A new bulk shallow-cumulus model and implications for penetrative entrainment feedback on updraft buoyancy

A new refinement of Albrecht et al.'s bulk model for shallow-cumulus convection is presented. It is used to illuminate fundamental aspects of oceanic shallow-cumulus boundary layer structure, including updraft buoyancy and vertical velocity scales, cumulus mass flux, temperature and humidity profiles, and trade-inversion height. The new model, like Albrecht et al.'s, includes a subcloud mixed layer and a cumulus layer with linear gradients of conserved thermodynamic variables, topped by a sharp capping inversion. Albrecht et al.'s model was not mathematically well posed, leading to an inconsistency between its heat and moisture balances at the inversion base. The new bulk model resolves this problem by diagnosing, rather than prognosing, the cumulus-layer gradients and introducing a penetrative entrainment closure to determine the growth of the cumulus layer into the overlying free troposphere. It uses more realistic assumptions about lateral cumulus entrainment and detrainment and a simplified sub-cloud-layer entrainment closure. When applied to a Barbados Oceanographic and Meteorological Experiment (BOMEX) trade-cumulus case, the steady state and the transient response of the bulk model to a sudden increase in sea surface temperature both compare favorably to a large-eddy simulation (LES). The new closure shows that penetrative entrainment constantly adjusts the cumulus-layer temperature and moisture profiles to keep the updraft buoyancy and vertical velocity small in inverse proportion to a penetrative entrainment efficiency A, estimated to be 5 from LES. Energy-balance arguments provide a skillful prediction of the steady-state bulk-model trade-inversion height. They also show that penetrative entrainment is driven by destabilization of the cumulus layer by radiative cooling and Lagrangian surface temperature increases.