Nonlinear stability of two-dimensional axisymmetric vortices in compressible inviscid medium in a rotating reference frame

TL;DR

This paper investigates the stability of 2D axisymmetric vortices in a compressible medium within a rotating frame, revealing that rotation can stabilize vortices that are otherwise unstable in non-rotating conditions.

Contribution

It demonstrates that the stability of compressible vortices in a rotating frame depends on the vorticity-to-Coriolis ratio, and identifies narrow conditions for stability using analytical and computer-aided methods.

Findings

Stability depends only on vorticity-to-Coriolis ratio.

Rotation can stabilize otherwise unstable vortices.

Narrow parameter range for vortex stability.

Abstract

We study the stability of the vortex in a 2D model of continuous compressible media in a uniformly rotating reference frame. As it is known, the axisymmetric vortex in a fixed reference frame is stable with respect to asymmetric perturbations for the solution of the 2D incompressible Euler equations and basically instable for compressible Euler equations. We show that the situation is quite different for a compressible axisymmetric vortex in a rotating reference frame. First, we consider special solutions with linear profile of velocity (or with spatially-uniform velocity gradients), which are important because many real vortices have similar structure near their centers. We analyze both cyclonic and anticyclonic cases and show that the stability of the solution depends only on the ratio of the vorticity to the Coriolis parameter. Using a very delicate analysis along with computer aided proof, we show that the stability of solutions can take place only for a narrow range of this ratio. Our results imply that the rotation of the coordinate frame can stabilize the compressible vortex. Further, we perform both analytical and numerical analysis of stability for real-shaped vortices.