What controls the decay of passive scalars in smooth flows?

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 gamma(2) is found as the eigenvalue of a linear operator. By analyzing the eigenvalue problem asymptotically in the limit of small diffusivity kappa, we establish that gamma(2) 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 gamma(2) 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 gamma(2) as kappa -> 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 kappa -> 0 of the eigenvalue identified with gamma(2): 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 gamma(2) differs between the two regimes, scaling like 1/log(2) kappa in regime (i), and like kappa(sigma) for some 0 < sigma < 1 in regime (ii). We confirm our theoretical results numerically both for Kraichnan and renewing flows. (c) 2005 American Institute of Physics.