Vortex pairs and dipoles on closed surfaces

TL;DR

This paper develops equations for point vortex dynamics on closed surfaces, proves the Kimura conjecture that vortex pairs move along geodesics in the dipole limit, and relates planar vortex motion to surface dynamics.

Contribution

It introduces a general framework for vortex motion on closed surfaces, including non-zero total vorticity, and provides a new proof of the Kimura conjecture.

Findings

Vortex pairs in the dipole limit follow geodesic paths.

The equations account for non-zero total vorticity.

Planar vortex motion is a special case of surface vortex dynamics.

Abstract

We set up general equations of motion for point vortex systems on closed Riemannian surfaces, allowing for the case that the sum of vorticities is not zero and there hence must be counter-vorticity present. The dynamics of global circulations which is coupled to the dynamics of the vortices is carefully taken into account. Much emphasis is put to the study of vortex pairs, having the Kimura conjecture in focus. This says that vortex pairs move, in the dipole limit, along geodesic curves, and proofs for it have previously been given by S.~Boatto and J.~Koiller by using Gaussian geodesic coordinates. In the present paper we reach the same conclusion by following a slightly different route, leading directly to the geodesic equation with a reparametrized time variable. In a final section we explain how vortex motion in planar domains can be seen as a special case of vortex motion on closed surfaces, and in two appendices we give some necessary background on affine and projective connections.