Abstract
It is well known that on long time scales the behaviour of tracer particles diffusing in a cellular flow is effectively that of a Brownian motion. This paper studies the behaviour on “intermediate” time scales before diffusion sets in. Various heuristics suggest that an anomalous diffusive behaviour should be observed. We prove that the variance on intermediate time scales grows like \(O(\sqrt{t})\). Hence, on these time scales the effective behaviour can not be purely diffusive, and is consistent with an anomalous diffusive behaviour.
Highlights
We study the behaviour of tracer particles diffusing in the presence of a strong array of opposing vortices (a.k.a. “cellular flow”)
Well known homogenization results show that on long time scales these particles effectively behave like a Brownian motion, with an enhanced diffusion coefficient
A purely diffusive process (e.g. Brownian motion) would have variance that is linear in t, and so the effective behaviour of the tracer particles at these time scales “must be anomalous”
Summary
We study the behaviour of tracer particles diffusing in the presence of a strong array of opposing vortices (a.k.a. “cellular flow”). Our approach to proving Theorem 1.1 is by estimating the expected number of times the process X crosses over the boundary layer Bδ. When X is in the boundary layer, it spends O(δ2) time near cell edges (where ∇h is non-degenerate) and O(δ2|ln δ|) near corners (where ∇h degenerates).
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