Lattice Models of Advection-Diffusion

R. T. Pierrehumbert


We present a synthesis of theoretical results concerning the probability distribution of the
concentration of a passive tracer subject to both diffusion and to advection by a spatially smooth
time-dependent flow. The freely decaying case is contrasted with the equilibrium case. A
computationally efficient 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 probability distribution for the freely
decaying case has fat tails, which have slower than exponential decay. The additively forced case
has a Gaussian core and exponential tails, in full conformance with prior theoretical expectations.
An analysis of the magnitude and implications of temporal fluctuations of the conditional diffusion
and dissipation is presented, showing the importance of these fluctuations in governing the shape of
the tails. Some results concerning the probability distribution of dissipation, and concerning the
spatial scaling properties of concentration fluctuation, are also presented. Though the lattice model
is applied only to smooth flow in the present work, it is readily applicable to problems involving
rough flow, and to chemically reacting tracers. © 2000 American Institute of Physics.

The evolution of the concentration field of a nonreacting chemical substance - a ''passive tracer'' ­ subject to rear-rangement by advection and by molecular diffusion pre-sents a rich variety of questions of deep theoretical inter-est. Advection-diffusion is also central to a variety of problems of considerable practical importance, such as combustion, and atmospheric chemistry. The probability distribution, or histogram of the tracer concentration field, provides much information about the mixing pro-cess, and has been the subject of much numerical and theoretical attention. We survey progress that has been made in understanding the PDF for the case of advection-diffusion by smooth flow, expose some remaining gaps in the current understanding, and point out a few aspects of the problem that have not hitherto been sufficiently ap-preciated. In addition, a computationally efficient lattice-based model problem suitable for exploratory inquiries into the subject is introduced. The utility of the method is illustrated through applications to spatially smooth ad-vection of a nonreactive tracer, and suggestions are made for extensions to problems where the theoretical under-pinnings are not so well developed.