Vortices on closed surfaces

Abstract

It was recognized, since the seminal papers of Arnold (Ann Inst Grenoble 16:319-361, 1966) and Ebin-Marsden (Ann Math Ser 2 92(1):102-163, 1970), that Euler's equations are the right reduction of the geodesic flow in the group of volume preserving diffeomorphisms. In 1983 Marsden and Weinstein (Physica D 7:305-323, 1983) went one step further, pointing out that vorticity evolves on a coadjoint orbit on the dual of the infinite dimensional Lie algebra consisting of divergence free vectorfields. Here we pursue a suggestion of that paper, namely, to present an intrinsic Hamiltonian formulation for a special coadjoint orbit, which contains the motion of N point vortices on a closed two dimensional surface S with Riemannian metric g. Our main results reformulate the problem on the plane, mainly C.C. Lin' s works (Lin, Proc Natl Acad Sci USA 27:570-575; Lin, Proc Natl Acad Sci USA 27:575-577, 1941) about vortex motion on multiply connected planar domains. Our main tool is the Green function Gg(s, so) for the Laplace-Beltrami operator of (S, g), interpreted as the stream function produced by a unit point vortex at so 2 S. Since the surface has no boundary, the vorticity distribution ! has to satisfy the global condition ffSw£2= 0, where £2 is the area form. Thus the Green function equation has to include a background of uniform counter-vorticity. As a consequence, vortex dynamics is affected by global geometry. Our formulation satisfies Kimura's requirement (Kimura, Proc R Soc Lond A 455:245-259, 1999) that a vortex dipole describes geodesic motion. A single vortex drifts on the surface, with Hamiltonian given by Robin's function, which in the case of topological spheres is related to the Gaussian curvature (Steiner, Duke Math J 129(1):63-86, 2005). Results on numerical simulations on flat tori, the catenoid and in the triaxial ellipsoid are depicted. We present a number of questions, intending to connect point vortex streams on surfaces with questions from the mathematical mainstream.