Consider a diffusive passive scalar advected by a two
dimensional incompressible flow. If the flow is cellular (i.e.\ has a
periodic Hamiltonian with no unbounded trajectories), then classical
homogenization results show that the long time behaviour is an effective
Brownian motion. We show that on intermediate time scales, the effective
behaviour is instead a fractional kinetic process. At the PDE level this
means that while the long time scaling limit is the heat equation, the
intermediate time scaling limit is a time fractional heat equation. We
will also describe the expected intermediate behaviour in the presence
of open channels.