Formation of vortex clusters on a sphere

This paper applies the Hamiltonian Approach (HA) to two-dimensional motions of incompressible fluid in curvilinear coordinates, in particular on a sph ere. The HA has been used to formulate governing equations of motion and to interpret the evolution of a system consisting of N localized two-dimensio nal vortices on a sphere. If the number of vortices N is large, N similar t o 10(2) - 10(3), 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 b een carried out for initial conditions taken in the form of a few axisymmet ric chains of point vortices distributed initially in fixed latitudes. The scheme of Runge-Kutta of 4th order has been used for simulating an evolutio n of resulting flows. The numerical analysis shows that the Kelvin-Helmholt z instability appears immediately formating initial disorganized structures which are developed and finally "bursted". The system evolves to a few sep arated vortex "spots" which exist sufficiently for a long time.