This article introduces an analytic formula for entraining convective available potential energy (ECAPE) with an entrainment rate that is determined directly from an environmental sounding, rather than prescribed by the formula user. Entrainment is connected to the background environment using an eddy diffusivity approximation for lateral mixing, updraft geometry assumptions, and mass continuity. These approximations result in a direct correspondence between the storm-relative flow and the updraft radius and an inverse scaling between the updraft radius squared and entrainment rate. The aforementioned 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, the ECAPE formula is modified to account for updraft enhancement via kinetic energy extracted from the cloudâ€™s background environment. CAPE and ECAPE can be viewed as predictors of the maximum vertical velocity_{max}in an updraft. Hence, these formulas are evaluated using_{max}from past numerical modeling studies. Both of the new formulas improve predictions of_{max}substantially over commonly used diagnostic parameters, including undiluted CAPE and ECAPE with a constant prescribed entrainment rate. The formula that incorporates environmental kinetic energy contribution to the updraft correctly predicts instances of exceedance of_{max}, 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.

Substantial mixing occurs between the upward-moving air currents in thunderstorms (updrafts) and the surrounding comparatively dry environmental air, through a process called entrainment. Entrainment controls thunderstorm intensity via its diluting effect on the buoyancy of air within updrafts. A challenge to representing entrainment in forecasting and predictions of the intensity of updrafts in future climates is to determine how much entrainment will occur in a given thunderstorm environment without a computationally expensive high-resolution simulation. To address this gap, this article derives a new formula that computes entrainment from the properties of a single environmental profile. This formula is shown to predict updraft vertical velocity more accurately than past diagnostics, and can be used in forecasting and climate prediction to improve predictions of thunderstorm behavior and impacts.