Mean-Field Limit of Point Vortices for the Lake Equations

TL;DR

This paper investigates the mean-field limit of point vortices in the lake equations, revealing how different interaction regimes lead to various limiting equations, using a modulated energy approach adapted for heterogeneous lake kernels.

Contribution

It extends mean-field analysis to the lake equations with non-uniform depth, introducing a novel adaptation of the modulated energy method for heterogeneous kernels.

Findings

Convergence to lake equations when self-interactions are negligible.

Convergence to forced lake equations when self-interactions are of order one.

Convergence to a transport equation when self-interactions dominate.

Abstract

In this paper we study the mean-field limit of a system of point vortices for the lake equations. These equations model the evolution of the horizontal component of the velocity field of a fluid in a lake of non-constant depth, when its vertical component can be neglected. As for the axisymmetric Euler equations there are non-trivial self interactions of the vortices consisting in the leading order of a transport term along the level sets of the depth function. If the self-interactions are negligible, we show that the system of point vortices converges to the lake equations as the number of points becomes very large. If the self-interactions are of order one, we show that it converges to a forced lake equations and if the self-interactions are predominant, then up to time rescaling we show that it converges to a transport equation.The proof is based on a modulated energy approach introduced by Duerinckx and Serfaty in (Duke Math. J., 2020) that we adapt to deal with the heterogeneity of the lake kernel.