The 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 velocity fluctuations produced by a random distribution of point vortices in two-dimensional turbulence. We show that the velocity probability density function (p.d.f) is pathological as it behaves in a manner which is intermediate between Gaussian and L\'evy laws. We introduce a function T(V) which 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. These results have a simple physical interpretation in the nearest neighbor approximation and in Smoluchowski (1916) theory concerning the persistence of fluctuations. We discuss the analogies with respect to the fluctuations of the gravitational field in stellar systems. As an application of these results, we determine an approximate expression for the diffusion coefficient of point vortices. When applied to the context of freely decaying two-dimensional turbulence, the diffusion becomes anomalous and we establish a relationship $\nu=1+{\xi\over 2}$ between the exponent of anomalous diffusion $\nu$ and the exponent $\xi$ which characterizes the decay of the vortex density. This relation is in good agreement with laboratory experiments and numerical simulations.