Vortices for lake equations (review with questions and speculations)
Jair Koiller
TL;DR
This paper surveys a geometric-mechanics perspective on the lake equations with bathymetry, formulating vortex dynamics via the Green function $G_{L_b}$ of the elliptic operator $L_b = -\operatorname{div}(\tfrac{1}{b}\nabla)$. It derives a vortex Hamiltonian with mutual interactions $\sum_{j<k} \Gamma_j \Gamma_k G_{L_b}(z_j,z_k)$ and a self-energy $Rich_b$, and discusses the need to include background/pseudoharmonic flows in multiply connected domains through a Hodge-type decomposition and a capacity matrix $P^b$, while outlining extensions to curved geometries and Riemann surfaces. The work connects rip-current physics to classical vortex-ring theory and sketches a program to develop coupled vortex–harmonic dynamics on closed surfaces via the Schottky double, addressing how to incorporate pseudopotential flows and boundary circulations. It highlights key mathematical questions about Green-function representations, operator choices, and rigorous vortex–pseudopotential coupling in complex geometries, setting a roadmap for future rigorous and applied developments.
Abstract
The `lake equation' on a planar domain D with bathymetry b(x,y) is given by $ \partial_t u + (u \cdot {\rm grad}) u= -{\rm grad}\, p \,, \,\,{\rm div} (b u) = 0 \,,\, \text{with}\,\, u \parallel \partial D.$ % \, \,\, \,\,\, \text{),}$$ We focus on Geometric Mechanics aspects, glossing over hard analysis issues. % related to the desingularization. Motivating example is a `rip current' produced by vortex pairs near a beach shore. For uniform slope beach there is a perfect analogy with \ Thomson's vortex rings. The stream function produced by a vortex is defined as the Green function of the operator $- {\rm div} ( {\rm grad} ψ/b)$ with Dirichlet boundary conditions. As in elasticity, the lake equations give rise to pseudoanalytical functions and quasiconformal mappings. Uniformly elliptic equations on closed Riemann surfaces could be called `planet equations'.