Collective behaviour of large number of vortices in the plane

Abstract

We investigate the dynamics of N point vortices in the plane, in the limit of large N . We consider relative equilibria , which are rigidly rotating lattice-like configurations of vortices. These configurations were observed in several recent experiments. We show that these solutions and their stability are fully characterized via a related aggregation model which was recently investigated in the context of biological swarms. By using this connection, we give explicit analytical formulae for many of the configurations that have been observed experimentally. These include configurations of vortices of equal strength; the N +1 configurations of N vortices of equal strength and one vortex of much higher strength; and more generally, N + K configurations. We also give examples of configurations that have not been studied experimentally, including N +2 configurations, where N vortices aggregate inside an ellipse. Finally, we introduce an artificial ‘damping’ to the vortex dynamics, in an attempt to explain the phenomenon of crystallization that is often observed in real experiments. The diffusion breaks the conservative structure of vortex dynamics, so that any initial conditions converge to the lattice-like relative equilibrium.