Formation of vortex clusters on a sphere

Abstract

This paper applies the Hamiltonian Approach (HA) to two-dimensional motions of incompressible fluid in curvi-linear coordinates, in particular on a sphere. The HA has been used to formulate governing equations of motion and to interpret the evolution of a system consisting of N localized two-dimensional vortices on a sphere. If the number of vortices N is large,  N ~ 102 - 103 , a small number of vortex collective structures (clusters) is formed. The surprise is that a quasi-final state does not correspond to completely disorganized distribution of vorticity. Numerical analysis has been carried out for initial conditions taken in the form of a few axisymmetric chains of point vortices distributed initially in fixed latitudes. The scheme of Runge-Kutta of 4th order has been used for simulating an evolution of resulting flows. The numerical analysis shows that the Kelvin-Helmholtz instability appears immediately formating initial disorganized structures which are developed and finally "bursted". The system evolves to a few separated vortex "spots" which exist sufficiently for a long time.