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 equilibr ium by an active role of convective heating against large-scale forcing (ph ysical 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 small er than the rate of change of the large-scale forcing (diagnostic convectiv e 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 convec tive quasi-equilibrium: however, theoretically, there are several alternati ve physical mechanisms that are also able to establish this diagnostic stat e. The paper examines the approach of the tropical atmospheric system to DCQ w ith increasing time-scale in order to distinguish various alternatives to P CQ. The latter predicts that the system approaches DCQ exponentially with a time-scale characteristic of convection. However, the alternatives conside red in the paper predict algebraic asymptotes to DCQ with increasing time-s care. First it is demonstrated that PCQ is not required to achieve DCQ by c onsidering a linear primitive-equation system with arbitrary convective hea ting, in which the roles of convective heating and large-scale forcing are completely reversed; algebraic asymptotes are achieved. An even simpler ana logue is to assume that the rate of generating CAFE is controlled by white- noise forcing. More generally, such an algebraic asymptote is obtained by a ny system with a power-law spectrum both for CAFE and large-scale forcing, although a restriction must be applied to ensure a decreasing asymptote wit h increasing time-scale. The approach to DCQ is examined for both the Maritime Continent Thunderstor m Experiment data and cloud-resolving model simulation data, and both indic ate no tendency for exponential adjustments in the short time limit.