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We study the stability of the vortex in a 2D model of continuous compressiblemedia in a uniformly rotating reference frame. As it is known, the axisymmetricvortex in a fixed reference frame is stable with respect to asymmetricperturbations for the solution of the 2D incompressible Euler equations andbasically instable for compressible Euler equations. We show that the situationis quite different for a compressible axisymmetric vortex in a rotatingreference frame. First, we consider special solutions with linear profile ofvelocity (or with spatially-uniform velocity gradients), which are importantbecause many real vortices have similar structure near their centers. Weanalyze both cyclonic and anticyclonic cases and show that the stability of thesolution depends only on the ratio of the vorticity to the Coriolis parameter.Using a very delicate analysis along with computer aided proof, we show thatthe stability of solutions can take place only for a narrow range of thisratio. Our results imply that the rotation of the coordinate frame canstabilize the compressible vortex. Further, we perform both analytical andnumerical analysis of stability for real-shaped vortices.

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