Collapse of point vortex lattices

Abstract

Configurations of several point vortices which collapse to a collision singularity in finite time are well known. In this paper, analogous configurations of singly- and doubly-periodic vortex lattices are shown to exist. The motion near the collision is asymptotic to the usual homographic collapse. Further, numerical evidence is presented for a new phenomemon: motions which both begin and end at finite times in collision singularities. These “oubly singular” motions involve the exchange of vortices between unit cells; exchange laws are described.