Nonlinear Solitary Rossby Waves With Heating External Source And β Effect Topographic Effect In Stratified Flows

Rossby waves are intrinsic to the large-scale systems in the fluids, so they are most important waves in the atmosphere and ocean. The theory and observation show that their basic characteristic is that they satisfy the quasi-geostrophic and quasi-static equilibrium approximations. In stratified fluids, some scholars discussed long waves in a homogenous atmosphere and obtained the KdV equation, but the analysis was limited to the case where the velocity shear was small compared with a basic uniform zonal motion and they gave no insight pertaining to the kinds of stream-line-flow patterns accompanying these waves. Here, the/?-plane approximation f= f0+β₀y （β₀ is a constant） is extended into f= f0+β{y)y, which includes a nonlinear function β（y） taking the place of β₀ in the β-plane approximation. Such approximation can depict the motion of the atmosphere and ocean more precisely, especially in the middle and high latitude regions. It generalizes the theory developed by Helfrich and Pedlosky for time-dependent coherent structures in a marginally stable zonal flow by including forcing. Such forcing could be due to topography or an external source. We take the basic flow is shear and the Vaisala-Brunt frequency N is a function of variable z. For the stratified fluids, based on the lower boundary with heating external source and topographic, and the quasi-geostrophic potential vorticity equation with heating external source, an inhomogeneous nonlinear Schr（?）dinger equation including topographic forcing and an heating external source is derived by using the perturbation method and stretching transforms of time and space. It is found that heating external source, β effect and topography effect are the important factors that induce the nonlinear solitary Rossby with inspection of the evolution of the amplitude of Rossby waves.