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

TL;DR

This paper provides a detailed statistical analysis of velocity and acceleration fluctuations caused by random point vortices in 2D turbulence, revealing their distribution laws, fluctuation durations, and implications for vortex diffusion and turbulence decay.

Contribution

It introduces a comprehensive statistical framework for velocity and acceleration fluctuations in vortex systems, including new analytical results and physical interpretations, especially for real vortices with finite cores.

Findings

Velocity p.d.f. is between Gaussian and Lévy laws.

Acceleration distribution follows a Cauchy law, transitioning to Gaussian for large fluctuations.

Derived an approximate diffusion coefficient and related anomalous diffusion exponent to vortex decay.

Abstract

This paper is devoted to a statistical analysis of the fluctuations of velocity and acceleration produced by a random distribution of point vortices in two-dimensional turbulence. We show that the velocity probability density function (p.d.f.) behaves in a manner which is intermediate between Gaussian and L\'evy laws while the distribution of accelerations is governed by a Cauchy 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 analytically that the distribution of accelerations makes a smooth transition from Cauchy (for small fluctuations) to Gaussian (for large fluctuations) passing probably through an exponential tail. 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 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.