Cellular flows: Homogenization, Averaging and Anomalous Diffusion

Stochastics Seminar
Thursday, February 28, 2013 - 15:05
1 hour (actually 50 minutes)
Skiles 006
Carnegie Mellon
I will talk about two model problem concerning a diffusion with a cellular drift (a.k.a array of opposing vortices). The first  concerns the expected exit time from a domain as both the flow amplitude $A$ (or more precisely the Peclet number) goes to  infinity, AND the cell size (or vortex seperation) $\epsilon$ approaches $0$ simultaneously. When one of the parameters is fixed, the problem has been extensively studied and the limiting behaviour is that of an effective "homogenized" or "averaged" problem. When  both vary simultaneously one sees an interesting transition at $A \approx \eps^{-4}$. While the behaviour in the averaged regime ($A \gg \eps^{-4}$) is well understood, the behaviour in the homogenized regime ($A \ll \eps^{-4}$) is poorly understood, and  the critical transition regime is not understood at all.      The second problem concerns an anomalous diffusive behaviour observed in "intermediate" time scales. It is well known that a passive tracer diffusing in the presence of a strong cellular flows "homogenizes" and behaves like an effective Brownian motion on large time scales. On intermediate time scales, however, an anomalous diffusive behaviour was numerically observed recently. I will show a few preliminary rigorous results indicating that the stable "anomalous" behaviour at intermediate time scales is better modelled through Levy flights, and show how this can be used to recover the homogenized Brownian behaviour on long time scales.