Description of a Turbulent Cascade by a Fokker-Planck Equation

From experimental data of a turbulent free jet we calculate the joint probability distribution $p\left(v_1{,L}_1{;v}_2{,L}_2\right)$ for two velocity increments $v_1{,v}_2$ of different length scales $L_1{,L}_2$. We present experimental evidence that the conditional probability distribution $p\left(v_2{,L}_2|v_1{,L}_1\right)$ obeys a Fokker-Planck equation. We calculate the corresponding drift and diffusion coefficients and discuss their relationship to universal behavior in the scaling region and to intermittency of the turbulent cascade. We explicitly present a stochastic process for the log-normal model of Kolmogorov and Oboukhov [A. M. Oboukhov, J. Fluid Mech. 13, 77 (1962); A. N. Kolmogorov, J. Fluid Mech. 13, 82 (1962)].