Toward a Stochastic Relaxation for the Quasi‐Equilibrium Theory of Cumulus Parameterization: Multicloud Instability, Multiple Equilibria, and Chaotic Dynamics

Abstract

The representation of clouds and organized tropical convection remains one of the biggest sources of uncertainties in climate and long‐term weather prediction models. Some of the most common cumulus parameterization schemes, namely, mass‐flux schemes, rely on the quasi‐equilibrium (QE) closure, which assumes that convection consumes the large‐scale instability and restores large‐scale equilibrium instantaneously. However, the QE hypothesis has been challenged both conceptually and in practice. Subsequently, the QE assumption was relaxed, and instead, prognostic equations for the cloud work function (CWF) and the cumulus kinetic energy (CKE) were derived and used. It was shown that even if the CWF kernel serves to damp the CWF, the prognostic system exhibits damped oscillations on a timescale of a few hours, giving parameterized‐cumulus‐clouds enough memory to interact with each other, with the environment, and with stratiform anvils in particular. Herein, we show that when cloud‐cloud interactions are reintroduced into the CWF‐CKE equations, the coupled system becomes unstable. Moreover, we couple the CWF‐CKE prognostic equations to dynamical equations for the cloud area fractions, based on the mean field limit of a stochastic multicloud model. Qualitative analysis and numerical simulations show that the CKE‐CWF‐cloud area fraction equations exhibit interesting dynamics including multiple equilibria, limit cycles, and chaotic behavior both when the large‐scale forcing is held fixed and when it oscillates with various frequencies. This is representative of cumulus convection variability, and its capability to transition between various regimes of organization at multiple scales and regimes of scattered convection, in an intermittent and chaotic fashion.

A Shallow‐Deep Unified Stochastic Mass Flux Cumulus Parameterization in the Single Column Community Climate Model

“…Here, we are mainly concerned with the issue of computing the bulk mass flux profile and the associated bulk entrainment and detrainment rates using a stochastic model; for the cloud base mass flux issue, below, we use a combined convective available potential energy (CAPE) and turbulent kinetic energy closures to treat simultaneously deep and shallow convection, respectively. An original idea on how to use the SMCM framework to relax the quasi-equilibrium closure, for the cloud base mass flux, is proposed in Khouider and Leclerc (2019) but this new closure is not employed here to focus on the effect of using the SMCM to change the entrainment and detrainment rates. Nonetheless, in a nutshell, the relaxed quasi-equilibrium closure using the SMCM framework amounts to proposing and using evolution equations (that can be either deterministic-mean field limit or stochastic) for the cloud area fractions (CAF) of multiple cloud types, to obtain closed prognostic equations for the cloud work function and cloud base mass flux of Pan and Randall (1998).…”

A shallow-deep unified stochastic mass flux cumulus parameterization in the single column Community Climate Model

“…For simplicity, we use a combined CAPE and turbulent kinetic closure to treat simultaneously deep and shallow convection respectively. An original idea on how to use the SMCM framework to break the relaxed quasi-equilibrium closure is proposed in Khouider and Leclerc (2019) but this new closure is not employed here to focus on the effect of the SMCM on the entrainment and detrainment rates. The inclusion of both the stochastic closure based on the SMCM and the SMCPCP, in the same parametrization scheme, will be the subject of future research.…”