bstract -For nonlinear dissipative systems with high frequency forcing the closeness of attractors of original and averaged systems is proved. As examples the equations of vorticity on the sphere and 2-D Navier-Stokes equations are considered. An analysis of the averaged Reynolds-type equations of vorticity on the sphere is given. Under certain assumptions one can determine constructive relations between the singular basis of the obtained linear operator and empirical orthogonal components of the low frequency atmospheric processes.In the past years one of the major problems in geophysical hydrodynamics has become the problem of studying the low frequency variability in the atmospheric processes.Here the difficulty is that the time spectrum of the atmospheric processes is continuous and does not have pronounced minima and maxima on the time intervals ranging from the characteristic synoptic scale to a period of several months.(Recall that the characteristic period of the low frequency variability comprises no less than 10 days while the time scale of the synoptic processes is 2-6 days).On the other hand, when studying the combined atmosphere-ocean system on large time intervals we commonly use the approach in which the atmosphere is considered to be a 'fast' medium while the ocean is considered to be a 'slow* medium. In this approach the problem of studying the ocean dynamics is reduced to a dynamic problem with fast oscillating external forcing. The main question here, from the viewpoint of the mathematical climate theory, is that of the closeness between the attractors of the initial and time-average systems of equations. In this work we study two problems that belong to the above problem of describing the low frequency variability of the atmospheric processes and to the general problem of averaging the systems of equations with fast oscillating forcing.When describing the low frequency variability of the atmospheric processes we have to make some assumptions as to the 'slowness' of the low frequency processes.In the context of these assumptions we may reduce the problem to a linear system of equations and study the possible relations between the singular functions of the corresponding linear operator and the empirical orthogonal components of the low frequency atmospheric processes.The paper consists of two parts. In Section 1 we consider a rather general class of nonlinear dissipative systems with fast oscillating forcing. We prove the theorem on closeness between the solutions of the initial and averaged systems on asymptotically * The work was supported by the Russian Foundation for the Basic Research (94-05-17715). ' Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow 117334, Russia Brought to you by | University of Queensland -UQ Library Authenticated Download Date | 6/15/15 6:04 AM
The paper deals with the formulation of the problem of estimating the stability of equivalently barotropic stationary solutions of equations describing the baroclinic atmosphere by the linear approximation of a nonlinear barotropic problem. Restrictions are imposed on the right-hand side for investigating the stability of the solutions of this class to continuously-acting perturbations.
The inertial manifold is proven to exist for the vorticity equation on a rotating sphere. The estimates of its dimension and the rate at which the trajectories are attracted to the manifold are given. The dependence of the properties of the equation on the parameter s (the degree of the Laplacian operator appearing in the term that describes the turbulent viscosity) is studied.
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