Correlation-based flux partitioning of water vapor and carbon dioxide fluxes: Method simplification and estimation of canopy water use efficiency

Agricultural and Forest Meteorology 279 (2019) 107732

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Correlation-based flux partitioning of water vapor and carbon dioxide fluxes: Method simplification and estimation of canopy water use efficiency

T

⁎

Todd M. Scanlona, , Daniel F. Schmidta, Todd H. Skaggsb a b

Department of Environmental Sciences, University of Virginia, Charlottesville, VA, United States U.S. Salinity Laboratory, USDA-ARS, Riverside, CA, United States

A R T I C LE I N FO

A B S T R A C T

Keywords: Water use efficiency Flux partitioning Eddy covariance

The partitioning of water vapor and carbon dioxide (CO2) exchange between vegetation and the atmosphere remains a current research priority. A technique that has been proposed to simultaneously partition these fluxes, based on the correlation between their high-frequency concentration time series, has been the subject of recent empirical evaluations and theoretical advances. The method assumes that flux-variance similarity can be applied separately to stomatal exchange (transpiration for water vapor and net photosynthesis for CO2) and non-stomatal exchange (direct evaporation for water vapor and soil and stem respiration for CO2). Here, we present a mathematical simplification of this approach, from which the partitioned fluxes can be derived from routine eddy covariance measurements. The simplification arises from the fact that the transpiration and net photosynthesis fluxes are linearly related in solution space with respect to variable canopy water use efficiency, W. Conditions that are amenable to successful partitioning can now be determined a priori for a given averaging period. The simplified framework also has the benefit of providing a means for estimating W based on optimization theory. This allow for the estimation of W without any preconceptions of how the intercellular CO2 concentration, ci, varies as a function of ambient conditions. The simplified partitioning framework is applied to eddy covariance measurements collected over a mixed deciduous forests for three growing season. Aside from being more computationally efficient, the partitioned results exhibit less scatter compared with prior implementations.

1. Introduction

have been developed with the objective of partitioning either CO2 or water vapor fluxes. The approach discussed here is somewhat unique in terms of its theoretical premise and its ability to partition both CO2 and water vapor fluxes based on statistics of measured high-frequency time series. By leveraging recent advances and insights, we present a simplified approach that can facilitate the more widespread use of this method while providing additional information about trade-offs between carbon uptake and water loss at the canopy scale. The leading methods for partitioning Fc are based on temperature sensitivities of ecosystem respiration, drawn from mechanistic relationships that are parameterized using either nighttime (Reichstein et al., 2005) or daytime (Lasslop et al., 2010) EC measurements. Objective assessment of these partitioning approaches is difficult, but isotopic evidence has suggested that leaf respiration may be inhibited by light, particularly early in the growing season for deciduous forests, which could lead to an overestimation of ecosystem photosynthesis derived from standard partitioning approaches (Wehr et al., 2016; Keenan et al., 2019). In dry ecosystems, respiration has

Over recent decades, knowledge of the Earth's terrestrial carbon and water cycles has been significantly advanced by information generated by a worldwide network of eddy covariance (EC) systems (Baldocchi, 2014), which measure the net ecosystem exchange of carbon dioxide and the total evapotranspiration flux over a wide variety of ecosystems at hourly to sub-hourly timescales. As this information has been collected, the need has persisted for these measured fluxes to be partitioned into their respective components, as these components represent distinct biophysical process, driven by distinct environmental forcings. For the land-atmosphere exchange of carbon dioxide (CO2), the eddy covariance-measured flux, Fc, is comprised of the difference between the downward flux of CO2 due to photosynthesis and the upward flux due to respiration. For water vapor, the eddy covariancemeasured flux, Fq, is the sum of two upward fluxes: direct evaporation from soil and vegetation surfaces and transpiration via a stomatal pathway. A number of approaches, both field-based and analytical,

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Corresponding author. E-mail address: [email protected] (T.M. Scanlon).

https://doi.org/10.1016/j.agrformet.2019.107732 Received 28 March 2019; Received in revised form 20 August 2019; Accepted 23 August 2019 0168-1923/ © 2019 Elsevier B.V. All rights reserved.

Agricultural and Forest Meteorology 279 (2019) 107732

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2. Methods

been found to be influenced by the abiotic process of photodegradation (Rutledge et al., 2010; Cleverly et al., 2013) and by transient pulses of soil moisture (Xu et al., 2004), neither of which are well captured using typical partitioning schemes. Further, the use of either air temperature or soil temperature in such approaches inevitably leads to biased estimates of respiration (Wohlfahrt and Galvagno, 2017), since respiration consists of both above-ground and below-ground components. Although the implementation of Fc flux partitioning has become routine and has resulted in improved quantification of the terrestrial carbon budget, the development of new methodologies that do not hinge on the accurate extrapolation of respiration fluxes during daytime conditions remains an active area of research. Among these are techniques that rely on additional measurements to isolate canopy photosynthesis, such as carbonyl sulfide (Wohlfahrt et al., 2012) and solar-induced fluorescence (Yang et al., 2015). Given the considerable degree of uncertainty with which Fq is partitioned into direct evaporation and transpiration at the ecosystem level (Schlesinger and Jasechko, 2014) and at the global scale (Jasechko et al., 2013; Coenders‐Gerrits et al., 2014; Good et al., 2015), improving our ability to partition water vapor fluxes is also a current research priority. A number of options are available for this at the site level, typically involving the installation of additional instrumentation such as microlysimeters, sap flux sensors, or sub-canopy EC systems (Kool et al., 2014), each with their specific strengths and weaknesses. In recent years, methods have been developed to partition Fq using EC measurements alone, drawing on relationships between the measured CO2 and water vapor fluxes. Simplifying assumptions are required for such approaches, such as Fq being comprised solely of transpiration during some part of the growing season (Zhou et al., 2016) or monthly values of bare soil evaporation being constant from year to year (Scott and Biederman, 2017). A promising approach has been introduced by Li et al. (2019), which partitions Fq based on separation of soil and canopy conductances. Like Zhou et al. (2016) and Scott and Biederman (2017), however, a prerequisite for its successful implementation is the accurate partitioning of the CO2 fluxes which, as noted above, involves its own uncertainty. An approach developed to simultaneously partition CO2 and water vapor fluxes based on flux-variance similarity theory (Scanlon and Sahu, 2008; Scanlon and Kustas, 2010) has been the subject of subsequent theoretical advances (Palatella et al., 2014; Skaggs et al., 2018). Despite its computational complexity and its requirement of high frequency time series for use in the analysis, this partitioning approach has been applied in a number of settings (e.g., Good et al., 2014; Sulman et al., 2016; Wang et al., 2016; Rana et al., 2018; PerezPriego et al., 2018; Peddinti and Kambhammettu, 2019), albeit with somewhat mixed results. It has been noted that the flux partitioning is sensitive to estimates of canopy level water use efficiency (W), the only input to the analysis that is not a direct measurement (Sulman et al., 2016; Klosterhalfen et al., 2019a). This is particularly true for the partitioning of Fc, since the magnitudes of the components are essentially unconstrained due to their opposite signs. This sensitivity can be particularly problematic for applications in settings where leaf-level measurements are not available to inform accurate estimates of W. The overarching purpose of this study is to simplify the partitioning technique and to leverage the improved framework to provide additional insights about the canopy-atmosphere exchange of CO2 and water vapor. Specifically, our objectives are (1) to perform a more formal analysis of the sensitivity of the partitioning results to W, (2) to develop a simplified mathematical scheme for calculating the partitioned fluxes as a function of W, and (3) to introduce an optimization hypothesis with the intended benefit of making W a derived, rather than a prescribed, variable used in the partitioning approach. The outcomes of these objectives are evaluated using eddy covariance data collected above a forest for three growing seasons.

2.1. A brief overview of the similarity-based partitioning procedure The mathematical framework for the partitioning procedure has been described in detail elsewhere (Scanlon and Sahu, 2008; Scanlon and Kustas, 2010; Palatella et al., 2014; Skaggs et al., 2018) so here we present a brief overview to highlight its essential features. The carbon dioxide (Fc) and water vapor (Fq) fluxes measured by eddy covariance are each comprised of two basic components that we seek to quantify. These components are:

Fc = w′c′ = Fphotosynthesis + Frespiration

(1a)

Fq = w′q′ = Ftranspiration + Fevaporation

(1b)

where c and q represent CO2 and water vapor concentrations, respectively, w is the vertical wind velocity, the overbar (–) represents temporal averaging, and the primes (′) represent departures from the mean values using Reynolds decomposition (e.g., c′ = c − c¯ ). With the eddy covariance approach, the values of c, q, and w are measured at high frequencies such as 10–60 Hz, and the usual assumptions apply (e.g., stationarity, planar homogeneity, negligible subsidence, negligible molecular diffusion with respect to turbulent fluxes) for the scalar fluxes to be independent of measurement height (e.g., Li et al., 2018). The underlying idea of Scanlon and Sahu (2008) and Scanlon and Kustas (2010) was to further decompose the high frequency concentration time series into perturbations arising from the distinct processes:

c′ = cp′ + cr′

(2a)

q′ = qt′ + qe′

(2b)

where cp′ and cr′ represent fluctuations in the high frequency time series imposed by the processes of photosynthesis and respiration, respectively, and qt′ and qe′ represent fluctuations due to transpiration and direct evaporation, respectively. The measured eddy covariance fluxes can then be divided into their respective components, here expressed in covariance form:

w′c′ = w′cp′ + w′cr′

(3a)

w′q′ = w′qt′ + w′qe′ .

(3b)

An implication of Monin–Obukhov similarity theory (Monin and Obukhov, 1954) is that scalar time series measured at the same position should exhibit perfect correlation (Hill, 1989), a statistical outcome that is consistent with the concept of flux-variance similarity. Instead of applying this concept to the measured c′ and q′ time series, however, we adopt a more conservative approach and apply this concept separately to the stomatal (cp′ and qt′) and non-stomatal (cr′ and qe′) components of the time series, owing to the better co-location of their respective source/sink distributions. Therefore, the correlation coefficient (ρ) for fluctuations in the high-frequency time series arising from stomatal exchange is ρcp, qt = −1 (negative because photosynthesis is a sink for CO2 while transpiration is a source for water vapor) and the correlation coefficient for non-stomatal exchange is ρcr , qe = +1 (positive because respiration and direct evaporation are both sources to the atmosphere) (Scanlon and Albertson, 2001; Thomas et al., 2008). Stomatal exchange is modulated by an important parameter, the canopy-level water use efficiency (W), which relates fluctuations in cp′ to those of qt′:

cp′ = Wqt′

(4a)

σcp = −Wσqt

(4b)

2

σcp =

W 2σqt 2

(4c)

where σ is the standard deviation and σ is the variance. W is also equal to (and is, in fact, typically defined as) the ratio of the photosynthesis 2

2

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flux (w′cp′ ) to the transpiration flux (w′qt′ ), which has a negative value due to the opposite signs of the fluxes (photosynthesis is downward, or negative, while transpiration is upward, or positive). There is no corresponding, biophysically meaningful metric to relate the fluctuations or fluxes involved in non-stomatal exchange. Strict similarity is not assumed between the disparate forms of exchange, i.e. between the stomatal and the non-stomatal exchange. The transfer efficiencies of the stomatal-derived scalars (i.e., ρw, cp and ρw, qt) are assumed to be greater than the efficiencies of the non-stomatal scalars (i.e., ρw, cr and ρw, qe), due to the sources of the latter being primarily situated in the subcanopy. Therefore, following Katul et al. (1995) and Bink and Meesters (1997), we can make the following approximations:

ρcp, cr ≈

ρqt , qe ≈

by:

2 σcp

ρw, qe ρw, qt

(7a)

⎛1 − ρ2 ⎟⎞ σ 2 σ 2 (F − F W )2 c q q c c, q ⎝ ⎠ = 2 2 . 2 2 2 (σq Fc − 2ρc, q σq σc Fq Fc + σc Fq )(σc − 2ρc, q σq σc W + σq2 W 2) ⎜

ρcp2 , cr

(7b) A posteriori confirmation is needed to ensure that the calculated fluxes are physically realistic (i.e. w′cp′ ≤ 0 and w′cr′ , w′qt′ , w′qe′ ≥ 0 ) and that the algebraic solutions are valid, as verified when the results are plugged back into the original equations. This algebraic solution represents a considerable advance with respect to its efficiency and is the basis for open-source software intended for flux partitioning (Skaggs et al., 2018). The complexity of the equations, having been reduced, is nevertheless considerable enough to obscure the sensitivity of the partitioning results to estimates of W.

ρw, cr ρw, cp

⎛⎜1 − ρ2 ⎞⎟ (σ σ W )2 (σ 2 F 2 − 2ρ σ σ F F + σ 2 F 2) q c q c c q c, q q c c q c, q ⎠ = ⎝ 2 [σc2 Fq + σq2 Fc W − ρc, q σq σc (Fc + Fq W )]

(5a)

, (5b)

with these correlations differing only by their sign, i.e. ρcp, cr = −ρqt , qe . Katul and Hsieh (1999) showed that such relationships, which result from flux-variance dissimilarity, can arise even in the presence of fluxgradient similarity. This approximation likely represents the largest source of uncertainty in the partitioning framework given its lack of empirical verification. Variance budgets for the CO2 and water vapor concentrations (see Eqs. (9a) and (9b) in Scanlon and Kustas (2010)) provide the final information needed for the flux partitioning. As such, the partitioning approach is conceptually simple. It is also parsimonious with respect to input requirements, needing only five commonly measured variables and one estimated variable. The measured variables are the CO2 flux (Fc), the water vapor flux (Fq), the standard deviations in CO2 and water vapor concentrations (σc and σq, respectively), and the correlation coefficient between these concentrations (ρc,q). The estimated variable is the canopy-level water use efficiency (W). Despite being conceptually straightforward, the application of the method, as originally conceived, presented some practical challenges. The system of non-linear equations that emerged from the combination of the root equations was somewhat unwieldy and difficult to solve. Scanlon and Kustas (2010) used a step-wise numerical procedure to arrive at an approximate solution to the equations, while noting that in some cases multiple solutions were found. In other cases the system of equations failed to converge on a solution. A gradient approach was used to estimate W, assuming a constant ratio of internal CO2 concentration (Ci) to external CO2 concentration (Ca).

2.3. Simplified expressions for flux partitioning The efficient procedures of Palatella et al. (2014) and Skaggs et al. (2018) allow for a more formal empirical exploration of the sensitivities of the partitioning to estimates of W. Both methods yield identical results, with an exception being that the numerical procedure of Palatella et al. (2014) fails to converge to a solution under certain circumstances. As noted by Palatella et al. (2014), the physically valid range for W is W ≤ Fc/Fq, which represents an important constraint that is best understood by considering the signs of the component fluxes:

w′cp′ ⎞ ⎛ Fc w′cp′ + w′cr′ ⎞ ⎛ ⎜W = w′q ′ ⎟ ≤ ⎜ F = w′q ′ + w′q ′ ⎟. t ⎠ t e ⎠ ⎝ ⎝ q

(8)

When the partitioned solutions are evaluated over this physically valid range, relationships like those shown in Fig. 1 are consistently found for half-hour periods. A surprising outcome, given the non-linear nature of Eqs. (6a), (6b) and (7a), (7b) is the observed linear relationship between the photosynthesis flux (w′cp′ ) and the transpiration flux (w′qt′ ) across values of W, as shown in Fig. 1c. This linearity is confirmed on an empirical basis, and a mathematical proof is offered in the Supplementary Information (this does not preclude the existence of a more concise proof of linearity). The linear relationship between w′cp′ and w′qt′ greatly simplifies the functional form of the flux partitioning. This relationship can be expressed as:

2.2. Subsequent improvements to the mathematical approach Palatella et al. (2014) revised the mathematical structure of the method, noting that Descartes’ rule of signs could be used to eliminate one branch of solutions, thereby reducing the complexity. They demonstrated the use of a globally convergent version of Newton's method to find a solution to the resulting non-linear equations. Later, Skaggs et al. (2018) found an algebraic solution to the equations, taking the form of:

w′cp′ = m w′qt′ + b

w′cp′ =

Wb W−m

(10a)

w′qe′

w′qt′ =

b , W−m

(10b)

w′qt′

−2 2 2 2 = −ρcp2 , cr + ρcp2 , cr 1 − ρcp , cr (1 − W σq / σcp )

w′cr′ 2 2 −2 = −ρcp2 , cr ± ρcp2 , cr 1 − ρcp , cr (1 − σc / σcp ) w′cp′

(9)

Where m is the slope and b is the photosynthesis flux as W→-∞. Given that W = w′cp′ / w′qt′ , the partitioned fluxes can be expressed as:

(6a)

thereby exhibiting a hyperbolic dependence on W. The non-stomatal partitioned fluxes (i.e. w′cr′ and w′qe′ ) can easily be calculated using Eqs. (3a) and (3b). A derivation of the equations used to calculate the m and b parameters is also included in the Supplementary Information. The final equations are:

(6b)

2 , (σcp

the variance in the high frequency CO2 time where the unknowns series caused by photosynthesis, and ρcp2 , cr , the square of the correlation coefficient between fluctuations in the high frequency CO2 time series caused by photosynthesis and respiration, respectively) are calculated

b=

3

σq2 Fc2 − 2ρc, q σq σc Fc Fq + σc2 Fq2 σq2 Fc − ρc, q σq σc Fq

(11a)

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precondition that needs to be met for the partitioned fluxes to have physically realistic signs (i.e. w′cp′ ≤ 0 and w′cr′ , w′qt′ , w′qe′ ≥ 0 ):

ρc−, q1

σc F σ ≤ c < ρc, q c σq Fq σq

Fc σ < ρc, q c Fq σq

for ρc, q < 0, and

for ρc, q > 0.

(13a)

(13b)

note that this precondition is wholly determined by the properties of the time series, not by the value of W used in the partitioning. As long as the condition given by Eqs. (13a) and (13b) is met, a W from the valid range (W ≤ Fc/Fq) will result in a physically valid solution. 2.4. Estimation of W: an optimization hypothesis The relationships shown in Fig. 1 confirms what others have noted: the partitioned fluxes are sensitive to estimates of W. Based on Eqs. (10a) and (10b), it can be seen that the sensitivities are amplified when W→ Fc/Fq and when the magnitude of b is large. Accurate estimates of W are therefore essential to the efficacy of the partitioning approach. One way to estimate W is to consider gradients in ambient and intercellular concentrations of CO2 and water vapor (e.g., Scanlon and Kustas, 2010; Sulman et al., 2016; Perez-Priego et al., 2018; Rana et al., 2018):

W=

m=−

σq2 Fc − ρc, q σq σc Fq

Mobj = −Fphotosynthesis − λFtranspiration ,

(15)

where a negative sign has been added to Fphotosynthesis to preserve the sign convention used here, thereby making this term positive, and λ is the marginal carbon gain per unit water loss. Katul et al. (2009) showed that λ (defined as ∂Fphotosynthesis/∂Ftranspiration) can be related to the fluxbased water use efficiency, W, by:

(11b)

thereby providing the necessary information to solve for the partitioned fluxes and evaluate their sensitivities to W. A constraint that appears in the proof of linearity and in the derivation of the m and b parameters is:

Fc σ < ρc, q c . Fq σq

(14)

where ca and qa are the ambient CO2 and water vapor concentrations, respectively, cι and qι are the intercellular CO2 and water vapor concentrations, respectively, and a is the ratio of the molecular diffusivities for water vapor and CO2 (with a value of 1.6). While this has been used because it has been best approach available, it is not without its limitations. Typically, ca and qa are estimated by extrapolation to the canopy level, with canopy boundary layer effects often being difficult to take into account. The value of qι is derived from estimates of leaf temperature, and the value of cι can be approximated as a constant or parameterized based on the cι / ca ratio for the particular vegetation. The cι / ca ratio may be well constrained for crops or for forest stands dominated by single species, but it can be difficult to estimate for mixed vegetation. Previous applications have noted some “spiky” behavior in the flux partitioning, particularly for the CO2 fluxes. This is caused, in part, by an estimate of W that is (1) inaccurate and/or (2) incompatible with the associated flux data used in the partitioning. Here we present an alternative way to estimate W. As shown in Fig. 2, Eqs. (10a), (10b) and (11a), (11b) yield solutions for how the photosynthesis (w′cp′ ) and transpiration (w′qt′ ) fluxes vary as a function of W. The true W associated with the canopy-atmosphere exchange is unknown, but we seek to identify this true W by searching for optimal behavior in the partitioning results. We adopt the framework of Cowan and Farquhar (1977), who argued that plants adjust their stomatal aperture to maximize carbon gain while minimizing water loss. The objective function (Mobj) to be maximized takes the form of:

Fig. 1. Partitioned fluxes of (a) water vapor, Fq, and (b) carbon dioxide, Fc, as a function of water use efficiency, W. A dashed vertical line representing Fc/Fq is also shown. (c) The relationship between the photosynthesis flux, w′cp′ , and the transpiration flux, w′qt′ exhibits linearity across values of W, with red circles representing the partitioning solutions for individual values of W. Solutions were found using the numerical procedure described by Palatella et al. (2014). Data were collected on June 17, 2011 at 8:00 a.m. at MMSF, with values of Fc = −0.08581 (µmol mol dry air−1) × (m s−1), Fq = 0.015259 (mmol mol dry air−1) × 0(m s−1), σc = 1.9931 µmol mol dry air−1, σq = 0.15577 mmol mol dry air−1, and ρc,q = 0.42737. Native units used in the flux partitioning procedure are reported in this example. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

σc2 Fq − ρc, q σq σc Fc

cι − ca a (qι − qa )

λ= (12)

W 2aD¯ ca

(16)

where D¯ is a water vapor deficit, equivalent to qι − qa . Therefore, Eq. (15) can be solved as a function of W by substituting Eqs. (10a), (10b) and (16) into Eq. (15), giving:

Indeed this, together with the requirement that m ≥ 0 (since negative values of m result in negative w′cr′ fluxes) leads to the following 4

Agricultural and Forest Meteorology 279 (2019) 107732

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Fig. 2. Diagram showing how the input variables can be used to determine how the partitioned fluxes vary with respect to W, and how the optimal water use efficiency (Wopt) can be inferred from this information.

Mobj = −

Wb W 2aD¯ b − . W−m ca W − m

cι =1− ca

(17)

This objective function exhibits optimal behavior, as shown in Fig. 2. The value of W where Mobj is maximal can be determined by taking the derivative of Eq. (17) with respect to W, setting this equal to zero, and solving for W, giving:

Wopt =

¯ − aDm

¯ (ca + aDm ¯ ) aDm , ¯ aD

¯ (ca + aDm ¯ ) − aDm ¯ aDm ca

.

(19)

The formulation for Mobj given by Eq. (17) results in the constraint cι / ca ≥ 0.5, which would not be a critical factor for C3 vegetation under typical field conditions, but would definitely impose a limitation for application to C4 vegetation. An alternative expression for Eq. (16), which relates W to λ, would likely be needed to bypass this constraint (see Section 4.1). The optimization model of Cowan and Farquhar (1977) has previously been criticized because λ is difficult to calculate and is assumed to remain constant over sufficiently long timescales (i.e. on the order of a day), but the present approach makes calculations of λ trivial (see Eq. (15)) and makes no such assumption about the timescale over which λ stays constant (other than over the half-hour period under consideration). Different from its originally intended use, here the optimization model is applied to the solution space of the partitioning approach under the hypothesis that the “true” solution corresponds to the state, among all the plausible outcomes, where the carbon-water economics of the canopy are optimal.

(18)

where a physically impossible root (which leads to Wopt > 0) has been discarded. If Wopt is in the valid range (Wopt ≤ Fc/Fq), then the derived Wopt may be used in Eqs. (10a) and (10b) to solve for the partitioned fluxes (Fig. 2). If Wopt > Fc/Fq, then Wopt is considered physically invalid and therefore cannot be used for flux partitioning. The use of Eq. (18) for estimating Wopt involves approximations of D¯ and ca which, as noted earlier in this section, typically entails extrapolation from the measurement height to the canopy height. However, it has the advantage over Eq. (14) in that estimates of cι are not needed since the parameter m provides this information. By combining Eqs. (14) and (18), the ratio of cι / ca can be calculated as: 5

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2.5. A note on the components of CO2 flux partitioning Up to this point, the general terms “photosynthesis” and “respiration” have been used to refer to the partitioned components of Fc. More precise definitions are warranted for these carbon accounting terms. The partitioning procedure is based on a canopy-level water use efficiency, W, which is the ratio of net photosynthesis flux to transpiration flux – a ratio that is commonly considered (e.g., Farquhar and Richards, 1984; Katul et al., 2000; Cernusak et al., 2007) because it is compatible with leaf-level gas exchange, as expressed in Eq. (14). Net photosynthesis is defined as the rate of carboxylation minus photorespiration and leaf respiration (Wohlfahrt and Gu, 2015). Since Fc is equivalent to net ecosystem exchange, this leaves stem and soil respiration as contributors to the “respiration” flux. Thus, by “photosynthesis” we mean net photosynthesis, and by “respiration” we mean stem and soil respiration. This form of partitioning is slightly different than the typical partitioning of Fc into gross primary production (GPP) and ecosystem respiration (Reco). GPP is defined as the rate of carboxylation minus photorespiration, while Reco consists of stem and soil respiration plus leaf respiration (Wohlfahrt and Gu, 2015). This means that the photosynthesis fluxes determined by the present partitioning approach would have a smaller magnitude than GPP (with the difference being the leaf respiration). Respiration fluxes would also be smaller in magnitude than Reco (with the difference again being the leaf respiration). These differences should be kept in mind when comparing results from partitioning approaches.

Fig. 3. Time series of (a) volumetric soil moisture, and (b) mid-day vapor pressure deficit (VPD) for the 2011–2013 growing seasons. Mid-day VPD represents the mean VPD from 10 a.m. to 2 p.m. local time.

3. A demonstration of the partitioning method 3.1. Site description

circumvents the need for the Webb–Pearman–Leuning (WPL) correction to be applied to the high frequency time series, which would otherwise be required if c and q were expressed in terms of densities (Detto and Katul, 2007). Low-frequency information from non-local processes such as dry air entrainment or large-scale advection can influence the scalar statistics in a manner inconsistent with the local land-atmosphere exchange processes (Scanlon and Sahu, 2008). In previous studies, wavelet analysis was used to remove this low-frequency information (Scanlon and Kustas, 2010; Sulman et al., 2016; Skaggs et al., 2018). Here, we undertook a more simplistic approach by applying a moving mean filter (with a time scale of 400 s) to the Reynolds decomposition of the w, c, and q high-frequency time series. The units for the five variables used in the partitioning were as follows: Fc (µmol CO2 mol dry air−1 × m s−1), Fq (mmol H2O mol dry air−1 × m s−1), σc (µmol CO2 mol dry air−1), σq (mmol H2O mol dry air−1), and ρc,q (unitless). Estimates of W from Eq. (18) require two additional variables, D¯ (mmol H2O mol dry air−1) and ca (µmol CO2 mol dry air−1), which describe the ambient conditions in the forest canopy. These variables were estimated from measurements of q¯ , c¯ , and air temperature (T¯ ) made at a height of 46 m, extrapolated to the canopy height using logarithmic flux-profile relationships (Brutsaert, 1982), similar to the approach described in Scanlon and Kustas (2010). This approach assumes that the estimated canopy air temperature is equivalent to the leaf temperature. Calculated values of Fc and Fq are likely to differ from the fluxes calculated by standard protocols due to the moving average filtering applied to the former and any additional operations (e.g., spectral corrections) applied to the latter. We assume that the partitioning results for Fc and Fq apply in the same relative proportions to these processed fluxes. For example, the portion of the latent heat flux (LE) that derives from transpiration (LEt), can be calculated as LEt = LE × w′qt′ / Fq . For CO2 fluxes, this scaling is somewhat more complex owing to the differences in signs for w′cp′ and w′cr′ . If the measured net ecosystem exchange (NEE) were to be proportioned in the

Correlation-based flux partitioning is applied to high-frequency time series collected at the Morgan-Monroe State Forest (MMSF) Ameriflux site in south-central Indiana, U.S.A., for three growing seasons (2011–2013). These years were selected due to the availability of leaf-level gas exchange measurements that were collected concurrent with the EC measurements (Sulman et al., 2016). The forest at the measurement site is characterized as mixed deciduous, with sugar maple (Acer saccharum), tulip poplar (Liriodendron tulipifera), sassafras (Sassafras albidum), white oak (Quercus alba), black oak (Quercus velutina), and red oak (Quercus rubra) representing the main species (Sulman et al., 2016). Meteorological conditions were distinct for the three growing seasons (Fig. 3). The 2012 growing season experienced the most severe drought due to the combination of low soil moisture and elevated vapor pressure deficit (VPD) for a prolonged period during the middle of the growing season. The 2011 growing season also experienced drought conditions, albeit for a shorter period of time and later in the growing season, while the 2013 growing season remained relatively wet and humid. 3.2. Data collection and data processing Eddy covariance measurements were collected at a height of 46 m, at a position well above the roughly 27-m forest. Wind velocities were measured with a sonic anemometer (CSAT3; Campbell Scientific, Logan, UT, USA), and CO2 and water vapor mixing ratios were measured with a closed-path infrared gas analyzer (IRGA; LI-7000, Li-Cor, Lincoln, NE, USA) located near the base of the tower, where it received the sampled air via a tube. For each half-hour period, the 10-Hz data were first despiked and then tube lag times were estimated from the lagged covariances between the vertical wind speed (w) and the CO2 mixing ratio (c). Measured mixing ratios had units of µmol CO2 mol air−1 and mmol H2O mol air−1; these were converted to µmol CO2 mol dry air−1 and mmol H2O mol dry air−1 by accounting for partial pressure of water vapor. Working with dry air mixing ratios 6

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same manner, the amount corresponding to photosynthesis (NEEp) would be calculated as NEEp = NEE /(1 + x ) , where x = w′cr′ / w′cp′ . If partitioned fluxes of photosynthesis and respiration are roughly the same magnitude (i.e. − w′cr′ ≈ w′cp′ , and therefore x ≈ −1), this can result in magnitudes of NEEp that are unreasonably high. To avoid this, we convert Fc into the same units as NEE (µmol m−2 s−1) by multiplying by the density of dry air (ρair,dry), and applying the partitioning to the differential between these fluxes. NEEp and NEEr, the portion of NEE corresponding to respiration, can then be calculated as:

NEEp = w′cp′ × ρair , dry + (NEE − Fc × ρair , dry ) ×

NEEr = w′cr′ × ρair , dry + (NEE − Fc × ρair , dry ) ×

w′cp′

2

2

w′cp′ + w′cr′

w′cr′

2

(20a)

2

2 2 w′cp′ + w′cr′

(20b)

which better constrains the magnitudes of the partitioned fluxes. 3.3. Leaf-level measurements Leaf-level measurements of W used in the current application are the same as those reported in Sulman et al. (2016). Briefly, a hand-held gas analyzer (LI-6400; Li-Cor, Logan, UT) was used to acquire leaf-scale measurements of photosynthesis and transpiration for 12 individual trees whose species together represented 75% of total basal area and 58% of total canopy leaf area at the site. Trees were accessed using an elevated platform with a maximum height of 24 m. Mid-day measurements were taken on an approximately weekly basis, with chamber conditions adjusted to match ambient conditions. Canopy-scale average W was estimated by weighting the species-specific W by their relative leaf areas, as determined by leaf litter collection in 2013. Five sunlit and five shaded leaves from each tree were measured during each the sampling day, and W was calculated as the ratio of photosynthesis to transpiration. 3.4. Results When applied to the 2011–2013 growing seasons, the scalar statistics used for the partitioning technique met the conditions given by Eqs. (13a) and (13b) for 73.5 to 92.5% of the half-hour periods (Table 1). With this condition met, the signs of the partitioned fluxes will be correct as long as W is in the valid range W < Fc/Fq. The optimal value of water use efficiency, Wopt, was sought using Eq. (18) for these half-hour periods. For the majority of these, the resulting Wopt fell within the valid range Wopt < Fc/Fq, while for others Wopt exceeded Fc/ Fq. Ultimately, the partitioned fluxes based on the optimized water use efficiency were available for 42.7–68.7% of the original half-hour periods (Table 1). Sulman et al. (2016) found that, of the options presented by Katul et al. (2000), leaf-level measurements of W could be best modeled using ci/ca ratios from the model of Leuning (1995), which has the form:

ci 1 − Γ/ ca ⎛ VPD ⎞ 1+ =1− ca mL ⎝ VPD0 ⎠ ⎜

Fig. 4. Comparison between leaf-level measurements of -W and (a) modeled estimates of -W, and (b) flux-variance similarity based estimates of -Wopt. Circles represent the mean mid-day values (10 a.m. to 6 p.m. local time) and bars represent the standard errors associated with multiple daily measurements.

where Γ is the leaf CO2 compensation point, and mL and VPD0 are empirical parameters. Despite this being the best available model, with three parameters fitted to match the observed leaf-level W, model performance was modest (Fig. 4a, which is adaptation of Fig. 3d from Sulman et al. (2016)). Comparison of the leaf-level W measurements with simultaneous eddy covariance Fc/Fq measurements (Supplementary Information, Figure S1) suggests that there is some potential for error in the leaf-level measurements and/or in the scaling of W to the landscape scale. In many cases leaf-level W exceeded Fc/Fq measurements which, according to Eq. (8), is physically impossible (this was the case for 69% of the leaf-level measurements in 2011, 38% in 2012, and 55% in 2013). Discarding these measurements that were deemed invalid, a comparison between Wopt from Eq. (18) and W measured at the leaf level exhibited improved agreement (Fig. 4b), especially with regard to capturing the temporal variance in W. Since D¯ exerts a strong control on W, the inherent water use efficiency of a canopy (Wi = WD¯ = (cι − ca )/ a) is sometimes considered (e.g., Keenan et al., 2013). This, together with ci and canopy conductance ( gc = w′qt′ /(aD¯ ) ) are shown for the 2011–2013 growing seasons in Fig. 5. In general, the two drier growing seasons (2011 and

⎟

(21)

Table 1 Percentage of half-hour time periods in each growing season meeting given conditions. Growing season

2011 2012 2013

Eqs. (13a) and (13b) condition met

92.5% 86.9% 73.5%

Eqs. (13a) and (13b) condition met and Wopt ≤

Fc Fq

68.7% 53.8% 42.7%

7

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Fig. 5. Time series of mid-day (a) inherent water use efficiency (Wi), (b) intercellular CO2 concentration, cι , and (c) canopy conductance, gc, for the 2011–2013 growing seasons. In panels (a) and (b) the solid lines represent the median values over a 7-day moving window and the shaded area represents the 25th and 75th percentiles. In panel (c), the solid line represents the 85th percentile and the shaded area represents the 75th and 95th percentiles, with this higher range chosen to isolate the maximum values of gc, when light is not limiting. Leaf-level measurements of cι are shown in panel (b), with circles representing mean daily values and bars representing the standard error.

the reduced magnitude of the NEE and latent heat fluxes (LE) for this season. A reduction in LE is also observed during the 2013 growing season as a result of the cloudy and humid ambient conditions. As noted in Section 2.5, the magnitude of Fphotosynthesis should be smaller than that of GPP, and the magnitude of Frespiration should be smaller than that of Reco, a situation that is borne out by the partitioning results. Frespiration differs from Reco by leaf respiration, with the difference between the two likely being exacerbated by potential overestimation of Reco during daytime conditions (Wehr et al., 2016; Keenan et al., 2019). Indeed, the partitioned Frespiration is similar in magnitude to the sub-canopy CO2 fluxes measured by Sulman et al. (2016), which were generally on the order of 1–2 µmol m−2 s−1. With regard to water vapor, the portion of LE that derives from direct evaporation, E, was found to exhibit remarkably consistent mean diurnal variability from year to year (Fig. 6d–f). The portion of LE that derives from transpiration was found

2012) exhibit higher Wi and lower ci compared with the relatively wet growing season (2013), but some differences are worth noting. Following the drought that occurred late during the 2011 growing season, Wi and ci never recovered to values that are common for wet conditions. In contrast, the 2012 drought, which peaked earlier in the growing season, was followed by a return to values that are comparable with those of the wet year. The inferred CO2 exchange characteristics are clearly related to the soil moisture status shown in Fig. 3a, both in terms of the long-term seasonal pattern as well as short-term deviations caused by the arrival of rainfall. The imprint of the 2011 and 2012 drought periods are also observable as departures from the seasonal arc of gc (Fig. 5c). Diurnal curves for the partitioned CO2 and water vapor fluxes are shown in Fig. 6 based on data from the middle of the growing seasons (Julian days 180–260). The impact of the 2012 drought is reflected in 8

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Fig. 6. Duirnal curves of (a)–(c) CO2 fluxes and (d)-(f) water vapor fluxes for the 2011–2013 growing seasons. Lines represent the median values for each half-hour and shaded areas span the 25th to 75th percentiles. Partitioned CO2 fluxes are shown along with estimated gross primary productivity (GPP) and ecosystem respiration (Reco) based on non-linear extrapolation of nighttime measurements to the daytime (median values only). Water vapor fluxes are expressed in terms of the latent heat flux (LE), with the portion deriving from direct evaporation (E) shown. These composite curves are from the middle of the growing season (Julian days 180–260).

partitioning may now be accomplished with just a handful of basic equations. A comparatively larger amount of computational exertion goes into the pre-processing of the high frequency w, c, and q time series from which the scalar statistics derive. If the c and q time series are measured as concentrations, then a WPL correction (Detto and Katul, 2007) should be applied when calculating the covariances, standard deviations, and correlation coefficients. It has also been a common practice to deal with the issue of surface heterogeneity and/or non-local processes impacting the scalar statistics (Detto et al., 2008; Scanlon and Sahu, 2008) by applying a high-pass filter to the high-frequency time series, whether this is a wavelet-based filter or a moving-average filter, as used in the current study. A drawback of this relates to the subsequent translation of the partitioned fluxes into the fluxes calculated with standard eddy covariance protocols (e.g. NEE, LE). For water vapor fluxes this is less of an issue, but for CO2 fluxes (since the components have different signs) this issue has contributed, in previous studies, to the presence of unrealistically high co-occurring values of photosynthesis and respiration. As noted in Section 3.2, this is particularly the case when w′cp′ and w′cr′ are of similar magnitude. One way to scale

to be 83.5% in 2011, 81.1% in 2012, and 78.9% in 2013. 4. Discussion 4.1. Theoretical advances and issues The simplification of the mathematical structure allows for (1) a priori determination of flux partitioning viability for each half-hour period, (2) efficient sensitivity analysis of the partitioned fluxes with respect to W, and (3) an analytical framework from which optimalitybased estimates of W may be inferred. Data requirements for the partitioning are minimal, consisting of five parameters derived from routine EC measurements: CO2 and water vapor fluxes (Fc and Fq, respectively), standard deviations in the CO2 and water vapor concentrations (σc and σq, respectively), and the correlation coefficient between these two concentrations (ρc,q). Two additional parameters, the mean CO2 concentration (ca ) and vapor deficit (D¯ ) at the canopy level, are needed for estimates of W. Formerly involving a complex step-wise routine (Scanlon and Sahu, 2008; Scanlon and Kustas, 2010) or a non-linear numerical solver (Palatella et al., 2014), correlation-based flux 9

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the partitioned CO2 fluxes, if values of NEE and Fc are comparable, is offered in Eqs. (20a) and (20b). An alternative option would be to bypass high-pass filtering altogether and instead use strict stationarity criteria for the w, c, and q time series to determine their eligibility for partitioning. Estimation of W requires estimates of canopy-level ca and D¯ since these are not typically measured directly. Of the two canopy-level variables, D¯ contributes the most uncertainty to W, as can be demonstrated by data from the current field study. Based on logarithmic fluxprofile relationships, ca at the canopy height (z = 27 m) was typically ∼1–3 µmol−1 lower than ca at the measurement height (z = 46 m). Although leaf boundary layer effects may enhance this difference, this amount is small relative to cι − ca , the numerator in Eq. (14), which is on the order of 100 µmol mol−1. Meanwhile, D¯ at the measurement height was typically 0–30 mmol mol−1 (but did reach as high as 52 mmol mol−1 in 2012). Any uncertainty within this range, which is amplified by the coefficient a (=1.6) in the denominator of Eq. (14), can propagate into large uncertainty in W, especially when D¯ approaches zero. Sensitivities of W and ci/ca to estimates of ca and D¯ are illustrated in the Supplementary Information, Figure S2. Contributing to the uncertainty is the fact that D¯ (= qι − qa ) is governed by two variables, humidity and temperature. Intercellular water vapor concentration, qa , is a function of leaf temperature, which can exceed air temperature by as much as 5–10 °C during extreme water-stressed conditions (Helliker and Richter, 2008). Therefore, an ideal field setup would include measurements of ca and qa in the vicinity of the canopy, plus infrared radiometric measurements of leaf skin temperature to better approximate qι . Regardless of if this information is available, the parsimonious flux partitioning expressions (Eqs. (10a) and (10b)) make it easy to discern how uncertainty in W can propagate into uncertainty in the component fluxes. Estimates of W using optimization theory, as given by Eq. (18), have the advantage of not requiring approximations of intercellular CO2 concentration, cι . Instead, information is supplied by the parameter m, which is derived from eddy covariance statistics (Eq. (11b)). This is significant because ci/ca models ranging from simple (e.g., constant ci or constant ci/ca) to more complex (such as Eq. (21)) typically do not account for the effects of soil moisture or leaf nutrient status with respect to their influence on the ci/ca ratio. By avoiding the need for a prescribed ci/ca model, this allows for such effects to be factored into the calculation of W and ultimately in the calculation of the partitioned fluxes. This is particularly advantageous for mixed canopies, which may be comprised of species that differ in their gas exchange characteristics and their responses to drought conditions. It is important to point out that Eq. (16) from Katul et al. (2009) is simplified and is based on a linear leaf-level Fc-ci relationship. This is more appropriate for Rubisco-limited photosynthesis, which occurs under light-saturating conditions. A more complete solution was offered by Katul et al. (2010) that allows for both ribulose-1,5-biphosphate (RuBP) limitations and Rubisco limitations to be active. The resulting expressions are somewhat more complex but remain analytical. Vico et al. (2013) proposed a model that accommodates the colimitations on photosynthesis and maintains the analytical tractability provided in Katul et al. (2010). Thus, more complex relationships could potentially be implemented in the optimality-based approach used to solve for W. Finally, the inclusion of mesophyll conductance as a limitation can become important during times of drought or salt stress (e.g. Volpe et al., 2011), as it is known to cause deviations from optimality theory.

from optimization theory, however, exceeded Fc/Fq for a considerable portion of these cases, leading to smaller subset (42.7–68.7%) of the original half-hour periods with successful solutions. When stomatal fluxes (i.e. w′cp′ and w′qt′ ) dominate, W is close to Fc/Fq, so random errors in the scalar statistics can lead to the derived Wopt sometimes exceeding Fc/Fq. Errors in the estimations of canopy-level ca and D¯ , which factor into Eq. (18), can also contribute to Wopt exceeding this threshold. Given these potential sources of error and the general noisiness of eddy covariance data, it is perhaps unsurprising that this theoretical expectation is not met for a considerable portion of the halfhour periods. Leaf-level measurements of W were scaled up to the canopy level by weighting according to the relative leaf areas of the dominant species (Sulman et al., 2016). This is a reasonable way to scale these measurements, but it is not without out potential pitfalls (differences in stomatal conductance between species could result in unequal contributions to the flux on a per leaf area basis, less common species representing the remaining 42% of the leaf area were not sampled, etc.). There is some evidence that the scaled-up leaf-level measurements may constitute an imperfect rendering of the canopy-scale W, most notably the fact that this value was commonly greater than simultaneous measurements of Fc/Fq (Supplementary Information, Fig. S1), which is technically impossible. Also, a calibrated model was only able to explain only 32% of the temporal variance in these scaled leaf-level measurements of W (Fig. 4a). Application of Eq. (18) to the flux data outperformed the calibrated model (Fig. 4b) which is a positive outcome, but some question remains regarding how much of the unexplained variance is due to errors in the scaled leaf-level measurements versus potential errors in the theoretical framework. In this sense it is not a true “validation” of this new technique for deriving canopyscale W from eddy covariance measurements, but it does suggest some degree of skill in matching observations. Leaf-level measurements from a homogeneous canopy, which are more readily scalable, would be better suited for comparison with the derived values of W. Canopy gas exchange parameters inferred from the flux partitioning (Fig. 5) reflect the general differences in water availability between the growing seasons. Drought conditions are known to result in lower intercellular CO2 concentration, ci (Brodribb, 1996; Dursma and Medlyn, 2012) and increased inherent water use efficiency, Wi (Chen et al., 2010), which is consistent with the observed differences between the drier (2011 and 2012) and wetter (2013) growing seasons. Intermittent rainfall events, seen by increases in volumetric soil moisture (Fig. 3a), are associated with short-term ci and Wi dynamics that are also consistent with these theoretical expectations. Maximum mid-day canopy conductance, gc, is also influenced by soil moisture status (Fig. 5c), but also reflects the seasonal trend in leaf area (Sulman et al., 2016). Values of gc are typically calculated by inversion of the Penman–Monteith equation, but partition-based estimates like those shown in Fig. 5c have the advantage of isolating the transpiration fluxes, which is the portion of the bulk evapotranspiration dictated by stomatal control (Scanlon and Kustas, 2012). Previous applications of the partitioning technique have noted that outliers are commonly encountered in the NEE partitioning, with cooccurring values of photosynthesis and respiration fluxes having unreasonably high magnitudes (Sulman et al., 2016; Rana et al., 2018; Klosterhalfen et al., 2019b). This was not encountered in the current application, which is likely the result of (1) the use of Wopt that was derived, in part, from the eddy covariance data, and (2) the strategy of translating the Fc partitioning to the NEE partitioning (Eqs. (20a) and (20b)), which differed from these previous studies. The CO2 partitioning (Fig. 6a–c) yielded results that were in agreement with expected constraints, with the magnitude of net photosynthesis being less than that of GPP, and were supported by ancillary measurements, with soil plus stem respiration flux estimates being similar to those measured by sub-canopy eddy covariance. Transpiration was found to account for roughly 80% of the total evapotranspiration flux during each of the

4.2. Application to MMSF eddy covariance time series For the 2011–2013 growing seasons, 73.5 to 92.5% of the half-hour periods met the conditions given by Eqs. (13a) and (13b), meaning that a given water use efficiency in the range W < Fc/Fq would result in partitioned fluxes with physically valid signs. The value of W derived 10

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three growing seasons, which is on par with that reported for forests with a similar leaf area index (Wang et al., 2014).

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4.3. Implications and future considerations The simplification of the correlation-based partitioning procedure allows for ready application to eddy covariance data. With this new framework, sensitivity analyses with respect to estimates of W are trivial. An optimality hypothesis is presented, which allows for W to be inferred, rather than drawn from a prescribed model of how ci is thought to respond to ambient conditions. Apart from its utility in flux partitioning, the ability to extract W from eddy covariance data would itself represent an advance. Using the assumption of negligible direct evaporation and extrapolation-based estimates of GPP, Keenan et al. (2013) found that the inherent water use efficiency, Wi, of forests has increased rapidly over the last two decades, more rapidly than would be predicted by land surface models. This finding, however, does not appear to be supported by isotope-based estimates from tree rings (Frank et al., 2015). An improved framework for estimating W is vital for characterizing this important trade-off between carbon gain and water loss and for understanding what drives the variability in W under a changing climate. The methodology described here is fairly simple in its theoretical construct, in that it assumes that fluxes derived from stomatal exchange (photosynthesis and transpiration) and non-stomatal exchange (respiration and direct evaporation) both adhere to flux-variance similarity. A recent study in which a large eddy simulation model was applied to a homogeneous canopy with synthetic fluxes (Klosterhalfen et al., 2019a) sheds some light on some aspects of this approach. First, the measurement level should not be so high as to eliminate the decorrelation between c and q that is imposed by the vertical separation between soil and canopy sinks/sources (although lateral heterogeneity may counteract this to a certain extent). Second, the approximations with regard to ratios of the transfer efficiencies given by Eqs. (5a) and (5b) may not be equalities, leading to errors in the partitioned fluxes. More complex relationships, similar to those derived for water vapor and temperature (Lamaud and Irvine, 2006; Moene and Schuttemeyer, 2008) may be needed to improve this approximation. The simplifications to the partitioning framework described here pave the way for such improvements to be included while maintaining the parsimonious nature of the overall approach. Acknowledgments This research was supported by NSF grant EAR-1562019. We thank B. Sulman and the principle investigator of the MMSF Ameriflux site, K. Novick, for use of the EC data and D. T. Roman for use of the leaf-level measurements. Data collection at the MMSF site was supported by the Ameriflux Management Project (managed by Lawrence Berkeley National Laboratory with support from the US Department of Energy). We thank Georg Wohlfahrt and an anonymous reviewer for their insightful comments on an earlier version of this manuscript. Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.agrformet.2019.107732. References Baldocchi, D., 2014. Measuring fluxes of trace gases and energy between ecosystems and the atmosphere – the state and future of the eddy covariance method. Glob. Change Biol. 20, 3600–3609. https://doi.org/10.1111/gcb.12649. Bink, N.J., Meesters, A.G.C.A., 1997. Comment on ‘Estimation of surface heat and momentum fluxes using the flux-variance method above uniform and non-uniform terrain’ by Katul et al., (1995). Bound.-Layer Meteorol. 84, 497–502. https://doi.org/10. 1023/A:1000427431944.

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