Log-Stable Distribution and Intermittency of Turbulence

The logarithm of the breakdown coefficient ε r /ε l , ε 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 ϕ( z | r / l )=( r / l ) (µ/2 α -2)[i z -( z e $ i π/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 ε r diverges as 1/ε r (-lnε r ) α+1 at the origin and decreases as exp [- A (lnε r ) α/(α-1) ], where A >0, as ε→∞. 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.