This book presents a geometric theory for incompressible flow and its applications to fluid dynamics. The main objective is to study the stability and transitions of the structure of incompressible flows, and applications to fluid dynamics and geophysical fluid dynamics. The development of the theory and its applications has gone well beyond the original motivation, which was the study of oceanic dynamics. One such development is a rigorous theory for boundary layer separation of incompressible fluid flows. This study of incompressible flows has two major parts, which are interconnected. The first is the development of a global geometric theory of divergence-free fields on general two-dimensional compact manifolds. The second is the study of the structure of velocity fields for two-dimensional incompressible fluid flows governed by the Navier-Stokes equations or the Euler equations. Motivated by the study of problems in geophysical fluid dynamics, the program of research in this book seeks to develop a new mathematical theory, maintaining close links to physics along the way. In return, the theory is applied to physical problems, with more problems yet to be explored.
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