  We consider a scalar field governed by an advection-diffusion equation (or a
more general evolution equation) with rapidly fluctuating, Gaussian distributed
random coefficients. In the white noise limit, we derive the closed evolution
equation for the ensemble average of the random scalar field by three different
strategies, i.e., Feynman-Kac formula, the limit of Ornstein-Uhlenbeck process,
and evaluating the cluster expansion of the propagator on an $n$-simplex. With
the evolution equation of ensemble average, we study the passive scalar
transport problem with two different types of flows, a random periodic flow,
and a random strain flow. For periodic flows, by utilizing the homogenization
method, we show that the $N$-point correlation function of the random scalar
field satisfies an effective diffusion equation at long times. For the strain
flow, we explicit compute the mean of the random scalar field and show that the
statistics of the random scalar field have a connection to the time integral of
geometric Brownian motion. Interestingly, all normalized moment (e.g.,
skewness, kurtosis) of this random scalar field diverges at long times, meaning
that the scalar becomes more and more intermittent during its decay.
