Cascade in turbulence and near-exponential distributions: a dynamical theory

A dynamical mechanism is considered which connects the cascade with non-gaussian statistics of velocity gradients. The analysis is based on the vorticity equation which describes time evolution of a fluctuation strained by a background field consisting of a regular component and a singular component of fractional scaling exponent. This field has a property consistent with the observed skewness of the longitudinal derivative in turbulence. Nonlinear model equations are derived to describe the cascade of the fluctuation. The present model predicts a difference in statistics between the lateral and longitudinal derivatives. The present analysis suggests an increase of the flatness factor with the order of derivative.