## 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 *θ*_{e}. 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 − CAPE is conserved for an adiabatically lifted parcel so long as its buoyancy is defined with respect to the hydrostatic state of the mean large-scale environment. We will show that this conservation is exact even in the presence of nonhydrostatic pressure-perturbation forces that convert CAPE to turbulent energy and wave energy in the environment. While entropy and equivalent potential temperature are conserved for adiabatic parcels in thermodynamic equilibrium, we emphasize that MSE − CAPE is conserved for adiabatic parcels that are either in equilibrium (e.g., a single condensed phase) or in disequilibrium (e.g., a nonisothermal mixed-phase stage, as occurs during the Bergeron process). In contrast, entropy and equivalent potential temperature are not conserved for disequilibrium processes. We argue, therefore, that MSE − CAPE is the true conserved variable for an adiabatically lifted parcel and that it—and not MSE, entropy, or equivalent potential temperature—should be used to calculate the parcel’s properties.

## 2. Theory

*p*is the pressure,

**u**is the velocity, and

*Q*is the external heating with dimensions of power per volume. Here,

*c*

_{υm}is the heat capacity at constant volume for moist air, which is given in terms of mass fractions and heat capacities of dry air (subscript

*a*), water vapor (

*υ*), liquid water (

*l*), and solid water (

*s*) as

*E*

_{0υ}is the difference in specific internal energy between water vapor and liquid at the triple-point temperature (

*T*

_{trip}= 273.16 K) and

*E*

_{0s}is the difference in specific internal energy between water liquid and solid at the triple-point temperature.

*c*is the heat capacity at constant pressure for moist air, and, with help from the continuity equation (

_{pm}*R*

_{m}equals

*p*

_{e}(

*z*) and

*ρ*

_{e}(

*z*); by hydrostatic balance, ∂

*p*

_{e}/∂

*z*= −

*ρ*

_{e}

*g*. Note that we do not require the flow to be hydrostatic at small scales or in the vicinity of the parcel. Instead, we allow there to be substantial pressure perturbations

*p*′, so that

*p*=

*p*

_{e}+

*p*′. This means that there may be significant pressure-perturbation forces (−

**∇**

*p*′) on the parcel. We will, however, assume that the pressure at the location of the parcel is equal to

*p*

_{e}; this is the usual assumption made when calculating the properties of a lifted parcel. With these two assumptions, we can write

*dp*/

*dt*as

*dp*

_{e}/

*dt*=

*w*

*∂*

*p*

_{e}/

*∂*

_{z}= −

*wgρ*

_{e}. Assuming the parcel is adiabatic (

*Q*= 0), dividing by

*ρw*, and writing

*gρ*

_{e}/

*ρ*as

*b*+

*g*, where

*b*=

*g*(

*ρ*

_{e}/

*ρ*− 1) is the parcel’s buoyancy, we get

*z*of the parcel to some fixed reference height

*z*

_{top},

The upper bound of integration in Eq. (4) is somewhat arbitrary. To be as consistent as possible with standard definitions of CAPE, it can be chosen to be a level of neutral buoyancy. For our purposes, all that really matters is that the upper bound is a constant—that is, that it does not change as the parcel is lifted. Note that Eq. (3) is used to calculate parcel ascent, and a constant upper bound drops out in that equation. Also, note that there is no unit step function of buoyancy in the integrand of Eq. (4) that would restrict the integral to regions of positive buoyancy, as is often the case in other definitions of CAPE.

The assumptions used in deriving Eq. (3) are no more restrictive than the assumptions used when calculating parcel properties with conservation of equivalent potential temperature *θ*_{e}. When using *θ*_{e} to calculate the parcel’s temperature at some height, the pressure of the parcel at that height must be known, and it is always assumed that *p* = *p*_{e}. As for the assumption of large-scale hydrostatic balance, this is essentially guaranteed for averages of environmental pressure and density taken over scales much larger than the scale height. Note that MSE − CAPE is conserved even if the parcel does not convert CAPE to its own kinetic energy (KE), but, instead, dissipates CAPE to environmental turbulence and wave energy. In other words, MSE + KE is not conserved, but MSE − CAPE is.

Why is MSE itself not conserved? Consider Eq. (2) for an adiabatic parcel. Dividing by *ρ*, the term (1/*ρ*)*dp*/*dt* can be written as the term *gz* in MSE only if the density of the parcel *ρ* equals the density of the mean large-scale environment *ρ*_{e}. If the parcel is buoyant (i.e., 1/*ρ* > 1/*ρ*_{e}), then (1/*ρ*)*dp*/*dt* is larger in magnitude than (1/*ρ*_{e})*dp*/*dt*, and this saps more specific enthalpy from the parcel than would be predicted by geopotential alone. In other words, for a given change in pressure, a parcel loses more specific enthalpy the lighter it is.

## 3. Examples

The adiabatic ascent of an air parcel is typically calculated using either conservation of MSE, entropy, or *θ*_{e}. Although *θ*_{e} has been written in many different ways with varying degrees of completeness and accuracy (e.g., Simpson 1978; Bolton 1980; Hauf and Höller 1987; Marquet 2011), *θ*_{e} is simply the exponential of entropy (Romps and Kuang 2010). Therefore, there are really just two distinct methods: conservation of MSE and conservation of entropy. We will now consider four examples that illustrate the errors in parcel buoyancy that are caused by incorrectly assuming that MSE is conserved or, for a parcel with disequilibrium mixed-phase microphysics, by assuming that entropy is conserved.

### a. Dry case with constant-lapse-rate environment

*T*

_{0}, the environmental temperature profile is

*T*

_{e}=

*T*

_{0}−

*gz*/

*c*

_{pa}. Note that this solution satisfies Eq. (7) with CAPE = 0, confirming that

*g*/

*c*

_{pa}is the lapse rate for an atmosphere in which displaced parcels are neutrally buoyant. For

*g*= 10 m s

^{−2},

*c*

_{pa}= 1000 J kg

^{−1}K

^{−1}, and

*T*

_{0}= 300 K, the temperature of the environment decreases linearly with height from 300 K at the surface to 0 K at the top of the atmosphere, which is at 30 km.

*g*/

*c*

_{pa}, thereby maintaining a constant temperature excess as it rises. This is incorrect. Defining the temperature excess of a buoyant parcel as Δ

*T*=

*T*−

*T*

_{e}, we can write Eq. (7) for the parcel as

*b*=

*g*Δ

*T*/

*T*

_{e}, we can solve for Δ

*T*to find

*T*

_{0}is the parcel’s temperature excess at the surface. Therefore, we see that the parcel’s temperature excess decreases linearly with height from Δ

*T*

_{0}at the surface to zero at the top of the atmosphere. For example, at 15 km, Δ

*T*is exactly half of its original value. This case is illustrated in the left panel of Fig. 1 for a large Δ

*T*

_{0}. This choice of Δ

*T*

_{0}is made so that both the temperature profiles and the parcel buoyancies are readily visible without the need for a skew-

*T*axis; nothing about Fig. 1 changes qualitatively for a smaller Δ

*T*

_{0}.

For a dry parcel, conservation of MSE − CAPE must give the same answer as conservation of potential temperature. This is easy to prove for this example. Both the environment and the parcel have a constant potential temperature (i.e., independent of height). At any given height, the parcel and environment have the same pressure, so the ratio of the parcel’s potential temperature and the environment’s potential temperature is simply (*T* + Δ*T*)/*T* = 1 + Δ*T*/*T*. Since the potential temperatures are constant, the ratio Δ*T*/*T* must be constant. Since *T* goes to zero linearly with height, so must Δ*T*. This completes the proof. In Fig. 1, the potential temperature is written as the equivalent potential temperature *θ*_{e} since *θ*_{e} equals *θ* for a dry parcel; the appendix provides the precise definition of *θ*_{e}.

### b. Dry case with isothermal environment

*T*

_{e}is independent of height and we can write Eq. (7) for the parcel as

*T*

_{0}is the parcel’s temperature excess at the surface. We see that the temperature excess decreases with height, but not linearly. This solution is illustrated in the right panel of Fig. 1. For a parcel that is buoyant at the surface (i.e., Δ

*T*

_{0}> 0), then the level of neutral buoyancy is

*T*

_{0}≪

*T*

_{e},

*z*

_{LNB}≈

*c*

_{pa}Δ

*T*

_{0}/

*g*. In general, the parcel’s temperature decays toward zero with a scale height of

*c*

_{pa}

*T*

_{e}/

*g*. We can compare this to the assumption of conservation of MSE alone, which would predict Δ

*T*= Δ

*T*

_{0}−

*gz*/

*c*

_{pa}as well as the nonsensical result that the parcel’s absolute temperature would be negative above a height equal to

*c*

_{pa}/

*g*times its temperature at the surface.

Of course, we can confirm in this case that conservation of MSE − CAPE gives the same result as conservation of potential temperature. Since the environmental pressure profile is *p*_{0} is the surface pressure, the parcel’s potential temperature at height *z* is *θ* = *T*_{e} + Δ*T*_{0}, this can be solved to give Eq. (10).

### c. Moist case without ice

For the first of two examples with moisture, we will simplify things by eliminating the solid phase of water. This is effectively accomplished by setting *c*_{υs} equal to *c*_{υl} and setting *E*_{0s} to zero in the definition of MSE and in the equations in the appendix. For the environmental profile, we choose a moist adiabat with *p* = 1 bar, *T* = 300 K, and RH = 1 at *z* = 0. This environmental profile is calculated using conservation of MSE − CAPE with CAPE set to zero (so that a parcel of air displaced from the environment will have zero buoyancy). In this case, since there is no ice and a thermodynamic equilibrium between water vapor and liquid is strictly obeyed, conservation of moist entropy gives the same environmental profile.

Next, consider a buoyant, saturated parcel of air that starts at the surface with *T* = 300.5 K. The left panel of Fig. 2 shows the profile of buoyancy (expressed as a virtual-temperature anomaly) for this parcel as it is lifted, calculated in three different ways: using conservation of MSE − CAPE (solid), conservation of moist entropy (dashed; overlapping the solid curve, so it is difficult to see), and conservation of MSE (dotted). Since the parcel is always in thermodynamic equilibrium, conservation of *θ*_{e} gives the correct answer, as seen from the overlap of the dashed and solid curves. Conservation of MSE, however, overestimates the buoyancy of the parcel by as much as 1 K, which is substantial for a parcel whose buoyancy is in the range of 0.5–1.5 K.

(left) Profiles of buoyancy for a buoyant moist parcel rising adiabatically through a moist-adiabatic environment in which neither has solid water (so that there is no mixed-phase thermodynamic disequilibrium). The profiles are shown as calculated correctly using either conservation of MSE − CAPE (solid) or conservation of *θ _{e}* (dashed) or calculated incorrectly using conservation of MSE (dotted). (right) As in the left panel, but for an atmosphere in which parcels have a mixed-phase stage between a temperature of 273.16 K and a temperature of 240 K. In this case, entropy is not conserved, so calculating the parcel ascent using conservation of entropy gives the wrong answer.

Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-15-0054.1

(left) Profiles of buoyancy for a buoyant moist parcel rising adiabatically through a moist-adiabatic environment in which neither has solid water (so that there is no mixed-phase thermodynamic disequilibrium). The profiles are shown as calculated correctly using either conservation of MSE − CAPE (solid) or conservation of *θ _{e}* (dashed) or calculated incorrectly using conservation of MSE (dotted). (right) As in the left panel, but for an atmosphere in which parcels have a mixed-phase stage between a temperature of 273.16 K and a temperature of 240 K. In this case, entropy is not conserved, so calculating the parcel ascent using conservation of entropy gives the wrong answer.

Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-15-0054.1

(left) Profiles of buoyancy for a buoyant moist parcel rising adiabatically through a moist-adiabatic environment in which neither has solid water (so that there is no mixed-phase thermodynamic disequilibrium). The profiles are shown as calculated correctly using either conservation of MSE − CAPE (solid) or conservation of *θ _{e}* (dashed) or calculated incorrectly using conservation of MSE (dotted). (right) As in the left panel, but for an atmosphere in which parcels have a mixed-phase stage between a temperature of 273.16 K and a temperature of 240 K. In this case, entropy is not conserved, so calculating the parcel ascent using conservation of entropy gives the wrong answer.

Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-15-0054.1

### d. Moist case with mixed-phase condensates

We now include ice by using the correct values for *c*_{υs} and *E*_{0s}. If a cloud parcel starts at a warm cloud base and rises through the atmosphere in thermodynamic equilibrium, then it experiences three stages. It begins in a state of thermodynamic equilibrium between water vapor and liquid. Then, at a temperature of 273.16 K, the parcel begins a stage of isothermal ascent in which vapor, liquid, and ice are in thermodynamic equilibrium at the triple point. The depth of this isothermal stage is approximately *q*_{l}*E*_{0s}/*g*, where *q*_{l} is the mass fraction of liquid water when the parcel first hits a temperature of 273.16 K. For *q*_{l} = 0.02, this is a depth of 600 m. At the end of the isothermal stage, all of the condensates have been converted to ice and the parcel enters its third stage, in which vapor and ice are in equilibrium.

In reality, this is not how the physics of clouds works. Instead of a ~600-m isothermal mixed-phase stage, there is a multikilometer nonisothermal mixed-phase stage in which vapor, liquid, and ice are in disequilibrium, typically with the vapor mass fraction in between saturation with respect to liquid and saturation with respect to ice. This causes vapor to diffuse down a vapor gradient, and that diffusion is a source of entropy (Pauluis and Held 2002). Therefore, adiabatic parcels that ascend with a multikilometer mixed-phase region do not conserve entropy. This means that entropy (or, equivalent potential temperature) should not be used to calculate adiabatic parcel ascent. Fortunately, MSE − CAPE is conserved in these cases. MSE − CAPE is conserved because it is based on conservation of energy, and the only sources and sinks of energy for an adiabatic parcel are pressure work (both expansion work and work against the pressure-perturbation drag force), and both of those are properly accounted for in MSE − CAPE no matter how out of thermodynamic equilibrium the parcel is. Therefore, MSE − CAPE is the true conserved variable for an ascending adiabatic parcel.

The right panel of Fig. 2 illustrates the failure of MSE conservation and *θ*_{e} conservation in the case of a parcel with a disequilibrium mixed-phase stage. The environmental profile is calculated from conservation of MSE − CAPE with CAPE set to zero (so that displaced environmental parcels are neutrally buoyant) and *p* = 1 bar, *T* = 300 K, and RH = 1 at *z* = 0. The mixed-phase region is defined to start at 273.16 K and end at 240 K. In between those two temperatures, the fraction of condensates in the ice phase transitions to one as a linear function of temperature. Similarly, the vapor mass fraction is set to a linear weighting of liquid and solid saturation vapor mass fractions; see the appendix for the relevant equations. The lifted parcel is subjected to these same rules of mixed-phase disequilibrium, and the parcel is initialized at *z* = 0 with *T* = 300.5 K and RH = 1.

As seen in the right panel of Fig. 2, the profiles of buoyancy calculated using conservation of MSE and conservation of *θ*_{e} are both erroneous, deviating from the correct profile (calculated by conservation of MSE − CAPE) by as much as 0.3–0.9 K. The profile calculated using conservation of MSE gives buoyancy that is too high because it misses the reduction of MSE due to CAPE. The profile of buoyancy calculated using conservation of entropy gives buoyancy that is too low because it misses the fact that the mixed-phase disequilibrium increases entropy. Note that the profile of buoyancy calculated by conservation of *θ*_{e} is correct up until the beginning of the mixed-phase layer, which begins around 6 km.

## 4. Summary

We have shown that MSE − CAPE, with CAPE calculated as the integral of parcel buoyancy up to the parcel’s height, is conserved for an adiabatically lifted parcel. The assumptions going into this conservation law are no more restrictive than the assumptions usually made for parcel calculations. In particular, the buoyancy must be calculated with respect to the density profile of the mean large-scale environment, which, by virtue of being large scale, is hydrostatic. As in any other method for calculating parcel properties, we assume that the pressure of the parcel is equal to the mean large-scale environmental pressure at the same height. Aside from this equality of pressures, no assumption is made about the pressure perturbations; arbitrary pressure-perturbation forces (e.g., form drag and wave drag) are allowed and have no impact on the conservation of MSE − CAPE. In addition, no assumption is made about thermodynamic equilibrium. Thermodynamic disequilibrium, such as a nonisothermal mixed-phase stage, will alter a parcel’s entropy, but has no effect on MSE − CAPE.

*d*/

*dz*of MSE − CAPE and setting it to zero, we get

*z*, given by either Eq. (5) or (6), and

*b*is the buoyancy of the parcel at height

*z*. If we know the state of the parcel at height

*z*(i.e., its temperature and its mixing ratios of water vapor, liquid, and solid), then we know MSE and

*b*. Equation (12) then tells us the MSE at height

*z*+ Δ

*z*, for some small Δ

*z*. Using pressure equality of the parcel and its environment, along with whatever assumptions we are making about the apportionment of water among the three phases [e.g., a mixed-phase transition from liquid condensates to solid condensates as a linear function of temperature as in Romps and Kuang (2010)], we can use this value of MSE to calculate the parcel’s temperature at

*z*+ Δ

*z*. We then know its buoyancy and can proceed to

*z*+ 2Δ

*z*. With this simple procedure, we can integrate the parcel’s properties upward in height.

Although the discussion in this paper has focused on adiabatic parcels, these results are just as important for entraining parcels or parcels that lose condensates by fallout. In the numerical calculation of such parcels, the ascent or descent is typically split into a sequence of serial processes. For example, the calculation of an entraining parcel with condensate fallout may proceed as a loop over the following processes with one iteration for each small height increment: entrainment at constant pressure, fallout at constant pressure, and adiabatic lifting. Regardless of the processes performed at constant pressure, the lifting from one pressure to the next should be performed using conservation of MSE − CAPE.

## Acknowledgments

This work was supported by the Scientific Discovery through Advanced Computing (Sci-DAC) program funded by the U.S. Department of Energy Office of Advanced Scientific Computing Research and Office of Biological and Environmental Research and by the U.S. Department of Energy’s Earth System Modeling, an Office of Science, Office of Biological and Environmental Research program under Contract DE-AC02-05CH11231. The author is grateful to three reviewers and the editor, all of whom provided input that improved this manuscript.

## APPENDIX

### Moist Thermodynamics

*r*

_{υ},

*r*

_{l}, and

*r*

_{s}are the mixing ratios (i.e., mass per mass of dry air) of vapor, liquid, and solid, respectively. The constant

*p*,

*T*, and total water (

*q*

_{υ}+

*q*

_{l}+

*q*

_{s}),

*ξ*prescribing a linear transition from liquid condensates to solid condensates between

*T*

_{trip}= 273.16 K and 240 K:

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