Physics – Condensed Matter – Statistical Mechanics
Scientific paper
1999-11-03
Phys. Rev. E 62, 490 (2000)
Physics
Condensed Matter
Statistical Mechanics
36 pages, submitted to Phys. Rev. E (improved version with extra material, see also cond-mat/9912222)
Scientific paper
10.1103/PhysRevE.62.490
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.
Chavanis Pierre-Henri
Sire Clément
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