Entrainment and Detrainment in a Simple Cumulus Cloud Model

A cumulus cloud's size, shape and internal properties can be predicted, provided that the rate of entrainment is determined by a suitable entrainment parameterization theory. A cumulus cloud model based on such a theory is analogous to the mixed-layer models of the planetary boundary layer (PBL) and the upper ocean. The entrainment rate is closely related to turbulent transport near the cloud boundary. The mixing-length theory suggested by Asai and Kasahara (1967) is examined in this light. An alternative theory is suggested, which completely removes the strong scale-dependence of the Asai-Kasahara model. Scale-dependence is reintroduced by including the perturbation pressure term of the equation of vertical motion. For a given sounding, the new model predicts deeper clouds than the Asai-Kasahara model. This results both from the entrainment assumption used, and from the effects of the perturbation pressure. The expected cloud-top entrainment rate is zero for the simple model considered, although finite-difference errors lead to a positive cloud-top entrainment rate in actual simulators. Lateral entrainment nevertheless dominates cloud-top entrainment. The need for a realistic parameterization of cloud-top entrainment is noted. The fractional entrainment rate for updrafts is shown to vary only slightly with height, and to decrease only slowly as the cloud radius increases. The fractional detrainment rate for updrafts increases with height. Downdrafts are found to entrain heavily near the PBL top, and to detrain primarily into the PBL, in agreement with the observations of Betts (1976).