This book provides an accessible introduction to a new set of methods for the analysis of Lagrangian motion in geophysical flows. These methods were originally developed in the abstract mathematical setting of dynamical systems theory, through a geometric approach to differential equations. Despite the recent developments in this field and the existence of a substantial body of work on geophysical fluid problems in the dynamical systems and geophysical literature, this is the first introductory text that presents these methods in the context of geophysical fluid flow. The book is organized into seven chapters; the first introduces the geophysical context and the mathematical models of geophysical fluid flow that are explored in subsequent chapters. The second and third cover the simplest case of steady flow, develop basic mathematical concepts and definitions, and touch on some important topics from the classical theory of Hamiltonian systems. The fundamental elements and methods of Lagrangian transport analysis in time-dependent flows that are the main subject of the book are described in the fourth, fifth, and sixth chapters. The seventh chapter gives a brief survey of some of the rapidly evolving research in geophysical fluid dynamics that makes use of this new approach. Related supplementary material, including a glossary and an introduction to numerical methods, is given in the appendices.

This book will prove useful to graduate students, research scientists, and educators in any branch of geophysical fluid science in which the motion and transport of fluid, and of materials carried by the fluid, is of interest. It will also prove interesting and useful to the appliedmathematicians who seek an introduction to an intriguing and rapidly developing area of geophysical fluid dynamics. The book was jointly authored by a geophysical fluid dynamicist, Roger M. Samelson of the College of Oceanic and Atmospheric Sciences at Oregon State University, USA and an applied mathematician, Stephen Wiggins of the School of Mathematics, University of Bristol, UK.

Introduction.- Steadily Translating Waves and Meanders.- Integrability of Lagrangian Motion.- Fluctuating Waves and Meanders.- Material Manifolds, Flow Regimes, and Fluid Exchange.- Lobe Transport and Flux.- Transport and Dynamics.- A Mathematical Properties of Fluid Trajectories.- B Action-Angle Coordinates.- C Numerical Methods.- D Finite-Time Material Manifolds: An Example.- E Glossary.- Index.

In many oceanic and atmospheric flows, the collection of trajectories of fluid elements the Lagrangian motion is of fundamental importance, determining the transport of fluid properties as well as basic characteristics of the flow itself. In the presence of large-scale, coherent structures, this motion can be analyzed using methods developed originally in the geometric theory of ordinary differential equations. The last two decades have seen the rapid development of this dynamical systems approach to Lagrangian transport for the analysis of engineering and geophysical fluid flows. Geophysical fluid problems presented a new set of mathematical and computational challenges for this approach; in turn, these methods have provided new perspectives on Lagrangian motion in geophysical flows. This is the first introductory textbook to describe this approach in the context of geophysical fluid dynamics. It develops the essential mathematical and physical concepts systematically and in parallel, in an accessible style, and will serve both to introduce the necessary mathematical framework to geophysical fluid dynamicists, and to introduce an active area of geophysical fluid research to applied nonlinear dynamicists. The text is suitable for graduate and advanced undergraduate students, and will also serve as a useful reference for researchers and scientists in either field.