Collective Dynamics of Vortex Clusters in Compact Fluid Domains: From Pair Interactions to a Quadrupole Description

TL;DR

This paper develops an analytical model for vortex cluster dynamics on compact fluid domains, revealing a universal quadrupole-based description that captures collective rotation and breathing behaviors, validated by numerical simulations.

Contribution

It introduces a novel quadrupole moment framework for describing vortex cluster dynamics on compact domains, extending understanding of collective vortex behavior.

Findings

Explicit formulas for vortex orbital rotation and dipole translation velocities.

Universal decomposition of cluster dynamics into planar, isotropic, and anisotropic modes.

Quantitative confirmation of the quadrupole-based reduced model through numerical simulations.

Abstract

Clusters of co-rotating vortices on compact fluid domains exhibit a simple collective dynamics, combining coherent global rotation with a slow breathing of the cluster size. In this work, we investigate an analytically tractable model of vortex interactions on a doubly periodic inviscid fluid domain, based on an exact representation in terms of the Schottky--Klein prime function and its $q$-representation. The two-vortex problem reduces to a single complex degree of freedom, from which explicit expressions for the orbital rotation frequency and dipole translation velocity are obtained. Building on this framework, we derive a small-cluster expansion that reveals a universal decomposition of the dynamics into planar interactions, isotropic torus corrections, and geometry-induced anisotropic modes. At leading order, the collective dynamics admits a description in terms of a single complex quadrupole moment: its real part governs corrections to the rotation rate, while its imaginary part controls the slow breathing of the cluster. These predictions are quantitatively confirmed by direct numerical simulations, establishing a reduced description of vortex clusters on the flat torus and compact fluid domains.