  We study the effect of impermeable boundaries on the symmetry properties of a
random passive scalar field advected by random flows. We focus on a broad class
of nonlinear shear flows multiplied by a stationary, Ornstein-Uhlenbeck (OU)
time varying process, including some of their limiting cases, such as Gaussian
white noise or plug flows. For the former case with linear shear, recent
studies \cite{camassa2019symmetry} numerically demonstrated that the decaying
passive scalar's long time limiting probability distribution function (PDF)
could be negatively skewed in the presence of impermeable channel boundaries,
in contrast to rigorous results in free space which established the limiting
PDF is positively skewed \cite{mclaughlin1996explicit}. Here, the role of
boundaries in setting the long time limiting skewness of the PDF is established
rigorously for the above class using the long time asymptotic expansion of the
$N$-point correlator of the random field obtained from the ground state
eigenvalue perturbation approach proposed in \cite{bronski1997scalar}. Our
analytical result verifies the conclusion for the linear shear flow obtained
from numerical simulations in \cite{camassa2019symmetry}. Moreover, we
demonstrate that the limiting distribution is negatively skewed for any shear
flow at sufficiently low P\'eclet number. We show that the long time limit of
the first three moments depends explicitly on the value of $\gamma$, which is
in contrast to the conclusion in \cite{vanden2001non} for the limiting PDF in
free space. We derive the exact formula of the $N$-point correlator for a flow
with no spatial dependence and Gaussian temporal fluctuation, generalizing the
results of \cite{bronski2007explicit}. The long time analysis of this formula
is consistent with our theory for a general shear flow. All results are
verified by Monte-Carlo simulations.
