Ring Configurations of Point Vortices in Polar Atmospheres

Abstract

This paper examines the stability and nonlinear evolution of configurations of equal-strength point vortices equally spaced on a ring of constant radius, with or without a central vortex, in the three-dimensional quasi-geostrophic compressible atmosphere model. While the ring lies at constant height, the central vortex can be at a different height and also have a different strength to the vortices on the ring. All such configurations are relative equilibria, in the sense that they steadily rotate about the $z$ axis. Here, the domains of stability for two or more vortices on a ring with an additional central vortex are determined. For a compressible atmosphere, the problem also depends on the density scale height $H$, the vertical scale over which the background density varies by a factor $e$. Decreasing $H$ while holding other parameters fixed generally stabilises a configuration. Nonlinear simulations of the dynamics verify the linear analysis and reveal potentially chaotic dynamics for configurations having four or more vortices in total. The simulations also reveal the existence of staggered ring configurations, and oscillations between single and double ring configurations. The results are consistent with the observations of ring configurations of polar vortices seen at both of Jupiter’s poles [1].