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:
\[ T_t = \gamma(t) x \frac{\partial T}{\partial y} + \Delta T \]
where $\gamma(t)$ 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 $T$.
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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