Rossby waves are intrinsic in the large-scale systems of fluids, so they are the most important waves in the atmosphere and ocean. Theory and observation show that their basic characteristic is to satisfy the quasi-geostrophic and quasi-static equilibrium approximations. In stratified fluids, we discuss the long waves in a homogenous atmosphere and obtain the KdV equation, but the analysis is limited to the case that the velocity shear is small compared with a basic uniform zonal motion, and it gives no insight pertaining to the kinds of stream-line-flow patterns accompanying these waves. Here, the β-plane approximation f = f0 +β0y (β0is a constant) is extended into f = f0 +β(y)y, which includes a nonlinear function β(y) taking the place of β in the β-plane approximation. Such an approximation can depict more precisely the motion of the atmosphere and ocean, 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 external source. We take the basic flow to be a shear and the Väisälä-Brunt frequency N a function of variable z. For the stratified fluids, based on the lower boundary with external heating source and topography, as well as the quasi-geostrophic potential vorticity equation with external heating source, an inhomogeneous nonlinear Schrödinger equation (including topographic forcing and an external heating source) is derived by using the perturbation method and stretching transforms of time and space. It is found that the external heating source,β effect and topography effect are the important factors that could induce the nonlinear solitary Rossby by inspection of the evolution of the amplitude of Rossby waves. On the assumption that nonlinear topographic effects and the dissipation of external heating source are balanced, an inhomogenous equation in which the coefficients depend on β(y), u(y, z) and N (z) is derived. Results show that the topography, external heating source and Rossby waves will interact with a basic stream function that has a shear. In stratified fluids, the inhomogeneous nonlinear Schrödinger equation is obtained for describing the evolution of the amplitude of solitary Rossby envelop solitary waves as the change of Rossby parameterβ(y) with latitude y, topographic forcing and the external heating source.
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