Exact solution to a nearly parallel vortex filament mean-field theory

TL;DR

This paper provides an exact solution for nearly parallel vortex filaments in a mean-field framework, revealing an infinite order phase transition absent in point vortex models, with implications for fluid dynamics and plasma physics.

Contribution

It introduces an exact solution for nearly parallel vortex filaments under angular momentum conservation, highlighting a novel phase transition.

Findings

Existence of an infinite order phase transition in the filament system

Differences between vortex filament and point vortex models

Analytical characterization of filament interactions

Abstract

Nearly parallel vortex filaments are a generalization of point vortices and describe many phenomena under conservation of angular momentum including vortices forming in deep ocean convection, magnetically confined plasmas, and the solar atmosphere. While point vortices represent perfectly straight, parallel lines of circulation, nearly parallel vortex filaments have some curvature due to internal viscosity. They interact logarithmically but have a kinetic self-energy as well. In this letter, I present an exact solution of a system of these filaments under angular momentum conservation in a mean-field theory. I show that the filaments have an infinite order phase transition not present in the point vortex model.