The analysis of the fully developed turbulence with the help of the binomial multiplicative process is reinvestigated from the standpoint of Tsallis nonextensive statistics in order to provide some insight in understanding the underlying physical meaning of the Tsallis ensemble. The formula is derived which relates Tsallis index q with the intermittency exponent that is a manifestation of multifractality of the sizes of eddies. It is shown that the probability density function of the local dissipation of turbulent kinetic energy can be represented by a distribution function of the type in Tsallis statistics with the index q determined by the experimentally observable quantity through the derived formula.

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In order to reveal the underlying statistics properly describing fully developed turbulence, the probability density function of the local dissipation is determined by taking the extreme of a generalized entropy (Tsallis entropy). With the density function, the scaling exponents ζ m of the velocity structure function are derived analytically. It is found that these scaling exponents are consistent with experimental data. The asymptotic expression of ζ m for m 1 has a log term.In a previous paper, 1) we showed that the Tsallis index q 2), 3) corresponding to the p-model 4) can be effectively determined by observed values of the intermittency exponent µ with the help of the scaling relation 5)where α max and α min represent the extremes of the multifractal spectrum f (α) within the p-model. We proposed in Ref. 1) a Tsallis-type distribution function for the probability density function of the local dissipation, and showed that the proposed distribution function with the derived Tsallis index q explains the binomial distribution function of the p-model very well.In this paper, we develop the program presented in Ref. 1) in order to reveal the underlying statistics properly describing fully developed turbulence. 6), 7) We seek an appropriate probability density function of the local dissipation by taking the extreme of a generalized entropy (called the Tsallis entropy 2), 3) ) under two constraints: normalization of the probability and fixing the intermittency exponent µ to a constant value. The latter constraint is a manifestation of the situation that the only information we have is the intermittency exponent µ. The Tsallis entropy includes the Boltzmann-Gibbs entropy as a special case. We determine the multifractal spectrum f (α) corresponding to the probability density function self-consistently in the sense that all quantities can be calculated by using the observed value of the intermittency exponent µ. There is no fitting parameter left. The probability density function we obtain reveals that the underlying statistics are not of the extensive Boltzmann-Gibbs-type but of the non-extensive Tsallis-type.With the help of the multifractal spectrum f (α), we derive the scaling exponents ζ m of the velocity structure function [see (5) below], and compare them with experimental data and with the curves given by the K41, log-normal, β-model, pmodel and log-Poisson theories. We show that the present result accounts for all the by guest on June 4, 2015 http://ptp.oxfordjournals.org/ Downloaded from

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