# Anomalous diffusion in passive scalar transport

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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.