Applied Mathematics Seminar
Paul Newton
Department of Aerospace Engineering and Center for Applied Math USC
Integrable Vortex Dynamics on a Sphere
Friday, May. 22
1:00-2:00 pm
South Hall 6623
Abstract:
The talk will describe the motion of point vortices moving on the
surface of a spherical shell.
The analysis is motivated by certain large scale geophysical vortex
structures, such as atmospheric cyclones or oceanographic eddies.
Because of the relatively large ratio of horizontal to vertical length
scales inherent in these structures, the two-dimensional Hamiltonian
formulation 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 vortices on a sphere, detailing the
Hamiltonian structure of the problem. I will describe our recent
solution to the integrable three vortex problem --
the system is more general than the one in the plane and reduces to it
in the limit as the radius of the sphere becomes large,
as long as the three vortices remain
sufficiently close to each other during the course of their motion.
I will describe all possible equilibria and relative equilibria on the
sphere, including some recent stability results,
as well as the `spherical collapse' process in which
the three vortices collide in a self-similar spiral in finite
time. Throughout the talk, I will emphasize the new effects
due to the spherical geometry that are not present in the corresponding
planar problem. The talk will finish with a brief description of other
important geophysical issues,
such as vertical density stratification and rotation, which can be dealt
with within the (non-integrable) Hamiltonian framework.