Colloquium Talk

Anomalous diffusion is the fundamental ansatz of phenomenological theories of passive scalar turbulence. As with the anomalous dissipation of kinetic energy in a turbulent fluid, the anomalous dissipation of passive scalar variance in a turbulent flow, as the Reynolds and Peclet numbers diverge, has been confirmed numerically and experimentally to an extraordinary extent. A satisfactory theoretical explanation of this phenomenon is however not available. 

In this talk, I will discuss a joint work with Scott Armstrong in which we construct a class of incompressible vector fields that have many of the properties observed in a fully turbulent velocity field, and for which the associated scalar advection-diffusion equation generically displays anomalous diffusion. We also propose an analytical framework in which to study anomalous diffusion, via a backward cascade of renormalized eddy viscosities. Our proof is by "fractal" homogenization, that is, we perform a cascade of homogenizations across arbitrarily many length scales.