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

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 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 introduced to achieve this. The resulting formulas are proportional to the standard deviation of angular frequencies, and depend in a simple way on the second and fourth moments of the power spectrum of the plane waves. Reconnections turn out to be 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.