ИСТИНА

Many flows in oceans and atmospheres are approximately two-dimensional. The vortex structures are their characteristic feature. Two-dimensional vortices also play an important role in tokamak-confined plasmas as well as in astrophysical situations such as accretion discs of neutron stars. Although the vortex dynamics can be complicated, it is natural to begin with the study of certain elementary processes. One example is the isolated circular free vortex in rotating fluids. Its stability/instability properties are of fundamental interest for refined models of general atmospheric circulation due to the presence of strain and shear in the ambient flow. We study the stability of the vortex in a 2D model of continuous compressible media in a uniformly rotating reference frame both analytically and numerically. The main questions are where rotation prevents a singularity formation and where it helps to keep an initial shape of the vortex. 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. Firstly, 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. The motion in this case can be described by a quadratically nonlinear system of ODEs. An analogous system arises when we want to study the behavior of the gas ellipsoid in vacuum. 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. We show that the stability of solutions can take place only for a narrow range of this ratio. In particular, the motion is stable in the "anticyclonic" domain. In the "cyclonic" domain the motion is stable "almost everywhere", may be except of discrete values of parameters. Our results imply that the rotation of the coordinate frame can stabilize the compressible vortex.