On the small-scale statistics of Lagrangian turbulence

We provide evidence that the small-scale statistics of the acceleration of a test particle in high-Reynolds number Lagrangian turbulence is correctly described by Tsallis statistics with entropic index q = 3 2 . We present theoretical arguments why Tsallis statistics can naturally arise in Lagrangian turbulence and why at the smallest scales q = 32 is relevant. A generalized Heisenberg-Yaglom formula is derived from the nonextensive model. 1 permanent address: School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS. Recently, methods from nonextensive statistical mechanics [1, 2, 3] have been successfully applied to fully developed turbulent flows [4, 5, 6, 7]. As a driven noneqilibrium system a turbulent flow cannot extremize the Boltzmann-Gibbs entropy — it is obvious that ordinary statistical mechanics fails to provide a correct description of turbulence. But there is experimental and theoretical evidence that the statistics of velocity differences is well described if one assumes that the flow extremizes the ‘second best’ information measures available. These are the Tsallis entropies [1] defined by