## 1. Introduction

It is difficult to know where to start when replying to a comment like that of Marquet (2016, hereafter M16), which is best summarized as a collection of general confusion, misconceptions, and numerical errors. Certainly, nothing in M16 is at all relevant to the claim by Romps (2015, hereafter, R15) that moist static energy minus convective available potential energy (MSE − CAPE) is the conserved thermodynamic variable for an adiabatically lifted parcel. For the sake of the reader, however, I will begin by summarizing the results from R15 in this section. I will then address the three components of M16 listed above—confusion, misconceptions, and errors—in the subsequent sections.

*b*from the parcel’s current height

*z*to some fixed reference height

This is not an entirely new idea, and it should not be a controversial one. Riehl and Malkus (1958) wrote down conservation of MSE − CAPE in their Eq. (10), although they made approximations to the moist thermodynamics and they believed (erroneously) that MSE − CAPE is approximately conserved only for small buoyancies. R15 showed that MSE − CAPE is exactly conserved even when accounting for the full details of moist thermodynamics (e.g., the temperature dependence of latent enthalpy and the dependence of heat capacity on water mixing ratios) and that it is exactly conserved even for large buoyancies.

## 2. General confusion

It is curious that Marquet spends very little time discussing MSE − CAPE in any of the versions of his comment, even though that is the topic of the paper on which M16 is ostensibly commenting. In the most recent version of his comment, which is the version published in this journal, Marquet claims that I have erred on the sign: he thinks the minus in MSE − CAPE should be a plus. As defined in Eq. (1) here or in Eq. (4) of R15, the convective *available* potential energy (CAPE) of a parcel at height *z* is the integral over height of that parcel’s buoyancy as it is lifted from its current height *z* to some reference height *z* as the integral of its buoyancy from its level of free convection *z*; this, however, is the parcel’s convective *expended* potential energy (CEPE), not its CAPE. We may describe the thermodynamic evolution of an adiabatically lifted parcel using either conservation of MSE − CAPE or MSE + CEPE; the sum of MSE and CAPE is not conserved.

In section 3 of M16, Marquet wonders aloud how the pressure

These points of confusion are all that M16 has to offer about MSE − CAPE. The rest of his comment harps on the third law of entropy, which is a common theme in his publications about entropy. Indeed, both the adiabatic and pseudoadiabatic parcel calculations presented by M16 are performed with zero parcel buoyancy, which is a poor choice for testing the conservation of MSE − CAPE since CAPE is identically zero. For adiabatic ascent, zero buoyancy implies that MSE should be exactly conserved, although Marquet seems unbothered by the fact that his calculation of adiabatic ascent violates this conservation in a rather spectacular way. This will be discussed further in section 4.

## 3. Misconceptions

*q*with subscripts denote the mass fractions of dry air (subscript

*a*), water vapor (subscript

*υ*), liquid water (subscript

*l*), and solid water (subscript

*s*) and

The main objection raised by M16, however, is that R15 and others (Emanuel 1994; Romps 2008; Romps and Kuang 2010; Pauluis et al. 2010) have made a grave mistake by defining the energy and entropy for moist air in such a way that the energy and entropy of liquid water and dry air are zero at a convenient reference temperature and pressure. M16 claims that the equations of moist thermodynamics can only be solved correctly using the additive offsets for the energy and entropy of dry air and liquid water that he prefers. This is a common refrain in Marquet’s publications (e.g., Marquet 2015) and it is demonstrably false. To claim that these overall energy and entropy offsets have any physical consequence is like claiming that one cannot calculate the trajectory of a falling ball using geopotential relative to the floor

Furthermore, M16 claims that these overall offsets must be defined in the way he prefers or else the resulting definitions of energy, enthalpy, and entropy will not give the right answer when considering open systems—for example, when there is a sink of water in a parcel due to precipitation fallout. This is also false. Note that M16 talks exclusively about definitions of energy, enthalpy, and entropy and not at all about the equations that govern them. When we go to write down the governing equations, we see that it is trivial to define the correct thermodynamic equations regardless of the choice of offsets. A pedagogical example is given in the appendix for a two-phase liquid system.

A secondary point made by M16 is that R15 added another adiabatically conserved variable, *h*—namely,

## 4. Numerical errors

M16 objects to the equivalent potential temperature ^{−6}–10^{−3} for MSE,

When calculated properly, all of these are conserved to within round-off error: fractional errors on the order of *s*, Marquet’s ^{5} and 10^{4} Pa. Heights of these pressure levels are integrated upward using hydrostatic balance; these integrated heights are used in the expression for MSE. The profiles displayed in Fig. 1 are subsampled from these solutions every 10 mb (1000 Pa). For context, the bottom row shows the profiles of parcel *T*, *s*,

Upon receiving a first draft of my reply, Marquet heavily edited his comment to remove his claim that only

Even more disconcerting is that the MSE of the adiabatically lifted parcel changes by 5000 J kg^{−1} at the freezing temperature in Fig. 2 of M16. Since Marquet is lifting a parcel with zero buoyancy, MSE should be exactly conserved. (Note that, when Marquet and I refer to MSE, we are both including the ice term; the resulting expression is what some people call the “frozen moist static energy,” and it should be conserved in the presence of fusion.) The fact that M16 is finding a nonconservation of MSE by such a large amount—equivalent to a 5-K anomaly in parcel temperature—means that his calculations are still suffering from grave numerical errors.

Marquet tries to justify the jumps in entropy and MSE that he finds at the freezing point for supposedly isentropic ascent by pointing to the jumps in buoyancy seen in Fig. 8 of Romps and Kuang (2010) and Fig. 2 of R15. Of course, this is a meaningless comparison of apples and oranges: a parcel can have a jump in buoyancy with no change in entropy. In Fig. 8 of Romps and Kuang (2010), the dark-blue curve is for an isentropic parcel, and its jump in buoyancy is not discontinuous but spread out over an isothermal triple-point layer. In the right panel of Fig. 2 of R15, the “hooks” in parcel buoyancy are not caused by discontinuous jumps in entropy as Marquet claims, but instead by the fact that the lifted parcels reach the temperatures bracketing the mixed-phase region at slightly different altitudes than the environment; this is as expected, since the parcels are warmer at every height than the environment.

In his section 8, Marquet turns his attention to the calculation of parcels lifted pseudoadiabatically. He begins by arguing that adiabatic ascent is not common in the real world. I agree. And neither is the pseudoadiabatic ascent that Marquet studies. In reality, cloud parcels lose condensed water by fallout and they dilute their MSE by entrainment. But, any ascent can be calculated numerically by subjecting the parcel to a repeated sequence of adiabatic ascent, fallout, and entrainment over sufficiently small height increments. The argument made by R15 is that the adiabatic-ascent part of this calculation must be performed conserving MSE − CAPE. The inclusion of fallout and entrainment does not alter the central message of R15: for any parcel lifted through the atmosphere (that suffers from any combination of drag, fallout, entrainment, etc.), there is a sink of MSE proportional to buoyancy times vertical velocity.

M16 attempts to show that

## 5. Conclusions

The concluding section of M16 is devoted to some lengthy and irrelevant discussion about the entropy of perfect crystals at absolute zero, Debye’s law, and a Nobel Prize presentation speech. I will, therefore, not spare any space to comment on it. Instead, I will summarize here the two most important lessons that have emerged from this discussion.

One lesson is that there are many ways to define the energy and entropy of a thermodynamic system. When choosing a set of definitions, it is important to give physical values to the differences between the energies and entropies of states between which matter transitions (e.g., between vapor and liquid and between liquid and solid). But, there is unlimited freedom to choose an overall additive energy constant for any set of states within which matter is conserved (e.g., an overall additive constant for the energies of vapor, liquid, and solid or an overall additive constant for the energy of dry air) and similarly for an overall additive entropy constant. This allows us to adopt some convenient conventions, such as setting the energy and entropy of both dry air and liquid water to zero at the triple-point temperature and pressure.

Another lesson is that there are many ways to define equivalent potential temperature. Loosely speaking, equivalent potential temperature is the exponential of entropy. Since there are several choices of factors when taking this exponential, and because the expression can be adorned with various instances of the dry-air mass fraction

This work was supported by the Scientific Discovery through Advanced Computing (SciDAC) program funded by the U.S. Department of Energy Office of Advanced Scientific Computing Research and Office of Biological and Environmental Research under Contract DE-AC02-05CH11231.

# APPENDIX

## Example: Two-Phase Liquid

Mathematically, the primary claim of M16 is that very precise values must be chosen for the specific energy of dry air at the triple point *ρ*. The equations for this system are simpler and, yet, contain all of the physics that is relevant for understanding energy and entropy offsets.

*T*is the temperature,

*i*,

*i*at

*e*, total specific entropy

*s*, and the heat capacity of the mixture

*c*arewhere

The equations for this system are explored in the subsections below. We begin in the first subsection by imagining that mass is never converted between these two types of liquid. In this case, all of the energy and entropy offsets can be chosen arbitrarily and independently of each other; this is analogous to dry air and liquid water, whose offsets can likewise be chosen arbitrarily and independently. In the second subsection, we imagine that mass is able to transition between these two phases of liquid. In this case, there is a phase boundary at a fixed temperature and the difference between the two energy offsets (and the difference between the two entropy offsets) must be chosen appropriately to give the right physics at that phase boundary; this is analogous to water vapor and liquid water, for which the relative energy offset must be chosen appropriately while leaving the overall offset for both phases (i.e., the liquid-water triple-point energy) free to be chosen arbitrarily, and similarly for the entropy offset. Finally, the third subsection adds sources of these two liquid phases to the equations to produce an open system. In this case, it is trivial to define the appropriate equations no matter what overall offsets have been chosen for energy and entropy.

### a. No conversion

*Q*is the heating rate (J m

^{−3}s

^{−1}). From these, we can deriveThe entropy equation follows immediately from Eqs. (A11)–(A13) applied to the definition of entropy given by Eqs. (A3), (A4), and (A6). This is an important point that was emphasized by Romps (2008): the entropy equation contains no information that is not already present in the governing equations for mass and energy. In practice, we find the equation for entropy by taking

*s*and rewriting it in terms of energy sources and mass sources using Eqs. (A11)–(A13). This producesFor adiabatic processes

*s*is conserved for all adiabatic transformations

### b. Conversion

^{−3}s

^{−1}), the governing equations becomeFrom these, we can deriveAs before, the entropy equation is obtained directly from Eqs. (A18)–(A20), yieldingThe term in square brackets is the difference in Gibb’s free energy between the two phases. In order for entropy to be conserved for adiabatic and reversible transformations, this difference in Gibb’s free energy must be zero at the phase boundary, which lies at a fixed temperature for these incompressible, same-density liquid phases. For simplicity, we can set the reference temperature

*s*is conserved for all transformations that are adiabatic

### c. Sources

*s*is conserved for all transformations that are adiabatic

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