MSE minus CAPE is the true conserved variable for an adiabatically lifted parcel

For an adiabatic parcel convecting up or down through the atmosphere, it is often assumed that its moist static energy (MSE) is conserved. Here, it is shown that the true conserved variable for this process is MSE minus convective available potential energy (CAPE) calculated as the integral of buoyancy from the parcel's height to its level of neutral buoyancy and that this variable is conserved even when accounting for full moist thermodynamics and nonhydrostatic pressure forces. In the calculation of a dry convecting parcel, conservation of MSE minus CAPE gives the same answer as conservation of entropy and potential temperature, while the use of MSE alone can generate large errors. For a moist parcel, entropy and equivalent potential temperature give the same answer as MSE minus CAPE only if the parcel ascends in thermodynamic equilibrium. If the parcel ascends with a nonisothermal mixed-phase stage, these methods can give significantly different answers for the parcel buoyancy because MSE minus CAPE is conserved, while entropy and equivalent potential temperature are not