Investigation of a generalized Obukhov model for turbulence

We introduce a generalization of Obukhov's model [A.M. Obukhov, Adv. Geophys. 6 (1959) 113] for the description of the joint position-velocity statistics of a single fluid particle in fully developed turbulence. In the presented model the velocity is assumed to undergo a continuous time random walk. This takes into account long time correlations. As a consequence the evolution equation for the joint position-velocity probability distribution is a Fokker–Planck equation with a fractional time derivative. We determine the solution of this equation in the form of an integral transform and derive a relation for arbitrary single time moments. Analytical solutions for the joint probability distribution and its moments are given.