Strange eigenmodes of advection-diffusion operators

Ray
Pierrehumbert

Dept. of the
Geophysical Sciences,
The
University of Chicago

We consider the long-time behavior of a
nonreacting chemical tracer under the action of advection by a
specified velocity field and del-squared diffusion.

This is a problem of considerable physical importance, and also serves
as a model problem for developing methods which may be suitable for
yet-more challenging problems involving nonlinear reactions or fluid
turbulence. Despite the simplicity of the advection-diffusion problem,
there are many open questions. The questions come down to the limiting
behavior of the eigenfunctions of operators of the form L + eps*M,
where L is an advection operator having invariant sets concentrated on
a a space of measure zero (e.g. streamlines) and M is a second-order
diffusion operator which regularizes the problem. A survey of what is
known about the subject from PDE theory and numerical experimentation
is given, together with a number of physical applications. A key open
question is that of the "zero limit," i.e. whether the decay rate of
the most slowly decaying eigenmode remains finite as eps approaches
zero. This class of problems is closely allied to problems
arising in dynamo theory and in quantum chaos.

Discrete forms of the problem are formulated on a lattice, and are used
to shed light on a few problems in ergodicity. The problem of
ergodicity is recast as one of finding the group generated by a family
of shift operators.