Strongly interacting travelling waves and quasiperiodic dynamics in porous medium convection

Abstract

Buoyancy-driven convection in a square container of porous medium, related to Rayleigh-Bénard convection, is studied in the parameter regime (Rayleigh parameter) where periodic and quasiperiodic dynamics occur. It is shown to have a branch structure determined by simple pairwise interactions of traveling waves. The simultaneous existence of two separate branches of traveling waves (stable or unstable) distinguished by temporal and spatial character (frequency and wavenumber) leads to a connecting branch of quasiperiodic motions. Furthermore, the spatial structure of the quasiperiodic convection is fixed by the wavenumbers of the two traveling wave structures to which it is connected. A simple bifurcation diagram built from several such pairwise interactions accounts for all the behavior, including stable tori, “reverse transitions” and hysteresis, reported in previous simulations and predicts behavior verified by simulations first reported here. The interaction picture for the square container is inferred from the behavior for a special shallow container (width/height near 2.5) where two traveling waves simultaneously come into existence. At this double Hopf bifurcation center manifold and normal form techniques with coefficients supplied by the “exact” equations (more than 100 ODEs) unravel the nonlinear interactions. It turns out that only a slight generalization of this weakly nonlinear picture is sufficient to infer the bifurcation diagram for the square box.