Mankin Mak Mak Mankin Mak Atmospheric Dynamicsm-mak/book.pdf · Atmospheric Dynamics Mankin Mak...

Atmospheric Dynamics

Mankin MakAtmospheric Dynamics

Mak Atm

ospheric Dynamics

Mankin Mak

Atmospheric dynamics is a core component of all atmospheric science curricula. It is concerned with how and why different classes of geophysical disturbances form, what dictates their structure and movement, how the Earth’s uneven surface impacts with them, how they evolve to mature stage, how they interact with the background flow, how they decay and how they collectively constrain the general circulation of the atmosphere.

Mankin Mak’s new textbook provides a self-contained course on atmospheric dynamics. The first half of the book is suitable for undergraduates, and develops the physical, dynamical and mathematical concepts at the fundamental level. The second half of the book is aimed at more advanced students who are already familiar with the basics. The contents have been developed from many years of the author’s teaching at the University of Illinois. The discussions are supplemented with schematics, weather maps and statistical plots of the atmospheric general circulation. Students often find the connection between theoretical dynamics and atmospheric observation somewhat tenuous, and this book demonstrates a strong connection between the key dynamics and real observations in the atmosphere, with many illustrative analyses in the simplest possible model settings. Physical reasoning is shown to be even more crucial than mathematical skill in tackling dynamical problems.

This textbook is an invaluable asset for courses in atmospheric dynamics for undergraduates as well as graduate students and researchers in atmospheric science, ocean science, weather forecasting, environmental science and applied mathematics. Some background in mathematics and physics is assumed.

Cover illustration:

© Science Photo Library.

Cover designed by Hart McLeod Ltd

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Comp. by: Kkavitha Stage: Proof Chapter No.: Front Matter Title Name: MAKPage Number: 0 Date:7/12/10 Time:20:17:02


Atmospheric dynamics is a core component of all atmospheric science curricula. It isconcerned with how and why different classes of geophysical disturbances form, whatdictates their structure and movement, how the Earth's uneven surface impacts with them,how they evolve to mature stage, how they interact with the background flow, how theydecay and how they collectively constrain the general circulation of the atmosphere.

Mankin Mak’s new textbook provides a self-contained course on atmospheric dynamics.The first half of the book is suitable for undergraduates, and develops the physical,dynamical and mathematical concepts at the fundamental level. The second half of thebook is aimed at more advanced students who are already familiar with the basics. Thecontents have been developed from many years of the author’s teaching at the Universityof Illinois. The discussions are supplemented with schematics, weather maps and statisticalplots of the atmospheric general circulation. Students often find the connection betweentheoretical dynamics and atmospheric observation somewhat tenuous, and this bookdemonstrates a strong connection between the key dynamics and real observations in theatmosphere, with many illustrative analyses in the simplest possible model settings.Physical reasoning is shown to be even more crucial than mathematical skill in tacklingdynamical problems.

This textbook is an invaluable asset for courses in atmospheric dynamics for under-graduates as well as graduate students and researchers in atmospheric science, oceanscience, weather forecasting, environmental science and applied mathematics. Some back-ground in mathematics and physics is assumed.

Mankin Mak is a Professor Emeritus in the Department of Atmospheric Sciences at theUniversity of Illinois at Urbana Champaign, where he has taught and researched onatmospheric dynamics for many years. He was born in Hong Kong, China, where hecompleted his high school education, before moving on to undergraduate study in Engin-eering Physics at the University of Toronto, Canada, and graduate study in Meteorology atthe Massachusetts Institute of Technology, USA. He has published extensively in variousinternational journals of atmospheric sciences and served as an editor of the Journal ofAtmospheric Sciences.

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Praise for this book

‘Professor Mak’s new text provides a comprehensive and self-contained introduction toatmospheric dynamics. The latter chapters provide invaluable material on a range of topicsnot found together in other texts: baroclinic lifecycles, the dynamics of stationary waves,and moist baroclinic instability. It presents a forceful case for the ability of dynamicalanalysis to continue to clarify the atmosphere’s behavior.’Dr Daniel Kirk-Davidoff, Department of Meteorology, University of Maryland

‘This book is a thoughtful distillation of decades of teaching. It includes figures fromillustrative computations that are custom-fitted to the mathematical development. Althoughit is brisk and condensed, with so much material to cover, the book has a clear anddistinctive voice, and offers some unique treatments and insights.’Dr Brian Mapes, Rosenstiel School of Marine and Atmospheric Science, Universityof Miami

‘This text neatly and logically outlines atmospheric dynamics, from the fundamentalconcepts in Chapter 1 and the development of the primitive equations in Chapter 2 towave dynamics in Chapter 5 and overturning circulations in Chapter 10. All the while,walking the reader through the derivations and theory, yet referencing the physicalapplications with maps and diagrams or model outputs. It would be a helpful referencefor any graduate student or scientist. Mak’s text would quickly educate someone newto atmospheric science, but ready to jump in to the fray at the graduate level. … It really isa nicely written text.’Dr Teresa Bals-Elsholz, Department of Geography and Meteorology, ValparaisoUniversity

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MANKIN MAKUniversity of Illinois at Urbana-Champaign

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cambridge university press

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore,São Paulo, Delhi, Dubai, Tokyo, Mexico City

Cambridge University PressThe Edinburgh Building, Cambridge CB2 8RU, UK

Published in the United States of America byCambridge University Press, New York

www.cambridge.orgInformation on this title: www.cambridge.org/9780521195737

# Mankin Mak 2011

This publication is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,

no reproduction of any part may take place withoutthe written permission of Cambridge University Press.

First published 2011

Printed in the United Kingdom at the University Press, Cambridge

A catalog record for this publication is available from the British Library

Library of Congress Cataloging-in-Publication Data

Mak, Mankin, 1939–Atmospheric dynamics / Mankin Mak.

p. cm.ISBN 978-0-521-19573-7 (Hardback)

1. Atmospheric physics–Textbooks. 2. Meteorology–Textbooks. I. Title.QC861.3.M335 2011

551.5105–dc222010029996

ISBN 978-0-521-19573-7 Hardback

Additional resources for this publication at www.cambridge.org/9780521195737

Cambridge University Press has no responsibility for the persistence oraccuracy of URLs for external or third-party internet websites referred to

in this publication, and does not guarantee that any content on suchwebsites is, or will remain, accurate or appropriate.

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To the memory of my parents who made it possible for me to pursue my dreams

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Contents

Preface page xi

1 Fundamental concepts and physical laws 11.1 Basic notions 11.2 Laws of mechanics 31.3 Equations of motion in a rotating reference frame 51.4 Forces 141.5 Conservation of mass 171.6 Thermodynamics and equation of state 191.7 Stratification and baroclinicity 231.8 Summary of the equations for a dry atmospheric model 25

2 Basic approximations and elementary flows 272.1 Sphericity of the Earth and thin-atmosphere approximation 272.2 Hydrostatic balance, implications and applications 282.3 Geostrophic balance 382.4 Thermal wind relation 402.5 Balanced flows 412.6 Kinematic properties of wind 492.7 Divergent wind and vertical motion 532.8 Summary: z-, p- and y-coordinates and equations of balance 54

3 Vorticity and potential vorticity dynamics 553.1 Vorticity and circulation of a three-dimensional flow 553.2 Relationship between vorticity and circulation 573.3 Kelvin circulation theorem 603.4 Dynamics of sea-breeze from the circulation perspective 613.5 Tendency of relative circulation 633.6 General vorticity equation 643.7 Vorticity dynamics of a large-scale flow 653.8 Potential vorticity dynamics 703.9 Impermeability theorem and generalized potential vorticity 79

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4 Friction and boundary layers 884.1 Scale and estimate of the frictional force 884.2 Concept of boundary layer 894.3 Reynolds averaging 904.4 Boussinesq approximation 914.5 Flux-gradient theory of turbulence 924.6 Types of boundary layer 934.7 Atmospheric Ekman layer 944.8 Oceanic Ekman layer 1034.9 Surface layer 1074.10 Mixed layer 108

5 Fundamentals of wave dynamics 1115.1 Preliminary remarks 1115.2 Physical nature of internal gravity waves 1135.3 Generic model of IGW 1135.4 Properties of prototype IGW 1195.5 Rudimentary characteristics of wave motions in large-scale flows 1295.6 Physical nature of Rossby waves in a simplest possible model 1315.7 Properties of prototype Rossby waves 1325.8 Forced orographic Rossby waves in a shallow-water model 1415.9 Some observed statistical properties of Rossby waves 1475.10 Edge waves 149

6 Quasi-geostrophic theory and two-layer model 1536.1 Observed features of a synoptic disturbance 1536.2 Scale analysis 1556.3 Quasi-geostrophic system of equations 1576.4 Diagnostic function of the QG theory 1606.5 Prognostic function of the QG theory 1656.6 Intrinsic wave modes in a QG model 1686.7 Evolution of a baroclinic jet streak in a quasi-geostrophic two-layer model 1726.8 Influences of the Earth’s sphericity in the QG theory 182

7 Dynamic adjustment 1877.1 Problem of rotational adjustment 1877.2 Rossby problem of geostrophic adjustment 1887.3 Complementary Rossby problem 1967.4 Gradient-wind adjustment problem 201

Concluding remarks 207

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8 Instability theories 209

Part 8A Small-scale and meso-scale instability 209

8A.1 Static instability and the impact of damping 2108A.2 Inertial instability and an application 2138A.3 Symmetric instability and an application 220

Part 8B Purely barotropic and purely baroclinic instability 225

8B.1 Historical highlights of past studies 2258B.2 Aspects of barotropic instability 2278B.3 Optimal growth of barotropic disturbance 2358B.4 Baroclinic instability in a two-layer QG model 2408B.5 Modal growth 2458B.6 Transient growth 2568B.7 Optimal growth 2598B.8 Wave-activity density and general necessary condition for instability 262


Part 8C Instability of jets 265

8C.1 Nature and scope of the problem 2658C.2 Barotropic-governor effect 2678C.3 Instability of baroclinic jets 2708C.4 Instability of a localized barotropic jet 2768C.5 Instability of a localized baroclinic jet 282


Part 8D Moist baroclinic instability 290

8D.1 Introductory remarks 2908D.2 A particular instability analysis 292


9 Stationary planetary wave dynamics 3099.1 Observed characteristics of stationary planetary waves 3099.2 Introductory remarks about the dynamics of stationary waves 3119.3 Impact of basic zonal flow on wave propagation 3139.4 Thermally forced stationary waves 3209.5 Orographically forced stationary waves 3339.6 Illustrative application: mean Asian monsoonal circulation 341


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10 Wave-mean flow interaction 35010.1 Eulerian mean meridional overturning circulation 35010.2 Lagrangian mean meridional overturning circulation 35510.3 Linear theory for the overturning circulation 35610.4 Non-Acceleration Theorem 37310.5 Stratospheric sudden warming 375

11 Equilibration dynamics of baroclinic waves 38211.1 Introductory remarks 38211.2 Rudiments of geostrophic turbulence in a two-layer model 38511.3 Life cycle of baroclinic waves 39311.4 Symbiotic relation between synoptic and planetary waves 40311.5 Relative intensity of the winter storm tracks 410

12 Nongeostrophic dynamics 41912.1 Surface frontogenesis 41912.2 Hadley circulation 42812.3 Non-supercell tornadogenesis 443

Appendix: Mathematical tools 458References 476Index 481The color plates are situated between pages 370 and 371.

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Preface

Atmospheric dynamics is the foundation for understanding the movement of air currents.As such, it is a core component of all atmospheric science curricula. Atmospheric dynam-ics is the discipline concerned with what different classes of geophysical disturbances aremade up of, how and why they form, what factors dictate their structure and movement,how the Earth’s uneven surface impacts them, how they evolve to their mature stage, howthey interact with the background flow, how they eventually decay and, above all, howthey collectively constrain the atmospheric general circulation as a whole. An analysis ofatmospheric dynamics can be process-oriented trying to address one or more of thequestions above. Its goal would be to establish quantitative understanding of the natureof those processes. An analysis of atmospheric dynamics can be also phenomenon-orientedwith the objective of developing a feel for why a specific phenomenon is as observed. Westrive to cover both aspects of atmospheric dynamics in this book.

This book is mainly intended to serve students of atmospheric sciences. The goal is tocome up with a book distinctly different from but complementary to the existing pool oftexts on atmospheric dynamics. This book aims at being a resource valuable to instructorsand self-explanatory to students as much as possible. It would be a bonus if weatherforecasters as well as some practitioners in atmospheric science, ocean science, environ-mental science and applied mathematics find this book to be a useful reference.

Progress in atmospheric dynamics would not have been possible without the introduc-tion of various insightful concepts and approximations. They greatly simplify the quanti-tative investigations of atmospheric disturbances with analytical and numerical methods.Computer graphics help visualize the structures and processes. For this reason, all of themreceive due emphasis in the discussions. We extensively use schematics to depict theconceptual issues. We use different means to strengthen the connection between theoreticaldynamics and atmospheric observations. For instance, real-time weather maps are gener-ated to illustrate the various dynamical characteristics of atmospheric disturbances. Thespatial and temporal statistical properties of various classes of disturbances are examinedas background information for related theoretical discussions of their dynamical roles.Analyses with idealized models are used to highlight the essence of the phenomena underconsideration.

The first half of the book discusses the physical concepts of atmospheric dynamics aswell as the mathematical methodology at the fundamental level. The emphasis is toelucidate the physical nature of dynamical processes. The materials are written withundergraduates in mind who might have only cursory exposure to atmospheric sciences.With the exception of several sections on the more advanced topics, the first five chapterscan be used in an introductory course on atmospheric dynamics. It would be helpful and/or

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necessary for an instructor to elaborate on the more technical aspects of the materials forthe benefit of the uninitiated students. The more advanced topics are best reserved for afollow-up course of atmospheric dynamics. A second undergraduate course may beginwith a review of the basic materials at a faster pace and proceed to cover a number ofadditional topics selected from the first eight chapters.

The second half of the book is suitable for graduate students who are already familiarwith the basics. It is a collection upward of twenty-five illustrative model analyses of awide variety of dynamical issues manifested in different interesting phenomena. Thematerials may be used as a graduate course sometimes called “Special Topics of Atmos-pheric Dynamics.” The overriding objective is to show how one might tackle problemsconcerning the dynamics of different types of disturbances with the use of simple models.I show the merit of different approaches and mathematical methods dependent on thenature of the problem under consideration. Through the sample analyses, I hope to conveythe idea that physical reasoning is even more crucial than mathematical skill in tacklingdynamical problems. The discussions of several topics extensively make use of publishedresearch articles. The other illustrative model analyses and computations are specificallyperformed for this book. The physical meanings of the model results are discussed withgenerous use of figures. I strive to present each model analysis in a self-contained manner,so that they need not be read in sequence. The format of presentation in the second half ofthe book is therefore quite different from that used in the first half. These analyses areconsiderably more involved than typical homework assignments. They are more akin toterm projects. I would encourage students, either individually or in groups, to reproducesome of the results as challenging exercises.

The book has twelve chapters starting with a synopsis in each:

• Chapter 1 introduces the basic concepts such as the continuum hypothesis of a fluid,Lagrangian versus Eulerian descriptions, non-inertial reference frame, stratification andbaroclinicity. It reviews all the laws of physics needed for investigating the dry dynamicsof atmospheric flows.

• Chapter 2 discusses the nature of different types of dynamical balance in atmosphericflows. The governing equations in different coordinate systems for atmospheric analysesare derived. We also elaborate on the general kinematic properties of atmospheric flows.

• Chapter 3 is a lengthy chapter focusing on the rotational properties of a geophysical flow.We discuss the concepts of vorticity, circulation and potential vorticity in conjunctionwith the related governing equations and mathematical theorems. The roles of differentphysical factors are delineated. This chapter also contains a detailed discussion of someadvanced topics such as the impermeability theorem for potential vorticity and the notionof generalized potential vorticity for succinctly representing the impact of boundary fromthe perspective of vorticity dynamics.

• Chapter 4 discusses the impacts of small turbulent eddies on a background flow as africtional force. The effect of such turbulence is discussed in the context of a simpleparameterization. We introduce the notion of boundary layer. We analyze the structuresof several different types of boundary layers and their implications, especially theatmospheric and oceanic Ekman layers.

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• Chapter 5 discusses the fundamentals of wave dynamics concerned with internal gravitywaves and Rossby waves. The characteristics of those free wave modes about theirstructure, propagation, dispersion and energetics are quantitatively examined. We alsodelineate the dynamical nature of the forced wave modes with two illustrative analyses.We examine some observed properties of Rossby waves in the format of generalcirculation statistics. We finally discuss the dynamics of edge waves.

• Chapter 6 presents the quasi-geostrophic theory for large-scale atmospheric flows, whichheralded the major advances of dynamic meteorology. After showing some real-timeweather maps of a representative event, we discuss the formulation of this theory. Theparticular form of this theory in a two-layer model setting is presented next. We illustrateall the dynamical concepts with an analysis of a baroclinic jet streak. The influences ofthe Earth’s sphericity in the QG theory are finally discussed.

• Chapter 7 discusses how and why the velocity and pressure fields of a large-scaleatmospheric flow would rapidly adjust towards a new balanced state whenever theirexisting balance is upset by unspecified causes. We analyze the adjustments from twocanonical forms of initial imbalance.

• Chapter 8 is a lengthy chapter divided into four parts for clarity. It covers a potpourri ofinstability theories for disturbances of widely different sizes arising from differentclasses of shear flows. The first part deals with the instability of small-scale and meso-scale disturbances. The second and third parts discuss the transient and modal dynamicsof cyclogenesis in the extratropics for basic flows of increasing structural complexity.The fourth part discusses the large-scale moist instability arising from self-inducedcondensational heating.

• Chapter 9 discusses the observed characteristics and the dynamics of stationary planetarywaves. We examine the propagation of such waves through a weak horizontal or verticalbackground shear flow. The similarities and differences in the responses to large-scalethermal and/or topographic forcing are then delineated. An analysis of the summer meanAsian monsoon as a forced circulation serves to illustrate the dynamical properties of thisimportant class of wave disturbances in a recognizable setting.

• Chapter 10 addresses the dynamics of interaction between a zonal mean flow and large-scale waves in the context of two intriguing phenomena. We begin by examining themean meridional overturning circulation in both Eulerian and Lagrangian sensesdeduced from a global dataset. We perform a linear model analysis of them with theuse of empirical forcing. Wave-mean flow interaction is then discussed in the contextof stratospheric sudden warming.

• Chapter 11 discusses the equilibration dynamics of nonlinear baroclinic waves. We beginwith a discussion of the rudiments of geostrophic turbulence in a two-layer modelsetting. It serves as background information for three specific analyses. The first analysisdelineates the dynamics of life cycle of baroclinic waves. The second analysis brings tolight a symbiotic relation between synoptic-scale waves and planetary-scale waves ofa forced dissipative system. The third analysis looks into the dynamical nature of therelative intensity of the two major winter storm tracks.

• Chapter 12 discusses the nature of nongeostrophic dynamics in the context of three verydifferent phenomena: surface frontogenesis, Hadley circulation and non-supercell

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tornadogenesis. We present three illustrative model analyses of them using very differentmathematical methods.

• An appendix summarizes the mathematical tools used in the book as a quick referencefor some readers who might not have used them for a while.

• Problems and exercises are posted online in the website of the Cambridge UniversityPress with the intention of updating them periodically in future. See www.cambridge.org/9780521195737

• Only the research articles and books that readily came to mind during the writing of thisbook are included. I have made no special effort to comprise a comprehensive bibliog-raphy. There is no doubt that the resulting list of references is quite incomplete.

The framework of this book has been gradually taking shape over the course of manyyears of teaching and research at the University of Illinois, Urbana-Champaign, Illinois,USA. Zhenhua Li, Xian Lu, Sara T. Strey, Lantao Sun and Andy Vanloocke provided mewith feedback on a number of chapters in the first half of the book from the students’perspective. Professors Ming Cai, Yi Deng and Lin Wang at different universities kindlygave me feedback on three chapters from the instructors’ perspective. A number ofanonymous reviewers solicited by the publisher provided some valuable suggestions. Allthis feedback has prompted significant revisions of the draft and for that, I am mostappreciative. Joseph Grim and David Wojtowicz helped produce the high-resolution real-time weather maps. Lusheng Liang made the diagnosis of a five-year global dataset andwrote the code of model calculation that produces the figures in Chapter 10. Their technicalcontributions are indispensable and gratefully acknowledged. Finally, I would like to thankthe editorial staff of Cambridge University Press – Matt Lloyd, Chris Hudson, Laura Clarkand Sabine Koch – for their inputs and suggestions that helped make writing this booka wonderful experience of going through the maze of publication protocol.

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