Linear stability and single‐column analyses of several cumulus parametrization categories in a shallow‐water model

Using a mass‐flux based approach, the thermodynamic cumulus parametrization problem is reformulated in a simple atmospheric model, which is an analogue of the shallow‐water equations. The objective is to investigate basic effects of elementary representations of several parametrization categories. In particular, a linear stability analysis and a single‐column experiment are performed to infer the characteristics of each parametrization as regards its ability to simulate the large‐scale organization or coherence of tropical convection. The moisture‐convergence closure (MC) scheme, which assumes that the ensemble of cumulus convection is controlled by the low‐level moisture convergence as in Kuo‐type scheme, predicts the largest growth at the smallest scale. Hence, although it ensures the generation of a coherent propagating structure, its scale always corresponds to the grid size. Furthermore, the MC tends to produce a catastrophic positive feedback of moist convection to the large‐scale convergence. In contrast, the statistical equilibrium scheme, which assumes an instantaneous adjustment of the large‐scale environment to a quasi‐equilibrium state, such as Arakawa‐Schubert and moist convective adjustment schemes, asymptotes to a constant growth rate at small scales. Hence, this type of parametrization tends to generate a field like white noise with no large‐scale coherence. The lagged‐adjustment (LA) schemes, which have a short time‐lag for the cumulus growth, as in the Betts–Miller scheme, feature a finite scale selection in the linear growth rate. This ensures a smooth large‐scale coherence that is independent of the grid size, and is consistent with the scale‐separation principle. A new type of parametrization is also tested. This convective life‐cycle (CLC) scheme represnets the life cycle of a common type of convective system made up of deep precipitating convection and a subsequent mesoscale response. It uses a buoyancy‐based closure. The growth‐rate curve is similar to the other LA schemes, but the behaviour in the zero‐dimensional (single‐column) version of the model is qualitatively different. Although the CLC scheme does not automatically saisfy the scale‐separation principle, its grid‐size dependence can be treated by a re‐normalization principle. The result are used to interpret some reported general‐circulation‐model results regarding the impact of different parametrization schemes on the tropical atmosphere at large scales.