Determinism for diffusing passive scalars advected by general unsteady random shear flows

We study the long time behavior of an advection-diffusion equation with a general random shear flow imposing no-flux boundary conditions on channel walls using Center Manifold Theory (CMT). Recent results have explicitly calculated using statistical moment closure the invariant measure for a diffusing passive scalar advected by a class of Gaussian random shear flows. Here we establish how center manifold theory can be used to greatly extend these theories to a much broader class of random (non-Gaussian) shear flows, particularly regarding their temporal statistics. In doing so, we can extend results which show how all the effective diffusion coefficients converge at long times to a deterministic value for this broader class of flows. Such results are important ergodicity-like results in that they assure an experimentalist need only perform a single realization of a random flow to observe the ensemble moment predictions at long time. Monte-Carlo simulations will be presented illustrating how the highly random behavior converges to the deterministic limit at long time.