Probability distribution function of self-organization of shear flows

We present a statistical theory of self-organisation of shear flows, modeled by a nonlinear diffusion equation driven by a stochastic forcing. A non-perturbative method based on a coherent structure is utilized for the prediction of the PDFs, showing strong intermittency with exponential tails. We confirm these results by numerical simulations. Furthermore, the results reveal a significant probability of supercritical states due to stochastic perturbation, which could have crucial implications in a variety of systems. To elucidate a crucial role of relative time scales of relaxation and disturbance in the determination of the 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 the statistical description of gradients, rather than their average value as has conventionally been done.