The random uniform shear layer: An explicit example of turbulent diffusion with broad tail probability distributions

Recent experimental and computational observations demonstrate the occurrence of large-scale intermittency for diffusing passive scalars, as manifested by broader than Gaussian probability distribution functions. Here, a family of explicit exactly solvable examples is developed which demonstrates these effects of large-scale intermittency at any positive time through simple formulas for the higher flatness factors without any phenomenological approximations. The exact solutions involve advection-diffusion with velocity fields involving a uniform shear flow perturbed by a random fluctuating uniform shear flow. Through an exact quantum mechanical analogy, the higher-order statistics for the scalar in these models are solved exactly by formulas for the quantum-harmonic oscillator. These explicit formulas also demonstrate that the large time asymptotic limiting probability distribution function for the normalized scalar can be either broader than Gaussian or Gaussian depending on the relative strength of the mean flow and the fluctuating velocity field.