Non-Gaussian invariant measures for the majda model of decaying turbulent transport

The problem of turbulent transport of a scalar field by a random velocity field is considered. The scalar field amplitude exhibits rare but very large fluctuations whose typical signature is fatter than Gaussian tails for the probability distribution of the scalar. The existence of such large fluctuations is related to clustering phenomena of the Lagrangian paths within the flow. This suggests an approach to turn the large-deviation problem for the scalar field into a small-deviation, or small-ball, problem for some appropriately defined process measuring the spreading with time of the Lagrangian paths. Here such a methodology is applied to a model proposed by Majda consisting of a white-in-time linear shear flow and some generalizations of it where the velocity field has finite, or even infinite, correlation time. The non-Gaussian invariant measure for the (reduced) scalar field is derived, and, in particular, it is shown that the one-point distribution of the scalar has stretched exponential tails, with a stretching exponent depending on the parameters in the model. Different universality classes for the scalar behavior are identified which, all other parameters being kept fixed, display a one-to-one correspondence with an exponent measuring time persistence effects in the velocity field.