An Analytical Model for Tropical Relative Humidity

Abstract

Abstract An analytical model is derived for tropical relative humidity using only the Clausius–Clapeyron relation, hydrostatic balance, and a bulk-plume water budget. This theory is constructed for radiative–convective equilibrium and compared against a cloud-resolving model. With some reinterpretation of variables, it can be applied more generally to the entire tropics. Given four variables—pressure, temperature, and the fractional entrainment and detrainment rates—the equations predict the relative humidity (RH) and the temperature lapse rate analytically. The RH is a simple ratio involving the fractional detrainment rate and the water-vapor lapse rate. When integrated upward in height, the equations give profiles of RH and temperature for a convecting atmosphere. The theory explains the magnitude of RH and the “C” shape of the tropospheric RH profile. It also predicts that RH is an invariant function of temperature as the atmosphere warms, and this behavior matches what has been seen in global climate models and what is demonstrated here with cloud-resolving simulations. Extending the theory to include the evaporation of hydrometeors, a lower bound is derived for the precipitation efficiency (PE) at each height: PE > 1 − RH. In a cloud-resolving simulation, this constraint is obeyed with the PE profile taking the shape of an inverted C shape.