Effects of background current and topography on the Rossby solitary waves via the Kadomtsev–Petviashvili approach

Theoretical investigations on the evolution of large-scale atmospheres and oceans benefit us a lot in predicting the changes in weather or climate. Solitary wave models proved to be sufficiently suitable in characterizing the propagations of nonlinear Rossby waves. The present paper focuses on understanding the dynamic mechanism of nonlinear Rossby waves using a new solitary wave model approach. Based on the classical barotropic potential vorticity conservation theory for large-scale motions in geophysics, by using the multiple scales and weak nonlinear analysis, we derive a Kadomtsev–Petviashvili equation to characterize the evolution of nonlinear Rossby wave amplitude. As we know, the Kadomtsev–Petviashvili equation was originally obtained for general shallow water problems, which did not include the effect of the earth's rotation. Further, the one-soliton and two-soliton solutions of the Kadomtsev–Petviashvili equation are presented by utilizing Hirota's bilinear method in order to investigate the evolutionary mechanism of nonlinear Rossby waves. Finally, the dynamical effects of typical background flows and topographies on Rossby waves are discussed by both qualitative analysis and quantitative simulations in detail according to the obtained Kadomtsev–Petviashvili hierarchy. The theoretical results will support real observations in some sense.