THE EQUATIONS OF OCEANIC MOTIONS Modeling and prediction of oceanographic phenomena and climate are based on the integration of dynamic equations. The Equations of Oceanic Motions derives and systematically classifies the most common dynamic equations used in physical oceanography, from those describing large-scale circulations to those describing small-scale turbulence. After establishing the basic dynamic equations that describe all oceanic motions, Müller then derives approximate equations, emphasizing the assumptions made and physical processes eliminated. He distinguishes between geometric, thermodynamic, and dynamic approximations and between the acoustic, gravity, vortical, and temperature salinity modes of motion. Basic concepts and formulae of equilibrium thermodynamics, vector and tensor calculus, curvilinear coordinate systems, and the kinematics of fluid motion and wave propagation are covered in appendices. Providing the basic theoretical background for graduate students and researchers of physical oceanography and climate science, this book will serve as both a comprehensive text and an essential reference. studied physics at the University of Hamburg. He received his Ph.D. in 1974 and his Habilitation in 1981. He worked at Harvard University before moving to the University of Hawaii in 1982, where he is now Professor of Oceanography in the School of Ocean and Earth Science and Technology. His research interests cover a broad range of topics in oceanography, climate dynamics, and philosophy, including wave dynamics, stochastic (climate) models, and foundations of complex system theories. He has published widely on these topics. He is co-author (with Hans von Storch) of the book Computer Modelling in Atmospheric and Oceanic Sciences: Building Knowledge. is the organizer of the Aha Huliko a Hawaiian Winter Workshop series and the chief editor of the Journal of Physical Oceanography. in this web service
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THE EQUATIONS OF OCEANIC MOTIONS PETER MÜLLER University of Hawaii in this web service
cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Mexico City The Edinburgh Building, Cambridge cb2 8ru, UK Published in the United States of America by, New York Information on this title: /9781107410602 P. Müller 2006 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of. First published 2006 First paperback edition 2012 A catalogue record for this publication is available from the British Library Library of Congress Cataloging in Publication Data isbn 978-0-521-85513-6 Hardback isbn 978-1-107-41060-2 Paperback has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. in this web service
Contents Preface page ix 1 Introduction 1 2 Equilibrium thermodynamics of sea water 10 2.1 Salinity 11 2.2 Equilibrium thermodynamics of a two-component system 11 2.3 Potential temperature and density 13 2.4 Equation of state 16 2.5 Spiciness 19 2.6 Specific heat 21 2.7 Latent heat 21 2.8 Boiling and freezing temperature 23 2.9 Chemical potentials 26 2.10 Measured quantities 29 2.11 Mixing 29 3 Balance equations 32 3.1 Continuum hypothesis 32 3.2 Conservation equations 33 3.3 Conservation of salt and water 34 3.4 Momentum balance 36 3.5 Momentum balance in a rotating frame of reference 37 3.6 Angular momentum balance 38 3.7 Energy balance 38 3.8 Radiation 40 3.9 Continuity of fluxes 40 4 Molecular flux laws 43 4.1 Entropy production 43 4.2 Flux laws 44 4.3 Molecular diffusion coefficients 46 v in this web service
vi Contents 4.4 Entropy production and energy conversion 48 4.5 Boundary conditions 49 5 The gravitational potential 53 5.1 Poisson equation 54 5.2 The geoid 56 5.3 The spherical approximation 57 5.4 Particle motion in gravitational field 60 5.5 The tidal potential 61 6 The basic equations 65 6.1 The pressure and temperature equations 66 6.2 The complete set of basic equations 67 6.3 Tracers 69 6.4 Theorems 69 6.5 Thermodynamic equilibrium 72 6.6 Mechanical equilibrium 74 6.7 Neutral directions 76 7 Dynamic impact of the equation of state 77 7.1 Two-component fluids 77 7.2 One-component fluids 78 7.3 Homentropic fluids 79 7.4 Incompressible fluids 81 7.5 Homogeneous fluids 81 8 Free wave solutions on a sphere 84 8.1 Linearized equations of motion 84 8.2 Separation of variables 86 8.3 The vertical eigenvalue problem 87 8.4 The horizontal eigenvalue problem 91 8.5 Short-wave solutions 98 8.6 Classification of waves 101 9 Asymptotic expansions 105 9.1 General method 106 9.2 Adiabatic elimination of fast variables 107 9.3 Stochastic forcing 110 10 Reynolds decomposition 112 10.1 Reynolds decomposition 112 10.2 Reynolds equations 114 10.3 Eddy fluxes 115 10.4 Background and reference state 116 10.5 Boundary layers 117 in this web service
Contents vii 11 Boussinesq approximation 119 11.1 Anelastic approximation 120 11.2 Additional approximations 121 11.3 Equations 122 11.4 Theorems 123 11.5 Dynamical significance of two-component structure 125 12 Large-scale motions 127 12.1 Reynolds average of Boussinesq equations 127 12.2 Parametrization of eddy fluxes 129 12.3 Boundary conditions 133 12.4 Boussinesq equations in spherical coordinates 135 13 Primitive equations 138 13.1 Shallow water approximation 138 13.2 Primitive equations in height coordinates 141 13.3 Vorticity equations 145 13.4 Rigid lid approximation 146 13.5 Homogeneous ocean 147 14 Representation of vertical structure 150 14.1 Decomposition into barotropic and baroclinic flow components 150 14.2 Generalized vertical coordinates 154 14.3 Isopycnal coordinates 157 14.4 Sigma-coordinates 159 14.5 Layer models 160 14.6 Projection onto normal modes 164 15 Ekman layers 169 15.1 Ekman number 169 15.2 Boundary layer theory 170 15.3 Ekman transport 171 15.4 Ekman pumping 172 15.5 Laminar Ekman layers 172 15.6 Modification of kinematic boundary condition 176 16 Planetary geostrophic flows 177 16.1 The geostrophic approximation 177 16.2 The barotropic problem 179 16.3 The barotropic general circulation 183 16.4 The baroclinic problem 185 17 Tidal equations 190 17.1 Laplace tidal equations 190 17.2 Tidal loading and self-gravitation 191 in this web service
viii Contents 18 Medium-scale motions 193 18.1 Geometric approximations 194 18.2 Background stratification 198 19 Quasi-geostrophic flows 201 19.1 Scaling of the density equation 201 19.2 Perturbation expansion 202 19.3 Quasi-geostrophic potential vorticity equation 203 19.4 Boundary conditions 205 19.5 Conservation laws 207 19.6 Diffusion and forcing 208 19.7 Layer representation 210 20 Motions on the f-plane 213 20.1 Equations of motion 213 20.2 Vorticity equations 214 20.3 Nonlinear internal waves 215 20.4 Two-dimensional flows in a vertical plane 216 20.5 Two-dimensional flows in a horizontal plane 216 21 Small-scale motions 218 21.1 Equations 218 21.2 The temperature salinity mode 220 21.3 Navier Stokes equations 221 22 Sound waves 225 22.1 Sound speed 225 22.2 The acoustic wave equation 226 22.3 Ray equations 230 22.4 Helmholtz equation 230 22.5 Parabolic approximation 231 Appendix A Equilibrium thermodynamics 233 Appendix B Vector and tensor analysis 250 Appendix C Orthogonal curvilinear coordinate systems 258 Appendix D Kinematics of fluid motion 263 Appendix E Kinematics of waves 275 Appendix F Conventions and notation 280 References 284 Index 286 in this web service
Preface This book about the equations of oceanic motions grew out of the course Advanced Geophysical Fluid Dynamics that I have been teaching for many years to graduate students at the University of Hawaii. In their pursuit of rigorous understanding, students consistently asked for a solid basis and systematic derivation of the dynamic equations used to describe and analyze oceanographic phenomena. I, on the other hand, often felt bogged down by mere technical aspects when trying to get fundamental theoretical concepts across. This book is the answer to both. It establishes the basic equations of oceanic motions in a rigorous way, derives the most common approximations in a systematic manner and uniform framework and notation, and lists the basic concepts and formulae of equilibrium thermodynamics, vector and tensor analysis, curvilinear coordinate systems, and the kinematics of fluid flows and waves. All this is presented in a spirit somewhere between a textbook and a reference book. This book is thus not a substitute but a complement to the many excellent textbooks on geophysical fluid dynamics, thermodynamics, and vector and tensor calculus. It provides the basic theoretical background for graduate classes and research in physical oceanography in a comprehensive form. The book is about equations and theorems, not about solutions. Free wave solutions on a sphere are only included since the emission of waves is a mechanism by which fluids adjust to disturbances, and the assumption of instantaneous adjustment and the elimination of certain wave types forms the basis of many approximations. Neither does the book justify any of the approximations for specific circumstances. It sometimes motivates but mostly merely states the assumptions that go into a specific approximation. The reason is that I believe (strongly) that one cannot justify any approximation for a specific oceanographic phenomenon objectively. The adequacy of an approximation depends not only on the object, the phenomenon, but also on the subject, the investigator. The purpose of the investigation, whether aimed at realistic forecasting or basic understanding, determines the choice of approximation as much as the phenomenon. The question is not whether a ix in this web service
x Preface particular approximation is correct but whether it is adequate for a specific purpose. This book is intended to help a researcher to understand which assumptions go into a particular approximation. The researcher must then judge whether this approximation is adequate for their particular phenomenon and purpose. All of the equations, theorems, and approximations covered in this book are well established and no attempt has been made to identify the original papers and contributors. Among the people that contributed to the book I would like to acknowledge foremost Jürgen Willebrand. We taught the very first Advanced Geophysical Fluid Dynamics course together and our joint encyclopedia article Equations for Oceanic Motions (Müller and Willebrand, 1989) may be regarded as the first summary of this book. Vladimir Kamenkovich s book Fundamentals in Ocean Dynamics helped me to sort out many of the theoretical concepts covered in this book. I would also like to thank Frank Henyey, Rupert Klein, Jim McWilliams, and Niklas Schneider for constructive comments on an earlier draft; Andrei Natarov and Sönke Rau for help with L A TEX; Martin Guiles, Laurie Menviel-Hessler, and Andreas Retter for assistance with the figures; and generations of students whose quest for rigor inspired me to write this book. in this web service