Phase Diagram for Turbulent Transport: Sampling Drift, Eddy Diffusivity and Variational Principles

We study the long-time, large scale transport in a three-parameter family of isotropic, incompressible velocity fields with power-law spectra. Scaling law for transport is characterized by the scaling exponent $q$ and the Hurst exponent $H$, as functions of the parameters. The parameter space is divided into regimes of scaling laws of different {\em functional forms} of the scaling exponent and the Hurst exponent. We present the full three-dimensional phase diagram. The limiting process is one of three kinds: Brownian motion ($H=1/2$), persistent fractional Brownian motions ($1/2<H<1$) and regular (or smooth) motion (H=1). We discover that a critical wave number divides the infrared cutoffs into three categories, critical, subcritical and supercritical; they give rise to different scaling laws and phase diagrams. We introduce the notions of sampling drift and eddy diffusivity, and formulate variational principles to estimate the eddy diffusivity. We show that fractional Brownian motions result from a dominant sampling drift.