On the Interplay Between Vortices and Harmonic Flows: Hodge Decomposition of Euler’s Equations in 2d

Let $\Sigma$ be a compact manifold without boundary whose first homology is nontrivial. The Hodge decomposition of the incompressible Euler 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 nonflat metric that shows typical features of nonintegrable 2 degrees of freedom Hamiltonians. In contradistinction, for flat tori the harmonic part is constant. Next, we turn to hyperbolic surfaces ($ g \geqslant 2$) having constant curvature $-1$, with discrete symmetries. Fixed points of involutions yield vortex crystals in the Poincaré 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 hydrodynamic Green function is clarified. A number of questions are suggested.