Vortices for lake equations: Review with questions and speculations

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Abstract
The ?lake equation? on a planar domain D with bathymetry b(x, y) is given by \partial_t u + (u \cdot grad)u = -grad p , \operatorname{div}(bu) = 0 , with u \parallel \partial D . It is well posed as a PDE, but when b \neq \operatorname{const} , justifying point vortex models requires the analyst?s attention. We focus on Geometric Mechanics aspects, glossing over hard analysis issues. The motivating example is a ?rip current? produced by vortex pairs near a beach shore. For a beach with uniform slope, 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 -\operatorname{div}(grad \psi / b) with Dirichlet boundary conditions. As in elasticity, the lake equations give rise to pseudo-analytical functions and quasi-conformal mappings. Uniformly elliptic equations on close Riemann surfaces could be called ?planet equations?.