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1. Introduction
One of the most fundamental and ubiquitous calculations in atmospheric science is the calculation of the properties of an adiabatically lifted air parcel—that is, a parcel lifted adiabatically, vertically, and rapidly enough so that the environment through which it rises can be considered time invariant. This calculation is performed thousands of times per day at weather centers around the world to quantify atmospheric instability and storm potential. It is also calculated many millions of times per day on supercomputers that are forecasting next week’s weather and next century’s climate. Despite the importance of this process, there is no agreement on how it should be calculated. The most common approach is to use conservation of moist static energy (MSE), which is defined as the sum of sensible enthalpy, latent enthalpy, and gravitational potential energy; see Eqs. (5) and (6) below for precise expressions. But, it is widely known that MSE is only approximately conserved for an adiabatically lifted parcel, and there seems to be no consensus on what alternative should be used.
An alternative that we explore here is the difference between MSE and convective available potential energy (CAPE) calculated as the integral of parcel buoyancy from the parcel’s height to its level of neutral buoyancy (LNB). Riehl and Malkus (1958) were the first to derive the conservation of MSE minus CAPE (MSE − CAPE) for an adiabatically lifted parcel [see their Eq. (10)] but the approximations in that derivation neglected the effect of water phase on density, pressure, and heat capacity. They also stated that MSE − CAPE is approximately conserved only when the parcel buoyancy is small; as shown here, this is incorrect.
Several years later, Madden and Robitaille (1970) and Betts (1974) observed that the integral of a parcel’s buoyancy can explain the difference in the level of its neutral buoyancy predicted using conservation of MSE versus conservation of equivalent potential temperature θe. Levine (1972) tried to refute this claim, but did so using a fallacious derivation (Madden and Robitaille 1972). Most importantly, none of these derivations accounted for the virtual effects of water and the dependence of heat capacity on composition.
The goal of this note is to show, with the full effects of water included, that MSE...