Stratified disordered media: exact solutions for transport parameters and their self-averaging properties

We investigate the transport of a passive tracer in a two-dimensional stratified random medium with flow parallel and perpendicular to the strata. Assuming a Gaussian random flow with a Gaussian correlation function, it is not only possible to derive exact expressions for the temporal behaviour of the dispersion coefficients characterizing the tracer transports but also to investigate the self-averaging properties of these quantities. We give explicit results for the dispersion coefficient and its mean square fluctuations as a function of time. As it turns out, the sample-to-sample fluctuations which are encoded in the latter quantity remain finite even in the asymptotic limit of infinite dimes, which implies that, the given two-dimensional model is not self-averaging.