Vortex reconnection rate, and loop birth rate, for a random wavefield

TL;DR

This paper derives exact formulas for the rates of vortex reconnection, loop birth, and death in a 3D Gaussian random wavefield, revealing reconnections are more frequent than births and deaths combined.

Contribution

It provides the first exact analytical expressions for vortex event rates in a 3D isotropic Gaussian wavefield based on the power spectrum.

Findings

Reconnection rate R is expressed in terms of the power spectrum moments.

Birth and death rates are equal and derived from the reconnection rate.

In 2D, vortex pair creation rate is also analytically calculated.

Abstract

A time dependent, complex scalar wavefield in three dimensions contains curved zero lines, wave 'vortices', that move around. From time to time pairs of these lines contact each other and 'reconnect' in a well studied manner, and at other times tiny loops of new line appear from nowhere (births) and grow, or the reverse, existing loops shrink and disappear (deaths). These three types are known to be the only generic events. Here the average rate of their occurrences per unit volume, R, B, and D is calculated exactly for a Gaussian random wavefield that has isotropic stationary statistics, arising from a superposition of an infinity of plane waves in different directions. A simplifying 'axis fixing' technique is used to achieve this. The resulting formulas are expressed in terms of the power spectrum of the ensemble plane waves: R=WSqrt[K4^3/(12 Pi^4 K2(K4-K2^2))], and B=D=(R/2)-(W/2)Sqrt[9 K2^3/(16 Pi^4)] where W is the standard deviation of angular frequencies, and K2 and K4 are the second and fourth moments of a wave vector component (say the x one). Thus reconnections are always more common than births and deaths combined. As an expository preliminary, the case of two dimensions, where the vortices are points, is studied and the average rate of pair creation (and likewise destruction) per unit area is calculated to be WSqrt[(K4-K2^2)/(4 Pi^4)].