Subgrid modeling for the filtered scalar transport equation

We consider passive scaler transport for the case of small diffusivity. In particular, we are concerned with deriving an evolution equation for a smoothed, filtered, or coarse-grained scalar field, i.e., we consider the subgrid modeling problem fur scalar transport. It is shown that for the case of velocity fields that are smooth but that induce chaotic advection a subgrid tensor diffusivity may be derived from first principles. However, this diffusivity can be locally strongly anti-diffusive, so that the filtered transport equation is ill-conditioned. A rationale for regularizing the equation is presented. In addition it is argued that analogous terms should appear in the filtered Navier-Stokes equations.