Kinetics of a Network of Vortex Loops in He II and a Theory of Superfluid Turbulence

TL;DR

This paper develops a kinetic theory for superfluid turbulence based on vortex loop dynamics, deriving a power-law distribution and connecting it to the Vinen equation for vortex line density.

Contribution

It introduces a novel kinetic model with an exact stationary solution describing vortex loop size distribution and fluxes, linking microscopic reconnection processes to macroscopic turbulence properties.

Findings

Derived an exact power-law distribution for vortex loops.

Connected vortex reconnection dynamics to the Vinen equation.

Analyzed the structure and evolution of vortex tangles in superfluid helium.

Abstract

A theory is developed to describe the superfluid turbulence on the base of kinetics of the merging and splitting vortex loops. Because of very frequent reconnections the vortex loops (as a whole) do not live long enough to perform any essential evolution due to the deterministic motion. On the contrary, they rapidly merge and split, and these random recombination processes prevail over other slower dynamic processes. To develop quantitative description we take the vortex loops to have a Brownian structure with the only degree of freedom, which is the length $l$ of the loop. We perform investigation on the base of the Boltzmann type kinetic equation for the distribution function $n(l)$ of number of loops with length $l$. By use of the special ansatz in the collision integral we have found the exact power-like solution to kinetic equation in the stationary case. This solution is not (thermodynamically) equilibrium, but on the contrary, it describes the state with two mutual fluxes of the length (or energy) in space of sizes of the vortex loops. The term flux means just redistribution of length (or energy) among the loops of different sizes due to reconnections. Analyzing this solution we drew several results on the structure and dynamics of the vortex tangle in the turbulent superfluid helium. In particular, we evaluated the mean radius of the curvature and the full rate of the reconnection events. We also studied the evolution of the full length of vortex loops per unit volume-the so-called vortex line density. It is shown this evolution to obey the famous Vinen equation. The properties of the Vinen equation from the point of view of the developed approach had been discussed.