Khor, Calvin (2019) Sharp fronts and almost-sharp fronts of a singular surface quasi-geostrophic equation. PhD thesis, University of Warwick.

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Abstract

In this thesis, we generalise results on sharp fronts and almost-sharp fronts by Fe↵erman, Luli, and Rodrigo [67], [68], [28], [26], [27], [19] to a singular variant of the Surface Quasi-Geostrophic Equation (SQG), where the velocity u = ∇⊥|∇|-1θ is replaced with the more singular velocity ∇⊥|∇|-1+αθ, for α ∈ (0, 1).

First, we derive the contour dynamics equation for a sharp front from the definition of a weak solution to our singular variant of SQG.

Then, we prove the existence of analytic sharp fronts to the sharp front equation using the abstract Cauchy–Kowalevskaya Theorem. This result is analogous to the result of Fefferman and Rodrigo in [27], which was a key result for proving the existence of analytic almost-sharp fronts whose existence time does not depend on the thickness of the transition region are transported by the velocity u = ∇⊥|∇|-1+αθ. This work generalises the result of [19] to our more singular equation.

Finally, we define a spine curve for the almost-sharp front analogously to the spine curve of SQG in the model where one space variable is periodised, defined in the work of Fefferman and Rodrigo. The spine evolves according to the sharp front equation modulo an O(δ2-a) error. As this does not vanish as α → 1, this formally suggests that the equation is in some sense not degenerate in this limit.

Item Type:	Thesis or Dissertation (PhD)
Subjects:	G Geography. Anthropology. Recreation > GC Oceanography Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Geostrophic currents -- Mathematical models, Fronts (Meteorology) -- Mathematical models
Official Date:	June 2019
Dates:	Date Event June 2019 UNSPECIFIED
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Rodrigo, José L.
Sponsors:	European Research Council
Format of File:	pdf
Extent:	viii, 112 leaves : illustrations
Language:	eng

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