Modeling Investigation of Diurnal Variations in Water Flux and Its Components with Stable Isotopic Tracers

The isotopic compositions of water fluxes provide valuable insights into the hydrological cycle and are widely used to quantify biosphere–atmosphere exchange processes. However, the combination of water isotope approaches with water flux components remains challenging. The Iso-SPAC (coupled heat, water with isotopic tracer in soil–plant–atmosphere-continuum) model is a useful framework for simulating the dynamics of water flux and its components, and for coupling with isotopic fractionation and mixing processes. Here, we traced the isotopic fractionation processes with separate soil evaporation (Ev) and transpiration (Tr), as well as their mixing in evapotranspiration (E) for simulating diurnal variations of isotope compositions in E flux (δE). Three sub modules, namely isotopic steady state (ISS), non-steady-state (NSS), and NSS Péclet, were tested to determine the true value for the isotope compositions of plant transpiration (δTr) and δE. In situ measurements of isotopic water vapor with the Keeling-plot approach for δE and robust eddy covariance data for E agreed with the model output (R2 = 0.52 and 0.98, RMSD = 2.72‰, and 39 W m−2), illustrating the robustness of the Iso-SPAC model. The results illustrate that NSS is a better approximation for estimating diurnal variations in δTr and δE, specifically during the alternating periods of day and night. Leaf stomata conductance regulated by solar radiation controlled the diurnal variations in transpiration fraction (Tr/E). The study emphasized that transpiration and evaporation, respectively, acted to increase and decrease the δ18O of water vapor that was affected by the diurnal trade-off between them.


Introduction
Isotopic compositions of water fluxes provide valuable insights into the hydrological cycle and are widely used to quantify biosphere-atmosphere exchange processes [1][2][3]. Evapotranspiration flux in terrestrial ecosystems is a combination of two or three different pathways (e.g., plant transpiration, soil evaporation, and canopy interception) of water vaporization [4,5]. Using the isotopic composition of soil evaporation (Ev), transpiration (Tr), and evapotranspiration (E) provided an independent approach to partitioning E in various ecosystems [6][7][8][9]. Combined measurements of the isotopic compositions of water vapor are often used to diagnose the local impacts on E, as well as its components (Ev and Tr) on atmospheric moisture [10,11]. However, studies on water flux and its component coupling with isotope fractionation and mixing processes remain few and challenging [12], in particular on a diurnal timescale [13].
Characterizing the land-atmosphere flux components of water and energy in response to available energy (e.g., radiation) and water input (e.g., precipitation or available soil water) is the primary task  Table 1 Summary of routine instrument and information of the dataset.

In Situ Isotopic and Supporting Dataset Measurements
On each day, in the morning (from 09:00 to 10:00 LST), at noon (from 12:00 to 14:00 LST), and in the afternoon (from 15:00 to 18:00 LST), the samples of different water pools (water vapor at three levels, plant stem, plant leaf, and soil) were collected for isotope analysis (Figure 2a). Traditional coldtrap methods were used to collect water vapor for isotopic analysis at the bottom (0.10 m), middle (1.6 m), and upper (2 m) levels ( Figure 2b).
Leaf, stem, and soil water samples were collected on a bihourly time interval. For extracting soil water, soil water samples were collected on each sampling day. Leaf and stem water samples were collected from the dominant species, named Miscanthus sinensis, with three replicates. For leaf sampling, the whole leaf was cut and divided into smaller sections and mixed into a test tube. The fresh weight of collected leaves and stems was about 4-5g. The leaf and stem water samples and surface soil water samples were extracted in the laboratory by a cryogenic distillation method ( Figure  2c). At the same time, supporting datasets were also collected. Air temperature and relative humidity were measured at the same level by a ventilated thermometer and a hygrometer for computing the mixing ratio. The leaf water content (W, kg m −2 ) was measured by weighing the masses of fresh leaves and dry leaves. Leaf area index (LAI) was measured by an automatic area meter by sampling at three

In Situ Isotopic and Supporting Dataset Measurements
On each day, in the morning (from 09:00 to 10:00 LST), at noon (from 12:00 to 14:00 LST), and in the afternoon (from 15:00 to 18:00 LST), the samples of different water pools (water vapor at three levels, plant stem, plant leaf, and soil) were collected for isotope analysis (Figure 2a). Traditional cold-trap methods were used to collect water vapor for isotopic analysis at the bottom (0.10 m), middle (1.6 m), and upper (2 m) levels ( Figure 2b).
Leaf, stem, and soil water samples were collected on a bihourly time interval. For extracting soil water, soil water samples were collected on each sampling day. Leaf and stem water samples were collected from the dominant species, named Miscanthus sinensis, with three replicates. For leaf sampling, the whole leaf was cut and divided into smaller sections and mixed into a test tube. The fresh weight of collected leaves and stems was about 4-5 g. The leaf and stem water samples and surface soil water samples were extracted in the laboratory by a cryogenic distillation method (Figure 2c). At the same time, supporting datasets were also collected. Air temperature and relative humidity were measured at the same level by a ventilated thermometer and a hygrometer for computing the mixing ratio. The leaf water content (W, kg m −2 ) was measured by weighing the masses of fresh leaves and dry leaves. Leaf area index (LAI) was measured by an automatic area meter by sampling at three spatially different areas following the day after each experiment day. Soil water content in the upper 10 cm of the soil was measured by amplitude domain reflectometry (ADR). The leaf temperature was also measured by a hand-held infrared thermometer during the sampling period. The details of the equipment information are summarized in Table 2. The isotopic ratios of the water samples were analyzed by using a liquid water isotope analyzer (L1102-i, Picarro, Santa Clara, CA, USA), which is a type of wavelength-scanned cavity ring-down spectroscopy, performed at the Center for Research in Isotopes and Environmental Dynamics (CRiED), the University of Tsukuba. The isotopic ratios of 18 O and D were expressed as δ 18 O and δD, respectively, normalized to the Vienna Standard Mean Ocean Water. The analytical errors were 0.1% for δ 18 O and 1% for δD [37].
Atmosphere 2019, 10, x FOR PEER REVIEW 4 of 19 spatially different areas following the day after each experiment day. Soil water content in the upper 10 cm of the soil was measured by amplitude domain reflectometry (ADR). The leaf temperature was also measured by a hand-held infrared thermometer during the sampling period. The details of the equipment information are summarized in Table 2. The isotopic ratios of the water samples were analyzed by using a liquid water isotope analyzer (L1102-i, Picarro, Santa Clara, CA, USA), which is a type of wavelength-scanned cavity ring-down spectroscopy, performed at the Center for Research in Isotopes and Environmental Dynamics (CRiED), the University of Tsukuba. The isotopic ratios of 18 O and D were expressed as δ 18 O and δD, respectively, normalized to the Vienna Standard Mean Ocean Water. The analytical errors were 0.1‰ for δ 18 O and 1‰ for δD [37].

Iso-SPAC Model
An Iso-SPAC model was used to estimate the water fluxes (E, Ev, and Tr) and their isotopic compositions (δE, δEv, and δTr) [23]. In this model, a SPAC model was used to simulate and partition water and energy fluxes. For the isotope budget, the Craig-Gordon model was used to estimate the δEv. Three sub-models with different complexities, including isotopic steady state (ISS), non-steadystate (NSS), and NSS Péclet models, were tested to determine the actual value for the isotopic composition of plant transpiration (δTr) ( Table 3), and the isotopic mass balance equation was used to

Iso-SPAC Model
An Iso-SPAC model was used to estimate the water fluxes (E, Ev, and Tr) and their isotopic compositions (δ E , δ Ev , and δ Tr ) [23]. In this model, a SPAC model was used to simulate and partition water and energy fluxes. For the isotope budget, the Craig-Gordon model was used to estimate the δ Ev . Three sub-models with different complexities, including isotopic steady state (ISS), non-steady-state (NSS), and NSS Péclet models, were tested to determine the actual value for the isotopic composition of plant transpiration (δ Tr ) ( Table 3), and the isotopic mass balance equation was used to calculate the δ E and was validated by measured values using the Keeling plot approach. The details of this model equations and parameters are described in Appendix A.

Sensitivity Analysis of Iso-SPAC Model
In order to better understand how the variation in the output of Iso-SPAC model can be attributed to variations of its input factors, the sensitivity analysis was conducted. According to the method proposed by Wang and Yamanaka [16], the sensitivity coefficient (S i ) is defined as: where p i is the i-th interested parameter, and has an effect on the outputs (O), such as E, transpiration fraction (Tr/E), or δ Tr and δ E . Negative S i indicates that a decrease in O results from an increase in p i , and vice versa. The coefficient value of 0.1 means that a 1% increase in p i would induce a 0.1% increase in O. S i provides a measure of the magnitude of O change associated with a unit change of a particular parameter or variable under the condition of a combination of possible ranges and other parameters. The sensitivities of interesting outputs (E, Tr/E, δ Ev , δ Tr , and δ E ) of our study to certain parameters and driving variables are summarized in Table 4. Among those parameters, the minimum leaf stomatal resistance (r st_min ) has the greatest influence on changing E; a 30% error in this parameter may result in an error of 8.1% in E on average. Additionally, relative humidity (h a ) was the most influential variable in changing E and can produce up to 6.5% error in E if there is 5% error in h a . Such an error range would be a minor problem in practice. For Tr/E, air tempreture (T a )and LAI were the most influential, although 5% errors in these parameters lead to only 1% and 1.2% errors in Tr/E, respectively. Among many parameters, r st-min is the most influential factor in changing δ E ; a 30% error in this parameter can introduce a 5.4% error in δ E . Additionally, thermal conductivity of surface soil (r ss ) is the most influential in changing δ Ev ; a 30% error in this parameter can introduce a 3.6% error in δ Ev . Further, r st-min are influential in changing δ Tr under NSS; a 30% error in this parameter can introduce a 15.9% error in δ Tr . Although this possible error range is not always negligible, it is not a serious problem. For measured variables, the isotope composition in water vapor (δ V ) is the most influential in changing both δ E and δ Ev , which can produce errors of up to 2.95% and 5.35% in δ E and δ Ev , respectively, if there is 5% error in δ V . The isotope composition in stem water (δ x ) is the most influential in changing δ Tr under both ISS and NSS simulation, and can produce up to 5% error in δ Tr if there is a 4.5% error in δ x under NSS. Such an error range would be a minor problem in practice.

Energy and Water Fluxes
In order to verify the reliability of the model's simulation of energy fluxes, we compared the observed and simulated values of diurnal variations of all energy fluxes (net radiation, sensible heat, latent heat, and soil heat fluxes) during each experimental day ( Figure 3). As summarized in Table 5, the diurnal variation of energy flux was well captured by the Iso-SPAC model. The indices of agreement (I) for the R n , lE, H, and G simulations were equal to 0.99, 0.98, 0.94, and 0.94, respectively, and the associated RMSD values were 29, 44, 42, and 12.6 W m −2 , respectively. The observed and simulated diurnal variation patterns of R n and lE fluxes were very similar, with respective R 2 values of 0.99 and 0.98, during the three experiment days. The peak values for Tr (Ev) that appeared at midday were 0.64 (0.05), 0.41 (0.14), and 0.40 (0.05) mm for DOY 203, 243, and 296, respectively ( Figure 4). Moreover, Sensitivity analysis also indicates how the uncertainty in the output of the Iso-SPAC model can be divided and allocated to different sources of uncertainty in its inputs. The value of Si for Tr/E is generally small, suggesting that Tr/E is insensitive to errors in assigned values of all of the parameters.      Our measurement results show a consistent pattern of the three observed days. We find that the isotopic composition of surface soil and leaf water deviated from LMWL, which implied that the isotopic kinetic fractionation process at the interface was caused by soil evaporation and plant transpiration. The stem water mainly lies within the LMWL, indicating that the plant water use origin was from precipitation. The isotopic composition of water vapor lies within or slightly deviated from the LMWL, suggesting a departure caused by local recycling effects. Based on the dataset of δV at three levels, the Keeling plot approach, which was used to estimate the δE at each experimental period, is summarized in Table 6. In the daytime (from 08:00 to 18:00 LST), the modeled δE seems to be controlled by LAI. Specifically, the average δE signatures based on the Keeling plot are −9.68‰ when LAI was 2.1 on DOY 203, −16.02‰ when LAI was 0.9 on DOY 243, and −12.17‰ when LAI was 1.5 at DOY 296. This may be because the transpiration fraction is higher when LAI is greater, inducing a high δE [40].  Our measurement results show a consistent pattern of the three observed days. We find that the isotopic composition of surface soil and leaf water deviated from LMWL, which implied that the isotopic kinetic fractionation process at the interface was caused by soil evaporation and plant transpiration. The stem water mainly lies within the LMWL, indicating that the plant water use origin was from precipitation. The isotopic composition of water vapor lies within or slightly deviated from the LMWL, suggesting a departure caused by local recycling effects. Based on the dataset of δ V at three levels, the Keeling plot approach, which was used to estimate the δ E at each experimental period, is summarized in Table 6. In the daytime (from 08:00 to 18:00 LST), the modeled δ E seems to be controlled by LAI. Specifically, the average δ E signatures based on the Keeling plot are −9.68% when LAI was 2.1 on DOY 203, −16.02% when LAI was 0.9 on DOY 243, and −12.17% when LAI was 1.5 at DOY 296. This may be because the transpiration fraction is higher when LAI is greater, inducing a high δ E [40].   Note: a day of year; b isotope composition in evapotranspiration flux; c isotope ratios of water vapor; d inverse of specific humidity of water vapor.

Isotope Composition in Water Flux
As shown in Figure 6, the temporal variations were reproduced reasonably by all three submodels for both days with strongly diurnal variations (i.e., DOY 203, DOY 296), as well as days with less variation (DOY 243). The statistics summary of the performances of the three sub-models is shown in Table 7, produced by comparing simulated δE with that of the derived Keeling plot. It is clear that NSS and NSS Péclet can capture the observed diurnal variations better than ISS, while NSS Péclet has few effects on improving the simulation of δE compared to NSS.   As shown in Figure 6, the temporal variations were reproduced reasonably by all three sub-models for both days with strongly diurnal variations (i.e., DOY 203, DOY 296), as well as days with less variation (DOY 243). The statistics summary of the performances of the three sub-models is shown in Table 7, produced by comparing simulated δ E with that of the derived Keeling plot. It is clear that NSS and NSS Péclet can capture the observed diurnal variations better than ISS, while NSS Péclet has few effects on improving the simulation of δ E compared to NSS.   Figure 7 shows the diurnal variations in transpiration fraction (Tr/E) calculated by Tr and E from the Iso-SPAC model, Tr/E estimated by the stable isotope method using δE from the Keeling plot approach, and δEv and δTr from the Iso-SPAC model. Tr/E from Iso-SPAC showed similar diurnal patterns among all three days. Moreover, the value of Si for Tr/E is generally small, suggesting that Tr/E is insensitive to errors in assigned values of all of the parameters. Additionally, day-to-day results show that higher contributions from transpiration from 09:00 to 17:00 LST are found when LAI is higher. Specifically, the mean (standard deviation) values of Tr/E are 0.91 (0.02), 0.70 (0.05), and 0.86 (0.03), with respective LAI of 2.1, 0.9, and 1.5. A lower fraction of less than 0.2 (close to 0) is found during nighttime for all three days. In the three sub-models, the diurnal variation of Tr/E was reasonably captured by the NSS and NSS Péclet. Differences between the isotope method and the Iso-SPAC model vary from 1% to 13%m with an average value of 7.6%. A previous report [41] on the differences between the isotope method and lysimeter plus sap flow measurements range from 4% to 26%, with an average of 15.6%. Considering the uncertainties in isotope measurements, the reasonable agreement between these two methods demonstrates the reliability of the present partitioning results.   Figure 7 shows the diurnal variations in transpiration fraction (Tr/E) calculated by Tr and E from the Iso-SPAC model, Tr/E estimated by the stable isotope method using δ E from the Keeling plot approach, and δ Ev and δ Tr from the Iso-SPAC model. Tr/E from Iso-SPAC showed similar diurnal patterns among all three days. Moreover, the value of S i for Tr/E is generally small, suggesting that Tr/E is insensitive to errors in assigned values of all of the parameters. Additionally, day-to-day results show that higher contributions from transpiration from 09:00 to 17:00 LST are found when LAI is higher. Specifically, the mean (standard deviation) values of Tr/E are 0.91 (0.02), 0.70 (0.05), and 0.86 (0.03), with respective LAI of 2.1, 0.9, and 1.5. A lower fraction of less than 0.2 (close to 0) is found during nighttime for all three days. In the three sub-models, the diurnal variation of Tr/E was reasonably captured by the NSS and NSS Péclet. Differences between the isotope method and the Iso-SPAC model vary from 1% to 13%m with an average value of 7.6%. A previous report [41] on the differences between the isotope method and lysimeter plus sap flow measurements range from 4% to 26%, with an average of 15.6%. Considering the uncertainties in isotope measurements, the reasonable agreement between these two methods demonstrates the reliability of the present partitioning results.  Figure 8 shows the relationship between hourly Tr/E and canopy conductance (=1/rc) over the three days. The Tr/E estimated by isotope approach was limited by rough time resolution and lacking the dataset during the night-time, therefore, the diurnal time series of Tr/E by SPAC model were analyzed. Since the LAI and θ were assumed to be constant within a given day in the present model, the diurnal variation in rC is controlled solely by the downward shortwave radiation (Sd). On the other hand, Figure 6b clarifies the day-to-day variation in the relationship between Tr/E and that the conductance is smaller if we consider the seasonal change in LAI. Nevertheless, the seasonal variation in Tr/E can be well represented as a function of LAI [16,42,43]. This may indicate that LAI regulates seasonal variation in Tr/E, not only via canopy stomata resistance but also via the distribution of radiation energy.

Diurnal Water Flux Partitioning
Partition flux can be realized by many approaches, such as the isotopic method and the SPAC model. Previous studies have shown that ISS is better than NSS for describing δTr and isotope composition of leaf water (δL,b) during the entire growing season (Table 8). In our study, more attention was focused on diurnal variations in δE and its components (δTr and δEv). It is worth mentioning that compared with the fixed constant input of δX in the Iso-SPAC model, the use of δX with daily variations (2h interval) can obtain better simulation results of δE. Although similar  Figure 8 shows the relationship between hourly Tr/E and canopy conductance (=1/r c ) over the three days. The Tr/E estimated by isotope approach was limited by rough time resolution and lacking the dataset during the night-time, therefore, the diurnal time series of Tr/E by SPAC model were analyzed. Since the LAI and θ were assumed to be constant within a given day in the present model, the diurnal variation in r C is controlled solely by the downward shortwave radiation (S d ). On the other hand, Figure 6b clarifies the day-to-day variation in the relationship between Tr/E and that the conductance is smaller if we consider the seasonal change in LAI. Nevertheless, the seasonal variation in Tr/E can be well represented as a function of LAI [16,42,43]. This may indicate that LAI regulates seasonal variation in Tr/E, not only via canopy stomata resistance but also via the distribution of radiation energy.  Figure 8 shows the relationship between hourly Tr/E and canopy conductance (=1/rc) over the three days. The Tr/E estimated by isotope approach was limited by rough time resolution and lacking the dataset during the night-time, therefore, the diurnal time series of Tr/E by SPAC model were analyzed. Since the LAI and θ were assumed to be constant within a given day in the present model, the diurnal variation in rC is controlled solely by the downward shortwave radiation (Sd). On the other hand, Figure 6b clarifies the day-to-day variation in the relationship between Tr/E and that the conductance is smaller if we consider the seasonal change in LAI. Nevertheless, the seasonal variation in Tr/E can be well represented as a function of LAI [16,42,43]. This may indicate that LAI regulates seasonal variation in Tr/E, not only via canopy stomata resistance but also via the distribution of radiation energy.

Diurnal Water Flux Partitioning
Partition flux can be realized by many approaches, such as the isotopic method and the SPAC model. Previous studies have shown that ISS is better than NSS for describing δTr and isotope composition of leaf water (δL,b) during the entire growing season (Table 8). In our study, more attention was focused on diurnal variations in δE and its components (δTr and δEv). It is worth mentioning that compared with the fixed constant input of δX in the Iso-SPAC model, the use of δX with daily variations (2h interval) can obtain better simulation results of δE. Although similar

Diurnal Water Flux Partitioning
Partition flux can be realized by many approaches, such as the isotopic method and the SPAC model. Previous studies have shown that ISS is better than NSS for describing δ Tr and isotope composition of leaf water (δ L,b ) during the entire growing season (Table 8). In our study, more attention was focused on diurnal variations in δ E and its components (δ Tr and δ Ev ). It is worth mentioning that compared with the fixed constant input of δ X in the Iso-SPAC model, the use of δ X with daily variations (2h interval) can obtain better simulation results of δ E . Although similar partitioning results were obtained by ISS, NSS, and NSS Péclet, the performance of ISS was lower than those of NSS and NSS Péclet. Additionally, δ Tr values simulated by ISS, NSS, and NSS Péclet at midday are close to the ISS assumption (i.e., δ Tr = δ X ), suggesting that ISS is reasonable at midday [44]. On the other hand, larger discrepancies between the simulated δ Tr and ISS and NSS or NSS Péclet were detected for both the periods in the morning and afternoon, when environmental conditions such as solar radiation and stomatal conductance change rapidly. The difference in Tr/E among the four sub-models indicates that there are still many uncertainties regarding the partitioning of E when using the isotope method in the diurnal timescale. These results indicate that for morning or afternoon, NSS is essential for simulating diurnal variations in δ Tr and δ E , because of the relatively rapid changes in environmental conditions. The comparison also shows little improvement of NSS Péclet (which further considers advection-diffusion) in δ Tr and δ E simulation compared to NSS. Nevertheless, non-steady-state behavior should be considered for δ Tr simulation.  Figure 6 shows the diurnal variations of in situ measured isotopic compositions of atmospheric water vapor (δ V ) and estimated isotopic compositions of evapotranspiration by the Keeling plot approach (δ E -Keeling plot, as well as the simulated isotopic compositions of soil evaporation (δ Ev ), isotopic compositions of transpiration under the state-state assumption (δ Tr -ISS) and non-steady-state assumption (δ Tr -NSS), and the simulated isotopic composition of evapotranspiration under ISS (δ E -ISS) and NSS (δ E -NSS) scenarios. The δ V has smaller diurnal variations because of rapid mixing with a larger background vapor pool. Soil evaporation has a depleted isotope signature because of the larger soil water pool. Transpiration shows an enriched isotope signature during the daytime because of fast transpiration flux with a rapid turnover time (less than 1 h) for the leaf water pool, whereas there is a larger deviation at night with a more considerable turnover time (more than 100 h). The pattern of δ Tr in the night varied day-by-day. At night-time, transpiration flux was much smaller (close to zero), and the leaf water content (W) was almost constant with lower sensitivity for the changes in δ Tr . The changes of δ Tr in the night was dynamic and mainly constrained by the night air temperature, relative humidity, leaf water enrichment during the daytime, and δ V of the background. The lowest temperature in the night-time was for DOY 296 with the smallest night-time transpiration, which resulted in drastic changes of δ Tr compared to others (Figures 4 and 6). The isotopic composition in the evapotranspiration flux was a mixture between the isofluxes of soil evaporation and plant transpiration. During the day, the isotope composition in the evapotranspiration flux was mainly controlled by the enriched isoflux of plant transpiration. During the night period, the variations in δ E were more complex, and it seemed that the δ E was mainly dominated by isoflux of soil evaporation, but there was a mixture both of isoflux from soil evaporation and transpiration after sunset. During the day, transpiration and evaporation, respectively, acted to increase and decrease the δ 18 O of water vapor, which was affected by the diurnal trade-off between them. At night, the situation was complex, where soil evaporation always acted to deplete the δ 18 O of water vapor and transpiration varied day-to-day. Clearly, the results indicate that for morning or afternoon, NSS should be considered because of relatively rapid changes in environmental conditions during the period of the day alternating with the night. High-frequency measurements in micrometeorology, fluxes and isotopic compositions in water vapor, even in transpiration, using bulk leaf water to simulate and constrain the diurnal variations in water fluxes, specifically for night-time, is promising.

Conclusions
The Iso-SPAC model was applied to simulate the diurnal variations in energy fluxes and isotopic compositions in E. Energy flux estimated by the Iso-SPAC model agreed very well with observed values. Isotopic compositions in E estimated by the Iso-SPAC model with NSS agreed well with those observed by the Keeling plot approach, while ISS is also applicable for times around midday. The Iso-SPAC model is a useful tool to partition E on a diurnal scale. The Tr/E is higher in the daytime and close to 0 at night. The isotopic method based on the Keeling plot approach is also a useful tool. To accurately capture the diurnal variations, the non-steady-state model is required for estimating δ Tr . The principal factor controlling the diurnal variation of Tr/E is stomatal conductance, which is mainly driven by the diurnal variation of solar radiation.
where R nV is the net radiation of the vegetation canopy (W m advection −2 ), H V is the sensible heat flux from the vegetation canopy (W m −2 ), Tr is the transpiration flux (kg m −2 s −1 ), f v is the permittivity of the vegetation canopy, α V is the albedo of the vegetation canopy, S d is the downward shortwave radiation (W m −2 ), L d is the downward longwave radiation (W m −2 ), σ is the Stefan-Boltzmann constant (=5.67 × 10 −8 W m −2 K −4 ), T G is the ground surface temperature ( • C), T L is the leaf temperature ( • C), R nG is the net radiation at the ground surface (W m −2 ), G is the ground heat flux (W m −2 ), H G is the sensible heat flux from the ground surface (W m −2 ), Ev is the evaporation flux (kg m −2 s −1 ), and α G is the albedo of the ground surface. Note that this model is not a patch model but is rather a layer model, and therefore, total flux per unit area is given as the sum of the vegetation canopy and ground surface components; that is, R n = R nV + R nG , H = H V + H G , and lE = l(Ev + Tr). Second, there is no need for radiometric temperature observations, as the model is separately solved by the Newton-Raphson scheme for vegetation canopy (T L ) and ground temperature (T G ), which can be easily integrated into the isotopic fractionation processes with H 2 O exchange in terrestrial ecosystems and has the advantage of enabling the long-term assessment of E partitioning, as well as isotopic enrichment output at the plant-atmosphere interface. Third, this model considers the nonlinear response of plant physiology to light penetration inside the canopy and ground surface by considering the interaction between them. Also, stomatal control of transpiration, including its dependence on soil moisture, is explicitly considered in the model, while some previous models do not consider it. More details are provided by Wang and Yamanaka [16].