DMS-9500804 Lebovitz The project consists of two principal parts. One is a series of explorations in fluid dynamics emphasizing parametric instabilities in flows with noncircular streamlines. These flows include many of interest in astrophysical and geophysical fluid dynamics. The other principal part is a mathematical study of slow evolution in finite-dimensional systems with two timescales (a fast and a slow timescale) that experience bifurcation in the limit in which the evolution on the slow timescale ceases. They are related in that the fluid-dynamical problems can, in some cases, be reduced to finite-dimensional problems of the kind described. The goal of the proposed research is to understand the behavior of fluids in natural settings, like stars, planets and earth's oceans and atmospheres. Certain simple, theoretically deduced flows, once thought to represent real flows in these natural settings, turn out to be impossible because they are unstable, i.e., they do not persist under the small disturbances to which all natural phenomena are subject. The goal is then to understand what flows are possible. For example, can one find stable flows by considering how an unstable flow changes with time? Many of the mathematical problems arising in this study arise already in the study of simpler dynamical systems, and part of the proposed research is to consider these simplified problems.
