Lattice models of advection-diffusion

We present a synthesis of theoretical results concerning the probability di stribution of the concentration of a passive tracer subject to both diffusi on and to advection by a spatially smooth time-dependent flow. The freely d ecaying case is contrasted with the equilibrium case. A computationally eff icient model of advection-diffusion on a lattice is introduced, and used to test and probe the limits of the theoretical ideas. It is shown that the p robability distribution for the freely decaying case has fat tails, which h ave slower than exponential decay. The additively forced case has a Gaussia n core and exponential tails, in full conformance with prior theoretical ex pectations. An analysis of the magnitude and implications of temporal fluct uations of the conditional diffusion and dissipation is presented, showing the importance of these fluctuations in governing the shape of the tails. S ome results concerning the probability distribution of dissipation, and con cerning the spatial scaling properties of concentration fluctuation, are al so presented. Though the lattice model is applied only to smooth flow in th e present work, it is readily applicable to problems involving rough flow, and to chemically reacting tracers. (C) 2000 American Institute of Physics. [S1054-1500(00)02201-1].