Rotating Rayleigh-Bénard Convection

Abstract

Rotating Rayleigh-Benard convection (rRBC) is studied as a paradigmatic example of pattern formation and spatiotemporal chaos. For large enough rotation rates, this system undergoes a supercritical bifurcation from the uniform state to a state known as domain chaos. In domain chaos, domains of straight parallel rolls change their orientation and size discretely. This roll switching causes an overall counterclockwise precession of the pattern. An additional mechanism of precession, glide-induced precession, is introduced here, by deriving the rRBC amplitude equation to higher order. New terms due to the rotation cause rolls to precess whenever there is an amplitude gradient in the direction parallel to the rolls. Hence, dislocations which are stationary in a nonrotating system will glide in a rotating frame, causing the overall precession. Theory that includes the Coriolis force but ignores the centrifugal force predicted scaling laws near the transition to domain chaos. However, experimenters found different scaling laws. The scaling laws are studied here by direct numerical simulations (DNS) for the exact parameters as experiments. When only the Coriolis force is included, the DNS scaling laws agree with theory. When the centrifugal force is also included, the DNS scaling laws agree better with experiment; hence the centrifugal force cannot be neglected from theory. The coefficients of the amplitude equation for the Complex Ginzburg-Landau equation (CGLE) are found for DNS of traveling waves. They agree well with experimental results. The CGLE is chaotic for certain values of the coefficients. However, for the parameters in the DNS, those chaotic regimes were not realized. Leading order Lyapunov exponents (LLE) and eigenvectors are computed for both rotating and nonrotating convection. For certain parameters, these systems are found to have positive LLEs; hence they are truly chaotic. For time-dependent systems, the leading eigenvector is characterized by localized bursts of activity which are associated with dynamical events. The short-time dynamics of the LLE is correlated with these dynamical events. However, contributions to the LLE are due to non-periodic events only. Lagrangian particle tracking methods are employed for rRBC. These systems exhibit chaotic advection in that initially localized particle trajectories explore the available phase space.