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Arakawa and Schubert proposed convective quasi equilibrium as a guiding principle for the closure of convection parameterization. However, empirical experiences from operational implementation efforts suggest that its strict application does not work well. The purpose of the present paper is to explain mathematically why this closure does not work in practice, and to suggest that problems stem from physically unrealistic assumptions. For this purpose, the closure hypothesis is examined in its original form, and without imposing a condition of a positiveness to the convective mass fluxes. The Jordan sounding with idealized large-scale forcing is used for diagnosis purposes. The question is addressed from several perspectives including the completeness of the entraining-plume spectrum, and a singular vector decomposition of the interaction kernel matrix. The main problems with the quasi-equilibrium closure are traced to (i) the relatively slow response of shallower convective modes to large-scale forcing and (ii) detrainment at convection top producing strong cooling and moistening. A strict application of the convective quasi-equilibrium principle leads to a singular response of shallow convection. An explicit coupling of convection with stratiform clouds would be crucial for preventing this unrealistic behavior, recognizing that the reevaporation of detrained cloudy air is a relatively slow process.

Why Does Arakawa and Schubert’s Convective Quasi-Equilibrium Closure Not Work? Mathematical Analysis and Implications