Log-Stable Distribution and Intermittency of Turbulence

The logarithm of the breakdown coefficient \varepsilon_r/\varepsilon_l, \varepsilon_r being the mean energy dissipation rate averaged over a sphere of radius r is shown, under a similarity assumption, to obey a stable distribution, the characteristic function of which is given by \varphi(z|r/l){=}(r/l)^{(μ/2^α}-2)[iz-(ze^iπ/2)^α]}, where μ{>0 and 0<α≤2. The scaling exponent of the p-th order moment of the energy dissipation rate is calculated to be μ_p{=}μ({p}^α-p)/(2^α-2), which is in excellent agreement with the experiments (Anselmet et al. 1984) when the intermittency parameter is μ{=}0.20 and the characteristic exponent of the distribution is α{=}1.65. The probability density function of \varepsilon_r diverges as 1/\varepsilon_r(-\ln \varepsilon_r)^α+1 at the origin and decreases as \exp [-A (\ln \varepsilon_r)^α/(α-1)], where A>0, as \varepsilon→∞. The present results include the log-normal theory for α{=}2 and coincide with the prediction of μ_p due to the β-model in the limit α→0.