Stochastic process for the dynamics of the turbulent cascade

The data on velocity increments over a distance l and turbulent energy dissipation on a box of size l are, in the inertial range, well described by the phenomenological multifractal models of fully developed turbulence. These quantities and models, however, do not specify the time correlations and therefore are not a complete parametrization for the dynamics of the turbulent cascade. An asymmetric stochastic process on a tree is proposed as a phenomenological model to parametrize the dynamics of the turbulent cascade. In the framework of this model, any concrete assumptions about the physical parameters of the cascade are easily related to the structure of the temporal correlations and, therefore, may be checked by comparison with experiment. As an example, we show how to relate the decay of the temporal correlations to the scale dependence of the lifetimes of the eddies at different length scales. From the hierarchical structure of the proposed process and the coexistence of many different time scales, we predict specific deviations from exponential behavior which may be checked by two-point correlation experiments.