ABSTRACT

Intermittency - the fact that the probability distribution function for a quantity transported by a turbulent flow is asymptotically broad - is an important phenomena in turbulence. We present some work (with R.M. McLaughlin - UNC Chapel Hill) on a model of passive scalar intermittency originally due to Majda: dT/dt=g(t) x dT/dy+DT, where g(t) is a random process and T is a passive scalar (for instance which is advected by the random (shear) flow). Majda was able to explicitly calculate moments of the distribution of the scalar T. Bronski and McLaughlin were able to calculate the large N asymptotics of the moments of the distibution and, by a large deviations/Tauberian type argument calculate the distribution of the quantity T. I will also talk about some recent work on a generalization of this model (originally proposed by E. Vanden-Eijnden). A similar calculation can be done for this generalized model, which involves calculating the asymptotics of a certain compact eigenvalue problem. As a by-product of this calculation one finds the (previously unknown) optimal constants in a certain probabilistic "small ball" estimate for the probability that a fractional Brownian motion stays in a small ball in L2.