ИСТИНА

In the last decades a huge interest in the emergence and persistence of vortices in two-dimensional flows has developed. The behavior of vortices is characterized by their tendency to assume axisymmetric shapes, their stability, and their evolution in a cascade to larger scales. Most of the results were obtained for the case of incompressible fluid. For example, as follows from [1], the axisymmetric form of vortex is stable with respect to asymmetric perturbations for the solution to the incompressible Euler equations. Results concerning the vortices in compressible media are rare. We are going to show that the response of the vortex to the symmetry breaking is quite different in the compressible case. In the present work we study nonlinear stability of axisymmetric vortex in barotropic compressible rotating fluid in two dimensional setting. The model is described by the full hyperbolic system of equations corresponding to the conservation of mass, momentum and energy. Our method is as follows: as in our previous works [2,3] we consider an exact solution to the system of compressible fluid with the velocity linear with respect to the space variables. This solution corresponds to the first terms in the expansion of the solution into the Taylor series near the center of vortex. The procedure allows us to reduce the problem to the study of equilibrium of certain nonlinear system of 7 ordinary differential equations. We show that the axisymmetric steady vortex is stable (non-asymptotically) with respect to small perturbation of symmetry if and only if its vorticity belongs to sufficiently narrow range depending on the Coriolis parameter. Moreover, as follow from our results, an increasing of rotation of the frame plane can stabilize the vortex and change the cyclonic behavior of the pressure to anticyclonic one. The proof of stability is very delicate: it reduces to the proof of existence of invariant tori for non-Hamiltonian system and uses the theory normal forms. We performed direct numerical computation and confirmed our theoretical results. Moreover, we studied numerically the character of breakdown of the vortex in the case of instability. Namely, it is shown that the vortex decays into several coherent structures. REFERENCES 1. D. A. Schecter, D. H. E. Dubin, A. C. Cass, C. F. Driscoll,I. M. Lansky,T. M. O'Neil // Inviscid damping of asymmetries on a two-dimensional vortex. Phys. Fluids. 2000. V.12. P. 2397-2412. 2 O.S. Rozanova, J-L. Yu, C-K. Hu //Typhoon eye trajectory based on a mathematical model: Comparing with observational data. Nonlinear Analysis: Real World Applications. 2010. V.11. P.1847--1861. 3. O.S. Rozanova, J-L. Yu, C-K. Hu // On the position of vortex in a two-dimensional model of atmosphere. Nonlinear Analysis: Real World Applications. 2012. V.13. P.1941 – 1954.