Large-scale effects on meso-scale modeling for scalar transport

The transport of scalar quantities passively advected by velocity fields with a component at small scale ℓℓ can be modeled at scales larger than ℓℓ by means of an effective drift and an effective diffusivity, which can be determined by means of multiple-scale techniques. We show that the presence of a weak flow at large scales L≫ℓL≫ℓ induces interesting effects on the scalar transport at the meso-scales (i.e. at scales intermediate between ℓℓ and LL). In particular, it gives rise to non-isotropic and non-homogeneous corrections to the meso-scale drift and diffusivity. We discuss an approximation that allows us to retain the second-order effects caused by the large-scale flow. This provides a rather accurate meso-scale modeling for both asymptotic and pre-asymptotic scalar transport properties. Numerical simulations in model flows are used to illustrate the importance of such large-scale effects.