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We consider a cellular flow with a random perturbation of size $\sqrt{\nu}$. On time scales of order $1$, a standard large deviations principle can be used to study the small noise limit. On the other hand, when $\nu$ is of order $1$, classical homogenization results show that the long time limit is an effective Brownian motion. Our aim is to study scaling regimes in between these two extremes. A recent result of Hairer, Koralov and Pajor-Gyulai establishes an averaging principle on time scales of order $1/\nu$. On even shorter time scales an anomalous diffusive effect is observed when the process starts from the separatrix. I will discuss asymptotic estimates for the variance in this situation, and ongoing work describing the effective process.