A new model equation for nonlinear Rossby waves and some of its solutions

In this paper, we investigate the $(2+1)$ dimensional nonlinear Rossby waves under non-traditional approximation. Using the asymptotic methods of multiple scales and weak nonlinear perturbation expansions, we derive a new modified Zakharov–Kuznetsov equation from the barotropic potential vorticity equation with the complete Coriolis parameter, the topography and the dissipation. Based on the new auxiliary equation method, new exact solutions of the new mZK equation are obtained when the dissipation is absent. However, the new auxiliary equation method fails to solve the new mZK equation with the dissipative term. Therefore, the weak nonlinear method and the homotopy perturbation method are developed to solve the obtained new mZK equation. Through numerical simulations, the results show the effects of different parameters on Rossby waves.