Intermittency and self-organization in turbulent flows

We present a statistical theory of self-organization of shear flows, modeled by a nonlinear diffusion equation with a stochastic forcing. A non-perturbative method based on a coherent structure is utilized for the prediction of probability distribution functions (PDFs), showing strong intermittency with exponential tails. We confirm these results by numerical simulations. Furthermore, the results reveal a significant probability of super-critical states due to stochastic perturbation. To elucidate the crucial role of relative time scales of relaxation and disturbance in the determination of PDFs, we present numerical simulation results obtained in a threshold model where the diffusion is given by discontinuous values. Our results highlight the importance of a statistical description of gradients.