An analytic description of the structure and evolution of growing deep cumulus updrafts

These expressions are a function of updraft radius, height, convective available potential energy (CAPE), and environmental relative humidity RH. They are quantitatively consistent with idealized three-dimensional moist updraft simulations with varying updraft sizes and in environments with differing RH.

In particular, the analytic expressions capture the rapid decrease of buoyancy with height due to entrainment for narrow updrafts in a dry environment despite large CAPE. In contrast to the standard entraining-plume model, the theoretical expressions also describe the effects of engulfment of environmental air between the level of free convection (LFC) and height of maximum buoyancy (HMB) required by mass continuity to balance upward acceleration of updraft air (i.e., dynamic entrainment).