Statistics of velocity fluctuations arising from a random distribution of point vortices: The speed of fluctuations and the diffusion coefficient

This paper is devoted to a statistical analysis of the fluctuations of velo city and acceleration produced by a random distribution of point vortices i n two-dimensional turbulence. We show that the velocity probability density function PDF behaves in a manner which is intermediate between Gaussian an d Levy laws, while the distribution of accelerations is governed by a Cauch y law. Our study accounts properly for a spectrum of circulations among the vortices. In the case of real vortices (with a finite core), we show analy tically that the distribution of accelerations makes a smooth transition fr om Cauchy (for small fluctuations) to Gaussian (for large fluctuations), pr obably passing through an exponential tail. We introduce a function T(V) wh ich gives the typical duration of a velocity fluctuation V; we show that T( V) behaves like V and V-1 for weak and large velocities, respectively. Thes e results have a simple physical interpretation in the nearest neighbor app roximation, and in Smoluchowski theory concerning the persistence of fluctu ations. We discuss the analogies with respect to the fluctuations of the gr avitational field in stellar systems. As an application of these results, w e determine an approximate expression for the diffusion coefficient of pain t vortices. When applied to the context of freely decaying two-dimensional turbulence, the diffusion becomes anomalous and we establish a relationship nu=1+(xi/2) between the exponent of anomalous diffusion nu and the exponen t xi which characterizes the decay of the vortex density.