Rotating Thermal Convection: Neosteady and Neoperiodic Solutions

Abstract

This paper considers the secondary and cascading bifurcation of two-dimensional steady and period thermal convection states in a rotating box. Previously developed asymptotic and perturbation methods that rely on the coalescence of two, steady convection, primary bifurcation points of the conduction state as the Taylor number approaches a critical value are employed. A multitime analysis is employed to construct asymptotic expansions of the solutions of the initial-boundary value problem for the Boussinesq theory. The small parameter in the expansion is proportional to the deviation of the Taylor number from its critical value. To leading order, the asymptotic expansion of the solution involves the mode amplitudes of the two interacting steady convection states. The asymptotic analysis yields a first-order system of two coupled ordinary differential equations for the slow-time evolution of these amplitudes. We investigate the steady states of these amplitude equations and their linearized stability. These equations suggest that the following sequenceof transitions may occur as the Rayleigh number R is increased, for a fixed value of the Taylor number near the critical value: First, the conduction state loses stability at a primary bifurcation point to steady convection states (rolls) characterized by a single wavenumber. Then, these states lose stability at a secondary bifurcation point to other steady convection states characterized by two different wavenumbers. Finally, these states become unstable at a tertiary Hopf bifurcation point, $R = R_H $, to time-periodic convection states, which are also characterized by the same two wavenumbers. At third order, the Hopf bifurcation is degenerate since for $R = R_H $ there is a one-parameter family of periodic solutions, which is bounded in the phase plane by a heteroclinic orbit. For $R \ne R_H $, but close to $R_H $, the center of the system’s phase plane trajectories is transformed into a focus which is unstable (stable) for $R > R_H ( < R_H )$. We find that, depending on the length of the observation time and on the initial conditions, the system may appear to be either in a single steady state, or in a periodic state, or in a transient state. If the observation time is sufficiently long, the trajectories ultimately spiral into the focus for $R < R_H $ and are thus captured by it; or they spiral out of the unstable focus for $R > R_H $, and thus escape from it. The new results presented here are: (a) an example of a semibounded convection system with a Hopf bifurcation branch generated by the interaction of two steady state branches; and (b) a detailed study of the transient motion near the “vertical” Hopf bifurcation branch. The “bending” of the branch is described in detail elsewhere.