Effective transport in random shear flows

We obtain an effective transport description for the superdiffusive motion of random walkers in stratified flow by projection of the process on the direction of stratification. The effective dimensionally reduced motion is shown to describe a correlated random walk characterized by the Lagrangian velocity correlation. We analyze the projected motion through exact analytical solutions for the distribution density for an arbitrary correlated Gaussian noise and derive an evolution equation for the one-point and conditional two-point displacement densities. The latter gives an explicit effective equation for superdiffusive transport in stratified random flow and demonstrates that the displacement density has a Gaussian scaling form for all times.