Point Process Analysis of Vortices in a Periodic Box

TL;DR

This paper models the dynamics of point vortices in a periodic box, using numerical simulations and spatial ecology point process theory to analyze vortex clustering and interactions over time.

Contribution

It introduces a novel application of point process theory to vortex dynamics, providing insights into clustering and vortex interactions in a periodic domain.

Findings

Clustering persists over time if initial conditions are clustered.

The $L$ function indicates positive clustering for vortices of both signs.

Vortex interactions include notable scattering and recoupling events.

Abstract

The motion of assemblies of point vortices in a periodic parallelogram can be described by the complex position $z_j(t)$ whose time derivative is given by the sum of the complex velocities induced by other vortices and the solid rotation centered at $z_j$. A numerical simulation up to 100 vortices in a square periodic box is performed with various initial conditions, including single and double rows, uniform spacing, checkered pattern, and complete spatial randomness. Point process theory in spatial ecology is applied in order to quantify clustering of the distribution of vortices. In many cases, clustering of the distribution persists after a long time if the initial condition is clustered. In the case of positive and negative vortices with the same absolute value of strength, the $L$ function becomes positive for both types of vortices. Scattering or recoupling of pairs of vortices by a third vortex is remarkable.