[Paper Review] An analytic formula for entraining CAPE in mid-latitude storm environments

John M. Peters, Daniel R. Chavas+3 more|arXiv (Cornell University)|Jan 11, 2023

Meteorological Phenomena and Simulations13 citations

TL;DR

This paper derives new analytic formulas for entraining CAPE (ECAPE) by linking entrainment to the storm environment, improving updraft velocity predictions over previous ECAPE approaches.

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 $\sqrt{2 ext{CAPE}}$ 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.

Motivation & Objective

• Develop an analytic ECAPE formula with entrainment rate determined from the storm environment rather than prescribed values.
• Improve the accuracy of predicting maximum updraft vertical velocity by incorporating entrainment effects.
• Provide a second formulation that includes environmental kinetic energy contribution to updrafts.
• Demonstrate the formulas against past numerical simulations and discuss potential forecasting applications.

Proposed method

• Derive a relation between updraft radius and fractional entrainment rate ε, showing ε ∝ R^{-2} with a fixed L_mix.
• Express ECAPE as an analytic function of ε and state variables by solving a moist static energy budget under entrainment.
• Derive an analytic buoyancy B as a function of ε and environmental moist static energy h0 and h0*, using a plume framework and adiabatic moist energy conservation.
• Obtain ECAPE by vertically integrating B from LFC to LNB and simplifying to an explicit formula (eq. 24) with NCAPE capturing environmental buoyancy dilution.

Experimental results

Research questions

• RQ1 Can ECAPE be expressed analytically in terms of state variables without specifying a fixed entrainment rate?
• RQ2 How does the environmental background influence ECAPE and the resulting updraft kinetic energy via an entrainment-dependent buoyancy formulation?
• RQ3 Does incorporating environmental kinetic energy improve predictions of w_max compared to CAPE and previous ECAPE formulations?
• RQ4 How accurate is the new ECAPE formulation when benchmarked against detailed parcel and environmental profiles?
• RQ5 Under what environmental conditions does the environmental dilution term NCAPE dominate ECAPE predictions?

Key findings

• The new ECAPE formulas remove the need to prescribe a constant entrainment rate by linking ε to the background environment and updraft radius.
• An improved buoyancy formulation (eq. 18) matches benchmark buoyancy profiles more closely than prior P20-based formulas across a range of updraft radii.
• A second formulation accounts for updraft enhancement from environmental kinetic energy, improving agreement with simulations that exceed sqrt(2 CACE) in w_max.
• ECAPE, as integrated from LFC to LNB, is expressed with explicit dependence on state variables and NCAPE, highlighting environmental dryness and temperature effects.
• The formulations, tested against 141 simulations from four numerical studies, show substantial improvement over undiluted CAPE, fixed-entrainment ECAPE, and the Peters et al. (2020a) ECAPE in predicting w_max.