# When does vapor pressure deficit drive or reduce evapotranspiration?

Adam Massmann Corresponding author<EMAIL_ADDRESS>Department of Earth
and Environmental Engineering, Columbia University, New York, NY 10027  Pierre
Gentine  Department of Earth and Environmental Engineering, Columbia
University, New York, NY 10027  Changjie Lin  Department of Earth and
Environmental Engineering, Columbia University, New York, NY 10027  State Key
Laboratory of Hydroscience and Engineering, Department of Hydraulic
Engineering, Tsinghua University, Beijing, CN 100084

###### Abstract

Increasing vapor pressure deficit (VPD) increases atmospheric demand for
water, and vapor pressure deficit is expected to rise with increasing
greenhouse gases. While increased evapotranspiration (ET) in response to
increased atmospheric demand seems intuitive, plants are capable of reducing
ET in response to increased VPD by closing their stomata, in an effort to
conserve water. Here we examine which effect dominates response to increasing
VPD: atmospheric demand and increases in ET, or plant physiological response
(stomata closure) and decreases in ET. We use Penman-Monteith, combined with
semi-empirical optimal stomatal regulation theory and underlying water use
efficiency, to develop a theoretical framework for understanding how ET
responds to increases in VPD.

The theory suggests that for most environmental conditions and plant types,
plant physiological response dominates and ET decreases with increasing VPD.
Plants that are evolved or bred to prioritize primary production over water
conservation (e.g. crops) exhibit a higher likelihood of atmospheric demand-
driven response (ET increasing). However for forest, grass, savannah, and
shrub plant types, ET more frequently decreases than increases with rising
VPD. This work serves as an example of the utility of our simplified framework
for disentangling land-atmosphere feedbacks, including the characterization of
ET response in an atmospherically drier, enriched CO2 world.

##  1 Introduction

Vapor pressure deficit (VPD) is expected to rise over continents in the future
due to the combination of increased temperature and, depending on region,
decreased relative humidity (Byrne and O’Gorman, 2013). Increases in VPD
increase the atmospheric demand for evapotranspirated water (Penman, 1948;
Monteith et al., 1965), but also stress plant stomata (Leuning, 1990; Medlyn
et al., 2011).

The opposing effects of increased atmospheric demand and higher stomatal
stress lead to two possible perspectives for how evapotranspiration (ET)
responds to shifts in VPD. The first, a hydrometeorological perspective, is
that higher VPD increases atmospheric demand for water from the land surface,
and this drives an increase in evapotranspiration (ET). This perspective is
particularly relevant because potential evapotranspiration (PET), which is
used in many drought indices and hydrometeorological studies (e.g., Heim Jr,
2002; Scheff and Frierson, 2015), typically only quantifies changes in
atmospheric demand and fails to account for ecosystem response (Swann et al.,
2016). In reality, plants’ stomata have evolved to optimally regulate the
exchange of water and carbon, and tend to partially close in response to
increased atmospheric dryness (Farquhar, 1978; Ball et al., 1987; Leuning,
1990; Medlyn et al., 2011). This leads to a plant physiology perspective, in
which an increase in VPD, particularly in well-watered soil conditions, may
actually correspond to a decrease in ET because of stomatal closure (e.g.
Rigden and Salvucci, 2017). In other words, the question “When does VPD drive
or reduce ET?” can be related to whether plant regulation or atmospheric
demand dominates ET response.

The ET response to changes in VPD alters water partitioning between the soil
and atmosphere. If ecosystem plant response reduces ET with atmospheric drying
then soil moisture will be better conserved. This represents a sensible
evolutionary strategy to cope with aridity: save water for periods when
atmospheric demand for water is relatively low, and atmospheric carbon can be
accessed with a relatively smaller cost in water loss. If instead stomata were
fully passive (similar to soil pores, e.g. Or et al., 2013), increased
atmospheric aridity would strongly reduce soil moisture (Berg et al., 2017).
This could further increase aridity as low soil moisture levels increases the
Bowen ratio, leading to increased temperature and atmospheric drying (Bouchet,
1963; Morton, 1965; Brutsaert, 1999; Ozdogan et al., 2006; Salvucci and
Gentine, 2013; Gentine et al., 2016; Berg et al., 2016). Therefore, passive
regulation and a lack of soil moisture conservation does not seem to be a
sensible strategy for plants from an evolutionary standpoint.

As a counterpoint, one may argue that increases in ET with increasing VPD
could increase the likelihood of precipitation (e.g., Findell et al., 2011).
However, increases in ET do not always guarantee an increased likelihood of
precipitation, which depending on environmental conditions could cause a
decrease in the likelihood of precipitation (Gentine et al., 2013).
Furthermore, any increases in precipitation are likely to be non-local, such
that plants giving up water to the atmosphere are not guaranteed to reap the
benefits of water returned from the atmosphere to the soil. Using this
subjective logic, from an ecosystem evolutionary perspective, water stored in
soil seems to be worth much more than the chance of water returned as
precipitation.

We can use intuition about plant water conservation strategy to hypothesize
about ET response to changes in VPD. Plants and ecosystems that evolved to
conserve water, such as arid shrubs or savannah, should be more likely to
reduce ET with increasing VPD, and plants that have evolved or have been
engineered to care little about water, such as crops, will be more likely to
increase ET with increasing VPD. Atmospheric conditions must matter as well.
At the ecosystem scale, there are limits to plant water conservation
strategies. As atmospheric demand for water (VPD) increases, ecosystems should
begin to reach their water conservation limits and might not be able to
entirely limit ET flux to the atmosphere. At this stage any further increase
in VPD will most likely drive a (limited) increase in ET, because the increase
in atmospheric demand for water overwhelms the limited plant response to
conserve water.

The objective of the present manuscript is to use reasonable approximations as
a tool to develop intuition for plant response to atmospheric drying and
evaluate the VPD dependence of ET. This intuition will aid interpretation of
observations, full complexity models, and facilitate the disentanglement of
complex land-atmosphere feedbacks. In the past, similar approaches were used
to understand interactions between stomatal conductance, evapotranspiration
and the environment (e.g., Jarvis and McNaughton, 1986; McNaughton and Jarvis,
1991). However, at the time researchers’ understanding of the form of VPD’s
effect on plant physiology was limited, so they could not explore the
sensitivity of ET to VPD, including VPD’s effect on stomatal conductance and
plant function.

Recent results have drastically improved our understanding of VPD’s impact on
physiology, especially at the leaf level. Medlyn et al. (2011) developed a
model for leaf-scale stomatal conductance ($g_{s}$), including VPD response,
by combining an optimal photosynthesis theory (Cowan and Farquhar, 1977) with
an empirical approach, and extended use of this model to the ecosystem scale
in Medlyn et al. (2017). Additionally, Zhou et al. (2014) demonstrated that a
quantity underlying water use efficiency
$\left(uWUE=\frac{GPP\;\sqrt{VPD}}{ET}\right)$ properly captures a constant
relationship between GPP, ET, and VPD over a diurnal cycle at the ecosystem
scale. uWUE is also remarkably well conserved in the growing season across
space and time, within a PFT (Zhou et al., 2015). While stomatal conductance
parameterizations and uWUE greatly simplify complex plant physiological
processes, they still capture leading order ecosystem behavior for vegetated
surfaces (Medlyn et al., 2017; Zhou et al., 2014), and are novel tools to
transparently develop intuition for the behavior of complex, multiscale
ecohydrologic systems.

In this manuscript, we leverage uWUE and recent developments in stomatal
conductance parameterizations (Medlyn et al., 2011) to derive the theoretical
one-way response of ET to VPD with other environmental variables properly
controlled for, i.e. we develop a framework for evaluating the partial
derivative of ET with respect to VPD. For the first time, we explicitly
include VPD’s full effect on stomatal conductance, including its leading order
impact on photosynthesis. Our theory is validated and tested at multiple eddy-
covariance stations spanning various climates and plant functional types.

##  2 Materials and Methods

###  2.1 Data

We use both meteorological and eddy-covariance data from the FLUXNET2015
database (data available at
https://fluxnet.fluxdata.org/data/fluxnet2015-dataset/ ), including all Tier 1
sites with at least four years of data and observations of the variables
described in the methods section (Sect. 2.2). Sixty-six sites met these
requirements, and were grouped into nine plant functional types (PFT)
according to the International Geosphere-Biosphere Programme vegetation
classification scheme (Loveland et al., 1999): cropland (CRO), grass (GRA),
deciduous broadleaf forest (DBF), evergreen broadleaf forest (EBF), evergreen
needleleaf forest (ENF), mixed forest (MF), closed shrub (CSH), savannah
(SAV), and woody savannah (WSA) (site locations and citations are in Fig. 1
and Appendix A, respectively).

Figure 1: Plant functional type and location of FLUXNET2015 sites used in this
analysis.

The purpose of this study is to examine ecosystem response to atmospheric
drying, focusing on the growing season. To accomplish this, we filter and
quality control the data using a similar procedure as Zhou et al. (2015):

  * •

Only measured or highest (“good”) quality gapfilled data, according to quality
control flags, are used.

  * •

To isolate the growing season, we only use days in which the average Gross
Primary Productivity (GPP) exceeds 10% of the observed 95th percentile of GPP
for a given site. GPP is calculated using the nighttime respiration
partitioning method.

  * •

We remove days with rain and the day following to avoid issues with rain
interception and sensor saturation at high relative humidity (Medlyn et al.
(2017)).

Additionally, as in Lin et al. (2018), we restrict data to the daytime, which
is identified when downwelling shortwave radiation is greater than 50 W m-2
and sensible heat flux is greater than 5 W m-2. To reduce the chance of sensor
saturation at high relative humidity, we remove all time steps for which VPD
is less than .01 kPa, and to reduce errors at low windspeeds we remove all
periods with wind magnitudes less than 0.5 m s-1. Timesteps with negative
observed GPP or ET are also removed, and we aggregate half hourly data to
hourly averages to reduce noise (Lin et al., 2018). After these quality
control procedures, 400,983 upscaled hourly observations remain.

###  2.2 Methods

The Penman-Monteith equation (hereafter PM, Penman, 1948; Monteith et al.,
1965) estimates ET as a function of observable atmospheric variables and
surface conductances:

$ET=\frac{\Delta
R_{net}+g_{a}\rho_{a}c_{p}VPD}{\Delta+\gamma(1+\frac{g_{a}}{g_{s}})},$ (1)

where $\Delta$ is the change in saturation vapor pressure with temperature,
given by Clausius-Clapeyron ($\frac{d\;e_{s}}{d\;T}$), $R_{net}$ is the net
radiation minus ground heat flux, $g_{a}$ is aerodynamic conductance,
$\rho_{a}$ is air density, $c_{p}$ is specific heat of air at constant
pressure, $\gamma$ is the psychometric constant, and $g_{s}$ is the stomatal
conductance (Table 1).

Table 1: Definition of symbols and variables, with citation for how values are calculated, if applicable. Variable | Description | Units | Citation
---|---|---|---
$e_{s}$ | saturation vapor pressure | Pa | -
$T$ | temperature | K | -
$P$ | pressure | Pa | -
$\Delta$ | $\frac{\partial e_{s}}{\partial T}$ |  Pa K-1 | -
$R_{net}$ | net radiation at land surface minus ground heat flux |  W m-2 | -
$g_{a}$ | aerodynamic conductance |  m s-1 | Shuttleworth (2012)
$\rho_{a}$ | air density |  kg m-3 | -
$c_{p}$ | specific heat capacity of air at constant pressure |  J K-1 kg-1 | -
$VPD$ | vapor pressure deficit | Pa | -
$\gamma$ | psychometric constant |  Pa K-1 | -
$g_{s-leaf}$ | leaf-scale stomatal conductance |  m s-1 | Medlyn et al. (2011)
$g_{s}$ | stomatal conductance |  m s-1 | Medlyn et al. (2017)
$g_{1-leaf}$ | leaf-scale slope parameter |  Pa0.5 | Medlyn et al. (2011)
$g_{1}$ | ecosystem-scale slope parameter |  Pa0.5 | Medlyn et al. (2017)
$R$ | universal gas constant |  J mol-1 K-1 | -
$R_{air}$ | gas constant of air |  J K-1 kg-1 | -
$\sigma$ | uncertainty parameter | - | -
$c_{a}$ |  CO2 concentration |  $\mu$ mol CO2 mol-1 air | -
$\lambda$ | marginal water cost of leaf carbon |  mol H2O mol-1 CO2 | -
$\Gamma$ |  CO2 compensation point | - | -
$\Gamma^{*}$ |  CO2 compensation point without dark respiration | - | -

Medlyn et al. (2011) developed a model for stomatal conductance ($g_{s}$) by
combining an optimal photosynthesis theory (Cowan and Farquhar, 1977) with an
empirical approach, which describes the dependence of $g_{s}$ to VPD. This
resulted in the following model for leaf-scale stomatal conductance:

$g_{s-leaf}=g_{0}+1.6\left(1+\frac{g_{1-leaf}}{\sqrt{VPD}}\right)\frac{A}{c_{a}},$
(2)

where $g_{1-leaf}$ is a leaf-scale “slope” parameter, A is the net CO2
assimilation rate, and $c_{a}$ is the atmospheric CO2 concentration at the
leaf surface. Medlyn et al. (2011) relate the slope parameter ($g_{1-leaf}$)
to physical parameters as:

$g_{1-leaf}=\sqrt{\frac{3\,\Gamma^{*}\,\lambda}{1.6}},$ (3)

where $\Gamma^{*}$ is the CO2 compensation point for photosynthesis (without
dark respiration), and $\lambda$ is the marginal water cost of leaf carbon
($\frac{\partial\;\text{transpiration}}{\partial\;A}$). So, $g_{1-leaf}$ is a
leaf-scale term reflecting the trade-off of water for carbon uptake. The
higher $g_{1-leaf}$, the more open the stomata and the more they release water
in exchange for carbon.

The Medlyn model for stomatal conductance has been shown to behave very well
across PFTs (Lin et al., 2015), and has been successfully adopted to ecosystem
scale analysis in Medlyn et al. (2017). In units of m s-1, the ecosystem scale
stomatal conductance is:

$g_{s}=\frac{R\,T}{P}\;1.6\left(1+\frac{g_{1}}{\sqrt{VPD}}\right)\frac{GPP}{c_{a}},$
(4)

where GPP is the ecosystem scale gross primary production, and $g_{1}$ is an
ecosystem scale analogue to $g_{1-leaf}$. We solve for $g_{1}$ following
Medlyn et al. (2017) (Eq. (5)), and take the median $g_{1}$ value to be
representative of each PFT (Table 2.2) instead of the mean to avoid extra
weighting of rare outliers.

Table 2: Plant functional types, their abbreviation, calculated Medlyn
coefficient, and calculated uWUE. uWUE from Zhou et al. (2015), including the
observed standard deviation, is shown for comparison. Note that uWUE from Zhou
et al. (2015) is calculated from a different set of sites, and that units are
converted such that the quantities work with Equations 1-8 and the variables
defined Table 1.

Abbreviation | PFT |  $g_{1}$ (Pa0.5) |  uWUE ($\mu$-mol [C] Pa0.5 J-1 [ET])
---|---|---|---
fitted | Zhou et al. (2015)
CRO | Crops | 140.67 | 2.85 |  3.80 $\pm$ 1.01
DBF | Deciduous Forest | 117.26 | 2.96 |  3.12 $\pm$ 0.52
EBF | Evergreen Broadleaf Forest | 101.92 | 3.12 | N/A
SAV | Savannah | 96.07 | 2.79 | N/A
GRA | Grass | 145.56 | 2.13 |  2.68 $\pm$ 0.61
MF | Mixed Forest | 79.23 | 3.68 |  2.99 $\pm$ 0.62
WSA | Woody Savannah | 117.36 | 2.22 |  2.88 $\pm$ 0.38
ENF | Evergreen Needleleaf Forest | 100.54 | 2.73 |  3.30 $\pm$ 0.91
CSH | Closed Shrub | 75.08 | 2.82 |  2.18 $\pm$ 0.44

While Eq. (3) can be used in PM (Eq. (1)), it will make analytical work with
the function intractable because $GPP$ is functionally related to ET itself.
Additionally, a perturbation to VPD should induce a physiological plant
response that will alter GPP and cause an indirect change in stomatal
conductance, in addition to the direct effect of VPD in Eq. (4). Therefore, in
order to derive the response of ET to VPD, we must account for the functional
relationship between GPP, ET, and VPD, and its effect on stomatal conductance.
We can use aforementioned semi-empirical results of Zhou et al. (2015) as a
tool to approach this problem. Zhou et al. (2015), showed that underlying
Water Use Efficiency (uWUE):

$uWUE=\frac{GPP\cdot\sqrt{VPD}}{ET}$ (5)

is relatively constant across time and moisture conditions within a plant
functional type, and correctly captures a constant relationship between GPP,
ET and VPD over a diurnal cycle (Zhou et al., 2014). The theoretical
derivation of the square root VPD dependence in $uWUE$ leverages the same
assumptions used in Medlyn et al. (2011) to derive the square-root VPD
dependence of the stomatal conductance model (Eq. (4)). We can use uWUE to
remove the $GPP$ dependence from $g_{s}$ in a way that makes PM analytically
tractable:

$g_{s}=\frac{R\,T}{P}1.6\left(1+\frac{g_{1}}{\sqrt{VPD}}\right)\frac{uWUE\;ET}{c_{a}\;\sqrt{VPD}}.$
(6)

Plugging Eq. (6) into Eq. (1) and rearranging gives a new explicit expression
for PM, in which dependence on $GPP$ is removed:

$ET=\frac{\Delta
R_{net}+\frac{g_{a}\;P}{T}\left(\frac{c_{p}VPD}{R_{air}}-\frac{\gamma
c_{a}\sqrt{VPD}}{R\;1.6\;\text{ uWUE
}(1+\frac{g_{1}}{\sqrt{VPD}})}\right)}{\Delta+\gamma}$ (7)

By accounting for photosynthesis changes in ecosystem conductance, with Eq.
(7) we have derived for the first time, using recent results (Medlyn et al.,
2011; Zhou et al., 2014, 2015; Medlyn et al., 2017), ET explicitly as function
of environmental variables and two plant-specific constants, the slope
parameter ($g_{1}$), and uWUE, both reflecting water conservation strategy.
The slope parameter is related to the willingness of stomata to trade water
for CO2 and to keep stomata open. uWUE is a semi-empirical ecosystem-scale
constant related to how WUE changes with VPD (specifically $VPD^{-1/2}$). It
is also roughly proportional to physical constants:

$uWUE\mathrel{\vbox{ \offinterlineskip\halign{\hfil$#$\cr\propto\cr\kern
2.0pt\cr\sim\cr\kern-2.0pt\cr}}}\sqrt{\frac{c_{a}-\Gamma}{1.6\lambda}},$

where $\Gamma$ is the CO2 compensation point (Eq. (5) in Zhou et al., 2014).
So uWUE is related to atmospheric CO2 concentration and compensation point,
and is inversely proportional to the marginal water cost of leaf carbon.

Given eddy-covariance FLUXNET2015 data (Sect. 2.1), every term in our new
version of PM (Eq. (7)) is observed, except for uWUE, which we fit by
calculating its expectation, given the model and FLUXNET2015 data.

However, eddy-covariance data are inherently noisy so we include a measure of
uncertainty in our analysis. To account for observational error, as well as
model uncertainty (e.g. temporal and spatial variations of uWUE and $g_{1}$),
we introduce an uncertainty parameter $\sigma$ modifying uWUE:

$ET=\frac{\Delta
R_{net}+\frac{g_{a}\;P}{T}\left(\frac{c_{p}VPD}{R_{air}}-\frac{\gamma
c_{a}\sqrt{VPD}}{R\;1.6\;\sigma\;\text{ uWUE
}(1+\frac{g_{1}}{\sqrt{VPD}})}\right)}{\Delta+\gamma}$ (8)

Now, from each FLUXNET2015 observation (i.e. for each hourly observation at
every time step) we can evaluate $\sigma$:

$\sigma=-\frac{g_{a}\gamma c_{a}\sqrt{VPD}L_{v}P}{\left(\text{ ET
}(\Delta+\gamma)-\Delta R_{net}-g_{a}\rho_{a}c_{p}VPD\right)1.6\;R\;T\;\text{
uWUE }(1+\frac{g_{1}}{\sqrt{VPD}})},$ (9)

So, with this uncertainty analysis we can evaluate departure from our theory
in observations, as a departure of $\sigma$ from unity. The variability of
$\sigma$ across sites and time provides a measure of uncertainty in our model,
assumptions, as well as in the FLUXNET2015 observations themselves. The
variability of $\sigma$ then propagates any uncertainty through to our partial
derivative of Eq. (8) with respect to VPD:

$\frac{\partial\;ET}{\partial\;VPD}=\frac{2\;g_{a}\;P}{T(\Delta+\gamma)}\left(\frac{c_{p}}{R_{air}}-\frac{\gamma
c_{a}}{1.6\;R\;\sigma\;\text{ uWUE
}}\left(\frac{2g_{1}+\sqrt{VPD}}{2(g_{1}+\sqrt{VPD})^{2}}\right)\right)$ (10)

With Eq. (10) we have an analytical framework for ecosystem response to
atmospheric demand perturbations with environmental conditions held fixed.
There are a few subtleties to taking the derivative in Eq. (10): $\Delta$
($\frac{de_{s}}{dT}$) and $VPD$ are functionally related, so while taking the
derivative we evaluate
$\frac{\partial\;ET}{\partial\;VPD}=\frac{\partial\;ET}{\partial\;e_{s}}\frac{\partial\;e_{s}}{\partial\;VPD}\Big{|}_{\text{RH
fixed}}+\frac{\partial\;ET}{\partial\;RH}\frac{\partial\;RH}{\partial\;VPD}\Big{|}_{\text{$e_{s}$
fixed}}$. $RH$ and $e_{s}$ are assumed to be approximately independent, which
is supported by the data (not shown).

This derivation relied either implicitly or explicitly on several assumptions.
First, we assume that VPD at the leaf surface is the same as VPD at
measurement height; physically this implies that leaves are perfectly coupled
to the atmosphere. In reality, for some conditions and plant types the leaves
can become decoupled from the boundary layer (De Kauwe et al., 2017; Medlyn et
al., 2017). Given our focus on the growing season, which is usually
characterized by relatively high insolation inducing instability and
convective boundary layers, we would expect the surface to be generally well
coupled. An additional assumption in the formulation of $uWUE$ (Zhou et al.,
2014, 2015) and Medlyn et al. (2017)’s stomatal conductance model is that
direct soil evaporation (E) contributions to ET remain small relative to
transpiration (T). This should be more true during the growing season. The
ratio of E to T may increase immediately after rainfall events due to high
soil moisture and ponding, but VPD is generally low anyways during these
times. However, some plant types allow for systematically larger contributions
of E in ET, particularly those with sparse canopies and smaller relative
amounts of transpiration. We therefore might expect that the theory will be
most applicable to forest PFTs, which will be most strongly coupled to the
boundary layer due to larger surface roughness, and will also generally have
the highest ratios of transpiration to evaporation. Finally, for the goal of
developing PFT-wide intuition, we assume that $g_{1}$ and $uWUE$ are constant
in space and time. Both quantities have been shown to be well conserved over
space and time within a PFT relative to inter-PFT variability. However, we
would expect them to vary somewhat with environmental conditions including
very low soil water content, temperature, and with intra-PFT plant-specific
characteristics like wood density for tree PFTs (Lin et al., 2015). Given
these strong assumptions made with the goal of understanding broad, leading
order plant behaviors, we take the extra care of including and analyzing
uncertainty (Sect. 3.4) and examining sensitivity to the exponent of the VPD
response (Sect. 3.7).

We note one final comment on our derivation which is relevant for drought
indices. If we approximate $c_{a}$ at a global mean CO2 concentration, then
the RHS of Eq. (7) is fully defined using commonly available weather station
data and the constants published here. This makes Eq. (7) useful in addition
to PET in drought indices and hydrometeorological analysis for vegetated
surfaces. Equation (7) better reflects the physics of water exchange at the
land surface and would only require fitting of uWUE and $g_{1}$, and can also
account for changes in CO2 concentration (assuming some relationship between
$\lambda$ and [CO2]), which is missing in other drought indices such as the
Palmer Drought Severity Index (PDSI) (Swann et al., 2016; Lemordant et al.,
2016, 2018). Lastly, all code and data used in this analysis, including those
used to generate the figures and tables, are publicly available at
https://github.com/massma/climate_et.

##  3 Results and Discussion

By construction, the variability in the $\sigma$ term (Eq. (9)) contains all
model and observational uncertainties. For an observation that perfectly
matches our model and constant uWUE assumption, $\sigma$ will be one.
Therefore, for our assumptions and framework to be reasonable $\sigma$ should
be close to 1. An additional concern is that $\sigma$ may in fact be
correlated with $VPD$, in which case the dependence would need to be accounted
for when taking the derivative. Fortunately, there is a very weak dependence
of $\sigma$ on VPD in their joint distribution, and $\sigma$ is indeed close
to unity i.e. $O(1)$ (Fig. 2). Given this weak dependence and the distribution
of $\sigma$ we have confidence in our model framework and the data quality.

Figure 2: The joint distribution of $VPD$ and $\sigma$, with outliers removed
(defined as lowest and highest 5% of $\sigma$). $\sigma$ exhibits a weak
dependence on $VPD$, and $\sigma$ is $O(1)$ for the bulk of the observations.

Before calculating the sensitivity of ET to VPD, we will consider the
functional form of Eq. (10). There are two main terms: a “scaling” term, which
modifies the magnitude but not the sign of the ET response to VPD
($\frac{\partial\;ET}{\partial\;VPD}$):

$\frac{g_{a}\;P}{T(\Delta+\gamma)},$ (11)

and a “sign” term, which determines whether ET increases or decreases with VPD
(i.e. atmospheric demand driven or physiologically controlled):

$\frac{c_{p}}{R_{air}}-\frac{\gamma c_{a}}{1.6\;R\;\text{ uWUE
}}\left(\frac{2g_{1}+\sqrt{VPD}}{2(g_{1}+\sqrt{VPD})^{2}}\right).$ (12)

All variables are positive, so the relative magnitude between the first term
and the second term in the sign term (Eq. (12)) will determine whether ET
increases or decreases with increasing VPD. If the second term is larger then
plant control dominates and ET decreases with increasing VPD. However, if the
first term is larger, then atmospheric demand dominates and ET increases with
increasing VPD.

###  3.1 Functional Form of the Sign Term

First, we explore the variables within the sign term to gain better intuition
on the driver of either the increase or reduction of ET with VPD. CO2
concentration ($c_{a}$) and the psychometric constant ($\gamma$) are
relatively constant over the dataset considered here so that the variability
is dominated by $\sigma$ and $VPD$. uWUE could vary with soil moisture but has
been shown to be relatively constant (Zhou et al., 2015). This then means that
the sign term only depends on VPD for a given PFT and is approximately just a
function of $VPD$. We can further determine a critical threshold separating an
increase from a decrease in ET, i.e. the threshold $VPD_{crit}$ such that the
derivative vanishes $\frac{\partial\;ET}{\partial\;VPD}=0$:

$VPD_{crit}=\frac{R_{air}}{4c_{p}}\left(\frac{\gamma
c_{a}}{1.6\;R\;uWUE}+\sqrt{\frac{\gamma c_{a}}{1.6\;R\;uWUE}\left(\frac{\gamma
c_{a}}{1.6\;R\;uWUE}+8g_{1}\frac{c_{p}}{R_{air}}\right)}-4g_{1}\frac{c_{p}}{R_{air}}\right),$
(13)

noting that $VPD_{crit}$ mostly depends on the PFT parameters uWUE and
$g_{1}$, and only varies weakly with climate as most other parameters related
to the environment are nearly constant. The calculated value of $VPD_{crit}$
for each PFT is shown in Table 3.1. For any values of $VPD$ less than
$VPD_{crit}$, ET will decrease with increasing VPD
($\frac{\partial\;ET}{\partial\;VPD}<0$), and for values of $VPD$ greater than
$VPD_{crit}$, ET will increase with increasing VPD
($\frac{\partial\;ET}{\partial\;VPD}>0$). In other words, ecosystems regulate
and mitigate evaporative losses up to the VPD limit, $VPD_{crit}$, above which
atmospheric demand is just too high to be entirely compensated by stomatal and
ecosystem regulation. We note however that even though ET increases again
above the critical threshold, $VPD_{crit}$, ET is still much lower than
potential evaporation as stomata are still strongly regulating vapor fluxes to
the atmosphere. However, even in the absence of soil pore evaporation ET
cannot go completely to zero at high VPD, because stomata are still slightly
open to perform some photosynthesis (Ball et al., 1987; Leuning, 1990; Medlyn
et al., 2011). In addition, upward xylem transport is necessary to maintain
phloem transport, as well as nutrient transport and thus carbon allocation (De
Schepper et al., 2013; Nikinmaa et al., 2013; Ryan and Asao, 2014).

Table 3: Values of $VPD_{crit}$, where $\frac{\partial\;ET}{\partial\;VPD}=0$,
evaluated at PFT average values for $R_{air}$, $\gamma$, and $c_{a}$. PFT-
specific constants ($g_{1}$, uWUE) are provided in Table 2.2. For values of
$VPD$ less than $VPD_{crit}$, $\frac{\partial\;ET}{\partial\;VPD}$ will be
negative, and for values of $VPD$ greater than $VPD_{crit}$,
$\frac{\partial\;ET}{\partial\;VPD}$ will be positive.

Differences in $VPD_{crit}$ are exclusively determined by uWUE and the slope
parameter ($g_{1}$) related to the plant functional type. A larger uWUE means
a smaller $VPD_{crit}$, and an ET response to increases VPD that is more
likely to be positive. At first glance this result is somewhat counter-
intuitive; we expect that plants with a higher water use efficiency would be
more water conservative. However, in reality uWUE determines how $WUE$ changes
with VPD:

$WUE=\frac{GPP}{ET}=\frac{uWUE}{\sqrt{VPD}}.$
$\frac{\partial\;WUE}{\partial\;VPD}=-\frac{uWUE}{2\;VPD^{3/2}}$

So, plants with a higher uWUE will have a greater decrease in ecosystem-scale
$WUE$ in response to increases in VPD. This decrease in $WUE$ causes more
water loss per unit carbon gain, and explains the relationship between high
uWUE and high likelihood of increases of ET in response to increasing
atmospheric drying (increases in VPD).

A tendency towards increasing ET response with increasing VPD can also be
caused by a high slope parameter ($g_{1}$), characteristic of plants that at
the leaf scale are more willing to trade water for access to atmospheric CO2.
Plants that are less conservative will be thus be more likely to increase ET
with increasing VPD. Both the aforementioned effects (large uWUE, $g_{1}$) can
amplify each other, and generally conspire to shift the sign term towards a
positive value for a given PFT.

This effect of uWUE and $g1$ on the sign term is most apparent by comparing
two extreme PFTs: water intensive crops (CRO) and water conservative closed
shrub (CSH). CRO has higher slope parameter and a slightly higher uWUE
($g_{1}=140.7$ Pa1/2; $2.85$ $\mu$-mol [C] Pa0.5 J-1 [ET]) compared to CSH
($g1=75.1\,Pa^{1/2}$, $uWUE=2.82$ $\mu$-mol [C] Pa0.5 J-1 [ET]). These
differences in PFT parameters cause opposite ET responses to changes in VPD
between CRO and CSH. ET theoretically always decreases with increasing VPD for
the more water conservative CSH, while ET frequently increases with increasing
VPD for the more water intensive CRO (Fig. 3). CROs evolved or were bred to
prioritize GPP and yield and are thus not water conservative. They are very
willing to trade water for photosynthesis and productivity, despite changes in
VPD, while CSH are very unwilling to trade water for more photosynthesis.

Figure 3: The functional form of the sign term, with $\sigma$ held fixed at 1,
and all terms except VPD set to PFT averages. For comparison, the observed
range of VPD for each PFT is plotted below the x-axis. Stars denote 25th,
50th, and 75th percentiles, and the range of the line spans the 5th-95th
percentiles of observed VPD. Vertical black lines denote the location of
$VPD_{crit}$ for each PFT, with the exception of CSH and WSA, for which
$VPD_{crit}$ is off-scale.

As expected, the slope parameter ($g_{1}$) is a primary determinant of the VPD
dependence for the sign term shown in Fig. 3. Plants that are more
conservative (small $g_{1}$) will tend to reduce ET with increasing VPD, and
will be very effective at reducing ET, especially at low VPD. However, at very
high VPD, gradients in vapor pressure at the leaf scale will become very
strong as stomata reach their limits of closure in response to VPD
(parameterized with $g_{1}$). As a result, ET response will begin to asymptote
towards a constant ecosystem-scale values as leaf-scale response to VPD
asymptotes towards zero. Therefore, plants with a low $g_{1}$ will have the
largest VPD dependence of ET response because the difference in ET response at
low VPD (leaf stomatal response dominates) and high VPD (VPD gradient
dominates) is largest. This is apparent in the strong VPD dependence of CSH,
which has the lowest slope parameter ($g_{1}=75.1$ Pa1/2) (Fig. 3).

To summarize our theoretical insights (Fig. 3 and Table 3.1), CROs are the
least water conservative and have the strongest overall tendency to increase
ET with increasing VPD, while CSH are the most water conservative and have the
strongest tendency to decrease ET with increasing VPD, as well as the
strongest VPD dependence of response. Fig. 3 clearly shows, according to our
theory, that for all PFTs except for crops there is frequent occurrence of a
negative (plant dominating) ET response to increases in VPD. Therefore, plants
are able in most atmospheric conditions to reduce ET in response to increased
VPD and thus to reduce water loss. To better illustrate this, the ranges of
observed environmental VPDs at the FLUXNET sites are plotted parallel to the
x-axis. For CSH and WSA, VPD is always less than VPDcrit (off scale) so that
the plant response dominates in typical environmental conditions, emphasizing
the water conservative strategy of those plants. For CRO on the other hand,
VPD is higher than VPDcrit for more than 50% of observations, emphasizing that
those plants operate with an aggressive water usage strategy, are water
intensive and were actually engineered for photosynthesis rather than water
saving. For DBF, EBF, MF, GRA, and SAV more than half of the observed VPD are
less than VPDcrit, i.e. in conditions where plant response dominates. SAV has
a more water conservative response than the forest, grass, and crop plan
types, but still responds by increasing ET with increasing VPD for about a
quarter of observations, due to the high aridity (VPD) of the SAV ecoclimate.
It is also important to note that for all PFTs, even when atmospheric demand
dominates, ET response to VPD is still far more negative than it would be for
potential evaporation $\partial PET/\partial VPD$, i.e. atmospheric demand
only, emphasizing that there is still a strong regulation of evaporative flux
by stomata and though the plant xylem. The sign term in the PET case would
just be a constant ($\frac{c_{p}}{R_{air}}\approx 3.5$), which is far larger
than any part of the curves for any PFT. Plants are always regulating water
exchange from the land surface, even when they reach the limits of they
ability to do so.

###  3.2 Functional Form of the Scaling Term

While the above discussion of the sign of $\frac{\partial\;ET}{\partial\;VPD}$
is important to answer our question of when ET response increases or decreases
with VPD, understating the overall magnitude of the ET response is important
to soil-plant-atmosphere water budgeting. So we now more closely examine the
terms that affect how the sign term is scaled:

$\frac{g_{a}\;P}{T(\Delta+\gamma)}.$ (14)

$\frac{P}{T}$ is an air-density term, which varies little compared to
aerodynamic conductance and Clausius-Clapeyron ($\Delta$). The psychometric
constant ($\gamma$) is also relatively constant, so the scaling term should be
primarily a function of aerodynamic conductance and temperature, through the
Clausius-Clapeyron relationship $\Delta$. This is as expected, given that the
aerodynamic conductance represents the efficiency of exchange between the
surface and the atmosphere. As aerodynamic conductance increases, any plant
response will be communicated more strongly to the atmosphere (and vice-
versa).

$\Delta$’s presence in the scaling term also matches physical intuition.
$\Delta$ (and also the approximately constant $\gamma$) control the
efficiencies with which surface energy is converted to latent and sensible
heat (Monteith et al., 1965). The functional from of $\Delta$ will be the same
across PFTs, but the temperature range may vary slightly. In contrast,
aerodynamic conductance will vary strongly with PFT due to the importance of
surface roughness for aerodynamic conductance. So most of the differences in
scaling between PFT should be in the aerodynamic conductance term.

The control of the scaling term variability between PFTs by aerodynamic
conductance is confirmed by data (Fig. 4). Differences between PFT are almost
entirely due to differences in aerodynamic conductance, rather than
differences in observed temperature ranges. The scaling term for the tree PFTs
(DBF, EBF, ENF, MF) is generally about double the scaling terms for other PFTs
which have lower surface roughness and generally smaller aerodynamic
conductance (GRA, CSH, CRO). The savannah (WSA, SAV) PFT’s scaling is
somewhere between GRA, CSH, and CRO, and DBF, EBF, ENF, and MF, due to higher
variability and surface roughness.

Figure 4: Primary sources of variability for the scaling term, as a function
of PFT. The 5th-95th percentile range of temperature is plotted at the 5th,
25th, 50th, 75th, and 95th percentiles of aerodynamic conductance, as observed
for each PFT.

Within each PFT, the scaling term variability is controlled both by
environmental temperature and aerodynamic conductance variability (Fig. 4).
While the observed variability of the aerodynamic conductance contributes more
to the scaling term variability than temperature, the temperature contribution
is non-negligible. Specifically, the scaling term is generally larger at low
temperatures when latent heat is relatively inefficient at moving energy away
from the surface. This effect amplifies the role of aerodynamic conductance
variability at low temperatures.

To summarize, variability between PFTs is mostly controlled by systematic
differences in aerodynamic conductance, due to differences in surface
roughness between each PFT, and possibly to a lesser extent wind conditions.
In contrast, variability within PFT is also controlled by temperature, through
Clausius-Clapeyron. But, aerodynamic conductance variability generally impacts
the scaling term more than temperature, even within PFTs.

###  3.3 Bulk statistics of ET response to VPD

In this section we consider direct observations of ET response with eddy-
covariance data, while including uncertainty with the $\sigma$ term (Sect.
2.2). These observational results of ET response (Table 3.3) largely confirm
our theoretical analysis, presented in the previous sections. For all PFTs,
mean ET response to increasing VPD is negative. However, ET response evaluated
at the average of all variables (e.g. $\sigma$, $T$, $c_{a}$, $VPD$) is
positive for CRO, and negative for all other PFTs. This difference in mean ET
response as compared to the ET response at mean environmental conditions is
due to the non-linear nature of the response, in which negative responses are
generally larger magnitude than positive responses (Fig. 3). Therefore, both
the mean ET response as well as the ET response at mean environmental
conditions matches our expectations from the theory (Sect. 3.1), with the
exception that CRO observations are shifted more towards a negative ET
response than we expect.

Table 4: Statistics of $\frac{\partial\;ET}{\partial\;VPD}$ as a function of
PFT.

Regarding the frequency of negative and positive ET response, all PFTs exhibit
a decreasing ET response with increasing VPD (physiologically controlled,
water conservative response) for the majority of observations. The more water
conservative PFTs generally exhibit higher frequency of negative ET response,
especially when one factors in the distribution of environmental VPD (e.g. SAV
and WSA grow in more arid climates, Fig. 3). In general the bulk statistics
match our theoretical expectations well, with the caveat that inclusion of
uncertainty shifts crops towards a slightly more negative response to VPD, and
shifts many of the other PFTs, which still exhibit a high frequency of
negative ET response to VPD, towards more frequent occurrence of positive ET
response than the theory and Fig. 3 would suggest. The bulk statistics
motivate a more thorough examination of the structure of uncertainty and more
sophisticated validation of our theory’s performance against observations.

###  3.4 Validation of theory at eddy-covariance sites

We now compare more sophisticated distributions of the observed response to
our simplified theory (Sect. 3.1). The observed distribution of the sign term,
as compared to what the theory would predict, is provided in Fig. 5. Our goal
was to capture the leading order behavior of the ET dependence on VPD. Given
the assumptions we made, and the uncertainties of flux tower observations
themselves, we expect a relatively large amount of noise when reproducing the
derivatives of ET. However, the data largely reproduces our theoretical
analysis.

Figure 5: Comparison of the sign term with model uncertainty included (box
plots) to the sign term as calculated with simplifying assumptions (blue line,
as in Fig. 3). Each box plot corresponds to 5% of the data, and the 5-95%
range of VPD is plotted.

This is particularly true for DBF and MF; the theory matches the leading order
behavior of the function when uncertainty is included, and the observations
match the theory with the addition of noise. The VPD dependence, given by the
slope parameter ($g_{1}$), follows the median values of each bin. Perhaps most
importantly, the x-intercept, and thus VPDcrit, matches nearly exactly between
the theory and the observations. Therefore the sign of the ET response to
increases in VPD should be well matched, subject to the unavoidable
constraints of noise, much of which comes from the observations themselves.
The uncertainty is non negligible; there are many observations in each bin for
which the the sign of observation is opposite the response predicted by the
theory, but to leading order our theory matches the observations well.

While CSH has a much different functional form of the sign term than DBF and
MF, CSH observations also match our theory to leading order, albeit with a bit
more variability as a function of VPD. Again, the VPD dependence mostly
determined from the slope parameter ($g_{1}$) closely matches the medians in
the observation bins. The VPD-independent, strongly negative response is also
captured. For CSH, there is rare occurrence of observed positive ET response
with VPD ($\approx 6$%), even with uncertainty, so the sign of the
observations almost always matches the sign of the theory, which states that
ET response should always be negative.

Biases between the theory and observations are similar for ENF, WSA, CRO and
SAV. At low VPD the theory and the observations match well. However, at high
VPD (upper 10-20% of observations) the theory is biased slightly towards
positive response as compared to the observations (Fig. 5). For ENF this
slight bias could be explained by a negative bias in $g_{1}$. However, for
WSA, CRO, and SAV the observations in the highest VPD bins exhibit a downturn
towards more negative response, which cannot be captured by the functional
form of the sign term (Fig. 3). Therefore, to explain the observations one of
the variables in the sign term must change at extreme VPD (upper 10-20%). The
most likely candidate is $uWUE$, which we have assumed constant to meet our
goal of developing intuition for leading order behavior, but might be expected
to decrease at extremely low SWC. This extremely low SWC should be correlated
with high VPD in very dry environments through sensible heating increases
(Gentine et al., 2016). The theory for EBF exhibits similar limitations as for
ENF, WSA, and SAV, except a larger portion of observations are biased negative
as compared to the theory ($\approx$ 35%, Fig. 5). However, in general, and
specifically for non-extreme VPD (VPD $<\approx$70-90th percentiles), the
theory matches the observations for the tree and savannah plant types well.

The theory for GRA suffers from similar, but much more severe, limitations as
for CRO, WSA, and SAV. GRA observations are characterized by a consistent
trend back towards negative ET response at higher VPD, which the functional
form of our theory is incapable of accounting for. As compared to CRO, WSA and
SAV, the divergence between the theory and the observations is far greater for
GRA, biasing 40-50% of observations at higher VPD. In addition for the
potential for soil moisture to alter $uWUE$, there are other sources of plant
heterogeneity specific to GRA (and some extent CRO) that may alter $uWUE$ (or
$g_{1}$) or invalidate other assumptions made in the methods section (Sect.
2.2). We do not account for variability in plant height and surface roughness,
or differences in C3 vs C4 photosynthesis and water strategies, which we might
expect to vary substantially across sites, years, and season for GRA. These
deficiencies could largely explain the inability of our theory to exactly
match the observations in croplands. For example, a superposition of sites
with C3 plants (at low environmental VPD) and C4 plants (at high environmental
VPD) would explain the observed shift back towards negative ET response at
high VPD when all sites are binned together, as in Fig. 5. We hypothesize that
the theory would validate against observations much better if these sources of
variability were accounted for, at a cost of increased complexity and
analytical opacity.

While the above discussion shows that our theory has some limitations when
applied to some PFTs, especially for grasslands, it does well for DBF, MF and
CSH PFTs, and captures the response at non-extreme VPD (VPD $<\approx$
70th-90th percentiles) for ENF, EBF, CRO, WSA, and SAV. In general, the
leading order behavior observed in the data is captured by the theory.
Departures between our theory and observations, specifically at high VPD,
could be explained and conceptualized with shifts in $g_{1}$ and/or $uWUE$ due
to site-specific plant type variability (e.g. more arid-adapted ecosystems at
more arid sites), or temporal variability for some environmental conditions
(e.g. decreases in $uWUE$ at extremely low soil moisture). To focus on general
behavior and develop intuition for PFT-scale response, we have ignored these
sources of variability in the present analysis. However, any site-specific or
temporal plant functional variability that can be conceptualized with shifts
to $g_{1}$ and $uWUE$ can be analyzed with our framework. This opens a door to
future analyses in which plant behavior in anomalous conditions can be
explained and analyzed using Eq. (7).

###  3.5 Observed ET response to VPD

Most of the results presented so far focus on the sign term, so now we turn to
observations of ET response with the scaling term included (Fig. 6). Until
now, in the interest of developing leading order intuition for ET response,
and to be conservative in our acknowledgment of model and observational error,
we’ve considered $\sigma$ variability to be a measure of uncertainty. An
alternative viewpoint is that $\sigma\cdot uWUE$ represents spatial and
temporal variability of uWUE, which may be expected within bounds (see Table
2.2, also Zhou et al., 2015). This is a less conservative view; some of the
$\sigma\cdot uWUE$ variability will be due to model and observational error,
so by viewing $\sigma\cdot uWUE$ as real variability we run the risk of
mistaking noise for signal. However, the advantage of this viewpoint is that,
from Zhou et al. (2014), we have very high confidence that uWUE fit at the
hourly timescale (as we do with $\sigma\cdot uWUE$) correctly captures the
relationship between ET, GPP, and VPD, and that our form of PM introduces
minimal error with its use of uWUE.

Figure 6: Scatter plots of observed $\frac{\partial\;ET}{\partial\;VPD}$,
including $\sigma$ variability, as a function of PFT, temperature and VPD.
Please note differences in the colorbar scale.

Therefore, with the caveat that some of the signal presented in Fig. 6 may in
fact be noise, we can interpret the observed distribution of ET response to
VPD. In general, the observed response matches the intuitive theory. ET
response to VPD shifts towards positive values as VPD increases (atmospheric
demand dominating). CRO exhibit the highest occurrence of positive ET
response, and the observations confirm that CRO are the most water intensive.
CSH are the most water conservative, with a strong negative response. DBF,
EBF, ENF, MF, SAV and WSA are also water conservative, but show some
occurrence of positive ET response to VPD, particularly at higher observed
VPD, as the theory predicts. GRA, while generally water conservative, does not
match the theory well, with increasing frequency of negative response at high
VPD. This is as expected, given previous discussion on how we might expect
more inter-site and inter-year variability for GRA (Sect. 3.4).

Fig. 6 also includes the impact of the scaling term. For a given VPD, the
magnitude of ET response does not vary strongly with temperature, confirming
that any impact of the scaling term on the magnitude of the response is
primarily due to changes in aerodynamic conductance. Intuitively this is
reasonable; aerodynamic conductance will control how dominant balances at the
land surface are communicated to the boundary layer. While $\Delta$ controls
the efficiency of energy conversion to latent heat, it appears this is a
second order term, relative to $g_{a}$, for scaling ET response.

As with Fig. 5, Fig. 6 matches our expectation based on simplified theory. The
sign term is most strongly scaled by $g_{a}$, and in general the occurrence of
positive ET response increases as VPD increases. The willingness with which a
given plant type evolved to use water dictates the occurrence of positive
versus negative ET response. Water conservative ecosystems are highly
effective at mitigating the effects of atmospheric demand, and can store water
for later use by reducing ET in response to increasing VPD.

###  3.6 Limitations of theory: very dry soil moisture conditions

In formulating our theory with Penman Monteith, we implicitly did not account
for very dry soil moisture conditions. For the majority of environmental
conditions observed at the eddy covariance sites used here, soil conditions
were not extremely dry so that we could assume a constant uWUE and $g_{1}$. We
posit that ecosystems will generally optimize to host plants living in
conditions which they evolved for. However, in extreme conditions and drought
scenarios soil water content (SWC) could become the limiting factor for ET
response to VPD, which our theory does not account for. In addition, low soil
moisture conditions themselves increase VPD through land-atmosphere feedback
(Bouchet, 1963; Morton, 1965; Brutsaert, 1999; Ozdogan et al., 2006; Salvucci
and Gentine, 2013; Gentine et al., 2016; Berg et al., 2016).

Within our framework, any systematic bias due to the failure to account for
SWC’s effects in very dry conditions should manifest itself in a functional
relationship between $\sigma$, our uncertainty measure, and SWC, which is
observed at the FLUXNET sites. Examining the relationship between $\sigma$ and
SWC will test to what extent our theory breaks down in very dry soil moisture
conditions.

If we again view $\sigma\cdot uWUE$ as, in addition to a measure of
uncertainty, a short time-scale observation of uWUE, we would expect
$\sigma\cdot uWUE$ to decrease at low SWC. If $\sigma\cdot uWUE$ is a very
strong function of SWC, then our theory should be conditioned more strongly on
well watered soil conditions. If $\sigma\cdot uWUE$ is weakly a function of
SWC, then our theory is more universal and independent of soil moisture
conditions.

Indeed, for all PFTs, there is some slight dependence of $\sigma\cdot uWUE$ on
SWC, especially at low SWC (Fig. 7). The portion of observations for which our
theory is biased by SWC-limitations varies by PFT, due to the nonlinear
threshold at which soil moisture availability limits plant function. For CRO,
MF and DBF, soil water content only matters for about the lowest 5% of
observations (each box is 5% of observations in Fig. 7). And, CRO, relative to
DBF, has a very weak dependence on soil moisture, which is reflective of the
high likelihood that CRO sites are optimally irrigated and not water stressed,
suggesting that observed departures from the theory for CRO are due to other
factors (e.g. different photosynthesis pathways; C3/C4) than soil moisture.
For ENF, EBF, SAV, and WSA, systematic SWC-induced biases in $\sigma\cdot
uWUE$ emerge in about the lowest 20-30% of SWC conditions, although large
variability in the SWC-$(\sigma\cdot uWUE)$ relationship hampers
interpretation for WSA. CSH presents a special case: the limited number of
sites (2) preclude comment on the exact PFT-wide relationship.

In contrast to ENF, CRO, and DBF, for GRA the relationship between
$\sigma\cdot uWUE$ and SWC is more linear and affects a greater portion of
observations; about 60% of observations. Clearly, for GRA soil water
frequently impacts plant function and alters ET response, and our theory is
limited for the majority of environmental conditions. It is therefore not
surprising that our theory tested poorly against the data for GRA, relative to
to the other PFTs (Sect. 3.4). For all other PFTs occurrence of soil moisture
impacts was rare enough to not manifest itself in bulk statistics and figures.

Figure 7: $\sigma\cdot uWUE$ as a function of PFT and SWC. Each box plot
represents 5% of all observations. Note that the highest 10% of SWC
observations are excluded to better resolve variability in the much more
narrow bins at lower SWC (e.g. SWC has long tails at high values).

The observed dependence of $\sigma\cdot uWUE$ on SWC for GRA would explain the
deficiencies of our theory compared to the observations in Fig. 5,
specifically the trend back towards negative ET response at high VPD.
Aforementioned feedbacks between the land surface and the atmosphere, which
are not accounted for due to our focus on the one-way response of the land
surface to atmospheric conditions, would cause high VPD to be correlated with
low SWC (Gentine et al., 2016; Berg et al., 2016). So, at high VPD
observations of low SWC are more likely, and this low SWC causes a lower uWUE
(Fig. 7). The lower uWUE at low SWC/high VPD then leads to the observed
downturn towards decreasing ET with increasing VPD, and the deviation between
our theory (based on a constant uWUE assumption) and observations. It is also
perhaps not coincidental that the portion of $\sigma\cdot uWUE$ affected by
low soil moisture observations is similar to the portion of observations that
do not match our theory at high VPD. Indeed, this may suggest that for all
PFTs except for GRA, coupling and feedbacks between SWC, VPD and plant
function are relatively rare. Future research will explore these relatively
rare feedbacks in extreme conditions, which due to analytical intractability
will require more opaque numerical analysis of many more complex processes,
including boundary layer growth and state and their relationship to surface
layer coupling and free-tropospheric lapse rates and humidity.

To summarize, our theory is limited by its inability to account for soil water
impacts on land surface response, and feedbacks between SWC and VPD.
Fortunately, for most PFTs SWC’s effect on ET is relatively rare ($<$30% of
observations) and does not manifest itself in the majority of observations and
bulk statistics. However for GRA, SWC decreases water use efficiency for the
majority of the observations. Soil moisture effects explain the deficiencies
of our theory in Sect. 3.4, particularly for GRA. By conceptualizing SWC
effects as a change in $uWUE$ (and/or $g_{1}$), it will be possible for future
analysis to explore the importance of soil moisture on plant response to VPD,
and feedbacks between plant function, SWC, and VPD.

###  3.7 Functional form of ET dependence on VPD and its relation to the VPD
exponent

The theory described in Sect. 3.1 indicates that for a given $uWUE$ and
$g_{1}$, the ET dependence on VPD should be concave upward, which is confirmed
by eddy covariance data across most PFTs. In other words, there should be some
local minimum in ET at a critical VPDcrit, assuming the scaling and plant
terms (e.g. aerodynamic conductance, $\Delta$, $g_{1}$ and $uWUE$) are held
fixed. This result warrants further investigation, because to our knowledge no
one has derived the theoretical ecosystem-scale relationship between ET and
VPD while controlling for other environmental conditions. In particular, from
personal communication, there is an apparent lack of consensus over whether
the shape of the ET-VPD curve should be concave upward (our result) or concave
downward in the absence of dramatic water stress. Given that understanding the
ET-VPD relationship of the one-way plant response is fundamental to
hypothesizing about any feedbacks between the land surface and the atmosphere,
we analyze why our derived ET-VPD relationship is concave upward, particularly
with respect to the exponent of VPD dependence in $uWUE$ and the Medlyn
unified stomatal conductance model.

There is a theoretical basis for the square root VPD dependence in both the
stomatal conductance model and $uWUE$ based on the assumption that stomata
behave to maximize carbon gain while minimizing water loss, which observations
also generally support (Lloyd, 1991; Medlyn et al., 2011; Lin et al., 2015;
Zhou et al., 2014, 2015; Medlyn et al., 2017). However, some purely empirical
results that fit the exponent of the VPD dependence to data have shown that it
may vary slightly from 1/2, suggesting that stomata, as well as ecosystem-
scale quantities based on stomata theory, may not always function optimally
(Zhou et al., 2015; Lin et al., 2018). Specifically with regards to $uWUE$,
one would not expect that this ecosystem scale WUE quantity will respond to
VPD exactly analogously to stomata. Direct soil evaporation’s contributions to
ET should shift the exponent of the VPD dependence, especially at conditions
of low GPP when we would expect a systematically larger portion of direct soil
evaporation contributions to ET, because we would also expect lower amounts of
transpiration at low GPP. Zhou et al. (2015)’s results corroborate this: they
found a mean empirically fit exponential VPD dependence of 0.55, varying
slightly from the theoretically optimal value of of 0.5 for AmeriFlux sites.
Interpreting Lin et al. (2018)’s results, which also show variance in the
empirical exponent of the VPD dependence of the stomatal conductance model, is
more difficult as Lin et al. (2018) do not handle GPP/A dependence of stomatal
conductance in a directly analogous manner to the optimal theory in Medlyn et
al. (2011) and Medlyn et al. (2017). Regardless, given that these recent
results on the relationship between VPD, GPP, and ET (Medlyn et al., 2011;
Zhou et al., 2014, 2015; Medlyn et al., 2017) form the backbone of our
analysis and are what allowed us to derive an explicit ET expression for the
first time (Eq. (7)), we will analyze if and how assumptions about the
exponent of the VPD dependence impacts the shape of the ET-VPD dependence.
This analysis is also important to understand whether the choice of stomatal
conductance model alters the fundamental behavior of the ET-VPD relationships,
as many commonly used models utilize a VPD exponent other than the 1/2
suggested by optimal theory (e.g. Leuning, 1990, which uses an exponent of 1).

By introducing $n$ and $m$ we can free our stomatal conductance model from
assumptions about VPD dependence:

$g_{s}=\frac{R\,T}{P}1.6\left(1+\frac{g*}{VPD^{m}}\right)\frac{*WUE\;ET}{c_{a}\;VPD^{n}},$
(15)

where:

$*WUE=\frac{GPP}{ET}VPD^{n},$

and $g*$ is a generic slope parameter of units $VPD^{m}$. To determine how the
exponent $n$ and $m$ alter the shape of the ET-VPD dependence we find the
roots of the second derivative of ET, using Eq. (15) for stomatal conductance
($g_{s}$), with respect to VPD:

$\frac{\partial^{2}\;ET}{\partial\;VPD^{2}}=0\quad\forall\quad\frac{VPD^{m}}{g*}=\frac{m\left(m-2n-\sqrt{m^{2}-4mn+2m-4n^{2}+4n+1}+1\right)}{2n\left(n-1\right)}-1.$
(16)

With this result we have defined the family of curves separating concave up
from concave down ET solutions (Fig. 8). These curves are only functions of
the exponent of the VPD dependence and a quantity we call non-dimensional VPD
($VPD^{m}/g*$). Several important relations reveal themselves from Eq. (16):

Figure 8:  Solutions corresponding to inflection points between concave up and
concave down ET-VPD curves (Eq. (16)) for three specific scenarios. Solutions
are defined in terms of a non-dimensional VPD ($VPD^{m}/g_{*}$), but to aide
physical interpretation the horizontal axis is additionally provided in terms
of dimensionalized VPD assuming $m=1/2$ and $g_{*}=110\;Pa^{1/2}$ (average of
all PFT $g_{1}$). The vertical axis has a different interpretation depending
on the solution curve. For the blue line ($m$ varying), it corresponds to $m$,
for the orange line ($n$ varying) it corresponds to $n$, and for the green
line it corresponds to the value of both $n$ and $m$ ($n=m$). Regions of the
parameter space that correspond to concave up and concave down results are
labeled.

  * •

For optimal behavior (n, m = 1/2) the ET-VPD curve will be concave up
regardless of the magnitude of the plant constants $g_{1}$ and $uWUE$.
Therefore, the general concave up nature of our results, given an assumption
of optimal behavior, is insensitive to plant type.

  * •

For all physically possible exponents of VPD dependence ($n,m$), whether the
solution is concave up or concave down does not depend on $uWUE$.

  * •

In general, increasing the exponent of VPD dependence increases the likelihood
of a concave down result. Additionally, as the exponent of VPD dependence
increases from the optimum value of 1/2, whether the curve is concave upward
or concave downward becomes a function of the plant specific slope parameter
$g_{*}$, through non-dimensional VPD ($VPD^{m}/g_{*}$). Because the exponent
of the VPD dependencies is capable of altering the fundamental shape of ET-VPD
dependence, future research investment in understanding the exact VPD
dependence of stomatal conductance, and further reconciliation of empirical
and theoretical stomatal and ecosystem behavior should be prioritized.

While it is possible that in the future some other form of VPD dependence is
derived, at present Medlyn et al. (2011) and Zhou et al. (2014) firmly
established n=m=1/2 as the most likely candidate given current theory and
empirical data. Additionally, we argue that a concave up result matches
physical intuition more than a concave down result. Plants must maintain
nutrient and sugar transport through the phloem and xylem. To accomplish this,
stomata must remain slightly open (De Schepper et al., 2013; Nikinmaa et al.,
2013; Ryan and Asao, 2014). Furthermore, even if complete stomatal closure
were possible, cuticular water loss and [at the ecosystem-scale] direct soil
evaporation are still sources of ET which increase with VPD, independent of
stomatal closure. Therefore, in the limit as VPD becomes large and we assume
plants are exercising all strategies to reduce ET, any further increase in VPD
should result in an increase in ET through cuticular water loss and/or direct
soil evaporation. This inevitable transition from conditions when stomata
respond strongly to VPD to conditions when stomata response is asymptoting
towards full closure would cause a concave up ET-VPD curve, which is matched
by the theory. In short, plant response becomes more limited as VPD increases,
while atmospheric demand monotonically increases with VPD, leading to the
result that atmospheric demand dominates plant response when atmospheric
demand is high.

This analysis allows us to understand the theoretical shape of the ET response
to VPD with environmental conditions held fixed. Accomplishing this with
purely statistical methods applied to flux observations would be very
difficult, given the relatively fast time scale of plant response and the non-
stationarity of [solar forced] environmental conditions over the relatively
coarse (half hourly) flux estimates (which is required to obtain robust eddy-
covariance statistics). Our results on the shape of the ET-VPD curve with
environmental conditions held fixed can be built upon with future work
examining how changes in VPD and environmental conditions (e.g. soil water
storage) feedback upon one another. In the soil water storage example, over
very long time scales extremely high VPD perturbations coupled with no
precipitation could result in decreases in soil water storage such that water
becomes limiting. This could be represented by an extension of our framework
in which uWUE is allowed to decrease with decreasing SWC, as observed in Sect.
3.6. Here, we focus our results by assuming constant PFT-wide conditions to
build baseline intuition for ET-VPD dependence. For most PFTs, the theory with
plant function held fixed matches the leading order behavior of the
observations where plant function varies (Sect. 3.4).

##  4 Conclusions

We derived a new form of Penman Monteith using the concept of semi-empirical
optimal stomatal regulation (Lin et al., 2015; Medlyn et al., 2011) and near
constant uWUE (Zhou et al., 2015) to remove the implicit dependence of
stomatal conductance on GPP and ET. With our new form of Penman Monteith we
developed a theory for when an ecosystem will tend to reduce or increase ET
with increasing VPD, which we evaluated against a range of eddy-covariance
data spanning different climates and plant functional types. The goal was to
capture the leading order behavior of the system to gain some intrinsic
knowledge for its behavior. This intuition can be used to disentangle land
atmosphere feedbacks in more complicated scenarios, and will aid
interpretation of observations and sophisticated models.

Our theory suggests that for a majority of environmental conditions, plants
will tend to conserve water and reduce ET with increasing VPD. Stomatal
regulation and plant physiological response strongly regulate ET, and this
regulation varies by PFT. CROs are the least water conservative, while DBF,
EBF, SAV, GRA, MF, WSA, ENF and CSH are progressively more water conservative
(more likely to reduce ET in response to increasing VPD). SAV and WSA exhibit
positive ET response to VPD not necessarily because of poor water conservation
strategies relative to other PFTs, but because of greater occurrence of high
atmospheric demand (VPD) relative to other PFTs. Observations of ET response
to VPD exhibit the same general behavior as the theory, with ET response
becoming more positive (atmospheric demand dominating) as environmental VPD
increases within a PFT, and more negative for PFTs that are adapted to arid
conditions and prioritize water conservation over primary production.

Our paper builds important intuition for how plants respond to VPD
perturbations. We show that given optimal stomatal function and fixed
environmental conditions, the ET-VPD dependence is theoretically concave
upward, with ET increasing with increasing VPD as VPD increases past some
critical value (Table 3.1). However future research should focus on fully
understanding the form of stomatal VPD dependence, as this result is sensitive
to the exponent of VPD dependence, which we currently believe is 1/2 (Medlyn
et al., 2011; Zhou et al., 2014). Indeed, this sensitivity to the exponent of
VPD dependence is an important result itself: land surface models, including
those used in earth system models for climate forecasts, employ different
assumptions about the exponent of VPD dependence in stomatal conductance
(e.g., Ball et al., 1987; Leuning, 1990; Medlyn et al., 2011), and these
assumptions can fundamentally change the relationship between ET and VPD from
one that is concave upward (local minimum in ET) to one that is concave
downward (local maximum in ET).

Our results are also applicable to understanding the impact of expected
increases in VPD induced by global change. Plant physiological responses to
direct CO2 effects (e.g., Swann et al., 2016; Lemordant et al., 2018) receives
more attention than physiological response to indirect effects like increased
VPD. Here, we provide broad PFT-focused results showing a likely decrease in
ET in response to positive VPD perturbations (atmospheric drying), which is
consistent with recent observational analysis (e.g., Rigden and Salvucci,
2017). Feedbacks between the land and the atmosphere may alter the net
response to a long-timescale global VPD perturbation, but our focus on the one
way plant response to a VPD perturbation in the atmospheric boundary layer is
an important first step to disentangling such feedbacks, both in observations
and model simulations of the present and future. By removing Penman Monteith’s
dependence on implicit relationships between GPP, VPD, and ET, we allow for
explicit, accurate future analysis of plant-VPD feedbacks in the atmospheric
boundary layer (Eq. (7)). Our approach can be extended to examine varying
plant response to more nuanced consideration of plant type and climate. Any
plant physiological heterogeneity or feedback that can be conceptualized with
shifts in $g_{1}$ (e.g. Lin et al., 2015; Medlyn et al., 2017) and/or uWUE
(e.g. Zhou et al., 2014) are representable within our framework, which opens
the door for a hierarchy of more sophisticated climate- and plant-specific
analysis of ET sensitivity to environmental variables (including VPD). We
argue that such simplified conceptual frameworks are critical tools for
disentangling land-atmosphere feedbacks at various scales, from diurnal to
seasonal and beyond, and to characterize ET response in a warmer,
atmospherically drier, and enriched CO2 world.

Acknowledgments This work used eddy covariance data acquired and shared by the
FLUXNET community, including these networks: AmeriFlux, AfriFlux, AsiaFlux,
CarboAfrica, CarboEuropeIP, CarboItaly, CarboMont, ChinaFlux, Fluxnet-Canada,
GreenGrass, ICOS, KoFlux, LBA, NECC, OzFlux-TERN, TCOS-Siberia, and USCCC. The
ERA-Interim reanalysis data are provided by ECMWF and processed by LSCE. The
FLUXNET eddy covariance data processing and harmonization was carried out by
the European Fluxes Database Cluster, AmeriFlux Management Project, and
Fluxdata project of FLUXNET, with the support of CDIAC and ICOS Ecosystem
Thematic Center, and the OzFlux, ChinaFlux and AsiaFlux offices. We would like
to thank Dr. Trevor Keenan for providing tools for citation of the FLUXNET2015
dataset, available at https://github.com/trevorkeenan/FLUXNET_citations. This
material is based upon work supported by the National Science Foundation
Graduate Research Fellowship under Grant No. DGE 16-44869. All code and data
used in this analysis, including those used to generate the figures and
tables, are publicly available at https://github.com/massma/climate_et

##  Appendix A FLUXNET2015 sites

Table A1: Metadata and citations for flux sites used in this analysis. All
data are gathered from www.fluxdata.org, and citations are aggregated using
tools available at https://github.com/trevorkeenan/FLUXNET_citations.

Site | PFT | Lat | Lon | Clim1 | Period | References
---|---|---|---|---|---|---
AT-Neu | GRA | 47.1167 | 11.3175 | Unk | 2002-2012 | Wohlfahrt et al. (2008)
AU-ASM | ENF | -22.2830 | 133.2490 | Unk | 2010-2013 | Cleverly et al. (2013)
AU-Cpr | SAV | -34.0021 | 140.5891 | Unk | 2010-2014 | Meyer et al. (2015)
AU-DaP | GRA | -14.0633 | 131.3181 | Aw | 2007-2013 | Beringer et al. (2011)
AU-DaS | SAV | -14.1593 | 131.3881 | Aw | 2008-2014 | Hutley et al. (2011)
AU-Dry | SAV | -15.2588 | 132.3706 | Unk | 2008-2014 | Cernusak et al. (2011)
AU-Gin | WSA | -31.3764 | 115.7138 | Unk | 2011-2014 | Beringer et al. (2016)
AU-How | WSA | -12.4943 | 131.1523 | Aw | 2001-2014 | Beringer et al. (2007)
AU-Rig | GRA | -36.6499 | 145.5759 | Unk | 2011-2014 | Beringer et al. (2016)
AU-Stp | GRA | -17.1507 | 133.3502 | Unk | 2008-2014 | Beringer et al. (2011)
AU-Tum | EBF | -35.6566 | 148.1517 | Cfb | 2001-2014 | Leuning et al. (2005)
AU-Whr | EBF | -36.6732 | 145.0294 | Unk | 2011-2014 | McHugh et al. (2017)
AU-Wom | EBF | -37.4222 | 144.0944 | Unk | 2010-2012 | Hinko-Najera et al. (2017)
BE-Lon | CRO | 50.5516 | 4.7461 | Cfb | 2004-2014 | Moureaux et al. (2006)
BE-Vie | MF | 50.3051 | 5.9981 | Cfb | 1996-2014 | Aubinet et al. (2001)
BR-Sa3 | EBF | -3.0180 | -54.9714 | Am | 2000-2004 | Wick et al. (2005)
CA-Qfo | ENF | 49.6925 | -74.3421 | Dfc | 2003-2010 | Bergeron et al. (2007)
CA-SF1 | ENF | 54.4850 | -105.8176 | Dfc | 2003-2006 | Mkhabela et al. (2009)
CA-SF2 | ENF | 54.2539 | -105.8775 | Dfc | 2001-2005 | Mkhabela et al. (2009)
CH-Cha | GRA | 47.2102 | 8.4104 | Unk | 2005-2014 | Merbold et al. (2014)
CH-Dav | ENF | 46.8153 | 9.8559 | Unk | 1997-2014 | Zielis et al. (2014)
CH-Fru | GRA | 47.1158 | 8.5378 | Unk | 2005-2014 | Imer et al. (2013)
DE-Geb | CRO | 51.1001 | 10.9143 | Unk | 2001-2014 | Anthoni et al. (2004)
DE-Gri | GRA | 50.9500 | 13.5126 | Cfb | 2004-2014 | Prescher et al. (2010)
DE-Hai | DBF | 51.0792 | 10.4530 | Unk | 2000-2012 | Knohl et al. (2003)
DE-Kli | CRO | 50.8931 | 13.5224 | Cfb | 2004-2014 | Prescher et al. (2010)
DE-Lkb | ENF | 49.0996 | 13.3047 | Unk | 2009-2013 | Lindauer et al. (2014)
DE-Obe | ENF | 50.7867 | 13.7213 | Cfb | 2008-2014 | –
DE-Seh | CRO | 50.8706 | 6.4497 | Unk | 2007-2010 | Schmidt et al. (2012)
DE-Tha | ENF | 50.9624 | 13.5652 | Cfb | 1996-2014 | Grünwald and Bernhofer (2007)
DK-Sor | DBF | 55.4859 | 11.6446 | Unk | 1996-2014 | Pilegaard et al. (2011)
FI-Hyy | ENF | 61.8474 | 24.2948 | Unk | 1996-2014 | Suni et al. (2003)
FI-Sod | ENF | 67.3619 | 26.6378 | Unk | 2001-2014 | Thum et al. (2007)
FR-Gri | CRO | 48.8442 | 1.9519 | Cfb | 2004-2013 | Loubet et al. (2011)
FR-LBr | ENF | 44.7171 | -0.7693 | Unk | 1996-2008 | Berbigier et al. (2001)
IT-Col | DBF | 41.8494 | 13.5881 | Unk | 1996-2014 | Valentini et al. (1996)
IT-Cpz | EBF | 41.7052 | 12.3761 | Unk | 1997-2009 | Garbulsky et al. (2008)
IT-Lav | ENF | 45.9562 | 11.2813 | Unk | 2003-2014 | Marcolla et al. (2003)
IT-MBo | GRA | 46.0147 | 11.0458 | Unk | 2003-2013 | Marcolla et al. (2011)
IT-Noe | CSH | 40.6061 | 8.1515 | Unk | 2004-2014 | Papale et al. (2014)
IT-Ren | ENF | 46.5869 | 11.4337 | Unk | 1998-2013 | Montagnani et al. (2009)
IT-Ro2 | DBF | 42.3903 | 11.9209 | Unk | 2002-2012 | Tedeschi et al. (2006)
IT-SRo | ENF | 43.7279 | 10.2844 | Unk | 1999-2012 | Chiesi et al. (2005)
IT-Tor | GRA | 45.8444 | 7.5781 | Unk | 2008-2014 | Galvagno et al. (2013)
NL-Loo | ENF | 52.1666 | 5.7436 | Unk | 1996-2013 | Moors (2012)
RU-Fyo | ENF | 56.4615 | 32.9221 | Unk | 1998-2014 | Kurbatova et al. (2008)
US-AR1 | GRA | 36.4267 | -99.4200 | Dsa | 2009-2012 | Raz-Yaseef et al. (2015)
US-AR2 | GRA | 36.6358 | -99.5975 | Dsa | 2009-2012 | Raz-Yaseef et al. (2015)
US-ARM | CRO | 36.6058 | -97.4888 | Cfa | 2003-2012 | Fischer et al. (2007)
US-Blo | ENF | 38.8953 | -120.6328 | Csa | 1997-2007 | Goldstein et al. (2000)
US-KS2 | CSH | 28.6086 | -80.6715 | Cwa | 2003-2006 | Powell et al. (2006)
US-MMS | DBF | 39.3232 | -86.4131 | Cfa | 1999-2014 | Dragoni et al. (2011)
US-Me2 | ENF | 44.4523 | -121.5574 | Csb | 2002-2014 | Irvine et al. (2008)
US-NR1 | ENF | 40.0329 | -105.5464 | Dfc | 1998-2014 | Monson et al. (2002)
US-Ne1 | CRO | 41.1651 | -96.4766 | Dfa | 2001-2013 | Verma et al. (2005)
US-Ne2 | CRO | 41.1649 | -96.4701 | Dfa | 2001-2013 | Verma et al. (2005)
US-Ne3 | CRO | 41.1797 | -96.4397 | Dfa | 2001-2013 | Verma et al. (2005)
US-SRG | GRA | 31.7894 | -110.8277 | Bsk | 2008-2014 | Scott et al. (2015)
US-SRM | WSA | 31.8214 | -110.8661 | Bsk | 2004-2014 | Scott et al. (2009)
US-Syv | MF | 46.2420 | -89.3477 | Dfb | 2001-2014 | Desai et al. (2005)
US-Ton | WSA | 38.4316 | -120.9660 | Csa | 2001-2014 | Baldocchi et al. (2010)
US-Var | GRA | 38.4133 | -120.9507 | Csa | 2000-2014 | Ma et al. (2007)
US-WCr | DBF | 45.8059 | -90.0799 | Dfb | 1999-2014 | Cook et al. (2004)
US-Wkg | GRA | 31.7365 | -109.9419 | Bsk | 2004-2014 | Scott et al. (2010)
ZA-Kru | SAV | -25.0197 | 31.4969 | Unk | 2000-2010 | Archibald et al. (2009)
ZM-Mon | DBF | -15.4378 | 23.2528 | Unk | 2000-2009 | Merbold et al. (2009)

[1ex]

1 Köppen Climate classification.

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