the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Relative humidity gradients as a key constraint on terrestrial water and energy fluxes

### Monica Garcia

### Laura Morillas

### Ulrich Weber

### T. Andrew Black

### Mark S. Johnson

Earth's climate and water cycle are highly dependent on terrestrial evapotranspiration and the associated flux of latent heat. Although it has been hypothesized for over 50 years that land dryness becomes embedded in atmospheric conditions through evaporation, underlying physical mechanisms for this land–atmosphere coupling remain elusive. Here, we use a novel physically based evaporation model to demonstrate that near-surface atmospheric relative humidity (RH) fundamentally coevolves with RH at the land surface. The new model expresses the latent heat flux as a combination of thermodynamic processes in the atmospheric surface layer. Our approach is similar to the Penman–Monteith equation but uses only routinely measured abiotic variables, avoiding the need to parameterize surface resistance. We applied our new model to 212 in situ eddy covariance sites around the globe and to the FLUXCOM global-scale evaporation product to partition observed evaporation into diabatic vs. adiabatic thermodynamic processes. Vertical RH gradients were widely observed to be near zero on daily to yearly timescales for local as well as global scales, implying an emergent land–atmosphere equilibrium. This equilibrium allows for accurate evaporation estimates using only the atmospheric state and radiative energy, regardless of land surface conditions and vegetation controls. Our results also demonstrate that the latent heat portion of available energy (i.e., evaporative fraction) at local scales is mainly controlled by the vertical RH gradient. By demonstrating how land surface conditions become encoded in the atmospheric state, this study will improve our fundamental understanding of Earth's climate and the terrestrial water cycle.

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Latent heat flux (LE) associated with plant transpiration and evaporation from soil and intercepted water (i.e., evapotranspiration, ET) links the water cycle with the terrestrial energy budget. More than half of the incoming radiation energy at the land surface is consumed as LE, making ET the second largest flux in the terrestrial water balance after precipitation (Oki and Kanae, 2006). Also, LE is a controlling factor for near-surface climatic conditions such as temperature and humidity (Ma et al., 2018; Byrne and O'Gorman, 2016). While most research has been devoted to developing and improving rate-limiting parameters constraining LE (e.g., García et al., 2013; Martens et al., 2017), exploring the governing physics of LE has received less attention following earlier pioneering work (Schmidt, 1915; Penman, 1948; Bouchet, 1963; Monteith, 1965; Priestley and Taylor, 1972). Nevertheless, improvement of the theoretical understanding of LE still remains an essential cornerstone to correctly simulate and predict climate and hydrological cycles (Emanuel, 2020).

Climatic conditions over the land surface have been getting not only warmer but also drier in recent decades (i.e., decrease in relative humidity) (Sherwood and Fu, 2014; Willett et al., 2014; Byrne and O'Gorman, 2018), but land–atmosphere feedback processes shaping the near-surface atmospheric state are still not well understood. In the early 1960s, Bouchet (1963) hypothesized that land surface dryness is coupled to the atmospheric state through LE, with the Bouchet hypothesis now widely accepted (Ramírez et al., 2005; Fisher et al., 2008; Mallick et al., 2014). However, the underlying physical mechanisms for this land–atmosphere coupling still remain elusive (McNaughton and Spriggs, 1989). Recently, McColl et al. (2019) introduced a novel theoretical perspective on land–atmosphere coupling which is referred to as “surface flux equilibrium (SFE)”. They hypothesized that relative humidity (RH) reaches a steady-state value in an idealized atmospheric boundary layer at daily to monthly timescales. Under steady RH conditions (i.e., the SFE state), LE can be determined using only the atmospheric state and radiative energy. Although this method performed well compared to actual LE observations for inland continental sites (McColl and Rigden, 2020; Chen et al., 2021), a further investigation is needed to understand how dynamics of turbulent heat fluxes in the atmospheric surface layer at sub-daily timescales evolve to the SFE state.

A traditional way to express the atmospheric surface layer processes is to partition LE into diabatic and adiabatic processes using the Penman–Monteith (PM) equation (Monteith, 1965), as proposed by Monteith (1981). The PM equation combines the energy balance equation with mass-transfer theory for water vapour and sensible heat, resulting in diabatic (radiative energy-related) and adiabatic (vapour pressure deficit-related) processes for a parcel of air in contact with a saturated surface (Monteith, 1981).

where *S* is the linearized slope of saturation vapour pressure versus
temperature (hPa K^{−1}), *γ* is the psychrometric constant (hPa K^{−1}), *ρ* is the air density (kg m^{−3}), *c*_{p} is the specific heat
capacity of air at constant pressure (MJ kg^{−1} K^{−1}), and *Q* is
available radiative energy (i.e., the difference between net radiation
(*R*_{n}) and soil heat flux (*G*) expressed in units of W m^{−2}). *e*^{∗}(*T*_{a}) is the saturation vapour pressure (hPa) corresponding to the air temperature (*T*_{a}) measured at a reference height (typically 2 m or eddy
flux measurement height), and *e*_{a} is vapour pressure (hPa) at the
reference height. ${e}^{\ast}\left({T}_{\mathrm{a}}\right)-{e}_{\mathrm{a}}$ is known as atmospheric vapour pressure deficit (VPD, expressed in units of hPa). *r*_{a} is aerodynamic resistance to heat and water vapour transfer (s m^{−1}), and *r*_{s} is surface
resistance to water vapour transfer (s m^{−1}) representing drying soil
and/or plant stomatal closure.

In principle, high *Q* and VPD at the reference height increase the diabatic
and the adiabatic terms respectively in the PM equation, and as such, *Q* and
VPD are the two primary drivers of evaporation (Monteith and Unsworth,
2013). Yet, this “high VPD leads to high LE” interpretation cannot be generalized because *r*_{s} increases with VPD due to stomatal closure by
vegetation under high-VPD conditions (Tan et al., 1978; Novick et al.,
2016; Massmann et al., 2019). While the PM equation is useful to explore
biological control of LE through *r*_{s} (Jarvis and McNaughton, 1986; Peng et al., 2019), physical mechanisms corresponding to each term in Eq. (1) are less intuitive due to the sensitivity of *r*_{s} to VPD. As a
result, how the atmospheric state affects evaporation and vice versa remains
ambiguous in the PM equation.

Is there a way to mathematically express the physical mechanisms of LE without requiring *r*_{s}? In this paper, we present a pair of equations expressing actual LE as a combination of diabatic and adiabatic processes without requiring *r*_{s}. Similar to the PM equation, our new equations are derived
by combining the energy balance equation with the flux gradient equations,
but crucially ours do not include *r*_{s}. The novel equations are applied empirically to eddy-covariance observation sites and a global LE dataset to
explore land–atmosphere coupling processes at various spatiotemporal scales.
To do this, we decomposed observed LE into adiabatic and diabatic components
and discuss how these patterns can help to understand land–atmosphere
interactions and potential responses under future climatic conditions.

## 2.1 A pair of evaporation equations for an unsaturated surface

In this section, we derive a pair of evaporation equations for an
unsaturated surface. In this derivation, we assume a horizontally homogenous
landscape where the sources of water vapour and heat are identical. Under
this idealized condition, aerodynamic resistances (*r*_{a}) to heat and water
vapour transfers are identical. Here, *r*_{a} is a parameterization of
turbulent mixing due to mechanical turbulence and buoyancy driven by surface
heating.

We first express LE using a flux gradient equation as $\mathit{LE}=\frac{\mathit{\rho}{c}_{p}}{\mathit{\gamma}}\frac{{e}_{\mathrm{s}}-{e}_{\mathrm{a}}}{{r}_{\mathrm{a}}}$, where *e*_{s} is the
surface vapour pressure. Here, the subscript *s* indicates the land surface
which is defined as an idealized plane specified as the sum of displacement
height and roughness length for heat (Knauer et al., 2018a; Novick and
Katul, 2020). If the land surface is saturated, *e*_{s} becomes equivalent to the saturation vapour pressure (i.e., ${e}_{\mathrm{s}}={e}^{\ast}\left({T}_{\mathrm{s}}\right)$). For an unsaturated land surface, however, relative humidity
should be introduced as ${e}_{\mathrm{s}}={\mathrm{RH}}_{\mathrm{s}}{e}^{\ast}\left({T}_{\mathrm{s}}\right)$, where RH_{s} is surface relative humidity, i.e., the ratio of
*e*_{s} to *e*^{∗}(*T*_{s}). For a vegetated
surface, RH_{s} as defined in this study represents relative humidity of the
foliage surface and is conceptually equivalent to surface water availability
in Li and Wang (2019). For a bare soil land surface, RH_{s}
represents soil surface relative humidity, which can be found using the
“alpha” method that is parameterized using soil moisture content or soil
water potential (Lee and Pielke, 1992; Wu et al., 2000; Cuxart and Boone,
2020). Using RH_{s}, LE can be written as $\mathit{LE}=\frac{\mathit{\rho}{c}_{p}}{\mathit{\gamma}}\frac{{\mathrm{RH}}_{\mathrm{s}}{e}^{\ast}\left({T}_{\mathrm{s}}\right)-{e}_{\mathrm{a}}}{{r}_{\mathrm{a}}}$ for an unsaturated surface condition.

In order to decompose LE into two individual fluxes related to temperature and relative humidity gradients, we add $-{\mathrm{RH}}_{\mathrm{s}}{e}^{\ast}\left({T}_{\mathrm{a}}\right)+{\mathrm{RH}}_{\mathrm{s}}{e}^{\ast}\left({T}_{\mathrm{a}}\right)$ (or $-{\mathrm{RH}}_{\mathrm{a}}{e}^{\ast}\left({T}_{\mathrm{s}}\right)+{\mathrm{RH}}_{\mathrm{a}}{e}^{\ast}\left({T}_{\mathrm{s}}\right)$) to the numerator of the flux gradient equation and rewrite LE as follows.

We then approximate ${e}^{\ast}\left({T}_{\mathrm{s}}\right)-{e}^{\ast}\left({T}_{\mathrm{a}}\right)=S({T}_{\mathrm{s}}-{T}_{\mathrm{a}})$ using the saturation vapour pressure slope
at the air temperature (*S*), and we introduce a flux gradient equation for
sensible heat flux (i.e., $H=\mathit{\rho}{c}_{p}\frac{{T}_{\mathrm{s}}-{T}_{\mathrm{a}}}{{r}_{\mathrm{a}}}$)
into Eqs. (2) and (3). Then, the energy balance equation is combined to
substitute *H* with *Q*−*LE*. As results, LE can be expressed as follows:

where LE_{Q} (and LE${}_{Q}^{\prime}$) is a diabatic component, and LE_{G} (and LE${}_{G}^{\prime}$) is an adiabatic component of latent heat flux. While the diabatic
component is mainly determined by available energy (*Q*), the adiabatic
component is driven by turbulent mixing and vertical gradient of RH.
Monteith (1981) originally suggested an equation equivalent to
Eq. (4) for the case when the surface does not reach saturation. To our
knowledge, Eq. (5) is derived for the first time here. Equations (4) and (5)
include RH_{s} to compensate for eliminating *r*_{s} from the original PM
equation.

Since the adiabatic processes in Eqs. (4) and (5) are controlled by the
vertical difference of RH, we refer to Eqs. (4) and (5) as the proposed
PM_{RH} model (Penman–Monteith equation expressed using RH) to distinguish
it from the original PM model. The two Eqs. (4) and (5) are
complementary to each other in that they represent distinct thermodynamic
paths, each of which will be discussed in the next section. Arguably,
applying PM_{RH} can provide new insights into the fundamental mechanisms
of LE, particularly when it is decomposed into its diabatic component
(LE_{Q} or LE${}_{Q}^{\prime}$) and its adiabatic component (LE_{G} or LE${}_{G}^{\prime}$). In the
following sections, we will discuss theoretical meanings of Eqs. (4) and (5)
in depth.

## 2.2 Generalized Penman equation

Before discussing PM_{RH} in depth, we revisit the Penman equation
(Penman, 1948) to help with the physical reasoning behind our
proposed framework. The widely recognized form of the Penman equation, which
was developed as an LE model for a saturated surface, is as follows:

We rearrange this formulation to derive Eq. (7) by factoring out
*e*^{∗}(*T*_{a}) and introducing
${\mathrm{RH}}_{\mathrm{a}}=\frac{{e}_{\mathrm{a}}}{{e}^{\ast}\left({T}_{\mathrm{a}}\right)}$
into the second term.

Equations (6) and (7) are mathematically equivalent, but their
interpretations are quite different. In Eq. (6), the adiabatic process is
controlled by VPD at the reference height. However, in Eq. (7), the
adiabatic process acts over the vertical RH gradient, i.e., the difference in
RH from the surface to the reference height (RH_{a}). Since the Penman equation is a model for saturated surfaces, 1−RH_{a} in Eq. (7)
indicates the difference in RH over the vertical distance between the ground
surface and the reference height. Arguably, Eq. (7) is more
thermodynamically sound compared to Eq. (6) since RH is an ideal-gas
approximation to the water activity (Lovell-Smith et
al., 2015) which represents the chemical potential of water (*μ*_{w})
(Monteith and Unsworth, 2013; Kleidon and Schymanski, 2008). When
the vertical gradient of RH dissipates owing to well-developed turbulence, the land surface and the atmosphere are in thermodynamic equilibrium
(Kleidon et al., 2009). Therefore, taking Eq. (7) instead of Eq. (6) allows us to view the adiabatic process of the Penman model as an
equilibration process driving land–atmosphere equilibrium by bringing the
surface *μ*_{w} to that of the atmosphere.

As with our interpretation of the Penman model, we can view Eqs. (4) and
(5) as a generalized form of the Penman model. Here, the LE_{G} (or LE${}_{G}^{\prime}$) term is an equilibration process between the land and the atmosphere when the land surface is not saturated. It is worth noting that LE_{G} can be negative when RH_{s} is less than RH_{a}. Thus, the LE_{G} term operated
by turbulent mixing acts to reduce the vertical RH gradient. This physical
interpretation is consistent with recent findings that the variance of the
RH gradient tends to be minimized over the course of the day, implying that
the difference between RH_{s} and RH_{a} is reduced (Salvucci and
Gentine, 2013; Rigden and Salvucci, 2015). The diabatic LE_{Q} (or
LE${}_{Q}^{\prime}$) term can be understood as equilibrium LE for an unsaturated surface,
which we discuss later in Sect. 2.4.

## 2.3 Thermodynamic paths

How can we interpret the two formulas of PM_{RH} in Eqs. (4) and (5)? To
explain the two forms, the psychrometric relationship is applied to a parcel
of air near an unsaturated land surface that is under constant pressure and
steadily receiving radiation energy. The psychrometric diagram in Fig. 1
describes the magnitude of turbulent flux (where the length of the arrow
corresponds to the magnitude) from the view point of a parcel of air located
at a reference height (an approach based on work by Monteith, 1981). Since the parcel of air receives heat and water vapour from the
land surface, the final state is represented by the surface condition, while
the initial state is represented by the atmospheric conditions at the
reference height. Therefore, the initial thermodynamic state of the air
parcel can be represented by its temperature and water vapour pressure such
as point A in Fig. 1. The initial state is changed by two processes as
follows: (1) equilibrating between the land surface (RH_{s}) and the air parcel (RH_{a}) and (2) increasing enthalpy forced by the incoming energy.
It should be noted that the changing process (i.e., thermodynamic path) from
the initial to the final states in this discussion should be understood as
the magnitude of the turbulent heat fluxes.

In the RH equilibrating process, the air parcel is adiabatically cooled (or
heated when RH_{s} < RH_{a}) due to turbulent mixing, while the
enthalpy of the parcel is not changed. Therefore, the increase (decrease) in
latent heat content in the parcel is exactly balanced by a decrease
(increase) in sensible heat (*A*→*B* in Fig. 1: trajectory along constant
enthalpy line). This process is equivalent to the LE_{G} term in Eq. (4).
Now, the air parcel is in thermodynamic equilibrium with the land surface
(point B in Fig. 1). Then, the air parcel receives energy while the
equilibrium is sustained (i.e., RH_{s} is steady), which increases both the temperature and absolute water vapour content of the air parcel (*B*→*C* in Fig. 1). This process can be expressed as LE_{Q} of Eq. (4). Consequently,
the thermodynamic state of the air parcel approaches point C in Fig. 1.

However, we should recognize that temperature and vapour pressure are
“state” variables, meaning that they do not depend on the thermodynamic
path by which the system arrived at its final state (Iribarne and Godson,
1981). In the above example, we conceptually followed the adiabatic process
first and then the diabatic process (Path 1 in Fig. 1), but one can imagine
the opposite order. If we choose Path 2 in Fig. 1, the diabatic process
comes first, and thus RH_{a} instead of RH_{s} is preserved while enthalpy increases (i.e., LE${}_{Q}^{\prime}$), and the adiabatic process is followed at temperature of *T*_{S} (i.e., LE${}_{G}^{\prime}$). Path 2 is described by Eq. (5).

Therefore, one can interpret the two forms of PM_{RH} in Eqs. (4) and (5)
as two thermodynamic paths where the diabatic and adiabatic processes occur
simultaneously. It should be noted that the diabatic and adiabatic processes
in PM_{RH} are “path” functions and thus they vary by path. For
instance, LE_{Q} is slightly higher than LE${}_{Q}^{\prime}$ when RH_{s} > RH_{a}. Also, the absolute magnitude of LE${}_{G}^{\prime}$ is always bigger than that of
LE_{G} when *Q*>0 (i.e., vector ${B}^{\prime}\to C$ is longer than vector *A*→*B* in Fig. 1).

## 2.4 Equilibrium LE for an unsaturated surface

Another distinct characteristic of the PM_{RH} model is the way it defines
equilibrium at the land–atmosphere interface. Unlike many previous studies
which focused on the steady state of VPD (McNaughton and Jarvis,
1983; Priestley and Taylor, 1972; Raupach, 2001), land–atmosphere equilibrium
is achieved in the PM_{RH} model when the vertical RH gradient (i.e., the *μ*_{w} gradient) dissipates. That is, if RH_{s}≈RH_{a}, then it follows that LE_{G} (or LE${}_{G}^{\prime}$) is zero and thus LE
becomes

We note that Eq. (8) is identical to the SFE theory recently introduced by
McColl et al. (2019). They hypothesized that in many
continental regions, the near-surface atmosphere is in a state of equilibrium,
where RH is steady with time in an idealized atmospheric boundary layer at
longer than daily timescales. Equation (8) successfully predicted observed
LE at daily and monthly timescales for inland regions (McColl and Rigden,
2020; Chen et al., 2021), which implies the vertical RH gradient tends to
evolve toward zero at longer timescales than the sub-daily scale. This is
logical in that LE_{G} itself diminishes the vertical RH gradient over the
course of a day.

From a different standpoint, if an observed LE is bigger or smaller than Eq. (8) at a longer timescale such as monthly, it may indicate that the land surface conditions are not completely embedded in the near-surface
atmospheric state due to highly wet or dry land conditions. Therefore,
LE_{G} (or LE${}_{G}^{\prime}$) value and sign at monthly timescales could be a useful
indicator reflecting land surface dryness relative to the atmosphere.

When both land surface and atmosphere are saturated (i.e., ${\mathrm{RH}}_{\mathrm{s}}\approx {\mathrm{RH}}_{\mathrm{a}}\approx \mathrm{1}$), Eq. (8) becomes classical equilibrium LE (i.e., $\mathit{LE}\approx \frac{S}{S+\mathit{\gamma}}Q$). This is consistent with one of the classical definitions of equilibrium LE that defines equilibrium LE as evaporation from a saturated surface into saturated air (Schmidt, 1915; Eichinger et al., 1996; Raupach, 2001; McColl, 2020). Therefore, we can regard Eq. (8) as a generalized equilibrium LE for an unsaturated surface.

In the following sections, we present a novel physical decomposition of LE from PM_{RH} into LE_{Q} and LE_{G} components to aid in understanding the governing physics of LE. Also, the proportion of net available energy consumed
in evapotranspiration, known as the evaporative fraction (EF)
$\left(\mathrm{EF}=\frac{\mathit{LE}}{Q}\right)$ is decomposed into $\frac{{\mathit{LE}}_{Q}}{Q}$ and
$\frac{{\mathit{LE}}_{G}}{Q}$. We conducted a detailed diagnostic analysis of the PM_{RH} model using the multi-year record of an eddy covariance (EC) flux observation site located in a wet–dry tropical climate. We also applied the PM_{RH} model to the 212 EC sites represented in the FLUXNET2015 dataset
(Pastorello et al., 2020) and to the FLUXCOM global LE product
(Jung et al., 2019). We describe the local and global datasets
and analysis methods here before presenting the results.

## 3.1 In situ EC flux observation

In situ half-hourly EC observations used in this study were made from 2015
to 2018 on a ratoon sugarcane farm in the province of Guanacaste, Costa Rica
(10^{∘}25^{′}07.60^{′′} N, 85^{∘}28^{′}22.22^{′′} W). The
site has a wet–dry tropical climate with a dry season from December to March
and a median monthly air temperature ranging from 27 to 30 ^{∘}C. The study site experienced a significant drought in 2015 as
the lowest precipitation rate in Fig. 2b (Hund et al., 2018; Morillas
et al., 2019). The site was irrigated occasionally during dry seasons via
furrow irrigation events, except for 2016 when there was no irrigation due
to crop replanting. Due to the ratooning practice (i.e., sugarcane cutting
each year followed by resprouting without replanting, detailed explanation
in the Supplement), the sugarcane growing seasons varied by year, which
provided an opportunity to explore distinct and varied combinations of land
surface vs. atmospheric aridity conditions.

The measured LE and sensible heat flux (*H*) were quality controlled following Morillas et al. (2019) (details in the Supplement). For the
study period, the surface energy balance closure (i.e.,
$\frac{\mathit{LE}+H}{{R}_{\mathrm{n}}-G}$) of 30 min data was 86 %, which is typical of high-quality eddy-covariance datasets (Wilson et al., 2002).
When canopy height was less than 1 m, the surface energy balance was almost
closed (97 %), whereas the closure was 83 % when canopy height was
higher than 1 m. It is expected that unmeasured canopy and soil heat
storages in this site are significant because the sugarcane canopy grew up
to 3.6 m tall with a dense canopy. For instance, Meyers and Hollinger (2004) showed that storage term comprised 14 % of net radiation for a
maize field with a 3 m canopy height and 8 % of net radiation for a
soybean field with a 0.9 m canopy height, implying larger heat storage
capacities for taller crop canopies. Also, since our study site is located
within a homogenous landscape (Fig. S1 in the Supplement), horizontal and vertical advective
flux divergence and the influence of secondary circulations on the energy
balance closure may be marginal (Mauder et al., 2020; Leuning et al.,
2012). Therefore, considering the homogenous landscape of the study site as
well as a possible significant role of unmeasured canopy and soil heat
storages, we did not force the energy closure. Consequently, we defined *Q* as the sum of LE and *H* instead of *R*_{n}−*G*. In doing so, we in effect attribute
the cause of the surface energy imbalance to unmeasured heat storage terms
following Moon et al. (2020).

In order to decompose LE into LE_{Q} and LE_{G}, we first estimated half-hourly aerodynamic resistance (*r*_{a}) by considering aerodynamic
resistance to momentum transfer and the additional boundary layer resistance
for heat and mass transfer (or excess resistance) (Thom, 1972; Knauer et
al., 2018a).

The first term on the right-hand side of Eq. (9) is the aerodynamic
component, and the second term is the boundary layer component. Here,
*u*_{∗} is friction velocity, *k* is the von Kármán constant (0.41),
*d* is the zero-plane displacement height (*d*=0.7*z*_{h}), *z*_{0 m} is the
roughness length for momentum (*z*_{0 m}=0.1*z*_{h}), and *ψ*_{h} is
the integrated form of the stability correction function. *z*_{h} is canopy
height based on manual measurements taken during regular maintenance visits.
*r*_{a} was estimated using the bigleaf R package (Knauer et
al., 2018a).

By rearranging Eq. (2), RH_{s} can be calculated using

Negative *H* and inaccurate *r*_{a} modelling sometimes yielded negative RH_{s} or values greater than 1, especially at nighttime. In these cases, RH_{s} was assigned the value of 1 following the approach described in the
bigleaf R package (Knauer et al., 2018a). We then
estimated LE_{Q} and LE_{G} from Eq. (4).

In order to explore the timescale of the covariances for LE ∼ LE_{Q} and LE ∼ LE_{G} in the frequency domain, we applied
wavelet coherence analysis using the WaveletComp R package (Roesch and
Schmidbauer, 2014). The package is designed to apply the continuous wavelet
transform using the Morlet wavelet, which is a popular approach to analyze
hydrological and micrometeorological datasets (Hatala et al.,
2012; Johnson et al., 2013). A total time series of half-hourly decomposed
LE for the 4-year measurement period was used to estimate localized coherence
and phase angle. The wavelet coherence can be interpreted as the local
correlation between two variables in the frequency–time domain (where red
indicates high correlation). A 0^{∘} phase angle (arrow pointing
right) indicates periods of positive correlation, while a 180^{∘}
phase angle (arrow pointing left) indicates periods of negative correlation.

## 3.2 FLUXNET2015

The daily-scale FLUXNET2015 dataset, which includes 212 empirical eddy-covariance flux tower sites around the globe (Pastorello et al., 2020), was used in this study. The turbulent heat fluxes, net radiation, soil heat flux, air temperature, relative humidity, wind speed, friction velocity, and barometric pressure were obtained from the dataset. For this analysis, we only included daily data for periods for which the quality control flag indicated more than 80 % half-hourly data were present (i.e., measured data in general, or good-quality gap-filled data in cases of partially missing data).

In order to decompose daily LE into LE_{Q} and LE_{G}, we estimated daily aerodynamic resistance (*r*_{a}) by Eq. (11) instead of Eq. (9) since canopy
and measurement heights are unknown (Thom, 1972; Knauer et al., 2018a).

where *u*(*z*_{r}) is reference height wind speed. *r*_{a} was estimated using the bigleaf R package (Knauer et al., 2018a), and RH_{s} was calculated from Eq. (8).

LE_{Q} and LE${}_{Q}^{\prime}$ were calculated using RH_{a} and RH_{s} following Eqs. (4)
and (5), and then LE_{G} and LE${}_{G}^{\prime}$ were calculated by subtracting
LE_{Q} and LE${}_{Q}^{\prime}$ from LE. To calculate LE_{Q} and LE${}_{Q}^{\prime}$, we define *Q* as LE + *H*, but it should be noted that this approach can include systematic uncertainty since the sum of LE and *H* measured by eddy covariance is typically lower than *R*_{n}−*G* (i.e., conditions referred to as the energy balance closure problem; Wilson et al., 2002). To investigate the effect of a lack of energy balance closure on resulting LE terms, we provide Fig. S2
that was generated by (1) defining *Q* as *R*_{n}−*G* and (2) correcting LE and *H* based on the assumption that the Bowen ratio ($B=H/\mathit{LE}$) is correct (Pastorello
et al., 2020).

## 3.3 FLUXCOM

The FLUXCOM dataset (Jung et al., 2019) is a global-scale
machine learning ensemble product which upscales FLUXNET observations
(Baldocchi et al., 2001) using Moderate Resolution Imaging
Spectroradiometer (MODIS) satellite data and reanalysis meteorological data.
In this study we used the monthly LE FLUXCOM dataset (0.5^{∘}
resolution) modelled using MODIS and ECMWF ERA5 reanalysis data
(Hersbach et al., 2020).

We obtained *Q* and LE from the FLUXCOM output, and air temperature and dew point temperature were retrieved from ERA5 monthly averaged data (2 m height). RH, *S*, and *γ* were calculated from ERA5 data, and then LE${}_{Q}^{\prime}$ was calculated. LE${}_{G}^{\prime}$ was then estimated by subtracting LE${}_{Q}^{\prime}$ from LE.

## 4.1 Decomposition analysis of in situ EC flux observation

Application results of the PM_{RH} model to the observed LE at an irrigated
sugarcane farm in Costa Rica are depicted in Fig. 2. The decomposition
analysis of observed LE shows that while LE_{Q} is the major component of LE, LE_{G} variability plays a non-negligible role in seasonal and interannual behaviour of *LE.* In terms of absolute magnitude, the LE_{Q} term can closely
approximate *LE,* and LE_{G} only represents 15 % of total evaporation (Fig. 2c). Also, positive coherence between LE and LE_{Q} was strong over the
entire period of observation, particularly at diurnal to multiday timescales (0.5–32 d), implying variability of LE is largely determined by LE_{Q} variability (i.e., red coloured regions in Fig. 2e).

Although absolute magnitude of LE_{G} was much smaller than that of LE_{Q}, the interannual variability of LE_{G} was larger than the interannual variability of LE_{Q} (Fig. 2c). Furthermore, LE and LE_{G} also had a strong positive correlation on longer timescales (32–365 d) (i.e., red coloured regions in Fig. 2f). Unexpectedly, a negative correlation between LE and LE_{G} at the diurnal timescale was observed in the wavelet analysis only when the land surface was
dry and there was little vegetation (i.e., after harvest) or during a year
in which there was no dry season irrigation applied. Also, we found that EF
variability is mostly determined by LE_{G} variability since the diurnal and seasonal signals of *Q* are removed from LE in EF. Interestingly, the annual mean LE_{G} was the highest in 2015, a drought year in which RH_{a} and precipitation were generally lower than for the other years, while the annual mean LE_{G} was close to zero in 2016 when there was no application
of dry season irrigation due to crop replanting.

To explore the diurnal behaviour of decomposed LE, we selected different surficial and atmospheric conditions when LE_{G} was zero, positive, or negative in Fig. 3. In the 2016 dry season, LE_{G} was close to zero as a daily average value, as a result of negative daytime and positive nighttime LE_{G} values due to dry air and dry soil conditions (no irrigation) and an undeveloped vegetation canopy (Fig. 3a). Daily LE_{G} was also close to
zero during wet season conditions (e.g., Fig. 3b). In this case,
LE_{G} was near zero during both daytime and nighttime periods due to near-saturated atmospheric and land surface conditions. These two cases show that
“dry land–dry air” or “wet land–wet air” conditions can each lead to
daily scale land–atmosphere equilibrium, although the diurnal pattern of
LE_{G} is starkly different for dry land–dry air vs. wet land–wet air
conditions.

Meanwhile, when RH_{a} was low and the canopy was well-developed, LE_{G} was found to be positive during both daytime and nighttime periods (Fig. 3c). On the other hand, during post-harvest conditions when vegetative canopy cover was minimal and air and soil moisture levels were low, daily LE_{G} was found to be negative as a result of negative daytime and positive nighttime LE_{G} (Fig. 3d). Diurnal variation in RH_{s} was maximized when daily LE_{G} was negative, implying a large diurnal fluctuation in surface temperature under the drier land surface conditions. Regarding the overall diurnal pattern, LE_{G} generally declined during the morning and
increased in the afternoon, which is consistent with the well-known diurnal
pattern of EF (Gentine et al., 2011, 2007) (Fig. 3e).

## 4.2 Decomposition analysis of the FLUXNET2015 dataset

Decomposition analysis of the daily FLUXNET2015 dataset is illustrated in
Fig. 4. In terms of absolute magnitude of each term, the majority of
LE_{G} (and LE${}_{G}^{\prime}$) values ranged from −50 to 50 W m^{−2},
with some exceptional values approaching ±100 W m^{−2}. On the
other hand, LE_{Q} (and LE${}_{Q}^{\prime}$) values ranged from 0 to 150 W m^{−2}.

One of the interesting findings from the decomposition analysis of the
FLUXNET2015 dataset was that differences between LE_{Q} and LE${}_{Q}^{\prime}$ are marginal at a daily timescale (the slope is close to 1 in Fig. 4a1).
This result implies that although the diabatic processes expressed by Eqs. (4) and (5) are different in magnitude due to the difference between
RH_{s} and RH_{a} (see Sect. 2.3), these differences are practically negligible. This is an important point since LE${}_{Q}^{\prime}$ can be determined
simply and directly using reference height meteorological measurements,
while LE_{Q} is required to know RH_{s}.

As for the adiabatic terms, LE${}_{G}^{\prime}$ is roughly 1.1 times LE_{G} at a daily
timescale (Fig. 4a2), which is consistent with the theory regarding
their respective thermodynamic paths. As we discussed in Sect. 2.3, the
absolute magnitude of LE${}_{G}^{\prime}$ must be bigger than that of LE_{G} when available energy is positive. Therefore, the empirical relationship between LE_{G} and LE${}_{G}^{\prime}$ in Fig. 4a2 is a consequence of a physical principle, and this result may provide the following empirical relationship.

Equation (12) reveals an emergent daily timescale relationship between temperature and relative humidity which has the potential to be used as a supplementary equation in future research.

Another important finding of the decomposition analysis is the global-scale
land–atmosphere equilibrium. Our analysis in Fig. 4d1 indicates that the
mean value of daily LE_{G} of all FLUXNET2015 sites is close to zero, implying the global mean RH gradient is near zero at a daily timescale. Importantly, LE is primarily determined by LE_{Q} (*R*^{2}=0.65) instead of LE_{G} (*R*^{2}=0.18) as depicted in Fig. 4b1 and b2. Nevertheless, FLUXNET2015 data also suggest that LE_{G} is the main driver
of local-scale variability of EF at the daily timescale (Fig. 4c1 and
c2). Although the mean value of daily EF is close to the mean value of
$\frac{{\mathit{LE}}_{Q}}{Q}$, the variation in daily EF depends more on the
variation in $\frac{{\mathit{LE}}_{G}}{Q}$ (Fig. 4d2). It should be noted that
Figs. 4 and S2 are almost identical (Fig. S2 repeats the presentation
shown in Fig. 4 using the value computed when forcing energy balance closure),
implying that the lack of surface energy balance closure for EC observations
does not significantly impact our analyses and interpretations.

## 4.3 Decomposition analysis of FLUXCOM dataset

We then applied the PM_{RH} model to the FLUXCOM dataset, a benchmark global LE data product (Jung et al., 2019). As shown in Fig. 5a and c, the spatial patterns of the annual mean LE and LE${}_{Q}^{\prime}$ were similar. For instance, both LE and LE${}_{Q}^{\prime}$ show the highest values around the Equator
at an annual timescale, which is mainly due to the energy available in this
region. Also, spatial variability of LE is mostly determined by LE${}_{Q}^{\prime}$ (*R*^{2}=0.85 and slope = 1) rather than by LE${}_{G}^{\prime}$ (*R*^{2}=0.18)
(Fig. 5f1 and f2). This result is consistent with Eq. (8) and the SFE
theory. In other words, the land surface is generally under thermodynamic
equilibrium with the atmosphere at the global–annual scale (i.e.,
RH_{s}≈RH_{a}). Furthermore, the monthly time series of global LE and its two components in Fig. 5e1 show that (i) LE${}_{G}^{\prime}$ is consistently
close to zero at the global scale and (ii) the seasonal variability of
global LE is primarily determined by LE${}_{Q}^{\prime}$.

However, while mean annual LE${}_{G}^{\prime}$ was close to zero in broad areas (particularly in high-latitude regions) as exemplified in Fig. 5e2, it
was distinctly positive or negative at the annual scale for many regions
(Fig. 5d). In humid tropical regions like the Amazon basin where moisture
convergence is large, LE${}_{G}^{\prime}$ was generally positive, whereas arid regions such as Australia were characterized by negative LE${}_{G}^{\prime}$ (Fig. 5e3 and e4). Here, positive LE${}_{G}^{\prime}$ (i.e., RH_{s} > RH_{a}) indicates
the land surface is wetter than the near-surface atmosphere while negative
LE${}_{G}^{\prime}$ (i.e., RH_{s} < RH_{a}) implies a drier land surface than the atmosphere. Therefore, the sign of LE${}_{G}^{\prime}$ in Fig. 5d can be interpreted as representing land surface dryness relative to the atmosphere
at an annual timescale.

The spatial pattern of LE${}_{G}^{\prime}$ is similar to the spatial pattern of EF, but differs from the spatial pattern of LE${}_{Q}^{\prime}$ (Fig. 5b). For example, EF
was high in not only tropical regions but also temperate climates such as
Mediterranean regions, and this spatial pattern is well matched with the
spatial pattern of LE${}_{G}^{\prime}$ but not LE${}_{Q}^{\prime}$. The finding that the spatial variation in EF is primarily controlled by LE${}_{G}^{\prime}$ instead of LE${}_{Q}^{\prime}$ was supported by correlation analyses in Fig. 5f3 and f4
(*R*^{2}=0.60 for EF ∼ $\frac{{\mathit{LE}}_{G}^{\prime}}{Q}$ and *R*^{2} = 0.28 for EF ∼ $\frac{{\mathit{LE}}_{Q}^{\prime}}{Q}$). This is
understandable in that EF is a reflection of the land surface dryness
(Gentine et al., 2011), and LE${}_{G}^{\prime}$ reveals the land surface
dryness relative to the atmosphere.

## 5.1 LE_{G} at sub-daily scale

Salvucci and Gentine (2013) found that the variance of the RH
gradient tends to be minimized over the course of the day. Based on this
empirical finding, they developed an approach to predict LE only using standard meteorological measurements, and this approach accurately predicted actual LE (Rigden and Salvucci, 2015, 2017). Our
PM_{RH} model provides theoretical support for their approach in that LE_{G} acts to reduce the RH gradient. Indeed, the U-shape diurnal cycles of LE_{G} in Fig. 3 (positive nighttime and negative daytime) show the
direction and the magnitude of the equilibration process of RH gradient at a
sub-daily scale resulting in a small gradient of RH on daily average.

We found positive nighttime LE_{G} values regardless of wet or dry
conditions, except when the atmosphere was fully saturated (Fig. 3). This result is a natural consequence since the land surface is close to saturation at night. This finding suggests that LE_{G} is a dominant
contributor to nighttime evaporation since available energy is close to zero
at night. It is important since nighttime evaporation is a non-negligible
component of total ET (Padrón et al., 2020).

Unlike nighttime LE_{G} values, the direction and the magnitude of the daytime LE_{G} are highly dependent on the land surface dryness. For example, the U-shape diurnal cycles of LE_{G} are apparent only when the land surface is dry, which is confirmed by the negative wavelet coherence between LE and LE_{G} at the diurnal timescale in Fig. 2f. When the land surface is wetter than the atmosphere, LE_{G} values were positive even in the daytime, and thus the U-shape diurnal cycles of LE_{G} did not appear
(Fig. 3c). The positive LE_{G} during daytime periods may be explained as
a consequence of warm and dry air entrainment at the top of the atmospheric
boundary layer and/or horizontal advection of sensible heat which may reduce
atmospheric relative humidity (Baldocchi et al., 2016; de Bruin et al.,
2016). Indeed, a strong entrainment effect and/or local advection of
sensible heat are common phenomena for irrigated agriculture (Baldocchi
et al., 2016; de Bruin and Trigo, 2019), and the irrigated sugarcane site
shows that the annual mean LE_{G} was always positive except for in 2016
when there was no application of dry season irrigation.

## 5.2 LE_{G} at daily to annual timescales

As described in the theory section, land–atmosphere equilibrium is achieved
when LE_{G} approaches zero and thus LE reduces to Eq. (8) at a timescale
longer than sub-daily. The decomposed terms derived from both the empirical
FLUXNET2015 and model-based FLUXCOM datasets show that the global mean for
LE_{G} is near zero, implying global-scale land–atmosphere equilibrium (Figs. 4 and 5). This result extends the SFE theory of McColl
et al. (2019). Although LE_{G} is not always near zero for timescales longer than sub-daily, moisture convergence and divergence at the
global scale tend to balance each other out, resulting in global-scale
land–atmosphere equilibrium on longer timescales (Fig. 5e1).

From a different perspective, non-zero LE_{G} value and its sign (+ vs. −)
at the local scale can be understood as an indicator reflecting land surface
dryness relative to the atmosphere. We found that LE_{G} clearly
distinguishes spatially wet and dry regions around the world (Fig. 5 d).
We also found that the spatiotemporal variability of EF was largely
explained by LE_{G} instead of LE_{Q} (Figs. 4c2 and 5f4). These results demonstrate the usefulness of LE_{G} to quantify land surface
dryness. Some previous studies introduced evaporative stress index or
evaporation deficit index based on the ratio (or difference) between
potential evaporation and actual evaporation (Anderson et al., 2015; Kim
and Rhee, 2016; Fisher et al., 2020; Baldocchi et al., 2021), but these
methods are highly dependent on the way one calculates potential
evaporation. Unlike potential evaporation (which is a theoretical value),
LE_{G} is a true physical quantity, and negative LE_{G} values directly
reflect water-limited land surface conditions. Therefore, we suggest
applying our decomposition method to better quantify evaporative stress.

## 5.3 Future applications

In this study, we present the PM_{RH} model and demonstrate its utility for exploratory and diagnostic purposes. However, the model has potential applications for other purposes. One possible application is to use
PM_{RH} to predict actual ET. Although the original PM equation is widely
used to predict evapotranspiration (e.g., Leuning et al., 2008; Mu et al.,
2011; Mallick et al., 2014), its accuracy often relies on parameterized
surface resistance models (Polhamus et al., 2013). Since
our PM_{RH} formulation does not include surface resistance, it could represent a good alternative. As shown in the results section, LE${}_{Q}^{\prime}$ can be calculated using typical meteorological data without additional surface parameters. Also, we found that LE${}_{Q}^{\prime}$ alone can be used effectively to approximate LE. If the remotely sensed land surface conditions are known (e.g., soil moisture and/or land surface temperature), actual LE may be more
accurately predicted by incorporating the LE_{G} term. To estimate the
LE_{G} term, RH_{s} may be estimated based on soil moisture and/or land
surface temperature data. For example, Eq. (12), which is well supported by
observations (Fig. 4a2) and on a physical basis, may be used to calculate
RH_{s}.

Another possible application is to study impacts of climate change and land
use and land cover change on surface energy partitioning and evaporation.
Changes in the atmospheric state such as temperature and relative humidity,
as well as changes in the land surface characteristics such as albedo and
aerodynamic roughness, can alter evaporation (Lee et al., 2011; Wang et
al., 2018). However, how these changes affect terrestrial energy
partitioning and ET is still unclear. Indeed, there is a large discrepancy
in long-term LE trends among current land surface models
(Pan et al., 2020). Since PM_{RH} makes it possible to physically decompose LE into adiabatic and diabatic thermodynamic components, the PM_{RH} approach can be useful to understand
how environmental changes affect surface energy partitioning. For instance,
trend analysis for the decomposed LE using PM_{RH} could contribute to
improve our understanding of earth's climate system and water cycle.

## 5.4 Potential caveats

Despite the insights it can offer, the PM_{RH} model shares
several limitations with the traditional PM model. First, PM-style equations
linearize the exponential relationship between saturation vapour pressure and
temperature (Clausius–Clapeyron relation), but the linearization can cause
bias when the temperature difference between surface and atmosphere is
substantial (Paw U and Gao, 1988; McColl, 2020). Second, the
PM_{RH} model assumes that aerodynamic resistance for heat and water vapour is identical, which implicitly relies on the assumption that the ratio of the turbulent Schmidt to Prandtl numbers is unity (Knauer
et al., 2018a). This similarity assumption cannot be held in some cases,
especially when advective flux divergence is significant (Lee
et al., 2004). The third potential caveat is that we define the land surface
as an idealized single plane that is equivalent to the “bigleaf”
representation of the traditional PM equation, but this approach ignores
profiles of temperature and humidity inside the canopy (Bonan
et al., 2021).

Another potential caveat concerns the surface energy balance. The surface
energy balance is a governing equation of the PM-style models, but it is not
satisfied in typical EC observations, which is referred to as the “surface
energy balance closure problem” (Wilson et al., 2002). This
closure problem could cause a systematic uncertainty in estimating *r*_{s} when using the PM equation (Knauer et al., 2018b; Wohlfahrt et al., 2009), and this issue may affect the diagnostic analyses using the PM_{RH}. In
this study, we did not force the energy balance closure and attributed the
cause of observed surface energy imbalances to unmeasured heat storage terms
for the Costa Rica EC site due to the possibly significant role of the heat storage
term (details in the Sect. 3.1). Wehr and Saleska (2021) recently
demonstrated that regardless of whether the lack of energy balance closure
of EC observations is due to *LE*+*H* or is due to *R*_{n}−*G*, applying the flux gradient equation to observed LE and *H* without energy balance correction is the best way to determine *r*_{s}. This is because applying the flux gradient equation to observed LE and *H* can dispense with the unnecessary assumption of
energy balance closure. They showed that bias introduced by underestimated
LE and *H* is smaller than the bias introduced by the energy balance closure assumption. This finding may be applied to our analysis in calculating RH_{s} instead of *r*_{s}.

As for the FLUXNET2015 dataset, we provide an alternate analysis using
energy-balance-corrected LE and *H* (Bowen ratio preserving method in Pastorello et al., 2020) in the Supplement. We found that the results
for corrected and uncorrected versions were almost identical, which can be
viewed as a natural consequence since in Eq. (10) LE and *H* are included in the numerator and denominator respectively. Multiplying the same ratio to LE and *H* in Eq. (10) to correct LE and *H* based on the Bowen ratio method does not significantly change the resulting RH_{s}. Therefore, the lack of surface
energy balance closure does not significantly impact our analyses and
interpretations unless the lack of energy balance is dominated by LE only or *H* only. If the lack of energy balance is dominated by LE only or *H* only, our
results and interpretation may include systematic bias.

Finally, there are several ways to calculate aerodynamic resistance, and the
chosen form for *r*_{a} may affect the results. However, the influence of this choice is expected to be marginal compared to the energy balance
problem. Knauer et al. (2018b) showed that uncertainty caused by different *r*_{a} values on surface conductance is low compared to the
energy balance closure problem. This finding can be applied to our analysis.
Specifically, in Eq. (10), *r*_{a} is multiplied by both denominator and numerator, and thus a small difference in *r*_{a} should not significantly affect the resulting RH_{s}.

We have shown that our novel PM_{RH} model provides a new opportunity to understand the governing physics of the terrestrial energy budget. Specifically, the PM_{RH} model helps to illustrate how the land surface conditions become encoded to the atmospheric state by partitioning LE into two
thermodynamic processes. “Dry land–dry air” or “wet land–wet air”
conditions can each lead to daily scale land–atmosphere equilibrium, although
the diurnal pattern of the equilibration process (i.e., LE_{G}) is starkly different. Our findings suggest that while LE_{G} is a primary component determining EF, spatiotemporal variability of LE_{Q} alone can adequately represent the variability of LE. We found global-scale land–atmosphere equilibrium at daily to annual scales, which implies that global LE can be
simply determined by the atmospheric state and radiative energy without any
surface constraint required to represent spatial heterogeneity and
physiological influences. From a different perspective, the non-zero LE_{G} value at a local scale can be understood as an indicator revealing land surface dryness. Questions remain regarding how LE_{Q} and LE_{G} will be
influenced in relation to changing climatic and land surface conditions and
how these changes might affect the climate system at differing spatial and
temporal scales through positive or negative feedbacks.

The FLUXNET2015 dataset is available in https://fluxnet.org/data/download-data/ (FLUXET community, 2019). The highlighted sugarcane eddy covariance site dataset will be available in AmeriFlux (by December 2021, https://ameriflux.lbl.gov/, AmeriFlux, 2021). The FLUXCOM dataset is available in http://www.fluxcom.org/EF-Download/ (FLUXCOM, 2021).

The supplement related to this article is available online at: https://doi.org/10.5194/hess-25-5175-2021-supplement.

YK and MSJ designed research; UW provided FLUXCOM data; YK, LM, and MSJ performed research; YK analyzed data; YK, MG, TAB, LM, and MSJ wrote the paper.

The authors declare that they have no conflict of interest.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We want to thank Iain Hawthorne, Pável Bautista, Silja Hund, Cameron Webster, Gretel Rojas Hernandez, Guillermo Duran Sanabria, Andrea Suarez Serrano, Ana Maria Duran, Martin Martinez, and Fermín Subirós Ruiz for field and logistical support. We also thank Martin Jung, the principal investigator of the FLUXCOM dataset. The authors would like to thank the EU and NSERC for funding, in the frame of the collaborative international Consortium AgWIT financed under the ERA-NET WaterWorks2015 Cofunded Call. This ERA-NET is an integral part of the 2016 Joint Activities developed by the Water Challenges for a Changing World Joint Programme Initiative (Water JPI).

This research has been supported by the Joint Programming Initiative Water challenges for a changing world (Agricultural Water Innovations in the Tropics).

This paper was edited by Anke Hildebrandt and reviewed by two anonymous referees.

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