Virial theorem for Onsager vortices in two-dimensional hydrodynamics

TL;DR

This paper derives a virial theorem for two-dimensional point vortices at statistical equilibrium, relating angular velocity, angular momentum, and temperature, applicable to multiple species and consistent with previous empirical relations.

Contribution

It provides a generalized virial theorem for 2D point vortices valid for multiple species and arbitrary vortex numbers, extending previous empirical findings.

Findings

Derived the virial theorem for 2D point vortices.

Established the relation between angular velocity, momentum, and temperature.

Validated the theorem against known empirical relations.

Abstract

We derive the virial theorem appropriate to two-dimensional point vortices at statistical equilibrium in the microcanonical and canonical ensembles. In an unbounded domain, it relates the angular velocity to the angular momentum and the temperature. Our expression is valid for an arbitrary number of point vortices of possibly different species. In the single-species case, and in the mean field approximation, it reduces to the relation empirically obtained by J.H. Williamson [J. Plasma Physics 17, 85 (1977)].