In 1973, Betts [1973] proposed a mass flux representation to couple a shallow cumulus layer with the mixed layer beneath (i.e., the subcloud layer). This scheme has been widely used. Recently, Albrecht [1993] extended it to include precipitation, and to show that even a small amount of precipitation can dry and warm the trade-wind cloud layer and also lower the inversion height. Wang et al. [1993] expanded Albrecht's model to a regional version that covers the eastern North Pacific to simulate both stratocumulus and shallow cumulus cloud regimes. The results compared reasonably to the satellite-derived results from FIRE, but were shown to be sensitive to the specification of large-scale divergence, drizzle, and shortwave radiation. The steady-state solution of Albrecht's one-dimensional model was analyzed by Bretherton [1993] to estimate the effect of external parameters (e.g., sea surface temperature, surface wind speed, etc.) on the internal structure and fractional coverage of trade cumulus. This study indicated that the model results depend strongly on the ad hoc entrainment parameter of cumulus clouds.
Many GCMs now use a simplified second-order closure scheme, i.e., solving the turbulent-kinetic-energy equation only, to represent the stratocumulus-topped PBL as well as the clear PBL. This type of modeling was modified by Bechtold et al. [1992] to include a subgrid-scale condensation scheme with a Gaussian probability density function. Their model successfully reproduced the behavior of several observed PBL cloud types, including those driven by shear, by buoyancy, and by a combination of shear and buoyancy.
Recognizing the importance of predicting the fractional cloud amount in GCMs, Randall et al. [1992] developed a second-order bulk boundary-layer model, which combines the mass-flux and second-order closure modeling concepts. This proposed model includes three new features: it relaxes the well-mixed assumption used in conventional bulk models, is capable of estimating the cloud amount, and matches the top and bottom boundary conditions with the interior circulations.
To study PBL equilibrium solutions over a tropical ocean, Betts [1992]
developed a one-dimensional model
that couples the convective PBL (based on a mixing line
representation) with a radiation model. He reported a
short time scale (
one day) and
a long time scale (
10 days) responses of the PBL depth
and surface fluxes to the sea surface temperature and surface
wind speed.