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