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Melanie Chanona Research Statement

Description of Research Projects - Melanie Chanona

Boson Stars and Black Holes in 2+1 Dimensions
 
In the summer of 2012, I investigated a special class of black hole and boson star solutions that only exist in a spacetime that has two spatial dimensions and one time dimension. Boson stars are theoretical objects that, in contrast to black holes, have no event horizon. Solutions in this type of spacetime are of considerable interest as it has developed into a fascinating test bed for understanding quantum gravity and has the potential to increase our understanding of the famous “AdS/CFT”
correspondence conjecture. My work focused particularly on finding solutions with helical symmetry that were hitherto unknown. I was successful in fully constructing numerical boson stars solutions and additionally was able to find strong evidence to support the existence of black hole solutions, which had been conjectured in earlier work by my supervisor Robert Mann and PhD student Sean Stotyn.
 
Angular Momentum Conservation in the Earth-Moon System: A Quasilocal Approach
 
In the summer of 2013, I switched my focus within the field of general relativity from numerical to theoretical work and helped develop a novel theory that expresses the fundamental properties of the universe such as
conservation of energy and momentum in a more elegant way than has been done before. The key concept in this approach is to enclose the system of interest in a two dimensional quasilocal rigid surface (called an RQF) in order to establish a mechanism for measuring the changing energy fluxes as the system evolves. My main goal was to demonstrate the applicability and efficacy of this approach by accurately predicting the rate of recession of the Moon, which is spiraling away from the Earth
due to tidal effects, thus causing a transfer of angular momentum. The RQF approach precisely reproduced the observed result of 3.8cm/year
and, most importantly, I was able to show this without requiring the restrictive conditions and assumptions of traditional methods. Thus my work was successful in motivating the use of RQFs and supported the
claim that it is a superior method for understanding and describing gravitational effects in a broad variety of astrophysical systems.
 
Linear Stability of Oceanic Fronts in a Shallow Water Model
 
In the summer of 2014, I explored the field of geophysical fluid dynamics. My research involved the growth of instabilities in currents under a variety of physical constraints. I began by looking at a three-layer shallow water model that simulated the Antarctic Shelf Front, which has striking
topographical features that affect the evolution of the current that travels across it. The complexity of this current motivated a closer examination of the dynamics of each layer independently and I was able to determine that
the most interesting features occurred when the surface layer was similar to a hyperbolic tangent profile. Using spectral collocation methods
to numerically solve the equations that govern the dynamics of the current, I was able to solve for both large scale and small scale
instabilities and see how the growth rates depended on the depth of the layer. Under specific conditions, we can allow the surface profile to ‘outcrop’ on one side, in other words, we can consider a vanishing layer depth. While previous work has speculated that a two-layer model was needed to find certain unstable modes, I was able to demonstrate that they in fact can develop in a one-layer model only. Finally, I examined a
parabolic profile and was able to categorize the instabilities into pairs that have peculiar asymmetries on either side of the current jet. This also
had not been addressed in earlier explorations of this type of setup.