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University of Sydney
School of Mathematics and Statistics
Applied Mathematics Seminar
Applied Mathematics 4th Year Honours Students, 2000
University of Sydney
Wednesday, October 25th, 2-4pm, Carslaw 173, moving to 275 (to be confirmed) at 3pm.
Timothy Schaerf: Nonlinear Evolution of Three Contour Rankine Vortices on the f-plane (Big Whirlpools in the Ocean)
Vast and gentle oceanic vortices are often observed travelling around the North Atlantic ocean. These vortices have cores of warm salty Mediterranean water and are approximately 100km in diameter. They are commonly referred to as Mediterranean water eddies, or meddies.
One of the simplest models relevant to the study of oceanic vortices, such as Mediterranean water eddies, is the Three Contour Rankine Vortex. When perturbed in azimuthal modes, three contour Rankine vortices exhibit a wide range of interesting behaviour. The development of satellite vortices, merger of like signed vortices and the exchange of dipole partners may all be observed.
During the course of my project work I ran many simulations of the evolution of three contour Rankine vortices on the f-plane. The fluid flow was modelled using the commonly employed quasigeostrophic approximation and simulated using the Contour-Advective Semi-Lagrangian algorithm.
In this seminar I will present the results of some of my numerical simulations, as well as the results of a simple analysis of the development of individual azimuthal modes.
Lei Zhang: Gravitational collapse and Einstein's field equations.
Gravitational collapse is the very last stage of stellar evolution. At this stage, a star will collapse due to its own gravitational force. The end point of collapse can be a white dwarf, a neutron star or a black hole. It is the gravity of the star's mass that drives stellar evolution and its rate from beginning to end.Spacetime is curved under strong gravitational field. This situation is described by Einstein's General Relativity Theory. K. Schwarzschild was the first to discover the exact solution of Einstein's field equation. Here the Schwarzschild exterior solution and interior solution are derived, and they are continous across the Schwarzschild radius.
The three classical tests, perihelion shift, bending of light and gravitational redshift confirm Einstein's General Relativity theory and are the foundation of modern astrophysics. The physical significance of these classical tests will be discussed.
Feraz Azhar: Ashtekar's variables in canonical gravity
In 1986 Abhay Ashtekar introduced new variables on the phase space of general relativity. This led to a considerable simplification of the constraints of general relativity and has since had important ramifications for the canonical approach to quantum gravity.
In this talk Ashtekar's approach to canonical gravity will be presented. A quick tour through the Hamiltonian formulation of general relativity will be followed by a discussion of an extension of classical general relativity known as the Einstein-Cartan theory. This will hopefully provide us with enough background to appreciate Ashtekar's work. The significance of the new variables in the context of quantum gravity will also be touched upon.