Simple examples with features of renormalization for turbulent transport

Abstract

Two simple exactly solvable models for turbulent transport are introduced and discussed here with complete mathematical rigour. These models illustrate several different facets of super-diffusion and renormalization for turbulent transport. The first model involves time dependent velocity fields with suitable long-range correlations and the complete renormalization theory is developed here in detail. In addition rigorous examples are developed by using variants of this model where the effective equation for the ensemble average at large scales and long times is diffusive despite the fact that each realization exhibits catastrophic large-scale instability. The second model introduced previously by the authors involves transport-diffusion in simple shear layers with turbulent velocity statistics. The theories of renormalized eddy diffusivity and higher-order statistics are surveyed here. An extreme limiting case of the theory involving turbulent velocity statistics with long-range spatial correlations but gaussian white noise in time is discussed in detail. Both the renormalized theory of eddy diffusivity and exact explicit equations for second-order correlations related to the pair distance function are developed in complete detail here in this instructive limiting case.