Velocity statistics for non-uniform configurations of point vortices

TL;DR

This paper analyzes how non-uniform, fractal distributions of point vortices affect the velocity fluctuation statistics, revealing power-law tails influenced by the spatial distribution exponent.

Contribution

It introduces a model for velocity fluctuations caused by fractally distributed vortices and derives the resulting power-law velocity distribution tail behavior.

Findings

Velocity fluctuations follow a power-law tail $P(V) \\sim V^{\\alpha-2}$.

The tail behavior depends on the fractal distribution exponent $\\alpha$.

The velocity distribution's scaling range is determined by vortex density and $\\alpha$.

Abstract

Within the point vortex model, we compute the probability distribution function of the velocity fluctuations induced by same-signed vortices scattered within a disk according to a fractal distribution of distances to origin $\sim r^{-\alpha}$. We show that the different random configurations of vortices induce velocity fluctuations that are broadly distributed, and follow a power-law tail distribution, $P(V)\sim V^{\alpha-2}$ with a scaling exponent determined by the $\alpha$ exponent of the spatial distribution. We also show that the range of the power-law scaling regime in the velocity distribution is set by the mean density of vortices and the exponent $\alpha$ of the vortex density distribution.