Abstract
In this paper, \((2+1)\)-dimensional nonlinear Rossby waves are considered with the generalized beta, the dissipation and the topography which includes both basic part and slowly varying part with time. Starting with a barotropic quasi-geostrophic potential vorticity equation, by using methods of multiscales and perturbation expansions, a generalized forced Zakharov–Kuznetsov equation is obtained in describing the evolution of Rossby wave amplitude. The effects of generalized beta, topography along with latitude and slowly variation with time are all included, indicating that the generalized beta is an essential factor in inducing the nonlinear Rossby solitary waves and the other two are both important factors for the evolution of Rossby wave amplitude. Periodic and solitary wave solutions of Zakharov–Kuznetsov equation are obtained by the elliptic function expansion method; meanwhile, solitary wave solution of generalized forced Zakharov–Kuznetsov equation is obtained by reduced differential transform method. At last, graphical presentations for solitary wave amplitude with different dissipations and slowly varying topographies with time are shown by the Mathematica.
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