The Betts-Miller scheme. Chapter 9 in "The Representation of Cumulus Convection in Numerical Models of the Atmosphere".
The impetus for the development of this simple lagged convective adjustment scheme came from the series of tropical field experiments in the decade 1969-79 (VIMHEX, the Venezuela International Meteorological and Hydrological Experiment in 1969 and 1972; GATE, the Global Atmospheric Research Program Atlantic Tropical Experiment in 1974 and MONEX, the Monsoon Experiment in 1979). Deep convection is the dominant vertical transport process in the tropics. In conjunction with the radiation field and the subsiding branches of the tropical circulations, convective processes maintain a vertical thermal structure, which is quite close to the moist adiabat through the equivalent potential temperature of the sub-cloud layer in the regions of deep convection. This was the basis of early cumulus parameterization schemes. Manabe (1965) proposed adjustment towards a moist adiabatic structure to remove conditional instability in large-scale models. Kuo (1965, 1974) proposed a simple cloud model for deep convection which adjusted the atmosphere towards the saturated moist pseudo-adiabat in the presence of grid-scale moist convergence. However, the mean tropical atmosphere is always cooler by several degrees in the middle troposphere than this reference moist adiabat, even in regions of vigorous convection (see Figs. 9.5-9.7). At the same time, the deep convective transports also maintain the water vapour and cloud distributions in the tropics, which in turn play a crucial role in the radiative fluxes.
This paper is a review and extension of Betts (1986) and Betts and Miller (1986).
Betts, A.K. and M.J. Miller, 1993. The Betts-Miller scheme. Chapter 9 in "The Representation of Cumulus Convection in Numerical Models of the Atmosphere". (Eds. K.A. Emanuel and D.J. Raymond.) Amer. Meteor. Soc., Meteor. Mon. 24, No 46, 107-121.