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We study the long time behavior of an advection-diffusion equation with arandom shear flow which depends on a stationary Ornstein-Uhlenbeck (OU) processin parallel-plate channels enforcing the no-flux boundary conditions. We derivea closed form formula for the long time asymptotics of the arbitrary $N$-pointcorrelator using the ground state eigenvalue perturbation approach proposed in\cite{bronski1997scalar}. In turn, appealing to the conclusion of the Hausdorffmoment problem \cite{shohat1943problem}, we discover a diffusion equation witha random drift and deterministic enhanced diffusion possessing the exact sameprobability distribution function at long times. Such equations enjoy manyergodic properties which immediately translate to ergodicity results for theoriginal problem. In particular, we establish that the first two Aris momentsusing a single realization of the random field can be used to explicitlyconstruct all ensemble averaged moments. Also, the first two ensemble averagedmoments explicitly predict any long time centered Aris moment. Our formulaequantitatively depict the dependence of the deterministic effective diffusionon the interaction between spatial structure of flow and random temporalfluctuation. This noteably contrasts the white noise case from the OU case.Further, this approximation provides many identities regarding the stationaryOU process dependent time integral. We derive explicit formulae for thedecaying passive scalar's long time limiting probability distribution function(PDF) for different types of initial conditions (e.g. deterministic andrandom). All results are verified by Monte-Carlo simulations.

Ergodicity and invariant measures for a diffusing passive scalar  advected by a random channel shear flow and the connection between the  Kraichnan-Majda model and Taylor-Aris Dispersion