Rossby waves under time-varying shear with the complete Coriolis force

Large-scale Rossby waves are of great significance in influencing the weather and climate system due to the Coriolis force. In this paper, we derive a variable coefficient extended Korteweg–de Vries model to describe large-scale Rossby waves. This model is obtained from the potential vorticity equation, which incorporates the complete Coriolis force, using the Gardner–Morikawa coordinate transformation and perturbation theory. Subsequently, we obtain N-soliton solutions and breather solutions with horizontal Coriolis parameters through the Hirota bilinear theory. Moreover, based on different basic flows, we analyze the evolution of the amplitudes of Rossby waves, encompassing solitons, breathers, and their interactions. The results reveal that the horizontal component of the Coriolis force impacts the propagation trajectory of the solitons, potentially transforming elevation solitons into depression solitons during their propagation. Furthermore, the horizontal Coriolis parameters can modify the polarity of Rossby wave amplitudes and cause blockages to move in a zonal direction. The findings also suggest that the beta effect, the background westerly wind, and the horizontal component of the Coriolis force all influence the strength and evolution of the dipole blockage. In particular, a cotangent time-varying shear flow can lead to the transition from monopole blockage to dipole blockage. It is hoped that these discoveries will be useful in the study of atmospheric and oceanic Rossby waves.