Effect of a traveling thermal wave on weakly nonlinear convection in a porous layer heated from below

The stability of weakly nonlinear convection in a porous layer heated from below is considered. Much is known about those cases where the boundary temperatures are uniform, or display steady small-amplitude variations about a uniform mean. In this study we consider the effects of small-amplitude traveling thermal waves on the ensuing weakly nonlinear convection. At sufficiently low Rayleigh numbers the induced flow follows the motion of the thermal wave. But at higher Rayleigh numbers this form of convection breaks down and there follows a regime where the flow travels more slowly on average and does not retain the forcing periodicity. It appears that the flow passes through all multiples of the forcing periodicity as the Rayleigh number increases. At much higher Rayleigh numbers within the weakly nonlinear regime two very distinct time scales appear in the numerical simulations, and these are described analytically using the method of multiple scales; comparisons with numerical simulations are excellent. A similar scenario exists for large wavespeeds but moderate values of the Rayleigh number.