Linear Stability of Penetrative Convection

Abstract

The linear stability of penetrative convection is investigated for three different undisturbed temperature profiles, each characterized by a stable layer on the top: case A) a two-layer profile with constant but different lapse rate in each layer, case B) a continuous and smooth mean temperature profile with the depth of the unstable layer fixed, and case C) a temperature profile determined by the conduction equation with the surface temperature undergoing a diurnal variation. The increase of the critical Rayleigh number Rc of the unstable layer with increasing stability number S of the top layer is prominent in case A, less appreciable in case B, and also less noticeable for case C. It appears that the relation between the temperature gradient in the unstable layer and that in the stable layer is more sensitive and meaningful than the relation between Rc and S for a smooth temperature profile, especially for case C in which the depth of the unstable layer is not kept constant. The results obtained from case C show that the depth of the unstable layer is about 55 m and the horizontal wavelength is about 250 m when convection starts. It is also shown that convection tends to produce a superadiabatic layer near the ground, a deep mixed layer above, and an inversion layer at the top of the mixed layer.