Stabilizing the atmosphere and ventilating the sub-cloud layer through vertical transport of moisture, moist convection is an important sub-grid scale process that still needs to be parameterized in general circulation models (GCMs) in the coming decades (Plant & Yano, 2016). The purpose of convection parameterization is to provide feedback on sub-grid scale convection to the large-scale fields as tendency terms. It is important to keep in mind that for achieving this goal, we do not have to introduce full details of convection into a parameterization. Rather, it must constitute a caricature of the reality of convection, as emphasized by Yano (2014b). Therefore, early convection schemes only considered the feedback without describing the convective clouds (e.g., the moist adjustment scheme of Manabe et al. (1965) or the moisture convergence scheme of Kuo (1965)). However, some shortcomings, including fixed tropical temperature profiles (Arakawa, 2004) and unreasonable convective moistening profiles (Emanuel, 1994), are related to such an oversimplified approach, indicating that an idealized caricature cannot encompass the collective effect of a series of complex processes during the convection. A practical way to mitigate this problem is to describe more basic features of convective clouds in detail in convection schemes. As a result, convection schemes started to adopt a bulk or spectrum mass-flux (plume) formulation (e.g.,