Point vortex motions with a center of symmetry

Abstract

The equations of motion for point vortices are well known to preserve certain discrete symmetries of the initial state. The case of a center of symmetry is considered in detail here, since this particular instance seems to have been overlooked in the classical literature. This symmetry provides a generalization of the early studies by Gröbli and Greenhill wherein several axes of symmetry are present, a case which leads to an effective one-body problem. The center of symmetry yields an effective two-body problem which is Hamiltonian and integrable. As an example the ‘‘double alternate ring’’ configurations, circular analogs of the vortex street introduced by Havelock, are considered. A fully nonlinear mode wherein these double rings asymptotically dissolve into freely moving vortex pairs is found analytically. The paper concludes with a discussion of the relevance of such modes to our understanding of the disintegration of vortex streets in two-dimensional flow.