Integrable two layer point vortex motion on the half plane

TL;DR

This paper derives and analyzes the equations governing two-layer point vortex motion on the half plane, demonstrating integrability, characterizing vortex behaviors, and exploring relative equilibria and streamline topologies.

Contribution

It introduces the integrable model of two-layer vortex motion on the half plane, characterizes all vortex motions for specific strengths, and identifies new relative equilibria configurations.

Findings

The two vortex problem is integrable on the half plane.

No equilibria exist when vortex strengths are opposite.

Only one relative equilibrium configuration exists when vortices are in different layers.

Abstract

In this paper we derive the equations of motion for two-layer point vortex motion on the upper half plane. We study the invariants using symmetry, including the Hamiltonian and show that the two vortex problem is integrable. We characterize all two vortex motions for the cases where the vortex strengths are both equal, $\Gamma_{1}=\Gamma_{2}=1$ and when they are opposite $\Gamma_{1}=-\Gamma_{2}=1$. We also prove that there are no equilibria for the two vortex problem when $\Gamma_{1}=-\Gamma_{2}=1$.\ We show that there is only one relative equilibrium configuration when $\Gamma_{1}=\Gamma_{2}=1$ and the vortices are in different layers. We also make observations concerning the finite-time collapse of two vortices in the half plane. We then compare the regimes of motion for both cases (motion on the half plane) with the case of the two-layer vortex problem on the entire plane. We also study several classes of streamline topologies for two vortices in different layers. We conclude with a Hamiltonian study of integrable two-layer 3 vortex motion on the half plane by studying integrable symmetrical configurations and provide a rich class of new relative equilibria.