Vortex Pairs and Dipoles on Closed Surfaces

• Main Author: Gustafsson, B. (Author)
• Format: Article
• Language: English
• Published: Springer 2022
• Subjects: Affine connection; Closed surfaces; Equations of motion; Geodesic curve; Geodesic curves; Geodesy; Green function; Greens function; Hamiltonian function; Hamiltonians; Point vortex; Point vortices; Projective connection; Robin function; Robin's function; Symplectic form; Symplectic forms; Vortex dipole; Vortex flow; Vortex pair; Vorticity
• Online Access: View Fulltext in Publisher

 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. © 2022, The Author(s).