Studies in Rotating Convection

This thesis presents an exploration and analysis of novel phenomena in RayleighBénard convection in a rotating cylinder using direct numerical simulation and equivariant bifurcation analysis. The numerical method used is a second order predictorcorrector method using Chebyshev collocation in the radial and axial directions and a Fourier-Galerkin discretization in the azimuthal direction. This numerical method is first applied to the problem of convection with a rotation rate sinuosoidally modulated about a non-zero mean in a parameter regime for which the onset of convection is to domain chaos. As laboratory experiments found the resulting flow to be axisymmetric with radially inward traveling waves an analysis of the axisymmetric subspace of the problem was conducted which showed that the emergence of the traveling waves resulted from a symmetry-restoring saddle-node-on-an-invariant-circle (SNIC) bifurcation. The exploration of the effect of modulated rotation on rotating convection was extended to the case of rapid rotation for which the onset of convection is wall-localized. It was found that the onset of convection could be delayed to thermal driving up to 20% beyond that required in the unmodulated case using modulation amplitudes of only 1% of the background rotation rate. Additionally, the resulting oscillatory boundary layers introduced a mean flow which can reverse the precession of the thermal plumes over a large range of imposed frequencies and amplitudes. Finally, an in-depth study of the origin of spatio-temporal chaos directly at onset is presented. In this study the case neglecting the centrifugal force (as is commonly done in theoretical studies) is compared to the more realistic case incorporating centrifugal force (which is necessarily evident in even the most carefully conducted laboratory experiment). The two cases lead to two different routes to complexity; a detailed generalized linear stability analysis is performed for each case. Novel patterns are observed including ratcheting states with and without roll switching and comparisons to existing laboratory experiments are drawn. iii ACKNOWLEDGEMENTS I would like to acknowledge the efforts of the many professors who contributed to my graduate education and especially Juan M. Lopez for helping me develop scientifically and for giving me professional advice. I would also like to thank my close collaborator and mentor Francisco Marques from the Departament de F́ısica Aplicada at the Universitat Politécnica de Catalunya for hosting me during my time in Catalunya and his enthusiasm for fluids research has been a welcome inspiration to me. I would like to thank our sister group in Barcelona for their warm hospitality and enlightening discussions, especially Isabel Mercader and Oriol Batiste whose numerical code made possible much of my research. Furthermore, I would like to thank the More Graduate Education at Mountain States Alliance (MGE@MSA) program and the Sloan Foundation for material support including funds for computer equipment and travel grants that greatly enhanced my studies. I would like to thank the Fulbright Program for the opportunity to study in Catalunya and for the realization that I am a citizen of the world as well as of the United States. Finally, I would like to acknowledge the support in terms of computing time and specialist help of the Ira A. Fulton High Performance Computing Initiative and the NSF Teragrid which made possible the laborious computations detailed in this thesis.