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.