Point vortex on surfaces with continuous symmetry
Yuuki Shimizu
TL;DR
The paper addresses the challenge of obtaining explicit hydrodynamic Green and Robin functions on curved surfaces with continuous symmetry (HKVF). It reduces the problem to conformal models on the 14 canonical Riemann surfaces, producing analytic, slip-boundary–satisfying formulas for $G$ and $R$ that unify previously known cases and extend to noncompact ends. A key outcome is the identification that single point vortex dynamics are governed by a Hamiltonian flow with velocity linked to curvature via $- riangle R=\frac{\kappa}{2\pi}+2c_{M}$, and a detailed torus example reveals linear and nonlinear curvature-response regimes. Collectively, the results provide a unified, practically useful toolkit for studying point-vortex dynamics and Euler–Arnol'd flows on symmetric surfaces, with broad implications for geometric fluid dynamics.
Abstract
We derive an analytic formula for the hydrodynamic Green function and the Robin function on every orientable surface admitting a hydrodynamic Killing vector field. Closed-form expressions are provided for all fourteen canonical Riemann surfaces, covering both compact and non-compact cases; the formulae satisfy the slip boundary condition and generate complete Hamiltonian vector fields. As an application, we clarify the mechanism whereby the curvature affects a point vortex in both qualitative and quantitative viewpoints. Qualitatively, we show a single point vortex is governed by a Hamiltonian flow whose vorticity is given by the curvature up to area constant. Quantitatively, on a rectangular torus with periodic curvature we use the analytic formula to describe two regimes: linear response that mirrors the curvature wave when the mean component is small, and a nonlinear response with amplitude resonance. The results supply a unified tool for detailed studies of point vortex dynamics and Euler-Arnold flows on surfaces with continuous symmetry.