Point vortex on surfaces with continuous symmetry
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.
Keywords
Cite
@article{arxiv.1810.09523,
title = {Point vortex on surfaces with continuous symmetry},
author = {Yuuki Shimizu},
journal= {arXiv preprint arXiv:1810.09523},
year = {2025}
}