On bulk models of shallow cumulus convection

Two-layer bulk models of the cloudy boundary layer are investigated, focusing on the consistency issues between common assumptions on the turbulent mixing and the thermodynamic profiles. Simplifications of the vertical structure of the conserved variables are shown to lead to inconsistent steady states. First, linear profiles in the cloud layer are incompatible with the common parameterizations of the turbulent mixing. Second, this approximation is inconsistent with simple models of an infinitely thin inversion layer. Simple mass-flux parameterizations of the turbulence are shown to produce reasonable profiles inside the cloud layer without solving the problem of the inversion. The cloud - environment difference and the vertical gradient of the conserved variables appear as interesting test variables for such models, as well as their ability to match the free-tropospheric boundary conditions. The latter certainly requires a unified approach for the cloud and inversion layers.