Generalized (2

We investigate a new higher-dimensional nonlinear dynamics model to describe the generation and evolution of Rossby waves. We derive a generalized (2 + 1)-dimensional modified Korteweg-de Vries (mKdV)-Burgers equation by considering the quasi-geostrophic potential vorticity equation under the generalized beta approximation with dissipation and external source in barotropic fluids, and by utilizing multiple scales and the perturbation expansion method. Qualitative analysis yields that the generalized beta and shear basic flow can induce the nonlinear Rossby solitary waves, and dissipation has effects on the propagation of Rossby waves as well. Then, we analyze the conservation laws for the mass and energy of Rossby solitary waves. Moreover, the asymptotic kink-shaped solitary wave solution for the generalized (2 + 1)-dimensional mKdV-Burgers equation is explored by modified hyperbolic function expansion method. The solitary wave solutions and figures are analyzed, the results show that the dissipation effect causes the speed and amplitude of Rossby solitary waves to decrease exponentially and the width of Rossby solitary waves to increase exponentially with time.