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.