ON PENETRATIVE CONVECTION AT LOW PECLET NUMBER

A new theoretical model is developed for the growth of a convecting fl uid layer at the base of a stable, thermally stratified layer when hea ted from below. The imposed convective heat flux is taken to be compar able to the heat flux conducted down the background gradient so that d iffusion ahead of the interface between the convecting and stable laye rs makes a significant contribution to the interfacial heat flux and t o the rate of rise of the interface. Closure of the diffusion problem in the stable region requires the interfacial heat flux to be specifie d, and it is argued that this is determined by the ability of convecti ve eddies to mix warmed fluid below the interface downwards. The inter facial velocity, which may be positive or negative, is then determined by the joint requirements of continuity of heat flux and temperature. A similarity solution is derived for the case of an initially linear temperature gradient and uniform heating. Solutions are also given for a heat flux that undergoes a step change and for a heat flux determin ed from a four-thirds power law with a fixed base temperature. Numeric al calculations show that the predictions of the model are in good agr eement with previously reported experimental measurements. Similar cal culations are applicable to a wide range of geophysical problems in wh ich the tendency for diffusive restratification is comparable to that for mixed-layer deepening by entrainment.