ИСТИНА

We study nonlinear stability of steady isolated vortices in two-dimensional compressible media in an uniformly rotating reference frame. First, we consider a vortex with linear profile of velocity. Its behavior can be completely described by a quadratically nonlinear system of ODEs. We find that the stability property depends only on one parameter, the ratio of relative vorticity of vortex to the Coriolis constant. We find the domain of this parameter ensuring nonlinear stability. Then we study a quasi-two-dimensional model of stratified gas with uniform deformation and show that the presence of the Coriolis force implies the existence of a family of gas clouds in a form of stationary vortex, which is nonlinearly stable. Both "pressureless" and general cases are studied. In the first case for any initial data we find the exact solution. Further, we consider more general class of isolated steady vortices, containing decaying at infinity and compactly supported vortices as particular cases. At every point of the plane this isolated steady vortex can be approximated by a solution with linear profile of velocity. Thus, at every point of the plane there arises a nonlinear system of ODEs with initial data generated by derivatives of the steady vortex state. It is hypothesized that if at every point the solution to this ODEs system falls in the domain of attraction of an equilibrium, then the steady vortex is nonlinearly stable. We compare this nonlinear stability hypothesis with Raleigh criterium of linearized stability with respect to radial perturbation and the results of numerical computations. Our results imply that the rotation of the coordinate frame stabilizes the compressible vortex.