Vortex Crystals in Fluids
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Abstract
It is common in geophysical flows to observe localized regions of enhanced vorticity. This observation can be used to derive model equations to describe the motion and interaction of these localized regions, or vortices, and which are simpler than the original PDEs.
The best known vortex model is derived from the incompressible Euler equations, and
treats vortices as points in the plane. A large part of this dissertation utilizes this particular model, but we also survey other point vortex and weakly viscous models. The main focus of this thesis is an object known as the vortex crystal. These remarkable configurations of vortices maintain their basic shapes for long times, while perhaps rotating or translating rigidly in space. We study existence and stability of families of vortex crystals in the special case where N vortices have small and equal circulation and one vortex has large circulation. As the small circulation tends to zero, the weak vortices tend to a circle centered on the strong vortex. A special potential function of this limiting problem can be used to characterize orbits and stability. Whenever a critical point of this function is nondegenerate, we prove that the orbit can be continued via the Implicit Function Theorem, and its linear stability is determined by the eigenvalues of the Hessian matrix of the potential. For general N, we find at least three distinct families of critical points, one of which continues to a linearly stable class of vortex crystals.
Because the stable family is most likely to be observed in nature, we study it extensively. Continuation methods allow us to follow these critical points to nonzero weak vortex strength and investigate stability and bifurcations. In the large N limit of this family, we prove that there is a unique one parameter family of distributions which minimize a "generalized" potential.
Finally, we use point vortex and weakly viscous vortex models to analyze vortex crystal configurations observed in hurricane eyes and related numerical simulations. We find striking numerical and analytical agreement, thus validating the use of simplified vortex models to describe geophysical phenomena.
Description
Thesis (Ph.D.)--Boston University
PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you.
PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you.