On the Smooth Transition to Convection

In the classical mathematical description of the convection of a viscous fluid heated from below, there is a sharp transition from conduction to convection as the Rayleigh number exceeds a critical value. The convection state bifurcates from the conduction state at the critical Rayleigh number. This is in contrast to experiments, where the transition is smooth as the Rayleigh number is increased. We employ a theory of singular perturbations of bifurcations [1], [2] to show that small imperfections, such as thermal noise in the applied temperatures, are sufficient to explain the smooth transition. Finally, we show that when the imperfections satisfy a special condition, the transition remains sharp, but it is slightly perturbed. Since the imperfections are arbitrary, smooth transitions are most likely to occur in experiments.