Formulation structure of the mass-flux convection parameterization

Structure of the mass-flux convection parameterization formulation is re-examined. Many of the equations associated with this formulation are derived in systematic manner with various intermediate steps explicitly presented. The nonhydrostatic anelastic model (NAM) is taken as a starting point of all the derivations. Segmentally constant approximation (SCA) is a basic geometrical constraint imposed on a full system (e.g., NAM) as a first step for deriving the mass-flux formulation. The standard mass-flux convection parameterization, as originally formulated by Ooyama, Fraedrich, Arakawa and Schubert, is re-derived under the two additional hypotheses concerning entrainment–detrainment and environment, and an asymptotic limit of vanishing areas occupied by convection. A model derived at each step of the deduction constitutes a stand-alone subgrid-scale representation by itself, leading to a hierarchy of subgrid-scale schemes. A backward tracing of this deduction process provides paths for generalizing mass-flux convection parameterization. Issues of the high-resolution limit for parameterization are also understood as those of relaxing various traditional constraints. The generalization presented herein can include various other subgrid-scale processes under a mass-flux framework.