Point-vortex cluster formation in the plane and on the sphere: an energy bifurcation condition.
The dynamics of a system of point vortices is considered in the plane and on the sphere. Particular attention is given to the formation of vortex clusters and to global vortex dynamics, especially in the spherical case. For integrable systems and systems with given symmetries, we show the existence of a critical energy above or below which (depending on the geometry of the surface) the system splits into clusters and vortex dynamics is confined to a particular region. The case of nonidentical vortices is of particular interest since we observe quite different global dynamics depending on the energy and the initial conditions. Furthermore we identify all the relative equilibria configurations as critical points of the reduced energy and we give an instability criterion to deduce instability for certain configurations.
Boatto
Stefanella
S
Departement de Mathematiques, Universite Paris 13, 99 Avenue J.-B. Clement, 93430 Villetaneuse, France. boatto@math.univ-paris13.fr
Laskar
Jacques
J