Swaters - Introduction to Hamiltonian Fluid Dynamics and Stability Theory

INTRODUCTION TO

HAMILTONIAN FLUID

DYNAMICS

AND

STABILITY THEORY

GORDON E. SWATERS

CHAPMAN & HALL/CRC

Monographs and Surveys in Pure and Applied Mathematics

Main Editors

H. Brezis, Universite de Paris

R.G. Douglas, Texas A&M University

A. Jeffrey, University of Newcastle upon Tyne (Founding Editor)

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R. Aris, University of Minnesota

G. I. Barenblatt, University of Cambridge

H. Begehr, Freie Universitt Berlin

P. Bullen, University of British Columbia

RJ. Elliott, University of Alberta

R.P. Gilbert, University of Delaware

R. Glowinski, University of Houston

D. Jerison, Massachusetts Institute of Technology

B. Lawson, State University of New York

B. Moodie, University of Alberta

S. Mori, Kyoto University

L.E. Payne, Cornell University

D.B. Pearson, University of Hull

I. Raeburn, University of Newcastle

G.F. Roach, University of Strathclyde

I. Stakgold, University of Delaware

W.A. Strauss, Brown University

J. van der Hoek, University of Adelaide

INTRODUCTION TO

HAMILTONIAN FLUID

DYNAMICS

AND

STABILITY THEORY

GORDON E. SWATERS

CRC Press

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Contents

About the author

Gordon E. Swaters is Professor of Mathematical Sciences and Director of the Applied Mathematics Institute at the UNIVERSITY OF ALBERTA. He is also an adjunct Professor in the Department of Earth & Atmospheric Sciences.

Gordon obtained the B. Math. (Hons.) degree from the UNIVERSITY OF WATERLOO and the Ph.D. from the UNIVERSITY OF BRITISH COLUMBIA.

Gordons research is focussed on understanding nonlinear wave and eddy processes in planetary-scale atmosphere and ocean dynamics. In particular, he has made contributions to understanding the destabilization of buoyancy and density-driven ocean currents for which he was awarded the 1994 Presidents Prize of the Canadian Meteorological and Oceanographic Society (CMOS) the most prestigious research award given by that society. Ten years earlier, in 1984, Gordon was awarded the Graduate Student Prize by CMOS for showing that the formation of large scale ocean eddies in the northeast Pacific ocean could be correlated to current changes forced by El Nino/Southern Oscillation events in the tropical Pacific.

In addition, Gordon has been a recipient of the Faculty of Science Research Award, McCalla Research Professorship and a Killam Annual Professorship at the UNIVERSITY OF ALBERTA.

Gordon was also a member of the Canadian Scientific Team of the Surface Velocity Drifter Program part of the World Ocean Circulation Experiment. This multinational research program made an important contribution in helping to understand the role of the oceans in Earths evolving climate.

Acknowledgments

Science is not a competition between individuals to see who gets the biggest grants or publishes the most papers. Blair Kinsmans eloquent preface to his book Water Waves expressed the point that Science, in addition to its obvious utilitarian importance in human affairs, is about a community of men and women engaged in conversation. There are two members of this community that I would like to publicly acknowledge here. The first is Lawrence Mysak who taught me that the give and take of this conversation is not about who is better, but is simply the process of discovering the truth. The second is Paul LeBlond who taught me not to confuse the language of this conversation for the conversation itself. It is the spirit of community in Science that I cherish the most and I thank Lawrence and Paul for introducing me to it.

I would also like to thank the Natural Sciences and Engineering Research Council of Canada, the Department of Fisheries and Oceans of Canada, and the Atmospheric Environment Service of Canada for their financial support of my research.

The study of the dynamics of fluids remains one of the most difficult areas in all of science. From the generation of water waves seen at the beach, to the formation of the Great Red Spot on Jupiter, there are no completely accepted explanations. In some respects this may seem quite remarkable since, so far as we are concerned here, the governing equations of fluid dynamics, the Navier-Stokes equations have been known for well over a hundred and fifty years (see the remarks by Lamb, 1932 concerning the contributions of Navier, 1827; Poisson, 1831; de Saint Venant, 1843; Stokes, 1845). The Navier-Stokes equations are a system of nonlinear partial differential equations in which the nonlinear terms play an essential role in determining the evolution of the flow. The enormous technical and conceptual difficulties associated with understanding nonlinear dynamical systems may help to explain why fluid dynamics temporarily lost its appeal as a fashionable research area in the middle third of the 20