Multi-length-scale theories for scale-up problem and renormalized perturbation expansion

Randomness is an essential feature for groundwater ecology and fluid turbulence. The scale-up problem refers to the situation in which the randomness occurs on all scales. Classical dispersion theories are not applicable in this situation due to the coupling between adjacent length scales. The difficulty in theoretical formulations lies in the closure problem - namely, the higher order random quantities emerge in the derivations for the dynamical equations of the lower order random quantities. In this paper, we consider tracer particles driven by a Gaussian random velocity field. We develop a renormalized perturbation expansion method for this problem. Our theory shows that the effective dispersivity has very complicated structures. We introduce a pairwise time ordering approximation to derive an approximate expression for a local diffusion operator. Under this approximation, we develop a multi-length-scale theory which gives analytical predictions for the effective diffusion coefficient and for the size of the mixing region along with its scaling exponent. Our theory has no adjustable parameter. The quantitative theoretical predictions are in excellent agreement with the results from full numerical simulations. © 1997 Elsevier Science Ltd. All rights reserved.