Linear And Nonlinear Rossby Waves In The Atmosphere

The basic features of the atmosphere are the spherical effect and rotational effect of the earth. In 1936, Rossby in his research "Relation between variations in the intensity of the zonal circulation of the atmosphere and the displacements of the semi-permanent centers of action"[1] proposed a proper method to study the dynamics of the observed large-scale waves in the atmosphere. He suggested that the motions of the atmosphere mainly presented horizontal due to the fact that the atmosphere is rather thin on the surface of the earth. Thus, only the local vertical component of the earth’s planetary vorticity of a certain plan f=2Ω sin φ become dynamically significant. The parameter f is the local component of the planetary vorticity normal to the earth’s surface and is called the Coriolis parameter, Ω,φ is the earth rotation angular velocity and latitude. Such the model, in which the effect of the earth’s sphericity is modeled by a linear variation of f in an otherwise planar geometry, is called β model (β plane approximation). However, the model, in which effect of the earth’s sphericity is modeled by the constant, is called f model (f plane approximation).Rossby waves are the large-scale waves which have the long life history and organized structure in the atmosphere. The generation of the Rossby waves due to the change of potential vorticity as the depth and latitude driving by the earth rotation and earth curvature. Pedlosky[2] gave the Rossby waves in shallow water model and pointed out that the Rossby waves could be generated if the Coriolis parameter and topography slope parameter are the constant. Redekop[3] proposed the reason of the generation of Rossby solitary waves by introducing a shear flow to the zonal flow. Based on the quasi-geostrophic potential vorticity equation, Charney and Straus[4] studied the factors to driven Rossby waves in a plane channel thermotropic model considering the topography, non-adiabatic heating and friction. Lv et al.[5] used the asymptotic expansion and elongation transformation of time and space to obtain the FKDV-Burgers equation including the topography and dissipation, which indicates that underlying surface can increase the amplitude obviously. Song et al.[6] made further efforts to study that, under the condition of shear basic flow, plane approximation of nonlinear topography change and external source can influence the evolution of the Rossby solitary waves. At the same time, they pointed that basic shear flow, nonlinear effect and topography are the important to produce Rossby solitary waves.Based on the results mentioned above, the dispersion relation of f plane small-amplitude motions and β plane small-amplitude motions are derived in shallow-water model by considering the topographic slope parameter changes with the latitude. The results show that Rossby waves can be driven when the topographic slope parameter and Coriolis parameter satisfies some certain relationship. Particularly, the dispersion relations are different are different as the change of the latitude in β plane.The effect of slowly varying underlying surface and dissipation on Rossby waves is an indispensible problem deserved consideration. A lot of observations and studies have shown that Rossby waves satisfy quasi-geostrophic and barotropic equations. In barotropic fluids, based on quasi-geostrophic vorticiy equation, the general Schodinger equation satisfied by the nonlinear Rossby envelop solitary waves affected by the slowly varying underlying surface and dissipation, is derived by the method of multiple scales and perturbation. Then we get the solution of the single Rossby envelope solitary waves. The solution shows that the Rossby waves affected by the slowly varying underlying surface and dissipation can present the shape of sech with time and space in the atmosphere. At the end of that section, we point out that the effect of slowly varying underlying surface and dissipation can change the velocity, wave number and the frequency of the Rossby envelope solitary waves through discussing the solution of the Rossby envelope solitary waves.In order to investigate the nonliearty of Rossby waves, we discussed the stability and the solutions of linear and nonlinear topographically. Variables transformation are used to the model and the result indicates that north to south and west to ease slopes have an obvious effect of the stability and periodic of the Rossby waves.