Vortex Pairs and Dipoles on Closed Surfaces

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.

Errata and Addenda to: “Hydrodynamic Vortex on Surfaces” and “The Motion of a Vortex on a Closed Surface of Constant Negative Curvature”

“…In a recent paper Björn Gustafsson [2] presented equations for the motion of hydrodynamic vortices on closed surfaces (compact and boundaryless). These equations, which will be referred as Gustafsson's equations, are different from those used by previous authors in the case of surfaces of genus g greater than zero.…”

“…In this note we present the equations of Gustafsson in the case of a single vortex of intensity one in a closed surface of a genus greater than zero. We will derive Gustafsson's equation not using a renormalized energy as done in [2], but our weak formulation of Euler's equation given in [6]. The reader must consult [2] for the Hamiltonian formulation of the equations of a system of vortices.…”

“…We will derive Gustafsson's equation not using a renormalized energy as done in [2], but our weak formulation of Euler's equation given in [6]. The reader must consult [2] for the Hamiltonian formulation of the equations of a system of vortices. The ideas in [2] can be used to correct the equations obtained in [6] for systems of vortices in surfaces with boundaries and/or ends and genus greater than zero.…”

“…The reader must consult [2] for the Hamiltonian formulation of the equations of a system of vortices. The ideas in [2] can be used to correct the equations obtained in [6] for systems of vortices in surfaces with boundaries and/or ends and genus greater than zero. Gustafsson's result do not change the equations in [6] when the surface has genus zero but it is not simply connected due to the presence of boundaries or ends, and this we will be explained below.…”

“…where G is the Green's function in equation (4).Following [2], if u satisfies Euler's equation ( 3), then the time derivative of the circulation along the curve a 1 must satisfy…”