Trapped slender vortex filaments in statistical equilibrium

TL;DR

This paper develops an explicit formula for the mean square vortex position in a system of nearly parallel vortex filaments, revealing how 3D effects influence confinement and length scale, validated through simulations.

Contribution

It introduces a mean-field approximation with a spherical constraint to analytically determine the vortex length scale, highlighting 3D effects in vortex filament systems.

Findings

3D vortex filaments resist confinement differently than 2D point vortices

High-density regimes show a shift in the length scale between quasi-2D and strictly-2D models

Analytical results align well with Path Integral Monte Carlo simulations

Abstract

The statistical mechanics of nearly parallel vortex filaments confined in the unbounded plane by angular momentum, first studied by Lions and Majda (2000), is investigated using a mean-field approximation to interaction and a spherical constraint to develop an explicit formula for the mean square vortex position or length scale of the system, $R$, verified with Path Integral Monte Carlo simulations. We confirm that 3D filaments resist confinement in a different way than 2D point vortices and that this results in a profound shift at high-densities for the length scale of quasi-2D versus strictly-2D models of vorticity fields in which angular momentum is conserved. Our analytical results correspond well with those of the Monte Carlo simulations and show a 3D effects contributing significantly to determination of the length scale.