (10.368) we have two terms acting as sources or sinks. The first is shear production, due to wave-momentum fluxes against or along the synoptic-scale velocity gradient. The second is buoyancy production due to wave-buoyancy fluxes against or along the synoptic-scale buoyancy gradient. A further helpful reformulation of the energy equations is possible if, as is the case here, the synoptic-scale flow is balanced. Geostrophy (10.277) and hydrostaticity (10.269) lead together to the thermal-wind relation p =

(2)

(0)

∇ X ,h B0 = − f 0 ez ×

∂U0 ∂Z

(10.372)

for the synoptic-scale flow, with its dimensional counterpart ∇h b = − f ez ×

∂u ∂z

(10.373)

Using these in the energy equations (10.367) and (10.368), respectively, leads to formulations    c p (0) (3) ∗ (0)  ∂ 1 (0) + U0 · ∇ X ,h E w + ∇ X · ρ (0) θ 1 V1 ∂T 2 R 1  (0) (0) ∗ (0)  (0) = − ρ U1 U1 · ·∇ X ,h U0 2 

 (0) ∂U0 1 ρ (0) (2) ∗ (0) 1  (0) (0) ∗ (0)  · − B U ρ W1 U1 + ez × f 0 (10.374) 1 2 2 ∂Z N02 1 and

 1 0 ∂ + u · ∇h E w + ∇ · c p ρθ π  v ∂t   0  1 0  1 ∂u ρ 0  1 = −ρ u u · ·∇h u − ρ w u + ez × f 2 b u · N ∂z



where no synoptic-scale buoyancy gradients appear any more.

(10.375)

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10 Gravity Waves and Their Impact on the Atmospheric Flow

Reformulation of the Wave-Energy Density In the next step we use the gravity-wave dispersion and polarization relations for expressing the non-dimensional wave-energy density (10.365) in terms of wave-buoyancy amplitude, frequency, and wavenumber. Due to the polarization relations (10.297) and (10.299), linking the gravity-wave wind amplitudes to its buoyancy amplitude, the kinetic-energy density of the waves is ⎛    ⎞  (0) 2  (0) 2 (0) U  W 1  ⎟ ρ ⎜ 1 + ε4 E w,kin ≡ ⎝ ⎠ 2 2 2 ⎡ = ρ (0) ⎣

ε4 ωˆ 2

− N02 ωˆ 2 − f 02

2

⎤   2 2 kh2 (ωˆ 2 + f 02 )  (2) 2 ωˆ  (2)  ⎦ B1  + ε4 B  4m 2 N04 4N04 1

(10.376)

where, by the gravity-wave dispersion relation (10.281), ε4 ωˆ 2 − N02 = ε4 ωˆ 2 − f 02 =

N02 kh2 + f 02 m 2 ε4 kh2 + m 2

N02 kh2 + f 02 m 2 ε4 kh2 + m 2

− N02 =

− f 02 =

(ε4 f 02 − N02 )m 2 ε4 kh2 + m 2

(N02 − ε4 f 02 )kh2 ε4 kh2 + m 2

(10.377) (10.378)

so that it can be rewritten, again using the dispersion relation,

E w,kin

  ⎤  (2) 2 ⎡ 2 2 2 + f 2) 2 B1  k ( ω ˆ m 0 h ⎣ = ρ (0) + ε4 ωˆ 2 ⎦ m2 4N04 kh2   

 (2) 2  2 2 B1  m m ωˆ 2 ε4 + 2 + 2 f 02 = ρ (0) 4N04 kh kh  2  (2)  B1  2 m 2 f 2 + N 2 k 2 0 0 h (0) =ρ 4N04 kh2

(10.379)

Inserting this into (10.365) yields, once more with the help of the dispersion relation,

Ew = ρ

   (2) 2 2 B1  f m 2 + N 2 k 2 0 0 h (0) 2 N04

N02 kh2

= ρ (0)

   (2) 2 2  4 2  B1  ωˆ ε k + m 2 h

2 N02

N02 kh2

(10.380)

10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow

471

Reformulation of the Energy Flux An interesting and important fact is that the pressure or energy flux is identical to the product of wave-energy density and intrinsic group velocity. Hence, the intrinsic group velocity indeed indicates where the energy is propagating, in the frame of reference moving with the synoptic-scale flow. In order to show this we first note that, via the dispersion relation (10.281) the horizontal and vertical parts of the intrinsic group velocity are, respectively, cˆ g,h = ∇k,h ωˆ = kh cˆg,z =

(N02 − ε4 f 02 )m 2 2  ωˆ ε4 kh2 + m 2

(ε4 f 2 − N02 )kh2 ∂ ωˆ =m  0 2 ∂m ωˆ ε4 k 2 + m 2

(10.381) (10.382)

h

so that, by multiplication with the wave-energy density in (10.380),

cˆ g,h E w =

(N 2 − ε4 f 02 )m 2 ωˆ  ρ (0) kh  40 2 ε kh + m 2 N02 kh2

   (2) 2 B1 

2 N02    (2) 2 4 f 2 − N 2 )ω B1  ˆ (ε 0  cˆg,z E w = ρ (0) m  4 20 ε kh + m 2 N02 2 N02

(10.383)

(10.384)

On the other hand, by the polarization relations (10.297) and (10.298), linking the gravitywave horizontal-wind and Exner-pressure amplitudes to the corresponding buoyancy amplitude, the horizontal pressure flux is 1  (0) c p (0) (3) ∗ (0)  θ 1 U1 ρ 2 R    (2) 2  

B1  i ε4 ωˆ 2 − N02 i ε4 ωˆ 2 − N02 k ωˆ + il f 0 (0) − =ρ l ωˆ − ik f 0 2 m N02 m N02 ωˆ 2 − f 02

(10.385)

Using herein (10.377) and (10.378), and comparing with (10.383) show that 1  (0) c p (0) (3) ∗ (0)  θ 1 U1 = cˆ g,h E w ρ 2 R

(10.386)

Furthermore, using the polarization relations (10.295) and (10.297), linking the gravity-wave Exner-pressure and vertical-wind amplitudes to the corresponding buoyancy amplitude, we find that the vertical pressure flux is     (2) 2  B1  i ε4 ωˆ 2 − N02 i ωˆ 1  (0) c p (0) (3) ∗ (0)  (0) =ρ θ 1 W 1 ρ − 2 R 2 m N02 N02

(10.387)

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10 Gravity Waves and Their Impact on the Atmospheric Flow

Using herein (10.377) and comparing with (10.384) verify that 1  (0) c p (0) (3) ∗ (0)  = cˆg,z E w θ 1 W 1 ρ 2 R

(10.388)

Hence one has from (10.386) and (10.388) in total the important identity 1  (0) c p (0) (3) ∗ (0)  θ 1 V1 = cˆ g E w ρ 2 R

(10.389)

with its dimensional counterpart 1 0 c p ρθ π  v = cˆ g E w

(10.390)

Reformulation of the Production Terms We now use the non-divergence of the leading-order synoptic-scale wind and the gravitywave dispersion and polarization relations to convert the production terms on the right-hand sides of the reformulations (10.374) and (10.375) of the energy equation into a compact form involving the wave-action density. First we consider the contributions involving horizontal gravity-wave momentum fluxes. We again make use of the polarization relation (10.299), linking the gravity-wave horizontalwind amplitudes to the corresponding buoyancy amplitude, to obtain (0) (0)  1  1  (0) (0) ∗ (0)  ∂U0 (0) ∗ (0) ∂ V0 ρ U1 U1 + ρ (0) V1 V1 2 ∂X 2 ∂Y  

 (2) 2 (0) (0) B1  (ε4 ωˆ 2 − N 2 )2 (0) 2 2 2 2 ∂U0 2 2 2 2 ∂ V0 0 =ρ + (l ωˆ + k f 0 ) (k ωˆ + l f 0 ) ∂X ∂Y 2 m 2 N04 (ωˆ 2 − f 02 )2 (10.391)

Because of the non-divergence (10.329) of the leading-order synoptic-scale horizontal wind we can rewrite the last bracket therein (k ωˆ

(0) 2 2 ∂U0 + l f0 )

(0) 2 ∂ V0 f0 )

+ (l ωˆ + k ∂X ∂Y   (0) (0) (0) (0) ∂U ∂U ∂ V ∂ V 0 = (k 2 ωˆ 2 + l 2 f 02 ) 0 + (l 2 ωˆ 2 + k 2 f 02 ) 0 − kh2 f 02 + 0 ∂X ∂Y ∂X ∂Y 2 2

(0)

= k 2 (ωˆ 2 − f 02 )

2 2

2

(0)

∂U0 ∂V + l 2 (ωˆ 2 − f 02 ) 0 ∂X ∂Y

(10.392)

10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow

473

Inserting this into (10.391) and using then (10.377) and (10.378) leads to (0) (0)  1  (0) (0) ∗ (0)  ∂U0 1  (0) ∗ (0) ∂ V0 ρ U1 U1 + ρ (0) V1 V1 2 ∂X 2 ∂Y      (2) 2 (0) (0) B1  (ε4 ωˆ 2 − N 2 )2 2 ∂U0 2 ∂ V0 0 k = ρ (0) + l ∂X ∂Y 2 m 2 N04 (ωˆ 2 − f 02 )  2    (2)  (0) (0) B1  (N 2 − ε4 f 2 )m 2 (0) 2 ∂U0 2 ∂ V0 0 0   k =ρ +l ∂X ∂Y 2 N04 ε4 kh2 + m 2 kh2

(10.393)

A look at the formulation (10.383) of the horizontal energy flux then shows that one has (0) (0)  1  (0) (0) ∗ (0)  ∂U0 1  (0) ∗ (0) ∂ V0 ρ U1 U1 + ρ (0) V1 V1 2 ∂X 2 ∂Y (0) (0) (0) (0) ∂U ∂V ∂U ∂V Ew Ew = cˆgx k 0 + cˆgy l 0 = Ak cˆgx 0 + Al cˆgy 0 ωˆ ∂X ωˆ ∂Y ∂X ∂Y

(10.394)

with its dimensional counterpart 0   1 ∂u 0   1 ∂v ∂u ∂v ρu u + ρv v = Ak cˆgx + Al cˆgy ∂x ∂y ∂x ∂y

(10.395)

We have also introduced the wave-action density

A=

Ew ωˆ

(10.396)

which has the same definition in non-dimensional and dimensional form. How important this field is will become clearer below. Moreover, the horizontal-wind polarization relation (10.299) also gives, again using (10.377) and (10.378),    (2) 2 B1  (ε4 ωˆ 2 − N 2 )2 1  (0) (0) ∗ (0)  0 = ρ (0) (kl ωˆ 2 − kl f 02 ) ρ U1 V1 2 2 m 2 N04 (ωˆ 2 − f 02 )2    (2) 2 B1  (ε4 ωˆ 2 − N 2 )2 0 kl = ρ (0) 2 m 2 N04 ωˆ 2 − f 02    (2) 2 B1  (N 2 − ε4 f 2 )m 2 kl 0 0 2  = ρ (0) ε4 kh + m 2 kh2 2 N04

(10.397)

Again also looking at the formulation (10.383) of the horizontal energy flux, one finds that 1  (0) (0) ∗ (0)  = Ak cˆgy = Al cˆgx ρ U1 V1 2

(10.398)

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10 Gravity Waves and Their Impact on the Atmospheric Flow

which is in dimensional form 0  1 ρu v = Ak cˆgy = Al cˆgx

(10.399)

(0)

Finally we simplify the flux term that multiplies ∂U1 /∂ Z in (10.374), and that we had obtained by application of the thermal wind. In its x-component one has via the gravitywave wind polarization relations (10.297) and (10.299)   1  (0) (0) ∗ (0)  1 ρ (0) (0) ∗ (2) − f0 V B1 ρ U1 W1 2 2 N02 1  

 (2) 2 B1  i ε4 ωˆ 2 − N02 ωˆ f 0 i ε4 ωˆ 2 − N02 (0) − =ρ (k ωˆ − il f 0 )i 2 − 2 (l ωˆ − ik f 0 ) 2 m N02 ωˆ 2 − f 02 N0 N0 m N02 ωˆ 2 − f 02    (2) 2 B1    N02 − ε4 ωˆ 2 = ρ (0) ˆ ωˆ + i f 0 (l ωˆ − ik f 0 ) (il f 0 − k ω) 4 2 2 2 m N0 (ωˆ − f 0 )  2    (2)   (2) 2 4 2 B1  N 2 − ε4 ωˆ 2 B1  (ε f − N 2 )mk 0 0 0 = ρ (0) k = ρ (0) (10.400) 2 mN4 2 N4 ε4 k 2 + m 2 0

0

h

where (10.377) has been used in the last step. Confronting this with the formulation (10.384) of the vertical energy flux shows that   1 ρ (0) (0) ∗ (2) 1  (0) (0) ∗ (0)  = Ak cˆgz − f0 V B1 (10.401) ρ U1 W1 2 2 N02 1 In an exactly analogous manner one also obtains for the y-component of the flux term of interest   1  (0) (0) ∗ (0)  1 ρ (0) (0) ∗ (2) = Al cˆgz + f0 U B1 (10.402) ρ V1 W1 2 2 N02 1 and hence in total, also using cˆgz = cgz ,   1  (0) (0) ∗ (0)  1 ρ (0) (2) ∗ (0) B U1 ρ W1 U1 + ez × f 0 = Akh cgz 2 2 N02 1

(10.403)

with its dimensional counterpart 1 1 0 ρ 0 ρ w u + ez × f 2 b u = Akh cgz N

(10.404)

We see that all relevant fluxes in the wave-energy equation (10.367) or (10.368) are fluxes, due to transport by the intrinsic group velocity, of the so-called wave pseudomomentum p h = Ak h

(10.405)

10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow

475

Reformulation of the Wave-Energy Theorem From the results above one can reformulate the wave-energy equation in a meaningful manner. We insert the identity (10.389) between pressure flux and energy flux, the helpful identity (10.394), the relation (10.398) between the meridional flux of zonal momentum (or vice versa) and the corresponding wave-pseudomomentum flux, and the same (10.403) for the vertical pseudomomentum flux, into the formulation (10.374) of the wave-energy equation. This yields     ∂ (0) (0) (10.406) + U0 · ∇ X ,h E w + ph cˆ g · ·∇ X U0 + ∇ X · cˆ g E w 0= ∂T Herein we have cˆ g = cg − U0(0) and, as expressed by (10.329), the leading-order synopticscale horizontal wind is non-divergent. Hence we obtain 0=

  ∂ Ew (0) + ∇ X · cg E w + ph cˆ g · ·∇U0 ∂T

(10.407)

  ∂ Ew + ∇ · cg E w + ph cˆ g · ·∇u ∂t

(10.408)

or in dimensional form 0=

Wave-Action Conservation From the reformulated wave-energy theorem it is only a small step to the central theorem for the prediction of the gravity-wave amplitudes. We replace E w = ωˆ A in (10.407). This yields       ∂ ∂A (0) (10.409) + ∇ X · cg A + A + cg · ∇ X ωˆ + Akh cˆ g · ·∇U0 0 = ωˆ ∂T ∂T For the along-ray derivative of the intrinsic frequency we apply the eikonal equations (0) (10.314) and (10.315), leading with ωˆ = ω − kh · U0 to   ∂ + cg · ∇ X ωˆ ∂T      ∂ ∂ (0) = + cg · ∇ X ω − + cg · ∇ kh · U0 ∂T ∂T       (0) ∂U0 ∂ ∂ (0) (0) = kh · − kh · + cg · ∇ X U0 − + cg · ∇ X kh · U0 ∂T ∂T ∂T    (0) (0) (0) = −kh cg · ·∇ X U0 + ∇ X U0 · kh · U0 (0)

(0)

(0)

(0)

= −kh cg · ·∇ X U0 + kh U0 · ·∇ X U0 = −kh cˆ g · ·∇ X U0

(10.410)

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10 Gravity Waves and Their Impact on the Atmospheric Flow

Inserting this back into (10.409) we finally obtain the important conservation law   ∂A + ∇ X · cg A = 0 ∂T

(10.411)

  ∂A + ∇ · cg A = 0 ∂t

(10.412)

in dimensional form

2 for the wave action d V A, with density A defined in (10.396). Predictions of the waveaction density via this equation, together with wavenumber and frequency via the eikonal equations yield the development of the absolute magnitude of the wave-buoyancy amplitude, e.g., and from the latter, via the polarization relations, up to a phase factor also of all other wave amplitudes.

10.3.7 Wave Impact on the Synoptic-Scale Flow Having settled above how the gravity-wave field responds to the synoptic-scale flow, the question remains whether and how it influences the synoptic-scale flow. The latter is governed by the horizontal momentum equation (10.354) and the buoyancy equation (10.341), both forced by gravity-wave flux convergence, the anelastic non-divergence constraint (10.332), and geostrophic equilibrium (10.277) and hydrostatic equilibrium (10.269), yielding together the thermal-wind relation (10.372). In the following these shall be used to derive a prognostic equation for the synoptic-scale quasigeostrophic potential vorticity. The derivation is much in parallel to the derivation of the conservation of quasigeostrophic potential vorticity in Chap. 6. What differs essentially is that we have the additional gravity-wave forcing terms that one must keep track of. It will turn out that they lead to a source or sink of quasigeostrophic potential vorticity, which can be expressed in terms of wave-pseudomomentum fluxes as well. To begin with, the vertical curl of the horizontal momentum equation (10.354) yields the quasigeostrophic vorticity equation with gravity-wave impact     1 c p (0) (2) ∂ (0) (1) + f 0 ∇ X ,h · U0 θ 0 + U0 · ∇ X ,h ∇ X2 ,h ∂T f0 R   ∂  (0) (0) ∗  1 ρ (0) ∂  (0) (0) ∗  − (10.413) = − (0) ∇ X · V1 V1 V1 U1 2 ∂X ∂Y ρ

10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow

477

Due to the anelastic non-divergence constraint (10.332) we have for the leading-order synoptic-scale horizontal divergence (1)

∇ X ,h · U0 = −

∂  (0) (1)  ρ W0 ρ (0) ∂ Z 1

(10.414)

Herein we can re-express the leading-order synoptic-scale vertical wind, using the buoyancy equation (10.341), as    (2)  (2) B0 ∂ 1 (1) (0) (0) ∗ B1 − ∇ X ,h · U1 W0 = − + U0 · ∇ X ,h (10.415) ∂T 2 N02 N02 leading to (1) ∇ X ,h · U0

 

 (0) (2) B0 ∂U0 1 ∂ ρ (0) (2) ∂ (0) = B + U0 · ∇ X ,h · ∇ + X ,h ∂T ∂Z N02 ρ (0) ∂ Z N02 0 

 (2) 1 ∂ ρ (0) (0) ∗ B1 + ∇ X ,h · (0) U1 2 N02 ρ ∂Z

' &   1 ∂ ρ (0) ∂  c p (0) (2)  ∂ (0) θ 0 = + U0 · ∇ X ,h ∂T ρ (0) ∂ Z N02 ∂ Z R 

 (2) 1 ∂ ρ (0) (0) ∗ B1 (10.416) + ∇ X ,h · (0) U1 2 N02 ρ ∂Z 

where we have used the thermal-wind relation (10.372) to eliminate the second term on the right hand, and where the first one has been reformulated applying the consequence      (2) c p (0) ∂0 ∂ ∂ (2) (10.417) θ , ∇ X ,h B0 = , ∇ X ,h ∂T ∂T R ∂Z of the hydrostatic equilibrium (10.269). Substitution of the leading-order horizontal divergence from (10.416) into the vorticity equation (10.413) results in a quasigeostrophic potential-vorticity equation with gravity-wave impact

'  &   ∂ f 0 ∂ ρ (0) ∂  c p (0) (2)  1 c p (0) (2) (0) 2 ∇ X ,h + (0) θ 0 θ 0 + U0 · ∇ X ,h ∂T f0 R ρ ∂ Z N02 ∂ Z R   ∂  (0) (0) ∗  ρ (0) ∂  (0) (0) ∗  1 − = − (0) ∇ X · V1 V1 V1 U1 2 ∂X ∂Y ρ 

 (2) ∂ ρ (0) f0 (0) ∗ B1 U1 − (0) ∇ X ,h · ∂Z 2 N02 ρ

478

10 Gravity Waves and Their Impact on the Atmospheric Flow

 ∂ 1  (0) (0) ∗  ∂ 1  (0) (0) ∗  − = −∇ X ,h · U1 V1 U1 U1 ∂X 2 ∂Y 2   !  (2) 1 ∂ ρ (0) ∂ (1) (0) ∗ (0) ∗ B1 − (0) W0 V1 + f 0 U1 2 ∂X N02 ρ ∂Z   (2) ( ∂ (1) (0) ∗ (0) ∗ B1 − W0 U1 − f 0 V1 ∂Y N02 

(10.418)

Herein the term under the horizontal divergence on the right-hand side is ∂ 1  (0) (0) ∗  ∂ 1  (0) (0) ∗  − U1 V1 U1 U1 ∂X 2 ∂Y 2   ∂ 2 1  (0) (0) ∗  ∂ 2 1  (0) 2  (0) 2 ∂ 2 1  (0) (0) ∗  − + V − = U V1 U1 U V   1 1 1 1 ∂ X2 2 ∂ X ∂Y 2 ∂Y 2 2 (10.419) Due to the horizontal-wind polarization relation (10.299) we have in the middle term therein    (2) 2  4 2 2      B   2 1 1   ωˆ − N02 1  (0) 2  (0) 2 2 2 2 U1  − V1  =  2 (k − l ) ωˆ − f 0 4 2 2 2 2 m N0 ωˆ 2 − f 0    (2) 2  2  1 B1  N0 −  4 f 02 (k 2 − l 2 )m 2  2  = (10.420) 2 N04  4 kh + m 2 kh2 where (10.377) and (10.378) have been used in the last step. Comparison with the horizontal energy flux (10.383) finally leads to    1  1  (0) 2  (0) 2 (10.421) − V1  = (0) cˆgx k A − cˆgy l A U1  2 ρ Inserting this together with the relation (10.398), between the meridional flux of zonal momentum (or vice versa) and the corresponding wave pseudomomentum flux, into (10.418) yields ∂ 1  (0) (0) ∗  ∂ 1  (0) (0) ∗  − U1 V1 U1 U1 ∂X 2 ∂Y 2       ∂2  ∂2  ∂2  ∂2  1 c ˆ k A = (0) − 2 cˆgx l A − cˆgy l A + cˆgx k A + gy ∂X ∂ X ∂Y ∂ X ∂Y ∂Y 2 ρ (10.422) Inserting this back into the quasigeostrophic potential-vorticity equation (10.418), and recognizing in the two last terms on the right hand-side the two components of the vertical pseudomomentum flux (10.403) we then obtain

10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow

479



' &    (0) ∂  c c ∂ 1 f ∂ ρ (0) (0) p p 0 (0) (2) (2) + (0) θ 0 θ 0 + U0 · ∇ X ,h ∇ X2 ,h ∂T f0 R ρ ∂ Z N02 ∂ Z R     ∂2  ∂2  ∂2  1 c ˆ − + l A l A c ˆ cˆgx k A = (0) − gx gy 2 ∂ X ∂ X ∂Y ∂ X ∂Y ρ     ∂2  ∂2  ∂2  c ˆ c ˆ c ˆ − + (10.423) l A l A k A + gy gz gz ∂Y 2 ∂ Z∂ X ∂ Z ∂Y



which is the prognostic equation 

     1 1 ∂ ∂ ∂ (0) ∇X · H + ∇X · G + U0 · ∇ X ,h πqg = − ∂T ∂ X ρ (0) ∂Y ρ (0)

(10.424)

with G = cˆ g ph,x = cˆ g k A and H = cˆ g ph,y = cˆ g l A the fluxes of zonal and meridional pseudomomentum, respectively, for the quasi geostrophic potential vorticity πqg =

∇ X2 ,h

c

p (0)

R

θ

(2) 0



f0 ∂ + (0) ρ ∂Z



ρ (0) ∂  c p (0) (2)  θ 0 N02 ∂ Z R

(10.425)

In the absence of gravity-wave fluxes it is conserved. Otherwise it is controlled by the curl of the vector of divergences of the fluxes of the components of the pseudomomentum ph = kh A. The dimensional form of this prognostic equation is 

     ∂ ∂ 1 ∂ 1 + u · ∇h πqg = − ∇ ·H + ∇ ·G ∂t ∂x ρ ∂y ρ

(10.426)

for the quasigeostrophic potential vorticity πqg = ∇h2 ψ + where ψ=

1 ∂ ρ ∂z



f 2 ∂ψ N 2 ∂z



1 1 c p θ δπ  = c p θ (π  − π) f f

(10.427)

(10.428)

is the synoptic-scale streamfunction so that in agreement with the dimensional form f ez × u = −c p θ ∇h δπ 

(10.429)

of the geostrophic equilibrium (10.277) the synoptic-scale wind is u = ez × ∇h ψ

(10.430)

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10 Gravity Waves and Their Impact on the Atmospheric Flow

10.3.8 Generalization to Gravity-Wave Spectra: Phase-Space Wave-Action Density The multiple-scale-asymptotic WKB theory as described so far is fully nonlinear. No assumptions have been used that wave amplitudes are small. They are actually assumed to be close to the static-instability threshold. What had helped avoiding such assumptions was that it assumes that at each location there is exactly one local phase (and hence wavenumber and frequency) and amplitude. Nonetheless, even when this is the case initially, this can lead to problems when rays cross, e.g., below a reflecting layer where rays approaching the reflecting layer or already having been reflected, pass each other. Another possibility is that rays with different group velocity catch up with each other. In such cases the numerical integration of the eikonal equations together with the wave-action conservation equation suffers from serious numerical instabilities. In true nature, however, be it by propagation effects as mentioned or by the emission by gravity-wave sources of whole spectra of waves, the locally monochromatic case is rather a conceptually useful exception. One needs to be able to also describe the dynamics of the interaction of gravity-wave spectra with a synoptic-scale flow. As long as the wave amplitudes are sufficiently weak, however, this is easily possible. All the results above on the wave properties, i.e., dispersion relation, polarization relations, eikonal equations, and wave-action conservation, have been obtained from wave equations which are linear in the wave amplitudes. This linearity had been facilitated by the elimination of the nonlinear advection terms in the wave equations by the orthogonality relation (10.249), and by the fact that even in the vicinity of the static-instability threshold the potential-temperature and Exner-pressure wave amplitudes are considerably weaker then the respective reference-atmosphere values. Had we linearized the equations beforehand we would have obtained the same results. Solutions from linear dynamical equations, however, can be superposed so that the resulting superposition is again a solution. Thus, in the weak-amplitude limit we can consider such a superposition of solutions, each indicated by an own value of a continuous three-dimensional index α, each moving in interaction with a geostrophically and hydrostatically balanced synoptic-scale flow, each satisfying its own wave-action conservation and eikonal equations,   ∂ Aα + ∇ · cgα Aα = 0 ∂t   ∂ + cgα · ∇ kα = −∇ (kα , x, t) ∂t

(10.431) (10.432)

We now define a wave-action density in phase space, spanned by location and wavenumber, 

N (x, k, t) =

d 3 α Aα (x, t) δ [k − kα (x, t)]

(10.433)

i.e., a superposition of wave-action densities from all separate solutions, and derive its prognostic equation. One first has

10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow

∂N = ∂t



 d α 3

∂kα ∂ Aα δ (k − kα ) − Aα · ∇k δ (k − kα ) ∂t ∂t

481

 (10.434)

By the wave-action conservation equations (10.431) this becomes      ∂kα ∂N = − d 3 α ∇ · cgα Aα δ (k − kα ) + Aα · ∇k δ (k − kα ) ∂t ∂t !    = − d 3 α ∇ · cgα Aα δ (k − kα ) − Aα cgα · ∇δ (k − kα ) + Aα

∂kα · ∇k δ (k − kα ) ∂t

( (10.435)

However, because ∂ [δ (k − kα )] /∂kαi = −∂ [δ (k − kα )] /∂ki , one has herein   ∂kα j ∂ ∂ δ (k − kα ) = −cgαi δ (k − kα ) cgα · ∇δ (k − kα ) = cgαi ∂ xi ∂k j ∂ xi    (10.436) = − cgα · ∇ kα · ∇k δ (k − kα ) so that one obtains, using the eikonal relations (10.432) and the fact that ∇k acts neither on dkα /dt = ∇ (kα , x, t) nor on Aα = Aα (x, t), ∂kα · ∇k δ (k − kα ) − Aα cgα · ∇δ (k − kα ) + Aα   ∂t    ∂kα  dkα Aα δ (k − kα ) + cgα · ∇ kα · ∇k δ (k − kα ) = ∇k · = Aα ∂t dt

(10.437)

This we insert into (10.435), and make use of the property that δ(k − kα ) is only nonzero where k = kα . Hence ( !     dkα ∂N Aα δ (k − kα ) = − d 3 α ∇ · cgα Aα δ (k − kα ) + ∇k · ∂t dt   dkα = −∇ · d 3 α cgα Aα δ (k − kα ) − ∇k · d 3 α Aα δ (k − kα ) dt   dk = −∇ · d 3 α cg Aα δ (k − kα ) − ∇k · d 3 α Aα δ (k − kα ) dt       dk d 3 α Aα δ (k − kα ) = −∇ · cg d 3 α Aα δ (k − kα ) − ∇k · dt     dk (10.438) N = −∇ · cg N − ∇k · dt We thus have the important result     ∂N + ∇ · cg N + ∇k · k˙ N = 0 ∂t

(10.439)

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10 Gravity Waves and Their Impact on the Atmospheric Flow

where we have introduced the more compact notation k˙ = dk/dt. This equation indicates that the integral of phase-space wave-action density over the total phase-space volume is conserved. Moreover, cg = ∇k  and k˙ = −∇ imply that the six-dimensional phase-space velocity is non-divergent, (10.440) ∇ · cg + ∇k · k˙ = 0 This has the helpful consequence that the conservation equation (10.439) can also be written ∂N + cg · ∇ N + k˙ · ∇k N = 0 ∂t

(10.441)

In contrast to wave-action density in position space, phase-space wave-action density is conserved along the rays in phase space satisfying dx = cg dt dk = k˙ dt

(10.442) (10.443)

The wave impact on the synoptic-scale flow can be handled as derived above, just that the results there have to be seen as impact from a single member of the superposition of weak-amplitude fields. In order to take the impact of all members into account, one uses the corresponding superposition. Thus, as an example, the flux of zonal pseudomomentum is understood to be, suppressing the dependence on space and time,    G = d 3 α cˆ g (kα ) kα Aα = d 3 α d 3 k δ (k − kα ) cˆ g (k) k Aα    (10.444) = d 3 k cˆ g (k) k d 3 α Aα δ (k − kα ) = d 3 k cˆ g (k) k N (k) or, including also the corresponding result for the flux of meridional pseudomomentum, 

G (x, t) = H (x, t) =



d 3 k cˆ g (k, x, t) k N (k, x, t)

(10.445)

d 3 k cˆ g (k, x, t) l N (k, x, t)

(10.446)

Likewise one proceeds with all other fluxes of interest, i.e., products between functions of wavenumber and the position-space wave-action density A are replaced by wavenumber integrals over products between the same function and the phase-space wave-action density N . In the remainder of this chapter we will take the more general spectral perspective expressed by the phase-space wave-action density and the fluxes derived from it. The locally monochromatic case is included in this as a special case.

10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow

483

10.3.9 Conservation Properties As we have seen above, neither is the gravity-wave energy conserved nor is the quasigeostrophic potential vorticity. In both cases the interaction between gravity waves and synoptic-scale flow leads to an exchange between the two so that the corresponding conserved quantity comprises contributions from both components. This will be shown in the following. Energy We begin with energy. We first note that the spectral form of the wave-energy equation (10.408) is, by the arguments above, 0=

  ∂ Ew + ∇ · Fw + ex G + e y H · ·∇u ∂t 

with Ew =

(10.447)

 d k ωˆ N 3

and

Fw =

d 3 k cg ωˆ N

(10.448)

being the wave-energy density and the wave-energy flux, respectively. It is the last term in (10.447) that describes the exchange of energy, by convective processes and horizontal and vertical shear production, with the synoptic-scale flow. A prognostic equation for the energy density of the latter can be obtained very much in parallel to the corresponding calculation in quasigeostrophic theory. One multiplies the prognostic equation (10.426) for the synoptic-scale quasigeostrophic potential vorticity by −ρψ, i.e., one forms        ∂ 1 ∂ ∂ 1 (10.449) + u · ∇h πqg = −ρψ − ∇ ·H + ∇ ·G − ρψ ∂t ∂x ρ ∂y ρ One then re-expresses the appearing products by divergences and spatial derivatives, typically vanishing if integrated over the total domain, by time derivatives contributing to the time derivative of the energy density of the synoptic-scale flow, and a term expressing exchange with the gravity waves. To begin with, one has from multiplying the time derivative on the left-hand side of (10.426), using the definition (10.427) of the quasigeostrophic potential vorticity,    ∂πqg ∂ 1 ∂ f 2 ∂ψ ∇h2 ψ + ρ 2 = − ρψ −ρψ ∂t ∂t ρ ∂z N ∂z     ∂ ∂∇h ψ f 2 ∂ 2ψ ∂ Es − (10.450) ρψ 2 = − ∇h · ρψ ∂t ∂t ∂z N ∂t∂z with



 ρ f 2 ∂ψ 2 2 |∇h ψ| + 2 Es = 2 N ∂z

(10.451)

484

10 Gravity Waves and Their Impact on the Atmospheric Flow

the energy density of the synoptic-scale flow. Multiplying the advection of quasigeostrophic potential vorticity yields, exploiting the non-divergence of the synoptic-scale flow, and the geostrophy (10.430) of the synoptic-scale horizontal wind,     −ρψu · ∇h πqg = − ρψ∇h · uπqg = −∇h · ρψuπqg + ρ∇h ψ · uπqg   (10.452) = − ∇h · ρψuπqg The product on the right-hand side of (10.449) is, again using the geostrophy (10.430),      ∂ 1 ∂ 1 − ρψ − ∇ ·H + ∇ ·G ∂x ρ ∂y ρ ∂ ∂ = (ψ∇ · H) − (ψ∇ · G ) − v∇ · H − u∇ · G ∂x ∂y ∂ ∂ = (ψ∇ · H) − (ψ∇ · G ) ∂x ∂y      − ∇ · u · ex G + e y H + ex G + e y H · ·∇u (10.453) Inserting (10.450), (10.452), and (10.453), into (10.449) yields the desired prognostic equation       ∂ ∂ Es ∂∇h ψ f 2 ∂ 2ψ − − ∇h · ρψuπqg ρψ 2 − ∇h · ρψ ∂t ∂t ∂z N ∂t∂z    ∂ ∂ = (ψ∇ · H) − (ψ∇ · G ) − ∇ · u · ex G + e y H ∂x ∂y   + ex G + e y H · ·∇u (10.454) for the energy density of the synoptic-scale flow. Combining this with the wave-energy equation finally yields      ∂ ∂ ∂∇h ψ f 2 ∂ 2ψ − ρψ 2 + uπqg (E s + E w ) − ∇h · ρψ ∂t ∂t ∂z N ∂t∂z    = −∇ · Fw − ex ψ∇ · H + e y ψ∇ · G + u · ex G + e y H

(10.455)

As discussed in Sect. 6.2.1, the volume integral of the flux convergences on the left-hand side vanishes in the case of an f -channel, and the same result also holds if the solid-wall boundary conditions in meridional direction are replaced by periodic boundary conditions, or if the synoptic-scale flow vanishes at the meridional boundaries, e.g., at infinity. Likewise, if the meridional boundaries are far enough away so that there are no gravity-wave pseudomomentum fluxes, and if there are also no gravity waves in contact with the vertical boundaries, the volume integral of the right-hand side vanishes as well. Then the volume integral of the total energy density E s + E w is conserved.

10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow

485

Potential Vorticity As will become clearer below, for the derivation of a potential-vorticity conservation property we first need a prognostic equation for pseudomomentum  ph = d 3 k kh N (10.456) For this purpose we begin with the phase-space wave-action equation (10.439). Because it will be of relevance later on, we also supplement it with a source or sink term S on the right-hand side that takes wave dissipation into account, e.g., by wave breaking due to static instability. We then multiply the equation by kh and integrate the result over the wavenumber space. This gives us       ∂ph (10.457) + d 3 k ∇ · cg N kh + kh ∇k · k˙ N = sh ∂t  where sh = d 3 k kh S is the pseudomomentum source or sink. In the first part of the integral on the left-hand side we replace cg = u + cˆ g and use the non-divergence of the synoptic-scale horizontal wind. This leads to         d 3 k ∇ · cg N kh = ∇ · d 3 k cg N kh = ∇ · u d 3 k N kh + ∇ · d 3 k cˆ g N kh   (10.458) = u · ∇ph + ∇ · G ex + He y By partial integration, assuming the wave-action density to vanish at infinite wavenumbers, the second part of the integral in (10.457) is     3 ˙ d k kh ∇k · kN = − d 3 k N k˙ · ∇k kh (10.459) With

 ∂ ∂ ∂  + e y + ez ) kex + le y = ex ex + e y e y ∂k ∂l ∂m and the wavenumber equation (10.319) this becomes       d 3 k kh ∇k · k˙ N = − d 3 k N k˙ · ex ex + e y e y     ∂u ∂u = d 3 k N ex · kh + e y · kh = d 3 k ∇h u · kh N ∂x ∂y ∇k kh = (ex

(10.460)

(10.461)

Using this together with (10.458) in (10.457) finally yields the desired pseudomomentum equation 

   ∂ + u · ∇ ph = −∇ · G ex + He y − ∇h u · ph + sh ∂t

(10.462)

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10 Gravity Waves and Their Impact on the Atmospheric Flow

In the next step we take the vertical curl of this equation. This gives   ∂ + u · ∇ (ez · ∇ × ph ) ∂t ∂ ∂ =− ∇ ·H+ ∇ · G + e z · ∇ × sh ∂x ∂y     ∂ ∂u ∂u ∂u ∂ ∂u − · ∇ phy + · ∇ phx − · ph + · ph ∂x ∂y ∂x ∂y ∂y ∂x ∂ ∂ (10.463) ∇ · G + e z · ∇ × sh =− ∇ ·H+ ∂x ∂y because the last four terms before the final result cancel each other due to ∇h · u = 0. Comparing the right-hand side of this equation finally with the one of the quasigeostrophic potential-vorticity equation (10.426) we see that one can formulate the prognostic equation 

 ∂ sh + u · ∇  = −ez · ∇ × ∂t ρ

 = πqg − ez · ∇ ×

ph ρ

(10.464)

for an extension  of quasigeostrophic potential vorticity that contains contributions from the synoptic-scale flow, that are linear in the synoptic-scale streamfunction, and the negative of the gravity-wave pseudovorticity ez · ∇ × ph /ρ, that is nonlinear in the gravity-wave amplitudes. In the absence of wave dissipation  is conserved. Non-Acceleration with Gravity Waves It is worthwhile to also revisit the non-acceleration theorem for the zonal mean of the synoptic-scale flow in Sect. 8.5. Comparing the prognostic equation (10.426) for quasigeostrophic potential vorticity with its counterpart (8.14) from quasigeostrophic theory one sees that they agree as long as in the latter equation all fields are understood to be the phaseaveraged mean fields in the former, and if the source and sink term on the right-hand side is understood to be the vertical curl of the vector of pseudomomentum-flux convergences     ∂ 1 ∂ 1 (10.465) ∇ ·H + ∇ ·G D=− ∂x ρ ∂y ρ Then the elliptic equation (8.263) applies that tells us that the zonal-mean wind is steady essentially if • the Rossby-wave amplitudes are weak, • the Rossby-wave amplitudes are steady, and • there is no forcing due to gravity waves.

10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow

Due to (10.463) the gravity-wave forcing is    ph sh ∂ − ez · ∇ × +u·∇ ez · ∇ × D= ∂t ρ ρ

487

(10.466)

where we have dropped the angle brackets indicating the phase average. Hence we can formulate the following supplement to the non-acceleration theorem: Gravity waves cannot influence the synoptic-scale wind if • their amplitudes and wavenumbers are steady, • their pseudovorticity does not vary horizontally, and • there are no sources or sinks active, e.g., due to wave breaking. Note that this is not formulated only in application to the zonal-mean synoptic-scale wind, because under the conditions above there is also no impact on zonal variations of the zonal wind. Of interest is that most present-day approaches to take in coarse-resolution models the effect of small-scale gravity waves into account neglect the effects of wave transience and horizontal variations in the gravity-wave amplitudes and wavenumbers. Hence they rely exclusively on wave breaking. We finally remark that the conditions above, while sufficient for no impact of gravity waves on the synoptic-scale wind, are not all necessary to guarantee a steady synoptic-scale flow. Consider for this steady waves (implying time-invariant pseudomomentum ph ) possibly subject to steady sources and sinks sh . The steady limit of the conservation equation (10.464) then yields, together with the definition (10.427) of quasigeostrophic potential vorticity and the streamfunction property (10.430) of the synoptic-scale wind, an equation   2    f ∂ψ ∂ψ ∂ 1 ∂ ph sh ∂ψ ∂ 2 ∇h ψ + − ez · ∇ × = −ez · ∇ × − ∂x ∂ y ∂x ∂ y ρ ∂z N 2 ∂z ρ ρ (10.467) that can be solved, with suitable boundary conditions, for a steady streamfunction ψ. Hence, Gravity waves do not preclude a steady synoptic-scale flow if • they are steady and • all their sources and sinks are steady as well.

488

10 Gravity Waves and Their Impact on the Atmospheric Flow

10.3.10 Summary The interaction between mesoscale gravity waves and the synoptic-scale flow is a typical multiscale problem that needs a bit more extensive a treatment, presented here for f -plane dynamics. • Revisiting synoptic scaling in quasigeostrophic theory one can show that the respective horizontal and vertical length scales, the time scale and also the scale of typical wind, entropy, and pressure fluctuations can all be expressed in terms of the Rossby number, the sound speed, and the Coriolis parameter. The mesoscale length and time scales are chosen to be shorter than the respective synoptic scales by a Rossby number. The mesoscale entropy fluctuations are chosen to be below but close to the threshold of static instability. With these decisions, and using estimates from the polarization relations from an atmosphere at rest, that are rederived later for the more general concept, the mesoscale fields can also be characterized by Rossby number, sound speed, and Coriolis parameter. • The compressible equations of motion are then non-dimensionalized using the identified mesoscales. As a result only the Rossby number appears as a factor multiplying the various terms, at various powers. The presence of synoptic-scale fields is taken into account by the introduction of compressed coordinates, where all non-dimensional coordinates are multiplied by the Rossby number that also has the role of a scale-separation parameter. A WKB ansatz is then introduced for the mesoscale fields, with amplitudes, frequency, and wavenumbers varying on the synoptic scales, while frequency and wavenumber themselves describe the chosen mesoscales. All fields, containing contributions from the reference atmosphere, with especially weakly varying potential temperature, from the synoptic-scale flow, and from the mesoscale wave field are then expanded in terms of the scale-separation or Rossby number. The ansatz assumes locally monochromatic fields. Higher harmonics arise from the dynamics as well that we do not consider here, however, because they can be shown not to contribute to the relevant orders of the scale separation parameter. A corresponding extension of the theory is to be found in Appendix I. • The field expansions are then inserted into the non-dimensional dynamical equations, and the resulting terms are sorted in terms of powers of the Rossby number and the exponential phase factor. Well-known properties of the phase-averaged mean flow are derived: It is to leading order horizontal and in geostrophic and hydrostatic equilibrium. For the mesoscale wave fields one obtains within this nonlinear theory a set of linear equations that lead to the same gravity-wave dispersion relation and polarization relations that we had already found for an atmosphere at rest, just that the frequency there is now replaced by the intrinsic frequency. As usual in WKB theory, the definition of frequency and wavenumber as time derivative and gradient of the wave phase, together with the dispersion relation, lead to a set of eikonal equations predicting the development of the frequency and wavenumber fields. A useful approach to these equations is a Lagrangian

10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow

•

•

•

•

489

perspective, where they are solved along so-called rays that are everywhere parallel to the instantaneous and local group velocity. In the next to leading order of the scale-separating Rossby number phase-averaged meanflow equations are found that are forced by gravity-wave flux convergences. The nextorder wave equations lead, via the Fredholm alternative, to a prognostic equation for the gravity-wave energy density. Both shear production and buoyancy production arise as sources or sinks of wave energy. Because the mean flow is in geostrophic and hydrostatic balance, these production terms and also the pressure or energy flux can all be formulated as fluxes of wave energy and wave pseudomomentum, where the group velocity turns out to be the transporting velocity. This leads in the end to the wave-action-conservation law. Here wave-action density is the ratio between wave-energy density and intrinsic frequency. Wave pseudomomentum is the product of wave-action density and horizontal wavenumber. Processing the mean-flow equations in a manner very analogous to what is done in quasigeostrophic theory, exploiting therein the geostrophic and hydrostatic balance of the synoptic-scale flow, one ends up with a prognostic equation for quasigeostrophic potential vorticity that is forced by the vertical curl of the vector of divergencies of the pseudomomentum flux. Of practical importance is that the locally monochromatic results derived can be generalized to the case of spectral gravity-wave distributions, provided the gravity-wave amplitudes are sufficiently weak. With this assumption one can superpose solutions from WKB theory, leading to a prognostic equation for the spectral wave-action density in phase space. A very useful result is that this phase-space wave-action density is conserved along gravity-wave rays, while this is not the case for wave-action density in position space. All gravity-wave fluxes and their relatives can also be written as integrals of the phase-space wave-action density multiplied by functions of wavenumber. The theory can be shown to respect energy conservation. Energy is exchanged between wave energy and the energy of the synoptic-scale flow but, in the absence of friction, heating, heat conduction, and wave dissipation, the sum of the two is conserved. Also a conservation law for a total potential vorticity can be formulated that is the sum of the quasigeostrophic potential vorticity and a negative wave pseudovorticity. One can also show that gravity waves are not able to influence the synoptic-scale flow if they are steady, their pseudovorticity does not vary horizontally, and there are no sources or sinks active, e.g., due to wave breaking. This supplements the non-acceleration theorem for Rossbywave fluctuations, but it also turns out that steady synoptic-scale flows are possible in the presence of gravity waves if those have steady amplitudes and wavenumbers, and also their sources and sinks are steady.

490

10.4

10 Gravity Waves and Their Impact on the Atmospheric Flow

Critical Levels and Reflecting Levels

The absolute magnitude of the gravity-wave intrinsic frequency lies between Coriolis frequency and Brunt–Väisälä frequency. When, in the course of propagation through a spatially and temporally varying synoptic-scale flow, the intrinsic frequency of a gravity-wave ray approaches one of these two limits, then a characteristic behavior sets in that is very much related to what happens with Rossby waves when they approach a critical line or a reflecting layer. In the case of gravity waves, critical levels are essential for understanding the gravitywave effect on the middle atmosphere. We here discuss both cases within WKB theory. As will become clear, this is a certain limitation but it is sufficient for understanding essential aspects.

10.4.1 Critical Levels At a critical level the intrinsic frequency of a ray approaches, e.g., due to variations of the synoptic-scale wind, the Coriolis frequency, ωˆ → ± f

(10.468)

From the dispersion relation (10.288) one sees that this implies m 2 → ∞. Because f 2  N 2 , many discussions of critical levels ignore the Coriolis effect. In this limit the intrinsic frequency vanishes at a critical level, ωˆ → 0 (10.469) If we define by

ω kh

(10.470)

u · kh kh

(10.471)

ch = a horizontal phase velocity and by u =

the synoptic-scale horizontal velocity projecting onto the horizontal wavenumber, then one can write   (10.472) ωˆ = ω − kh · u = ch − u  kh which tells us that a critical level is approached if u  → ch ∓

f kh

(10.473)

where the part due to f is often neglected. In the following analysis of the behavior of a ray close to a critical level we assume, for the sake of simplicity, that horizontal gradients and time dependence of the synoptic-scale flow can be ignored and that the stratification is locally constant, i.e.,

10.4

Critical Levels and Reflecting Levels

491

u = u(z)

(10.474)

N = const.

(10.475)

2

Then the eikonal equations (10.319) and (10.320) tell us that frequency and horizontal wavenumber do not vary along a ray, 0=

dk dl dω = = dt dt dt

(10.476)

so that also kh and ch do not vary. The vertical wavenumber, however, satisfies ∂u  dm ∂u = −kh · = −kh dt ∂z ∂z

(10.477)

Due to variations of u  , the intrinsic frequency ωˆ = ω − kh u  varies as well. Vertical wavenumber and intrinsic frequency are connected by the dispersion relation (10.288). Solving for the vertical wavenumber yields m 2 = kh2

N 2 − ωˆ 2 ωˆ 2 − f 2

(10.478)

Close to a critical level at z = z c we can expand the intrinsic frequency in z, obtaining    du   (z − z c ) + O (z − z c )2 ωˆ = ω − kh u  = ω − kh u  (z c ) − kh  dz z c    du   = ± f − kh (10.479) (z − z c ) + O (z − z c )2  dz zc

Hence ωˆ 2 − f 2 = ∓2 f kh

   du   (z − z c ) + O (z − z c )2  dz z c

(10.480)

which leads in (10.478) to m 2 = ∓kh

N2 − f 2 1  + ... 2 f du  /dz z z − z c

(10.481)

c

This shows how the vertical wavenumber diverges as the critical level is approached. Negative m 2 indicate unphysical cases. In those a ray would be at a location where ωˆ 2 − f 2 < 0, which is forbidden by the dispersion relation. Actually it is never possible that a ray moves from allowed locations where ωˆ 2 − f 2 > 0 to forbidden locations where ωˆ 2 − f 2 < 0 because it cannot cross critical levels. While approaching a critical level a ray will experience a decay in its group velocity so that it gets asymptotically trapped by the critical level. Taking the derivative of the dispersion

492

10 Gravity Waves and Their Impact on the Atmospheric Flow

relation (10.288) with respect to the vertical wavenumber, one finds that the vertical group velocity is ( f 2 − N 2 )kh2 m ( f 2 − N 2 )kh2 cgz = −→ 0 (10.482)  2 2 −→ ωˆ k + m 2 ωm ˆ 3 m 2 →∞ m 2 →∞ h

A ray approaching a critical level will have a vertical group velocity opposite in sign to z − z c , hence  cgz = −

  8 f kh  du  3 z − z c |z − z c |3/2 + . . . N 2 − f 2  dz z c |z − z c |

(10.483)

The slowing down of the vertical propagation of a ray has the consequence that, within WKB theory, it never reaches a critical level. To see this we assume the example of an upward propagating ray approaching a critical level from below, i.e., cgz > 0 and z < z c . Then one has    dz 8 f kh  du  3 3/2 α= (10.484) = cgz = α(z c − z) + . . . dt N 2 − f 2  dz z c whence dt =

1 dz + ... α (z c − z)3/2

or, by integration from a position z 0 taken at time t0 ,   1 2 1 + . . . −→ ∞ − t − t0 = z→z c α (z c − z)1/2 (z c − z 0 )1/2

(10.485)

(10.486)

so that the time it would take to reach the critical level at z c is infinitely long. Figure 10.11 illustrates this behavior. While a ray is approaching a critical level its wave-action density is diverging as well. For illustration, and in line with the assumptions on the synoptic-scale flow made here, we take the wave-action density to be steady and independent of horizontal location. Wave-action conservation (10.432) then implies, dropping the index α as we have already done with regard to the wavenumber, ∂ (10.487) 0= (cgz A) ∂z or

A∝

1 1 ∝ cgz |z − z c |3/2

(10.488)

Hence, also wave-energy density and amplitude diverge. Moreover, from the dimensional equivalent of the wave-energy-density dependence (10.380) on wavenumber and buoyancy amplitude,

10.4

Critical Levels and Reflecting Levels

493

z

z

zc zc x

ω

cgz = ∂ω > 0 ∂m

x

N cgz < 0

f

m –f

cgz < 0

cgz > 0

–N

z

z zc

zc x

x

Fig. 10.11 Illustration of the approach of a critical level by a gravity-wave ray. Shown are the dependence of intrinsic frequency on vertical wavenumber, and the trajectory of a ray in x − z-plane, assuming that k > 0 and l = 0. The sign of the vertical group velocity, i.e., the derivative of intrinsic frequency with respect to vertical wavenumber, is indicated as well. Depending on the sign of the intrinsic frequency and that of the vertical wavenumber, there are four different cases. In each of these the vertical wavenumber diverges. Hence the group velocity gets negligibly small so that the ray is trapped in the vicinity of the critical level.

E w = ωˆ A = ρ one obtains |mb| = N2



|b|2 ωˆ 2 (kh2 + m 2 ) 2N 2 N 2 kh2

2kh2 Am 2 ρ|ω|(k ˆ h2

∝

+ m 2 ) z→z c

1 |z − z c |3/4

(10.489)

(10.490)

As we have argued above, static instability can arise when |mθ  | > dθ /dz or, multiplying by g/θ , |mb| > N 2 (10.491)

494

10 Gravity Waves and Their Impact on the Atmospheric Flow

Whenever this happens, convection leads to the onset of turbulence and the gravity wave is reduced. As we see, this instability arises near a critical level. As a matter of fact the horizontal-wind shear diverges as well, resulting in additional shear instabilities. Two comments are in place here: First of all, close to a critical level WKB theory does not really hold! A basic assumption that we have made has been that the wavenumber varies only weakly over a wavelength. However, close to the critical level        λz ∂m  1 1  ∝ 1  ∂m  ∝ |z − z c |  = −→ ∞ (10.492)  m ∂z  m 2  ∂z  3/2 |z − z c | |z − z c |1/2 z→z c Hence very near to the critical level WKB theory breaks down. Nonetheless, a more detailed treatment shows that the reasoning above is still qualitatively correct. A small part of the gravity wave is, however, not absorbed at the critical level, but rather passes it within linear theory. Second, one should recall that we have always assumed here that the time dependence of the synoptic-scale flow can be neglected. Significant transience of the latter can change the results considerably so that critical levels are sometimes even removed.

10.4.2 Reflecting Levels Another type of level occurs where a ray approaches ωˆ → ±N

(10.493)

implying that its vertical wavenumber vanishes, i.e., m → 0. As will be shown below, it actually changes sign when a ray touches a reflecting level, and then also the resulting group velocity changes sign, so that rays are reflected at such a level. This situation can be caused by variations either in the synoptic-scale wind or in the stratification. For a discussion of the ray properties near a reflecting level we again assume timeindependent winds only depending on the vertical and constant stratification. Then (10.478) again describes the dependence of the vertical wavenumber. Near the position z = zr of the reflecting level we expand  du   (z − zr ) + . . . (10.494) ωˆ = ±N − kh dz zr whence m2 ≈ ±

 2N kh3 du   (z − zr ) −→ 0 z→zr N 2 − f 2 dz zr

(10.495)

In a manner analogous to the discussion of critical levels one can obtain from this for the vertical group velocity close to the reflecting level

10.4

Critical Levels and Reflecting Levels

cgz =

495

m ( f 2 − N 2 )kh2 m( f 2 − N 2 ) −→ 0  2 2 −→ ωˆ k + m 2 ωk ˆ h2 m 2 →0 m 2 →0 h

or

1

cgz ∝ |z − zr | 2

(10.496)

(10.497)

This implies that the reflecting level is reached in finite time. Consider, e.g., the case of a ray approaching the level from below. Then dz √ ∝ zr − z dt whence t − t0 ∝

√

zr − z 0 −

√

zr − z −→ z→zr

(10.498) √

zr − z 0 < ∞

(10.499)

Once the reflecting level is reached, however, the vertical wavenumber will keep on varying according to (10.477), pass through zero, and change sign. This is accompanied by a corresponding change in sign of the vertical group velocity so that the ray is reflected and begins moving away from the reflecting level. Figure 10.12 illustrates the four possible cases. Here as well, however, one should be careful: m → 0 contradicts WKB-theory , since this corresponds to infinite wavelengths. Qualitatively, however, the treatment above indicates the right behavior. A rigorous treatment would imply the use of Airy functions. An aspect that WKB theory completely misses is that waves can tunnel through forbidden layers where ωˆ 2 > N 2 .

10.4.3 Summary When the intrinsic frequency on a gravity-wave ray, subject to changes by variations of the synoptic-scale wind and the Brunt–Väisälä frequency, approaches one of the two bounding limits, i.e., the Coriolis or the Brunt–Väisälä frequency, characteristic behavior sets in that is of relevance for the impact of gravity waves on the atmospheric circulation. • The approach of the Coriolis frequency happens close to critical levels. Near these the vertical wavenumber diverges and the vertical group velocity gets negligibly small. In consequence rays are trapped by critical levels and the gravity-wave amplitude diverges. The latter causes wave dissipation, either by static or by shear instability. • The approach of the Brunt–Väisälä frequency causes the vertical wavenumber to pass though zero and the group velocity to change sign. Hence one speaks in this context of a reflecting level. • Strictly speaking WKB theory loses its validity close to these levels but it is still able to predict essential aspects correctly that are borne out by more exact linear treatments.

496

10 Gravity Waves and Their Impact on the Atmospheric Flow

z

z zc

zc x

ω

cgz = ∂ω > 0 ∂m

x

N f

cgz < 0 m

cgz < 0 z

–f

cgz > 0

–N

z

zc zc x

x

Fig. 10.12 Illustration of reflecting levels, assuming k > 0 and l = 0: Depending on the sign of the intrinsic frequency and that of the vertical wavenumber, there are four different cases. In all of them a ray will change the sign of its vertical wavenumber while passing ωˆ = ±N , thereby also change the sign of its vertical group velocity, and hence be reflected by the level.

10.5

The Middle-Atmosphere Gravity-Wave Impact

We are now ready for discussing the middle-atmosphere gravity-wave effects outlined in Sect. 10.1. For this purpose first the transformed Eulerian mean will be extended by the effect of gravity waves. It will become clear that the zonal pseudomomentum flux introduced above supplements the Eliassen–Palm flux. Then the effect of critical levels will be invoked to understand the observed mean circulation.

10.5 The Middle-Atmosphere Gravity-Wave Impact

497

10.5.1 Extension of the TEM by Gravity-Wave Effects Formulation of the Extended TEM Equations For an analysis of the impact of gravity waves on the interaction between zonal-mean winds and temperatures and the residual circulation, we have to go back to the momentum equation (10.355), and the buoyancy equation (10.347) supplemented by a heat source Q. In order to keep the notation simple, and in line with the discussions in Sect. 8.4 above for the interaction of a zonal-mean flow (indicated by angle brackets) with synoptic-scale fluctuations (indicated by primes), we drop the angle brackets indicating the gravity-wave phase average, indicate gravity-wave fluctuations by hats, and denote phase-averaged gravity-wave fluxes by an overbar, i.e., we write them5 

   ∂ 1 + u · ∇ u + f ez × u = − c p θ∇h π − ∇ · ρ vˆ uˆ ∂t ρ   ∂ + u · ∇ b + N 2 w = − ∇ · uˆ bˆ + Q ∂t

(10.500) (10.501)

Horizontal wind and buoyancy are connected by the thermal-wind equation (10.373) that we now write ∂u (10.502) ∇h b = − f e z × ∂z Finally the leading-order non-divergence (10.329) of the horizontal wind and the next-order anelastic divergence condition (10.332) can be combined to the dimensional formulation ∇ · (ρv) = 0

(10.503)

By the same arguments as in Sect. 8.4.1 their zonal means, indicated by angle brackets, yield to leading order, keeping all gravity-wave fluxes, ∂ 3 4 1 ∂  3 4 ∂u ∂ 0  1 uˆ vˆ − ρ uˆ wˆ − f v = − uv − ∂t ∂y ∂y ρ ∂z ∂b ∂ 3 ˆ4 ∂ 0  1 vˆ b + Q + N 2 w = − vb − ∂t ∂y ∂y ∂b ∂u =− f ∂y ∂z ∂ ∂ 0= (ρ v) + (ρ w) ∂y ∂z

(10.504) (10.505) (10.506) (10.507)

On the right-hand side of the zonal-mean zonal-momentum and buoyancy equation we see first the fluxes due to synoptic-scale Rossby waves (indicated by primes) and then the gravity-wave fluxes (indicated by hats). 5 ρ and θ still denote density and potential temperature of the reference atmosphere.

498

10 Gravity Waves and Their Impact on the Atmospheric Flow

We then recall from Sect. 8.4.3 that the mass-weighted mean meridional wind agrees with the residual-mean meridional wind from the transformed Eulerian mean. All that had been assumed there was that the wave amplitudes are sufficiently weak, and that the leadingorder pressure and potential-temperature fluctuations are negligibly small in comparison with the reference-atmosphere mean. This holds with gravity waves as well. Then, because the residual mean circulation must satisfy the anelastic divergence constraint as well, it is given by   ρ 0   1 3 ˆ 4 1 ∂ v + b v ˆ b (10.508) v∗ =v − ρ ∂z N 2 1 ∂ 0   1 3 ˆ 4 v b + vˆ b w∗ =w + 2 (10.509) N ∂y i.e., the Rossby-wave Stokes drift is supplemented by a, typically weak, gravity-wave Stokes drift so that the anelastic divergence constraint   ∇ · ρv∗ = 0

(10.510)

is satisfied. Finally we replace in the zonal-mean zonal-momentum and buoyancy equation, respectively, the Eulerian mean circulation by its relation with the transformed Eulerian mean, yielding   1 ∂u f 0 ∂ 0  1 1 ∂ ρ 2 v  b − f v∗ = − uv + ∂t ∂y ρ ∂z N  4 3 4 3 ∂ 1 ∂ f 3 4 − uˆ vˆ + ρ 2 vˆ bˆ − ρ uˆ wˆ (10.511) ∂y ρ ∂z N ∂b (10.512) + N 2 w∗ = Q ∂t We now see, by comparison with the definition (8.113) of the Rossby-wave Eliassen–Palm flux F that the two Rossby-wave terms on the right-hand side of the zonal-mean zonalmomentum equation can be subsumed under   1 f 0  1 ∂ 0  1 1 ∂ ρ 2 v b = ∇ ·F (10.513) uv + − ∂y ρ ∂z N ρ Moreover, comparing with the relationships (10.399) and (10.404) between gravity-wave fluxes and pseudomomentum-fluxes one also sees that, for each locally monochromatic subcomponent of the gravity-wave field and hence also for the total gravity-wave spectrum, ρ uˆ vˆ = G y f ˆ ρ uˆ wˆ − ρ 2 vˆ b = Gz N

(10.514) (10.515)

10.5 The Middle-Atmosphere Gravity-Wave Impact

and therefore −

 3 4 f 3 4 1 ∂ 3 4 1 ∂ uˆ vˆ + ρ 2 vˆ bˆ − ρ uˆ wˆ = − ∇ · G  ∂y ρ ∂z N ρ

499

(10.516)

so that the zonal-mean zonal-momentum equation can be written 1 ∂u − f v∗ = ∇ · (F − G ) ∂t ρ

(10.517)

One sees that in the presence of gravity waves the Rossby-wave Eliassen–Palm flux is supplemented by the negative zonal-mean gravity-wave zonal-pseudomomentum flux. Therefore the total is termed Eliassen–Palm flux as well. In summary, the TEM equations with gravity-wave impact are ∂u 1 − f v∗ = ∇ · (F − G ) ∂t ρ ∂b + N 2 w∗ = Q ∂t ∂b ∂u =− f ∂y ∂z   ∗ ∇ · ρv = 0

(10.518) (10.519) (10.520) (10.521)

Basically all that has been said about this system in Sects. 8.4 and 8.5 applies here as well. Non-Acceleration Revisited Because the gravity-wave effect in the TEM, besides the weak gravity-wave Stokes drift, is expressed by the pseudomomentum-flux divergence ∇ · G , it is worthwhile asking under which conditions it will not contribute. This can be seen by taking the zonal component of the pseudomomentum equation (10.462), exploiting the leading-order non-divergence of the horizontal wind, ∂u ∂ phx + ∇ · (u phx ) = −∇ · G − · ph + shx ∂t ∂x

(10.522)

The zonal mean of this yields ∇ · G  = −

6 5  ∂u ∂ 0   1 ∂  phx  − v phx − · ph + shx  ∂t ∂y ∂x

(10.523)

Herein one has, by partial integration and again by the non-divergence of the horizontal wind,

500

10 Gravity Waves and Their Impact on the Atmospheric Flow

5

8 6 5  6 5  6 5  6 7  ∂ p ∂u  ∂u  ∂v  ∂v  hy + =− − v · ph = p p p ∂x ∂ x hx ∂ x hy ∂ y hx ∂x 8 7    ∂ phy ∂ phx ∂ 0   1 =− v phx + v  − ∂y ∂y ∂x

(10.524)

so that (10.523) becomes ∇ · G  = −

1 ∂  phx  0  + v ez · ∇ × ph + shx  ∂t

(10.525)

Here the second term on the right-hand side represents the meridional flux, by synopticscale winds, of synoptic-scale variations of gravity-wave pseudovorticity ez · ∇ × ph /ρ multiplied by the reference-atmosphere density. Hence the gravity-wave contribution to the Eliassen–Palm-flux divergence vanishes if • the gravity waves are steady, • there is no meridional flux (by synoptic-scale wind fluctuations) of synoptic-scale fluctuations of gravity-wave pseudovorticity, as would be the case, e.g., if there were no synoptic-scale zonal fluctuations either in the synoptic-scale flow or in the gravity-wave amplitudes, and • there is no gravity-wave source or sink.

10.5.2 The Gravity-Wave Effect on the Residual Circulation and on the Zonal-Mean Flow The gravity-wave effects described in Sect. 10.1 can now be understood as follows. We consider the solsticial climatological mean of the TEM equations (10.518)–(10.521), i.e., all time derivatives vanish6 and the zonal means are to be understood as solsticial climatological means as well. Moreover, we choose as heat source a relaxation Q = −(b − br )/τ toward a zonally symmetric radiative-equilibrium buoyancy distribution br (y, z), with a characteristic time scale τ . Hence we have 1 − f v∗ = ∇ · (F − G ) ρ b − br 2 ∗ N w = − τ ∂u ∂b =− f ∂y ∂z   ∗ ∇ · ρv = 0

(10.526) (10.527) (10.528) (10.529)

6 This would not be the case under equinoctial conditions, where one has a climatological-mean time

derivative describing the transition from summer to winter and vice versa.

10.5 The Middle-Atmosphere Gravity-Wave Impact

501

A notion of what the gravity-wave part of the Eliassen–Palm flux looks like can be obtained by considering the effect of critical levels. For this we neglect the effect of Rossbywave winds, that are typically weak in the upper stratosphere and in the mesosphere, and focus on the zonal-mean zonal wind, i.e., we assume that u = ex u. Under these conditions a gravity wave experiences a critical level when its intrinsic frequency approaches the inertial frequency ωˆ → ± f , i.e., with ωˆ = ω − kh · u = ω − ku = (c − u) k

(10.530)

where c = ω/k is a zonal phase velocity, a critical level is approached as u → c ∓

f ≈c k

(10.531)

neglecting f in the last step. Hence, a gravity wave emitted by some process in the troposphere will not be able to pass a critical level where the zonal-mean wind agrees with its phase velocity. This has consequences for the spectrum of transmitted waves reaching the mesosphere. The typical situation is illustrated in Fig. 10.13. The climatological zonal-mean winds sketched there are representing the midlatitude situation visible in Fig. 10.2. Essentially one has westerlies in the winter middle atmosphere and easterlies in summer. The spectrum of gravity waves emitted from the troposphere is quite broad. In coarse approximation the range of phase velocities emitted agrees with the range of synoptic-scale zonal winds in the troposphere. The critical-level filtering process, however, only allows gravity waves to pass into the middle atmosphere that have positive intrinsic phase velocities on the summer side and negative intrinsic phase velocities on the winter side, i.e.,

z

z

u

u

c

c 0

a summer

u ,c

0

u ,c

b winter

Fig. 10.13 Effect of critical-level filtering of gravity waves emitted from the troposphere on the gravity-wave spectrum reaching the mesosphere. In summer (a) the transmitted waves have negative intrinsic phase velocities ω/k ˆ = c − u, while in winter (b) the transmitted waves carry negative intrinsic phase velocities.

502

10 Gravity Waves and Their Impact on the Atmospheric Flow

& > 0 in summer ωˆ = c − u k < 0 in winter. Because the zonal pseudomomentum carried by each spectral component is d phx = d 3 k N k = k d A =

k d Ew ωˆ

(10.532)

the sign of the transmitted zonal pseudomomentum agrees with the sign of the transmitted intrinsic phase velocities, &  > 0 in summer phx = d phx < 0 in winter. Eventually, most typically in the upper mesosphere, wave amplitudes will be too large for the waves to be stable anymore, and they become unstable, e.g., by static instability. The hence ensuing wave breaking reduces the wave amplitude, representing a sink for the transported pseudomomentum. Therefore, in the wave-breaking zone in the upper mesosphere the convergence of the flux of zonal pseudomomentum is & > 0 in summer −∇ · G  < 0 in winter. The consequences for the solsticial climatology of the mesosphere can be deduced from this finding in a straightforward manner. The residual-mean meridional wind is obtained from the mean zonal-momentum equation. With good justification we can neglect there to leading-order the Rossby-wave effect, but we must heed that the Coriolis parameter differs in sign between the southern hemisphere, where it is negative, and the northern hemisphere, where it is positive. Hence one finds that the residual-mean meridional wind & > 0 during northern-hemispheric winter 1 ∗ v = ∇ · G  is on both hemispheres f < 0 during northern-hemispheric summer, i.e., it is always directed from the summer pole to the winter pole. Mass conservation then implies that the residual-mean vertical wind & ∗ > 0 over the summer pole w < 0 over the winter pole, as illustrated in Fig. 10.14. Hence air masses are subject to adiabatic cooling (heating) over the summer (winter) pole. From the climatological-mean buoyancy equation (10.527) follows correspondingly that the resulting buoyancy is

10.5 The Middle-Atmosphere Gravity-Wave Impact

503

km 70 60 50 40 30 20

60°S

30°S

0°

30°N

60°N

latitude Fig. 10.14 Middle-atmosphere residual circulation during northern-hemispheric winter. Due to gravity-wave forcing it is directed in the mesosphere from the summer to the winter pole. Mass conservation implies that air masses rise over the summer pole and sink over the winter pole. Reprinted from Dunkerton (1978).

∗

b = br − τ N w 2

& < br over the summer pole > br over the winter pole.

This explains the cold summer mesopause and the warm winter stratopause in Fig. 10.2, implying that buoyancy (or potential temperature) increases from the summer pole to the winter pole. Hence, by the thermal-wind relation (10.528) one has & 1 ∂b > 0 at the summer mesopause ∂u =− ∂z f ∂ y < 0 at the winter mesopause, which is in agreement with the zonal-mean zonal-wind reversal visible in the upper mesosphere in Fig. 10.2, a feature that could not be explained within radiative equilibrium, as one sees in Fig. 10.1.

504

10 Gravity Waves and Their Impact on the Atmospheric Flow

10.5.3 Summary With the theoretical preparations above the impact of gravity waves on the middle atmosphere can be understood. • Extending the TEM description of the interaction between zonal-mean climatology and Rossby waves for the gravity effect shows that, after the incorporation of the gravitywave Stokes drift, the gravity-wave impact is captured by supplementing the Rossbywave Eliassen–Palm flux by the negative zonal mean of the flux of zonal gravity-wave pseudomomentum. One can show that its divergence does not affect the mean flow when the gravity waves are steady, there is no meridional flux (due to synoptic-scale wind fluctuations) of synoptic-scale fluctuations of gravity-wave pseudovorticity, and there is no gravity-wave source or sink, e.g., by wave breaking. • Gravity-wave breaking turns out to be very important in the mesosphere. This together with the effect of filtering by critical levels is decisive for an explanation of the cold summer mesopause, the warm winter stratopause, and the zonal-wind reversal in the upper mesosphere.

10.6

References and Recommendations for Further Reading

Between the fully compressible equations and the Boussinesq approximation there are two approximative equation sets that are useful for the analysis of gravity-wave dynamics. In contrast to the compressible equations they do not have sound waves as fundamental wave modes, but describe gravity waves at good accuracy, while also allowing for the density effect of the gravity-wave amplitudes. These are the anelastic equations (Lipps 1990, Lipps and Hemler 1982, Ogura and Phillips 1962) and the pseudo-incompressible equations (Durran 1989). Further discussions on this topic can be found in Durran and Arakawa (2007), Klein et al. (2010), and Achatz et al. (2010). Atmospheric and oceanic waves are discussed in depth by Lighthill (1978), and Sutherland (2010) gives a very broad account of many aspects of gravity-wave dynamics, perhaps leaning a bit more to the ocean than to the atmosphere. Both books also discuss the limits of WKB theory. The German book by Pichler (1997) is a very good reference on various aspects of atmospheric dynamics, among these also waves, and the same definitely also holds for the extremely broad and very insightful book by Vallis (2006). Nappo (2002) gives an introduction into atmospheric gravity waves, especially the wave generation by flow over mountains. The WKB theory used here goes back to Bretherton (1966) and Grimshaw (1975b) and has been extended by Achatz et al. (2010, 2017) to include stratospheric conditions, higher harmonics, and the geostrophic mode. The concept of wave action in phase space is described by Dewar (1970), Dubrulle and Nazarenko (1997), Hertzog et al. (2002) and Muraschko

10.6

References and Recommendations for Further Reading

505

et al. (2015). WKB on the sphere needs special care that has been discussed by Hasha et al. (2008) and Ribstein et al. (2015). Booker and Bretherton (1967) discuss the linear theory of gravity-wave propagation close to a critical level, which has later on been extended to include rotation by Jones (1967) and Grimshaw (1975a). The effect of time-dependent mean flows on critical levels is discussed by Broutman and Young (1986) and Senf and Achatz (2011). Nonlinear interactions between gravity waves are discussed by Müller (1976) and references therein, and the book by Nazarenko (2011) gives an excellent introduction into the corresponding theory. Eden et al. (2019) discuss the interaction between gravity waves and geostrophic modes. None of these theories fully includes a leading-order mean flow. A very important body of theory on the interactions between waves and mean flows is the generalized Lagrangian-mean theory that has been introduced by Andrews and McIntyre (1978a, b), and Bühler (2009) is the best reference on this theory and on the corresponding developments following from these seminal papers. Lindzen (1981) is a seminal publication on the impact of breaking gravity waves on the middle atmosphere, and the book by Lindzen (1990) is a worthwhile reference in this regard as well. Useful review papers on atmospheric gravity waves and their parameterization in global models are Alexander et al. (2010), Fritts and Alexander (2003), Kim et al. (2003), and Sutherland et al. (2019).

11

Appendices

11.1

Appendix A: Useful Elements of Vector Analysis

Vector analysis is an important tool of fluid mechanics. Here we summarize some of its important elements.

11.1.1 The Gradient The gradient ∇ of a scalar field  is a vector field. It is defined most generally in terms of the variation of  along an arbitrary spatial path, via  x2  (x2 ) −  (x1 ) = dx · ∇ (11.1) x1

In Cartesian coordinates one obtains ∇ =

∂ ∂ ∂ ex + ey + ez ∂x ∂y ∂z

(11.2)

11.1.2 The Divergence and the Integral Theorem from Gauss The divergence ∇ · b of an arbitrary vector field b is a scalar field which is defined in most general terms via the integral theorem from Gauss:   b · dS = dV ∇ · b (11.3) S

V

© Springer-Verlag GmbH Germany, part of Springer Nature 2022 U. Achatz, Atmospheric Dynamics, https://doi.org/10.1007/978-3-662-63941-2_11

507

508

11 Appendices

The integral of the divergence over a volume is identical with the integral of the flux of the corresponding vector field through the volume surface, i.e., the surface integral of the projection of the vector field onto the outwardly directed surface normal. In Cartesian coordinates the divergence is ∂b y ∂bx ∂bz ∇ ·b= + + (11.4) ∂x ∂y ∂z

11.1.3 The Curl and the Integral Theorem from Stokes The curl ∇ × b of an arbitrary vector field b is again a vector field, defined in most general terms via the integral theorem from Stokes:   dS · ∇ × b = b · dx (11.5) A surface integral of the curl is identical with the path integral of the corresponding vector field, in projection, around the surface boundary. Seen from above, i.e., from the direction into which the surface normal points, the direction of integration in this is anti-clockwise. In Cartesian coordinates the curl is       ∂b y ∂b y ∂bx ∂bz ∂bx ∂bz ex + ey + ez (11.6) − − − ∇ ×b = ∂y ∂z ∂z ∂x ∂x ∂y

11.1.4 Some Identities For arbitrary scalar fields  and  and vector fields a and b, one has the following identities: 1  = − 2 (∇ − ∇)   ∇ · (b) = ∇ · b + ∇ · b ∇

(11.7) (11.8)

∇ · (∇ × b) = 0

(11.9)

∇ × ∇ = 0

(11.10)

∇ × (b) = ∇ × b + ∇ × b ∇ × (a × b) = a∇ · b + (b · ∇)a − b∇ · a − (a · ∇)b b·b − b × (∇ × b) (b · ∇) b = ∇ 2

(11.11) (11.12) (11.13)

11.2

Appendix B: Rotations

509

11.1.5 Recommendations for Further Reading Here and for all other mathematical appendices, textbooks on mathematical methods for physicists are a good support, e.g., Arfken et al. (2012).

11.2

Appendix B: Rotations

Rotations of coordinate systems are described mathematically via linear maps. Consider, e.g., the rotation of a coordinate system with basis vectors e1 , e2 , e3 about the e3 -axis (Fig. 11.1). A vector x shall be represented in both coordinate systems. It is x = x1 e1 + x2 e2 + x3 e3 = x1 e1 + x2 e2 + x3 e3

(11.14)

The mathematical representation of the rotation yields (x1 , x2 , x3 ) in terms of (x1 , x2 , x3 ). As visible in Fig. 11.2, one has e1 = cos α e1 − sin α e2 e2 = sin α e1 + cos α e2

(11.15)

e3

Fig. 11.1 Rotation of a coordinate system about the e3 -axis.

x e'2

α α

e1 e'1

e2

510

11 Appendices

e2 e'2 e'1

α

2

–α α

e1

Fig. 11.2 Projection of the original basis vectors e1 and e2 onto their rotated counterparts.

Inserting this into (11.14) yields, after comparison of the coefficients to e1 and e2 , and trivially for x3 , x1 = cos α x1 + sin α x2 x2 = − sin α x1 + cos α x2 x3 = x3 or

with the rotational matrix

(11.16)

⎛

⎛ ⎞ ⎞ x1 x1 ⎝ x  ⎠ = Rz (α) ⎝ x2 ⎠ 2 x3 x3

(11.17)

⎛

⎞ cos α sin α 0 Rz (α) = ⎝ − sin α cos α 0 ⎠ 0 0 1

(11.18)

In analogy one obtains matrices for rotations by an angle β about the e2 -axis or an angle γ about the e1 -axis: ⎛ ⎞ cos β 0 − sin β (11.19) R y (β) = ⎝ 0 1 0 ⎠ sin β 0 cos β ⎛ ⎞ 1 0 0 Rx (γ ) = ⎝ 0 cos γ sin γ ⎠ (11.20) 0 − sin γ cos γ Each arbitrary spatial rotation can be represented as a sequence of such elementary rotations. For this one can proceed as follows:

11.2

Appendix B: Rotations

511

Fig. 11.3 Illustration of the first two of three elementary rotations constituting together a general rotation.

e3

e'3

e 2'

e'1

α

β

e2

e1 1. First one brings e1 into the desired position. Let α be the angle between the projection of e1 onto the plane spanned by e1 and e2 on the one hand and e1 on the other. Let β be the angle between e1 and this plane (Fig. 11.3). a) First one performs a rotation by the angle α about the e3 -axis, yielding the rotated coordinates xi = Rz (α)i j x j . b) Then one rotates the coordinate system by β about the e2 -axis, thus obtaining the transformed coordinates xi = R y (β)i j x j = R y (β)i j Rz (α) jk xk 2. Finally e2 and e3 are rotated by an angle γ about the e1 -axis in such a way that they become identical with the desired e2 and e3 . One obtains xi = Rx (γ )i j x  j or x = Rx

(11.21)

R = Rx (γ )R y (β)Rz (α)

(11.22)

where An important property of rotational matrices is their orthogonality, i.e. R −1 = R t

(11.23)

512

11 Appendices

For the elementary rotations one can check this directly. For example, one has Rz (α)−1 = Rz (−α) ⎛ ⎞ cos α − sin α 0 = ⎝ sin α cos α 0 ⎠ 0 0 1 = Rz (α)t

(11.24)

The general orthogonality property is thus obtained from (11.22). The orthogonality is a direct expression of the fact that rotations do not change the norm of a vector. One also has for infinitesimally small rotations R = I + δR (11.25) where the infinitesimally small matrix satisfies δ R t = −δ R

(11.26)

For this one notes that I = R −1 R = R t R = I + δ R + δ R t + δ R t δ R −−−→ I + δ R + δ R t δ R→0

(11.27)

leading to (11.26).

11.2.1 Recommendations for Further Reading Here and for all other mathematical appendices, textbooks on mathematical methods for physicists are a good support, e.g., Arfken et al. (2012).

11.3

Appendix C: Isotropic Tensors

An nth-rank tensor T is isotropic if it is invariant under rotations, i.e., Ril R jm · · · Rkn Tlm...n = Ti j...k

(11.28)

where R is a rotational matrix (Appendix C). In the following we will develop the most general form of isotropic tensors of rank up to four.

11.3

Appendix C: Isotropic Tensors

513

11.3.1 Isotropic Tensors of Rank One There is no isotropic tensor of rank one. In order to see this consider an infinitesimally small rotation R = I + δ R, where δ R is antisymmetric. An isotropic tensor T must satisfy Ti = (δi j + δ Ri j )T j = Ti + δ Ri j T j

(11.29)

δ Ri j T j = 0

(11.30)

which is only possible if for all possible antisymmetric δ Ri j . Thus δ R11 T1 + δ R12 T2 + δ R13 T3 = 0 δ R21 T1 + δ R22 T2 + δ R23 T3 = 0 δ R31 T1 + δ R32 T2 + δ R33 T3 = 0

(11.31)

Due to the antisymmetry of δ R one has δ R11 = δ R22 = δ R33 = 0, while δ R21 = −δ R12 , δ R32 = −δ R23 , and δ R13 = −δ R31 . Thus one obtains T1 = T2 = T3 = 0

(11.32)

so that there is no isotropic tensor of rank one, with the exception of the null tensor.

11.3.2 Isotropic Tensors of Rank Two If T is an isotropic tensor of rank two, one has for all i and k, to lowest order in the infinitesimally small δ Ri j Tik = (δi j + δ Ri j )(δkl + δ Rkl )T jl = Tik + δ Ri j δkl T jl + δ Rkl δi j T jl = Tik + δ Ri j T jk + δ Rkl Til

(11.33)

so that δ Ri j T jk + δ Rk j Ti j = 0

(11.34)

For the case of unequal i and k we choose as example i = 1 and k = 2. Since δ R11 = δ R22 = 0 one has 0 = δ R12 T22 + δ R13 T32 + δ R21 T11 + δ R23 T13 = δ R12 (T22 − T11 ) + δ R13 T32 + δ R23 T13

(11.35)

514

11 Appendices

and thus T32 = T13 = 0; T11 = T22

(11.36)

In analogy one can show in general that Tik = 0 if i  = k, while T11 = T22 = T33 . With this one has (11.34) also for identical i and k. For example, let i = k = 1. Then one must have δ R12 T21 + δ R13 T31 + δ R12 T12 + δ R13 T13 = 0

(11.37)

This is indeed the case since all terms are null. Thus the only isotropic tensors of rank two are products of a scalar with δik .

11.3.3 Isotropic Tensors of Rank Three If T is an isotropic tensor of rank three, one has for all i, k, and m Tikm = (δi j + δ Ri j )(δkl + δ Rkl )(δmn + δ Rmn )T jln

(11.38)

δ Ri j T jkm + δ Rk j Ti jm + δ Rm j Tik j = 0

(11.39)

and thus Consider first i = k = 1: δ R12 T21m + δ R13 T31m + δ R12 T12m + δ R13 T13m + δ Rm1 T111 + δ Rm2 T112 + δ Rm3 T113 = 0 (11.40) Now choose m = 2. Thus one also has δ Rm2 = 0. Summarizing all terms to δ R12 , δ R13 , and δ R23 one obtains T212 + T122 = T111 (11.41) T312 + T132 = 0 T113 = 0 Via analogous results for the last of the three equations, with two identical indices each, one finds in total that Tikm = 0 if two indices are equal and the third is different. From the first of the equations then follows that Tiii = 0 for arbitrary i. The second of the three equations finally shows that (11.42) Tikm = −Tkim if all indices differ from each other. In analogy one finds via equating two other indices from i, k, m in (11.39) that Timk = −Tkmi

(11.43)

Tmik = −Tmki

(11.44)

In total this also leads to (11.40) being satisfied with m = 1.

11.3

Appendix C: Isotropic Tensors

515

If finally in (11.39) all i, k, m differ from each other, one has T jkm = 0 unless j = i, which however means that δ Ri j = 0 so that (11.39) is satisfied. From this follows that every isotropic tensor of rank three is a product of ikm with a scalar.

11.3.4 Isotropic Tensors of Rank Four For an isotropic tensor T of rank four one obtains in analogy to the above δ Ri j T jkmp + δ Rk j Ti jmp + δ Rm j Tik j p + δ R pj Tikm j = 0

(11.45)

There are only three possible values for i, k, m, p, so that two of them must be identical. In the following we consider the four different cases that (a) only two indices are equal, (b) three indices are equal and the last differs from them, (c) two indices are identical each, and (d) all indices are identical. Only Two Identical Indices Choose i = k = 1, m = 2, and p = 3. Then 0 = δ R12 T2123 + δ R13 T3123 + δ R12 T1223 + δ R13 T1323 + δ R21 T1113 + δ R23 T1133 + δ R31 T1121 + δ R32 T1122

(11.46)

Due to the antisymmetry of δ R this leads to T2123 + T1223 − T1113 = 0 T3123 + T1323 − T1121 = 0

(11.47)

T1133 − T1122 = 0

(11.48)

and Further results are obtained by interchanging two indices which are identical already, and by cyclic permutation of the axes (1 → 2, 2 → 3, 3 → 1). Thus (11.48) yields T1133 = T1122 = T2233 = T2211 = T3322 = T3311 ≡ λ

(11.49)

and also T1313 = T1212 = T2323 = T2121 = T3232 = T3131 ≡ μ + ν

(11.50)

T3113 = T2112 = T3223 = T1221 = T2332 = T1331 ≡ μ − ν

(11.51)

In this λ, μ, and ν are free parameters.

516

11 Appendices

Three Identical Indices Choose i = k = m = 1, and p = 2, so that 0 = δ R12 T2112 + δ R13 T3112 + δ R12 T1212 + δ R13 T1312 + δ R12 T1122 + δ R13 T1132 + δ R21 T1111 + δ R23 T1113

(11.52)

The last term leads to T1113 = 0

(11.53)

Via interchanging indices one finds this way that in general Tikmp = 0 if only three indices are identical (type a). Moreover the terms to δ R13 lead in (11.52) to T3112 + T1312 + T1132 = 0

(11.54)

Due to the result derived immediately above, the last term in (11.47) vanishes so that T2123 + T1223 = 0

(11.55)

By cyclic axis permutation one obtains from this T1312 + T3112 = 0

(11.56)

T1132 = 0

(11.57)

so that via (11.54) Thus also all tensor elements with only two identical indices vanish (type b). The terms to δ R12 in (11.52) yield together T1111 = T2112 + T1212 + T1122

(11.58)

so that tensor elements with four identical indices (type d) can be expressed via tensor elements with two pair-wise identical indices (type c). Thus the remaining nonvanishing coefficients are of types (c) and (d). Consideration of the corresponding transformation relations according to (11.45) does not yield any new results. For example, if i = k = 1 and m = p = 2, replacement of either i or k in the first two terms by j only leads to non-vanishing tensor elements if j = 1. Then, however, δ Ri j = 0 or δ Rk j = 0. In analogy also the two last terms in (11.45) vanish automatically so that the equation is satisfied. Similar considerations hold for i = k = m = p. Because of (11.49) – (11.51) one can rewrite (11.58) as T1111 = T2222 = T3333 = λ + 2μ

(11.59)

There are thus three independent isotropic tensors of rank four. These are obtained by setting one of the three parameters λ, μ, and ν to one and all others to zero. In the λ-tensor one

11.4

Appendix D: Spherical Coordinates

517

has Tikmp = 1 if i = k and m = p. All other components are zero. It is thus a multiple of the tensor with components δik δmp . As one sees from (11.49) and (11.50), the μ-tensor has Tikmp = 1 if i = m and k = p, or if i = p and k = m where always i  = k. If also i = k, one sees from (11.59) that the component is 2. All others are zero so that Tikmp = δim δkp + δi p δkm

(11.60)

In the ν-tensor one has Tikmp = 1 if i = m and k = p, and Tikmp = −1 if i = p and k = m. In all other cases one has Tikmp = 0. This also holds for the case i = k = m = p. Thus we obtain (11.61) Tikmp = δim δkp − δi p δkm The general isotropic tensor of rank four is thus Tikmp = λδi j δmp + μ(δim δkp + δi p δkm ) + ν(δim δkp − δi p δkm )

(11.62)

where λ, μ, ν are free parameters.

11.3.5 Recommendations for Further Reading Much more on tensors can be found, e.g., in Hess (2015).

11.4

Appendix D: Spherical Coordinates

11.4.1 The Local Basis Vectors The approximate sphericity of the earth suggests the use of an adapted coordinate system. Spherical coordinates, as illustrated in Fig. 1.12, fulfill this purpose. The geographic longitude λ is in the range 0 ≤ λ ≤ 2π . The geographic latitude is φ with − π2 ≤ φ ≤ π2 . These are supplemented by the radial distance r from the center of earth. The position vector in Cartesian coordinates is obtained from the spherical coordinates via (1.93). We define unit vectors pointing into the direction of change of location under variation of one of the coordinates:



∂x

1 ∂x with h λ =



eλ = h λ ∂λ

∂λ

∂x

1 ∂x eφ = with h φ =



(11.63) h φ ∂φ

∂φ

∂x

1 ∂x er = with h r =



h r ∂r ∂r

518

Inserting (1.93) yields, e.g.,

11 Appendices

⎛ ⎞ −r cos φ sin λ ∂x ⎝ = r cos φ cos λ ⎠ ∂λ 0

(11.64)

h λ = r cos φ

(11.65)

and thus the metric factor so that

⎛

⎞ − sin λ eλ = ⎝ cos λ ⎠ 0

(11.66)

After analogous calculations also for the two other coordinates one obtains in summary ⎛

⎞ − sin λ eλ = ⎝ cos λ ⎠ h λ = r cos φ 0 ⎛ ⎞ − sin φ cos λ eφ = ⎝ − sin φ sin λ ⎠ hφ = r cos φ ⎛ ⎞ cos φ cos λ er = ⎝ cos φ sin λ ⎠ hr = 1 sin φ

(11.67)

(11.68)

(11.69)

In contrast to their Cartesian counterparts the unit vectors depend on location. Nonetheless they are orthogonal, i.e., eλ · eφ = eλ · er = eφ · er = 0 (11.70)

11.4.2 The Gradient in Spherical Coordinates For a reformulation of the equations for atmospheric dynamics in spherical coordinates one must rewrite the various differential operators for the new coordinate system with its local unit vectors. Here this shall first be done for the gradient. Desired are, for an arbitrary spatially dependent function , factors gλ , gφ , and gr so that ∇ = gλ eλ + gφ eφ + gr er For this we use that in general for arbitrary changes of location   ∂x ∂x ∂x d = ∇ · dx = ∇ · dλ + dφ + dr ∂λ ∂φ ∂r = ∇ · (h λ eλ dλ + h φ eφ dφ + h r er dr )

(11.71)

(11.72)

11.4

Appendix D: Spherical Coordinates

519

Setting now dφ = dr = 0 and using (11.71) we obtain d = ∇ · eλ h λ dλ = gλ h λ dλ so that gλ =

1 ∂ h λ ∂λ

(11.73)

(11.74)

Analogous calculations also for φ and r thus yield ∇ =

1 ∂ 1 ∂ 1 ∂ eλ + eφ + er h λ ∂λ h φ ∂φ h r ∂r

(11.75)

so that, employing (11.67) – (11.69), ∇ =

1 ∂ 1 ∂ ∂ eλ + eφ + er r cos φ ∂λ r ∂φ ∂r

(11.76)

11.4.3 The Divergence in Spherical Coordinates For a determination of the divergence of an arbitrary vector field b in spherical coordinates we use the integral theorem from Gauss (11.3) in the limit of an infinitesimally small integration volume and thus obtain  1 b · dS (11.77) ∇ · b = lim V →0 V S We now consider a volume enclosed by isosurfaces, each corresponding to one of the spherical coordinates λ, φ, or r (Fig. 11.4). The length of the line segment AB is to leading order in the infinitesimal differentials dλ, dφ, and dr



∂x

AB = dλ

∂λ = h λ dλ (11.78) In analogy we obtain AD = h φ dφ

(11.79)

E A = h r dr

(11.80)

The total volume thus is V = (h λ dλ)(h φ dφ)(h r dr ) = h λ h φ h r dλdφdr

(11.81)

520

11 Appendices

Fig. 11.4 An infinitesimally small volume enclosed by isosurfaces with constant value of one of the spherical coordinates, respectively.

hr dr

A

h

dλ λ

B(λ + dλ, ϕ + dϕ, r + dr) C D(λ, ϕ, r + dr)

F G

E

H(λ, ϕ, r)

The normal vectors of the isosurfaces are either parallel or antiparallel to the corresponding unit vectors so that for b = bλ eλ + bφ eφ + br er , again to leading order in dλ, dφ, and dr  b · dS = br (λ, φ, r + dr )d S ABC D − br (λ, φ, r )d S E F G H S

+ bφ (λ, φ + dφ, r )d S E AB F − bφ (λ, φ, r )d S H DC G + bλ (λ + dλ, φ, r )d S BC G F − bλ (λ, φ, r )d S E AD H

(11.82)

Here one has, e.g., br (λ, φ, r + dr )d S ABC D − br (λ, φ, r )d S E F G H  = br (λ, φ, r + dr ) h λ h φ )(λ, φ, r + dr dλdφ − br (λ, φ, r )(h λ h φ )(λ, φ, r )dλdφ ∂ = (11.83) (br h λ h φ )(λ, φ, r )dλdφdr ∂r After analogous calculations for the two other pairs of surfaces we obtain  ∂ ∂ ∂ b·dS = (h φ h r bλ )dλdφdr + (h λ h r bφ )dλdφdr + (h λ h φ br )dλdφdr (11.84) ∂λ ∂φ ∂r s Inserting (11.81) and (11.84) into (11.77) yields

 ∂ 1 ∂ ∂ ∇ ·b= (h φ h r bλ ) + (h λ h r bφ ) + (h λ h φ br ) h λ h φ h r ∂λ ∂φ ∂r

(11.85)

and, using the metric factors from (11.67) to (11.69), ∇ ·b=

1 ∂ 1 ∂bλ 1 ∂  + cos φ bφ + 2 (r 2 br ) r cos φ ∂λ r cos φ ∂φ r ∂r

(11.86)

11.4

Appendix D: Spherical Coordinates

521

11.4.4 The Curl in Spherical Coordinates The curl of an arbitrary vector field b in spherical coordinates can be determined using the theorem from Stokes (11.5) via the relationship  1 b · dx (11.87) (∇ × b) · n = lim S→0 S C where the geometry is shown again in Fig. 11.5. Seen from above, the direction of integration in the path integral is anticlockwise. The components of the curl in spherical coordinates are defined as (11.88) ∇ × b = (∇ × b)λ eλ + (∇ × b)φ eφ + (∇ × b)r er Thus (∇ × b)λ = (∇ × b) · eλ

(11.89)

(∇ × b)φ = (∇ × b) · eφ

(11.90)

(∇ × b)r = (∇ × b) · er

(11.91)

The determination of the scalar products is done via (11.87), where the integration surface is chosen such that the corresponding surface normal coincides with one of the three unit vectors n = eλ , eφ , or er .

n

S C Fig. 11.5 An infinitesimally small surface S with surface normal n, for the determination of the component of the curl parallel to n via the integral theorem from Stokes.

522

11 Appendices

ϕ + dϕ

hϕdϕ

eλ

r + dr dϕ

r, ϕ

hrdr

Fig. 11.6 Surface for the determination of (∇ × b)λ via the theorem from Stokes.

As an example we show this for the component (∇ × b)λ . As integration surface we choose an infinitesimally small surface, lying in the plane of fixed λ, and enclosed by curves with constant φ or r (Fig. 11.6). The surface area is S = (h r dr )(h φ dφ)

(11.92)

The path integrals are   b · dr = bφ h φ dφ (λ, φ, r ) + (br h r dr ) (λ, φ + dφ, r ) C  − bφ h φ dφ (λ, φ, r + dr ) − (br h r dr ) (λ, φ, r )

  ∂ = bφ h φ (λ, φ, r ) dφ + (br h r ) (λ, φ, r ) + (br h r ) (λ, φ, r ) dφ dr ∂φ

  ∂  − bφ h φ (λ, φ, r ) + bφ h φ (λ, φ, r ) dr dφ − (br h r ) (λ, φ, r ) dr ∂r

  ∂ ∂ = (11.93) bφ h φ dφdr (br h r ) − ∂φ ∂r Inserting (11.92) and (11.93) into (11.87) thus yields 1 (∇ × b)λ = hr h φ

∂ ∂  h φ bφ (h r br ) − ∂φ ∂r

 (11.94)

In close analogy one also obtains 1 (∇ × b)φ = h λ hr and 1 (∇ × b)r = hλhφ



∂ ∂ (h λ bλ ) − (h r br ) ∂r ∂λ



∂  ∂ h φ bφ − (h λ bλ ) ∂λ ∂φ

(11.95)  (11.96)

11.5

Appendix E: Fourier Integrals and Fourier Series

523

Employing the metric factors from (11.67) to (11.69) finally yields

 1 ∂br ∂  − r bφ (∇ × b)λ = r ∂φ ∂r

 ∂ ∂br 1 (∇ × b)φ = (r cos φ bλ ) − r cos φ ∂r ∂λ

 ∂bφ 1 ∂ − (∇ × b)r = (cos φ bλ ) r cos φ ∂λ ∂φ

(11.97) (11.98) (11.99)

11.4.5 Recommendations for Further Reading Here and for all other mathematical appendices, textbooks on mathematical methods for physicists are a good support, e.g., Arfken et al. (2012).

11.5

Appendix E: Fourier Integrals and Fourier Series

11.5.1 Fourier Integrals Especially in the treatment of wave phenomena, but not only there, the Fourier transform is an indispensable tool. We here summarize some of its most important aspects.

Definition Let f : R → C, x  → f (x) be a function with the following properties: • f (x) is piecewise continuous. • f (x) is differentiable. ∞ • f (x) is absolutely integrable, i.e., −∞ d x| f (x)| < ∞, which only can be satisfied if f (x) vanishes at infinity. Under these conditions the Fourier transform of f at the wavenumber k is 1 f˜(k) = 2π



∞ −∞

d xe−ikx f (x)

(11.100)

The original function can be obtained from the Fourier transform via the inverse Fourier transform  ∞ f (x) = dkeikx f˜(k) (11.101) −∞

524

11 Appendices

One thus can see f (x) as a superposition of waves with wave number k, or wavelength λ = 2π/k, while the corresponding amplitude is f˜(k)dk. The higher-dimensional generalization for functions f : Rn → C, x  → f (x) is conceivable as successive Fourier transforms over all variable xi . One obtains  ∞  ∞  ∞ 1 d x d x ... d xn e−ik·x f (x) 1 2 (2π )n −∞ −∞ −∞  ∞  ∞  ∞ f (x) = dk1 dk2 ... dkn e−ik·x f˜(k)

f˜(k) =

−∞

−∞

−∞

(11.102) (11.103)

With regard to time a slightly different definition is customary. For a time-dependent function f (t) the Fourier transform at a frequency ω is 1 f˜(ω) = 2π and its inverse

 f (t) =



∞

−∞

∞

−∞

dteiωt f (t)

dωe−iωt f˜(ω)

(11.104)

(11.105)

For spatially and time-dependent functions we obtain in direct generalization  ∞  ∞  ∞  ∞ 1 dx dy dz dte−i(k·x−ωt) f (x, t) (2π )4 −∞ −∞ −∞ −∞  ∞  ∞  ∞  ∞ f (x, t) = dk x dk y dk z dωei(k·x−ωt) f˜(k, ω)

f˜(k, ω) =

−∞

−∞

−∞

−∞

(11.106) (11.107)

The function is thus decomposed into waves at frequency ω and wavenumber k.

Some Properties The Fourier transform is a linear operation, i.e.,

leads to

g(x, t) = a f (x, t) + bh(x, t)

(11.108)

˜ g(k, ˜ ω) = a f˜(k, ω) + bh(k, ω)

(11.109)

Very important is the property that the Fourier transform of a derivative is the product of the Fourier transform of the non-differentiated function and the wavenumber. For example, one obtains for g(x) = d f /d x via partial integration

11.5

Appendix E: Fourier Integrals and Fourier Series

 ik ∞ df 1  −ikx ∞ d xe f + d xe−ikx f = ik f˜(k) = e −∞ dx 2π 2π −∞ −∞ (11.110) which results from the consequence of absolute integrability that f vanishes at infinity. In analogy one has in direct generalization 1 g(k) ˜ = 2π

g(x, t) =



525

∞

−ikx

∂ m ∂ n ∂ o∂ p f ∂ x m ∂ y n ∂z o ∂t p

g(k, ˜ ω) = (ik x )m (ik y )n (ik z )o (−iω) p f˜(k, ω)

⇔

(11.111)

11.5.2 Fourier Series In the special case of periodic functions or functions with vanishing boundary values the Fourier integral becomes a Fourier series over wavenumbers corresponding to the basic wavelength and its higher harmonics.

Periodic Functions Every function f : R → C, x  → f (x), periodic over length L so that f (x + L) = f (x), can be written as Fourier series: ∞ 

f (x) =

f n eikn x

with

kn = n

n=−∞

2π L

(11.112)

The corresponding Fourier coefficients are 1 fn = 2π



L 2

d x f (x)e−ikn x

(11.113)

d x f (x)e−ikn x = f n

(11.114)

− L2

For real functions one has ∗ f −n

1 = L



L 2

− L2

so that f (x) =

−1 

f n eikn x + f 0 +

n=−∞

= f0 +

∞  n=1

∞ 

f n eikn x =

n=1

( f n eikn x + f n∗ e−ikn x )

∞  n=1

f −n e−ikn x + f 0 +

∞ 

f n eikn x

n=1

(11.115)

526

11 Appendices

With this we can show easily that for real functions ∞

c0  + [cn cos(kn x) + sn sin(kn x)] 2 n=1  L 2 2 cn = d x f (x) cos(kn x) L − L2  L 2 2 d x f (x) sin(kn x) sn = L − L2

f (x) =

(11.116)

(11.117) (11.118)

Here the cosine part is symmetric with regard to x = 0, and the sine part is antisymmetric.

The Series Representation of a Function with Vanishing Boundary Values Fourier series can also be used for the representation of a real-valued function f : [0, L] → R; x  → f (x) on a finite interval, with vanishing boundary values f (0) = f (L) = 0. One constructs from f , via point reflection at x = 0, a function ⎧ ⎨ f (x) if x ≥ 0 g : [−L, L] → R; x  → g(x) = ⎩ − f (−x) if x < 0 which is antisymmetric with respect to x = 0. This again can be seen as part of an antisymmetric function R → R with period 2L. Thus g(x) =

∞ 

∞

sn sin n

n=1

with 1 sn = L



 2π nπ sn sin x= x 2L L

(11.119)

n=1

L

nπ 2 d x g(x) sin x= L L −L



L

d x g(x) sin 0

nπ x L

(11.120)

since g(x) is, just as the sine, antisymmetric with respect to x = 0. Because the functions f and g are identical for 0 ≤ x ≤ L, one can write f : [0, L] → R; x  → f (x) as f (x) =

∞ 

sn sin

n=1

sn =

2 π



L 0

nπ x L

d x f (x) sin

(11.121) nπ x L

(11.122)

11.6 Appendix F: Zonally Symmetric Rossby Waves in the Quasigeostrophic…

527

Since in the case of complex-valued functions f : [0, L] → C; x  → f (x) with vanishing boundary values a corresponding separate treatment of real and imaginary part is possible, the representation (11.121) also holds for these.

11.5.3 Recommendations for Further Reading Here and for all other mathematical appendices, textbooks on mathematical methods for physicists are a good support, e.g., Arfken et al. (2012).

11.6

Appendix F: Zonally Symmetric Rossby Waves in the Quasigeostrophic Two-Layer Model

As described in Chapter 6.3.1, infinitesimally small perturbations of a barotropic zonal flow U in the quasigeostrophic two-layer model satisfy the equations   ∂ ∂ψ  ∂ +U ∇h2 ψ  + β =0 (11.123) ∂t ∂x ∂x and



∂ ∂ +U ∂t ∂x





∂τ  ∇h2 τ  − κ 2 τ  + β =0 ∂x

(11.124)

where ψ  and τ  are the streamfunctions of the barotropic and baroclinic mode, respectively, which can be determined from the streamfunctions in the two layers via 1  (ψ + ψ2 ) 2 1 1 τ  = (ψ1 − ψ2 ) 2

ψ =

(11.125) (11.126)

Due to their periodicity in x each of these can be represented as a Fourier series ψi (x, y, t) =

∞  n=−∞

ψin (y, t) eikn x ; kn = n

2π Lx

(11.127)

The treatment of the parts with n  = 0 is discussed in Chapter 6.3.1. We here focus on the zonally symmetric case n = 0. Due to (6.211) one has on each layer for the respective streamfunction  ∂ ∂ψi0 (11.128) =0 y = 0, L y ∂t ∂ y

528

11 Appendices

That is, at the meridional boundaries of the β-channel the meridional streamfunction gradients are time independent. We now decompose   ∂ψi0 y ∂ψi0 y ∂ψi0  L y + φi (y, t) = 1− (0) + ∂y Ly ∂y Ly ∂y

(11.129)

where from the definition follows that the time-dependent part vanishes at the meridional boundaries:  (11.130) y = 0, L y φi = 0 It can thus be written as Fourier series φi (y, t) =

∞ 

φim (t) sin (lm y) ;

lm = m

m=1

π Ly

(11.131)

Integration in y yields 

ψi0

y2 = C (t) + y − 2L y



∞ ∂ψi0 y 2 ∂ψi0   0m Ly + ψi (t) cos (lm y) (11.132) (0) + ∂y 2L y ∂ y m=1

with a steady integration constant C and ψi0m = −

φim lm

(11.133)

Since only the spatial gradients of the streamfunction are of any relevance, we can assume without loss of generality C = 0 .

11.7

Appendix G: Explicit Solution of the Initial-Value Problem of Baroclinic Instability in a Quasigeostrophic Two-Layer Model

From the eigenvalue equations (6.338) and (6.341) one obtains for the eigenvectors of the baroclinic-instability problem in a quasigeostrophic two-layer model, and the corresponding adjoint problem, the form     1 −α/(ωψ − ωˆ j ) nm nm j = bj (11.134)  j = aj −α/(ωτ − ωˆ ∗j ) 1 The factors a j and b j must be chosen so that the normalization condition (6.346) is satisfied, i.e., ωψ + ωτ − 2ωˆ j † nm  j = −b∗j a j α (11.135) 1 = nm j (ωψ − ωˆ j )(ωτ − ωˆ j )

11.8 Appendix H: Polarization Relations of the Geostrophic Mode …

529

or, using the characteristic equation (6.337), 1 = −b∗j a j

ωψ + ωτ − 2ωˆ j α−γ

(11.136)

A possible choice is bj = 1

(11.137)

α−γ aj = − ωψ + ωτ − 2ωˆ j

(11.138)

so that the eigenvectors become  nm j

α−γ =− ωψ + ωτ − 2ωˆ j



−α/(ωψ − ωˆ j ) 1



 nm j

=

1 −α/(ωτ − ωˆ ∗j )

 (11.139)

With this result one can use (6.348) to determine the expansion coefficients of the initial state with regard to the eigenvectors. Applying (6.325) one obtains  Anm j

=

1 −α/(ωτ − ωˆ j )

= K nm ψ

nm

t 

nm  K nm ψ (0) 2 + κ 2 τ nm (0) K nm

 2 + κ2 α K nm (0) − τ nm (0) ωτ − ωˆ j



(11.140)

Finally we also represent the eigenvectors in terms of the corresponding barotropic and baroclinic streamfunctions. From (11.139) and (6.325) follows directly ⎛ ⎞ α/K nm   ⎜ ωψ − ωˆ j ⎟ ψ nm α−γ j ⎜ ⎟ (11.141) =− nm ⎠ τj ωψ + ωτ − 2ωˆ j ⎝  1 2 + κ2 K nm This used together with (11.140) in (6.350) makes the general solution of the instability problem explicit, where one has to use (6.370) for the eigenfrequencies.

11.8

Appendix H: Polarization Relations of the Geostrophic Mode and all f -Plane Modes Without Buoyancy Oscillations

Geostrophic Mode The geostrophic mode has zero eigenfrequency, as can be seen from (10.74), so that from (10.65) follows that it has no vertical wind, wˆ = 0

(11.142)

530

11 Appendices

(10.62) – (10.64) yield the relations of geostrophic and hydrostatic equilibrium, l cs pˆ f ρ0 k cs vˆ = i pˆ f ρ0   cs ˆb = im − 1 + 1 pˆ Hθ 2H ρ0

uˆ = −i

(11.143) (11.144) (11.145)

linking horizontal wind and buoyancy to pressure. For the latter one obviously has an amplitude Pˆg so that (11.146) p(k, ˆ ω) = Pˆg (k)δ(ω) whence, by a procedure analogous to that used for gravity and sound waves, using the back-transformations (10.61) and (10.45), the pressure field is   p = d 3 k eik·x−z/2H Pg (k) (11.147) with

Pg = cs Pˆg

(11.148)

p  ∈ R also here implies Pg (−k) = Pg∗ (k). Moreover, (11.148) together with (11.142) – (11.145) leads, again via (10.61) and (10.45), to      v 3 ik·x+z/2H Vg = d ke (k) (11.149) b Bg with l Pg f ρ0 k Pg Vg = i f ρ0 Wg = 0   Pg 1 1 + Bg = im − Hθ 2H ρ0 Ug = −i

(11.150) (11.151) (11.152) (11.153)

Of course we have also here Vg (−k) = V∗g (k) and Bg (−k) = Bg∗ (k). Geostrophic Mode without Buoyancy Variations In the case of the modes without buoyancy fluctuations not only the frequency is fixed by the dispersion relation, given the wavenumber, but also the vertical wavenumber, given the horizontal wavenumber, via (10.89). Hence, with kh = kex + le y the horizontal wavenumber and xh = xex + ye y the horizontal position,

11.8 Appendix H: Polarization Relations of the Geostrophic Mode …

z ik · x + = ikh · xh + 2H and likewise ik · x − where



1 1 − Hθ 2H



z = ikh · xh + 2H

1 1 − = Hθ H



R −1 cp

z z = ikh · xh + 2H Hθ

z+





1 1 − Hθ H

531

(11.154)

 z

1 cV 1 =− H cp H

(11.155)

(11.156)

Following procedures analogous to the ones used for the modes with buoyancy oscillations one then obtains for the dynamical fields of the geostrophic mode with dispersion relation (10.91)      v VgL 2 ikh ·xh +z/Hθ = d ke (kh ) (11.157) b BgL and p =



d 2 k eikh ·xh −(cV /c p )z/H PgL (kh )

(11.158)

where PgL is the pressure amplitude at a given horizontal wavenumber, and the corresponding wind and buoyancy amplitudes follow therefrom by l PgL f ρ0 k PgL =i f ρ0 =0

UgL = −i

(11.159)

VgL

(11.160)

WgL

(11.161)

BgL = 0

(11.162)

Lamb Wave Formally, for the Lamb wave there are the possible frequencies  = ± ω2L ω± L

(11.163)

so that its dynamical fields are   ν i(kx+ly)+ Hz  θ v = d 2k e VνL e−iω L t

(11.164)

ν

b = 0  c i(kx+ly)− cVp  p = d 2k e

(11.165) z H

 ν

ν

PLν e−iω L t

(11.166)

532

11 Appendices

where the wind-field amplitudes can be obtained from the pressure amplitude via kωνL + il f ωL 2 − f 2 lων + ik f VLν = L2 ωL − f 2

U Lν =

PLν ρ0 PLν ρ0

W Lν = 0

(11.167) (11.168) (11.169)

However, again all fields are real, yielding PL−ν (k) = PLν∗ (−k)

(11.170)

and analogous relations for all other variables. We thus obtain    2 ikh ·xh +z/Hθ + −iω+ t −iω− L Lt v = 2 d k e VL e = 2 d 2 k eikh ·xh +z/Hθ V− (11.171) Le   + − p  = 2 d 2 k eikh ·xh −(cV /c p )z/H PL+ e−iω L t = 2 d 2 k eikh ·xh −(cV /c p )z/H PL− e−iω L t (11.172)

11.9

Appendix I: The Higher Harmonics of a Gravity-Wave Field in WKB Theory

A comprehensive treatment of the interaction between a locally monochromatic gravitywave field and a mean flow in Section 10.3 must also allow for the existence of its higher harmonics. As we have seen in the discussion of the advection terms in the entropy equation, they lead in (10.335) and (10.336) not only to a forcing of the mean flow, but also of a higher harmonic with twice the phase of the basic wave. This harmonic will again induce the quadruple phase, and in interaction with the basic wave, also the triple phase. Continuing this argument one sees that the most general ansatz for the non-dimensional decomposition of all dynamical fields would be to replace (10.237) – (10.239) by

11.9 Appendix I: The Higher Harmonics of a Gravity-Wave Field in WKB Theory

v=

∞ 

ε

j

( j) V0 (X, T ) + 

j=0









∞ 

ε

j=0

1  j=0



εjθ

( j)

(Z ) +





reference

π=

1  j=0



reference

j=2



α=1

Vα( j) (X, T )eiαφ(X,T )/ε 

( j)

ε j 0 (X, T ) +  



∞ 





∞  j=2



( j)

ε j 0 (X, T ) +  



synoptic−scale part



∞ 

εj

α=1

j=2

∞  j=3

εj

(11.173)

 (αj) (X, T )eiαφ(X,T )/ε 

wave + higher harmonics

synoptic−scale part

ε j π ( j) (Z ) + 

∞ 

∞ 

wave + higher harmonics

synoptic−scale part

θ=

j

533

∞  α=1

(11.174)



(αj) (X, T )eiαφ(X,T )/ε (11.175) 

wave + higher harmonics



Here in all sums over α the basic wave is given by α = 1, while α = 2, 3, . . . correspond to the second, third, etc., harmonic. The analysis of the leading and next orders of the dynamical equations in Subsections 10.3.4 and 10.3.5 is then changed by replacing everywhere the product of wave amplitude and phase exponential by the corresponding sum over all harmonics. As we will see, all results that have been derived so far under the neglect of the higher harmonics stay the same in the generalized approach. On top of this, however, one can also determine these higher harmonics themselves.

11.9.1 Leading-Order Results The leading-order results in Subsection 10.3.4 can be replaced in a rather straightforward manner. To begin with, the leading O(1) of the non-dimensional Exner-pressure equation (10.220) is, instead of (10.248), ∞

R (0)  π  ik · Vα(0) eiαφ/ε = 0 cV

(11.176)

α=1

yielding

k · Vα(0) = 0

α = 1, 2, . . .

(11.177)

As in the locally monochromatic case without higher harmonics this relation is decisive in removing all leading-order gravity-wave self-advection terms. The leading O(ε2 ) in the non-dimensional entropy equation (10.218) is, instead of (10.252),

534

11 Appendices (0) dθ W0

(1)

dZ

+

∞  α=1

+

∞  α=1

 −iαω(2) α

Vα(0) eiαφ/ε · 

∞  β=1

+ iαk

(0) · V0 (2) α

+

dθ Wα(0)

(1)

dZ

(2)

iβkβ eiβφ/ε = 0

 eiαφ/ε

(11.178)

Herein the last term on the left-hand side vanishes due to the orthogonality relation (11.177). The mean-flow part of the remainder yields the old result (10.256) that the leading-order synoptic-scale wind has no vertical component. The remaining wave contributions, now also including the higher-harmonic parts, yield − iα ωB ˆ α(2) + Wα(0) N02 = 0 where Bα( j) =

α = 1, 2, . . .

(11.179)

( j)

α θ

(11.180)

(0)

is the O(ε j ) non-dimensional wave-buoyancy amplitude of the αth harmonic. The leading O(ε) and O(ε2 ) of the non-dimensional vertical-momentum equation (10.216) remain (10.263) and (10.265), respectively, leading to the hydrostatic equilibrium of the reference atmosphere as expressed by (10.264) and (10.266). The next order O(ε3 ) is the first with non-zero wave contributions, again absorbing the O(ε7 ) terms from the Eulerian time derivative and the advection term. Taking also the orthogonality condition (11.177) into account, this yields, instead of (10.267), ε  4

∞  α=1

! (0) −iαωWα(0) + iαk · V0 Wα(0) eiαφ/ε

" (2) (1) (0) c p (0) ∂0 (1) dπ (2) dπ θ +θ + 0 R ∂Z dZ dZ   # ∞ (0)  (2) dπ iαφ/ε e + iαm(3) +  α α dZ

=−

(11.181)

α=1

With the definition (11.180) of the wave buoyancy this leads, taking the hydrostatic equilibrium (10.264) of the leading-order reference-atmosphere Exner pressure into account, to the linear vertical-momentum equations − ε4 iα ωW ˆ α(0) − Bα(2) + iαm

c p (0) (3) θ α = 0 R

α = 1, 2, . . .

(11.182)

while the mean-flow part yields again the hydrostatic equilibrium (10.269) of the synopticscale flow. Finally, the leading O(ε3 ) of the non-dimensional horizontal-momentum equation (10.213) is, instead of (10.275),

11.9 Appendix I: The Higher Harmonics of a Gravity-Wave Field in WKB Theory



∞  α=1

−iαωUα(0)

+ iαk

(0) · V0 Uα(0)

(0)

∞ 



+ f 0 ez × U0 + 

α=1

! e

iαφ/ε



+

∞  α=1

Vα(0) eiαφ/ε

·

∞  β=1

535 (0)

iβkUβ eiβφ/ε

Uα(0) eiαφ/ε

  ∞  R (0) (2) (3) iαφ/ε =− θ iαkh α e ∇ X ,h 0 +  cp

(11.183)

α=1

The nonlinear self-advection term vanishes due to the orthogonality relation (11.177), leaving the wave contributions − iα ωU ˆ α(0) + f 0 ez × Uα(0) +

c p (0) θ iαkh (3) α =0 R

α = 1, 2, . . .

(11.184)

and the mean-flow part (10.277) expressing the geostrophic equilibrium of the synoptic-scale flow. Hence we see that the leading-order mean-flow results are the same as from the theory without higher harmonics. The wave equations, however, are now supplemented by counterparts for all of the higher harmonics. In total these are (11.177), (11.179), (11.182), and (11.184), which we summarize under ⎞ ⎞⎛ ⎛ (0) Uα −iα ωˆ − f 0 0 0 iαk ⎟ ⎜ (0) ⎜ f ⎟ Vα 0 0 iαl ⎟ ⎟⎜ ⎜ 0 −iα ωˆ ⎟ ⎜ ⎟ ⎜ (0) 4 ⎟=0 ⎜ (11.185) −N0 iαm ⎟ ⎜ Wα 0 −iα ωε ˆ ⎜ 0 ⎟ ⎟ ⎜ (2) ⎜ ⎝ 0 0 N0 −iα ωˆ 0 ⎠ ⎝ Bα /N0 ⎟ ⎠ c p (0) (3) iαk iαl iαm 0 0 θ  α     R   Mα = M(αk, αω) Z(0) α (0)

Non-trivial basic-wave amplitudes Z1  = 0 still require det M1 = 0, yielding the nondimensional dispersion relations (10.280) and (10.281) for geostrophic modes and gravity waves, respectively. Corresponding results for the higher harmonics α > 1, however, differ substantially between the gravity-wave and the geostrophic-mode case. One can convince oneself easily that if ω and k are related by the gravity-wave dispersion relation (10.281) then for α > 1 this is not possible for αω and αk. Hence det Mα  = 0, i.e., Mα is invertible, so that one has α>1 (11.186) gravity waves : Z(0) α =0 In other words Higher harmonics do not contribute to the gravity-wave fields to leading order.

536

11 Appendices

Such a statement cannot be made about geostrophic modes because (αω, αk) satisfy the geostrophic-mode dispersion relation (10.280) for all α.

11.9.2 Next-Order Results It turns out that the second harmonic of the gravity-wave field can be determined from the next order of the equations of motion. Beyond this all results from the theory without higher harmonics can be rederived. To see this we have to go through all of the equations again, now gathering the next-order terms and focussing on gravity waves, i.e., leaving the geostrophic-mode case aside. To begin with, the O(ε) of the Exner-pressure equation (10.220) is, replacing (10.324),   ∞ (0)  (0) (0) iαφ/ε dπ W0 +  Wα e dZ α=1 $ % " ∞ !  R (0) iαk · Vα(1) + ∇ X · Vα(0) eiαφ/ε π (0) ∇ X · V0 +  + cV α=1 # ∞ !  (11.187) + π (1)  iαk · Vα(0) eiαφ/ε = 0 α=1

We now recall the definition (10.244) of P¯ (0) and the orthogonality relation (11.177). Moreo(0) ver, we remember that due to (11.186) one has Vα = 0 for all α > 1. Hence the wave contributions yield ! R (0) R π (0) ¯ (0) Vα(0) = 0 π iαk · Vα(1) + δα,1 ∇ · P X cV cV P¯ (0)

(11.188)

whence, and because P¯ (0) differs from the leading-order reference-atmosphere density ρ (0) (0) only by a constant factor 1/θ , (1) iαk · Vα(1) = Rπ,α ≡−

δα,1

ρ (0)

∇ X · ρ (0) Vα(0)

! (11.189)

Note that the right-hand side is only nonzero for α = 1, so that also the next-order higherharmonic wind-field amplitudes are orthogonal to the wavenumber vector. Moreover, the mean-flow parts of (10.324) and (11.187) agree with each other so that the result (10.329) on the non-divergence of the leading-order horizontal synoptic-scale flow is obtained here as well. Finally, from the next O(ε2 ) terms of the Exner-pressure equation (10.220) we again only use the mean-flow part which yields the same anelastic divergence constraint (10.332) as in the theory neglecting the higher harmonics.

11.9 Appendix I: The Higher Harmonics of a Gravity-Wave Field in WKB Theory

537

The O(ε3 ) contributions to the entropy equation (10.218) are, instead of (10.333),   (1) ∞ ∞ (2)   ∂α iαφ/ε (1) (1) iαφ/ε dθ − e + W0 +  Wα e ∂T dZ α=1 α=1 α=1  $ %  ∞ ∞ !   (0) (0) (2) (3) (2) iαkα + ∇ X α eiαφ/ε Vα eiαφ/ε · ∇ X 0 +  + V0 +  (2)

∞ 

∂0 ∂T

 (1)

+ V0 + 

(3) iαωα eiαφ/ε + 

α=1 ∞ 

 (1)

Vα eiαφ/ε · 

α=1

∞ 

α=1

! (2) iαkα eiαφ/ε = 0

(11.190)

α=1 (0)

Herein all leading-order higher-harmonic amplitudes vanish due to (11.186), i.e., both Vα = (2) 0 and α = 0 for all α > 1. Because of this and also the orthogonality relation (10.249) one has moreover ∞ ∞   iαφ/ε  Vα(0) eiαφ/ε ·  iαk(3) =0 (11.191) α e α=1

α=1

but from the self-advection of the wave two other nonzero products appear, where we also use the orthogonality (11.189) of the next-order higher-harmonic wind amplitudes on the wavenumber vector to obtain, instead of (10.335) and (10.336), ! ! (0) (2)  V1 eiφ/ε ·  ∇ X 1 eiφ/ε ! 1 ! 1 (0) (2) (0) (2) ∗ =  V1 · ∇ X 1 e2iφ/ε +  V1 · ∇ X 1 2 2 ∞ ! ! !  (1) (2) (1) (2)  Vα eiφ/ε ·  ik1 eiφ/ε =  V1 eiφ/ε ·  ik1 eiφ/ε

(11.192)

α=1

=

! 1 ! 1 (1) (2) (1) (2) ∗  V1 · ik1 e2iφ/ε −  V1 · ik1 2 2

(11.193)

The same mean-flow terms arise as in the theory neglecting the higher harmonics. Hence the mean-flow part of (11.190) leads to the same prognostic equation (10.341) for the synopticscale buoyancy that we had already determined before, with a forcing by the wave-buoyancy flux convergence. Now, however, we do not simply neglect the second-harmonic contributions. Keeping them the wave parts of (11.190) are found to be, instead of (10.337), (3)

− iα ωB ˆ α(3) + N02 Wα(1) = Rb,α

α = 1, 2, . . .

(11.194)

538

11 Appendices

with (1)

 ∂ (0) (1) (2) (0) (2) B1 − V1 · ∇ X B0 + U0 · ∇ X + ik · V0 ∂T ! 1 V1(0) · ∇ X + ik · V1(1) B1(2) =− 2 =0 α>2 

Rb,1 = − (1)

Rb,2 (1)

Rb,α

(11.195) (11.196) (11.197)

(1) (1) In the second-harmonic right-hand side Rb,2 the term ik · V1 can be calculated via (11.189) (0)

(1)

from V1 so that Rb,2 is completely determined by the leading-order basic-wave amplitudes. The analysis of the next-order terms in the momentum equation works in a completely analogous manner to what has been described above, so that we only give the final results. Into the terms of O(ε4 ) in the vertical momentum equation (10.216) we again also incorporate the material derivative, now by its O(ε8 ) terms. The wave parts then yield, instead of (10.350), − iα ωε ˆ 4 Wα(1) − Bα(3) + iαm with

c p (0) (4) (1) θ α = Rw,α R

α = 1, 2, . . .

 ∂ (0) (1) (0) =−ε + U0 · ∇ X + ik · V0 W1 ∂T   (3) (1) cp (0) ∂1 (1) (3) (2) ∂π − θ + θ im1 + 1 R ∂Z ∂Z ! ε4 (1) (0) (1) (0) V1 · ∇ X + ik · V1 W1 Rw,2 = − 2 (3) =0 α>2 Rw,α

(11.198)



(1) Rw,1

4

(11.199) (11.200) (11.201)

The terms of O(ε4 ) in the horizontal-momentum equation (10.213) yield the wave parts, instead of (10.352), − iα ωU ˆ α(1) + f 0 ez × Uα(1) + iαkh θ with (1) Ru,1

(0) c p

R

(1) (4) α = Ru,α

α = 1, 2, . . .

 ! ∂ (0) (1) (0) (0) (0) + U0 · ∇ X + ik · V0 U1 − V1 · ∇ X U0 ∂T ! c p (0) (1) (3) (3) θ ∇ X ,h 1 + θ ikh 1 R ! 1 (0) (1) (0) V1 · ∇ X + ik · V1 U1 2 α>2

(11.202)

 =− −

(1)

Ru,2 = − (1) =0 Ru,α

(11.203) (11.204) (11.205)

11.9 Appendix I: The Higher Harmonics of a Gravity-Wave Field in WKB Theory

539

while the mean-flow part leads again to the prognostic equation (10.354) for the synopticscale horizontal wind with forcing by the gravity-wave momentum-flux convergence that we had obtained already from the theory without higher harmonics. In summary we have rederived in the general theory all the mean-flow equations that we had already obtained under the neglect of the higher harmonics. The wave parts we had obtained there now take the more general forms (11.189), (11.194) – (11.197), (11.198) – (11.201), and (11.202) – (11.205) that can be summarized under (1) Mα Z(1) α = Rα

with

α = 1, 2, . . .



t Z(1) α

Rα(1)

t

(3) Bα c p (0) (4) = , θ α N0 R   (1) Rb,α (1) (1) t (1) = Ru,α , Rw,α , , Rπ,α N0 t Uα(1) , Wα(1) ,

(11.206)  (11.207)

(11.208)

As in the theory without higher harmonics, M1 is singular, leading to the gravity-wave dispersion and polarization relations. This singularity again implies that the null space of (1) M1 does not project onto R1 , i.e., (10.362) is rederived, whence one obtains again the prognostic equation (10.367) for the wave-energy density and the wave-action conservation equation (10.411) that we had already obtained without higher harmonics. These findings are now supplemented by results on the higher-harmonic amplitudes. As (1) we have seen above, for all α > 1 the matrix Mα is invertible. Hence, and because Rα = 0 for all α > 2, one finds that (1)

(1)

Z2 = M2−1 R2

(11.209)

Z(1) α =0

(11.210)

α>2

(1)

Inspection of the elements of R2 in (11.189), (11.196), (11.200), and (11.204), and (1) (0) (1) noting that the term ik · V1 can be calculated via (11.189) from V1 , one sees that R2 is completely determined by the leading-order basic-wave amplitudes, so that (11.209) tells us how second harmonics can be calculated from the latter. Moreover, by (11.210) the third and higher harmonics do not even contribute to the wave field to the next to leading order.

11.9.3 Recommendations for Further Reading The treatment is based on Achatz et al. (2010, 2017). The second text also relaxes the assumptions on the reference-atmosphere stratification so that the theory is also applicable under stratospheric conditions.

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Index

A Absolute circulation, 103 Absolute vorticity, 99 frozen-in property, 113 Anelastic equations, 250–253 Angular-momentum conservation, 77–79 Available potential energy, 209, 212

B Balanced flows, 87–95 Baroclinic instability, 223–248 β-effect, 232–234 continuous stratification, 236–246 Eady problem, 242–246 heat transport, 232 initial-value problem, 225–229 mechanisms, 235–236 Rayleigh theorem, 239–242 scales, 230, 243–245 structure, 229–234, 245–246 two-layer model, 224–236 westward tilt, 232 Baroclinic vector, 106 Barotropic flow, 106 β-plane shallow-water dynamics, 137–140 stratified atmosphere, 184–186 Boundary layer, 249–299 Ekman layer, 286–299 Ekman pumping, 296–298 Ekman spiral, 288–296

instabilities, 256–269 Prandtl layer, 281–286 influence of stratification, 283–285 logarithmic wind profile, 282–283 Reynolds equations, 271–273 turbulence, 269–299 and mean flow, 269–271 buoyancy flux, 272 diffusivity, 274 gradient ansatz, 273–275 kinetic energy, 275–280 mixing length, 273–275 stress tensor, 272 viscosity, 274 viscous sub-layer, 249 Boussinesq equations, 253–255 Brunt-Väisälä frequency, 70 Buoyancy oscillations, 69 Buoyancy production, 469

C Carnot process, 50 efficiency, 51, 52 Centrifugal acceleration, 21 Change of state adiabatic, 44–45 irreversible, 37 isothermal, 45 quasi-static, 37 reversible, 37 Circulation, 103–108

© Springer-Verlag GmbH Germany, part of Springer Nature 2022 U. Achatz, Atmospheric Dynamics, https://doi.org/10.1007/978-3-662-63941-2

547

548 absolute, 103 and baroclinicity, 106 and Coriolis force, 104 relative, 103 Circulation theorem, 104 Clausius inequality, 55 Continuity equation, 8–10 in a rotating reference frame, 22 in arbitrary vertical coordinates, 85 in isentropic coordinates, 122 in pressure coordinates, 86 in spherical coordinates, 26 shallow-water dynamics, 134 β-plane, 140 synoptic scale analysis, 183 Coriolis acceleration, 21 Cyclic process, 37 Cyclostrophic flow, 91

D Dispersion relation external gravity waves, 154, 159 geostrophic mode, 418, 420, 455 internal gravity waves, 419, 425 internal gravity waves in synoptic-scale flow, 456 Lamb wave, 420 Rossby waves baroclinic, 219 barotropic, 217 continuous stratification, 222 meridionally varying basic flow, 311 shallow-water dynamics, 161 sound waves, 419, 425 Dust devil, 91 Dyer–Businger function, 284

E Eady problem, 242–246 Efficiency Carnot process, 51, 52 heat engine, 48 Eikonal equations internal gravity waves, 458–460 Rossby waves, 312 Ekman layer, 286–299 Ekman pumping, 296–298

Literature Ekman spiral, 288–296 scale asymptotics, 291–296 thickness, 295 Ekman number, 289 Eliassen-Palm flux internal gravity waves, 497–499 mid-latitudes, 402 Rossby waves, 317–324 Energy conservation, 75–77 linear compressible dynamics, 411–414 shallow-water dynamics, 135 stratified atmosphere, 205–214 continuous stratification, 206–211 two-layer model, 211–213 synoptic-scale flow and internal gravity waves, 483–484 Energy equation, 64 Energy theorem, 76 Enstrophy, 305 eddy enstrophy, 305–306 enstrophy equation, 305–306 Enthalpy equation, 65 Entropy, 52–59 change, 55, 56, 59 definition, 55 equivalence with potential temperature, 57 prediction, 66 Entropy equation, 66 scale asymptotics, 188–191 synoptic scale analysis, 188–191 Equation of state, 38 Equations of motion, 1–34 Equations of motion on the sphere, 23–28 Equilibrium geostrophic, 29 hydrostatic, 31 Euler equations, 74 Eulerian perspective, 2 Exner pressure, 431–432

F f -plane, 153 Ferrel cell, 377–380, 399 Fjørtoft theorem, 264 Fluid element, 1 Flux-gradient ansatz, 273–275 Fredholm alternative, 467 Friction, 13

Literature Frictional heating, 64

G Geostrophic adjustment, 166–177 Geostrophic equilibrium, 29, 454 Geostrophic flow, 89 Geostrophic mode compressible dynamics, 418, 420, 455 shallow-water dynamics, 158 Geostrophic streamfunction, 192–194 pressure coordinate, 201 shallow-water dynamics, 147 Geostrophic wind, 30, 193 pressure coordinates, 198 shallow-water dynamics, 147 Gradient wind, 92–94 Gravity waves internal, 264–266, 407–505 critical levels, 494 dispersion elation, 425 dispersion relation, 419, 456 eikonal equations, 458–460 energy conservation, 483–484 energy flux, 472 group velocity, 427 higher harmonics, 532–539 impact on synoptic-scale buoyancy, 464 impact on synoptic-scale entropy, 464 impact on synoptic-scale momentum, 466 impact on the middle atmosphere, 407–409, 496–504 impact on the quasigeostrophic potential vorticity, 476–479 interaction with synoptic-scale flow, 431–495 non-acceleration theorem, 486–490, 499–500 phase velocity, 427 polarization relations, 420–426, 456–458 potential vorticity, 485 pseudomomentum, 474, 485 pseudomomentum flux, 473–475, 482 pseudovorticity, 486 ray equations, 458–460 reflecting levels, 494–495 scale analysis, 437–441 transformed Eulerian mean, 497–499

549 wave action, 466–476, 480–482 wave-action density, 473, 480 wave-energy theorem, 467–469, 475 WKB theory, 431–495 shallow-water dynamics, 154 Group velocity external gravity waves, 156 internal gravity waves, 427

H Hadley circulation, 344–377 Held-Hou model, 344–364 Hide’s theorem, 350–353 summer-winter asymmetry, 364–367 wave impact, 367–376 Heat capacities, 43–44 Heat conduction, 62 Heat engine, 47 Carnot process, 50 efficiency, 48 Heat pump, 48 Heat transfer, 38 Hide’s theorem, 350–353 Horizontal momentum equation in arbitrary vertical coordinates, 83 in isentropic coordinates, 121 in natural coordinates, 89 in pressure coordinates, 85 Hydrostatic equilibrium, 31, 451–453 in arbitrary vertical coordinates, 83 in isentropic coordinates, 122 in pressure coordinates, 86

I Inertial flow, 90 Initial-value problem linear shallow-water dynamics, 166–170 linear two-layer model, 225–229, 528–529 Internal energy, 42 Irreversible change of state, 37 Isentropic coordinates, 120–122

K Kelvin theorem, 108 Kelvin–Helmholtz instability, 259–261

550 L Lagrangian perspective, 2 Lamb wave, 420 Land-sea circulation, 107 Laws of thermodynamics, 41–60 first law, 42 second law, 46–50 according to Clausius, 47 according to Kelvin, 47 equivalence of the formulations according to Kelvin and Clausius, 48 mathematical formulation, 56 Linear dynamics shallow-water dynamics, 150–177 initial-value problem, 166–170 Logarithmic wind profile, 282–283

M Mass streamfunction Eulerian, 329 residual, 330 Mass-weighted mean circulation, 332–337 Material derivative, 3–7 geostrophic, 148, 193 in arbitrary vertical coordinates, 83 in isentropic coordinates, 120 in pressure coordinates, 85 in spherical coordinates, 24 of a scalar, 3 of a vector field, 4 of volume integrals, 5 Meridional circulation Eulerian mean, 324–329 Hadley circulation, 344–377 Held-Hou model, 344–364 Hide’s theorem, 350–353 summer-winter asymmetry, 364–367 wave impact, 367–376 in mid-latitudes, 377–405 barotropic jet stream, 380–387, 398 Eliassen-Palm flux, 402 Ferrel cell, 377–380, 399 residual circulation, 396 surface winds, 380–387, 398, 402 in the tropics, 344–377 Held-Hou model, 344–364 Hide’s theorem, 350–353 summer-winter asymmetry, 364–367

Literature wave impact, 367–376 in troposphere and stratosphere, 343–405 mass-weighted mean circulation, 332–337 middle atmosphere, 407–409, 496 residual-mean circulation, 330–337 stratosphere, 341 transformed Eulerian mean, 324–338 transformed Eulerian mean with gravity waves, 497–499 Momentum equation, 10 in a rotating reference frame, 20 in spherical coordinates, 26 scale analysis, 34 shallow-water dynamics, 133 β-plane, 140 synoptic scale analysis, 28–183 Monin–Obukhov length, 284 Mountain ridge, flow over a, 125

N Natural coordinates, 87–89 Navier–Stokes equations, 74 Non-acceleration theorem internal gravity waves, 486–490, 499–500 Rossby waves, 339–341

P Perturbation approach, 151 Phase velocity external gravity waves, 155 internal gravity waves, 427 Planetary vorticity, 100 Polarization relations geostrophic mode, 529–531 internal gravity waves, 420–426 internal gravity waves in synoptic-scale flow, 456–458 Lamb wave, 531–532 sound waves, 420–426 Potential temperature, 57–59 and static stability, 67–71 definition, 58 prediction, 65 Potential vorticity, 116 and the circulation theorem, 117 conservation property, 117 primitive equations, 125

Literature prognostic equation, 116 shallow-water dynamics, 136 synoptic-scale flow and internal gravity waves, 485 Prandtl layer, 281–286 influence of stratification, 283–285 logarithmic wind profile, 282–283 Prandtl number, 279 Prediction of temperature, 60–65 Pressure coordinate, 85–86 Pressure-gradient force, 11 Pressure velocity, 85 Primitive equations, 80–86 in isentropic coordinates, 120–122 in pressure coordinates, 82–86 potential vorticity, 125 vortex dynamics, 120–129 vorticity equation, 125 Pseudomomentum, 474, 485 Pseudomomentum flux, 473–475, 482 Pseudovorticity, 486

Q Quasi-static change of state, 37 Quasigeostrophic potential vorticity gravity-wave forcing, 479 pressure coordinate, 201 relation with potential vorticity shallow-water dynamics, 148 stratified atmosphere, 195–197 shallow-water dynamics, 148 stratified atmosphere, 193 two-layer model, 204 Quasigeostrophic theory baroclinic instability, 223–248 continuous stratification, 236–246 two-layer model, 224–236 energy conservation, 205–214 continuous stratification, 206–211 two-layer model, 211–213 pressure coordinate, 198–201 shallow-water dynamics, 137–150 stratified atmosphere, 179–248 vorticity equation pressure coordinate, 199 stratified atmosphere, 188

551 R Ray equations internal gravity waves, 458–460 Rossby waves, 312 Rayleigh equation, 258 Rayleigh theorem baroclinic instability, 239–242 shear-layer instabilities, 263 Relative circulation, 103 Relative vorticity, 100 Residual-mean circulation, 330–337 middle atmosphere, 500–503 stratosphere, 341 Reversible change of state, 37 Reynolds equations, 271–273 Richardson criterion, 266–267 Richardson number flux Richardson number, 278 gradient Richardson number, 267 Rossby deformation radius external, 143 internal, 193 Rossby number, 29 Rossby waves Eliassen-Palm flux, 317–324 Eliassen-Palm relationship, 319–320 enstrophy equation, 305–306 influence on the zonal-mean flow, 339–341 Eulerian mean, 324–329 non-acceleration, 339–341 transformed Eulerian mean, 324–332 mean-flow interaction, 301–342 propagation, 306–316 critical lines, 315 reflection, 314 shallow-water dynamics, 160–165 stratified atmosphere, 214–223 continuous stratification, 221–222 two-layer model, 214–221 stratosphere, 313–315, 341 wave action, 319–324 WKB theory, 306–316 critical lines, 315 dispersion relation, 311 eikonal equations, 312 Eliassen-Palm flux, 320–324 ray equations, 312 reflection, 314 wave action, 320–324

552 zonally symmetric, 527–528 Rotating reference frame, 19–23 thermodynamics, 66 Roughness length, 283

S Scale asymptotics Ekman layer, 291–296 entropy equation, 188–191 Fredholm alternative, 467 shallow-water dynamics, 144–148 stratified atmosphere continuity equation, 186–188 momentum equation, 186–188 synoptic scale and mesoscale, 442–479 Shallow-water dynamics, 131–177 continuity equation, 134 β-plane, 140 energy conservation, 135 external gravity waves, 154 dispersion relation, 154, 159 group velocity, 156 phase velocity, 155 geostrophic adjustment, 166–177 geostrophic mode, 158 geostrophic streamfunction, 147 geostrophic wind, 147 linear dynamics, 150–177 initial-value problem, 166–170 momentum equation, 133 β-plane, 140 potential vorticity, 136 quasigeostrophic potential vorticity, 148 quasigeostrophic theory, 137–150 Rossby waves, 160–165 dispersion relation, 161 scale asymptotics, 144–148 synoptic scale analysis, 140–144 vortical mode, 157 vorticity equation, 136 wave solutions, 150–165 Shear production, 278, 469 Shear-layer instabilities, 256–269 neutral stratification, 258–264 Fjørtoft theorem, 264 instability conditions, 261–264 Kelvin–Helmholtz instability, 259–261 Rayleigh theorem, 263

Literature Richardson criterion, 266–267 Shear-stress velocity, 274 Sound waves dispersion relation, 419, 425 polarization relations, 420–426 Stability parameter, 191, 193, 201 Static instability, 264–266 Static stability, 67–71 Stratification, 69 Streamfunction, 152 Stress tensor turbulenter, 272 viscous, 14 Synoptic scale analysis continuity equation, 183 entropy equation, 188–191 momentum equation, 28–183 shallow-water dynamics, 140–144 together with mesoscale, 432–441

T Taylor–Goldstein equation, 256–258 Thermal wind, 95–97, 194, 469 pressure coordinate, 201 Thermodynamic energy equation, 64 Thermodynamic equilibrium, 36 Thermodynamic system, 35 closed, 36 energy change, 38 isolated, 36 open, 36 Thermodynamics, 35–71 in a rotating reference frame, 66 in spherical coordinates, 66 laws of thermodynamics, 41–60 first law, 42 second law, 46–50 Tornado, 91 Traditional approximation, 80 Transformed Eulerian Mean (TEM), 324–338, 497–499 Turbulence, 269–299 and mean flow, 269–271 buoyancy flux, 272 diffusivity, 274 Ekman layer, 286–299 Ekman pumping, 296–298 Ekman spiral, 288–296

Literature flux Richardson number, 278 gradient ansatz, 273–275 kinetic energy, 275–280 dissipation, 278 prognostic equation, 275–278 shear production, 278 sources and sinks, 278–280 mixing length, 273–275 Prandtl layer, 281–286 influence of stratification, 283–285 logarithmic wind profile, 282–283 Reynolds equations, 271–273 stress tensor, 272 viscosity, 274 Turbulent diffusivity, 274 Turbulent disipation, 278 Turbulent viscosity, 274 Two-layer model, 202–204

V Velocity in spherical coordinates, 24 Velocity potential, 152 Vertical coordinate arbitrary, 82–85 isentropic coordinate, 120–122 pressure coordinate, 85–86 Viscous stress tensor, 14, 18 Volume forces, 10 Volume heating, 61 Volume work, 39–40 Vortex dynamics, 99–129 primitive equations, 120–129 Vortex line, 100 Vortex tilting, 112 Vortex tube definition, 100 strength, 100 stretching, 111 vorticity flux, 100 Vortical mode shallow-water dynamics, 157 Vorticity, 99–102 absolute, 99 planetary, 100 relative, 100 Vorticity equation, 111 primitive equations, 125 shallow-water dynamics, 136

553 Vorticity flux, 101

W Water spout, 91 Wave action internal gravity waves, 466–476 Rossby waves, 319–320 spectral wave-action density, 480–482 wave-action density in phase space, 480–482 Wave solutions compressible dynamics, 409–431 shallow-water dynamics, 150–165 Weather prediction, 74 WKB theory Fredholm alternative, 467 geostrophic mode, 455 internal gravity waves, 431–495 critical levels, 494 dispersion relation, 456 eikonal equations, 458–460 energy conservation, 483–484 energy flux, 472 higher harmonics, 532–539 impact on synoptic-scale buoyancy, 464 impact on synoptic-scale entropy, 464 impact on synoptic-scale momentum, 466 impact on the quasigeostrophic potential vorticity, 476–479 non-acceleration theorem, 486–490 polarization relations, 456–458 potential vorticity, 485 pseudomomentum, 474, 485 pseudomomentum flux, 473–475, 482 ray equations, 458–460 reflecting levels, 494–495 wave action, 466–476, 480–482 wave-action density, 473, 480 wave-energy theorem, 467–469, 475 Rossby waves, 306–316 critical lines, 315 dispersion relation, 311 eikonal equations, 312 Eliassen-Palm flux, 320–324 ray equations, 312 reflection, 314 wave action, 320–324 Work, 39