Statistical Equilibrium of trapped slender vortex filaments - a continuum model

TL;DR

This paper investigates the statistical equilibrium of slender vortex filaments using a continuum model, revealing divergences from point vortex behavior at high temperatures and developing a free energy approach to match Monte Carlo simulations.

Contribution

It introduces a continuum model for vortex filaments, analyzes their statistical equilibrium, and develops a free energy framework incorporating mean field effects.

Findings

Divergence in mean square vortex position at high temperature.

Development of a free energy equation matching Monte Carlo results.

Comparison with point vortex model of Onsager.

Abstract

Systems of nearly parallel, slender vortex filaments in which angular momentum is conserved are an important simplification of the Navier-Stokes equations where turbulence can be studied in statistical equilibrium. We study the canonical Gibbs distribution based on the Klein-Majda-Damodaran (KMD) model and find a divergence in the mean square vortex position from that of the point vortex model of Onsager at high temperature. We subsequently develop a free energy equation based on the non-interacting case, with a spherical constraint, which we approximate using the method of Kac-Berlin, adding a mean field term for logarithmic interaction. This free energy equation, we use to predict the Monte Carlo results.