Velocity statistics for non-uniform configurations of point vortices

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.