the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Impacts of non-ideality and the thermodynamic pressure work term *p*Δ*v* on the surface energy balance

### William J. Massman

Present-day eddy-covariance-based methods for measuring the energy and mass exchange between the earth's surface and the atmosphere often do not
close the surface energy balance. Frequently the turbulent energy fluxes (sum of sensible and latent heat) underestimate the available energy (net
incoming radiation minus the soil conductive heat flux) by 10 % to 20 % or more. Over the last 3 or 4 decades several reasons for this
underestimation have been proposed, but nothing completely definitive has been found. This study examines the contribution of two rarely discussed
aspects of atmospheric thermodynamics to this underestimation: the non-ideality of atmospheric gases and the significance the water vapor flux has
for the sensible heat flux, an issue related to the pressure work term *p*Δ*v*. The results were not unexpected; i.e., these effects are too
small to account for all of the imbalance between the sum of the turbulent fluxes and the available energy. Together they may contribute 1 %–3 % of
the difference (or 10 % to 15 % of the percentage imbalance).

*p*Δ

*v*on the surface energy balance, Hydrol. Earth Syst. Sci., 24, 967–975, https://doi.org/10.5194/hess-24-967-2020, 2020.

This paper was written and prepared as part of my official duties as a U.S. Government employee. It is, therefore, in the public domain and may not be copyrighted.

The microclimate at any given location on the earth's surface is determined by a balance between the incoming and outgoing energy. Documenting and
measuring these energy flows is fundamental to micrometeorology and to the understanding of the functioning of the earth's ecosystems
(e.g., Geiger et al., 2003). In its simplest form the surface energy balance (SEB) is composed of four terms: ${R}_{\mathrm{n}}={L}_{\mathrm{v}}E+H+G$, where *R*_{n}
(W m^{−2}) is net radiation (incoming radiation minus reflected and outgoing infrared radiation), *L*_{v}*E* (W m^{−2}) is the latent
heat flux or the energy required to evaporate (and transpire) moisture, *H* (W m^{−2}) is the sensible heat flux associated with heated air
currents as they move upward and away from the surface, and *G* (W m^{−2}) is the heat conducted into the components of the surface (soil,
tree branches and trunks). For the purposes of the present study all other terms of the SEB, which tend to be small, can be ignored. But despite
decades of effort micrometeorologists worldwide have not been able to achieve a fully satisfactory level of closure to the SEB
(e.g., Twine et al., 2000; Oncley et al., 2007; Leuning et al., 2012).

There have been many studies that have proposed explanations for the often observed imbalance, but the present study focuses on only two,
Paw U et al. (2000, Appendix C) and Kowalski (2018), which are centered exclusively on *L*_{v}*E* and *H*. The authors of both of these studies seek at
least a partial “solution” to the energy imbalance problem by suggesting that the pressure work term, *p*Δ*v* (J kg^{−1}), that part of
the first law of thermodynamics that accounts for the work done on a system or by a system during the physical expansion or compression of that
system, has not been incorporated correctly into micrometeorological theory underpinning the measurements of *L*_{v}*E* and *H*. Kowalski (2018) argued
(incorrectly) that the enthalpy of vaporization, *L*_{v} (J kg^{−1}), did not include *p*Δ*v*. So he proposed adding *p*Δ*v*, which by
his analysis was equal to the term *R*_{d}*T*_{v} (where *R*_{d} is the specific gas constant for dry air ($\mathrm{J}\phantom{\rule{0.125em}{0ex}}{\mathrm{kg}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}$) and *T*_{v} is the virtual
temperature of the air (K)), to correct *L*_{v}, yielding in turn a 3 %–4 % increase in *L*_{v}. But, as pointed out by the reviewers and
commenters on Kowalski's study, adding *p*Δ*v* to *L*_{v} is incorrect because *p*Δ*v* is by definition a component of *L*_{v}, and so adding it to
*L*_{v} would be double-counting it. Furthermore, as noted by another commenter, *p*Δ*v* can be computed directly for the evaporative process, and it
is not equal to, nor numerically the same as, *R*_{d}*T*_{v} (also see Fig. 1 and the related discussion below).

Paw U et al. (2000, Appendix C), on the other hand, take a different approach to the *p*Δ*v* term. They do not apply their correction directly to
*L*_{v}*E* in the SEB equation. Rather they apply their correction to the heat flux, *H*, based on a change in density of an air parcel associated with
mixing newly transpired or evaporated water vapor with the air contained within that air parcel. They pose their correction in terms of an equivalent
temperature perturbation, such that after evaporation has occurred the (turbulent + diffusive) transport-driven expansion of the water vapor into the
atmosphere surrounding the source of water vapor (e.g., plant stomatal pores and the porous soil) results in a change in the atmospheric density that
is associated with a concomitant change in the atmospheric temperature. So in effect Paw U et al. (2000) are using the first law of thermodynamics
(expressed in terms of atmospheric processes and the pressure work term) to argue that *H* should be adjusted to include a small term that is
proportional to the mass flux of water vapor, *E* ($\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}$).

The present paper employs “classical” thermodynamics to examine (a) the influence that the non-ideality of atmospheric gases can have on the SEB and
(b) the methods and conclusions of Paw U et al. (2000, Appendix C) regarding the first law of thermodynamics and the pressure work term's influence on the
turbulent heat flux and ultimately the SEB as well. Although it is true that what I develop herein is not necessarily “new” science, some of the
theory I employ may well be new to the general environmental and geo-biophysical communities. The present study is divided into two parts. The first
examines and quantifies how mixing of air and water vapor as non-ideal (or real) gases, rather than as ideal gases, can impact *L*_{v} and the specific
heat of moist air. In the second part the first law of thermodynamics is employed to derive the influence water vapor has on potential temperature,
which in turn gives rise to an expression, different from that developed by Paw U et al. (2000, Appendix C), relating how the kinematic heat flux is
influenced by the mass flux of water vapor, *E*. In summary, this study shows that any potential corrections to the SEB from either of these two
sources are likely to be negligible and certainly much smaller than either Kowalski (2018) or Paw U et al. (2000) propose.

The next three sections are a purely theoretical argument intended to estimate the influence that a mixture of non-ideal gases (water vapor and dry
air) can have on the SEB near standard pressure and temperature (STP) by comparing the enthalpy of vaporization of water and the specific heat of
moist air associated with ideal gases and non-ideal gases. Here “near STP” will be understood as pressures between about 70 and 105 kPa and
temperatures between about 0 and 100 ^{∘} C or so – or an atmospheric state typical of near-surface conditions on earth.

## 2.1 Enthalpy of vaporization

The enthalpy of vaporization for pure water into an atmosphere of pure water vapor (see either Wagner and Pruß, 2002, or Harvey and Friend, 2004) is expressed as

where ${L}_{\mathrm{v}}^{\ast}$ (J kg^{−1} or J mol^{−1}) is the enthalpy of vaporization for pure liquid water into an atmosphere of pure
saturated water vapor, ${h}_{\mathrm{v}}^{\ast}$ (J kg^{−1} or J mol^{−1}) is the specific enthalpy of saturated vapor, and ${h}_{\mathrm{l}}^{\ast}$
(J kg^{−1} or J mol^{−1}) is the specific enthalpy of pure liquid water. Note (a) that the asterisk superscript (^{∗}) will be
used to denote a pure quantity (as opposed to a mixture which will not be superscripted) and (b) that the researchers cited above essentially employ
the Clausius–Clapeyron equation to determine ${h}_{\mathrm{v}}^{\ast}-{h}_{\mathrm{l}}^{\ast}$. Of course, liquid water under near-earth-surface conditions will not be
composed solely of pure liquid water. Rather it will be a mixture of pure liquid water and, e.g., dissolved atmospheric gases (O_{2}, CO_{2}, CH_{4},
etc.) and possibly any number of dissolved organic and inorganic compounds (e.g., mineral salts, organic acids). But for the present study, it
is unnecessary to consider this additional complexity. Figure 1 includes plots of ${L}_{\mathrm{v}}^{\ast}$ as a function of temperature, *T*_{K}
(K), computed using the formulations of Wagner and Pruß (2002) (red line), and a linear approximation to it (black line) over the plotted temperature
range.

Also included in Fig. 1 are the two components of the specific enthalpy of vaporization (i.e., the change in internal energy,
d*u*^{∗}, and the pressure work term, ${p}^{\ast}\mathrm{\Delta}{v}^{\ast}$), where accordingly ${L}_{\mathrm{v}}^{\ast}=\mathrm{d}{u}^{\ast}+{p}^{\ast}\mathrm{\Delta}{v}^{\ast}$. For this
figure d*u*^{∗} is calculated as the difference ${L}_{\mathrm{v}}^{\ast}-{p}^{\ast}\mathrm{\Delta}{v}^{\ast}$ and ${p}^{\ast}\mathrm{\Delta}{v}^{\ast}$ is estimated as follows:

where *p*_{v} (Pa) is the vapor pressure, *ρ*_{v} (kg m^{−3}) is the vapor density and *ρ*_{l} (kg m^{−3}) is the density of
liquid water. The numerical algorithms used for *p*_{v}, *ρ*_{v} and *ρ*_{l} are from Wagner and Pruß (2002). But since *ρ*_{l}≫*ρ*_{v} it follows that
${p}_{\mathrm{v}}(\mathrm{1}/{\mathit{\rho}}_{\mathrm{v}}-\mathrm{1}/{\mathit{\rho}}_{\mathrm{l}})\approx {p}_{\mathrm{v}}/{\mathit{\rho}}_{\mathrm{v}}$. In turn the ideal gas law yields ${p}_{\mathrm{v}}/{\mathit{\rho}}_{\mathrm{v}}=R{T}_{\mathrm{K}}/{M}_{\mathrm{w}}$ – also shown in Fig. 1 –
where *R* ($\mathrm{J}\phantom{\rule{0.125em}{0ex}}{\mathrm{mol}}^{-\mathrm{1}}{\mathrm{K}}^{-\mathrm{1}}$) is the universal gas constant and *M*_{w} (kg mol^{−1}) is the molecular mass of water. The three
quantities d*u*^{∗}, ${p}^{\ast}\mathrm{\Delta}{v}^{\ast}$, and *R**T*_{K}∕*M*_{w} are included in Fig. 1 primarily for the sake of completeness
and to give some sense of their relative contributions to ${L}_{\mathrm{v}}^{\ast}$. Figure 1 indicates that
${p}^{\ast}\mathrm{\Delta}{v}^{\ast}\approx {L}_{\mathrm{v}}^{\ast}/\mathrm{15}$, meaning that ${p}^{\ast}\mathrm{\Delta}{v}^{\ast}$ is a relatively small component of ${L}_{\mathrm{v}}^{\ast}$.

Next, consider a system of *N*_{d} mols of dry air and *N*_{l} mols of pure liquid water separated from one another by an impermeable
membrane. Both are at the same temperature *T*_{K,init} and the pressure of the dry air is *p*_{d} (Pa). Further assume that this dry
air–liquid water system is isolated, i.e., that it cannot exchange mass or energy or interact mechanically with its surroundings. The total enthalpy of
this system is ${N}_{\mathrm{d}}{h}_{\text{d,init}}^{\ast}+{N}_{\mathrm{l}}{h}_{\text{l,init}}^{\ast}$ (J). This will now be considered the initial state of the system.

After removing the membrane the final state of the system occurs after *N*_{v} mols of liquid has evaporated and diffused throughout the volume
of dry air to the point of saturation, where of course *N*_{v}≤*N*_{l}, to ensure that there is enough liquid to achieve saturation. Note: it is
possible to calculate *N*_{v}, because *N*_{v}=*N*_{v,sat}, but for the present purposes this is not necessary. The final state now comprises
*N*_{d} mols of dry air, *N*_{l}−*N*_{v} mols of pure liquid water and *N*_{v} mols of water vapor. For an ideal gas the final pressure
is *p*_{v,sat} (Pa), but for a non-ideal gas the saturated vapor pressure is *f**p*_{v,sat} (Hyland and Wexler, 1983; Goff, 1949), where
$f=f({T}_{\mathrm{K}},{p}_{\mathrm{a}})$ is termed the enhancement factor and $\mathrm{1}<f<\mathrm{1.006}$ near STP (Hyland and Wexler, 1983; Nelson and Sauer, 2004). Consequently, the final pressure of the
water vapor will exceed *p*_{v,sat} by a small amount. On the other hand, the final pressure of the dry air, *p*_{d,final} (Pa),
will be slightly less that *p*_{d} because the final gas volume of the system will be slightly greater than the initial volume due to the decrease in
the volume of liquid with the evaporative loss of *N*_{v} mols of liquid. In the present scenario this difference between the final and initial
pressures is small: ≈0.001*p*_{d}. Because both *f* and this relative pressure difference are so small and they tend to compensate for one
another, it is reasonable to ignore both effects and approximate the final total pressure, *p*_{a} (Pa), as simply as
${p}_{\mathrm{a}}={p}_{\mathrm{d}}+{p}_{\text{v,sat}}$, meaning that for present purposes evaporation occurring within an isolated system can be considered an archetypical
constant pressure process. Nonetheless, it is also worth emphasizing that, in fact, evaporation in the present isolated system (as well as within the
atmospheric surface layer) is neither a constant volume process nor a constant pressure process. Rather it is a combination or hybrid of the two processes.

The total enthalpy of the final state of the system is $\left({N}_{\mathrm{d}}+{N}_{\mathrm{v}}\right){h}_{\mathrm{a}}+\left({N}_{\mathrm{l}}-{N}_{\mathrm{v}}\right){h}_{\mathrm{l}}^{\ast}$, where *h*_{a} (J mol^{−1}) is
the specific enthalpy of the resulting moist air. But because of evaporative cooling, the temperature of the final state of the system, *T*_{K}
($\mathrm{273.16}\phantom{\rule{0.125em}{0ex}}\mathrm{K}<{T}_{\mathrm{K}}\le \mathrm{373.15}\phantom{\rule{0.125em}{0ex}}\mathrm{K}$), is less than *T*_{K,init}. This change in temperature of the system, *δ**T* (K), is
defined as $\mathit{\delta}T={T}_{\mathrm{K}}-{T}_{\text{K,init}}<\mathrm{0}$. The Appendix examines this temperature difference in more detail. With this last simplification in
mind, the change in total enthalpy of the system, Δ*H*_{s} (J), is

where ${h}_{\mathrm{a}}={\mathit{\chi}}_{\mathrm{d}}{h}_{\mathrm{d}}^{\ast}+{\mathit{\chi}}_{\mathrm{v}}{h}_{\mathrm{v}}^{\ast}+{I}_{B}$ (e.g., Hyland and Wexler, 1983) and ${\mathit{\chi}}_{\mathrm{d}}={N}_{\mathrm{d}}/({N}_{\mathrm{d}}+{N}_{\mathrm{v}})={p}_{\mathrm{d}}/{p}_{\mathrm{a}}$ is the dry air molar fraction
(mol mol^{−1}) of the moist air, ${\mathit{\chi}}_{\mathrm{v}}={N}_{\mathrm{v}}/({N}_{\mathrm{d}}+{N}_{\mathrm{v}})={p}_{\text{v,sat}}/{p}_{\mathrm{a}}$ is the vapor molar fraction (mol mol^{−1}) of the
moist air, and *I*_{B} is the excess enthalpy of mixing (e.g., Wormald et al., 1977; Sattar, 2000) that arises because of the non-ideality of the gases
(e.g., Hyland and Wexler, 1983).

After some algebraic manipulation the following simplified expression for Δ*H*_{s} results:

where $\mathit{\delta}{h}_{\mathrm{d}}^{\ast}={h}_{\mathrm{d}}^{\ast}-{h}_{\text{d,init}}^{\ast}$ and $\mathit{\delta}{h}_{\mathrm{l}}^{\ast}={h}_{\mathrm{l}}^{\ast}-{h}_{\text{l,init}}^{\ast}$. Because both
${h}_{\mathrm{d}}^{\ast}$ and ${h}_{\text{d,init}}^{\ast}$ are functions of temperature, i.e., ${h}_{\mathrm{d}}^{\ast}={h}_{\mathrm{d}}^{\ast}\left({T}_{\mathrm{K}}\right)$ and
${h}_{\text{d,init}}^{\ast}={h}_{\text{d,init}}^{\ast}\left({T}_{\text{K,init}}\right)$, and *δ**T* is small in comparison to either *T*_{K,init} or
*T*_{K} (Appendix A), it is reasonable to approximate $\mathit{\delta}{h}_{\mathrm{d}}^{\ast}$ as $(\partial {h}_{\mathrm{d}}^{\ast}/\partial {T}_{\mathrm{K}})\mathit{\delta}T$. Similar results hold for
$\mathit{\delta}{h}_{\mathrm{l}}^{\ast}$, so that the ${N}_{\mathrm{d}}\mathit{\delta}{h}_{\mathrm{d}}^{\ast}+{N}_{\mathrm{l}}\mathit{\delta}{h}_{\mathrm{l}}^{\ast}$ component of Δ*H*_{s} can be reasonably assumed to be a function of
both temperature and the temperature difference. On the other hand, the ${N}_{\mathrm{v}}({h}_{\mathrm{v}}^{\ast}-{h}_{\mathrm{l}}^{\ast})+({N}_{\mathrm{v}}+{N}_{\mathrm{d}}){I}_{B}$ component of Δ*H*_{s} is
a function only of the final temperature, *T*_{K}, and is not influenced by *δ**T*. This allows the following identification to be made:
$\mathrm{\Delta}{H}_{\mathrm{s}}=\mathrm{\Delta}{H}_{\mathrm{s},L}+\mathrm{\Delta}{H}_{\mathrm{s},T}$, where $\mathrm{\Delta}{H}_{\mathrm{s},L}={N}_{\mathrm{v}}({h}_{\mathrm{v}}^{\ast}-{h}_{\mathrm{l}}^{\ast})+({N}_{\mathrm{v}}+{N}_{\mathrm{d}}){I}_{B}$ results from the change in phase
associated with evaporation and $\mathrm{\Delta}{H}_{\mathrm{s},T}=\left[{N}_{\mathrm{d}}\right(\partial {h}_{\mathrm{d}}^{\ast}/\partial {T}_{\mathrm{K}})+{N}_{\mathrm{l}}(\partial {h}_{\mathrm{l}}^{\ast}/\partial {T}_{\mathrm{K}}\left)\right]\mathit{\delta}T$ results from
the change in temperature. The principal interest of this study is Δ*H*_{s,L}. Therefore, dividing Δ*H*_{s,L} by *N*_{v} yields

At this point it is important to note that except for the non-ideality of water vapor and dry air, the enthalpy of vaporization of water would be
completely independent of the presence of dry air, i.e., ${L}_{\mathrm{v}}\equiv {L}_{\mathrm{v}}^{\ast}$. In other words, if not for the non-ideal behavior of these gases
*L*_{v} would be the sole property of water and would otherwise not be influenced by the presence or absence of dry air.

In general *I*_{B} is expressed in terms of the second and third virial coefficients (Hyland and Wexler, 1983; Wagner and Pruß, 2002), which are defined by the virial equation
of state (Hyland and Wexler, 1983; Sattar, 2000) as follows:

where the subscript “*i*” refers to water vapor (*i*= v), dry air (*i*= d), or moist air (*i*= a); *B*_{i} (m^{3} mol^{−1}) is the second
virial coefficient, *C*_{i} (m^{6}mol^{−2}) is the third virial coefficient, and in general *B*_{i} and *C*_{i} are both functions of temperature,
*T*_{K}; *p*_{i} is the gas pressure (Pa) and *v*_{i} is the molar volume (m^{3} mol^{−1}) of the gas. For this study it is sufficient to
consider only the second virial coefficients. For dry air and water vapor *B*_{i}=*B*_{i}(*T*_{K}) is determined by empirical curve fitting of observed data.
For this study *B*_{v}(*T*_{K}) is taken from Eq. (6) of Harvey and Lemmon (2004) and *B*_{d}(*T*_{K}) is taken from Eq. (10) of Hyland and Wexler (1983). Because moist air is
a mixture of dry air and water vapor the second virial coefficient for moist air takes the form ${B}_{\mathrm{a}}={\mathit{\chi}}_{\mathrm{v}}^{\mathrm{2}}{B}_{\mathrm{v}}+\mathrm{2}{\mathit{\chi}}_{\mathrm{v}}{\mathit{\chi}}_{\mathrm{d}}{B}_{\mathrm{vd}}+{\mathit{\chi}}_{\mathrm{d}}^{\mathrm{2}}{B}_{\mathrm{d}}$
(Sattar, 2000), where *B*_{vd} (m^{3} mol^{−1}) is the cross virial coefficient for moist air. For the present study *B*_{vd}(*T*_{K}) is taken
from Eq. (15) of Hyland and Wexler (1983). Once the equation of state has been specified, the general expression for *I*_{B} can be derived
(e.g., Sattar, 2000), yielding

The final step is to specify whether the enthalpic change occurs at constant pressure or at constant volume. Although assuming a constant pressure
pathway for modeling evaporation into the atmosphere is likely to be more appropriate than assuming a constant volume pathway, both pathways need to
be considered here because any evaporation occurring on the earth's surface is going to lie somewhere between these two (bounding) pathways. This is
equivalent to specifying *p*_{a} and *p*_{d} at the initial and final states. At a constant pressure *p*_{a} is held constant, so that
${p}_{\mathrm{d}}\left(\text{final state}\right)={p}_{\mathrm{d}}\left(\text{initial state}\right)-{p}_{\mathrm{v}}\left(\text{final state}\right)$, where *p*_{d}(initial state)=*p*_{a} and
*p*_{v}(final state)=*p*_{v,sat}, and (for the sake of completeness it should also be noted that) *p*_{v}(initial state)=0. In this
case *p*_{a} is arbitrarily assigned a value of 101.325 kPa. To evaluate ${L}_{\mathrm{v}}^{\ast}$ at a constant volume *p*_{d} is held constant, so
${p}_{\mathrm{a}}\left(\text{final state}\right)={p}_{\mathrm{a}}\left(\text{initial state}\right)+{p}_{\mathrm{v}}\left(\text{final state}\right)$. In this case *p*_{d} is arbitrarily assigned a value of
101.325 kPa. The only difference between these two cases is that the final molar values of *N*_{v} and *N*_{a}($={N}_{\mathrm{v}}+{N}_{\mathrm{d}}$) can be different, so
that the term *p*_{a}*χ*_{d} in Eq. (7) can vary slightly depending on whether the evaporation is occurring at a constant pressure or
a constant volume.

The results of evaluating Eq. (7) for these two different processes are shown in Fig. 2. Note that beginning
with this figure and henceforth Δ*L*_{v} will be used as shorthand for *I*_{B}∕*χ*_{v}. With the exception of sublimation of ice or snow, these
results suggest that surface energy fluxes associated with ET measured at temperatures commonly encountered with micrometeorological techniques
(i.e., between about 275 and 325 K) could be underestimated by 1 % to 2.5 % solely on the basis of using an estimate for the enthalpy of
vaporization, ${L}_{\mathrm{v}}^{\ast}$, that does not allow for the fact that dry air and water vapor are non-ideal gases. Categorically then this underestimate
is at least an order of magnitude less that the often observed surface energy imbalance mentioned in the introduction.

## 2.2 Specific heat

But in many micrometeorological studies of the SEB *L*_{v}*E* is only half the story. There is also the sensible or convective heat flux,
$H={\mathit{\rho}}_{\mathrm{A}}{C}_{p\mathrm{a}}\stackrel{\mathrm{\u203e}}{{w}^{\prime}{T}^{\prime}}\equiv {\mathit{\varrho}}_{\mathrm{A}}{c}_{p\mathrm{a}}\stackrel{\mathrm{\u203e}}{{w}^{\prime}{T}^{\prime}}$, where *ρ*_{a} (kg m^{−3}) and
*ϱ*_{a} (mol m^{−3}) are the density of the ambient moist air (in mass or molar units) and *C*_{pa} ($\mathrm{J}\phantom{\rule{0.125em}{0ex}}{\mathrm{kg}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}$) and
*c*_{pa} ($\mathrm{J}\phantom{\rule{0.125em}{0ex}}{\mathrm{mol}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}$) are the specific heat of moist air at constant pressure (in units corresponding to the moist air
density). $\stackrel{\mathrm{\u203e}}{{w}^{\prime}{T}^{\prime}}$ is the kinematic heat flux, which is obtained directly from eddy covariance measurements. Assuming ideal
gases, ${c}_{p\mathrm{a}}^{\ast}={c}_{p\mathrm{a}}^{\ast}\left({T}_{\mathrm{K}}\right)={\mathit{\chi}}_{\mathrm{v}}{c}_{p\mathrm{v}}^{\ast}\left({T}_{\mathrm{K}}\right)+{\mathit{\chi}}_{\mathrm{d}}{c}_{p\mathrm{d}}^{\ast}\left({T}_{\mathrm{K}}\right)$ is the weighted sum of the specific heats of pure
water vapor (subscript v) and pure dry air (subscript d). For the present study ${c}_{p\mathrm{a}}^{\ast}$, ${c}_{p\mathrm{v}}^{\ast}\left({T}_{\mathrm{K}}\right)$, and ${c}_{p\mathrm{d}}^{\ast}\left({T}_{\mathrm{K}}\right)$
are obtained from Eq. (6) of Bücker et al. (2003).

Using Δ to denote the departure from ideality, the derivation of $\mathrm{\Delta}{c}_{p\mathrm{a}}\equiv {\mathit{\chi}}_{\mathrm{v}}\mathrm{\Delta}{c}_{p\mathrm{v}}+{\mathit{\chi}}_{\mathrm{d}}\mathrm{\Delta}{c}_{p\mathrm{d}}$ begins with
the following (standard thermodynamic) relation $d{L}_{\mathrm{v}}/\mathrm{d}T={c}_{p\mathrm{v}}-{c}_{p\mathrm{l}}$ (e.g., Curry and Webster, 1999, Eq. 4.29), where *c*_{pv} and *c*_{pl} are
the specific heats at constant pressure for water vapor (subscript v) and liquid water (subscript l). Combining this relationship, which is valid
for both ideal and non-ideal gases, with Eq. (5), it is straightforward to show that

For the present purposes it can be assumed that liquid water always remains pure (or ideal) and, therefore, ${c}_{p\mathrm{l}}={c}_{p\mathrm{l}}^{\ast}$. Then, identifying
Δ*c*_{pv} as ${c}_{p\mathrm{v}}-{c}_{p\mathrm{v}}^{\ast}$ and using Eq. (7) above, it follows from Eq. (8) that

To complete the estimate of Δ*c*_{pa} it is necessary to determine *χ*_{d}Δ*c*_{pd}, which is easily deduced from the dry air term
(d^{2}*B*_{d}∕d*T*^{2}) in Eq. (9). In this last equation, as well as in Eqs. (5) and (7),
the dry air term (any *B*_{d} term) is basically meant to account for the effects of dry air interacting with itself. Consequently, it is fairly
straightforward to conclude from Eq. (9) that

Combining this last expression for *χ*_{d}Δ*c*_{pd} with that for *χ*_{v}Δ*c*_{pv} yields the final results for
$\mathrm{\Delta}{c}_{p\mathrm{a}}/{c}_{p\mathrm{a}}^{\ast}$ as a function of *T*_{K}, which is shown in Fig. 3 overlaying $\mathrm{\Delta}{L}_{\mathrm{v}}/{L}_{\mathrm{v}}^{\ast}$ from
Fig. 2.

## 2.3 Consequences for the surface energy balance

Implications for the SEB of mixing the two non-ideal gases (water vapor and dry air) during evaporation can now be estimated by combining the results
for $\mathrm{\Delta}{L}_{\mathrm{v}}/{L}_{\mathrm{v}}^{\ast}$ and Δ*C*_{p}∕*C*_{p}. For example, assuming a Bowen ratio of approximately unity (i.e., the magnitudes of *H* and *L*_{v}*E*
are approximately the same) and a temperature between say 275 and 325 K, then the term *L*_{v}*E*+*H* in the SEB could be underestimated
between 1 % and 1.5 % with micrometeorological techniques due to the non-ideality of water vapor and dry air. Allowing for different values of the
Bowen ratio would imply a somewhat broader range of percentage underestimates. But even so, it is unlikely that non-ideality could cause *L*_{v}*E*+*H* to
be underestimated by more than 2 %, which, at best, is an order of magnitude less than required to account for the imbalance of the SEB.

*p*Δ

*v*and the surface fluxes of sensible heat and water vapor

This section examines the issue Paw U et al. (2000) address, viz., the “energy associated with evaporation into the atmosphere, necessary for the
expansion of eddy parcels against an approximately constant pressure”. In essence the authors are proposing a correction to eddy covariance
measurements of turbulent temperature fluctuations (*T*^{′}) that account for the density change of an air parcel associated with the mixing of
a relatively dense fluid (ambient air), with a relatively less dense fluid (water vapor). The following is a slight reformulation of their approach.

For an adiabatic process the first law of thermodynamics can be expressed as

where *c*_{v} ($\mathrm{J}\phantom{\rule{0.125em}{0ex}}{\mathrm{mol}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}$) is the molar specific heat of moist air at constant volume and *v*_{a} (m^{3} mol^{−1}) is the specific
volume of air, which by definition is the reciprocal of the molar air density, *ϱ*_{a} (mol m^{−3}). Switching from differential notation
to perturbation notation, Eq. (11) can be written as ${c}_{\mathrm{v}}{T}^{\prime}+{p}_{\mathrm{A}}{v}_{\mathrm{a}}^{\prime}=\mathrm{0}$. By definition, ${v}_{\mathrm{a}}=\mathrm{1}/{\mathit{\varrho}}_{\mathrm{a}}$, so
it also follows that ${v}_{\mathrm{a}}^{\prime}=-{\mathit{\varrho}}_{\mathrm{a}}^{\prime}/{\mathit{\varrho}}_{\mathrm{a}}^{\mathrm{2}}$, which combined with the ideal gas law *p*_{A}*v*_{a}=*R**T*_{K} yields the following
equivalent expression for Eq. (11):

where ${T}_{\mathrm{e}}^{\prime}$ is defined by Paw U et al. (2000) as “the temperature perturbation equivalent to the energy needed for expansion”. Next they assume that the change in molar air density, ${\mathit{\varrho}}_{\mathrm{a}}^{\prime}$, is due to the mol per mol displacement of moist air by water vapor, so that for present purposes ${\mathit{\varrho}}_{\mathrm{a}}^{\prime}=-{\mathit{\varrho}}_{\mathrm{v}}^{\prime}$, from which is follows that

where the second expression on the right is expressed in mass units (kg) rather than mols; i.e., *μ*=1.609 is the ratio of the
molecular mass of dry air to the molecular mass of water vapor, *ρ*_{v} (kg m^{−3}) is the mass density of water vapor, and *ρ*_{a}
(kg m^{−3}) is the mass density of the ambient atmosphere. Equation (13) is a rephrasing of the principal result – Eq. (C3) – of
Appendix C of Paw U et al. (2000).

Before proceeding with an alternative approach to deriving an expression for ${T}_{\mathrm{e}}^{\prime}$, it is insightful to examine an apparent sign error made
by Paw U et al. (2000) in their mathematical development of ${T}_{\mathrm{e}}^{\prime}$ and its equivalent heat flux – their Eq. (C4). First, they pose their Eq. (C2)
as the antecedent to their expression for ${T}_{\mathrm{e}}^{\prime}$ by asserting that ${v}_{\mathrm{a}}^{\prime}\propto -{\mathit{\rho}}_{\mathrm{v}}^{\prime}$ when displacing heavier dry air
molecules by lighter water vapor molecules, meaning that the specific volume perturbation should decrease; i.e., ${v}_{\mathrm{a}}^{\prime}<\mathrm{0}$. But this
contradicts the fact that the specific volume should increase when the density of the (formerly dry) air parcel decreases when displacing heavier
molecules by lighter ones. From the discussion in the paragraph immediately preceding the present one – ${v}_{\mathrm{a}}=\mathrm{1}/{\mathit{\varrho}}_{\mathrm{a}}$ implies
${v}_{\mathrm{a}}^{\prime}=-{\mathit{\varrho}}_{\mathrm{a}}^{\prime}/{\mathit{\varrho}}_{\mathrm{a}}^{\mathrm{2}}$ combined with the displacement assumption ${\mathit{\varrho}}_{\mathrm{a}}^{\prime}=-{\mathit{\varrho}}_{\mathrm{v}}^{\prime}$ – it follows
that ${v}_{\mathrm{a}}^{\prime}\propto {\mathit{\rho}}_{\mathrm{v}}^{\prime}>\mathrm{0}$, in agreement with expectations. Interestingly, despite this sign error in Eq. (C2), Paw U et al. (2000)
have the same sign for their ${T}_{\mathrm{e}}^{\prime}$ – their Eq. (C3) – as Eq. (13) above; i.e., both expressions yield ${T}_{\mathrm{e}}^{\prime}<\mathrm{0}$.
Nonetheless, and even more puzzling, is that Paw U et al. (2000) reverse the sign again when they proceed to their Eq. (C4), the succedent to their Eq. (C3).
In this step of their development of the heat flux *H*_{e}, generally defined such that ${H}_{\mathrm{e}}\propto {T}_{\mathrm{e}}^{\prime}<\mathrm{0}$, they suggest that
${H}_{\mathrm{e}}\propto -{T}_{\mathrm{e}}^{\prime}>\mathrm{0}$. The reason for reversing the sign a second time is not discussed, nor is how this might relate to the pressure work
term. But if their goal is to determine an equivalent heat flux associated with a change in density of an air parcel due to the partial displacement
of a heavier gas in that air parcel by a lighter gas, then it is reasonable to expect that *H*_{e}>0 because that air parcel would be positively
buoyant relative to the surrounding (drier and heavier) air. Assuming this conjecture is true, then this contradiction in the sign of *H*_{e} suggests
seeking an alternative approach to determine *H*_{e}. The remaining portion of this study outlines such an alternative.

The final portion of this study attempts to clarify the nature of *H*_{e} and the role of the work term and whether the surface sensible heat flux
includes a water vapor term similar to that suggested by Paw U et al. (2000) and recast as Eq. (13) above. I begin with the time-dependent
version of the first law of thermodynamics expressed as the conservation law for potential temperature, *θ* (K), for atmospheric
processes:

where d*q*∕d*t* represents the heat flow associated with diabatic atmospheric processes, ** u** (m s

^{−1}) is the atmospheric velocity, and ∇ is the vector gradient operator. Here the term associated with compressible effects,

*θ*∇⋅

**, is included for completeness, although it is not important for the present purposes. Equation (14) is standard and in and of itself is not novel. Nonetheless it does imply an important take-away, which is that the turbulent surface sensible heat flux is more correctly expressed in terms of potential temperature, $\stackrel{\mathrm{\u203e}}{{w}^{\prime}{\mathit{\theta}}^{\prime}}$, rather than in terms of temperature, $\stackrel{\mathrm{\u203e}}{{w}^{\prime}{T}^{\prime}}$. This is principally because $\stackrel{\mathrm{\u203e}}{{w}^{\prime}{\mathit{\theta}}^{\prime}}$ explicitly includes the effects of any change in ambient pressure and the concomitant work done on or to the atmosphere during turbulent atmospheric processes. Note: for the sake of completeness**

*u**w*(m s

^{−1}) is the vertical velocity and the

^{′}notation is standard and refers to Reynolds averaging. Having identified potential temperature as the key variable for discussion, the next step is to examine the influence moisture has on

*θ*.

Including the effects of water vapor on potential temperature yields the following relation (e.g., Curry and Webster, 1999, Eq. 2.66):

where *p*_{00}=100 kPa is a constant reference pressure; $\mathit{\kappa}={R}_{\mathrm{d}}/{C}_{p\mathrm{d}}$, for which *C*_{pd} is the specific heat for dry air and
consequently $\mathit{\kappa}=\mathrm{2}/\mathrm{7}$ is an extremely good approximation; and ${q}_{\mathrm{v}}={\mathit{\rho}}_{\mathrm{v}}/{\mathit{\rho}}_{\mathrm{a}}$ (kg kg^{−1}) is the specific humidity of moist air.
Equation (15) clearly indicates that *θ* is dependent on moisture. Although this dependency is extremely weak, the purpose here is
to assess the influence of ${\mathit{\rho}}_{\mathrm{v}}^{\prime}$ on the *θ*^{′} using Eq. (15) and to compare the result with
Eq. (13). A sketch of the derivation follows.

Linearize Eq. (15) first by noting that near-surface atmospheric conditions (i.e., *q*_{v}<0.04 and $\mathrm{log}({p}_{\mathrm{00}}/{p}_{\mathrm{a}})<\mathrm{0.35}$ or
0.26*q*_{v}<0.011 and $\mathit{\kappa}\mathrm{log}({p}_{\mathrm{00}}/{p}_{\mathrm{a}})<\mathrm{0.1}$) are sufficient to guarantee that $\mathrm{0.26}\mathit{\kappa}{q}_{\mathrm{v}}\mathrm{log}({p}_{\mathrm{00}}/{p}_{\mathrm{a}})<\mathrm{0.26}{q}_{\mathrm{v}}\ll \mathrm{1}$ and second by
assuming that the perturbation quantities are small compared to their background levels (which will be denoted by an overbar). This yields

where $\mathit{\alpha}=\mathrm{0.26}\mathit{\kappa}{q}_{\mathrm{v}}\mathrm{log}({p}_{\mathrm{00}}/{p}_{\mathrm{a}})$, ${\mathit{\alpha}}^{\prime}=-\mathrm{0.26}\mathit{\kappa}\stackrel{\mathrm{\u203e}}{{q}_{\mathrm{v}}}({p}_{\mathrm{a}}^{\prime}/\stackrel{\mathrm{\u203e}}{{p}_{\mathrm{a}}})+\mathrm{0.26}\mathit{\kappa}\mathrm{log}({p}_{\mathrm{00}}/\stackrel{\mathrm{\u203e}}{{p}_{\mathrm{a}}}){q}_{\mathrm{v}}^{\prime}$, and for later use
$\stackrel{\mathrm{\u203e}}{\mathit{\alpha}}=\mathrm{0.26}\mathit{\kappa}\stackrel{\mathrm{\u203e}}{{q}_{\mathrm{v}}}\mathrm{log}({p}_{\mathrm{00}}/\stackrel{\mathrm{\u203e}}{{p}_{\mathrm{a}}})$. Substituting *α*^{′} into Eq. (16) yields

where $\mathit{\gamma}=\mathrm{0.26}\mathit{\kappa}\mathrm{log}({p}_{\mathrm{00}}/\stackrel{\mathrm{\u203e}}{{p}_{\mathrm{a}}})<\mathrm{0.028}\ll \mathrm{1}$. Next is the evaluation of ${q}_{\mathrm{v}}^{\prime}$ by expanding and linearizing ${q}_{\mathrm{v}}={\mathit{\rho}}_{\mathrm{v}}/{\mathit{\rho}}_{\mathrm{a}}$ in terms of ${\mathit{\rho}}_{\mathrm{v}}^{\prime}$ and ${\mathit{\rho}}_{\mathrm{a}}^{\prime}$. This yields ${q}_{\mathrm{v}}^{\prime}={\mathit{\rho}}_{\mathrm{v}}^{\prime}/\stackrel{\mathrm{\u203e}}{{\mathit{\rho}}_{\mathrm{a}}}-\stackrel{\mathrm{\u203e}}{{q}_{\mathrm{v}}}({\mathit{\rho}}_{\mathrm{a}}^{\prime}/\stackrel{\mathrm{\u203e}}{{\mathit{\rho}}_{\mathrm{a}}})$. The ideal gas law for ambient air yields ${\mathit{\rho}}_{\mathrm{a}}^{\prime}/\stackrel{\mathrm{\u203e}}{{\mathit{\rho}}_{\mathrm{a}}}={p}_{\mathrm{a}}^{\prime}/\stackrel{\mathrm{\u203e}}{{p}_{\mathrm{a}}}-{T}^{\prime}/\stackrel{\mathrm{\u203e}}{{T}_{\mathrm{K}}}$ and, therefore, ${q}_{\mathrm{v}}^{\prime}={\mathit{\rho}}_{\mathrm{v}}^{\prime}/\stackrel{\mathrm{\u203e}}{{\mathit{\rho}}_{\mathrm{a}}}-\stackrel{\mathrm{\u203e}}{{q}_{\mathrm{v}}}({p}_{\mathrm{a}}^{\prime}/\stackrel{\mathrm{\u203e}}{{p}_{\mathrm{a}}})+\stackrel{\mathrm{\u203e}}{{q}_{\mathrm{v}}}({T}^{\prime}/\stackrel{\mathrm{\u203e}}{{T}_{\mathrm{K}}})$. Substituting this last expression for ${q}_{\mathrm{v}}^{\prime}$ into Eq. (17) yields

or

where $\mathit{\beta}=\mathrm{0.26}\stackrel{\mathrm{\u203e}}{{q}_{\mathrm{v}}}[\mathrm{1}+\mathrm{log}({p}_{\mathrm{00}}/\stackrel{\mathrm{\u203e}}{{p}_{\mathrm{a}}}\left)\right]<\mathrm{0.015}\ll \mathrm{1}$. At this point it is important to reiterate that for near-surface conditions $\stackrel{\mathrm{\u203e}}{\mathit{\alpha}}<\mathrm{0.26}\stackrel{\mathrm{\u203e}}{{q}_{\mathrm{v}}}<\mathit{\beta}<\mathrm{0.015}\ll \mathrm{1}$.

Equation (19) suggests that water vapor contributes two different “corrections” to the kinematic heat flux. First, the middle
term on the right-hand side of this equation is due to the overall presence of water vapor, $\stackrel{\mathrm{\u203e}}{{q}_{\mathrm{v}}}$, and second, the last term on the
right-hand side of Eq. (19) and the term of interest in this study, results from fluctuations in water vapor,
${\mathit{\rho}}_{\mathrm{v}}^{\prime}$. Although Eqs. (13) and (19) have somewhat different definitions of heat flux, it is still
possible to assess the appropriateness of the displacement assumption made by Paw U et al. (2000) by numerically comparing the dimensionless coefficient
*μ**R*∕*c*_{v} in Eq. (13) with *γ* in Eq. (19). Noting that $R/{c}_{\mathrm{v}}=\mathrm{2}/\mathrm{5}$, then
$\mathit{\mu}R/{c}_{\mathrm{v}}\approx \mathrm{0.644}\approx \mathrm{24}\mathit{\gamma}$. In other words, $\mathit{\gamma}\ll \mathit{\mu}R/{c}_{\mathrm{v}}$ and, therefore, the approach followed by Paw U et al. (2000) predicts
significantly more turbulent heat flux associated with the water vapor flux than does the approach based on potential temperature (initiated above
with Eq. 14). Even allowing for the difference between potential temperature and *T*_{K} does not really change this result by more
than 10 % because $({p}_{\mathrm{00}}/{p}_{\mathrm{a}}{)}^{\mathit{\kappa}}<\mathrm{1.1}$ for conditions being considered here.

This difference between Paw U et al. (2000) and the present result is made more explicit by comparing the next two expressions. The first expression derives from combining Eq. (13) for ${T}_{\mathrm{e}}^{\prime}$ with the equivalent heat flux, ${H}_{\mathrm{e}}=\stackrel{\mathrm{\u203e}}{{\mathit{\rho}}_{\mathrm{a}}}{C}_{p}\stackrel{\mathrm{\u203e}}{{w}^{\prime}{T}_{\mathrm{e}}^{\prime}}$, from Paw U et al. (2000). This yields the following generalization of the Paw U et al. (2000) result:

where *R*_{d}=287 $\mathrm{J}\phantom{\rule{0.125em}{0ex}}{\mathrm{kg}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}$. The second expression results by identifying the equivalent potential temperature, ${\mathit{\theta}}_{\mathrm{e}}^{\prime}$,
associated with the water vapor perturbation, ${\mathit{\rho}}_{\mathrm{v}}^{\prime}$, in Eq. (19), i.e.,
${\mathit{\theta}}_{\mathrm{e}}^{\prime}=-\stackrel{\mathrm{\u203e}}{\mathit{\theta}}(\mathit{\gamma}{\mathit{\rho}}_{\mathrm{v}}^{\prime}/\stackrel{\mathrm{\u203e}}{{\mathit{\rho}}_{\mathrm{a}}})$, and combining it with the expression for the equivalent heat flux,
*H*_{e}, appropriate to Eq. (14), i.e., ${H}_{\mathrm{e}}=\stackrel{\mathrm{\u203e}}{{\mathit{\rho}}_{\mathrm{a}}}{C}_{p}\stackrel{\mathrm{\u203e}}{{w}^{\prime}{\mathit{\theta}}_{\mathrm{e}}^{\prime}}$. This yields

It is not possible to reconcile these two expressions, which brings into question the validity of the displacement assumption of Paw U et al. (2000) (i.e., ${\mathit{\varrho}}_{\mathrm{a}}^{\prime}=-{\mathit{\varrho}}_{\mathrm{v}}^{\prime}$), on which Eq. (20) is based. But to truly assess the cogency of this assumption and any enthalpic changes associated with mixing of the dry air and water vapor requires a better description of the physical processes and the initial and final states involved than Paw U et al. (2000) provide. Nevertheless, since they are addressing evapotranspiration, it seems reasonable to assume they are envisioning the final state of the evaporative process. In this case the work done to/by the atmosphere associated with the expansion of water vapor into the atmosphere is appropriately included in the enthalpy of vaporization as previously discussed and, consequently, the displacement assumption would result in over-counting the work term. On the other hand, if they are describing the enthalpic changes associated with rising plumes of warm moist air associated with density differences between very moist air near the surface and drier and therefore, denser air above the near surface, then the methods and results outlined by Eqs. (14), (15), and (21) above are more appropriate.

The present study has explored some of the issues involving the surface energy balance (SEB) and the thermodynamics of evaporation of water into the
atmosphere. Specifically I have looked at (a) the influence that molecular interactions between water vapor and dry air (non-ideality of atmospheric
gases) could have on estimates of *L*_{v} and *C*_{p} and the SEB and (b) the impact that fluctuations of atmospheric water vapor could have on the
surface heat flux. At typical atmospheric temperatures (275–325 K), the influence of the first effect is probably on the order of about
1 %–2 % and the second is about 0.4 %. Consequently, these phenomena acting either independently or in consort are too small to be of any real
significance in explaining the lack of closure of the SEB. This result should not be surprising, but because these issues may not be well known to the
micrometeorological and geo-biophysical communities, it seemed worthwhile to attempt to verify this supposition quantitatively.

This Appendix derives the relationship between *δ**T* and *N*_{v}, *N*_{d} and *N*_{l} appropriate to evaporative cooling of the isolated thermodynamic
system discussed in Sect. 2.1 of the main text. Achieving this requires an approach similar to that used when calculating the wet bulb temperature
(e.g., Curry and Webster, 1999). The formal expression for the first law of thermodynamics for the system under consideration is

where the subscripts f and i refer to the final and initial states; *Q*_{f}−*Q*_{i} (J) is the total heat exchanged by the system and its
environment, which must be 0 since the system is isolated from its environment; *C*_{ps} (J K^{−1}) is the bulk heat capacity of the composite
system (vapor + dry air + pure liquid water) at constant pressure so that the change in the heat content of the system,
(*C*_{ps}*T*_{K})_{f}−(*C*_{ps}*T*_{K})_{i}, must exactly cancel the change in the enthalpy of the system $({N}_{\mathrm{v}}{M}_{\mathrm{v}}{L}_{\mathrm{v}}^{\ast}{)}_{\mathrm{f}}-({N}_{\mathrm{v}}{M}_{\mathrm{v}}{L}_{\mathrm{v}}^{\ast}{)}_{\mathrm{i}}$, which is
expressed here in terms of the water vapor component. ${L}_{\mathrm{v}}^{\ast}$ assumes that water vapor is an ideal gas (an assumption that is sufficient for the
present purposes) and *M*_{v} (kg mol^{−1}) the molecular mass of water vapor. Simplifying this expression begins by identifying

and

where *N* refers to the number of mols of any particular component (subscript d for dry air, v for vapor, and l for liquid); *M* refers
to the molecular mass of that component; *c*_{p} ($\mathrm{J}\phantom{\rule{0.125em}{0ex}}{\mathrm{kg}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}$) refers to the specific heat at constant pressure of that component; and
*M*_{l}=*M*_{v} has been used for the right-hand side of the last expression.

Combining these last two expressions with *N*_{vi}=0 and after dividing the resulting expression by *M*_{v} yields

where the *δ**T* term on the left-hand side and the term ${N}_{\mathrm{v}}{L}_{\mathrm{v}}^{\ast}$ on the right-hand side are evaluated at *T*_{K} (the final temperature), and
the first term on the right-hand side, $-{N}_{\mathrm{v}}({c}_{p\mathrm{v}}-{c}_{p\mathrm{l}}){T}_{\mathrm{Ki}}$, is evaluated at the initial temperature *T*_{Ki}. The order of magnitude
calculation is facilitated by dividing the last expression by *c*_{pl}*T*_{Ki}, by noting that ${M}_{\mathrm{d}}/{M}_{\mathrm{v}}\approx \mathrm{1.6}$ and that
${c}_{p\mathrm{l}}\approx \mathrm{2}{c}_{p\mathrm{v}}\approx \mathrm{4}{c}_{p\mathrm{d}}$ and by ignoring the relatively weak temperature dependency of the various *c*_{p}s. This yields

The last step to deriving $\mathit{\delta}T=\mathit{\delta}T({N}_{\mathrm{v}},{N}_{\mathrm{d}},{N}_{\mathrm{l}})$ assumes that ${L}_{\mathrm{v}}^{\ast}\approx \mathrm{2.5}\times {\mathrm{10}}^{\mathrm{6}}$ J kg^{−1} and requires noting
that for the temperature range $\mathrm{295}\phantom{\rule{0.125em}{0ex}}\mathrm{K}\le {T}_{\mathrm{Ki}}\le \mathrm{325}\phantom{\rule{0.125em}{0ex}}\mathrm{K}$, ${c}_{p\mathrm{l}}\approx \mathrm{4.186}\times {\mathrm{10}}^{\mathrm{3}}$ $\mathrm{J}\phantom{\rule{0.125em}{0ex}}{\mathrm{kg}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}$ and
${c}_{p\mathrm{l}}{T}_{\mathrm{K}}\approx \left(\mathrm{1.24}\text{\u2013}\mathrm{1.36}\right)\times {\mathrm{10}}^{\mathrm{6}}$ J kg^{−1}. These last conditions yield the final result:

For the isolated thermodynamic system discussed in the present study it is also valid to assume that *N*_{v}≪*N*_{d}, which in turn is sufficient to
guarantee that *δ**T*≪*T*_{Ki}, meaning that the temperature change associated with evaporative cooling should be quite small.

The computer code used in this study was developed using MatLab version 2017b and is publicly available along with any output data at the Forest Service Research Data Archive: https://doi.org/10.2737/RDS-2019-0042 (Massman, 2020).

The author declares that there is no conflict of interest.

The author would like to thank Allan Harvey for his insights and his significant time and efforts involving Sect. 2 of this work. I also gratefully acknowledge Kyaw Tha Paw U, Grant Petty, Jim Wilczak, and Ned Patton for their comments during the development of this work.

This paper was edited by Stan Schymanski and reviewed by Andrew Kowalski, Grant Petty, and one anonymous referee.

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