Nonlinear Rossby Waves In Geophysical Fluid

In this dissertation, based on the quasi-geostrophic potential vorticity equation, we investigate the evolution of the amplitude of Rossby waves with the generalized beta-plane approximation by employing the reduc-tive perturbation method and multiple scale method (including topography problems; slowly changing topography problems; external source and dissi-pation problems; stratification effect), equatorial envelope Rossby solitons and action of the horizontal component of the earth’s rotation on nonlinear Rossby waves in barotropic fluid and stratified fluid.The basic features of geophysical fluid are spherical effect and ro-tational effect of the earth’s. In1939, Rossby in his article "Relation between variations in the intensity of the zonal circulation of the atmo-sphere and the displacements of the semi-permanent centers of action"[1] suggested the study of a homogeneous sheet of fluid on a sphere as the simplest relevant model for the dynamics of the observed large-scale waves in the earth’s atmosphere, namely that the effect of spherical earth so that only the local vertical component of the earth’s planetary vorticity another plane f=2Ω sin φ simulation mode, but it is dynamically significant. Such the model, in which the effect of earth’s sphericity is modeled by a linear variation of f in an otherwise planar geometry, is called the β model (β plane approximation), where the parameter f is the local component of the planetary vorticity normal to the earth’s surface and is called the Cori-olis parameter, Ω, φ is the earth rotation angular velocity and latitude. He pointed out that the atmospheric long wave is due to f with latitude φ change(i.e. β effect). In order to confirm Rossby’s this argument is rationali-ty, Many scholars[13,21,56-60]have studied the Rossby waves problems in β plane approximation, they give Rossby waves amplitude satisfy Korteweg-de Vries(KdV)equation, the modified Korteweg-de Vries(mKdV) equation for a class of nonlinear equations. These research greatly enriched the geo-physical fluid dynamics theory, and have achieved many significant results. For example, Kartashova and L’vov[2] proposed the physical mechanism of resonance of the nonlinear Rossby waves intraseasonal oscillation; Luo De-hai[22,23]used KdV or mKdV equation explains the atmospheric blocking phenomenon.In this dissertation, based on the results mentioned above, using the quasi-geostrophic vorticity equation, we will consider the approximate Rossby parameter β vary with latitude β plane under the generalized non-linear equations, nonlinear Rossby wave amplitude of the various effect, and influence of Rossby waves under the horizontal component of the earth’s rotation.Firstly, we consider nonlinear Rossby waves under the generalized β plane approximation in barotropic fluids problem. Without considering the effect of topography condition, by using the reductive perturbation method for nonlinear equations is derived for nonlinear Rossby waves, we obtain Rossby solitary waves satisfy mKdV equation in barotropic fluids. Then only consider the influence of topography on Rossby waves, getting Rossby waves amplitude satisfies the mKdV equation under the effect of linear and nonlinear topography. Further, under the generalized β plane approximation condition, we investigated the quasi-geostrophic vorticity equation with external source, dissipation and topography, obtaining Ross-by solitary waves satisfied inhomogeneous Boussinesq equation dan mKdV equation.Because of the characteristics of the geophysical fluid with multi tem-poral and spatial scales, hence we use the method of multiple scales is derived for the Rossby waves satisfy nonlinear Schrodinger equation. By considering topography varies slowly with time, we give the amplitude non-linear Rossby waves satisfy inhomogeneous Benjamin-Davis-Ono(BDO) e-quation results, and analysis of the topographic forcing and dissipation effect Rossby solitary wave quality, energy change.At the same time, under the generalized β plane approximation con-dition, the dissertation also studied Rossby solitary wave interaction effect of topography and stratification effect in stratified fluid, using the pertur-bation method is used to deduce the Rossby wave amplitude satisfies the mKdV equation, we give topograph force in stratified fluid also caused the Rossby solitary waves of mass and energy change. In addition, we consid-ered equatorial envelope solitary Rossby wave, in barotropic fluids, based on the equatorial potential vorticity equation, the nonlinear equatorial en-velope solitary Rossby wave are investigated by the asymptotic method of multiple scales. The inhomogeneous Schrodinger equation, satisfied by Rossby solitary waves packet is derived.Finally, this dissertation also studies the influence of the horizontal component of earth rotation approximation of Rossby wave in β plane promotion, starting from the atmospheric equations, using Taylor series method to derive the Rossby wave to satisfy KdV and KdV-mKdV equa-tions of the conclusion.This dissertation contains eight parts:1. The background and the main results of study problem;2. Nonlinear Rossby waves under the generalized beta-plane approxima- tion in barotropic fluid;3. Nonlinear Rossby waves with topography effect in barotropic fluid;4. Nolinear Rossby waves and envelop Rossby solitary waves problems with beta effect and topography synergism in barotropic;5. Rossby solitary waves with beta effect, topography effect and stratifica-tion effect in stratified fluid;6. Nonlinear Rossby waves problem excited slowly changing topography and dissipation in barotropic;7. Equatorial envelope solitary Rossby waves under the generalized beta-plane approximation;8. Action of the horizontal component of the earth’s rotation on nonlinear Rossby waves.