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.