THE EQUATIONS OF OCEANIC MOTIONS

Transcription

1 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 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

2 in this web service

3 THE EQUATIONS OF OCEANIC MOTIONS PETER MÜLLER University of Hawaii in this web service

4 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: / 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 Hardback isbn 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

5 Contents Preface page ix 1 Introduction 1 2 Equilibrium thermodynamics of sea water Salinity Equilibrium thermodynamics of a two-component system Potential temperature and density Equation of state Spiciness Specific heat Latent heat Boiling and freezing temperature Chemical potentials Measured quantities Mixing 29 3 Balance equations Continuum hypothesis Conservation equations Conservation of salt and water Momentum balance Momentum balance in a rotating frame of reference Angular momentum balance Energy balance Radiation Continuity of fluxes 40 4 Molecular flux laws Entropy production Flux laws Molecular diffusion coefficients 46 v in this web service

6 vi Contents 4.4 Entropy production and energy conversion Boundary conditions 49 5 The gravitational potential Poisson equation The geoid The spherical approximation Particle motion in gravitational field The tidal potential 61 6 The basic equations The pressure and temperature equations The complete set of basic equations Tracers Theorems Thermodynamic equilibrium Mechanical equilibrium Neutral directions 76 7 Dynamic impact of the equation of state Two-component fluids One-component fluids Homentropic fluids Incompressible fluids Homogeneous fluids 81 8 Free wave solutions on a sphere Linearized equations of motion Separation of variables The vertical eigenvalue problem The horizontal eigenvalue problem Short-wave solutions Classification of waves Asymptotic expansions General method Adiabatic elimination of fast variables Stochastic forcing Reynolds decomposition Reynolds decomposition Reynolds equations Eddy fluxes Background and reference state Boundary layers 117 in this web service

7 Contents vii 11 Boussinesq approximation Anelastic approximation Additional approximations Equations Theorems Dynamical significance of two-component structure Large-scale motions Reynolds average of Boussinesq equations Parametrization of eddy fluxes Boundary conditions Boussinesq equations in spherical coordinates Primitive equations Shallow water approximation Primitive equations in height coordinates Vorticity equations Rigid lid approximation Homogeneous ocean Representation of vertical structure Decomposition into barotropic and baroclinic flow components Generalized vertical coordinates Isopycnal coordinates Sigma-coordinates Layer models Projection onto normal modes Ekman layers Ekman number Boundary layer theory Ekman transport Ekman pumping Laminar Ekman layers Modification of kinematic boundary condition Planetary geostrophic flows The geostrophic approximation The barotropic problem The barotropic general circulation The baroclinic problem Tidal equations Laplace tidal equations Tidal loading and self-gravitation 191 in this web service

8 viii Contents 18 Medium-scale motions Geometric approximations Background stratification Quasi-geostrophic flows Scaling of the density equation Perturbation expansion Quasi-geostrophic potential vorticity equation Boundary conditions Conservation laws Diffusion and forcing Layer representation Motions on the f-plane Equations of motion Vorticity equations Nonlinear internal waves Two-dimensional flows in a vertical plane Two-dimensional flows in a horizontal plane Small-scale motions Equations The temperature salinity mode Navier Stokes equations Sound waves Sound speed The acoustic wave equation Ray equations Helmholtz equation 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

9 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

10 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