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Wave packets and their bifurcations in geophysical fluid dynamics. (English) Zbl 0744.76010

Applied Mathematical Sciences. 85. New York etc.: Springer-Verlag. vii, 247 p. (1991).
The book has eight chapters. Chapter 1 is the shortest one and presents some background in geophysical fluid dynamics, from the basic laws to various approximations of such equations as the Boussinesq approximation, the earth’s \(f\)-plane and \(\beta\)-plane approximations, and the earth’s \(\delta\)-surface approximation. In chapter 2 the theory of wave packet is developed systematically. The chapter begins with the wave packet representation and then discusses the asymptotic solution of wave equations and the WKB method and their relationship. General theorems of the wave packet are also addressed, including an example of a Rossby wave packet, the eikonal equation, the theory in stationary and moving media, wave-action conservation, and the energy exchange with basic flow.
Chapters 3 to 6 are devoted to discussions of the evolution of the wave packet in barotropic basic flows, with the emphasis on the structure and structural change of the wave packet. Chapter 3 introduces the inviscid shallow-water model and potential vorticity equation. Equations governing the evolution of the Rossby wave packet are derived. Integral properties of a wave packet are discussed, including extensions of the Rayleigh theorem and the Fjørtoft theorem. Chapter 4 introduces the concept of the wave packet’s structural vacillation (or oscillation) and considers its relevance to vacillations observed in geophysical fluids. Chapters 5 and 6 consider the bifurcation behavior of wave packets by the use of bifurcation diagrams and phase space diagrams. The dynamic behavior of the structure of a wave packet is discussed.
Chapter 7 extends earlier studies in barotropic basic flows to the case of three-dimensional wave packets in stratified baroclinic basic flows. The structures of developing and decaying wave packets and bifurcation properties are investigated in such cases as the stably stratified and unstably stratified purely barotropic basic flows, the purely baroclinic basic flows, and the barotropic and baroclinic basic flows. The last chapter deals with the theory of wave packet propagation associated with teleconnections observed in geophysical fluids.

MSC:

76-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to fluid mechanics
76Bxx Incompressible inviscid fluids
37G99 Local and nonlocal bifurcation theory for dynamical systems
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
86A10 Meteorology and atmospheric physics
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