A cumulus parameterization with a prognostic closure

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Abstract
The paper describes the introduction of a prognostic cumulus kinetic energy (CKE) as a replacement for the quasi‐equilibrium closure hypothesis of Arakawa and Schubert (AS).In the original version of the AS parameterization, the cloud work function, a measure of the convective available potential energy, is assumed to be maintained at ‘small’ values through a quasi‐equilibrium between the cumulus convection and the ‘large‐scale forcing’. It is argued here, however, that the distinction between the convective and large‐scale processes is ambiguous and subjective. It is demonstrated that the need for such a distinction can be avoided by relaxing the quasi‐equilibrium assumption, through the introduction of a prognostic CKE; referred to as prognostic closure. A dimensional parameter, α, is introduced to relate the CKE to the square of the cloud‐base convective mass flux. It is shown that ‘adjustment time’ defined by AS is related to α, so that when the adjustment time approaches zero the prognostic closure reduces to quasi‐equilibrium closure. A second dimensional parameter, τD, is used to determine the rate at which the CKE is dissipated. In the limit of small α and τD, the convective mass flux is formally independent of both α τD if the environmental sounding is assumed to be given, but in reality the results of a prognostic model do depend on these two parameters because they affect the time‐dependent sounding.For simplicity, a single constant value of α is used for all cloud types in tests with a general‐circulation model, and this gives reasonably good results. Larger values of α lead to more frequent shallow cumulus convection and a cooler and more humid troposphere, in which stratiform condensation is more active and more large‐scale precipitation can reach the surface. A longer dissipation time‐scale leads to a warmer tropical troposphere. The interactions between stratiform cloudiness and convection prove to be quite important, leading to the conclusion that the convection parametrization really cannot be evaluated independently of the stratiform cloud parametrization with which it interacts.