What controls the decay of passive scalars in smooth flows?

Abstract

The exponential decay of the variance of a passive scalar released in a homogeneous random two-dimensional flow is examined. Two classes of flows are considered: short-correlation-time (Kraichnan) flows, and renewing flows, with complete decorrelation after a finite time. For these two classes, a closed evolution equation can be derived for the concentration covariance, and the variance decay rate γ2 is found as the eigenvalue of a linear operator. By analyzing the eigenvalue problem asymptotically in the limit of small diffusivity κ, we establish that γ2 is either controlled (i) locally, by the stretching characteristics of the flow, or (ii) globally, by the large-scale transport properties of the flow and by the domain geometry. We relate the eigenvalue problem for γ2 to the Cramer function encoding the large-deviation statistics of the stretching rates; hence we show that the Lagrangian stretching theories developed by Antonsen et al. [Phys. Fluids 8, 3094 (1996)] and others provide a correct estimate for γ2 as κ→0 in regime (i). However, they fail in regime (ii), which is always the relevant one if the domain scale is significantly larger than the flow scale. Mathematically, the two types of controls are distinguished by the limiting behavior as κ→0 of the eigenvalue identified with γ2: in the local case (i) it coincides with the lower limit of a continuous spectrum, while in the global case (ii) it is an isolated discrete eigenvalue. The diffusive correction to γ2 differs between the two regimes, scaling like 1∕log2κ in regime (i), and like κσ for some 0<σ<1 in regime (ii). We confirm our theoretical results numerically both for Kraichnan and renewing flows.