Tsallis statistics and turbulence

In order to reveal the underlying statistics describing properly the fully developed turbulence, the probability density function of the local dissipation is derived by taking extremal of a generalized entropy (Tsallis entropy) under the two constraints, i.e., one is the normalization of probability and the other is to fix the intermittency exponent being constant. The generalized entropy includes the Boltzmann–Gibbs entropy as a special case where the Tsallis index q is equal to 1. The multifractal spectrum f(α) corresponding to the probability density function is determined self-consistently in the sense that all quantities can be determined by the observed value of the intermittency exponent. It is shown that the scaling exponents ζm of velocity structure function derived by making use of f(α) explains experimental data very well. It is also revealed that the asymptotic expression of ζm for m 1 has a log term. The Tsallis index q turns out to be 0.380 which manifests itself that the system of fully developed turbulence has a nonextensive character.