On the parameterization of convective entrainment: Inherent relationships among entrainment parameters in bulk models

Abstract

In this paper, the equilibrium entrainment into a shear-free, linearly stratified atmosphere is discussed under the framework of bulk models, namely, the zero-order jump model (ZOM) and the first-order jump model (FOM). The parameterizations for the dimensionless entrainment rate versus the convective Richardson number in the two models are compared. Based on the assumption that the parameterized entrainment rates in ZOM and FOM should be the same, the inherent relationships among the entrainment parameters in the bulk models are revealed. These relationships are supported by tank experiments and large-eddy simulations. The validity of these inherent relationships indicates that, for a convective boundary layer growing into a linearly stratified free atmosphere, the only dominant factors of the growth rate are the turbulent buoyancy in the mixed layer and the stratification in the free atmosphere. In the point of the similarity view, the former is characterized by turbulent temperature and mixing length scales (mixed layer depth), and the latter is characterized by the lapse rate of potential temperature in the free atmosphere. Thus, the commonly-used Richardson number scheme for the parameterization of the entrainment rate is just as an equivalent description. The variability of the total entrainment flux ratio in FOM, which is connected with the entrainment zone thickness, can implicitly describe the effect of the stratification in the free atmosphere, but the entrainment zone thickness is not an independent parameter. These results demonstrate the validity of the hypothesis that there exists a similarity limit in which the mixed layer depth is the only lengthscale.