Vortex motion for the lake equations

Abstract

The lake equations $$\left\{\begin{aligned} \nabla \cdot \big( b \, \mathbf{u}\big) &= 0 & & \text{on}\ \mathbb{R}\times D,\\ \partial_t\mathbf{u} + (\mathbf{u}\cdot \nabla)\mathbf{u} &= -\nabla h & & \text{on}\ \mathbb{R}\times D ,\\ \mathbf{u} \cdot \boldsymbol{\nu} &= 0 & & \text{on}\ \mathbb{R}\times\partial D . \end{aligned}\right.$$ model the vertically averaged horizontal velocity in an inviscid incompressible flow of a fluid in a basin whose variable depth $b : D \to [0, + \infty)$ is small in comparison with the size of its two-dimensional projection $D \subset \mathbb{R}^2$. When the depth $b$ is positive everywhere in $D$ and constant on the boundary, we prove that the vorticity of solutions of the lake equations whose initial vorticity concentrates at an interior point is asympotically a multiple of a Dirac mass whose motion is governed by the depth function $b$.