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The complex flows in the atmosphere and oceans are believed to be accurately modelled by the Navier-Stokes equations of fluid mechanics together with classical thermodynamics. However, due to the enormous complexity of these equations, meteorologists and oceanographers have constructed approximate models of the dominant, large-scale flows that control the evolution of weather systems. The simplifications often result in models that are amenable to solution both analytically and numerically. This volume and its companion explain why such simplifications to Newton's second law produce accurate, useful models and, just as the meteorologist seeks patterns in the weather, mathematicians seek structure in the governing equations. They show how geometry and analysis facilitate solution strategies.
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LARGE-SCALE ATMOSPHERE–OCEAN DYNAMICS, I The Isaac Newton Institute of Mathematical Sciences of the University of Cambridge exists to stimulate research in all branches of the mathematical sciences, including pure mathematics, statistics, applied mathematics, theoretical physics, theoretical computer science, mathematical biology and economics. The research programmes it runs each year bring together leading mathematical scientists from all over the world to exchange ideas through seminars, teaching and informal interaction.
LARGE-SCALE ATMOSPHERE–OCEAN DYNAMICS Volume I Analytical Methods and Numerical Models edited by
John Norbury University of Oxford and
Ian Roulstone Met Office
published by the press syndicate of the university of cambridge The Pitt Building, Trumpington Street, Cambridge, United Kingdom cambridge university press The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 33207, Australia Ruiz de Alarc´ on 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org c
Cambridge University Press 2002
This book 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 Cambridge University Press. Printed in the United Kingdom at the University Press, Cambridge Typeset in 11pt Computer Modern A catalogue record for this book is available from the British Library ISBN 0 521 80681 X hardback
To the memory of Rupert Ford
Contents
Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ix Preface
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
J.C.R. Hunt, J. Norbury and I. Roulstone Introduction and Scientific Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii 1. A view of the equations of meteorological dynamics and various approximations A.A. White . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Extended-geostrophic Euler–Poincar´e models for mesoscale oceanographic flow J.S. Allen, D.D. Holm and P.A. Newberger . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3. Fast singular oscillating limits of stably-stratified 3D Euler and Navier–Stokes equations and ageostrophic wave fronts A. Babin, A. Mahalov and B. Nicolaenko . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .126 4. New mathematical developments in atmosphere and ocean dynamics, and their application to computer simulations M.J.P. Cullen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 5. Rearrangements of functions with applications to