Evolution of the Convection Rolls in Low Prandtl Number Fluids

Time evolution of the RayleighBenard convection in a small-aspect-ratio rectangular box is investigated for the cases of the Prandtl number 0=0.2 and 0.5. The basic equations of motion for convection are assumed to obey the Boussinesq approximations. The Galyorkin method is employed to obtain from them a truncated model system of ordinary differential equations governing time evolution of the convection rolls. With the aid of numerical time integration and linear stability analysis thereof, it is found that there are two types of instabilities of the steady convection rolls which lead to the periodic oscillatory motion: (i) The normal Hopf bifurcation and (ij) a kind of global bifurcation initiated with a real mode instability. For the latter case, the frequency tends to vanish when the Rayleigh number is decreased down to the steady convection state. Numerical time integration of the system shows that for both cases the periodic motion turns chaotic with further increase of the Rayleigh number.