Collective behaviour of large number of vortices in the plane

TL;DR

This paper analyzes the dynamics and stability of large vortex systems in the plane, providing explicit formulas for observed configurations and exploring how damping influences vortex crystallization.

Contribution

It introduces a novel connection between vortex dynamics and biological swarm models, enabling explicit characterization of various vortex configurations and their stability.

Findings

Explicit formulas for vortex configurations including equal strength and mixed strength cases.

Demonstration of configurations not yet observed experimentally, such as vortices inside an ellipse.

Artificial damping leads to convergence to lattice-like equilibria, explaining vortex crystallization.

Abstract

We investigate the dynamics of $N$ point vortices in the plane, in the limit of large $N$. We consider {\em relative equilibria}, which are rigidly rotating lattice-like configurations of vortices. These configurations were observed in several recent experiments [Durkin and Fajans, Phys. Fluids (2000) 12, 289-293; Grzybowski {\em et.al} PRE (2001)64, 011603]. We show that these solutions and their stability are fully characterized via a related {\em aggregation model} which was recently investigated in the context of biological swarms [Fetecau {\em et.al.}, Nonlinearity (2011) 2681; Bertozzi {\em et.al.}, M3AS (2011)]. By utilizing this connection, we give explicit analytic formulae for many of the configurations that have been observed experimentally. These include configurations of vortices of equal strength; the $N+1$ configurations of $N$ vortices of equal strength and one vortex of much higher strength; and more generally, $N+K$ configurations. We also give examples of configurations that have not been studied experimentally, including $N+2$ configurations where $N$ vortices aggregate inside an ellipse. Finally, we introduce an artificial ``damping'' to the vortex dynamics, in an attempt to explain the phenomenon of crystalization that is often observed in real experiments. The diffusion breaks the conservative structure of vortex dynamics so that any initial conditions converge to the lattice-like relative equilibrium.