1. Introduction
The strong pressure dependence of many state variables can complicate attempts to compare the properties of atmospheric air parcels. For dry air, approximated as an ideal gas, the potential temperature θ elegantly describes an air parcel’s state. It does so by accounting for the effect of pressure on the state of the air parcel, which then facilitates comparisons of the properties of air parcels independent of their ambient pressure.
For a variable composition fluid, even for the limiting case of an ideal mixture of ideal gases, the situation is more complicated. Admitting condensable phases for the minor constituent, what we call moist air, complicates matters further. Earth’s atmosphere is, however, fundamentally composed of moist air—it cannot be understood without considering the water it contains. This makes it necessary to address these complications, and explains the rich literature that has developed, proposing one or the other generalization of the idea of the potential temperature to moist air. As it turns out, these moist potential temperatures all measure slightly different quantities, and while this point is generally well understood (see Pauluis 2018), physical understanding of exactly what they measure remains rudimentary.
Given the prominence of a literature that has made statements to the contrary (cf. Emanuel 1994; Pauluis et al. 2008; Raymond 2013; Romps 2015), it may come as a surprise that the equivalent potential temperature θ_{e} does not measure the entropy of an air parcel—not even approximately. Figure 1, which presents vertical profiles of the liquidwater potential temperature θ_{ℓ} the entropy potential temperature θ_{s}, and θ_{e} as calculated from thermodynamic measurements made during the recent Elucidating the Role of Clouds‐Circulation Coupling in Climate (EUREC^{4}A) field study, substantiates this point. All quantities are invariant for isentropic transformations of closed air parcels (viz., with a constant total water content), but this adiabatic invariance, as the figure demonstrates, does not mean that their differences (e.g., with altitude) are indicative of differences in entropy, nor does it guarantee that their isopleths are isentropes. Were this the case, then θ_{e} or θ_{ℓ} could not take on different values for the same value of θ_{s}.
As it turns out, only
2. Terminology and definitions
a. Moist air
We idealize the atmosphere as moist air, i.e., as a mixture of dry air and water, allowing a portion of the latter to condense as conditions dictate. In equilibrium, the thermodynamic state of the moist air is completely specified by three thermodynamic coordinates. For these we adopt the temperature T, the pressure P, and the water mass fraction (totalwater specific humidity) q_{t}. A guide to the subscript notation adopted is given in Table 1.
Subscript notation for the specification of particular states.
To arrive at an analytically tractable description, and to facilitate precise statements, we make four further assumptions: (i) the specific heats are approximated as constant, i.e., not varying with temperature; (ii) the noncondensate phase (gas/vapor) is approximated to behave as an ideal mixture of ideal gases; (iii) the contribution of the condensate to the total volume is negligible; and (iv) only a single condensate phase is admitted, and this is treated as an ideal liquid, whose mass fraction is denoted q_{l}.
Assumptions (i) to (iii) are a common starting point for atmospheric thermodynamics (Emanuel 1994; Stevens and Siebesma 2020; Romps 2021), which facilitates analytic work (Ooyama 1990; Raymond 2013; Romps 2017). Approximation (iv) is adopted because including the ice phase introduces formal complexity that is not relevant to our arguments. Approximations (i) and (ii) can be relaxed by using the variable values of specific heat and nonideal effects based on International Association for the Properties of Water and Steam (IAPWS) and International Thermodynamic Equation Of Seawater—2010 (TEOS10) tools (IAPWS 2010; Feistel 2018), but sacrifices analytic clarity for accuracy.
b. Reference states and notation
Many thermodynamic state functions, such as the entropy or the enthalpy are defined with respect to some referencestate value. For icefree moist air in thermal equilibrium, a reference state can be fully characterized by specifying a reference temperature T_{r}, and the referencestate composition {P_{d,r}, q_{v,r}, q_{l,r}}. Here P_{d} denotes the partial pressure of the dry air, and the roman subscript “r” denotes a reference value. A description in terms of three (rather than two) additional state variables anticipates the possibility of mechanical disequilibrium.^{2} This possibility is required to accommodate the derivation of some of the moist potential temperatures in the proposed framework.
The specification of the reference state already illustrates how notation can be a challenge. In the present manuscript, subscripts are used to give specificity to a class of variables. For instance, roman subscripts d, v, and l are used to distinguish properties of dry air versus water vapor or liquid water. Roman subscript t is used to denote total water, whereby in an icefree system q_{t} = q_{v} + q_{l}. In addition, we introduce the roman subscript r to identify a referencestate value, and x to denote quantities associated with a particular choice of referencestate composition. As a rule, and as summarized in Table 1, numeric subscripts are used to distinguish different airparcel states (with 0 denoting standard values), and letters are used to denote a particular disposition of matter.
Three special compositions of the reference state are defined as special cases of x. These correspond to endmember (or limiting) situations whereby:

e state: denotes the “equivalent” composition of the reference state, whereby x → {P_{d,0}, 0, q_{t}}, and hence is vapor free;

ℓ state: denotes the “liquidless” composition of the reference state, whereby x → {P_{d,0}, q_{t}, 0}, and hence is condensate free;

s state: denotes the “entropic” composition of the reference state, whereby x → {P_{d,0}, 0, 0}, and hence is waterfree (dry).
c. The entropy temperature ϑ
d. Potential temperatures
Given a reference state whose composition is denoted by x, the potential temperature θ_{x} is the temperature (T_{r}) this reference state must adopt to have the same entropy as the given state.
Mathematically this defines θ_{x}, implicitly to satisfy s − s_{x}(θ_{x}) = 0, for some given specification of x. It follows that θ_{x} = ϑ_{x}. The adjective “potential” describes how θ_{x} is the temperature the system would adopt were it brought to the reference state without changing its entropy. The potential temperature as defined above is thus a generalization of the Hauf and Höller (1987) entropy temperature.
By definition, θ_{x} is invariant for any isentropic transformation that does not imply a change in the referencestate composition. For dry air, the reference state is completely specified by P_{r} the referencestate pressure, usually taken to be standard pressure P_{0}. A stricter form of the above definition, and one satisfied by the dryair potential temperature θ would additionally require the transformations to be closed and reversible, but by this definition there can be at most one moist potential temperature.
The three moist potential temperatures θ_{ℓ}, θ_{s}, and θ_{e} are shown below to correspond to the three limiting reference states (e, ℓ, s) described above. Each is illustrated schematically in Fig. 2, whereby all of the points connected by lines in the figure share the same entropy, but the transformations that bring them to their respective reference states differ. Reversible transformations of the closed system are shown along the solid line, and the dashed–dotted lines show either nonequilibrium transformations (for instance associated with the condensate for x = e), or open transformations as associated with removing the water substance at constant entropy, for x = s.
3. Moist potential temperatures
a. The equivalent potential temperature θ_{e}
The oldest, and most familiar, moist potential temperature θ_{e} was introduced by Rossby (1932) as the value of θ for a parcel undergoing an infinite pseudoadiabatic ascent toward P = 0, with all the water removed by precipitation (Fig. 2). Hence it measures the potential temperature required of dry air, such that following an adiabatic expansion its temperature asymptotically approaches that of moist air expanded pseudoadiabatically—herein lies the modern idea of equivalence.^{3}
In the present paper, and following contemporary usage, θ_{e} is defined as the temperature of a vaporfree (equivalent) reference state, with the same entropy, but for which all the water is in the condensate phase at the standard pressure P_{0}. This differs from Rossby’s definition by virtue of being isentropic (condensate is not precipitated from the parcel), hence “equivalence” is being drawn to a system in which the specific heat of the water mass is retained. Retaining the condensate maintains a closed system, but comes at the cost of the reference state being in a state of mechanical disequilibrium [
Physically, θ_{e} measures the temperature air would have if all of its vaporization enthalpy were used to warm the parcel (accounting for the specific heat of the condensate) at standard pressure. It does not satisfy our stricter definition of a potential temperature as the reference state is not in mechanical equilibrium, and complete condensation through expansional cooling can only be realized asymptotically.
b. The liquidwater potential temperature θ_{ℓ}
Physically, θ_{ℓ} measures the temperature the air would have were any (here liquid) condensate evaporated through a process of isentropic warming by compression. To the extent its reference state is in equilibrium, it is thus a potential temperature in the same (strict) sense as θ, provided that all condensed water can be evaporated in this reference state.
c. The entropy potential temperature θ_{s}
Despite frequent statements to the contrary, neither θ_{ℓ} nor θ_{e} are indicative of the specific entropy s of moist air. To address this shortcoming, Marquet (2011) introduced the entropy potential temperature θ_{s}. The insight required to ensure that θ_{s} measures entropy, is the necessity to completely standardize the referencestate composition, denoted by x. For a multicomponent system, doing so introduces a dependency on the absolute entropies, and a role for the third law in atmospheric physics (appendix B).
Thermodynamic constants calculated with dry air composed with a CO_{2} concentration of 420 ppmv.
d. Reference states and pseudoentropies
A substantial and enduring body of literature (Pauluis et al. 2008, 2010; Raymond 2013; Romps 2015), which dates back to Emanuel (1994), introduces the moist potential temperatures, θ_{e} and θ_{ℓ}, as a measure of the entropy that would arise if the reference entropies in Eqs. (8) and (15)—respectively depending on s_{e}(T_{r}) and s_{ℓ}(T_{r})—were assumed to be zero. By adopting this approach one can arrive at expressions for θ_{e} and θ_{ℓ} that are equivalent to Eqs. (11) and (16), with the seemingly attractive property that θ_{e} ∝ T_{r} exp(s/c_{e}), and equivalently θ_{ℓ} ∝ T_{r}exp(s/c_{ℓ}). This has led many authors to conclude that θ_{e} and θ_{ℓ} measures the entropy, or at least a closely related quantity which Pauluis (2018) calls the “relative” entropy.^{4}
A difficulty with defining the moist air entropy as a “relative” entropy, in the sense of Pauluis (2018), is that it is then measured relative to a reference state that varies with the composition of the system, so that comparing “relative” entropies of fluid parcels invariably conflates differences in their referencestate entropies. In a single component fluid, where the composition is fixed, this problem vanishes. To finesse this difficulty, some of the above cited studies have asserted that s_{v,0} − s_{d}_{,0}, which defines
To circumvent these difficulties Marquet (2011) derived θ_{s}. In terms of the present interpretative framework, θ_{s} can be understood as the result of an openprocess that transforms the moistair to a dryair reference state by removing the water while heating to maintain constant entropy—isentropic desiccation. This then defines θ_{s} in terms of a reference state whose composition can be fixed absolutely (q_{r,t} = 0), thereby fixing s_{r,}_{x} and c_{x} independently of the state of the parcel, and recovering the desired property whereby θ_{s} ∝ T_{r,}_{s} exp(s/c_{s}), as described by Eq. (21).
e. Simplified expressions
The different magnitudes of the moist potential temperatures reflect the different degree to which temperature has to compensate differences in the composition of the chosen reference state to maintain the same entropy. Comparing θ_{e} to θ_{ℓ}, for instance, shows that a system with all its water in the liquid phase must be much warmer, than the same system with all its water in the vapor phase, if it is to have the same entropy.
Keeping in mind that each of the moist potential temperatures describe the same system, with the same entropy, Eq. (25) shows how, due to q_{t}, none of the moist potential temperatures are proportional to one another. And although each describes (approximately) a system with the same entropy, at most one can actually be proportional to entropy, which is a state function whose difference between two points (i.e., states) takes a unique value.
4. Properties of the moist potential temperatures
By virtue of their derivation, the moist potential temperatures, θ_{ℓ}, θ_{s}, and θ_{e} are all potential temperatures in the weak sense of the term, i.e., being the temperature of a reference system with the same entropy as the actual system. Only θ_{ℓ} qualifies as a potential temperature in the strict sense of the term, i.e., corresponding to a reference temperature accessible by an isentropic and closed transformation of a system in equilibrium.
For θ_{s} the reference state has a different composition and thus cannot be attained by a closed system. For θ_{e} the reference state is in mechanical (phase) disequilibrium. Even for θ_{ℓ}, mechanical equilibrium of the reference state is only guaranteed for under or justsaturated water vapor pressure at T_{r} = θ_{ℓ}, which corresponds to
a. Entropy
Equation (30) shows that, even after standardizing the referencestate pressures the relationship between Δs and Δ(lnθ_{ℓ}) is modified by Φ_{ℓ}, whose value depends on differences in the composition of the two states. Because Φ_{ℓ}(q_{t,2}, q_{t,1}) = 0 only if q_{t,2} = q_{t,1}, differences in θ_{ℓ} can only measures the entropy differences of systems with the same composition (q_{t,2} = q_{t,1}). The same is true for θ_{e} although the form of Φ_{e} differs from that of Φ_{ℓ}.
b. Enthalpy
Equations (36) and (37) demonstrate how the dependence of the referencestate enthalpy on q_{t} conflates the relationship between Δ(h) in the reference state and Δ(c_{e}θ_{e}). The situation for θ_{ℓ} is no different. However, for many purposes (e.g., measuring temperature changes from mixing) differences in reference temperatures and enthalpies play no role—knowledge of differences in “relative” enthalpies and q_{t} is sufficient. From Eq. (35) we note that in the reference state Δ(h_{e}) → Δ(c_{e}θ_{e}) and Δ(h_{e}) → Δ(c_{ℓ}θ_{ℓ}). This gives a weak form of correspondence between θ_{e} or θ_{ℓ} and their dry air counterpart θ.
Because θ_{s} defines the temperature dry air must have to have the same entropy as the moist system, differences in Δθ_{s} measure differences in the enthalpy of dry air with the same entropy as the moist systems being compared, but the meaning of this enthalpy is not especially informative. This is not unexpected given that Δ(θ_{s}) was designed to measure changes in entropy, not enthalpy.
c. Linear mixing
Entropy S and enthalpy H for a given mass (m) are both extensive variables, whereas specific values for both entropy (s = S/m) and enthalpy (h = H/m) are intensive variables. The total entropy and enthalpy embodied in two parcels of air of mass m_{1} and m_{2} is the sum of the entropy and enthalpy of each parcel, respectively. When the parcels mix, the total entropy increases because the process is irreversible, but the total enthalpy does not change. Therefore, the specific enthalpy is linearly mixing, but the specific entropy is not.
From the discussion of the previous section, this would seem to be the case for x ∈ {e, ℓ}. However, as pointed out there, Δ(h_{x}) → c_{x}θ_{x} only for the reference state. For mixing of air in a different state it is additionally required that the work done to move the mixed system from its reference state to the given state is the same as the work done on the component systems to move them to their reference state, that this is not generally satisfied is also why the moist static energies do not mix linearly (Bretherton 1987).
5. Examples
In this section we present several examples chosen to further illustrate the properties of various choices of θ_{x}. The first compares the structure of the tropical atmosphere as seen through profiles of θ_{e}, θ_{ℓ}, and θ_{s}. The second explores the ability of θ_{x} to measure changes in the state of the atmosphere resulting from the isobaric mixing of air parcels, using a challenging but relevant example of cloudtop mixing. The third compares the ASTEX observed vertical profiles of θ_{e}, θ_{ℓ}, and θ_{s} to study the transition from stratocumulus to cumulus.
a. Contrasting the wet and dry tropics
For the first example we compare the representation of the thermodynamic state in the troposphere in terms of θ_{e}, θ_{ℓ} and θ_{s}. Composite temperature and humidity profiles are derived from global stormresolving (2.5 km) simulations from the Dynamics of the Atmospheric general circulation Modeled On Nonhydrostatic Domains (DYAMOND) project (Stevens et al. 2019) using the Icosahedral Nonhydrostatic (ICON) model (Hohenegger et al. 2020). The composite soundings are taken points over the ocean within the deep (10°S–10°N) tropics. Two soundings are constructed, the first by compositing over regions drier than the 10th percentile of precipitable water, the second by compositing over columns moister than the 99th percentile of precipitable water (to capture the very moistest convective regions). They thereby contrast the thermodynamic structure of the dry and wet tropics, the latter being indicative of regions of active convection.
Figure 3 complements Fig. 1 to more generally show how different expressions for θ_{x} have neither the same values, nor even the same structure. If each expression for θ_{x} were proportional to the entropy (or the entropy as measured relative to some reference), as is sometimes maintained, then how in the case of the moist atmosphere (solid lines) could θ_{s} increase in the lower atmosphere (between 800 and 600 hPa) while θ_{e} decreases. Likewise, how can θ_{s} decrease below 800 hPa in the dry sounding (left panel) where θ_{ℓ} increases. This provides a vivid example of how differences in θ_{e} and θ_{ℓ}, measure differences in the entropy of the reference states of each profile, rather than differences in the entropy of the actual state. Put another way, if two gas quanta have the same entropy, but differ in composition, their values of θ_{e} and θ_{ℓ} will vary to reflect these differences in composition.
In contrast, by virtue of being defined relative to an absolute reference state, θ_{s} is proportional to
As evident from Eq. (25) θ_{e} is consistent with constant θ_{s} only for the case of constant q_{t}. Homogenizing θ_{e} while reducing q_{t}, as the moist profiles in Fig. 3 show to be the case in the convective state, increases θ_{s}.
b. Cloudedge isobaric mixing
For the second example we compare isobaric mixing between two air masses at a cloudtop interface. The mixing of saturated and unsaturated air is nonlinear, so this provides a challenging but relevant test of the properties of the various forms of θ_{x}. The case we explore is based on measurements of marine stratocumulus made as part of the DYCOMS II field study, wherein a stratocumulus layer was topped by warmer and much drier air (Stevens et al. 2003). The conditions sampled during the first research flight satisfied the buoyancy reversal criteria, whereby the air aloft, which we designated by subscript 1, had a higher density temperature, T_{ρ} than the air in the cloud, designated by subscript 2. This situation, whereby T_{ρ}_{,1} > T_{ρ}_{,2}, corresponds to a stable stratification in the absence of mixing. For the observed conditions, mixtures of the warmer drier air aloft with the cooler saturated air in the cloud layer, would (for a range of mixing fractions) result in air parcels denser than the air in the cloud layer. This is a mixing instability whose importance for the dynamics of marine stratocumulus continues to be debated (Deardorff 1980; Randall 1980; Mellado 2017).
Figure 4 confirms our earlier arguments that none of the formulations for θ_{x} linearly mix. Although our particular example involves phase changes, the structure of the error in Fig. 4 (right panel), which is on the order of 5%–10% and maximizes (near η = 0.6) for unsaturated mixtures, is primarily due to the effect of Δq_{t} rather than from phase changes.
c. Stratocumulus–cumulus transition
For the third example we study profiles of θ_{x} for the 43 observed sounding profiles of the first ASTEX Lagrangian experiment described in Bretherton and Pincus (1995) and de Roode and Duynkerke (1997). The profiles are shown for the respective values of θ_{x} in Fig. 5. The sets of profiles are subjectively associated with different cloud regimes. Stratocumulus profiles (colored blue) are associated with mixing from cloud top to the surface and have extensive cloud cover. Profiles associated cumuliform cloud regimes, are colored black. The transition between the two, often associated with stratocumulus whose thermodynamic properties are differentiated (decoupled) from the thermodynamic state of the subcloud layer, are colored red.
As a consequence of the changing profile of q_{t}, the cloud transition admits very different interpretations depending on which form of θ_{x} it is viewed from. Transition profiles are associated with a weakening of the negative θ_{ℓ} gradients in the hydrolapse^{5} regions that demarcates the top of the marine (moist) layer, and a reversal above a certain threshold value of the gradient as measured by θ_{e}. The latter is the basis for the cloudtop entrainment instability hypothesis (Randall 1980). The behavior of θ_{s} is somewhat different, as the transition is better demarcated by a homogenization of θ_{s} in the lower troposphere and almost a null topPBL jump. Whether this is the cause as once suggested by Richardson (1919), or an effect, of increased lower tropospheric mixing is difficult to say, particularly given the strong entropy sources and sinks in this region of the atmosphere. Nonetheless the observation, whereby θ_{s} gradients tend to vanish as stratocumulus gives way to shallow cumulus, has recently been used by Marquet and Bechtold (2020) to introduce an index for demarcating regions of stratocumulus from cumulus.
6. Conclusions
Our main conclusion is that it is hard to avoid accounting for composition when comparing air parcels whose composition varies. While this might seem trivial, a poor recognition of this fact can, and has, led to considerable confusion—for instance, the idea that somehow θ_{e} measures entropy.
In retrospect it seems obvious that composition matters for varied airparcel properties in ways that the introduction of a single moist potential temperature cannot account for—a point also emphasized by Pauluis et al. (2008). Recognizing this fact raises the question as to whether the different moist potential temperatures measure the same thing, and if not, then what precisely do they measure?
We answer these questions first by showing that the equivalent potential temperature (θ_{e}) of Rossby (1932), the liquidwater potential temperature (θ_{ℓ}) of Betts (1973), and the entropy potential temperature (θ_{s}) of Marquet (2011) all share the property of describing the temperature air in some specified reference state would need to have, to have the same entropy as the air parcel they characterize. Each of these adopt standard pressure for the reference state, but differ in the disposition of the variable component. The reference state for θ_{s} is waterfree, the reference state for θ_{ℓ} is condensate free, and the reference state for θ_{e} is vaporfree.
Even if it is not crucial to the validity of its definition, only the θ_{ℓ} reference state is attainable through an isentropic, reversible, and closed transformation, as is the case for the dry potential temperature, θ, and then only in the case when the mass fraction of the water mass in the air parcel is less than the saturated mass fraction at the referencestate temperature and pressure. The reference state for θ_{e} is one of mechanical (phase) disequilibrium of the water phase, and the reference state for θ_{s} can only be accessed by an open process (to remove the water mass entirely).
The reference states that define θ_{ℓ} and θ_{e} are variable, which means they depend on the composition of the parcel which they characterize. In contrast, the reference state of θ_{s} is absolute; it is independent of the composition of the air parcel it characterizes. The latter is a necessary condition for a moist potential temperature to measure entropy. Put differently, θ_{e} (and θ_{ℓ}) only measures entropy differences of air parcels with the same composition, hence in a variable composition atmosphere, only isopleths of θ_{s} coincide with isentropes. Compositional contributions to the entropy are substantial and can only be accounted for by accounting explicitly for the entropy difference between dry air and water vapor (via Λ), similar to the wellappreciated fact that condensational effects can only be accounted for by explicitly accounting for entropy differences between water vapor and condensate, which in equilibrium is proportional to ℓ_{v}, the vaporization enthalpy. This is why ln(θ_{e}/θ_{ℓ}) ∝ q_{t}ℓ_{v} and why ln(θ_{s}/θ_{ℓ}) ∝ Λq_{t}, with Λ measuring the difference between the entropy of water vapor and dry air.
It should come as no surprise that each of the moist potential temperatures are useful for precisely measuring something, and each usefully approximates several airparcel properties, but none usefully approximate all important properties. θ_{ℓ} and θ_{e} are poor measures of entropy, but accurately measure the referencestate “relative” enthalpy. In the case of θ_{e} whose reference state has already valorized the vaporization enthalpy of its water, the addition or removal of condensate, has a relatively minor effect. Likewise θ_{ℓ} is relatively insensitive to changes in vapor. This explains the popularity of θ_{e} as a basis for tracking air parcels in the presence of precipitation, or the use of θ_{ℓ} in studies more interested in isolating an air mass’s thermal properties—for instance, as a component of a mixing diagram. In contrast, θ_{s} measures the entropy of moist air. None of the moist potential temperatures mix linearly, and the errors encountered by assuming they do so can be substantial (ranging from a few to 10%).
Several examples are explored as a basis for exploring tradeoffs in the use of different forms of θ_{x} to interpret the structure of the tropical atmosphere. These examples show how θ_{s} is generally better mixed through the tropical troposphere than is either θ_{e} or θ_{ℓ}, and that the transition from stratocumulus to cumulus is associated with a transition of the troposphere to a state where θ_{s} becomes mixed through the lower troposphere, despite considerable gradients in moisture—whether or not this structure, which is also corroborated by many other observations (see Marquet 2011), is indicative of a process that acts to homogenize entropy, or occurs by chance, is an open question.
For the treatment of the thermodynamics it is common to neglect changing CO_{2} and O_{2} from burning fossil fuels.
What we call mechanical equilibrium, which is a force (pressure) balance between phases, is sometimes referred to as phase equilibrium.
Marquet and Dauhut (2018) traces the idea of an “equivalent” potential temperature, to Normand (1921), who introduced it as a generalization of Schubert (1904, p. 18) and Knoche (1906, p. 3), and ultimately von Bezold’s concept of (“higher” or “supplemented” or “complete”) “equivalent” temperature (T_{e}). Normand’s equivalent potential temperature (θ_{e}) was defined using
As pointed out by Marquet and Dauhut (2018), this terminology risks confusion with the paper where the Shannon (1948) entropy is defined, but with another different quantity with the same name of “relative” entropy.
The term hydro–lapse is used to demarcate the trade wind inversion region as the falloff of moisture with height is often more pronounced than the increase of temperature at the top of the trade wind cloud layer.
Acknowledgments.
The ideas were developed jointly by the authors and hence ordered alphabetically by last name. The research was made possible by generous public support for the scientific activities of the Max Planck Society, MétéoFrance, and the CNRS—sometimes research is still possible without third party funding. The authors thank Dave Raymond, Martin Singh, and an anonymous reviewer, as well as the editor William Boos, for their constructive and critical comments, which led to substantial improvements in the presentation of our ideas.
Data availability statement.
Profiles used for the cumulus to stratocumulus transition are available at http://www.atmos.washington.edu/∼breth/astex/lagr/README.hourly.html and ftp://eos.atmos.washington.edu/pub/breth/astex/lagr/lagr1/hourly/. EUREC^{4}A data for Fig 1 are available from the JOANNE dataset as cited. The temperature soundings used Fig. 2 are provided courtesy of the DYAMOND project; it and a notebook containing the calculations presented in the manuscript are made available by the host authors corresponding institutions by contacting publications@mpimet.mpg.de.
APPENDIX A
Numerical Evaluation
The approximations given by Eqs. (22)–(24) introduce errors on the order of 1%, or about 4 K. The left panel of Fig. A1 shows how
The chosen forms for
The results shown in this appendix are in agreement with errors on the order of 0.6 K or 0.2% shown in Fig. 1. The larger errors in Fig. A1 are due to cumulative effects during the vertical ascents.
APPENDIX B
Historical Notes on the Application of the Third Law
The dependence of θ_{s} on the absolute entropy, through the factor Λ in Eq. (18), arises because from the need to characterize a multicomponent system whose relative composition (in our case between dry air and water vapor) is allowed to vary.
The recognition that the absolute value of the entropy are important for reacting, or multicomponent systems, dates to Le Chatelier (1888), who first described the need to know the absolute values of entropy of reactants and products in order to be able to predict the stability of all chemical processes. Then Nernst (1906) derived his “theorem of heat,” but it is Planck (1914, 1917) who really derived what is nowadays known as the Boltzmann equation S = k ln(W) with k the Boltzmann constant. The absence of an additive constant corresponds to cancelling the entropy of all perfect crystalline state at zero Kelvin temperature (third law of thermodynamics), due to the unique remaining number of configuration W = 1 at 0 K. Pauling (1935) and Nagle (1966) computed the residual entropy for ice at 0 K (ΔS ≈ 189 J K^{−1} kg^{−1}), which must be taken into account for computing the entropy of water at any finite positive absolute temperature. The link between the third law of Planck and the principle of unattainability of absolute zero temperature derived by Nernst (1912) and studied by Simon (1927) has been recently clarified by Masanes and Oppenheim (2017).
Values of absolute reference entropy of atmospheric gases (N_{2}, O_{2}, Ar, H_{2}O, CO_{2}) used in Hauf and Höller (1987), Marquet (2011), and Stevens and Siebesma (2020) were already available in Kelley (1932), Lewis and Randall (1961), and Gokcen and Reddy (1996). They are now accurately determined and available in the National Institute of Standards and Technology Joint Army–Navy–Air Force (NISTJANAF) tables (Chase 1998). The agreement between the various way to compute the absolute entropies can be fairly appreciated in Fig. B1, where the “calorimetric” and “statisticalphysics” methods lead to the same results in the range of atmospheric temperatures up to better than ±0.6% for H_{2}O and N_{2} and better than ±0.1% for O_{2}, Ar, and CO_{2}. The accuracy of the NISTJANAF tables are indicated as being better than one tenth of the differences between calorimetric and the statistical methods.
It can be recalled that, if the third law is applied to 0 K, the consequences of this hypothesis impact the atmospheric temperatures domain via the calorimetric method and the integrations made between 0 K and any temperature T. The same is true for the statistical physics method, where the partition function Z is computed with the hypothesis S = k ln(W) with no additive constant and with S = 0 because W = 1 at 0 K.
The impacts of the hypotheses (i) and (ii) made at the end of section 2a concerning the constancy of the specific heats and the deviations from the ideal gas aspects remain small when compared to the data computed by the IAPWS and TEOS10 software (not shown). Moreover, the absolute values of the entropies can easily be computed with TEOS10 if one takes into account the data from the thermodynamic tables (Lewis and Randall 1961; Chase 1998), or at least the liquidwater and dryair absolute entropies given by Millero (1983) and Lemmon et al. (2000), respectively.
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