On penetrative convection at low Peclet number

Abstract

A new theoretical model is developed for the growth of a convecting fluid layer at the base of a stable, thermally stratified layer when heated from below. The imposed convective heat flux is taken to be comparable to the heat flux conducted down the background gradient so that diffusion ahead of the interface between the convecting and stable layers makes a significant contribution to the interfacial heat flux and to the rate of rise of the interface. Closure of the diffusion problem in the stable region requires the interfacial heat flux to be specified, and it is argued that this is determined by the ability of convective eddies to mix warmed fluid below the interface downwards. The interfacial 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 determined from a four-thirds power law with a fixed base temperature. Numerical calculations show that the predictions of the model are in good agreement with previously reported experimental measurements. Similar calculations are applicable to a wide range of geophysical problems in which the tendency for diffusive restratification is comparable to that for mixed-layer deepening by entrainment.