Dynamics of vortex clusters on a torus

We investigate the collective dynamics of multivortex assemblies in a two dimensional toroidal fluid film of distinct curvature and topology. The incompressible, inviscid nature of the fluid permits a Hamiltonian description of the vortices, along with a self-force of geometric origin. The Hamiltonian dynamics is constructed in terms of q-digamma functions Ψq(z), closely related to the Schottky–Klein prime function known to arise in multiply connected domains. We show the fundamental motion of the two-vortex system and identify five classes of geodesics on the torus for the special case of a vortex dipole, along with subtle distinctions from vortices in quantum superfluids. In multivortex assemblies, we observe that a randomly initialized chiral cluster of vortices remains geometrically confined on the torus, while undergoing an overall drift along the toroidal direction. A cluster of fast and slow vortices also shows the collective toroidal drift, with the fast ones predominantly occupying the core region of the revolving cluster. Achiral clusters show unconfined dynamics and scatter all over the surface of the torus. A chiral cluster with an impurity in the form of a single vortex of opposite sign also shows similar behavior as a pure chiral cluster, with occasional “jets” of dipoles leaving and reentering the revolving cluster. The work serves as a step toward analysis of vortex clusters in models that incorporate harmonic velocities in the Hodge decomposition.