Dynamics and Mechanisms of Time-Dependent Natural Convection in Porous Media

Porous media and Hele-Shaw cells are attractive systems for studying nonlinear dynamics in natural convection systems heated from below, because vorticity transport and inertial effects can be eliminated, leaving only thermal effects. In this thesis, we study time-dependent convection in these systems at small aspect ratio in regimes where the flow varies from periodic to weakly chaotic. Emphasis is placed on phase space behavior, physical mechanisms and the relationship between temporal and spatial structures. The primary tools for this computational and theoretical study are continuation with the AUTO subroutine package and a pseudo-spectral initial value problem solver. In the course of the study, approximate inertial manifold techniques are tested and found to yield a moderate reduction in the order of the differential system needed to resolve bifurcation points. It is found that 3D convection in a cube of saturated porous medium undergoes Hopf bifurcations with mechanism and boundary layer structure analogous to what is found in 2D. Symmetry considerations show the equivalence between the 3D flow we consider and an orthogonal pair of 2D flows. In 2D convection in rectangular boxes, Hopf bifurcations interact to yield windows of stable periodic and quasiperiodic flow. Rigorous demonstration of this interaction mechanism is performed in the neighborhood of a double Hopf bifurcation. At higher Rayleigh numbers, periodic flows (born at a Hopf bifurcation) are found which obey the classical asymptotic boundary layer scaling as plumes form within and break free from the boundary layers. This plume formation process drives parametric instabilities that lead to windows of quasiperiodic or subharmonic behavior, and at sufficiently high Rayleigh number, to weak chaos. In experimental observations of convection in Hele-Shaw cells, a "diagonal" oscillation is found, which is not governed by a classical boundary layer mechanism. We show that this oscillation originates in a symmetry-breaking Takens-Bogdanov bifurcation of a four-vortex (two on two) steady flow. The mechanism for oscillation is the instability of an internal layer, leading to buoyancy driven competition between pairs of isolated and merged vortices. The spatial symmetry of the underlying steady state plays an important role in the instability.