title: Point vortex interactions on a toroidal surface.
abstract: Owing to non-constant curvature and a handle structure, it is not easy to imagine intuitively how flows with vortex structures evolve on a toroidal surface compared with those in a plane, on a sphere and a flat torus. In order to cultivate an insight into vortex interactions on this manifold, we derive the evolution equation forN-point vortices from Green's function associated with the Laplace-Beltrami operator there, and we then formulate it as a Hamiltonian dynamical system with the help of the symplectic geometry and the uniformization theorem. Based on this Hamiltonian formulation, we show that the 2-vortex problem is integrable. We also investigate the point vortex equilibria and the motion of two-point vortices with the strengths of the same magnitude as one of the fundamental vortex interactions. As a result, we find some characteristic interactions between point vortices on the torus. In particular, two identical point vortices can be locally repulsive under a certain circumstance.
authors: Department of Mathematics , Kyoto University , Sakyo-ku, Kyoto 606-8502, Japan.
authors: Takashi
authors: 0000-0002-4290-0942
authors: T
authors: Sakajo
authors: Department of Mathematics , Kyoto University , Sakyo-ku, Kyoto 606-8502, Japan.
authors: Yuuki
authors: Y
authors: Shimizu
keywords: Hamiltonian dynamical system; hydrodynamic Green function; ring torus; symplectic geometry; vortex dynamics; vortex interactions