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Water Resources Research
RESEARCH ARTICLE
10.1002/2013WR014333
Special Section:
Climate, Hydrology and Water
Resources of Eastern Africa
d2H isotopic flux partitioning of evapotranspiration over a
grass field following a water pulse and subsequent dry down
Stephen P. Good1,2, Keir Soderberg2,3, Kaiyu Guan2,4, Elizabeth G. King5, Todd M. Scanlon6,
and Kelly K. Caylor2
1
Key Points:
Isotopic ﬂux partitioning separates
evaporation and transpiration from
bulk ﬂux
Results compare well with scaled
leaf-level observations and
ﬂux-variance similarity theory
Results differ from ﬂux-variance
similarity theory during stressed
periods
Correspondence to:
S. P. Good,
[email protected]
Citation:
Good, S. P., K. Soderberg, K. Guan, E. G.
King, T. M. Scanlon, and K. K. Caylor
(2014), d2H isotopic ﬂux partitioning of
evapotranspiration over a grass ﬁeld
following a water pulse and
subsequent dry down, Water Resour.
Res., 50, doi:10.1002/2013WR014333.
Received 27 JUNE 2013
Accepted 1 FEB 2014
Accepted article online 8 FEB 2014
Department of Geology and Geophysics, University of Utah, Salt Lake City, Utah, USA, 2Department of Civil and
Environmental Engineering, Princeton University, Princeton, New Jersey, USA, 3S. S. Papadopulos and Associates, Inc.,
Bethesda, Maryland, USA, 4Department of Environmental Earth System Science, Stanford University, Stanford, California,
USA, 5Odum School of Ecology, University of Georgia, Athens, Georgia, USA, 6Department of Environmental Sciences,
University of Virginia, Charolottesville, Virginia, USA
Abstract The partitioning of surface vapor ﬂux (FET) into evaporation (FE) and transpiration (FT) is theoretically possible because of distinct differences in end-member stable isotope composition. In this study, we
combine high-frequency laser spectroscopy with eddy covariance techniques to critically evaluate isotope
ﬂux partitioning of FET over a grass ﬁeld during a 15 day experiment. Following the application of a 30 mm
water pulse, green grass coverage at the study site increased from 0 to 10% of ground surface area after 6
days and then began to senesce. Using isotope ﬂux partitioning, transpiration increased as a fraction of total
vapor ﬂux from 0% to 40% during the green-up phase, after which this ratio decreased while exhibiting hysteresis with respect to green grass coverage. Daily daytime leaf-level gas exchange measurements compare
well with daily isotope ﬂux partitioning averages (RMSE 5 0.0018 g m22 s21). Overall the average ratio of FT
to FET was 29%, where uncertainties in Keeling plot intercepts and transpiration composition resulted in an
average of uncertainty of 5% in our isotopic partitioning of FET. Flux-variance similarity partitioning was
partially consistent with the isotope-based approach, with divergence occurring after rainfall and when the
grass was stressed. Over the average diurnal cycle, local meteorological conditions, particularly net radiation
and relative humidity, are shown to control partitioning. At longer time scales, green leaf area and available
soil water control FT/FET. Finally, we demonstrate the feasibility of combining isotope ﬂux partitioning and
ﬂux-variance similarity theory to estimate water use efﬁciency at the landscape scale.
1. Introduction
Knowledge of ecosystem controls on hydrologic cycling is needed to understand both leaf and ﬁeld-scale
biophysical processes [Rodriguez-Iturbe et al., 1999; Caylor et al., 2006; Wang et al., 2010] as well as mesoscale
water resource and climate dynamics [Lawrence et al., 2007]. A diverse set of measurement networks such
as FluxNet and National Ecological Observatory Network (NEON) are directed at estimating ﬁeld-scale bulk
evapotranspiration ﬂux, FET, across various ecosystems [Baldocchi et al., 2001; Keller et al., 2008]. However,
the partitioning of surface ﬂuxes into constituent components of evaporation (FE) and transpiration (FT)
remains predominately model driven in practice [Stoy et al., 2006; Aubinet et al., 2012]. Theoretical
approaches such as those of Shuttleworth and Wallace [1985] are able to model FET and its components FE
and FT, yet they require detailed knowledge of soil and leaf conductances, parameters that are difﬁcult to
measure both accurately and continuously during long studies [Stannard, 1993]. The development of methods to accurately partition FET thus remains a focus of contemporary hydrologic research.
Measurement of plant water use at the scale of individual leaves or plants is readily accomplished through
leaf-level gas exchange [Huxman et al., 2004] and sap ﬂux techniques [Williams et al., 2004; Raz-Yaseef et al.,
2012]. Similarly, microlysimeters and high-precision soil moisture monitoring equipment provide measurements of evaporation from bare soils at small scales [Raz-Yaseef et al., 2010; Zhang et al., 2011]. A number of
hydrologic studies have been directed at partitioning FET utilizing combinations of point measurements of
FE and/or FT with landscape measurements of FET [Williams et al., 2004; Raz-Yaseef et al., 2010, 2012]. Combinations of point and other direct mass loss measurements [Wang et al., 2010] have demonstrated the ability
to partition ﬂuxes, however, many experimental artifacts are introduced though these observation
GOOD ET AL.
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strategies. The exclusion of roots from lysimeters alters soil water dynamics and may not provide a representative evaporate ﬂux, while plant chamber methods inﬂuence biophysical process. The problems of scaling individual point measurements to the landscape scale remains a challenge due to heterogeneity in the
spatial conﬁguration of plant canopies, diversity in species composition, and differences in biophysical processes within landscapes. Furthermore, the equipment necessary to continuously monitor ecosystems with
point-scale measurements presents serious logistical challenges such as uninterrupted sample collection or
maintenance of multiple instruments, and often only a small subset of the community can be monitored.
Current research issues require evapotranspiration partitioning methods operating with continuous observations from ﬁeld-deployed instrumentation. The instruments, in turn, require calibration, and the overall
methods require independent validation because standard methods are still being established. Moran et al.
[2009] proposed a method to partition FET based on surface temperature measurements. This semiempirical
method requires long time series of temperature and ﬂux measurements to estimate a daily ﬂux partition.
Scanlon and Sahu [2008] and Scanlon and Kustas [2010] have proposed utilizing ﬂux-variance similarity
theory to partition ﬂuxes. This method has the ability to partition both carbon dioxide and water vapor
ﬂuxes continuously based on high-frequency observations typical of eddy covariance systems by making
use of correlations between water vapor and carbon dioxide concentrations. Flux-variance similarity partitioning, however, requires the a priori estimation of the ecosystem water use efﬁciency, a parameter difﬁcult to assess beyond the leaf scale, particularly in heterogeneous environments.
Differences in the stable isotope composition of transpiration, dT [see methods for isotope terminology and
units], and evaporation, dE, provide the opportunity to partition FET at the landscape scale. Yakir and Sternberg [2000] theorized that the end-member difference (i.e., dT 2 dE), in combination with the observation of
the bulk ﬂux composition, dET, can be used for isotope ﬂux partitioning. Studies by Wang and Yakir [2000]
and Williams et al. [2004] employed cold traps to sample vapor isotope proﬁles over wheat and olive canopies to estimate isotope ﬂux partitioning. Yepez et al. [2003, 2005] utilized a combination of cold trapped
vapor proﬁles and chamber methods at a study site in the semiarid southwest of the United States to separate FE and FT. A highly controlled vegetation chamber with all exhaust gasses trapped and analyzed was
used by Rothfuss et al. [2010] for isotope ﬂux partitioning. Most recently, Zhang et al. [2011] partitioned FET
over a wheat ﬁeld and estimated the depth of water uptake in the root zone. All the above studies utilized
methods to capture and store vapor, then analyzed the collected liquid through traditional laboratory techniques. This collection and analysis step has limited the temporal resolution of past isotope ﬂux partitioning
studies to relatively long sampling intervals.
The development of stable isotope laser spectroscopy allows for continuous monitoring of ﬂux isotopic
composition at spatial and temporal scales relevant to landscape-scale processes [Lee et al., 2005, 2006,
2007; Grifﬁs et al., 2010, 2011; Sturm et al., 2012; Good et al., 2012]. Welp et al. [2008] used high-frequency
measurements of dET above a canopy to assess leaf water enrichment over a soybean ﬁeld with the assumption of no evaporation. Isotope ﬂux partitioning at the ﬁeld scale (e.g., 100s of m2) has been successfully
employed by Wang et al. [2010] in the controlled environment of Biosphere 2 with potted plants in different
conﬁgurations. Haverd et al. [2011] assimilated high-frequency isotope measurements through a Bayesian
framework to estimate the added information isotope ﬂux partitioning provides to a soil-vegetationatmosphere transfer modeling scheme. In a tall forest canopy where 85% of FET was derived from transpiration, Haverd et al. [2011] concluded that isotope observations did not improve partitioning estimates. However, in arid and semiarid regions, evaporation is a substantial component of the hydrologic budget, and
isotopic differences between FT and FE may provide valuable information about environmental processes.
Thus, further demonstration and application of isotope ﬂux partitioning is required before adoption by the
larger ﬂux community.
In this paper, we present the results of a ﬁeld campaign directed at continuous observation of the isotopic
composition of evapotranspiration with the objective of partitioning FET across a variety of vegetation conditions in African grassland during the dry season. A high-frequency (1 Hz) laser-based isotope analyzer
intake tube was colocated with eddy covariance equipment over a homogenous C4 grass ﬁeld, which was
watered with a total of 30 mm applied over 3 days. We made measurements over a 2 week dry-down
period, and demonstrate here that isotope ﬂux partitioning can capture shifts in transpiration and evaporation ﬂuxes. We compared our partitioning estimates to scaled leaf-level measurements of transpiration as
well as ﬂux-variance similarity partitioning. The hydrologic drivers of evapotranspiration partitioning were
GOOD ET AL.
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investigated over both the diurnal cycle and the entire 2 week experiment. Finally, we combined isotope
ﬂux partitioning with ﬂux-variance similarity theory and used the isotope-based partitioning results to estimate landscape water use efﬁciency.
2. Methodology
The stable isotopic composition of water is expressed in d notation, d5ðR=RVSMOW 21Þ, where R is the ratio
of rare to abundant isotopes (i.e., 2H and 1H respectively) and RVSMOW refers to the same ratio in Vienna
Standard Mean Ocean Water (VSMOW) [De Laeter et al., 2003]. The d values are reported in per mil [&] notation, which is equivalent to 1 3 103 [Coplen, 2011]. We applied a 30 mm water pulse in a desiccated perennial grass ﬁeld (Figure 1) and utilized isotope ﬂux partitioning to understand the mechanisms controlling FT
and FE following grass green-up and subsequent leaf senescence.
2.1. Isotope Flux Partitioning Theory
Isotope ﬂux partitioning is directed at determining the amount of transpiration ﬂux, FT [g m22 s21], in the
bulk evapotranspiration ﬂux, FET [g m22 s21]. The ratio of these ﬂuxes, fT=ET, is found via measurement of the
composition of water molecules derived from ﬂuxes with distinct isotopic signatures. fT=ET is calculated as
fT=ET 5
FT dET 2dE
5
;
FET dT 2dE
(1)
where dET [&] is the isotopic composition of the bulk evapotranspiration, dE [&] is the isotopic composition
of evaporated water, and dT [&] is the isotopic composition of transpired water. The linear mixing model
in (1) has been utilized in previous studies to partition water vapor ﬂuxes [Yakir and Sternberg, 2000;
Williams et al., 2004; Yepez et al., 2005; Wang et al., 2010].
Off-axis integrated cavity
output spectrometer
Tripod eddy
covariance setup
Leaf-level
gas exchange
Watering treatment
extent (13m radius)
Dominant wind
direction (NE)
Figure 1. Experimental setup during the ﬁeld campaign, with an eddy covariance system positioned in the center of the wetted area. The
intake tube of an off-axis integrated cavity output spectrometer (DLT-100 Los Gatos Research Inc., Mountain View, CA) was colocated with
the eddy covariance system at 40 cm above a 5 cm sparse grass canopy. Leaf-level gas exchange measurements were also made with
hand-held equipment (Li-6400XT, LI-COR Biosciences, Lincoln, NE, USA) throughout the ﬁeld campaign. Prevailing winds were predominantly from the North-East.
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The isotopic composition of the FET is determined through the use of a Keeling mixing model [Keeling, 1958;
Yakir and Sternberg, 2000; Yepez et al., 2005; Good et al., 2012]. Assuming that background vapor concentration (qB) and composition (dB) are constant over short time periods (e.g., 30 min), the instantaneous isotopic
composition in the air above a surface, dA [&], is given by
dA 5ðdB 2dET Þ
qB
1dET ;
qA
(2)
which is a liner mixing between isotopic end-members dB and dET. Utilizing equation (2), a regression
between dA and the reciprocal of its vapor concentration, qA [g m23], produces in a line whose y intercept is
equal to the surface ﬂux composition dET. Further analysis of this linear relationship allows for a simple
assessment of the uncertainties of surface ﬂux composition estimates [Pataki et al., 2003; Zobitz et al., 2006;
Good et al., 2012].
The value of dT is measured through direct observation with a simple leaf chamber following Wang et al.
[2012]. A transparent leaf chamber, placed around leaf samples, captures transpired water that is then
drawn into a water vapor isotope analyzer. Assuming background vapor composition and concentrations
are constant during a leaf-sampling event (3 min) allows for the unmixing of ambient water vapor and
leaf transpiration compositions [Haverd et al., 2011; Wang et al., 2012]. The value of dT is then calculated as
dT 5
qM dM 2qA dA
;
qM 2qA
(3)
where qM [g m23] and dM [&] are the water vapor concentration and isotopic composition measured when
the chamber is clamped on a leaf [Wang et al., 2012].
The isotopic composition of soil evaporation is estimated using the Craig and Gordon [1965] model in conjunction with measurements of the soil liquid water isotopic composition, dS [&]. The Craig-Gordon (CG)
model estimates the combined effect of equilibrium and kinetic fractionation during the phase change of
water from liquid to vapor. Atmospheric relative humidity, hA, soil temperature, TS [ C], as well as the gradient between dA and the vapor in equilibrium with dS in the soil, determine the composition of soil evaporation [Gat, 1996; Horita et al., 2008]. Further adjustments to the CG model are made following Soderberg et al.
[2012], accounting for the effect of soil moisture, hS [m3 m23], on soil relative humidity, hs, and soil vapor
transport via the kinetic fractionation factor ej. The value of dE is calculated as
dE 5
ða hS dS 2h0A dA Þ2ðhS e 1ej Þ
:
ðhS 2h0A Þ1ej
(4)
where the equilibrium vapor-liquid fractionation factor a 5ð12e Þ is calculated as a function of TS following
Majoube [1971]. Note that atmospheric relative humidity is renormalized to the surface temperature,
h0A 5hA eA =eS , with eA representing the saturated vapor pressure [kPa] in the atmosphere and eS representing
the saturated vapor pressure at the soil surface. Because of the extremely dry conditions at the study site,
soil vapor saturation is not assumed and instead the relative humidity within the soil is adjusted as function
of the soil water potential via the Kelvin equation, hS 5exp ðMwS =RTS Þ [Mathieu and Bariac, 1996; Soderberg
et al., 2012]. Finally, the kinetic effect is incorporated as
Di
;
ej 5nðhS 2h0A Þ 12
D
(5)
where the ratio of molecular diffusion coefﬁcients of water vapor in dry air Di/D is taken as 0.9757 from Merlivat [1978]. The aerodynamic diffusion parameter, n varies from 1/2 when the soil is saturated ðhS 5hsat Þ, to
1 when the is soil dried to residual moisture levels ðhS 5hres Þ, as [Mathieu and Bariac, 1996; Soderberg et al.,
2012]
1 hS 2hres
:
(6)
n512
2 hsat 2hres
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See Appendix A and Soderberg et al. [2012] for an expanded derivation of the Craig and Gordon [1965]
model for unsaturated soils.
2.2. Experimental Setup
The objective of this ﬁeld experiment was to characterize fT=ET over the green-up and senescence of a typical dry land grass patch in response to a large pulse of soil moisture. The ﬁeld observations were conducted
from 7th to 21st February 2011, in the middle of the dry season (December through March) at the Mpala
Research Center in central Kenya (0.3229 N, 36.9028 E). A grass ﬁeld devoid of green leaf area which had
not received rainfall for 49 days, and only 12 mm in the previous 3 months, was selected and is shown in
Figure 1. A circular region with a radius of 13 m (area 5 530 m2) was then watered for 3 days using water
with a d2H of 4.25& (average precipitation d2H for this region is 27.6& [Soderberg et al., 2013]). For each of
these 3 days, a volume of water equivalent to a 10 mm rainfall event was applied by hand with watering
cans, resulting in 16,000 L applied uniformly within the treatment. The ﬁeld received additional rainfall of
6.7 mm (d2H of rain water was 9.44&) in the predawn period 8 days after the ﬁrst watering as shown in Figure 2. Watering began on the 8th of February 2011, and throughout this article this date is referred to as
day 0, with all other dates expressed relative to the ﬁrst day of watering. No data were collected on day 9
due to equipment and logistical problems.
The grass patch consisted of sparse cover of a C4 herbaceous layer dominated by species from the Cynodon
genus. Two 26 m line transects [Floyd and Anderson, 1987] were surveyed by counting the millimeters of
green leaf contacting a tape-measure edge on days 5–13 to quantify green leaf fractional coverage (Pc),
with no green leaf area observed before or during watering. Leaf area index (LAI) is estimated based on the
relationship between plant canopy gap frequencies and LAI, where LAI52ln ð12Pc Þ=GðhÞ [Nilson, 1971],
with G(h) the projection of unit foliage area on the plane perpendicular to the view direction. Because all
green leaf area consisted of new leaf blades spreading on the ground from nodes in grass stolons, the leaf
angle distribution is treated as horizontal in this very low sparse canopy, i.e., Gð0Þ 1. The time series of
the grass green-up and die-off is shown in Figure 2.
A set of two 10 3 10 3 30 cm soil columns were extracted and used to determine soil texture and root
distribution. Of the collected dry root biomass, 87% was found to occur in the upper 20 cm of soil. Each
day, three to six sets of three soil cores were extracted from random locations within the patch to collect
soils from depths of 1, 5, 10, and 20 cm using a hand auger. Temperature measurements (Extech IR Thermometer) at these depths were used to ﬁt the amplitude and phase of a sinusoidal diurnal model of soil
temperature and heat ﬂow using Fourier’s law of heat transfer [Campbell and Norman, 1998]. This model
was then used to estimate soil skin temperatures. Extracted soils were weighed and dried to determine
gravametric water content [g g21], which was converted to volumetric soil moisture, hS [m3 m23], using a
bulk density of 1457 kg m23 and a porosity, /, of 0.45 [Franz et al., 2011]. Subsamples of the extracted soils
were analyzed for water potential, wS ½MPa , using a potentiomGreen leaf coverage [%]
eter (WP4-T, Decagon Devices,
15
Water pulse & rain event [mm]
Pullman WA). A Campbell [1974]
3 −3
Volumetric water content [m m ] × 100
type power law wS -hS curve was
developed
for the red sandy
10
loam soils at the study site, with
jwS j5jwae jð/=hÞb ; where
wae 5 21.71 3 1022 MPa and
5
b 5 3.1 based on 41 aggregated
samples (r2 5 0.83). The time
series of average volumetric
0
water content in the upper 20
−1 0 1 2 3 4 5 6 7 8 9 10 11 12 13
cm of soil is provided in Figure 2.
Time after 1st water pulse [Days]
Figure 2. Greening up and drying down of the irrigated grass treatment. Three 10 mm
water pulses were applied from 8 to 10 February and an additional light precipitation
event of 6.7 mm occurred before dawn 8 days later (16 February). The average daily volumetric soil moisture in the upper 20 cm of the soil column is also shown.
GOOD ET AL.
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V
From 1 day preceding, the ﬁrst
wetting event (day 21) through
2 weeks afterward (day 13) an
eddy covariance system was
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10.1002/2013WR014333
mounted at 40 cm on a tripod in the center of the circular region. The eddy covariance setup consisted of a
three-dimensional sonic anemometer (CSAT-3, Campbell Scientiﬁc, Logan UT) and an infrared gas analyzer
(Li-7500, LiCor Biosciences, Lincoln NB). Water vapor and carbon dioxide concentrations as well as vertical,
stream wise, and crosswind component wind speeds were measured at 10 Hz. Net radiation (NR-LITE,
Campbell Scientiﬁc, Logan UT), relative humidity (HMP45C, Campbell Scientiﬁc, Logan UT), and air temperature (HMP45C, Campbell Scientiﬁc, Logan UT) were also measured at 1.2 m. Postprocessing of the highfrequency data involved removal of data spikes larger than 43 the standard deviation of the time series.
Coordinate rotation was applied such that mean vertical and crosswind velocities are zero. Standard corrections were applied to the 10 Hz open path ﬂux estimates to account for the dynamic response of sensors,
30 min block averaging, lateral sensor separation, vector and scalar path averaging, and digital ﬁltering
[Moore, 1986; Lee et al., 2004]. Density corrections (Webb-Pearman-Leuning (WPL)) were applied to the
water vapor and carbon dioxide ﬂuxes [Lee et al., 2004]. Average correction factors for the sensible heat
ﬂux, latent heat ﬂux, and CO2 ﬂux were 1.58 6 0.13, 1.19 6 0.22, and 1.19 6 0.22, respectively.
2.3. Isotope Measurements
Isotope measurements were conducted utilizing an off-axis integrated cavity output spectroscopy system
(ICOS; DLT-100 Los Gatos Research Inc., (LGR) Mountain View, Calif.). Air was drawn into the ICOS system via
Teﬂon tubing (14 m long, 1/400 outer diameter) with its intake colocated with the CSAT-3 and Li-7500 at a
ﬂow rate of 500 mL min21. Calibration of the ICOS system was performed before, during, and after the 2
week experiment with a liquid water nebulizer (Water Vapor Isotopic Standard Source (WVISS), Los Gatos
Research, Inc., manufactured in 2009). All standard waters measured on this instrument within a year of the
ﬁeld campaign have a standard deviation of 2.1& for d2H and 0.78& for d18O (n 5 203; with Los Gatos
Research Working Standards 1, 3, and 5). Raw values for analyses of a single working standard (LGR-WS5,
d2H 5 29.8&) performed before and during the experiment (2, 13, 15, and 17 February) had a standard
deviation of 1.8& (n 5 4), comparable to the long-term standard deviation with this standard of 1.9&
(n 5 89). Intensive calibrations were also performed on this instrument (26 August 2011 to 13 September
2011) using three standards covering a range of 188& for d2H. On eight separate days during this period,
multiple calibrations were conducted (average n 5 8 standards ran per day), with single-day calibration
standard deviations ranging from 0.5& to 3.3&, with a mean of 1.9& across all d2H standards. Raw values
for each standard had standard deviations of 1.4& (n 5 16), 1.3& (n 5 17), and 1.3& (n 5 18). No trends
were observed in the measured values relative to standard composition, and instrument drift is assumed
minor and treated as random instrumental noise adding uncertainty in dA values. The effect of this uncertainty on the overall FET partitioning is explored in section 3.2.
The dependency of ICOS measurements on water vapor mixing ratios for Los Gatos Research systems has
been show to be instrument speciﬁc [Sturm and Knohl, 2010; Rambo et al., 2011], and has been previously
reported to be negligible for this particular instrument (ICOS2 of Good et al. [2012]). The linear relationship
between either d2H or d18O and mixing ratios in this ICOS was determined by Good et al. [2012] to have r2
values of 0.00 and 0.03, respectively. (n 5 36 over a range of 20,000 ppmv). However, given the possibility
of bias at very low mixing ratios all individual ambient vapor measurements below 5000 ppmv were
removed from analysis (less the 0.6% of all data). The correlation between d18O or d2H and cavity temperature has also been found to be minor, with r2 values of 0.09 and 0.13, respectively (n 5 149 over a temperature range of 0.5 ). The optical cell is maintained at a temperature of 49 C and a pressure of 52 hPa;
additionally, the case in which the instrument rests is fan cooled. All 30 min blocks during which the optical
cell had conditions farther than 60.5 C away from the operating temperature or 60.1 hPa away from the
operating pressure were removed from the analysis. Details of the ICOS system optical cell and function are
presented in Baer et al. [2002].
The ICOS system was utilized to estimate the isotopic composition of the aggregated evapotranspiration
ﬂux, dET. The value of dET was found though a Keeling mixing model (equation (2)) for each half-hour ﬂux
averaging block (138 half-hour blocks). The ICOS system reports isotopic composition and water vapor concentration measurements at 1 Hz. In each 30 min block, 1800 pairs of dA and qA data points were used to
assess dET. Uncertainty assessments
following Good et al. [2012] were performed on each 30 min estimate
pﬃﬃﬃﬃ
of dET, where rdET rdK =ðcv NÞ, with rdK the standard deviation of differences between individual dA measurements and each Keeling mixing line, cv the coefﬁcient of variability for qA in each 30 min block, and N
the number of data points in each 30 min block. If measurement errors in dA are the only source of error in
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the Keeling plot then rdK should approach the previously assessed instrument uncertainty (2.1& for this
ICOS system). Throughout the course of this experiment, values of rdK ranged from 1.3& to 8.9&, with an
average value of 2.8&. Any increase in rdK values beyond that of the measurement uncertainty is attributed
to unwanted footprint contributions, source water variability, and any other unknowns. Those periods
where rdET was greater than 10& or operational conditions were not met were removed (94 blocks remain).
Periods where dET fell below dE by more than 2 times the calculated dET uncertainty and periods where dET
fell above dT by more than 2 times the calculated dET uncertainty were also removed (78 remain). Half-hour
periods where dET was beyond the end-member value but within acceptable error (2 3 dET) were set as fully
transpiring or fully evaporating. Estimates of dET for each day and associated standard deviations are shown
in Figure 3.
Isotopic composition, δ 2H [‰]
Estimates of the isotopic composition of transpired water (dT) were obtained by direct measurements with
the ICOS system [Wang et al., 2012]. Four times each day between 10:00 AM and 5:00 PM the ICOS system
was connected to a custom leaf chamber (part no. 6400-05, LI-COR Biosciences, Lincoln, NE, USA) which
was placed around a small clump of green leaves. After the isotopic composition and vapor concentrations
stabilized, a 100 s sampling period was used to determine values of dM and qM for equation (3). Ambient values of water vapor concentration (qA) and isotopic composition (dA) were obtained by letting the chamber
sit open for 90 s adjacent to the sampling point immediately before clamping the chamber onto a leaf.
Measurements during which chamber concentrations did not stabilize were not used, nor were measurements with vapor concentration spikes after the leaf removal, indicating condensation (12.5% of leaf measurements). Small clumps (area 4 cm2) of grass were dug up, separated into aboveground and
belowground parts, sealed in glass Vacutainer vials, and frozen until distillation. Water within the plant
100
2
2
Irrigated water δ H=4.25
Additional rain water δ H=9.44
75
50
25
0
-25
-50
-75
-100
-125
δ
ET
(Keelling Plot)
-150
δE (Craig Gordon)
-175
δT (Chamber Measurments)
δS (Distilations)
F Isotope flux partitioning
0.08
T
FT Leaf-level measurments
F Flux-variance similariy partitioning
0.06
T
0.04
0.02
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
Average flux [g m-2 s-1]
0.1
FE Isotope flux partitioning
0
Time after 1st water pulse [Days]
Figure 3. Isotopic composition of bulk evapotranspiration ﬂux, dET (red squares), evaporation ﬂux, dE (blue triangles), transpiration ﬂux, dT
(inverted green triangles), and soil water composition, dS (ﬁlled circles), are used to partition evapotranspiration into FT and FE between
10:00 AM and 5:00 PM. The value of dET relative to dE and dT determines the transpiration and evaporation ﬂuxes (bar graph). Error bars
depict the standard deviation for dET, dE, dT, and partitioned ﬂuxes. Estimates of FT obtained from scaled leaf-level observations (white
crosses) and ﬂux-variance similarity partitioning (smaller red circles) are also shown.
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material was extracted via vacuum distillation on a glass line following West et al. [2006]. The distilled water
was then analyzed using the WVISS and ICOS system. Due to the possibility of contamination of the ICOS
measurements from the water obtained through distillation of organic material [West et al., 2010; Schultz
et al., 2011], only the direct measurements of dT with the chamber method were utilized. Trends were not
observed in the four measurements of dT collected each day and therefore these values were averaged to
obtain a daily dT value (average daily standard error of dT 5 5.9&), which was applied to all 30 min blocks
for that day. Estimates of dT from chamber measurements are shown in Figure 3.
Estimates of the isotopic composition of soil evaporation dE were obtained through the Craig and Gordon
[1965] model given in equation (4). A portion of the soils extracted for water content analysis were vacuum
distilled in the same manner as the plant material. Organic contamination of extracted soil water is not signiﬁcant [West et al., 2010], and soil dS values were utilized in equation (4) to estimate dE. Values of dS and wS
were assumed constant each day and a different dE was calculated for each 30 min block based on changes
in atmospheric conditions (dA, hA, TA) and soil temperature (see equations (4)–(6) and Appendix A). Average
daily soil liquid isotope composition from samples collected from depths of 1–20 cm and the estimated
evaporation composition are shown in Figure 3. On days when soil water isotope composition was not
measured, this value was linearly interpolated.
2.4. Leaf-Level Observations and Flux-Variance Similarity Theory
Direct measurements of leaf-level gas exchange were taken on days 3–8 and 10–13 using a chamber-based
hand-held photosynthesis system (Li-6400XT, LI-COR Biosciences, Lincoln, NE, USA). Estimates of leaf transpiration FT(leaf) [g m22 s21], leaf net photosynthesis, FP [g m22 s21], and estimates of the intercellular to
atmospheric CO2 concentration, ci/cA, where obtained from measurements of three to ﬁve leaves with the
Li-6400XT system at approximately hourly intervals between 10 AM and 5 PM local time, resulting in a total
of 484 measurements of each variable over the entire period. Transpiration rates per unit leaf area were
scaled by measurements of LAI based on the transect surveys to estimate transpiration values per square
meter within the watered area.
Following the approach of Neftel et al. [2008], we scaled FT(leaf) by percent of ﬂux arriving at our eddy covariance system originating from within the watered area (pF, see section 2.5) to determine the transpiration
ﬂux equivalent to that estimated with the isotope approach. In this approach, we assumed FET beyond the
watered area is equal to ﬂuxes measured before irrigation. Thus, FT 5pF FTðleaf Þ LAI1ð12pF ÞfT=ETðdryÞ FETðdryÞ ,
with the transpired fraction in the nonirrigated area fT=ETðdryÞ set as both 0 and 1 to give approximate
bounds. Because most ﬂux originates from within watered area, the difference between assuming all ﬂux
from beyond the irrigated area is due to transpiration versus evaporation is minor at 0.0016 g m22 s21, and
we present the average value (i.e., with fT=ETðdryÞ 50:5).
Flux-variance similarity theory [Scanlon and Sahu, 2008; Scanlon and Kustas, 2010] was also used to obtain
another estimate of FT and fT=ET. This approach has previously been used to partition ﬂuxes over wheat
[Scanlon and Sahu, 2008], a C3 crop, and over corn [Scanlon and Kustas, 2010, 2012], which is a C4 crop. Similarity theory is based on the assumption that the transfer efﬁciencies of stomatal scalars are greater than
those of nonstomatal scalars. From this assumption and the work of Katul et al. [1995] and Bink and Meesters
[1997], the correlation between evaporated water and transpired water is approximated by the ratio of the
transfer efﬁciencies of the evaporation ﬂux to that of the transpiration ﬂux. A similar assumption is made
with respect to photosynthesis, FP, and respiration, FR, giving
qqT ;qE qw;qE
qw;cR
and qcP ;cR ;
qw;qT
qw;cP
(7)
where q is the correlation coefﬁcient, and subscript q refers to water vapor, c refers to carbon dioxide, and
w refers to vertical wind speed. The above assumptions imply that transpiration and photosynthesis are perfectly anticorrelated ðqqT ;cP 521Þ, evaporation and respiration are perfectly correlated ðqqE ;cR 51Þ, and the
correlation between evaporation and transpiration is the negative of the correlation between photosynthesis and respiration ðqqT ;qE 52qcP ;cR Þ.
Combining these statements leads to a system of equations in which the ﬂuxes (FT, FE, FP, and FR), standard
deviations of the scalar concentrations from each source ðrqT ; rqE ; rcP ; and rcR Þ, and the correlations
GOOD ET AL.
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between ﬂux-derived scalars ðqqT ;qE and qcP ;cR Þ are 10 unknowns. Ten equations (WUE5FP =FT , and equations
(5a), (5b), (8a), (10a), (10b), (11c), (12a), (12b), (18) of Scanlon and Sahu [2008]) are then solved simultaneously to estimate the partitioning of water vapor ﬂux and carbon dioxide ﬂux. A key input to the ﬂuxvariance similarity partitioning method is the water use efﬁciency, WUE [mg CO2 g21 H2O], of the vegetation. Water use efﬁciency is the ratio of carbon assimilation to water lost to transpiration ðWUE5FP =FT Þ and
is important indicator of biophysical functionality [Bacon, 2004]. WUE is modeled-based gradients between
interstomatal and leaf surface concentrations of carbon dioxide and water vapor [Campbell and Norman,
1998]. Leaf surface concentrations are estimated based on stability corrected logarithmic concentration proﬁles. Following Scanlon and Sahu [2008], interstomatal water vapor is assumed saturated at the measured
air temperature and interstomatal carbon dioxide is estimated based on an average ratio of internal to
ambient CO2. This value, ci =cA 5:21, was obtained directly from the Li-6400 leaf-level measurements and is
slightly lower the typical reported values of 0.3–0.4 [Jones, 1992; Bunce, 2005], though values less then this
have been reported under stressed conditions.
2.5. Footprint Considerations and Experimental Artifacts
Our experimental framework is aimed at providing a detailed analysis of the ecohydrologic processes occurring in a natural savanna during different stages of growth. Conducting this experiment at an open ﬁeld at
the Mpala Research Centre ensures realistic conditions found in an actual dry land, but also brings extreme
difﬁculties in work environment. Because of the possibility of equipment damage from wildlife, the eddy
covariance system operated during daytime hours only (10:00 AM to 5:00 PM local time) and was
removed from the ﬁeld at night. The site location is within a very dry region with a mean annual precipitation of 600 mm, and the irrigation water was trucked in from a nearby river, thus limiting the extent to
which we could irrigate the experimental setting. Due to the limited size of the watered region, turbulent
eddies carrying dry air from beyond the experiment were also sampled at the eddy covariance tripod location. Furthermore, the irrigation of a small patch within a very arid region may lead to oasis conditions
wherein the turbulent regime above the watered area differs from that beyond the irrigated extent. Finally,
because of power and other logistical difﬁculties, air was only sampled at a single height, and only a temporal Keeling plot approach, as opposed to the more reliable multiheight ﬂux-gradient technique [Wang and
Yakir, 2000; Grifﬁs et al., 2004, 2005; Santos et al., 2012], is available for estimation of source isotopic
composition.
Given the heightened possibility of experimental artifacts being introduced by our manipulation and the
large footprint associated with concentration measurements needed for Keeling plot analysis, a Lagrangian
stochastic dispersions model [Wilson and Sawford, 1996; Thomson, 1987] is employed to determine the distribution of locations for water vapor molecules that have left the surface and arrived at our collector. The
Lagrangian stochastic dispersion model, fully described in Appendix B, tracks the position and velocity of a
large collection of ﬂuid elements through a Markov process and permits footprint modeling in complex
environments. Lagrangian stochastic models are the most powerful tool available for investigating atmospheric transport [Wilson and Sawford, 1996] and are particularly suited to complex environments with inhomogeneities where higher-order analytical models are known to fail [Thomson, 1987]. As employed here,
the Lagrangian stochastic dispersion model allows for incorporation of differential surface ﬂuxes and turbulence regimes within and beyond the watered area. Simulations were run for each 30 min block separately,
given the surface sensible heat ﬂux (FH [W m22]), latent heat ﬂux (kvFET [W m22]), and friction velocity (u* [m
s21]) within and beyond the irrigated area. Conditions beyond the irrigated area were based on the diurnal
cycle observed before the irrigation was applied.
The Lagrangian stochastic dispersion model demonstrates that the large majority of evapotranspired source
vapor and ﬂuxes arriving at our collector location originated within the irrigated area (Figure 4). After watering, an average of 76% of the vapor arriving at our collector originated from within the irrigated area, and
an average of 81% of the evapotranspiration ﬂux at our collector originated within the watered area. The
particle fraction from within the irrigated area is positively correlated with FET (q 5 0.83) and atmospheric
stability f 5 z/L (q 5 0.50), and negatively correlated with FH (q 5 20.60). Wind speed demonstrated little
correlation with source origin (q 5 0.06) and wind direction varied little during the experiment (standard
deviation of average wind direction was 17 ). When the same model is run with uniform conditions, only
40% of vapor and 57% of the ﬂux originated within the irrigated area. Our collector (the LGR ICOS system)
measured a mixture of background and irrigated source vapor, with a small percentage of nonirrigated
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water also measured. The source
composition identiﬁed with Keeling plot analysis for each time
block therefore includes the
minor inﬂuence of water vapor of
unknown composition from
beyond the irrigated area. The
additional source water will add
variance to the dA values measured, degrade the dv -1=qA regression, and add uncertainty to the
ﬁnal intercept dET estimate.
Therefore, careful estimation of
uncertainty bounds is required
which includes the inﬂuence of
the interrelated distributions of
qA and dA on dET.
100
Irrigrated Area Contribution [%]
90
80
70
60
50
40
30
20
Concentraion Contribution (Different wet/dry conditions)
Concentraion Contribution (Uniform conditions)
Flux Contribution (Different wet/dry conditions)
Flux Contribution (Uniform conditions)
10
0
0
0.02
0.04
F
0.06
-2 -1
0.08
0.1
(watered) [g m s ]
ET
3. Results
Figure 4. Contribution of the irrigated area to concentration and ﬂux measurements
based on Lagrangian stochastic dispersion model. Model runs with different u*, FH, and
FET within the irrigated area versus outside that area demonstrate that the large majority
of vapor concentration and ﬂux arriving at the collector originated within the watered
area (ﬁlled symbols), with an increasing percentage when evapotranspiration within the
watered area increases. When the same model is run with uniform conditions and ﬂuxes
(open symbols), a much smaller fraction of vapor concentration and ﬂux at the collector
originate from within the watered area.
Estimates of fT=ET were obtained
from isotope ﬂux partitioning for
10 days and are shown in Figure
3. Before the watering treatment,
FET was 0.006 g m22 s21 (0.5
mm d21). Subsequent to the
watering, FE increased rapidly
and peaked on the last day of watering at 0.070 g m22 s21 (6 mm d21). Green leaf area ﬁrst appeared on
day 3, but FE continued to dominate the surface ﬂux, with fT=ET equal to 16%. Figure 5 depicts the timeline
of isotope ﬂux partitioning results from day 3 to day 13. After day 3, FE decreased while FT increased steadily, with fT=ET rising to 40% on day 6. At day 8, when the 6.7 mm event occurred, both FT and FE peaked, and
afterward began to decline.
3.1. Partitioning Intercomparison
Transpiration estimated from isotope ﬂux partitioning compares well with scaled leaf-level measurements,
with best agreement obtained from daily average values (Figure 5). Half-hour FT results give a root mean
square error of 0.0056 g m22 s21 between leaf level and isotope ﬂux partitioning methods, with a regression slope of 0.22 and an intercept of 0.006 g m22 s21. Analysis of partitioning results averaged to a daily
estimate of FT (Figure 6) give a root mean square error of 0.0018 g m22 s21, with a regression slope of 0.40
and an intercept of 0.004 g m22 s21 (r2 5 0.64). These low regression slopes may result from the temporal
patterns of fT=ET, with more evaporation occurring during periods when dET is more accurately measured.
Flux-variance similarity partitioning results are partially consistent with the isotope ﬂux partitioning results,
yet show divergence on days 8, 12, and 13 (Figure 5). On days 0 and 1 (shown in Figure 3), the ﬂux-variance
method estimates nonzero transpiration although no green leaf area had yet appeared. On day 8, the ﬂuxvariance method estimates a sharp drop in fT=ET, resulting in a large increase in FE and a decrease in FT. Isotope ﬂux partitioning and leaf-level estimates suggest an increase in both FT and FE on day 8, with the ﬂuxvariance estimate indicating an increase in FE but a decrease in FT. On days 12 and 13, a period when green
leaf area began to senesce and soil moisture was limited, the ﬂux-variance similarity partitioning estimated
a higher amount of transpiration than isotope ﬂux partitioning.
3.2. Uncertainty in Evapotranspiration Partitioning
Uncertainty in the results of a single isotope two-source mixing model is directly related to the uncertainty
in end-member and source estimates ðrdE ; rdT ; and rdET Þ as well as the magnitude of the difference in
end-members (dT 2 dE) and the ﬁnal partitioned value of fT=ET. Assuming source and end-members are
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10.1002/2013WR014333
independently measured, a ﬁrstorder Taylor series expansion of
the uncertainty in the ﬁnal partitioned value is calculated as [Phillips and Gregg, 2001]
Transpiration [g m-2 s-1]
Isotopic flux partitioning
0.03
Scaled leaf-level measurments
0.02
-1
r2fT =ET 5
0.01
r2d
T
1ðf
Þ
T=ET
ðdT 2dE Þ2
ðdT 2dE Þ2
r2dET
1ð12fT =ET Þ
0
Evaporation [g m s ]
Flux-varaince similairy partitioning
0.03
3
4
5
6
7
8
9
10
11
12
13
r2d
E
ðdT 2dE Þ2
:
(8)
Time after 1st water pulse [Days]
-2
In (8), the three terms on the
right-hand side represent the
contribution of uncertainties in
0.02
rdET ; rdT ; and rdE to the total
variance. Values of rdET are found
by assessing the Keeling plot lin0.01
ear regression [Good et al., 2012],
which includes the possible inﬂuence of moisture from beyond
0
3
4
5
6
7
8
9
10 11 12 13
the irrigated area. A single value
of rdT is used for all blocks within
Time after 1st water pulse [Days]
that day, with the value of rdT
Figure 5. Timeline of average daily transpiration, 10:00 AM and 5:00 PM, (top) ﬂux and
estimated as the standard error
(bottom) evaporation ﬂux. Results from isotope ﬂux partitioning (solid blue line) are comof all chamber-based dT measurepared with those of ﬂux-variance similarity partitioning (dashed red line) and scaled leafments made on that particular
level measurements (crosses). Error bars depict the standard error of the mean of all
observations for that day. Days with less than three measurements are marked with open
day (because all dT values were
circles.
averaged to a give a single daily
value). The value of rdE is calculated based on the estimated instrument uncertainty propagated though
the Craig-Grodon equation for each half-hour block.
Across all days, the average uncertainty in the bulk evapotranspiration ﬂux isotope composition was 3.1&.
This uncertainty is smaller than that determined for dT (6.0&) and larger than that of dE (2.2&). On average,
the partition value, fT=ET, and its uncertainty, rfT=ET , is estimated as 29 6 5%. As is evident from (8), the results
of a partitioning calculation are more sensitive to a particular end-member’s uncertainty (i.e., rdT or rdE ) as
fT=ET approaches either the fully transpiring or fully evaporating state. Figure 7 depicts the fractional contribution to total variance of uncertainties in bulk ﬂux and end-member composition. At low values of fT=ET,
partitioning uncertainty is due mostly to uncertainty in bulk ﬂux composition because dET is more poorly
quantiﬁed then dE. As fT=ET increases, uncertainties in dT and dET both degrade the ﬁnal partitioning estimate
accuracy. Because dT was only measured four times each day, uncertainty in this term is large. Uncertainties
in dET are due to both measurement errors as well as contributions from beyond the watered area, source
water variability, and any other unknown errors. A ﬁnal average uncertainty of 5% on an average partition
value of 29% represents a relative uncertainty of 17%. Further improvement in the uncertainties of dET and
dT is necessary for isotope ﬂux partitioning to separate ﬂuxes of transpiration and evaporation with a high
degree of accuracy.
3.3. Environmental Controls on Partitioning
Isotope ﬂux partitioning results are used to examine the factors controlling FET and fT=ET at multiple time
scales. Isotope ﬂux partitioning estimates of FT and FE are strongly impacted by the average diurnal cycle of
net radiation, air temperature, and relative humidity (Figure 8). Total evapotranspiration was highly correlated with net radiation ðqRn ;FET 5:64Þ, with both Rn and FET peaking midday. Average air temperature continued to rise throughout the day and peaked in the early evening. Conversely, relative humidity peaked in
the morning and again slightly in the late afternoon. Water availability in the root zone showed little
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0.03
10.1002/2013WR014333
variation at the subdaily time
scale (with the obvious exception
of days with watering or rainfall).
Throughout the course of the
experiment, diurnal variation in
FET was predominantly driven by
the total amount of available
energy to drive
evapotranspiration.
〈F 〉 RMSE=0.0018, r2=0.64
T
T
-1
FT from isotope flux partitioning [g m s ]
F RMSE=0.0056, r2=0.04
-2
1:1 line
0.02
Fractional contribution to uncertainty
The diurnal cycles of transpiration and evaporation are different than those of total
evapotranspiration. Based on the
0.01
averaged diurnal cycle, FE is the
largest ﬂux component and is
also highly correlated with net
radiation ðqRn ;FE 5:57Þ, however,
FT is less correlated with radiation
ðqRn ;FT 50:30Þ. For both relative
humidity and air temperature,
0
0
0.01
0.02
0.03
the correlations between total
FT from scaled leaf-level partitioning [g m-2 s-1]
ﬂux and these forcing variables
(qhr ;FET 50:33 or qTemp ;FET 52:40)
Figure 6. Comparison of isotope ﬂux partitioning to leaf-level measurements. Both, averare close to that of the transpired
age daily estimates, hFT i (solid blue line, solid blue circles), and half hour, FT (dashed blue
ﬂux (qhr ;FT 5:39 or
line, open blue circles), comparisons are provided. Periods with both leaf level and isoqTemp ;FT 52:44), while the relatope partitioning observations are shown for the half-hour plot, while all available data
for each method were averaged each day then compared in the daily plot. Error bars
tionships between these variashow the standard error of the mean of all observations for each day.
bles and the evaporation ﬂux is
weaker (qhr ;FE 5:15 or
qTemp ;FE 52:19). Over the average diurnal cycle, isotope ﬂux partitioning indicates that FT peaks in the late morning (11:00 AM–12:30 PM)
and again in the late afternoon (3:30 PM–4:30 PM), while showing depressed FT values in the early afternoon
(2:00 PM–3:30 PM). The early afternoon period corresponds to both high net radiation and high temperatures. Given the low green leaf area (LAI 0.1), photosynthesis is not expected to be radiation limited.
Instead, plant regulation of stomatal conductance appears to
σ
lower FT during periods of high
δ
1
temperatures and high net radiaσδ
tion as may be expected from
σ
0.8
δ
leaf physiology models [Jones,
1992].
ET
T
E
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Transpired Fraction,fT/ET
Figure 7. Fractional contribution to partitioning uncertainty of bulk ﬂux composition
uncertainty ðrdET Þ and end-member uncertainty (rdT and rdE ) as a function of the transpired fraction of evapotranspiration, fT=ET. Fractional contribution calculated as the three
separate terms of equation (8) divided by the total partition uncertainty rfT=ET and
smoothed with a moving average.
GOOD ET AL.
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V
Over the time scale of multiple
days, large shifts of fT=ET are
observed despite similar meteorological cycles. Daily average isotope ﬂux partitioning results
comparing the transpired fraction to green leaf coverage, the 5
cm soil water potential, and the
center of mass of soil water are
shown in Figure 9. At multiday
time scales, a complex combination of green leaf area and available water controls fT=ET. Green
12
600
10.1002/2013WR014333
Net Radiation
400
200
0
Relative Humidity
Temperature
50
40
30
20
Humidity [%] & Temp [ oC]
-2 -1
Net Radiation [W m s ]
Water Resources Research
leaf area and fT=ET are initially
very low (Figure 9, left). As the
green leaf area increases, so does
the portion of FET derived from
transpiration. After day 8, fT=ET
and green leaf area decline, yet
not along the same trajectory as
their prior respective increases
from day 3–8. This hysteresis
with respect to leaf area is
strongest on day 12, where a similar amount of green leaf as on
day 6 results in a fT=ET substantially smaller.
The availability of water for transpiration and evaporation is
related to the soil water potential
F Isotope flux partitioning
E
0
and the vertical location of water
F Isotope flux partitioning
T
in the soil column as represented
F Leaf-level measurments
T
0.03
by the center of mass of soil
water, zhCM [cm] (Figure 9). As
long as water potential at 5 cm is
above 25 MPa, FT increases as a
portion of the total ﬂux up to a
0.02
maximum fT=ET of 40% on day 6.
fT=ET remains above 25% from
days 6 to 8 while green leaf continues to increase. Once the 5 cm
potential falls below 25 MPa,
0.01
fT=ET begins to decrease along
with green leaf area. These
dynamics suggest 25 MPa as a
likely wilting point for the grasses
within our study area. The 6.7
0
10
11
12
13
14
15
16
17
mm rainfall between days 7 and
Diurnal Time [Hour]
8 increased the 5 cm soil water
potential, but this increase was
Figure 8. Average diurnal cycle of (top) net radiation (yellow line), (middle) relative
insufﬁcient to cause further
humidity (red line) and air temperature (orange dash), and (bottom) isotopically partitioned transpiration, FT (green bars), and evaporation, FE (blue bars). Shaded areas and
increases in green leaf area. The
error bars depict one standard error of the mean value of all measurements collected
center of mass of soil water was
during that speciﬁc hour over the 13 day experiment. The average scaled leaf-level measnear the surface (7.3 cm below
urements of FT are also depicted (white crosses).
ground level) following the 30
mm watering, and reached a maximum depth of around 10.5 cm by day 13. The center of mass shifted
from 10 to 6.8 cm on day 8 in response to the rain pulse, but returned to 9.4 cm by day 10. Transpiration is
most strongly correlated to green leaf area ðqFT ;LAI 50:64Þ while higher evaporation is most closely correlated with water centered at a shallow depth in the soil column ðqFE ;zhCM 52:72Þ.
-2 -1
Average Flux [g m s ]
10
3.4. Combining Isotope Flux Partitioning and Flux-Variance Similarity Theory to Estimate Landscape
Water Use Efficiency
WUE is governed by leaf physiochemical functioning and may be an indicator of plant water stress. Plants
utilizing the C4 pathway typically have higher WUE and are able to survive in arid environments more readily than C3 species [Jones, 1992; Bacon, 2004]. Knowledge of WUE characteristics is important for understanding and potentially improving ecosystem and agricultural productivity, yet WUE is difﬁcult to measure
at the ﬁeld scale because of the confounding effects of evaporation and respiration. To accurately estimate
GOOD ET AL.
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Transpired Fraction × 100 [%]
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50
50
50
45
45
45
6
40
8
35
4
5
10
7
13
30
6
40
30
11
25
8
35
13
11
20
12
3
4
4
30
20
12
3
10
10
10
5
5
5
4
6
8
Green Leaf
Coverage [%]
10
0
2
-10
1
0
-10
-10
12
3
15
2
13
25
15
0
5
10 7
11
15
0
8
35
5
10
7
25
20
6
40
-1
-10
5cm Soil Water
Potential, ψ [MPa]
S
0
6
8
10
12
Soil Water Center of
Mass, [cm below surface]
Figure 9. Isotope ﬂux partitioning with respect to (left) green leaf area, (middle) 5 cm soil water potential, and (right) soil water center of mass. The number of days after the ﬁrst watering event is shown next to each point. The dashed vertical line at wS 5 25 MPa indicates a likely wilting point for the vegetation.
vegetation WUE at the landscape scale, nonstomatal ﬂuxes of evaporation and soil respiration need to be
removed from the bulk ﬂuxes (FET and FC). By combining isotope ﬂux partitioning and ﬂux-variance similarity theory, estimates of the carbon dioxide partitioning and landscape WUE are obtained based on knowledge of the evapotranspiration partitioning and the covariance structure of water vapor and carbon
dioxide.
Flux-variance similarity partitioning consists of a system of interconnected equations and unknowns as outlined in section 2.4. Scanlon and Sahu [2008] and Scanlon and Kustas [2010] model WUE based on concentration proﬁles and use it as an input to the ﬂux-variance similarity partitioning method with the objective
of partitioning water vapor and carbon dioxide ﬂuxes. Alternatively, WUE may be considered an unknown
and instead the isotope ﬂux partitioning FT estimates may be considered an input. In this case, the water
vapor partitioning results may be used in a inversion implementation of ﬂux-variance theory to derive the
partitioning of the carbon ﬂux. To obtain the CO2 partitioning, the solution to the set of ﬂux-variance similarity equations proceeds in the same manner as in Scanlon and Sahu [2008], with an initial guess at the
WUE as estimated in section 2.4. The resulting partitioned estimates are compared to those obtained from
isotope ﬂux partitioning and the WUE is iteratively adjusted until the two methods’ estimates of fT=ET converge. The same criteria for realistic ﬂux results as the original implementation ðFT 0; FE 0; FP 0; and
FR 0Þ are used to ﬁlter out invalid solutions. The unconstrained nonlinear optimization routine of Lagarias et al. [1998] was implemented in the MATLAB programming environment (Math-Works Inc., Natick USA,
version R2012a) by minimizing the difference between the ﬂux-variance similarity fT=ET estimate and that
obtained from isotope ﬂux partitioning. We found that when FT values became very small the results
become unstable. This is expected because FT forms the denominator of WUE such that WUE ! 1 as
FT ! 0. The absolute difference between the leaf level and isotope ﬂux partitioning with similarity theory
becomes very large below a value of approximately 0.005 g m22 s21. Therefore, all individual 30 min blocks
where FT 0.005 g m22 s21 were removed for this method. After ﬁltering FT values below this threshold,
the half-hour RMSE difference between the leaf level and inversion ﬂux-variance similarity WUE estimates
was 21.9 mg CO2 g21 H2O.
Leaf-level measurements show a gradual increase in WUE as the experiment continued, likely due to
decreasing soil water potential (Figure 10). The modeled values of WUE based on log-proﬁles of temperature, water vapor, and carbon dioxide are highly variable [Scanlon and Kustas, 2010, equation 11], particularly on day 8 when the WUE 5 196 mg CO2 g21 H2O. When the similarity partitioning method is inverted to
estimate the WUE and the results are averaged to daily estimates the RMSE was 8.7 mg CO2 g21 H2O when
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Inversion of similariy eqns. based on isotopic partitioning
Modeled based on measured Ci/Ca ratio
Leaf-level measurments
40
-1
-Water Use Efficiency, WUE [mg g ]
10.1002/2013WR014333
30
20
10
0
3
4
5
6
7
8
9
10
11
12
13
Time after 1st water pulse [Days]
Figure 10. Water use efﬁciency, WUE, is estimated by inversion of ﬂux-variance similarly partitioning equations based on isotopic partitioning of water vapor ﬂux (blue line). The WUE input used to partition FET with the ﬂux-variance method is iteratively adjusted until the partitioning matches that obtained from the isotopic technique. WUE is also modeled based on the gradient of interstomatal CO2 and H2O to
external CO2 and H2O concentrations (red dashed line). Interstomatal H2O is assumed to be saturated at air temperature, and internal CO2
is estimated based on a constant ci/cA ratio, while external concentrations are estimated based on stability corrected logarithmic proﬁles
[Scanlon and Sahu, 2008]. Leaf-level measurements suggest an increase in WUE (crosses) as soils dry at the end of the observation period.
Error bars show the standard error of the mean of all observations for that day. Days with less than three measurements are marked with
open circles.
FT values below 0.005 g m22 s22 are removed. Thus, based on leaf-level measurements, the combined isotope and ﬂux-variance method provides better estimates of WUE than the straightforward modeling. The
isotope ﬂux partitioning and inverse ﬂux-variance calculations produce trends in WUE estimates (solid line
Figure 10: slope 5 0.15, p value 5 0.85) with values slightly higher than the leaf-level measurements. The
trend of increasing WUE that is observed in the leaf-level measurements (dotted line Figure 10:
slope 5 0.54, p value 5 0.06) is qualitatively matched by the results of the inverse ﬂux-variance implementation but is not statistically signiﬁcant in either linear regression.
4. Discussion
The results presented here indicate that isotope ﬂux partitioning is able to estimate the transpired fraction
of surface vapor ﬂux with an uncertainty of 5% during periods of diurnal change and daily shifts in green
leaf area. Validation of landscape level partitioning methods remains challenging because of the difﬁculty
of accurately assessing FT over large areas. Our comparison of isotope ﬂux partitioning with direct leaf-level
transpiration measurements over a homogeneous area is a rigorous test of partitioning methodology. However, even the scaling of leaf-level gas exchange data potentially encompasses signiﬁcant errors including
small sample size, ﬁne-scale heterogeneity in vegetation structure, soil heterogeneity, variability in plant
ecophysiology across the ﬂux footprint, and the relative amount of transpiration arriving at the eddy covariance system. The inherent difﬁculties in scaling from individual leaf measurements to ﬁeld-scale estimates
indicates the need for landscape-scale methods such as isotope ﬂux partitioning, and at the same time
accentuates the difﬁculty in validating them.
Logistical difﬁculties limited the transportation and application of water over large areas at the Kenya
research station. As a consequence, the areal extent of the watering treatment was restricted to a 13 m
radius circle. Sunrise and sunset at the ﬁeld site was approximately 6:30 AM and 6:30 PM local time, so a
small portion daily evapotranspiration likely occurred outside our measurement windows. Furthermore,
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evaporation likely continued throughout the night at low rates. Thus, our partitioning estimates only
address the daytime hours during which our equipment was running. The eddy covariance system was
placed low in order to minimize the amount of turbulent eddies originating from beyond the watered area
observed at the sonic anemometer. The grass canopy was extremely short (5 cm) and placement of the
eddy covariance system at 40 cm resulted in a instrument height 8 3 the canopy height, well beyond the
typical 4 3 canopy height suggested in literature [Aubinet et al., 2012; Burba and Anderson, 2010]. The low
placement of the system results a measurement height 4 3 the sonic anemometer path length of 10 cm.
This is within the range where Kristensen and Fitzjarrald [1984] found 1-D line averaging did not signiﬁcantly
affect surface ﬂuxes; however, van Dijk [2002] found larger ﬂux attenuation with 3-D sonic anemometers.
We have corrected for cospectral attenuation of ﬂux through frequency response corrections [Moore, 1986]
and after these corrections ﬁnd the energy budget at the site is closed to within 12.8% with a RMSE of 74.9
W m22 s21. Furthermore, the total mass of water applied (30 mm per square meter) is consistent with the
sum of evapotranspired water based on an average ﬂux of 0.02 g m22 s21 over 14 days (24 mm per square
meter).
Because of the limited extent of the watered surface area, our system measured turbulent eddies carrying
ﬂuxes originating from beyond the treatment area. However, as shown by Neftel et al. [2008] with EC ﬂux
measurements of adjacent but distinct ﬁelds, proper consideration of the ﬂux footprint allows for the reconstruction of measured eddy covariance ﬂuxes based on separate ﬂux information from distinct sources. Our
footprint calculations show that on average 81% of the observed ﬂux originated from within the watered
area while 19% originated from beyond the watered area. While this current conﬁguration is not the ideal
setup for measuring ﬂuxes over a grass surface, the conﬁguration we chose represents a compromise
between ﬂux observation and experimental limits. Because of the similarity between the bulk mass ﬂux
measured and water applied, closure of the energy budget to 13%, and excellent matching of scaled leaf
level and isotope ﬂux partitioning transpiration measurements, we have conﬁdence in both our estimates
of surface ﬂuxes and the isotope partitioning methodology.
The isotope ﬂux partitioning technique is predicated on separate characterization of dT, dE, and dET, as well
as a distinct difference between dE and dT. Measurements of the isotopic composition of transpiration were
obtained through direct measurement in this study and assumed to be unchanging over the diurnal cycle.
Large diurnal shifts are known to occur in plant liquid water isotope values over the course of the day and
these shifts in plant water affect the composition of transpired water [Farquhar and Cernusak, 2005; Farquhar et al., 2007]. Our measured dT values fall above the extracted soil water compositions, indicating nonsteady state conditions during the day. As noted by Dongmann et al. [1974], over longer periods transpired
water composition must approach that of water entering the leaf. However, this applies only when leaf
water content is not changing, a condition not present in the green-up and dry down of this experiment.
Clear trends were not observed in the four measurements of dT collected each day, and therefore, a single
daily average value of dT was used.
The Craig and Gordon [1965] model has been utilized in previous studies to estimate the isotopic composition of evaporated water [Yepez et al., 2003; Williams et al., 2004; Yepez et al., 2005; Wang et al., 2010, 2013].
As noted by Wang et al. [2013], the ﬁnal value of fT=ET is most sensitive to changes in dS, while shifts in a due
to temperature are relatively small in comparison. Recently, Soderberg et al. [2012] reviewed methods of
characterizing reservoirs and ﬂuxes of stable isotopes of water in the vadose zone and found that adjusting
the Craig and Gordon [1965] framework in dry soils can lead to either enhanced or diminished fractionation.
For the soils at the study site in Kenya, the consideration of relative humidity less than 100% in the vadose
zone and increased vapor diffusion in unsaturated soils results in an enhancement of the kinetic fractionation and an average shift in ﬁnal ﬂux composition estimates of 29.5& in the overall fractionation factor relative to the classical form of the Craig and Gordon [1965] model (see Appendix A for details). This shift was
highly consistent with a standard deviation of 0.43& across all days. With the exception of day 21 when
the soil was extremely dry (wS 5 2125 MPa) all soil water potentials fell between 220 and 20.5 MPa, the
range in which dE is least sensitive to wS values [see Soderberg et al., 2012, Figure 2]. The n values (equation
(14)) determined by Mathieu and Bariac [1996] are based on experiments with drying soils not open water
as is typically considered. Calculation of dE (average of 266.3&) with these n values is consistent with our
observed dET estimates (average of 270.19&) during low LAI days 0–3 when no transpiration was likely
occurring and thus provided further validations for these soil moisture-based n values.
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Currently there is no standard method for measuring the isotopic composition of evapotranspiration ﬂux
and a variety of techniques have been applied in recent studies. The most reliable method, the ﬂuxgradient technique [Wang and Yakir, 2000; Grifﬁs et al., 2004, 2005; Good et al., 2012; Santos et al., 2012], was
not able to be applied in this situation, though its use would likely decrease uncertainty in our dET estimates.
However, the Keeling mixing model has been shown to produce reliable results which can approach the
ﬂux-gradient technique [Good et al., 2012; Santos et al., 2012] if the quality control ﬁltering is applied to
remove periods with large uncertainty. Furthermore, application of Keeling plots over short time intervals
(less then 1 h) reduces complications that arise due to the nonstationarity of surface and micrometeorological conditions and also minimizes the impact of possible shifts in background and source isotopic composition [Good et al., 2012].
Throughout the experiment relatively constant midday meteorological conditions and strong surface heating
suggest that the planetary boundary layer is expected be high ( 2.5km [Culf, 1992]). In that situation, any
entrained vapor must mix with a large volume of boundary layer moisture and thus any effects on planetary
boundary layer background composition are dampened. Any dET values with uncertainties larger than 10&
were excluded; however, even with this ﬁltering, days such as 21 still show large variability, likely due to low
ﬂux rates. Retained uncertainties in dET were 3.1& on average, which is much less than the average dT–dE difference of 94&. The presence of vapor originating from beyond the watered area will affect the Keeling plot
[Grifﬁs et al., 2007], thus our analysis reﬂects partitioning based on the observed bulk ﬂux composition over
the entire footprint and end-members assessed for the plot. However, as demonstrated by the Lagrangian
stochastic dispersion model, the concentration (and ﬂux) footprints are strongly weighted toward the watered
area; the similarity of these footprints thereby allows an intercomparison of these methodologies.
Flux-variance similarity theory provides estimates of evaporation and transpiration similar to isotope ﬂux
partitioning during the majority of the experiment. Prediction of nonzero FT values in the initial days of the
experiment (0–2) likely arise due to numerical instabilities caused by near-zero terms in the denominator of
equations. On day 8, the similarity method indicates a suppression of FT, while isotope ﬂux partitioning
does not. As noted in Scanlon and Kustas [2012], rainfall may suppress transpiration and increase evaporation as the total FET ﬂux approaches potential evapotranspiration. These effects are particularly relevant
when leaf area index is larger and when substantial amounts of water may be stored in the canopy during
and immediately after rainfall [Scanlon and Kustas, 2012]. In our study, leaf area index remained very low
ðLAI 1Þ. Potential evapotranspiration was on average 8.7 times greater then that of FET, and three times
greater than peak FET observed on day 8. We neither expect transpiration to be suppressed in this setting,
nor do we observe lower FT from our leaf-level measurements. This drop in ﬂux-variance similarity FT may
be due to a wet canopy altering the relationship between transfer efﬁciencies, though more research into
the ﬂux-variance method is required.
During the ﬁnal 2 days of the experiment, the isotope partitioning and the ﬂux-variance approach diverge,
with isotope ﬂux partitioning and leaf-level results both below the estimates of the similarity partitioning
method. The very negative soil water potential and decrease in green leaf area during the later half of the
experiment imply stressed conditions under which the assumption that leaf temperature and air temperature are the same may break down. During stressed conditions, as leaves lose the ability to maintain their
temperature via evaporation, this WUE modeling approach may not produce reliable results [Scanlon and
Kustas, 2010]. However, even during the initial phase of the experiment, the modeled WUE values, based on
an average ci /cA ratio, are about 10 mg g21 larger than those obtained from the leaf-level observations, indicating that the ﬂux-variance FT is too large. Furthermore, when the similarity partitioning method is inverted
to estimate WUE, the results are similar to hand-held observations with an average absolute difference of 8.7
mg g21 for daily averages (when FT values below 0.005 g m22 s22 are removed). Finally, the short grass in this
study results in very small vertical separation between the transpiring canopy and the evaporating surface.
This small spatial separation may diminish differences in transport efﬁciencies between stomatal and
nonstomatal-derived scalars. The possibility of combining ﬂux-variance similarity assumptions and isotopic
methods to determine WUE, FT, FE, FP, and FR simultaneously is an exciting avenue for further research.
5. Conclusion
We have, for the ﬁrst time, tested the landscape-scale partitioning of water vapor ﬂuxes with isotopic methods against the ﬂux-variance similarity theory approach. At the daily time scale, we demonstrate that
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isotope ﬂux partitioning is able to partition the total ﬂux of water vapor into its constituent components of
transpiration and evaporation, with an estimated average uncertainty of 5% of a total transpired fraction of
29%. Our isotope ﬂux partitioning results compare well with scaled leaf-level results and taken together
form a comprehensive demonstration of the isotopic approach to ﬂux partitioning. However, large uncertainties and instrumental requirements suggest that improvement in techniques, such as gradient-based
dET estimates, is required to accurately partition ﬂuxes for long periods of time without continuous inperson ﬁeld observations. Improvements in dET measurements will lead to the most direct improvement in
partitioning estimates. Additional research is required into methods for automated measurement of soil
water and transpired water composition in order to consistently partition FET over seasonal and annual time
scales.
The portion of water vapor leaving the surface via the stomatal pathway was shown to be dependent on
both the green leaf area, vegetation status, and the vertical location of moisture within the soil column. Our
method was able to identify stomatal control of transpiration during the course of the diurnal cycle as well
as a critical soil water potential wilting point. Consideration of plant water stress will be required in modeling and predicting speciﬁc surface vapor ﬂuxes in future settings, particularly for ﬂux-varaince similarity
theory. Finally, the combined ﬂux-variance similarity with isotope techniques can be used to estimate WUE
and carbon dioxide ﬂux partitioning. The ability to assess WUE at landscape scales opens the possibility of
investigating leaf physiology and productivity in a wide variety of settings if partitioning estimates can be
made reliably.
Appendix A: Craig-Gordon Equation for Unsaturated Soils
The Craig and Gordon [1965] formulation for the isotopic composition of evaporated water is derived following Gat [1996], but with the relative humidity in the soil column, hS, taken to be less than saturation. This
approach is functionally the same as the use of the activity of water by Soderberg et al. [2012] to incorporate
soil water status into the Craig-Gordon model. Relative humidity in the soil is a function of the soil water
potential following the Kelvin equation, hS 5exp ðMwS =RTS Þ [Mathieu and Bariac, 1996; Soderberg et al.,
2012]. The difference between the normalized atmospheric humidity h0A 5hA eA =eS and hS for both the rare
and bulk ﬂux are multiplied by the conductance (1/r, where r is the resistance) to calculate the separate
ﬂuxes.
The ratio of the ﬂux of the rare isotopologue FEi to the bulk FE evaporation is taken as the ratio of two Fickian diffusion equations dominated by the vapor pressure gradient between the soil, eS [kPa], and the atmosphere, eA, as
1
RE 5
FEi ri ða eS RS 2eA RA Þ
:
5
1
FE
r ðeS 2eA Þ
(A1)
Equation (A1) is then written in terms of relative humidity and converted to d notation as
0
dE 5
ða hS dS 2h0A dA Þ1ða hS 2h0A Þ2ðhS 2hA Þri =r
:
0
ðhS 2hA Þri =r
(A2)
We deﬁne vapor-liquid equilibrium enrichment, e ð12a Þ, and kinetic enrichment,
0
ej ðhS 2hA Þðri =r21Þ. Making these substitutions gives us
dE 5
ða hS dS 2h0A dA Þ2ðhS e 1ej Þ
:
0
ðhS 2hA Þ1ej
(A3)
Equation (A3) simpliﬁes to the classical Craig and Gordon [1965] formulation when hS 5 1.
The vapor-liquid equilibrium fractionation factor a* and enrichment e* are calculated as a function of TS by
inverting the empirical liquid-vapor equation of Majoube [1971]. The resistance to water ﬂux for both the
rare isotopologue and the bulk ﬂux consists of a diffusive layer, rðDÞ and a turbulent layer, rðTÞ . These
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resistances are added in series ðr5rðDÞ 1rðTÞ ; ri 5riðDÞ 1riðTÞ Þ and the turbulent resistance is assumed to be
equal for both ﬂuxes (i.e., riðTÞ 5rðTÞ ). Therefore, ej is may be rearranged to
ej 5ðhS 2h0A Þ
r
ðDÞ
r
r
iðDÞ
rðDÞ
21 :
(A4)
The diffusive ﬂux in the laminar layer is proportional to the molecular diffusivity coefﬁcient rðDÞ / D2n ,
where n is an aerodynamic parameter. The ratio rðDÞ =r 1, but may reach lower values in strongly evaporating humid environments [Gat, 1996; Horita et al., 2008]. The kinetic enrichment is then
2n Di
21
D
Di
:
ﬃ nðhS 2h0A Þ 12
D
ej 5ðhS 2h0A Þ
(A5)
Note that dividing equation (A3) by hS, produces the classical Craig and Gordon [1965] formulation with
atmospheric relativity adjusted for soil moisture content as in Soderberg et al. [2012]. The aerodynamic diffusion parameter, n varies from 1/2 when the soil is saturated ðhS 5hsat Þ, to 1 when the soil is dried to residual
moisture levels ðhS 5hres Þ, as [Mathieu and Bariac, 1996]
n512
1 hS 2hres
:
2 hsat 2hres
(A6)
Appendix B: Lagrangian Stochastic Dispersion Model
The Lagrangian stochastic dispersion model employed here is based on Wilson and Sawford [1996] and
Thomson [1987] as outlined in Hsieh et al. [2000]. Within the model, the turbulence regime is expressed as a
function of the momentum ﬂux, FM, and sensible heat ﬂux, FH, via the friction velocity (u*) and Obukhov
length (L). Given the turbulence regime, ﬂuid parcels are advected and dispersed following a memoryless
Markov process by tracking the location (x, z) and velocity (u, w) of all individuals separately.
For each simulation, a large number (N) of ﬂuid particles are released from random locations on the model
surface. The number of particles released from the watered and unwatered portions is weighted relative to
their estimated FET ﬂux and area. Latent heat, sensible heat, and momentum ﬂux outside the watered area
are based on the diurnal cycle observed before the irrigation. The domain is 1000 m in length and 100 m in
height, with the watered area centered 900 m from the upwind edge. A 10 cm square collector region is
located 40 cm above the surface in the center of the watered region. The concentration footprint,
fC ðx; xm ; zm Þ, for a collector located at xm and height zm from a location x on the surface is given by
fC ðxjxm ; zm Þ5
1 dðnx Þ
;
Nm dx
(B1)
where nx is the number of particles that passed though the collector originating from x and Nm is the total
number of particles released that passed though the collector. Similarly, for the footprint of the measured
ﬂux, fF ðxjxm ; zm Þ, at the collector is
fF ðxjxm ; zm Þ5
1 dðnx" 2nx# Þ
:
Nm
dx
(B2)
Where nx" and nx# are the number of upward and downward moving particles that passed through the collector originating at x. The percent of the measured concentration (pC) or ﬂux (pF) at the collector from the
Ð
Ð
irrigated area is then calculated by integrating (B1) or (B2) as pC 5 fc dA and pF 5 fF dA, where A is the irrigated area.
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The location of a ﬂuid particle evolves from time step n to time step n 1 1 as a function of horizontal and
vertical winds as
xn11 5xn 1un Dt;
(B3)
zn11 5zn 1wn Dt:
(B4)
and
Here the horizontal mean wind proﬁle is calculated from Monin-Obukhov similarity theory [Kaimal and Finnigan, 1994] as
un 5
u
ðln ðzn =zo Þ2wm ðzn =LÞÞ:
k
(B5)
The roughness length is zo 50:1h, where grass height h is 10 cm, and k is the von Karman constant (50.4).
Under the assumption of a Markov process, the evolution of vertical windspeed is expressed as
wn11 5wn 1aðzn ; wn ; tl ÞDt1bðzn ; wn ; tl ÞDe;
(B6)
where De is a Gaussian random number with mean zero and variance Dt. In (B6), the drift coefﬁcient a and
random acceleration coefﬁcient b are calculated as
w 1 @r2w
w2
11 2 ;
a52 1
rw
tl 2 @z
(B7)
2 12
2rw
:
b5
tl
(B8)
and
The discrete time step Dt is set at 0.1 tl, where the Lagrangian time scale, tl, is estimated as
tl 5
kzu
:
/h r2w
(B9)
The Eulerian vertical velocity variance proﬁle is calculated as
(
r2w 5
1:25u ð123z=LÞ1=3
for z=L < 0
1:25u
for z=L 0
:
(B10)
The stability correction functions for the velocity (wm) and heat (/n) proﬁles are:
8
2 >
< 2 ln y11 1ln y 11 12 tan21 12y
for z=L < 0
2
2
11y
wm 5
;
>
:
2bz=L
for z=L 0
(
0:32ð0:0372z=l Þ21=3 for z=L < 0
/h 5
;
115z=L
for z=L 0
(B11)
(B12)
with y5ð12az=LÞ1=4 ; a516, and b55.
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Acknowledgments
Stephen Good, Keir Soderberg, and
Kelly Caylor acknowledge the National
Science Foundation for ﬁnancial
support of this work (NSF CAREER
Award EAR-0847368). Todd Scanlon
acknowledges the National Science
Foundation for ﬁnancial support of
this work (NSF CAREER Award
EAR-0645697). Kaiyu Guan was
supported by the Consortium of
Universities for the Advancement of
Hydrologic Science (CUAHSI)
Patherﬁnder Fellowship. Elizabeth King
acknowledges support from the
Princeton Environmental Institute. The
assistance of Anthony Lekutaas, John
Gitonga, and the staff at the Mpala
Research Center during the
experiment are also greatly
appreciated.
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10.1002/2013WR014333
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