Evolution of a Network of Vortex Loops in the Turbulent Superfluid Helium; Derivation of the Vinen Equation

TL;DR

This paper models the evolution of vortex loop networks in turbulent superfluid helium using a rate equation approach, deriving the Vinen equation and analyzing the interplay of deterministic and stochastic processes.

Contribution

It introduces a comprehensive rate equation framework that combines deterministic mutual friction and stochastic fusion/breakdown processes to derive the Vinen equation.

Findings

The evolution of vortex line density obeys the Vinen equation.

The approach clarifies the roles of deterministic and stochastic processes in vortex dynamics.

The properties of the Vinen equation are discussed within this new framework.

Abstract

The evolution a network of vortex loops due to the fusion and breakdown in the turbulent superfluid helium is studied. We perform investigation on the base of the "rate equation" for the distribution function $n(l)$ of number of loops in space of their length $l$. There are two mechanisms for change of quantity $n(l)$. Firstly, the function changes due to deterministic process of mutual friction, when the length grows or decreases depending on orientation. Secondly, the change of $n(l)$ occurs due to random events when the loop crosses itself breaking down into two daughter or two loops collide merging into one larger loop. Accordingly the "rate equation" includes the "collision" term collecting random processes of fusion and breakdown and the deterministic term. Assuming, further, that processes of random colliding are fastest we are in position to study more slow processes related to deterministic term. In this way we study the evolution of full length of vortex loops per unit volume-so called vortex line density ${\cal L}(t)$. It is shown this evolution to obey the famous Vinen equation. In conclusion we discuss properties of the Vinen equation from the point of view of the developed approach.