Diffusive and nondiffusive limits of transport in nonmixing flows

We study the passive scalar transport in a class of nonmixing Markovian flows with power-law spectra and correlation times. We establish a new diffusion regime under an optimal condition ( convergent Kubo formula) on the spatial/temporal structure of this family of flows. Under such a condition, the Peclet number of the problem may be infinite. We propose a general criterion for the diffusion regime that takes into account of both the effect of molecular diffusion and the spatial/temporal structure of the velocity field. We conjecture the criterion to be applicable in general for temporally ergodic, reversible Markovian flows. We show heuristically that the violation of this criterion may lead to a nondiffusive scaling limit.