Asymptotic stability of a polar vortex perturbed by harmonic waves describing atmospheric gravity waves circulating in an equatorial plane of a spherical planet

Department of Mathematics and Physics, University of Wisconsin-Parkside, 900 Wood Rd, Kenosha, WI 53144, USA

^* Corresponding author: ibrranis@gmail.com

Received: 15 November 2017
Accepted: 25 January 2018

Abstract

We examine stability of the vortex that represents one particular class of exact solution of a a nonlinear shallow water model describing atmospheric gravity waves circulating in an equatorial plane of a spherical planet. The mathematical model is represented by a two-dimensional free boundary Cauchy–Poisson problem on the nonstationary motion of a perfect uid around a solid circle with a sufficiently large radius so that the gravity is directed to the center of the circle. It is shown that the model admits two functionally independent nonlinear systems of shallow water equations. Two essential parameters that control stability of the vortex for both systems are identified. The order of their importance is analyzed and it is shown that one of the systems is more resistant to small perturbations and remains stable for larger range of these two parameters.