An analytic formula for entraining CAPE in mid-latitude storm environments

Abstract

This article introduces an analytic formula for entraining convective available potential energy (ECAPE) with an entrainment rate that is determined directly from the storm environment. Extending previous formulas derived in Peters et al. (2020), entrainment is connected to the background environment via an analytic manipulation of the equations of motion that yields a direct correspondence between the storm relative flow and the updraft radius, and an inverse scaling between the updraft radius squared and entrainment rate. These concepts, combined with the assumption of adiabatic conservation of moist static energy, yield an explicit analytic equation for ECAPE that depends entirely on state variables in an atmospheric profile and a few constant parameters with values that are established in past literature. Using a simplified Bernoulli-like equation, a second formula is derived that accounts for updraft enhancement via kinetic energy extracted from the cloud's background environment. CAPE and ECAPE can be viewed as predictors of the maximum vertical velocity $w_{max}$ in an updraft. Hence, these formulas are evaluated using $w_{max}$ from past numerical modeling studies. Both of the new formulas improve predictions of $w_{max}$ substantially over undiluted CAPE, ECAPE with a prescribed entrainment rate, and the ECAPE formula from Peters et al. (2020). The formula that incorporates environmental kinetic energy contribution to the updraft correctly predicts instances of exceedance of $\sqrt{2\text{CAPE}}$ by $w_{max}$ in simulations, and provides a conceptual explanation for why such exceedance is rare among past simulations. These formulas are potentially useful in nowcasting and forecasting thunderstorms and as thunderstorm proxies in climate change studies.