A parametrization for the entrainment rate in shear-free convective cloud-capped boundary layers is derived. The only information it requires is the external mixed-layer turbulence forcings (the surface buoyancy flux and the net radiative-divergence profile), the inversion jumps of temperature and humidity, the cloud-top liquid water mixing ratio and the cloud and mixed-layer depths. Despite this simplicity it is found to compare well against both a wide range of large-eddy simulations and observations of stratocumulus.
The parametrization is an extension of those derived previously for the idealised cases of smoke clouds, where turbulence is driven by combinations of surface heating and radiative cooling, and liquid water clouds driven solely by buoyancy reversal. The radiative forcing is specified as an indirect forcing, through the buoyant production of turbulence within the boundary layer and a direct forcing, which promotes deepening of the boundary layer when undulations in the cloud top cause part of the cooling to occur within the horizontally averaged inversion. Condensation of water in saturated air reduces the strength of both these terms but otherwise, for radiatively driven liquid water clouds (where evaporative cooling of entrained air does not generate buoyancy reversal), the parametrization is unchanged from that for smoke clouds.
Where evaporative cooling of entrained air is strong enough to generate buoyancy reversal, and therefore drive convective motions, not only does it provide an additional turbulence source, but it is also found to compensate for the reduction in strength of the radiative-forcing terms by the presence of saturated air. Allowing for this enhancement, the entrainment rate is predicted with remarkable accuracy by the sum of previously derived parametrizations for the rates that would have been generated by radiative cooling and buoyancy reversal acting in isolation.