PAUL NEWTON
Departments of Aerospace Engineering and Mathematics
University of Southern California, Los Angeles
Los Angeles, CA 90089-1191
http://ae-www.usc.edu/bio/newton.html
newton@spock.usc.edu
Abstract:
The talk will describe the evolution of vorticity on a sphere and the corresponding velocity fields and streamline patterns that are produced. Our analysis is motivated by the paradox that despite the fact that large scale geophysical vortex structures such as atmospheric cyclones or oceanographic eddies usually generate turbulent flowfields, many typical averaged streamline patterns produced by these structures are relatively simple. Because of the large ratio of horizontal to vertical length scales inherent in these structures, the {\it two-dimensional} Hamiltonian formulation of the Euler equations can be used. Because the structures typically survive for long times and are capable of transporting passive scalars such as heat, environmental pollutants, or biota over large distances, the spherical geometry of the Earth's surface becomes important.
The first part of the talk will introduce the basic equations of motion for point vortices on a sphere based on the two dimensional Euler equations in spherical geometry, emphasizing the Hamiltonian structure of the problem and the special role played by triad interactions. I will briefly describe relevant aspects of our recent solution to the integrable three vortex problem, including equilibrium configurations and their nonlinear stability, and non-equilibrium configurations (periodic and quasi-periodic solutions and finite time collapsing states).
The main part of the talk will focus on our recent categorization of all generic integrable streamline patterns produced by the vortices. In particular, we have shown that there are exactly 12 topologically distinct instantaneous streamline patterns produced by three vortices, which then can be continuously deformed to form an additional 23 distinct but homotopically equivalent structures. Using these 35 patterns as templates, we show how spherical streamline plots from atmospheric weather data can be decomposed and their patterns identified. We finish the talk with speculation on why such relatively simple patterns might be generic in real atmospheric flows, based on a statistical mechanical argument.
Future challenges include understanding the {\it evolution} of streamline patterns on the sphere and the role that the instantaneous structures play in the finite time mixing and transport of passive particles, as well as the addition of other effects such as the added complication of coastlines and topography for oceanographic flows.