Competition of the Two Unstable Modes in the Rayleigh-Benard Convection

Abstract

A model dynamical system which governs time evolution of the convection roBs in a rectangular box is investigated. With increase of the vertical temperature gradient R, either the real mode or the complex mode instability triggers the transition of the steady convection state depending on whether the side length of the box perpendicular to the roll axis ry is less or greater than a certain critical value. In the neighborhood of this critical value a variety of dynamical behavior originating from competition of these real and complex modes are observed. With further increase of R it is found by numerical time integration of the system that the bifurcation sequences leading to chaos take various forms depending on the value of ryIn addition, temporal evolution of the spatial patterns of convection is presented using the isopleths of the temperature and the velocity fields for several different time-dependent states. In a previous paper referred to as I below/) the author studied the dynamical properties of the Rayleigh-Benard convection for the cases where the Prandtl number is less than unity. The model dynamical system was obtained with the aid of the Galyorkin procedure to describe the dynamics of the convection rolls for a Boussinesq fluid confined in a rectangular box. For a variant of the model system dealt with in Case C of I, it was found that the real mode instability gives rise to the periodic motion with increase of the vertical temperature gradient or the Rayleigh number R. It is usually presumed that the periodic oscillations of the convection cells appear as a result of the Hopf bifurcation (the complex mode instability). Hence, it is of some interest to clarify the mechanism operative at the real mode instability leading to the periodic motion. As will be shown in this paper, this is the manifestation of the saddle-saddle loop in the phase space arising due to the competition of the two unstable modes: The one is real while the other is complex. When R is increased in this model, these two leading modes become unstable almost simultaneously. Therefore, if one more external parameter is suitably controlled, it is possible that the instability of these two modes occurs precisely at the same point in the parameter space. In this paper, we study in some detail this codimension-2 bifurcation phenom­ ena exhibited by our model choosing the aspect ratio of the box perpendicular to the roll axis 0 as the second external parameter. Codimension-2 bifurcation phenomena have recently attracted some interest among both mathematicians and physicists. Normal forms of the dynamical systems which describe complicated dynamical behavior near the codimension-2 bifurcation point were obtained for each type of the two leading unstable modes. 2 ),3). Further­