Multiple-scale analysis and renormalization for preasymptotic scalar transport

Preasymptotic transport of a scalar quantity passively advected by a velocity field formed by a large-scale component superimposed on a small-scale fluctuation is investigated both analytically and by means of numerical simulations. Exploiting the multiple-scale expansion one arrives at a Fokker-Planck equation which describes the preasymptotic scalar dynamics. This equation is associated with a Langevin equation involving a multiplicative noise and an effective (compressible) drift. For the general case, no explicit expression for either the affective drift on the effective diffusivity (actually a tensorial field) can be obtained. We discuss an approximation under which an explicit expression for the diffusivity (and thus for the drift) can be obtained. Its expression permits us to highlight the important fact that the diffusivity explicitly depends on the large-scale advecting velocity. Finally, the robustness of the aforementioned approximation is checked numerically by means of direct numerical simulations.