  We provide a concise PDE-based proof of anomalous diffusion in the Kraichan
model -- a stochastic, white-in-time model of passive scalar turbulence. That
is, we show an exponential rate of $L^2$ decay in expectation of a passive
scalar advected by a certain white-in-time, correlated-in-space,
divergence-free Gaussian field, uniform in the initial data and the diffusivity
of the passive scalar. Additionally, we provide examples of correlated-in-time
versions of the Kraichnan model which fail to exhibit anomalous diffusion
despite their (formal) white-in-time limits exhibiting anomalous diffusion. As
part of this analysis, we prove that anomalous diffusion of a scalar advected
by some flow implies non-uniqueness of the ODE trajectories of that flow.
