Vortex motion of the Euler and Lake equations

TL;DR

This paper surveys vortex motion in Euler and lake equations, proves non-collision for 2-vortex systems, shows nonintegrability for N>2 in the half-plane, and compares vortex dynamics across different models and geometries.

Contribution

It provides rigorous proofs of non-collision and nonintegrability in vortex systems, and draws novel comparisons between Euler, lake equations, and geometric vortex flows.

Findings

2-vortex system in half-plane is non-collision

N-vortex system in half-plane is nonintegrable for N>2

similarities between vortex motions in different models and geometries

Abstract

We start by surveying the planar point vortex motion of the Euler equations in the whole plane, half-plane and quadrant. Then we go on to prove non-collision property of 2-vortex system by using the explicit form of orbits of 2-vortex system in the half-plane. We also prove that the $N$-vortex system in the half-plane is nonintegrable for $N>2$, which is suggested previously by numerical experiments without rigorous proof. The skew-mean-curvature (or binormal) flow in $\mathbb R^n,\;n\geq3$ with certain symmetry can be regarded as point vortex motion of the 2D lake equations. We compare point vortex motions of the Euler and lake equations. Interesting similarities between the point vortex motion in the half-plane, quadrant and the binormal motion of coaxial vortex rings, sphere product membranes are addressed. We also raise some open questions in the paper.