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Description

Intermittency is one of the important phenomena in turbulence. Simply put intermittency is the fact that the probability distribution functions (PDF's) for quantities transported by a turbulent flow are asymptotically broad - wider than a Gaussian distribution. We present some work (with R.M. McLaughlin (UNC)) on a model of passive scalar intermittency originally due to Majda: \[ T_t = \gamma(t) x \frac{\partial T}{\partial y} + \D\elta T \] where $\gamma(t)$ is a random process, and $T$ is a passive scalar (for instance a dye) which is advected by the random (shear) flow. Majda was able to explicitly calculate moments of the distribution of the scalar $T$. McLaughlin and B. were able to calculate the large $N$ asymptotics of the moments of the distribution 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. 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 $L_2$.

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