the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Using phase lags to evaluate model biases in simulating the diurnal cycle of evapotranspiration: a case study in Luxembourg

### Claire Brenner

### Kaniska Mallick

### Hans-Dieter Wizemann

### Luigi Conte

### Ivonne Trebs

### Jianhui Wei

### Volker Wulfmeyer

### Karsten Schulz

### Axel Kleidon

While modeling approaches of evapotranspiration (*λ**E*) perform
reasonably well when evaluated at daily or monthly timescales, they can show systematic
deviations at the sub-daily timescale,
which results in potential biases in modeled *λ**E* to global climate
change. Here we decompose the diurnal variation of heat fluxes and
meteorological variables into their direct response to incoming solar
radiation (*R*_{sd}) and a phase shift to *R*_{sd}. We analyze data from an
eddy-covariance (EC) station at a temperate grassland site, which experienced a
pronounced summer drought. We employ three structurally different modeling
approaches of *λ**E*, which are used in remote sensing retrievals, and
quantify how well these models represent the observed diurnal cycle under
clear-sky conditions. We find that energy balance residual approaches, which
use the surface-to-air temperature gradient as input,
are able to reproduce the reduction of the phase lag from wet to dry conditions. However, approaches
which use the vapor pressure deficit (*D*_{a}) as the driving gradient
(Penman–Monteith) show significant deviations from the observed phase lags,
which is found to depend on the parameterization of surface conductance to
water vapor. This is due to the typically strong phase lag of 2–3 h
of *D*_{a}, while the observed phase lag of *λ**E* is only on the order of
15 min. In contrast, the temperature gradient shows phase differences in
agreement with the sensible heat flux and represents the wet–dry difference
rather well. We conclude that phase lags contain important information on
the different mechanisms of diurnal heat storage and exchange and, thus,
allow a process-based insight to improve the representation of
land–atmosphere (L–A) interactions in models.

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Evapotranspiration and the corresponding latent heat flux (*λ**E*)
couple the surface water and energy budgets and are of high relevance for
water resources assessment. *λ**E* is generally limited by four physical
factors: (i) the availability of energy mostly supplied by solar radiation,
(ii) the availability of and the access to water, (iii) the plant
physiology, and (iv) the atmospheric transport of moisture away from the
surface (Brutsaert, 1982). These different limitations have led to different
approaches on how to model *λ**E*.

Key approaches either focus on the surface energy balance where the
surface-to-air temperature gradient dominates the flux or approaches which
focus on the moisture transfer limitation where vapor pressure gradients
dominate the flux. It is critical to recognize that these two limitations
are not independent of each other but rather are shaped by land–atmosphere
heat and water exchange and thus covary with each other. The diurnal
variation of incoming solar radiation (*R*_{sd}) causes a strong diurnal
imbalance in surface heating leading to the pronounced diurnal cycles of
surface states and fluxes (Oke, 1987; Kleidon and Renner, 2017). This heat
exchange of the surface with the lower atmosphere thus influences the
near-surface air temperature (*T*_{a}), skin temperature (*T*_{s}), vapor
pressure (*e*_{a}), soil or canopy saturation water pressure (*e*_{s}), vapor
pressure deficit (*D*_{a}), and wind speed (*u*), which are being regarded as
important controls on *λ**E* (e.g., Penman, 1948). These interactions are
particularly dominant at the diurnal timescale (e.g., De Bruin and Holtslag,
1982) and depend on meteorological as well as on surface conditions
(Jarvis and McNaughton, 1986; van Heerwaarden et al.,
2010). Ignoring the interdependence of the surface variables may lead to
biases in model parameterizations and compensating errors when evaluating
the model performance only with respect to a single variable (Matheny et
al., 2014; Best et al., 2015; Santanello et al., 2018).

There is a strong need to investigate and to derive metrics based on comprehensive observations that characterize the whole land-surface–atmosphere system (Wulfmeyer et al., 2018). Several authors proposed different multivariate metrics to better evaluate land–atmosphere (L–A) interactions in observations and models. Generally, these metrics explore internal relationships between state variables to better characterize key processes and to guide a more systematic exploration and understanding of model deficiencies. A number of metrics focus on the diurnal evolution of the heat and moisture budgets in the planetary boundary layer (e.g., Betts, 1992; Santanello et al., 2009, 2018). Also statistical metrics exploring the strength of linear relationships between surface heat fluxes and states to surface radiation components have been employed to evaluate the performance of reanalysis with observations (Zhou and Wang, 2016; Zhou et al., 2017, 2018).

Furthermore, there are pattern-based metrics which focus on nonlinear
interactions at the diurnal timescale. Wilson et al. (2003) proposed the
method of a diurnal centroid to measure the timing of the surface heat
fluxes and their timing difference, which was more recently used by Nelson
et al. (2018) to quantify the timing of evapotranspiration under different
dryness condition for the FLUXNET dataset. In contrast, Matheny et al. (2014)
and Zhang et al. (2014) explored the diurnal relationship of the
latent heat flux to vapor pressure deficit showing a pronounced hysteresis
loop. Zheng et al. (2014) also included air
temperature and net radiation as reference variables and showed that the
hysteresis loops of *λ**E* to *D*_{a} or *T*_{a} are large, while there
are only small hysteresis effects when *R*_{n} was used. Hysteresis loops
have also been found when heat fluxes are plotted against net radiation (Camuffo
and Bernardi, 1982; Mallick et al., 2015), with many studies showing
hysteretic loops of the soil heat flux against net radiation (Fuchs and
Hadas, 1972; Santanello and Friedl, 2003; Sun et al., 2013). The presence of
a hysteresis loop indicates that there is a time-dependent nonlinear
control on the variable of interest, typically induced by heat storage
processes. Camuffo and Bernardi (1982) showed that the magnitude and
direction of such hysteretic loops can be estimated by a multilinear
regression of the variable of interest against the forcing variables and its
first-order time derivative. This simple model allows estimating storage
effects on diurnal (Sun et al., 2013) to seasonal timescales (Duan and Bastiaansen, 2017).

Here, we choose the Camuffo and Bernardi (1982) model because it provides an objective measure of the magnitude of hysteresis loops and it allows for an assessment of statistical significance. We extend the Camuffo and Bernardi (1982) model in two ways.

First, we use incoming solar radiation (*R*_{sd}) as a reference variable
instead of net radiation to estimate the phase lag of surface heat flux
observations and models. And secondly, we use a harmonic transformation of
the Camuffo and Bernardi (1982) regression model to estimate the phase lag
in time units. This extension allows us to compare the diurnal phase lag
signatures of the different model inputs and how these influence the
resulting diurnal course of the latent heat flux estimate.

We specifically choose incoming solar radiation *R*_{sd} as the reference for
the phase-shift analysis, since *R*_{sd} can be regarded as an independent
forcing of the surface energy balance (e.g., Ohmura, 2014):

with surface albedo *α*, incoming longwave radiation *R*_{ld}, sensible
heat flux *H*, latent heat flux *λ**E*, the conductive soil heat flux *G*, the
outgoing longwave radiation *σ**T*^{4}, and storage terms of the surface
layer summarized in *m*. This formulation of the surface energy balance
provides the direction of the energy exchange processes at the surface,
illustrating that the terms on the right-hand side depend on heat fluxes on
the left-hand side of Eq. (1) (Ohmura, 2014). As a consequence, the term net
radiation *R*_{n}, which resembles the radiation budget of the shortwave and
longwave components, ${R}_{\mathrm{n}}={R}_{\mathrm{sd}}(\mathrm{1}-\mathit{\alpha})+{R}_{\mathrm{ld}}-\mathit{\sigma}{T}^{\mathrm{4}}$,
cannot be regarded as an independent surface forcing. Consequently,
we choose *R*_{sd} instead of *R*_{n} or *R*_{n}−*G* as the reference variable for
the phase-shift analysis of the latent heat flux and the main input
variables of evapotranspiration model approaches.

We focus on two different approaches to estimate *λ**E*. The first
approach is based on the energy limitation of *λ**E*, using the
equilibrium evaporation concept (Schmidt, 1915) as formulated by Priestley
and Taylor (1972) for potential evaporation. For actual evaporation we focus
on one-source and two-source energy balance schemes (OSEB and TSEB,
respectively) which derive *λ**E* as the residual term of the surface energy
balance and parameterize the sensible heat flux by a resistance description
of the surface-to-air temperature gradient (Kustas and Norman, 1999). The second approach
is based on the Penman–Monteith
(PM hereafter) approach (Monteith, 1965), which adds water vapor pressure deficit
as a driving gradient (referred to as the “vapor-gradient scheme”). We use the
widely used Food and Agriculture Organization of the United Nations (FAO)
Penman–Monteith formulation (Allen et al., 1998) for
potential or reference evapotranspiration. For actual evapotranspiration we
use a modified PM approach which was formulated by Mallick et al. (2014,
2015, 2016, 2018) (see also Bhattarai et al., 2018) and is termed as a
the Surface Temperature Initiated Closure (STIC). STIC is based on finding the
analytical solution of the surface and aerodynamic conductances in the PM
equation while simultaneously constraining the surface and aerodynamic
conductances through both surface temperature and vapor pressure deficit.

Several inter-comparison studies evaluated the performance of these schemes
using observations from different landscapes. OSEB and TSEB, which are often
used in remote sensing retrievals of *λ**E*, have been found to perform
comparably well in reproducing tower-based energy flux observations
(Timmermans et al., 2007; Choi et al., 2009; French et al., 2015). Yang et
al. (2015) compared temperature-gradient approaches (including TSEB) with the
Penman–Monteith approach (based on vapor pressure gradient only) employed by
the MODIS evapotranspiration product (MOD16, Mu et al., 2011) and found
strongly reduced capability of MOD16 to estimate spatial variability of
evapotranspiration. They concluded that the moisture availability
information obtained from the relative humidity and vapor pressure deficit of
the air is not able to capture the surface water limitations as reflected in surface temperature.

In this study, we focus on the ability of these different evapotranspiration
models to reproduce the diurnal cycle of *λ**E* under wet and dry
conditions. In particular, we assess if significant nonlinear relationships
in the form of hysteretic loops exist, if these change under different wetness
conditions, and if temperature-gradient and vapor-gradient approaches such as PM
are able to reproduce this behavior. Further, we evaluate which input
variables of the evapotranspiration schemes show a hysteretic pattern and
how these patterns influence the flux estimation. To address these
questions, we analyze observations and models with respect to internal
functional relationships (pattern-based) and use solar radiation as
an independent driver of land–atmosphere exchange. We focus on wet vs. dry
conditions since this is another critical deficiency identified in previous
analyses (e.g., Wilson et al., 2003; Matheny et al., 2014; Zhou and Wang,
2016). To ensure similar radiative forcing and avoid variability due to
cloud cover we focus the evaluation on clear-sky days. We illustrate our
approach on a grassland site in a temperate semi-oceanic climate using
surface energy balance observations.

The analysis will shed light on the capabilities of process-based
evapotranspiration schemes to capture the dynamics of diurnal
land–atmosphere exchange. We show that the phase lag of surface states and
fluxes reveals important imprints of heat storage processes and how this
guides the evaluation of the different approaches for modeling *λ**E*.
This is important for applications in remote sensing with respect to the
choice of observational input variables. In doing so, we provide a further,
pattern-based metric to assess land–atmosphere interactions and, thus, guide
process-based improvements and calibration of land-surface schemes.

## 2.1 Diurnal patterns and hysteresis loop quantification

We first illustrate the pattern-based evaluation of the diurnal cycle using
two hypothetical variables *Y*_{1} and *Y*_{2}, as shown in Fig. 1. If a
variable (*Y*_{1}) is in phase with *R*_{sd}, it shows a linear behavior when
plotted against *R*_{sd} (Fig. 1b). However, if a variable (*Y*_{2}) has a
time lag with respect to *R*_{sd}, showing a significant difference between
morning and afternoon values, it results in a hysteretic loop. The area
inside the loop indicates the magnitude of the phase difference, while the
direction of the loop, marked by an arrow at the morning rising limb in Fig. 1b,
indicates if a variable is preceding or lagging *R*_{sd} in time. If a
variable shows consistently larger values during the afternoon as compared
to the morning, this will appear as a counterclockwise (CCW) hysteretic
loop indicating a positive phase lag with respect to *R*_{sd}. A negative
phase lag appears as a clockwise (CW) loop.

To obtain a quantitative measure of the hysteretic pattern, we use the
Camuffo–Bernardi equation (Camuffo and Bernardi, 1982), which relates the
time series of the response variable *Y*(*t*) to the forcing variable *R*_{sd}(*t*)
and its first-order time derivative d*R*_{sd}(*t*)∕d*t*:

Using multilinear regression, we obtain the coefficients *a*, *b*, and *c* assuming a
normal distribution of the residuals *ε*(*t*). If *Y* is linear
with *R*_{sd}, the parameter *c* should be zero. However, if a consistent pattern
such as a hysteretic loop exists, then parameter *c* should be significantly
different from zero. Hence, by using regression analysis we can determine if
a significant hysteretic relationship between two variables exists and if
the inclusion of such a nonlinear term (with *c*≠0) would improve the model fit.

Although significance testing of the coefficient *c* is an advantage, it is
clear from Eq. (2) that the magnitude of *c* depends on the units and magnitude
of the response variable *Y*. In order to estimate a comparable estimate of the
phase lag we employ a harmonic transformation of the regression model.
Assuming that *R*_{sd} is a harmonic function with an angular frequency *ω*,
the phase difference *φ* can be estimated from the two regression coefficients *b* and *c*:

To derive the first-order time derivative of solar radiation, we use a
simple difference between time steps. Since the data we use is available in
30 min time steps (see below), we have 48 time steps per day; thus $\mathit{\omega}=\mathrm{48}/\left(\mathrm{2}\mathit{\pi}\right)$.
To obtain a phase lag between *Y* and *R*_{sd} as a time lag *t*_{φ} (min) we use

Note that the phase lag estimate *t*_{φ} is somewhat similar to the
relative diurnal centroid metric proposed by Wilson et al. (2003) for the
analysis of the timing of heat and mass fluxes. The diurnal centroid
identifies the timing of the peak of a variable with respect to local time.
Since the peak of *R*_{sd} is at noon local time, both metrics are
qualitatively comparable.

## 2.2 Field site and observations

The study area is a grassland site in Petit-Nobressart, Luxembourg, situated
on a gentle east-facing slope. The grassland is used as a hay meadow and had
short vegetation of about 10–15 cm as the grass was mowed before the start
of the experiment. An eddy-covariance (EC) station (with the setup described
in Wizemann et al., 2015) was installed at the grassland close to the village
of Petit-Nobressart (Fig. 2; exact coordinates: 49^{∘}46.77^{′} N,
05^{∘}48.22^{′} E). The EC station was operated from 11 June until
23 July 2015. The three-dimensional wind and temperature fluctuations were
measured at 2.41 m above ground by a sonic anemometer (CSAT3, Campbell
Scientific Inc., Logan, USA) facing to the mean wind direction of
290^{∘}. A fast-response open-path CO_{2}∕H_{2}O infrared gas analyzer
(IRGA LI-7500, LI-COR, USA) installed at a lateral distance of 0.2 m to the
sonic path was used to measure CO_{2} and H_{2}O fluctuations. The high-frequency
signals were recorded at 10 Hz by a CR3000 data logger and the TK3 software
was used to compute turbulent fluxes of sensible heat (*H*), latent
heat (*λ**E*), and CO_{2} (Mauder and Foken, 2015).

Downwelling and upwelling shortwave and longwave radiation were obtained by a four-component net-radiation sensor (NR01, Hukseflux, the Netherlands). The meteorological variables (air temperature, humidity, and precipitation) were monitored with a time resolution of 30 min. Soil heat flux was measured by heat flux plates (two in 8 cm depth; HFP01, Hukseflux, the Netherlands), soil temperature was measured at 2, 5, 15, 30 cm depth (model 107, Campbell Scientific Inc., UK), water content at 2.5, 15, 30 cm depth (CS616, Campbell Scientific Inc., UK), and matric potential at 5, 15, 30 cm depth (model 253, Campbell Scientific Inc., UK). All soil sensors were installed between the turbulence and radiation measurement devices.

Unfortunately, the two upper-temperature probes and soil-matric-potential
sensors showed data gaps and erroneous values from 30 June until excavation
on 23 July 2015. Thus, the ground heat flux was calculated by the heat flux
plate method with correction for heat storage (Massman, 1992) only for the
period from 11 to 30 June 2015. To still obtain soil heat fluxes for
the entire measuring period, additionally harmonic wave analysis (Duchon and
Hale, 2012) of the heat flux plate data was applied. The harmonic wave
analysis calculates the wave spectrum at the soil surface from the Fourier
transform of the soil heat flux measured by the heat flux plates in a
few centimeter depth (here: 8 cm) by correcting for wave amplitude damping and phase shift.
The surface ground heat flux is then obtained by an inverse Fourier
transformation of the corrected wave spectrum. The method has a dependence
on soil moisture affecting the damping depth. The dependence is, however,
weak for clayey soils with soil water contents >10 % (Jury and
Horton, 2004) as observed at the site. The damping depth was obtained by the
exponential decay of the soil temperature amplitude measured at the various
depths. Differences in the damping depth between wet and drier soil moisture
conditions only yielded differences in *G* smaller than 10 W m^{−2}.
Therefore, we used a constant damping depth for the whole period.

Both methods for deriving the total soil heat flux agreed well for the period before 30 June, so the latter method should provide reliable ground heat flux values for the entire period until 23 July. Table 1 lists the variables obtained from the EC station and used in this work. For more details on instrumentation and EC data processing, see Ingwersen et al. (2011) and Wizemann et al. (2015).

### 2.2.1 Derived meteorological variables

We derived the saturated water vapor pressure *e*_{s} (hPa) using the empirical
Magnus equation (Magnus, 1844) as a function of air temperature *T* (^{∘}C)
with empirical coefficients from Alduchov and Eskridge (1996):

Then, the water vapor pressure of the air *e*_{a} (hPa) was obtained by using
air temperature *T*_{a} and relative humidity (RH):

To assess the moisture conditions of each date of the site we used the
evaporative fraction *f*_{E}:

Since daily averages can be influenced by single large values of the
turbulent fluxes and contain missing values, we estimated a daily *f*_{E}
based on the 30 min values of each day using the following linear regression:

where *f*_{E} is the slope of the linear regression, *β* its
intercept, and *ε*_{R} the residuals. Since we use the fluxes of *H*
and *λ**E* without energy balance closure correction, we obtain the upper
range of *f*_{E}.

Since the sonic anemometer measures friction velocity (*u*^{*}) and the absolute
value of wind speed $u=\sqrt{({U}^{\mathrm{2}}+{V}^{\mathrm{2}})}$, we estimate the aerodynamic
conductance for momentum (${u}^{*\mathrm{2}}/u$) and the aerodynamic conductance (*g*_{ah})
for heat including the excess resistance to heat transfer using an empirical
formula by Thom (1972):

We chose to use this formula for its simplicity and similar performance than more recent, complex parameterizations (Knauer et al., 2018; Mallick et al., 2016). Also note that effects of atmospheric stability are accounted for in the first term of Eq. (5).

### 2.2.2 Energy balance closure gap correction

Most EC measurements show that the sum of the observed turbulent heat fluxes
is smaller than the available energy and thus does not close the energy
balance, leaving an energy balance closure gap (*Q*_{gap}) (Foken, 2008):

For our site we observed on average a slope of ($H+\mathit{\lambda}E)\sim ({R}_{\mathrm{n}}-G)=\mathrm{0.81}$
(by linear regression) with an average
gap of 37 W m^{−2} over the whole duration of the field campaign. These
values are in the typical range of what is commonly found for grassland sites
(Stoy et al., 2013).

To correct the turbulent fluxes for the energy balance closure gap
(evaluated at the 30 min time steps), we use a correction based on the Bowen
ratio (*B*_{R}) (Twine et al., 2000), which is directly related to the
evaporative fraction ${f}_{\mathrm{E}}=\mathrm{1}/({B}_{\mathrm{R}}+\mathrm{1})$ to obtain corrected fluxes:

and

The correction is applied at 30 min time steps using the daily *f*_{E} estimates.
We use these corrected fluxes in the further analysis.

### 2.2.3 Clear-sky day classification

In order to achieve comparable conditions with respect to incoming solar
radiation, we identified clear-sky conditions. A clear-sky day was defined
by its daily sum of incoming solar radiation being larger than 85 % of the
potential surface radiation (*R*_{sd,pot}), which is a function of latitude
and day of year (using R package REddyProc, function fCalcPotRadiation):

where *t* corresponds to each time step of measurement and with *f*_{diff}=0.78
being a constant factor taking into account atmospheric extinction of solar radiation.

## 2.3 One- and two-source energy balance models

Thermal-remote-sensing-based models estimate evapotranspiration by solving
the surface energy balance and rely on land-surface temperature (*T*_{s})
information as a key boundary condition (Kustas and Norman, 1999). A bulk
layer formulation of the soil-plus-canopy sensible heat flux is employed and
*λ**E* is derived by enforcing the surface energy balance. Hence
*λ**E* is written as

where *ρ* is the density of air, *c*_{p} is the specific heat of air at
constant pressure, and *g*_{ah} is the effective aerodynamic conductance of
heat that characterizes the transport of sensible heat between the surface
and the atmosphere. We obtained *T*_{s} from the observed longwave emission
of the surface ${T}_{\mathrm{s}}=({R}_{\mathrm{lu}}/(\mathit{\sigma}{\mathit{\epsilon}}_{\mathrm{s}}){)}^{\mathrm{1}/\mathrm{4}}$
with $\mathit{\sigma}=\mathrm{5.67}\times {\mathrm{10}}^{-\mathrm{8}}$ W K^{−4} (the Stefan–Boltzmann constant)
and a surface emissivity *ε*_{s}=0.98, which is typical
for a grassland and agrees with Brenner et al. (2017).

We use two different approaches which are generally classified as one- and
two-source models with regard to the implemented treatment of the energy
exchange with the surface. While one-source energy balance models
treat the surface as a uniform layer, two-source energy balance models
partition temperatures as well as radiative and energy fluxes into a soil and
vegetation component. The one-source approach (OSEB) parameterizes the
aerodynamic conductance *g*_{ah} as follows (e.g., Kalma et al., 2008;
Tang et al., 2013):

where *z*_{u} and *z*_{t} are the measurement heights of wind and air
temperature, respectively; *z*_{0m} and *z*_{0h} are roughness lengths for
momentum and heat, respectively; *k* is the von Kármán constant; *d* is
the displacement height; *u* is the wind speed; and Ψ_{m} and
Ψ_{h} are the the integrated Monin–Obukhov (MO) similarity functions which
correct for atmospheric stability conditions (Brutsaert, 2005; Jiménez et
al., 2012). For the investigated grassland site, *d* and *z*_{0m} were
calculated as fractions of the vegetation height, *h*_{c}, with
*d*=0.65*h*_{c} and *z*_{0m}=0.125*h*_{c}. The roughness length for
heat *z*_{0h} was set using the dimensionless parameter $k{B}^{-\mathrm{1}}=\mathrm{ln}({z}_{\mathrm{0}\mathrm{m}}/{z}_{\mathrm{0}\mathrm{h}})$,
which was set to 2.3 in accordance with Bastiaanssen et
al. (1998). Note that this parameterization of aerodynamic conductance does
not explicitly distinguish between bare soil and canopy boundary layer
conductance, as it is done in two-source approaches.

In addition to OSEB we applied the two-source energy balance model
developed by Norman et al. (1995) and Kustas and Norman (1999).
For both the soil and canopy components, a separate energy balance (with different
component temperatures) and bulk resistance scheme with different
aerodynamic conductance are formulated. Then the energy balance equations
are solved iteratively. It starts by assuming that a fraction of the canopy
(described by vegetation greenness fraction *f*_{g}) transpires at a
potential rate as described by the Priestley–Taylor equation (Priestley and Taylor, 1972):

where *α*_{PT} is the Priestley–Taylor coefficient (1.26), *s* is the
slope of the saturation water vapor pressure curve, and *γ* is the
psychrometric constant. However, the canopy latent heat flux
*λ**E*_{c}=*f*_{g}*λ**E*_{PT} might be too large and the soil
component would become negative (condensation at the soil surface), which is
unlikely during daytime conditions. To avoid condensation at the soil
surface, the *α*_{PT} coefficient is reduced incrementally until the
soil latent heat flux becomes zero or positive. Once this condition is met,
all other energy balance components are updated accordingly to satisfy the
energy balance equation. For this study we used a constant vegetation
fraction of *f*_{c}=0.9 and a greenness fraction *f*_{g} which was derived
from close-up pictures taken at the beginning and the end of the field
campaign and linearly interpolated in-between.

## 2.4 Penman–Monteith approach

In the Penman–Monteith approach (Monteith, 1965) the inclusion of
physiological conductance (*g*_{s}) imposes a critical control on *λ**E*:

In Eq. (9), the transfer of moisture is linked to a supply–demand reaction
where the net available energy (*R*_{n}−*G*) is the supply energy for
evaporation and the vapor pressure deficit of the air *D*_{a} [$={e}_{\mathrm{s}}\left({T}_{\mathrm{a}}\right)-{e}_{\mathrm{a}}$]
is the demand for evaporation from the atmosphere.
In the PM approach, the two conductances, the aerodynamic conductance *g*_{av}
and the surface conductance *g*_{s}, to water vapor are
unknown. A widely used approach to obtain a reference evapotranspiration
estimate from meteorological data is the FAO Penman–Monteith reference
evapotranspiration (Allen et al., 1998). It defines the two conductances for
a well-watered grass surface with a standard height of *h*=0.12 m. The
aerodynamic conductance is obtained by a bulk approach (Eq. 7) with wind
speed *u* measured at 2 m above the surface, $d=\mathrm{2}/\mathrm{3}h$, *z*_{0m}=0.123*h*,
*z*_{0h}=0.1*z*_{0m}, yielding ${g}_{\mathrm{av}}=u/\mathrm{208}$ (Box 4 in Allen et al., 1998). Surface
conductance is fixed at a constant ${g}_{\mathrm{s}}=\mathrm{1}/\mathrm{70}$ m s^{−1}. Here, we use the latter
definitions of the conductances and use direct measurements for the other
input variables of Eq. (9) to obtain the FAO Penman–Monteith estimate.
While the FAO estimate is typically intended for estimates of the reference
evaporation for well-watered grass on a daily basis, we use it here as a
reference for comparison on a sub-daily scale. In order to understand the
effect of the aerodynamic conductance parameterizations we add another
reference evapotranspiration estimate in which the aerodynamic conductance
is given by Eq. (5) using observations of friction velocity and wind speed,
but keeping *g*_{s} fixed.

### Penman–Monteith-based Surface Temperature Initiated Closure (STIC) (version STIC1.2)

In order to estimate an actual evapotranspiration rate from meteorological
data we employ a method (STIC1.2 hereafter referred to as STIC), which is
based on the PM equation, but which in addition integrates surface
temperature information. The STIC methodology is based on finding analytical
solutions for the two unknown conductances to directly estimate *λ**E*
(Mallick et al., 2016, 2018). STIC is a one-dimensional physically
based surface energy balance model that treats the vegetation–substrate complex as a single unit
(Mallick et al., 2016; Bhattarai et al., 2018). The fundamental assumption
in STIC is the first-order dependency of *g*_{a} and *g*_{s} on soil moisture
through *T*_{s} and on environmental variables through *T*_{a}, *D*_{a}, and
net radiation. Therefore, surface temperature is assumed to provide
information on water limitation which is linked to the advection–aridity
hypothesis (Brutsaert and Stricker, 1979). In STIC, no wind speed is required
as input data, as opposed to the temperature-gradient approaches, but vapor
pressure of the air and its saturation value become critical input
variables; see Table 2 for an overview. A detailed description of STIC
version 1.2 is available in Mallick et al. (2016, 2018) and Bhattarai et al. (2018).

## 3.1 Daily clear-sky and moisture classification

The field campaign was conducted during an exceptionally warm and dry period
characterized by clear-sky conditions with remarkably high air temperatures
with daily maxima above 30 ^{∘}C and little precipitation. Compared
to the climatic normal (1981–2010) the precipitation deficit in this region
was −44 % in June and −41 % in July, respectively (source:
meteorological station Arsorf, Administration des services techniques de
l'agriculture – ASTA). The air temperature anomaly was higher in July
(1.9 ^{∘}C) than in June (0.7 ^{∘}C) (source: meteorological
station Clemency, ASTA). The soil water content decreased and parts of the
site, especially the upper part, showed clear signs of vegetation water
stress (see Brenner et al., 2017, for an analysis of the spatial
heterogeneity of water limitation). However, the dry period was interrupted
by a few but strong rainfall events, which significantly changed soil
moisture and thus *f*_{E} with time (Fig. 3a). Based on the
observed *f*_{E} we classified dry days with *f*_{E}<0.5 and wet days with
*f*_{E}>0.6. This separation of dry and wet days is also reflected in
the top soil moisture conditions (measured at 5 cm depth) as shown in Fig. 3b.

Based on the classification of wet and dry days under clear-sky conditions
we computed composites of the diurnal cycle for each hour. By using only
sunny days we aim to achieve similar conditions with respect to downwelling
shortwave radiation (*R*_{sd}). Figure 4a confirms that *R*_{sd} and net
radiation (*R*_{n}) had very similar diurnal cycles and magnitudes for the
wet and dry days. However, the downwelling longwave radiation *R*_{ld} and the
soil heat flux were somewhat higher under wet conditions (Fig. 4a). The
higher *R*_{ld} is related to higher air temperatures and air vapor
pressures observed under wet conditions (Fig. 4b), which may explain the
greater value of *R*_{ld} by affecting the atmospheric emissivity for
longwave radiative exchange. This has also an impact on the minimum
temperatures both for air and skin temperature, which are higher under wet
conditions and lower under dry conditions (Fig. 4b). Hence, although we
achieve fairly similar conditions for shortwave radiation under wet and dry
conditions, we observed a small but significant difference in the longwave
radiative exchange.

## 3.2 Diurnal cycle of evapotranspiration under wet and dry conditions

Next, we evaluate how the different evapotranspiration schemes are able to
reproduce the fluxes during wet and dry conditions under similar *R*_{sd}
forcing. Figure 5 shows the average diurnal cycle of observations and models
for *λ**E*. The observations showed a significant difference in *λ**E*
between dry and wet conditions, with a maximum value of *λ**E*
of about 200 W m^{−2} for dry and 350 W m^{−2} under wet
conditions, which amounts to a mean difference of 100 W m^{−2} for
daylight conditions (Table 3). As reference, we also included two common
formulations of potential evapotranspiration, the Priestley–Taylor potential
evapotranspiration (PT) and the FAO Penman–Monteith reference
evapotranspiration (FAO-PM). Both do not account for water limitation and
show a marginal difference of 10 W m^{−2} between wet and dry conditions.
While FAO-PM yielded lower mean conditions than PT, it showed lower
correlation and RMSE as compared to PT (Table 3). We find that all models
for actual *λ**E* (rather than PT or FAO-PM) showed differences in *λ**E*
between wet and dry conditions. Both OSEB and TSEB showed a
tendency to overestimate *λ**E* under dry conditions but captured the
high *λ**E* values under wet conditions. In contrast, STIC captured the
low *λ**E* magnitude under dry conditions (*f*_{E}<0.5) but
underestimated *λ**E* under wet conditions (for *f*_{E}>0.6).

Table 3 shows the statistical metrics of the model performances with respect
to the Bowen-ratio-corrected *λ**E*. In general, both OSEB and TSEB
produced mean *λ**E* values within the range of 96 %–98 %
(255 and 259 W m^{−2}) of the observed *λ**E* (264 W m^{−2}) in
wet conditions, while mean *λ**E* from STIC was within 83 % (218 W m^{−2})
of observed *λ**E* for the same conditions. However, for the
dry conditions, simulated *λ**E* from STIC (180 W m^{−2}) was 91 %
of the observed mean *λ**E* (164 W m^{−2}), while the simulated
mean *λ**E* from OSEB and TSEB was 77 %–78 % of the observed
mean *λ**E*. Overall, the three models captured 86 % (OSEB), 88 % (TSEB),
and 95 % (STIC) of the observed mean *λ**E*. Results show that, under
wet conditions, RMSE of the OSEB–TSEB models is well within the range of
the errors when compared with the uncorrected *λ**E*, whereas STIC
showed relatively higher RMSE. However, under dry conditions the RMSE of
OSEB–TSEB models was found to be larger than for STIC. For the entire
observation period the three models produced comparable RMSE (41–46 W m^{−2})
but with different correlation. STIC produced relatively low
correlation (*r*^{2}=0.72) as compared to the other two models (*r*^{2}=0.84–0.85).
Therefore, we find that the correlation of the schemes is
distinctly larger under wet conditions as compared to dry conditions. The
correlations under wet conditions of OSEB and TSEB are in the range of the
correlation of the uncorrected *λ**E* (*r*^{2}=0.91), whereas STIC
and FAO-PM showed lower correlation. Under dry conditions the correlation
was significantly lower than the correlation of the uncorrected *λ**E*
(*r*^{2}=0.93). While OSEB–TSEB explained 62 % of the observed
uncorrected *λ**E* variability in dry conditions (STIC explained
44 %), both models produced higher RMSE (57–58 W m^{−2}) as compared
to STIC (45 W m^{−2}) under these conditions.

## 3.3 Diurnal patterns of evapotranspiration

The evaluation of the diurnal cycle shows that *λ**E* was strongly
related to the incoming solar radiation, emphasizing that *R*_{sd} is the
dominant driver of *λ**E* (Fig. 6). However, under wet conditions we
found a marked and consistent difference between morning and afternoon
in *λ**E*,
forming a CCW hysteresis loop (Fig. 6b). Using the
Camuffo–Bernardi regression we found a significant phase lag for the
Bowen-ratio-corrected flux (*λ**E*_{BRC}) with a mean *t*_{φ}=15 min
under wet conditions and no significant lag under dry conditions (Fig. 7 and
Table 4). The uncorrected observations showed only a slightly lower wet–dry
difference, highlighting that the method to close the energy balance closure
gap does not significantly influence the estimated phase lag.

The two potential evapotranspiration estimates showed large differences in
their phase lag. While the PT estimate showed a small hysteretic loop with a
phase lag between *t*_{φ}=6–9 min, the FAO-PM estimate showed a
substantial loop with a phase lag of *t*_{φ}=31 min. This large
phase lag of the FAO-PM estimate is very similar to the phase lag when we
use a constant *g*_{s} in the PM equation but with *g*_{av} obtained
from Eq. (5) using friction velocity observations (Table 4). The temperature-gradient
schemes (OSEB and TSEB) reproduced the observed phase lag relatively well
(mean *t*_{φ}=9 min for wet and around 0 for dry conditions).
However, the temperature- and vapor-gradient scheme (STIC) showed relatively
larger phase lags under both dry and wet conditions (*t*_{φ}=14–20 min) (Fig. 7, Table 4).

Since all evapotranspiration schemes use *R*_{n}−*G* as forcing, we also
computed the phase lags with *R*_{n}−*G* as a reference variable (see Table 4).
The differences to *R*_{sd} as reference are, however, rather small with
slightly lower phase lags and in the range of the standard deviation of the
daily estimates. This small difference can be attributed to a negligible
phase lag between *R*_{sd} and *R*_{n} as well as the rather small magnitude
and the phase lag of the soil heat flux.

## 3.4 Diurnal patterns of observed fluxes and states

In order to understand the diurnal patterns of *λ**E* we also analyzed
the hysteresis loops of the observed surface energy balance components
[$\mathit{\lambda}E=({R}_{\mathrm{n}}-G)-H$] with respect to *R*_{sd} (Fig. 8).
Generally, there was only a small hysteresis in the available energy
(*R*_{n}−*G*) (Table 4). The turbulent heat fluxes showed significant
hysteresis under wet conditions but not under dry conditions. Interestingly,
under wet conditions the CCW hysteresis of *λ**E* with a phase lag (mean
*t*_{φ}=15 min) was mostly compensated for by a CW hysteresis of *H*
(mean ${t}_{\mathit{\phi}}=-\mathrm{22}$ min) (Fig. 8 and Table 4). This compensation
is an outcome of net available energy (*R*_{n}−*G*) showing little hysteresis
for both conditions.

We next analyzed the bulk sensible heat flux formulation used in the OSEB
and TSEB models to understand how the observations of temperature and the
inferred aerodynamic conductances are related to each other. The diurnal
patterns of both air and surface temperature revealed a strong CCW
hysteresis with *R*_{sd} (Fig. 9). Air temperature showed a more pronounced
hysteretic loop than surface temperature, and with a triangular shape with
higher values during the afternoon when solar radiation reduces.
Interestingly, the surface-to-air temperature gradient, being the driving
gradient for the sensible heat flux, showed much lower hysteretic behavior.
The hysteresis is in a clockwise direction, with a higher gradient in the
morning hours compared to the afternoon. It had a similar phase lag to *H* (see Table 4).

We further analyzed different formulations of the aerodynamic conductance (*g*_{a})
directly inferred from measurements and from how these are
represented in the models evaluated here (FAO-PM, OSEB, TSEB, STIC). We
inferred the aerodynamic conductance from observations in three different
ways: firstly, we used the eddy-covariance measurements of friction velocity (*u*^{*})
and wind speed (*u*) to estimate the aerodynamic conductance for momentum
(${g}_{\mathrm{am}}={u}^{*\mathrm{2}}/u$). We then used the empirical formula by Thom (1972) to
calculate the aerodynamic conductance for heat including the excess
resistance to heat transfer (Eq. 5). Thirdly, we inferred the aerodynamic
conductance from the observed sensible heat flux (*H*_{BRC}) and temperature
gradient (*T*_{s}−*T*_{a}) by inverting *H*_{BRC} using Eq. (6). The FAO-PM
describes the aerodynamic conductance with a simple linear relationship to
wind speed. OSEB and TSEB estimates the aerodynamic conductance to heat (*g*_{ah}),
while STIC estimates the conductance to water vapor (*g*_{av}).
Thus by comparing these different conductance estimates we assume similarity
between the fluxes.

The different estimates for the aerodynamic conductance are compared to each
other in Fig. 10 for midday conditions. Although the three observation-based
estimates show some variations in the absolute value of the aerodynamic
conductance, they consistently showed a significantly greater conductance
for dry days compared to wet days, suggesting a stronger aerodynamic
exchange between the surface and the atmosphere under dry conditions. This
difference in aerodynamic conductance is partly reproduced by the simple
FAO-PM scheme, which means that the median wind speed was higher under the
drier conditions. The temperature-gradient schemes (OSEB and TSEB) reproduce
the wet–dry difference rather well, and they also use wind speed but rely on
Monin–Obukhov similarity theory and stability correction. STIC, which does not use wind
speed, did not show any significant differences in *g*_{av} between wet and dry conditions.

Finally we analyze the diurnal patterns of the vapor pressure deficit
${D}_{\mathrm{a}}={e}_{\mathrm{s}}\left({T}_{\mathrm{a}}\right)-{e}_{\mathrm{a}}$, which is a critical driver of the latent
heat flux in the PM equation. Since *D*_{a} is derived from the observations,
we analyzed its diurnal patterns in Fig. 11. We found that the vapor
pressure in the air remained fairly constant during the day; hence it did
not co-vary with *R*_{sd} and only showed a small CW hysteresis with higher
vapor pressure during the morning compared to during the afternoon. The saturation
vapor pressure, which is a function of air temperature, however, showed a
distinct and large CCW hysteresis loop with respect to *R*_{sd}, which is
consistent with the large hysteresis in air temperature (Figs. 9 and 12). As
a consequence, *D*_{a} also showed a distinct and large CCW hysteresis with a
large phase lag of ${t}_{\mathit{\phi}}=\sim \mathrm{150}$ min (see Table 4).
This large hysteresis and phase lag is consistent with the respective
characteristics of air temperature, but not with those of the temperature
gradient (see Fig. 9). Furthermore, we note that the phase lag in *D*_{a} did
not show any significant influence of wetness, while the phase lag of the
temperature gradient became more negative under wet conditions (Fig. 12,
Table 4). It would thus seem that the bias in PM-based estimates identified
here may relate to a too-pronounced role of *D*_{a} in the evapotranspiration estimate.

## 4.1 Dominant controls of *λ**E* at the diurnal cycle

Our analysis of the diurnal cycle showed that *λ**E* follows the diurnal
course of incoming solar radiation, explaining most of the variance in *λ**E*.
However, a significant nonlinearity in the form of a phase lag
between *λ**E* and *R*_{sd} was detected, which showed larger *λ**E*
for the same *R*_{sd} in the afternoon as in the morning. We found that the
lag in *λ**E* is accompanied by a preceding phase lag of the sensible
heat flux, while the other surface energy balance components (e.g., net
radiation and soil heat flux) revealed very small phase lags with *R*_{sd}.
Hence, there is compensation between the phase shifts of sensible and latent
heat fluxes, which becomes more apparent under the wet conditions. Our
results are consistent with the comprehensive FLUXNET studies of Wilson et al. (2003)
and Nelson et al. (2018) which used a different metric (median
centroid) for assessing diurnal phase shifts. Wilson et al. (2003) found
that *H* precedes *λ**E* at most sites, with the exception of sites in a
Mediterranean climate. Using the FLUXNET2015 dataset, Nelson et al. (2018)
found that the median centroid of *λ**E* occurs predominantly in the
afternoon across all plant functional types when *f*_{E}>0.35,
while for very dry conditions (*f*_{E}<0.2) a shift of the
*λ**E* centroid towards the morning was found. This indicates that our
results are not just applicable to Luxembourg, but are a general phenomenon
which justifies a wider interpretation within temperate climates.

It is important to emphasize here that the observed phase lags are not
dominated by diurnal heat storage changes below the surface, since both the
diurnal magnitude and the phase lag of the soil heat flux were relatively
small compared to the turbulent heat fluxes. The models employed here use
available energy (*R*_{n}−*G*) as input to estimate *λ**E*. However, the
phase lag of the latent heat flux would only reduce by about 3 min when
choosing *R*_{n}−*G* instead of *R*_{sd} as the reference variable to calculate the
phase lags. Hence, the observed phase lags of *λ**E* and *H* to *R*_{sd} are
not an artifact of the analysis, but can be considered as an imprint of L–A interaction.

The obtained phase lags of the surface fluxes and variables allow for a process-based insight into the diurnal heat exchange of the surface with the atmosphere. Since there is only limited heat storage in the surface layer itself, which explains the small phase lags of the heat fluxes, the heating imbalance caused by solar radiation must be effectively redistributed. Over land it is the lower atmosphere which acts as efficient heat storage to buffer most of the diurnal imbalance caused by solar radiation, because the heat storage of the subsurface is limited by the relatively slow heat conduction into the soil. Thus, the lower atmosphere is effectively heated by surface longwave emission and the sensible heat flux, which in combination with the diurnal cycle of vertically transported turbulent kinetic energy (TKE) leads to the development of the convective planetary boundary layer (CBL) (e.g., Oke, 1987). The changes in heat storage in the CBL are reflected by the very large phase lags for air temperature and longwave downwelling radiation, which both have a phase lag of about 2.5 h. This large phase lag of air temperature then shapes (i) the vertical surface-to-air temperature gradient, which drives the sensible heat flux; and (ii) the vapor pressure deficit of the air. Despite the complexity of processes within the convective boundary layer, including the morning transition and entrainment at its top, we find that all surface energy components correlate strongly with solar radiation (Table 4). What this suggests is that the state of the surface–atmosphere system is predominantly shaped by fluxes, particularly by solar radiation as its primary driver, with the state in terms of temperatures and humidity gradients adjusting to these fluxes, rather than the reverse, where the state (in terms of temperature and humidity) drives the fluxes.

We also found that the phase lag of the turbulent heat fluxes is affected by soil water availability. This is most clearly seen for the surface-to-air temperature gradient and the sensible heat flux, whose phase lag is 2 times larger for wet than for dry days. This means that for the same solar radiation forcing we find higher values of the sensible heat flux in the morning than in the afternoon. The effect of water availability is also seen for the phase lag of the latent heat flux and to a lesser extent for the soil heat flux.

Our findings agree well with studies which use the diurnal centroid method,
showing that moisture limitation decreases the lag in timing of *λ**E*
(Wilson et al., 2003; Xiang et al., 2017; Nelson et al., 2018). The phase
shift of *D*_{a} might enhance evaporation at the cost of the sensible heat
flux during the afternoon under sufficient moisture availability. However,
under drier conditions, our findings suggest that the surface heats more
strongly and generates more buoyancy, which is reflected by higher
aerodynamic conductances as compared to the wet conditions (Fig. 10). The
larger aerodynamic conductance would then enable a more effective sensible
heat exchange and would thus lower the phase difference between the
sensible and latent heat fluxes.

Note that our interpretation disregards the effects of horizontal advection
of moisture and temperature. Events of strong advection, e.g., of
temperature, can add heat to the surface energy balance and thus alter the diurnal cycle.
Similarly, events of dry air advection may enhance local *λ**E* at the
cost of the sensible heat flux. Since we used composite averages and
statistics over a set of days we aim to reduce the impact of such advective
events. We expect that it is unlikely that such events occurred throughout
all wet–dry days in a consistent manner.

## 4.2 Using phase lags to identify model biases

Our comparison of different modeling approaches shows that by using phase
lags one can identify biases in evapotranspiration parameterizations and
relate these towards processes for a better understanding of
surface–atmosphere interactions under different conditions of water
availability. One of our main findings is that the surface energy balance
fluxes and the temperature gradient have a comparatively small phase lag to
the incoming solar radiation, while air temperature and vapor pressure
deficit have substantial phase lags. This difference in phase lags can then
be used to infer biases in estimates of evapotranspiration. In our
application of this approach to observations of one site in a temperate
climate we found that evapotranspiration exhibits a comparatively small
phase lag, indicating that it was dominantly driven by solar radiation and
temperature gradients, and not by the water vapor pressure deficit. Our
findings are in line with observations of a near-linear relationship of *λ**E*
to *R*_{sd} by Jackson et al. (1983) which stimulated remote-sensing-based spatial mapping of *λ**E* (Crago, 1996). Also the
semi-empirical Makkink equation to estimate potential evapotranspiration
(Makkink, 1957; de Bruin and Lablans, 1998; de Bruin et al., 2015) uses *R*_{sd} as the main driver.

Further support of the argument is given by the successful application of
equilibrium evapotranspiration (Schmidt, 1915; Priestley and Taylor, 1972;
Miralles et al., 2011; Renner et al., 2016) which uses *R*_{n} and air
temperature as key inputs.

Our interpretation is consistent with studies of non-water-stressed evapotranspiration that is best represented by potential evapotranspiration schemes which are primarily driven by net radiation, as demonstrated for FLUXNET observations by Maes et al. (2018) and for climate model simulations by Milly and Dunne (2016). Milly and Dunne (2016) interpreted these findings in terms of strong feedbacks between the surface and the atmosphere, which couple the surface variables and which result in a top-down energy constraint that is well captured by energy-only formulations.

Our analysis allows for the better understanding of the relevance of the feedbacks
which occur at a sub-daily timescale. These feedbacks are driven by the
redistribution of heat gained by absorption of solar radiation at the
surface, which causes a significant co-variation of the input variables to
incoming solar radiation (Table 4). This is especially important for the
vapor pressure deficit of the air which acts as a driver of *λ**E* and
is also known to affect the stomatal conductance (Jarvis, 1976; Jarvis and
McNaughton, 1986). De Bruin and Holtslag (1982) showed that a positive
correlation between *D*_{a} and *R*_{n} allows simplifying the complex
PM equation to a form similar to equilibrium evaporation (Eq. 8) with net
radiation as the dominant driver. Therefore, simpler, energy-based
formulations for *λ**E* show similar performance to PM-based approaches,
but with less input parameters (De Bruin and Holtslag, 1982; Beljaars and
Bosveld, 1997). The challenge of the PM equation is then a parameterization
of the conductances, which must capture the feedbacks included in the input
data. Since the co-variation originates from the diurnal redistribution of
heat, a mismatch would then clearly be seen at the sub-daily timescale.
Hence by focusing on the internal relationship of the modeled *λ**E*
to *R*_{sd} at the sub-daily timescale we found that (i) the Penman–Monteith-based
approaches showed a consistently larger phase lag than what was actually
observed and (ii) these approaches did not show a reduction of the phase lag
under dry conditions. The PM approaches use the vapor pressure deficit as an
input which showed a substantial hysteresis loop on the order of 2.5 h
lagging *R*_{sd}. This is due to the temperature dependency of the saturation
vapor pressure, while the actual vapor pressure shows no relationship with *R*_{sd}.
Besides *D*_{a} and *s*, all other input variables to the Penman–Monteith
approaches used here (both FAO and STIC) showed minor phase lags with
respect to *R*_{sd}. Since the surface conductance in the FAO Penman–Monteith formulation is
fixed with time, the resulting prediction of potential *λ**E* showed a
significant and large phase lag on the order of 0.5 h. Even when we use the
observed aerodynamic conductance as input, the effect remains the same,
which emphasizes that a constant surface conductance results in biases in
the diurnal cycle of *λ**E*. In contrast to assuming a constant *g*_{s},
STIC computes *λ**E* through analytical estimation of *g*_{s}
and *g*_{av} from the information of both the surface-to-air temperature gradient
and the vapor pressure deficit. This dynamic treatment of *g*_{s} reduced the
phase lag to values similar to the observations under the wet conditions.
However, under dry conditions STIC still showed significant phase lags,
which may be related to the lag of *D*_{a} to *R*_{sd} which was similar for
both dry and wet conditions (Fig. 12). Hence, our analysis indicates that
the PM-based approaches used here overestimated the effect of water vapor
deficit on actual evapotranspiration, which, in the end, reflects the
estimation of the surface and the aerodynamic conductance to water vapor.

The temperature-gradient approaches used here (OSEB and TSEB) are
structurally different from the PM approaches, since they infer *λ**E* from
the residual of the surface energy balance and thus do not explicitly deal
with the aerodynamic and surface conductance of water vapor. The phase lag
analysis of the environmental variables used to drive the predictive models
of *λ**E* helped to identify an important benefit of the
temperature-gradient approaches over the Penman–Monteith-based approach.

The temperature-gradient approaches employ the vertical temperature gradient
(*T*_{s}−*T*_{a}) which showed a significant counterclockwise, i.e., a
leading, hysteretic loop, which is on the order of the phase shift detected
for the sensible heat flux (Fig. 12). In addition, there is a distinct and
significant increase in the phase shift in both the temperature gradient and
the sensible heat flux under the wet conditions. Hence, the temperature
gradient as input contains valuable information on water limitation in terms
of the magnitude (i.e., the slope of (*T*_{s}−*T*_{a}) to *R*_{sd}) and the
diurnal phase lag (see Table 4).

While the PM approaches must identify two conductances simultaneously, the
temperature-gradient approaches only need to parameterize the aerodynamic
conductance to heat (*g*_{ah}) using wind speed as input. Therefore, we found
that these approaches agreed well with the approximated *g*_{ah} from the
EC tower, which shows an enhanced conductance under dry conditions. In contrast,
the diagnosed *g*_{av} from STIC did not show substantial differences between
dry and wet conditions, pointing to the difficulty of the analytical
approach and its associated assumptions to identify two bulk conductance
parameters from the available radiometric and meteorological data (Mallick
et al., 2018) for the climatic conditions in which these were evaluated here.

Note that we evaluated a temperate grassland site which experienced an
exceptional summer drought. Therefore, the evaporative fraction did not
decline below 0.3. In semi-arid ecosystems the evaporative fraction may
decrease substantially below 0.3 and Nelson et al. (2018) showed that there
is a morning shift of *λ**E* (analogous to a negative phase lag) under
very dry conditions (*f*_{E}<0.2). This points towards a different
stomatal regulation changing the diurnal course of surface conductance.
While it was shown by Bhattarai et al. (2018) that STIC performs well also
under semi-arid conditions, temperature-gradient approaches can show larger
biases under semi-arid conditions (Morillas et al., 2013). The difficulty of
temperature-gradient approaches is predominantly in the parameterization of
aerodynamic conductance of heat which becomes more challenging under these
very dry conditions (Kustas et al., 2016).

The relevance of the diurnal timescale for the problem of surface
conductance parameterizations was already highlighted by Matheny et al. (2014).
However, they and others evaluated the diurnal patterns of the
hysteretic loops between *λ**E* and *D*_{a} (see also Zhang et al.,
2014; Zheng et al., 2014). Given that solar radiation is the cause of the
strong L–A feedbacks at the diurnal timescale we believe that solar
radiation is better suited as a reference variable than *D*_{a}. Our results
show that the new metric of the phase lag of heat fluxes and surface states
to incoming solar radiation reveals important biases of evapotranspiration
schemes. These biases may well be compensated for at a longer timescales
(Matheny et al., 2014) but would lead to biased sensitivities with respect
to climate change (Milly and Dunne, 2016). Here, we applied the phase lag
metric to observationally driven evapotranspiration schemes. In the future,
we plan to apply these new metrics based on hysteretic loops to model
outputs of land-surface models (such as NOAH-MP, Niu et al., 2011) as well
as of fully coupled surface–atmosphere simulations in order to detect and to
identify errors in the parameterization of state-of-the-art LSMs.

We analyzed the relationship of surface heat fluxes and states to incoming
solar radiation at the sub-daily timescale for a temperate grassland site
which experienced a summer drought. Most variables showed significant
hysteresis loops which we objectively quantified by a linear component and a
nonlinear phase lag component using multiple linear regression and harmonic
analysis. We then compared these diurnal signatures obtained from
observations of an eddy-covariance station with commonly used but
structurally different approaches to model actual and potential
evapotranspiration. The models have been forced by the observational data
such that the differences to observations can be attributed to model
formulation and signals contained in the input data. Our analysis guides
model selection with a preference for the temperature-gradient approaches,
because the vertical temperature gradient contains relevant signals of soil
moisture limitation as opposed to the vapor pressure deficit of the air.
Furthermore, schemes which use vapor pressure deficit as additional input
(such as the Penman–Monteith formulation) require a dynamic, i.e., time-dependent,
characterization of surface conductance to account for the strong
phase lag in vapor pressure deficit. Hence, our results suggest that
simpler *λ**E* approaches based on the surface energy balance and surface
temperature may be more suitable to estimate evapotranspiration from
observational data (e.g., remote sensing data) in climates without
substantial water stress. Apparently, the surface observations already
contain the imprint of land–atmosphere interactions, whereas in the case of
coupled land-surface–atmosphere models these interactions are explicitly
resolved. Hence, detailed models of aerodynamic and surface conductance and
its interaction with the environment are of crucial importance for skillful
climate predictions including the carbon cycle (Prentice et al., 2014; Wolf
et al., 2016; Konings et al., 2017).

We suggest that an evaluation of these schemes should be based on the sub-daily timescale, because a land–atmosphere exchange scheme must accomplish a balance between the surface energy balance with small imprints of heat storage changes and the lower atmosphere with strong imprints of heat storage changes (Kleidon and Renner, 2017). Although a mismatch of the diurnal patterns may not be detected at the aggregated timescales of days and months, it may lead to biased model sensitivities (Matheny et al., 2014). For example, an overly sensitive formulation of evapotranspiration to vapor pressure deficit and thus to temperature would predict larger changes in potential evapotranspiration under global warming (Milly and Dunne, 2016). Here, we analyzed observationally driven evapotranspiration schemes and their inputs, which revealed an apparent energy constraint. This constraint, which appears as a strong correlation of surface fluxes and gradients to incoming solar radiation should be correctly represented by any land-surface model which resolves the land–atmosphere interaction. While this may sound trivial, recent benchmarking studies showed that current state-of-the-art land-surface models have difficulties in representing the strong link of turbulent heat fluxes to solar radiation (Best et al., 2015; Haughton et al., 2016). Our findings provide an explanation of this model deficiency and we suggest that further information is gained by evaluating land-surface schemes in terms of phase lags in surface fluxes and states such as the sensible and soil heat flux including the diurnal dynamics of surface and air temperatures. Correctly representing these metrics will lead towards a more accurate representation of the diurnal heat and mass exchange of the land with the atmosphere.

The data analysis was performed with the open-source environment R (R Core Team, 2015). Functions to calculate the phase lag are provided as R package “phaselag” (Renner, 2019a), which is available from GitHub at https://github.com/laubblatt/phaselag, 2019. The script to perform data analysis and figures is published on Zenodo (Renner, 2019b) and can also be obtained from https://github.com/laubblatt/2018_DiurnalEvaporation, 2019. Code to perform OSEB and TSEB simulations is published as Brenner (2019) and can also be found at https://github.com/ClaireBrenner/pyTSEB_Renner_et_al2018, 2019. Code to simulate STIC1.2 simulations is available upon request from Kaniska Mallick (LIST, kaniska.mallick@list.lu).

Data of observations (Wizemann et al., 2018, http://doi.org/10.5880/fidgeo.2018.024) and model output (Renner et al., 2018, http://doi.org/10.5880/fidgeo.2018.019) used in this study can be obtained from the research data repository GFZ Data Services (http://dataservices.gfz-potsdam.de).

MR and AK conceived the analysis of the diurnal cycle. VW, KS, and IT designed the field campaign. HW carried out the EC measurements and EC data processing. CB performed OSEB–TSEB simulations. KM performed STIC1.2 simulations. MR merged the data and performed the data analysis. MR and LC developed the phase lag computation. JW provided ancillary simulation data. IT provided climate information. MR prepared the manuscript with contributions from all co-authors.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Linking landscape organisation and hydrological functioning: from hypotheses and observations to concepts, models and understanding (HESS/ESSD inter-journal SI)”. It is not associated with a conference.

This study was supported by the German Research Foundation (DFG) through funding
of the research unit “From Catchments as Organised Systems to Models based on
Dynamic Functional Units – CAOS” (FOR 1598) within the sub-project
“Understanding and characterizing land-surface–atmosphere exchange and feedbacks”
(project number 182331427). Maik Renner and Axel Kleidon were funded
by DFG grant number KL 2168/2-1. Claire Brenner was supported by the Austrian
Science Fund (FWF) through funding of the CAOS Research Unit (I 2142-N29).
Kaniska Mallick was supported by the Luxembourg Institute of Science and
Technology (LIST) through the project BIOTRANS (grant number 00001145), CAOS-2
project grant INTER/DFG/14/02 funded by Fonds National de la Recherche (FNR)–DFG,
and HiWET project funded by the Belgian Science Policy (BELSPO)–FNR under
the program STEREO III (INTER/STEREOIII/13/03/HiWET; contract no. SR/00/301).
IT was supported by the Luxembourg Institute of Science and Technology (LIST)
through the project BIOTRANS (grant number 00001145).

The article processing charges for this open-access

publication
were covered by the Max Planck Society.

Edited by: Shraddhanand Shukla

Reviewed by: Chunlüe Zhou and one anonymous referee

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