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Infinite-dimensional dynamical systems in atmospheric and oceanic science. (English) Zbl 1303.86002

Hackensack, NJ: World Scientific; Hangzhou: Zhejiang Science and Technology Publishing House (ISBN 978-981-4590-37-2/hbk; 978-7-5341-5935-0/hbk; 978-981-4590-39-6/ebook). x, 318 p. (2014).
The long-term numerical weather prediction and the development of atmospheric circulation and climate models are very complicated tasks of the geophysical fluid dynamics. They are demanding for good models of different factors driving weather and climate, as the Earth’s surface, the wind systems, the heat and radiation transport, the atmospheric chemistry and the occurrence of eddy turbulence. All these factors have to be joint on to the general system of hydrodynamic equations, that means to nonlinear partial differential equations with initial-boundary value conditions.
The aim of the present book is the introduction of new results of studies of partial differential equations and infinite-dimensional dynamical systems into the geophysical fluid dynamics. In doing so, the authors focus on large-scale phenomena in the Earth’s atmosphere and the oceans.
The book consists of five chapters. In chapter 1, the basic hydrodynamic equations of the atmospheric and the ocean motion are introduced. These are the atmospheric continuity and momentum conservation equations taking into account gravity, molecular viscosity of the air, and the Earth’s rotation. Further, the atmospheric thermodynamic equations of dry air and of moist air are given. As basic hydrodynamic equations of the ocean, continuity, momentum conservation, thermodynamic equation, salinity conservation equation and the equation of state are considered. Then all the ocean equations are written in Boussinesq approximation (the mass density is not constant only in the gravity term and in the equation of state), and the equation of state is replaced by a temperature and entropy dependent empirical formula.
Based on this system of equation further mathematical relations are derived. The equations of the atmosphere and the Boussinesq approximation of the ocean equations are presented in spherical coordinates. Under the condition that the hydrostatic vertical momentum approximation is applicable, the isobaric surface coordinate system is discussed. For practical cases when the topography of the Earth’s surface is of importance or boundary conditions at the surface have to be considered, the equations of the atmosphere are given in the topography coordinate frame. The equations are also represented in a local rectangular coordinate system. Further, the expressions of the atmosphere and the ocean for gravity stable systems are given, including the dry atmospheric primitive equations. Finally, useful boundary conditions of the atmosphere and the oceans are explained.
As the basic and primitive hydrodynamic equations introduced in chapter 1 are yet very complicated, researchers have to find yet simpler models, which reflect nevertheless some physical essence. This is valid for the quasi-geostrophical models of the atmospheric and ocean motions, which are widely used in geophysics, despite the enormous development of computer technique. Quasi-geostrophic behaviour of a fluid means, that the advective derivative terms in the momentum equation are an order of magnitude smaller than the Coriolis and the pressure gradient forces. Thus in chapter 2, various quasi-geostrophic models are discussed. The author starts with the introduction of the (hyperbolic) barotropic (or shallow-water) model, a model describing the dynamics of a homogeneous incompressible inviscid rotating shallow fluid with a free surface. Then the two-dimensional quasi-geostrophic system of hydrodynamic equations is found based on the barotropic model. For the derivation of the three-dimensional quasi-geostrophic equation the model of the adiabatic frictionless dry atmosphere in the local rectangular frame is applied. It follows the description of two-layer quasi-geostrophic models and of the surface quasi-geostrophic model. Some numerical results of the surface quasi-geostrophic models are explained, e.g.the existence of local smooth solutions and the global well-posedness of the critical dissipative surface quasi-geostrophic equation.
Chapter 3 considers the initial value boundary problem for the three-dimensional viscous primitive equations of the large-scale moist atmosphere and oceanic science. There a global existence of weak solutions and of trajectory attractors of the primitive equations of the moist atmosphere are obtained. Further, the global existence, uniqueness, stability and long-time behavior of the strong solutions to the initial boundary value problem of the primitive equations of the large-scale atmosphere are discussed. In doing so, a very important concept of an infinite-dimensional dynamical system, i.e.the global attractor based on semigroup is explained, and two theorems about the existence of global attractors are recalled. The global well-posedness of the equations of the large-scale dry atmosphere and of the viscous three-dimensional equations of large-scale oceans are discussed.
Solar radiation and effects of the Earth’s surface on the atmosphere are random. Thus, in the past decade, researchers began to pay attention to the study of stochastic climate models. Therefore, chapter 4 introduces the qualitative theory of the random dynamic system of the atmosphere and oceans. The two-dimensional quasi-geostrophic equation with random external force (i.e.with wind stress on the upper surface of the homogeneous fluid), friction and dissipation are derived. The global well-posedness of the primitive equations of large-scale oceans with random external forces and the existence of the global attractors of the corresponding infinite-dimensional random dynamic systems are studied for the first time (ibid.,p. 166). Also the qualitative theory of the ocean primitive equations with random boundary conditions, representing the random atmospheric influence, are discussed.
Chapter 5 is devoted to the theory of stability and instability of atmospheric and oceanic waves. Phenomena like neutral, linear, formal, and nonlinear stability in geophysics fluid dynamics are explained. Several methods of the stability analyses as the normal mode approach, the energy principle, and the variational method are applied. For instance, the stability of internal gravity waves (without Coriolis force) and internal inertial gravity waves (that means the influence of the Coriolis force is taken into account additionally) is studied using the normal mode approach. Necessary conditions for the instability of Rossby waves, which occur due to shear in rotating fluids, are derived using the normal mode ansatz and the energy principle. Growth rates of barotropic instable Rossby waves (the density is pressure dependent) and instabilities of the simplest baroclinic Rossby waves (the density is pressure and temperature dependent) are discussed. The stability of two-dimensional quasi-geostrophic flows is studied. Finally, critical Rayleigh numbers for Rayleigh-Bénard convection with respect to a nonlinear instability, are given.
The present work is intended for mathematicians and physicists developing climate and circulation models of planetary atmospheres and oceans.

MSC:

86-02 Research exposition (monographs, survey articles) pertaining to geophysics
86A05 Hydrology, hydrography, oceanography
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
00A06 Mathematics for nonmathematicians (engineering, social sciences, etc.)
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