On the dynamics of nonlinear Rossby solitary waves via the Ostrovsky hierarchy

The impact mechanisms of large-scale atmospheric and ocean dynamics on weather and climate change have long been a focus of attention. In this paper, based on the generalized beta-plane approximation with turbulence dissipation and forcing terms, we derived the Ostrovsky equation describing the evolution of Rossby wave amplitudes using multiscale and perturbation expansion methods. This is the first derivation of the Ostrovsky equation from the quasi-geostrophic potential vorticity conservation equation. A detailed analysis was conducted on the evolution of Rossby waves under the influence of multiple physical factors. We investigated the evolution of flow fields and Rossby wave amplitudes under conditions of weak shear in the background flow and discussed the effects of physical factors such as Rossby parameter beta 0 and turbulence dissipation on the evolution of dipole blocking and Rossby wave amplitudes. The results indicate that an increase in the Rossby parameter slows down the evolution of dipole blocking and amplitudes, while an increase in turbulence dissipation and background flow shear accelerates these evolutions. Additionally, we conducted comparative analyses on the evolution of relative vorticity and perturbed relative vorticity, further enriching the theoretical achievements in atmospheric dynamics.