A Steady-State Model for the Relationship Between Humidity, Instability, and Precipitation in the Tropics

A simple steady-state model for the thermodynamic structure of a convecting atmosphere under the influence of large-scale dynamics is derived based on a bulk-plume representation of convection. Given profiles of the large-scale vertical velocity and convective mass flux, the model predicts the steady-state temperature and environmental relative humidity profiles as a function of the convective entrainment rate and a parameter representing the importance of condensate reevaporation. The bulk-plume model determines the environmental relative humidity through a balance between the moistening effect of convective detrainment and the drying effect of subsidence in the cloud environment. As the convective mass flux is increased, the importance of detrainment moistening increases, leading to a higher environmental relative humidity. Since the precipitation rate also increases with the convective mass flux, this leads to a steady state in which the precipitation is a strongly increasing function of environmental relative humidity. The bulk-plume model also predicts that the atmospheric stability increases with large-scale upward motion, with the tropospheric lapse rate being most unstable under large-scale descent, when the environmental relative humidity is low. The above relationships are reproduced in a set of simulations with a cloud-system resolving model (CRM) run to equilibrium with imposed large-scale vertical velocity profiles. The results provide insight to the steady-state limit of the well-known problem of determining the relationships between large-scale conditions and the convective state of the atmosphere.