Asymptotic approaches to convective quasi-equilibrium

The physical principle of convective quasi-equilibrium proposed by Arakawa and Schubert states that the atmosphere is effectively adjusted to equilibrium by an active role of convective heating against large-scale forcing (physical convective quasi-equilibrium, or PCQ). A simple consequence of this principle is that the rate of change of the thermodynamic field (typically measured by the convective available potential energy (CAPE)) is much smaller than the rate of change of the large-scale forcing (diagnostic convective quasi-equilibrium, or DCQ). Such a diagnostic state is generally observed in the tropical atmosphere at the synoptic-scale, and this is often taken as a proof for the physical mechanisms behind Arakawa and Schubert's convective quasi-equilibrium: however, theoretically, there are several alternative physical mechanisms that are also able to establish this diagnostic state. The paper examines the approach of the tropical atmospheric system to DCQ with increasing time-scale in order to distinguish various alternatives to PCQ. The latter predicts that the system approaches DCQ exponentially with a time-scale characteristic of convection. However, the alternatives considered in the paper predict algebraic asymptotes to DCQ with increasing time-scale. First it is demonstrated that PCQ is not required to achieve DCQ by considering a linear primitive-equation system with arbitrary convective heating, in which the roles of convective heating and large-scale forcing are completely reversed; algebraic asymptotes are achieved. An even simpler analogue is to assume that the rate of generating CAPE is controlled by white-noise forcing. More generally, such an algebraic asymptote is obtained by any system with a power-law spectrum both for CAPE and large-scale forcing, although a restriction must be applied to ensure a decreasing asymptote with increasing time-scale. The approach to DCQ is examined for both the Maritime Continent Thunderstorm Experiment data and cloud-resolving model simulation data, and both indicate no tendency for exponential adjustments in the short time limit.