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

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.