This paper addresses an equation for moist entropy in the framework of cloud-resolving models. After rewriting the energy equation with moist entropy in the place of temperature, an equation for moist entropy is obtained. The equation expresses the internal and external sources of moist entropy explicitly, providing a basis for the use of moist entropy as a prognostic variable in long-term cloud-resolving modeling. In addition, a precise formula for the surface flux of moist entropy from the underlying surface into the air above is derived.The equation for moist entropy is used to express the Neelin-Held model for the diagnosis of large-scale vertical velocity. After applying the model to a tropical oceanic atmosphere with mean annual soundings, the paper shows the sensitivity of large-scale vertical circulations to the radiative cooling rate and the surface flux of moist entropy, which demonstrates the necessity for a precise equation for moist entropy in the analysis and modeling of large-scale tropical circulations.

A cumulus ensemble model is used to simulate the interaction between tropical atmospheric convection and the large-scale tropical environment in the context of Sobel and Bretherton's (2000) weak temperature gradient approximation. In this approximation, gravity waves are assumed to redistribute buoyancy anomalies over a broad area of the tropics, thus maintaining the local virtual-ternperature profile close to the large-scale mean. This result is implemented in the model by imposing the advective effects of a hypothetical mean vertical velocity which is just sufficient to counteract the local heating induced by convection and radiation. The implied vertical advection in the moisture equation and entrainment of air from the surrounding environment have major effects on the evolution of convection in the model.The precipitation produced by the model mimics the results of a very simple model of tropical precipitation introduced by Raymond (2000), in that the mean rainfall rate predicted by the cumulus ensemble model is, to a good approximation, a function only of the mean column precipitable water. The evolution of the precipitable water, and hence the precipitation rate, is a result of the imbalance between the surface flux of moist entropy into the domain and the radiative loss of entropy out of the top of the domain. This evolution leads to a statistically steady solution in which the resulting precipitation rate is a unique function of the entropy flux imbalance. These results support the hypothesis that tropical precipitation averaged over distance scales of a few hundred kilometres and time scales of a day is a consequence only of local thermodynamic factors.