On the interplay between vortices and harmonic flows: Hodge decomposition of Euler's equations in 2d

TL;DR

This paper explores the dynamics of point vortices on various compact surfaces using Hodge decomposition, revealing complex behaviors and symmetries, and connecting hydrodynamical Green functions with electrostatic ones.

Contribution

It extends vortex dynamics analysis to compact Riemann surfaces, computes phase space structures, and clarifies Green function relations, introducing new insights into vortex motion on curved surfaces.

Findings

Phase space for vortex motion on tori and hyperbolic surfaces analyzed.

Surface of section reveals non-integrable behavior for non-flat tori.

Green function relations between hydrodynamics and electrostatics clarified.

Abstract

Let $\Sigma$ be a compact manifold without boundary whose first homology is nontrivial. Hodge decomposition of the incompressible Euler's equation in terms of 1-forms yields a coupled PDE-ODE system. The $L^2$-orthogonal components are a `pure' vorticity flow and a potential flow (harmonic, with the dimension of the homology). In this paper we focus on $N$ point vortices on a compact Riemann surface without boundary of genus $g$, with a metric chosen in the conformal class. The phase space has finite dimension $2N+ 2g$. We compute a surface of section for the motion of a single vortex ($N=1$) on a torus ($g=1$) with a non-flat metric, that shows typical features of non-integrable 2-dof Hamiltonians. In contradistinction, for flat tori the harmonic part is constant. Next, we turn to hyperbolic surfaces ($ g \geq 2$), having constant curvature -1, with discrete symmetries. Fixed points of involutions yield vortex crystals in the Poincar\'e disk. Finally we consider multiply connected planar domains. The image method due to Green and Thomson is viewed in the Schottky double. The Kirchhoff-Routh hamiltonian given in C.C. Lin's celebrated theorem is recovered by Marsden-Weinstein reduction from $2N+2g$ to $2N$. The relation between the electrostatic Green function and the hydrodynamical Green function is clarified. A number of questions are suggested.