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Title: Localised solutions in the magnetorotational Taylor-Couette flow with a quartic marginal stability curve
Author: Bentley, David Christopher
Awarding Body: University of Leeds
Current Institution: University of Leeds
Date of Award: 2012
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This thesis is motivated by the observation in the magnetorotational Taylor-Couette flow that, for certain configurations of an externally applied magnetic field, there is competition between two different wavelengths at the thresh• old of instability. Moreover, for a particular magnetic field configuration the two critical wavelengths coalesce, such that the marginal stability curve has a quartic minimum. By perturbing about this quartic minimum, we recover competition between two similar wavelengths. This competition suggests the possibility that the secondary flow may exhibit localised patches of Taylor vortices of one wavelength embedded in a background of Taylor vortices with the other wavelength. In this thesis we develop a model equation that displays qualitatively the aforementioned behaviour, based on the Swift- Hohenberg equation [75]. A weakly nonlinear analysis is performed, in the manner of the Ginzburg-Landau derivation from the Swift- Hohenberg equation [41]. The resultant amplitude equation is, under certain restrictions on the parameters, the complex SwiftHohenberg equation [32J. We next extend the recently developed [24] techniques for finding localised solutions of the Swift- Hohenberg equation to the model equation; in particular the use of a conserved quantity to identify the constituent wavelengths of the localised solutions, and numerical continuation to compute the bifurcation diagrams for the model equation. We also compute the normal form for the bifurcation at the quartic minimum, following the similar analysis of the Hamiltonian-Hopf bifurcation relevant to the Swift- Hohenberg equation. The extension does not carry forward the integrals we might expect, however. Finally, we present preliminary numerical simulations of the nonlinear Taylor-Couette system in the appropriate parameter regime. The derivation and analysis of the model equation in this thesis represents a significant advance in the development of a framework for understanding localised solutions consisting of regions of two similar wavelengths.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available