A model for the onset of penetrative convection

Abstract

The stability of an S-shaped, cubic temperature profile, maintained by internal heating, is considered as a model for circumstances in which an unstably stratified layer of fluid is bounded by two stable layers. Critical Rayleigh numbers are computed for the cases of an infinitely deep layer, and for a layer of finite depth with symmetrically placed free or rigid boundaries. It is found that the introduction of boundaries can reduce the stability of the system. A weakly nonlinear analysis shows that the bifurcation is supercritical and that rolls are preferred to squares for all values of the Prandtl number. This result prompts a re-examination of the model of penetrative convection in water above ice, in which the bifurcation is subcritical, in order to understand the difference between the two models.