Dynamics of several point vortices for the lake equations

TL;DR

This paper rigorously derives the asymptotic behavior of point vortices in lake equations, showing their concentration and motion along depth level lines, with implications for vortex filamentation and general initial conditions.

Contribution

It provides a rigorous derivation of vortex dynamics for the lake equations with general initial data and no symmetry assumptions, extending previous 3D vortex ring results.

Findings

Vorticity remains concentrated around vortex centers over time.

Vortex motion follows the depth function's level lines at leading order.

The analysis includes boundary effects and general initial concentration properties.

Abstract

The global asymptotic dynamics of point vortices for the lake equations is rigorously derived. Vorticity that is initially sharply concentrated around $N$ distinct vortex centers is proven to remain concentrated for all times. Specifically, we prove weak concentration of the vorticity and in addition strong concentration in the direction of the steepest ascent of the depth function. As a consequence, we obtain the motion law of point vortices following at leading order the level lines of the depth function. The lack of strong localization in the second direction is linked to the vortex filamentation phenomena. The main result allows for any fixed number of vortices and general assumptions on the concentration property of the initial data to be considered. No further properties such as a specific profile or symmetry of the data are required. Vanishing topographies on the boundary are included in our analysis. Our method is inspired by recent results on the evolution of vortex rings in 3D axisymmetric incompressible fluids.