The presence of cloud capping the atmospheric boundary layer introduces a number of additional processes that contribute to the turbulence kinetic energy of the layer and strongly influence the evolution of the cloud sheet. The most significant of these are cloud-top radiative cooling and evaporative cooling associated with mixing across cloud-top which can, under certain circumstances, lead to buoyancy reversal of the entrained air. The negatively-buoyant parcels generated by these processes will sink and drive turbulent circulations. These circulations both maintain a well-mixed layer and also drive entrainment at cloud-top.
Within the framework of a first-order closure, a method for parametrizing the turbulent fluxes, developed specifically for cloud-free boundary layers, is to specify the eddy-diffusivity profile as a fixed function of height over the diagnosed depth of the boundary layer, with its magnitude proportional to a velocity scale representative of the strength of the surface forcing (as in Holtslag and Boville, 1993, for example). The results from a wide range of high resolution large-eddy simulations of convective boundary layers suggest that this method can be generalised to the cloud-capped case using representative velocity scales for radiative cooling and buoyancy reversal derived by dimensional arguments. Good agreement with the simulations is then obtained with an approximately upside-down version of the surface-heated profile and analogous counter-gradient terms.
The fluxes across cloud-top can be specified using a parametrization for the entrainment rate (as in Beljaars and Betts, 1992, for example, although again they only considered the cloud-free surface-heated case). Across the range of simulations performed in this study, it is found that a constant fraction of the total energy supplied to the boundary layer turbulence is available to drive entrainment. Furthermore, the different sources appear to act largely independently so that their combined forcing can be expressed as a cubic sum of the same representative velocity scales used in the eddy-diffusivity profiles. In addition, radiative cooling is found to promote entrainment both indirectly, through the buoyant production of turbulence, and directly, when undulations in the cloud-top cause part of the cooling to occur within the horizontally-averaged inversion