It is a well documented, experimental fact that while single point velocity measurements in turbulent fluids admit nearly Gaussian statistics, measurements of other quantities including velocity increments, vorticity, pressure, and passively transported quantities admit strongly non-Gaussian statistics. There has been an intense and important phenomenological effort attempting to explain this behavior in the context of a passive scalar using a variety of closure approximations and other ad-hoc approximations. Even in the context of a passive scalar, the question of inherited statistics is extremely difficult, requiring the consideration of partial differential equations with variable coefficients, in very large dimension.

In this lecture, we study the model introduced by Majda which concerns a passive scalar decaying in the presence of a rapidly fluctuating, Gaussian linear shear profile. In joint work with Richard M. McLaughlin (UNC - Chapel Hill), we present the first explicit construction of the limiting asymptotics for the moments of the normalized scalar, and use this information to rigorously deduce the pdf tail. This elementary field theory shows the pdf tail ranging from Gaussian through stretched exponential as a parameter is varied, and we additionally obtain an explicit relation between the tail of the scalar and the tail of the scalar gradient which explicitly demonstrates how derivatives may become more intermittent.

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