Abstract
Stable isotopes (
15N and 
18O) of the greenhouse gas N2O provide information about the sources and processes leading to N2O production and emission from aquatic ecosystems to the atmosphere. In turn, this describes the fate of nitrogen in the aquatic environment since N2O is an obligate intermediate of denitrification and can be a by-product of nitrification. However, due to exchange with the atmosphere, the 
values at typical concentrations in aquatic ecosystems differ significantly from both the source of N2O and the N2O emitted to the atmosphere. A dynamic model, SIDNO, was developed to explore the relationship between the isotopic ratios of N2O, N2O source, and the emitted N2O. If the N2O production rate or isotopic ratios vary, then the N2O concentration and isotopic ratios may vary or be constant, not necessarily concomitantly, depending on the synchronicity of production rate and source isotopic ratios. Thus prima facie interpretation of patterns in dissolved N2O concentrations and isotopic ratios is difficult. The dynamic model may be used to correctly interpret diel field data and allows for the estimation of the gas exchange coefficient, N2O production rate, and the production-weighted 
values of the N2O source in aquatic ecosystems. Combining field data with these modelling efforts allows this critical piece of nitrogen cycling and N2O flux to the atmosphere to be assessed.
Introduction
Nitrous oxide (N2O) is a powerful greenhouse gas, 298 times more potent than CO2 over a 100-year time line [1]. Atmospheric N2O concentrations have been increasing at a rate of 0.25%/year over the last 150 years [2]. Consequently, the global N2O budget has been the subject of intensive research efforts over the past few decades. N2O is produced through multiple microbial pathways: hydroxylamine oxidation during nitrification and as an obligate intermediate during denitrification and nitrifier–denitrification. Because these pathways of N2O production have different stable isotopic enrichment factors, isotopic analysis of N2O can potentially distinguish N2O produced through different pathways or from different sources [3]. Identifying N2O sources will provide insights on the fate of N at the ecosystem-scale (e.g., [4]–[6]). The isotopic ratios of N2O produced in soil environments (e.g., [7]–[11]), and in aquatic environments (e.g., [12]–[18]) have been measured to some extent. Although N2O production in rivers and estuaries is a significant portion of the global N2O budget (approximately 1.5 TgN/year, [19]), few studies report isotopic data for rivers [5], [20], [21].
In ice-free aquatic ecosystems, the 
15N and 
18O of dissolved N2O is affected by gas exchange with the atmosphere. As a result, the isotopic ratios of dissolved N2O are not equal to those of the N2O produced within the aquatic ecosystem and continue to change as atmospheric exchange (both ingassing and outgassing) occurs. In addition, isotopic fractionation during influx and efflux causes the isotopic ratios of N2O flux emitted to the atmosphere to be different than that of the dissolved N2O [22]. Thus, the simple method of calculating the instantaneous isotopic ratios of the N2O flux by taking measured dissolved isotopic ratios, adding an equilibrium isotope fractionation, and applying them to measured flux rates is inappropriate. Adjustments of measured isotopic ratios are necessary to understand the isotopic ratios of both produced and emitted N2O.
In this paper, we present a dynamic model of the stable isotopic composition of both the dissolved and emitted N2O in aqueous systems. We apply this model to two different measured diel patterns of the isotopic ratios of N2O in an aquatic ecosystem. We use the model to elucidate the relationship between the isotopic ratios of source, dissolved, and emitted N2O, to allow for improved interpretation of dissolved N2O isotope data. Ultimately, a process-based understanding on N cycling with aquatic ecosystems may be developed based on interpretation of N cycling processes.
Materials and Methods
Stable Isotopes of N2O
N2O is an asymmetric molecule: the most abundant isotopologues of N2O are 
, 
, 
and 
. The isotopic ratios, 15N: 14N and 18O: 16O, are:
 |
(1) |
 |
(2) |
where 
, 
, 
and 
represent the concentrations of the various N2O isotopologues. Note that 15R is the bulk 15N: 14N ratio and represents an average ratio of the two 15N isotopomers and isotopic ratios are reported as 
15N relative to air and 
18O relative to VSMOW. Although the isotopic ratio of the 15N isotopomers can be measured (e.g., [23]–[25]), the gas exchange fractionation factors are not affected by the intramolecular distribution of 15N [22]. Many laboratories cannot measure the intramolecular distribution of 15N and analysis of the bulk 15N: 14N ratio of N2O is more common [26]. Here, we confine our analysis to bulk 15N: 14N ratios and use 15N2O to represent the average abundance of the two 15N isotopomers. The same approach could easily be extended to consider each isotopologue separately.
Dynamic Isotope Model for Dissolved N2O
A simple three box model (SIDNO, Stable Isotopes of Dissolved Nitrous Oxide) was created using Stella modelling software (version 9.1.4, http://www.iseesystems.com) in order to study the relationships between the isotopic ratios of source, dissolved and emitted N2O (model file is available at https://github.com/jjvenky/SIDNO and by contacting the corresponding author). This model is an adaptation of the isotopic gas exchange portion of the PoRGy model [27], which successfully modelled diel isotopic ratios of O2 resulting from photosynthesis, respiration, and gas exchange in aquatic ecosystems. One key difference is photosynthetically produced O2 in PoRGy has a 
18O value fixed by the H2O molecules, whereas SIDNO has N2O production 
15N and 
18O values that can vary independently of each other and of N2O production rate in order to simulate variability in nitrification and denitrification.
One box in SIDNO is used for the total mass of dissolved N2O and two additional boxes for the dissolved masses of the two heavy isotopologues (15N2O and N2
18O). The boxes are open to the atmosphere for gas exchange, are depth agnostic, and each box can gain N2O via a production term; there is no N2O consumption term since the 
values of N2O are largely controlled by the production pathways [28], [29] though certain waters can exhibit significant N2O reduction to N2
[30], [31]. The masses and magnitude of the flows of 15N2O and N2
18O relative to bulk N2O are used to calculate the isotopic composition of source, dissolved, and emitted N2O. Although isotopic ratios are used in the model, we discuss 
values that are common for reporting isotopic ratios. N2O production rate and its 
values are user-defined and can be adjusted for diel patterns in N2O production that may be caused by variable O2 levels [32]–[37].
Stable Isotope Dynamics of Gas Exchange
The 
values of the net gas exchange flux are controlled by the kinetic fractionation factors for evasion (
, 0.9993 for 
15N and 0.9981 for 
18O) and invasion (
, 1.0000 for 
15N and 0.9992 for 
18O) [22]. These two 
values are related to the equilibrium fractionation factor: 
(0.99925 for 
15N and 0.99894 for 
18O) and are independent of temperature over the range of 0
to 44.5
[22].
The 
values of tropospheric N2O are 6.72 ‰
0.12‰ for 
15N and 44.62‰ 
0.21‰ for 
18O [38]. Therefore, at equilibrium, dissolved N2O has dissolved 
values slightly greater than these at 7.48‰ and 45.73‰, respectively.
In the model, net N2O flux between the atmosphere and dissolved phase was calculated using the thin boundary layer approach as:
 |
(3) |
where the N2O flux is calculated in mol/m2/h, 
is the user-modifiable gas exchange coefficient (m/h), is the partial pressure of tropospheric N2O (assumed to be 320 ppbv from data provided by the ALE GAGE AGAGE investigators, [39], [40]), 
is the Henry constant for N2O (mol/atm-m3), and 
is the dissolved concentration of N2O (mol/m3). 
is a function of water temperature [41]:
 |
(4) |
where 
is temperature in kelvins.
Gas exchange is a two-way process. The net N2O flux rate (the difference between the invasion and evasion rates) depends on the dissolved N2O concentration. When a solution is at equilibrium with the atmosphere, the invasion and evasion rates will be equal, and the net flux will be zero.
As with the bulk N2O flux, the flux of the heavy isotopologues (15N2O and N2 18O) can be calculated by including the kinetic fractionation factors for N2O (adapted from [27]):
 |
(5) |
 |
(6) |
where 
and 
are the partial pressures of 15N2O and N2
18O.
Results
Test of Model Performance
To test the ability of SIDNO to reproduce observed isotopic data, input parameters (N2O production rate, N2O 
values, and 
) were set to replicate a series of experiments designed to derive fractionation factors for N2O gas exchange [22]. In these experiments, degassed water was exposed to N2O gas of known isotopic ratios in a sealed container to varying degrees of saturation.
Modelled dissolved N2O concentration and 
values increased in response to gas exchange (Figure 1). The model fit to the experimental data is comparable to the original best-fit derivations (
for 
15N and 
for 
18O for both the original fit [22] and the SIDNO fit) (Figure 1). The initial isotopic composition of dissolved N2O was identical to the gas phase 
15N value, but the 
18O of dissolved N2O was slightly less than the gas phase 
18O value. Ultimately, at 100% saturation the 
values of the dissolved N2O were greater than those of the gas phase as a result of 
. The model successfully simulated the kinetic and equilibrium fractionations during gas exchange under the experimental conditions.
Figure 1. Comparing the model output to the experimental data of [22].
The coefficient of determination for experimental data and SIDNO model outputs were comparable to those of [22], 
for 
15N and 
for 
18O. Precision of measurements for the experimental data was 
0.05‰ for 
15N and 
0.1‰ for 
18O.
Next, SIDNO was used to provide insight into the effect of degassing on the 
values of dissolved and emitted N2O. Here the results of two model runs with the same initial N2O concentration but different initial 
values of dissolved N2O were compared (Figure 2). As N2O saturation declined both the dissolved 
15N values and instantaneous 
15N values of the emitted N2O remained relatively constant, dissolved 
18O values and instantaneous 
18O values of the emitted N2O varied by about 10, when the solution was very supersaturated (
300% saturation). The 
values rose quickly as the system approached 100% saturation. Because the light isotopologue diffuses out of solution faster than the heavy isotopologue, the instantaneous 
values of the emitted N2O were always less than the concomitant 
values of dissolved N2O. The isotopologues of N2O reached equilibrium independently of each other and therefore the total mass emitted for each isotopologue and rate of change depended on the initial concentration and 
values. The retention of N2O in the dissolved phase caused the 
values of the mass emitted to differ from those of total mass production. However, when initial dissolved N2O concentrations were high (
1000% saturation) the 
values of the total N2O emitted were similar to the 
values of dissolved N2O because the mass of N2O lost is very much larger than the N2O that remained dissolved. The value of 
did not affect the gas exchange trajectories only the speed at which the system reached equilibrium.
Figure 2.
15N and 
18O trajectories for dissolved and emitted N2O in two supersaturated solutions with zero N2O production.
Initial dissolved isotopic values for the two dissolved N2O solutions were 
15N = −50‰, 
18O = 10‰, and 
15N = −10‰, 
18O = 30‰. Both runs used an initial dissolved N2O concentration of 1500% saturation. Note that in the 
18O versus 
15N plot, the dissolved N2O curves do not pass through the tropospheric N2O value due to the small equilibrium isotope effect.
N2O isotope data are often plotted as 
18O versus 
15N to elucidate relationships between the various sources and tropospheric N2O [38]. The trajectories on these plots (Figure 2C) were dictated by the 
values of the source relative to the constant atmospheric value and the 
values. Note that some plots in the literature differ due to different reference materials for the 
18O scale (VSMOW and atmospheric O2).
Modelling Scenarios with Steady State Production of N2O
The SIDNO model can be used to probe the stable isotope dynamics of N2O in a variety of situations that may be encountered in aquatic environments to elucidate the relationship between the N2O source (a function of N cycling processes), dissolved (the easily measured component), and emitted (of consequence for greenhouse gas production and global N and N2O cycle).
In the steady-state production of N2O (constant rate and 
values), by definition, the 
values of N2O production must be the same as those of the emitted N2O. As a result, the 
values of the dissolved N2O cannot equal that of the source (or emitted) N2O at steady state because the dissolved N2O must be offset from the emitted N2O by at least the 
values. As the steady-state production rate was increased, the steady-state N2O concentration increased and the dissolved 
values approached but did not equal the source (Figure 3). Even at moderate supersaturations (
1000%) the effect of atmospheric N2O equilibration on the 
values of dissolved N2O cannot be ignored.
Figure 3. The relationship between 
15N, 
18O and N2O concentration in a system at steady state with constant N2O production and open to gas exchange with the atmosphere.
The point marked with a 
represents the minimum difference between the isotopic composition of dissolved and source N2O. The point at 100% saturation is the equilibrium value, the 
15N and 
18O of this point is controlled by the isotopic composition of tropospheric N2O and the equilibrium enrichment factors.
At steady state, the 
values of the emitted N2O must be equal to the source; the large difference between source/emitted and dissolved N2O underscores the importance of adjusting the measured 
values of dissolved N2O in order to determine aquatic contributions of N2O to the atmosphere or N2O sources. This is critical when using dissolved measurements of N2O to constrain the global isotopic N2O budget, but not been done in most studies, e.g., [16], [42]–[44] but see [45].
Modelling Scenarios with Variable Production of N2O
The relationship between the 
values of source, dissolved, and emitted N2O are much more complicated when N2O production is variable rather than when it is constant. N2O production may vary with respect to production rate and/or 
values; in many aquatic environments, N2O production is not likely to be constant. The N2O production processes, nitrification and denitrification, are sensitive to redox conditions, which can be highly variable, due to diel changes in dissolved O2 concentration, flow regime, etc. For example, [34] observed diel changes in the denitrification rate in the Iroquois River and Sugar Creek (Midwestern USA) and found that the denitrification was consistently greater during the day than night. The relative importance of nitrification and denitrification can change in response to the diel oxygen cycle: e.g., [46] observed a change from daytime nitrification to nighttime denitrification in a subtropical eutrophic stream. Coupling of N2O and O2 diel cycles has been observed in agricultural and waste-water treatment plant (WWTP) impacted rivers [36]. Since fractionation factors and substrates are different for nitrification and denitrification, ecosystem-scale fractionation factors may be rate and process dependent, and the 
values of N2O production in a given ecosystem may not be constant over a diel cycle.
To simulate the diel variability, various scenarios were modelled by adjusting either production rate and/or the associated 
values. The variabilities in these input parameters were driven by a sine function with a 24 h period similar to a dissolve O2 curve. In all scenarios, the chosen range of production rates was based on published N2O flux rates (Table 1) and varied from 1 to 5 mol/m2/h1 (Table 2), which was between the diel variation in N2O flux observed by [33] and [46]. Temperature was held constant at 20
. The value of 
was varied between 0.1 and 0.3 m/d (Table 2), within the range observed in other river studies (Table 1). The combination of production rates and 
values were chosen to produce N2O between 150% and 500% saturation (Table 2) coinciding with the range of published data (Table 1). The range of 
values used for the N2O source (Table 2) was within published values from various field studies [47]. For scenarios where the 
values of source N2O was variable, the sine function for the 
values was synchronized so that maxima and minima 
15N values coincided with those of 
18O. This was done for simplicity, and because, in general, nitrification yields N2O with lower 
15N and 
18O values than denitrification (e.g., [10], [48]). Nevertheless, scenarios with greater amounts of N2O reduction to N2 can be modelled by increasing the source 
15N and 
18O values to those appropriate for any given ecosystem. Model scenarios were run until the output parameters (i.e., N2O saturation and the 
values of dissolved, source, and emitted N2O) reached dynamic steady state: model output was not constant over 24 h but the diel patterns on successive days were repeated.
Table 1. Summary of relevant published data on N2O production in aquatic environments.
| Location | Range of N2O Saturation (%) | Range of N2O Flux ( mol/m2/h1) |
Range of  values (m/h) |
Reference |
| Ohio River, OH, US | 95 to 745 | 5 to 90 | — | [12] |
| 5 agricultural streams, ON, CA (over 2–3 years) | 14 to 1700 | −1 to 91.7 | 0.002 to 0.59 | [5] |
| 10 agricultural streams, ON, CA (over 17 diel cycles) | 30 to 2570 | −0.33 to 52.1 | 0.004 to 0.30 | [45] |
| Bang Nara River, TH | 170 to 2000 | — | — | [20] |
| LII River, NZ | 201 to 404 | 1.35 to 17.9 | 0.13 to 0.82 | [58] |
| LII River, NZ | 402 to 644 | 0.46 to 0.89 | 14.76 | [33] |
| Seine River, FR | — | 2.2 to 5.2 | 0.04 to 0.06 | [59] |
| Canal Two, Yaqui Valley, MX | 100 to 6000 | 0 to 34.9 | 0.3 to 0.6 | [46] |
| agricultural stream, UK | 100 to 630 | 0 to 37.5 | — | [60] |
| Grand River, ON, CA | 38 to 8573 | −1.4 to 173.6 | 0.06 to 0.35 | [36], [37] |
Table 2. Summary of input parameters for the SIDNO model scenarios for non-steady state production of N2O.
| Scenario # | Results Figure |
|
N2O Source Production Rate | N2O Source 
15N and 
18O Values |
| (m/k) | ( mol/m2/h1) |
(‰) | ||
| Variable Production Rate, Constant Isotopic Composition of Source | ||||
| 1 | 4 | 0.3 | 1 to 5 | −50, 10 |
| Constant Production Rate, Variable Isotopic Composition of Source | ||||
| 2 | 5 | 0.3 | 3 | −50, 10 to −30, 10 |
| 3 | 6 | 0.1 | 3 | −50, 10 to −30, 10 |
| Variable Production Rate, Variable Isotopic Composition of Source | ||||
| 4* | 7 | 0.3 | 1 to 5 | −50, 10 to −30, 10 |
| 5** | 8 | 0.3 | 1 to 5 | −50, 10 to −30, 10 |
| 6* | 9 | 0.1 | 1 to 5 | −50, 10 to −30, 10 |
*Maximum production rate coincides with the lowest source 
values.
**Maximum production rate coincides with the highest source 
values.
Model Scenario #1: Variable Production Rate, Constant Isotopic Composition of Source
In scenario #1 (Table 2), the 
values of source N2O were held constant and the production rate was variable. An example of such a system may be N2O production via denitrification in river sediments with abundant 
. Denitrification rates in rivers have been observed to fluctuate in response to the diel O2 cycle [34]. If the fractionation factors for denitrification are not rate dependent, the resulting N2O production rate would be variable but the source 
values of N2O values could be constant.
Here, the maximum concentration lagged approximately 2.75 h behind the maximum N2O production rate, a function of the magnitude of the gas exchange coefficient, cf. [49]. The 
values for the instantaneously emitted N2O were relatively constant and very similar to the N2O source (within 0.4‰ for 
15N and 1.1‰ for 
18O, Figure 4, Table 3). However, the 
values of dissolved N2O were more variable, spanning 16‰ for 
15N and 10‰ for 
18O. Thus, a change in the 
values of dissolved N2O can be driven simply by a change in production rate and not necessarily a change in the 
values of the source. Since the system was at dynamic steady state, the average 
values of the emitted N2O were identical to the average 
values of the source. This must be true in all steady-state cases to conserve the mass of the N2O isotopologues.
Figure 4. Model scenario #1 – isotopic composition of dissolved and emitted N2O with a variable production rate and constant isotopic composition of the source.
Note, in panel D, the data points for emitted N2O are masked by the data point for source N2O.
Table 3. Summary of SIDNO output for model scenarios simulating non-steady state production of N2O.
| Dissolved N2O | Emitted N2O | ||||||
| Scenario # | Saturation |
15N, 
18O |
15N |
18O |
15N, 
18O |
15N |
18O |
| (%) | (‰) | (‰) | (‰) | (‰) | (‰) | (‰) | |
| Variable Production Rate, Constant Isotopic Composition of Source | |||||||
| 1 | 153 to 263 | −27.8, 24.6 to −12.1, 34.4 | 37.9 | 24.4 | −49.6, 9.5 to −50.2, 11.1 | 0.2 | 0.5 |
| Constant Production Rate, Variable Isotopic Composition of Source | |||||||
| 2 | 208 | −19.6, 29.4 to −3.7, to 37.4 | 30.4 | 19.4 | −45.3, 12.3 to −14.7, 27.7 | 4.7 | 2.3 |
| 3 | 423 | −26.1, 24.8 to −15.1, 30.3 | 23.9 | 14.8 | −37.2, 16.4 to −22.8, 23.6 | 12.8 | 6.4 |
| Variable Production Rate, Variable Isotopic Composition of Source | |||||||
| 4* | 153 to 263 | −25.8, 25.6 to −1.2, 39.8 | 24.2 | 15.6 | −46.9, 11.3 to −18.0, 26.5 | 8 | 3.5 |
| 5** | 153 to 263 | −11.2, 33.8 to −5.0, 36.1 | 38.8 | 23.8 | −41.6, 14.6 to −13.4, 28.1 | 8.4 | 4.6 |
| 6* | 345 to 501 | −31.6, 21.6 to −18.4, 29.3 | 18.4 | 11.6 | −42.1, 13.7 to −29.4, 20.6 | 19.4 | 9.4 |
Temporally variable parameters are given as a range. 
15N and 
18O (
) are the maximum difference between the range of source N2O and the range for the model output parameter.
*Maximum production rate coincides with the lowest source 
values.
**Maximum production rate coincides with the highest source 
values.
In some aquatic systems, the N2O production rate may remain constant with time but the 
values of the source may change with time. In rivers or lakes without a strong diel O2 cycle, sediment denitrification may produce N2O at an approximately constant rate. Denitrification rate may also be independent of water column 
concentration if limited by factors other than diffusion in the sediments. The 
values of the source N2O may thus change if the 
values of the 
substrate changed with time. For example, many studies have shown that the 
values of residual 
increase during denitrification [50]. Similarly, 
from WWTPs may have different 
values than agricultural runoff and diel changes in WWTP release may result in changing 
values of 
. Changes in N cycling may also vary on a diel basis but result in fortuitously similar N2O production rates due to, for example, changes in the N2O:N2 ratio of denitrification or changes in the relative importance of nitrification and denitrification. Thus changes in 
values of the N2O source do not necessarily indicate changes in N2O production rates.
Model Scenario #2: Constant Production Rate, Variable Isotopic Composition of Source
In scenario #2, when the N2O production rate was held constant and the 
values of the source varied with time (from −50‰ to −10‰ for 
15N and from 10‰ to 30‰ for 
18O), the 
values of the dissolved N2O was also much farther from that of the source than the dissolved N2O due to the effects of atmospheric exchange and the emitted N2O varies linearly between the two source values. In contrast, the dissolved N2O is parallel but offset from the line connecting the two sources (Figure 5, Table 3). The maximum difference between emitted and source N2O was 4.7‰ for 
15N and 2.3‰ for 
18O. The dissolved and emitted 
values also lagged 2.75 h behind the source as a result of gas exchange (as above). Since the system was at dynamic steady state, the average 
values of the emitted N2O were identical to the average 
values of the source.
Figure 5. Model scenario #2 – Isotopic composition of dissolved and emitted with a constant production rate and variable isotopic composition of the source.
Model Scenario #3: Constant Production Rate, Variable Isotopic Composition of Source
To examine the effects of varying 
on the scenario of constant N2O production with variable isotopic signature of the source, 
was reduced from 0.3 m/h (scenario #2) to 0.1 m/h (scenario #3; Figure 6, Table 3). The 
values for the emitted N2O were centred between the sources N2O values, but dissolved N2O 
values were farther from tropospheric N2O than the high-
scenario #2 (Figure 6 D).
Figure 6. Model scenario #3 – Isotopic composition of dissolved and emitted N2O with a constant production rate and variable isotopic composition of the source.
is reduced from 0.3 m/h to 0.1 m/h.
The effect of reducing 
was an increase in N2O concentration with the same production rate and a shift in the 
values of dissolved N2O toward the source values. Reducing 
also dampened the response between the instantaneous 
values of the emitted N2O and the 
values of the source. As above, the lag time between the 
values of the source and emitted N2O increased as 
decreased. The total range of the source and emitted 
values decreased. The difference between the source and emitted 
values was 12.8‰ for 
15N and 6.4‰ for 
18O.
To simulate a system alternating between two N2O production processes, such as differing relative contributions of nitrification and denitrification, with different rates of N2O production and 
values, the model was run with both production rate and its 
values variable with time (scenarios #4, #5, and #6). The production rate and 
values were adjusted so that the maximum rate coincided with the lowest source 
values in scenarios #4 and #6 and so that maximum rate coincided with the highest source 
values in scenario #5.
Model Scenario #4: Variable Production Rate, Variable Isotopic Composition of Source
For scenario #4, the resulting N2O concentrations were identical to those in model scenario #1, with the maximum concentration lagging approximately 2.75 h behind the maximum production rate (Figure 7, Table 3). The relationship between the 
values of the dissolved and emitted N2O was more complex than in other scenarios. The lag time between the maximum source 
values and those of dissolved and emitted N2O (when the production rate was minimum) was 3.75 h; however, the lag time between the minimum source 
values and those of the dissolved and emitted N2O (when the production rate was maximum) was only 2.25 h. The difference between the emitted and source N2O was 3.1‰ to 8.0‰ for 
15N and 1.3‰ to 3.4‰ for 
18O. The 
values of emitted N2O were closer to those of the source during periods of high production rates (and thus higher concentrations) than periods of low production rates. However, the flux-weighted average 
values of emitted N2O were equal to the average production-weighted source 
values because the system was at dynamic steady state.
Figure 7. Model scenario #4 – Isotopic composition of dissolved and emitted N2O with a variable production rate and variable isotopic composition of the source.
Maximum production rate is in sync with the lowest 
15N and 
18O values of the source.
Model Scenario #5: Variable Production Rate, Variable Isotopic Composition of Source
The isotopic counterpoint to scenario #4 is adjusting the timing of maximum N2O production to coincide with the highest 
values of production (scenario #5). All other parameters were the same as scenario #4 (Table 3). The resulting pattern for the 
values of dissolved N2O was very different than scenario #4 (Figure 8, Table 3). While the dissolved N2O concentrations were identical to the model scenario #4, the 
values of dissolved N2O were nearly constant with time. The relationship between the 
values of emitted and source N2O was similar to scenario #4, although the instantaneous difference in 
values were slightly greater. The 
values of the dissolved N2O were greatly dampened by the fact that maximum production rate coincided with source 
values that were closest to tropospheric N2O. In scenario #4, the high rates of N2O production at 
values very different than tropospheric N2O increased the amplitude of the 
values of dissolved N2O.
Figure 8. Model scenario #5 – Isotopic composition of dissolved and emitted N2O with a variable production rate and variable isotopic composition of the source.
Maximum production rate is in sync with the greatest 
15N and 
18O values of the source.
Model Scenario #6: Variable Production Rate, Variable Isotopic Composition of Source
To determine the effects of a lower 
on model scenario #4, 
was reduced from 0.3 m/h from 0.1 m/h for scenario #6. As shown above, lower 
increased the dissolved N2O concentrations and dampened the diel range of 
values of both dissolved and emitted N2O (Table 3, Figure 9). Lower 
also increased the lag time between the 
values of emitted and source N2O and increased the difference between the 
values of emitted and source N2O (Figure 9). As in all scenarios, the flux-weighted average 
values of emitted N2O were equal to the average production-weighted source 
values.
Figure 9. Model scenario #6 – Isotopic composition of dissolved N2O and emitted N2O with a variable production rate and variable isotopic composition of the source.
Maximum production rate is in sync with the lowest 
15N and 
18O values of the source. 
is reduced from 0.3 m/h to 0.1 m/h.
Grand River
The ability of SIDNO to reproduce measured patterns of N2O concentration and 
values in a human-impacted river was also assessed. The Grand River is a seventh-order, 300 km long river that drains 6800 km2 in southern Ontario, Canada, into Lake Erie, see [36], [37], [51]. There are 30 WWTPs in the catchment and their cumulative impact can be observed via the increase in artificial sweeteners in the river [52].
Samples were collected approximately hourly for 28 h at two sites in the central, urbanized portion of the river: sites 9 and 11 in [51], [52]. The upstream site, Bridgeport, is where the river enters the urban section of the river at the city of Waterloo and is immediately above that city's WWTP. Blair is 26.6 km downstream of Bridgeport and below the cities of Waterloo and Kitchener. It is also 5.5 km downstream of the Kitchener WWTP. Average river depth at both sites was 30 cm. Values of 
were determined by best-fit modelling of diel O2 and 
18O-O2 values at the sites [36]. N2O concentration analyses were performed on a Varian CP-3800 gas chromatograph with an electron capture detector and isotopic ratio analyses were performed on a GV TraceGas pre-concentrator coupled to a GV Isoprime isotope ratio mass spectrometer, see [5] for analytical details.
Data from upstream and downstream of large urban waste-water treatment plants on the Grand River show diel patterns in N2O saturation and 
values (Figures 10 and 11). At the Bridgeport site, the diel patterns of N2O saturation and 
15N values were opposite of each other, that is, when N2O saturation was highest around sunrise the 
15N values were lowest and when when N2O saturation was lowest around before sunset the 
15N values were greatest. 
values between field and model data for N2O saturation, 
15N, and 
18O values are 0.83, 0.68, and 0.30. Model results reproduce the range and sinusoidal patterns of the field data though the 
18O fit was poor in the second half of the field data. The diel pattern in 
18O values was similar to that of 
15N but was shifted earlier by about 4 h. These patterns were similar to those of scenario #4 (variable N2O production and variable 
values of the source N2O coinciding when maximum production rates coincided with lowest source 
values) and the result of consistent diel five-fold variability in N2O production and variability in the 
18O of the N2O produced in the river.
Figure 10. Diel variability in N2O concentration and 
values at Bridgeport in the Grand River, Canada.
The time axis begins at 00∶00 on 2007-06-26. Maximum production rate is in sync with the greatest 
18O values of the source, while 
15N of the source was constant. 
values between field and model data for N2O saturation, 
15N, and 
18O values are 0.83, 0.68, and 0.30. This is similar to model scenario #4 (Figure 7).
Figure 11. Diel variability in N2O concentration and 
values at Blair in the Grand River, Canada.
The time axis begins at 00∶00 on 2007-06-26. Maximum production rate is in sync with the lowest 
15N and 
18O values of the source. 
values between field and model data for N2O saturation, 
15N, and 
18O values are 0.78, 0.53, and 0.03. This is similar to model scenario #5 (Figure 8).
At the downstream Blair site, both 
15N and 
18O values were much lower and exhibited a greater range than at Bridgeport. 
values between field and model data for N2O saturation, 
15N, and 
18O values are 0.78, 0.53, and 0.03. Model results reproduce the range and peak-and-trough pattern of the N2O saturation and 
15N data. Model results reproduce the range of 
18O values but the pattern is not well reproduced. While all data at Bridgeport exhibited smooth, sinusoidal diel changes, the data at Blair show rapid changes. The diel patterns of N2O saturation and 
15N values were opposite of each other, that is, when N2O saturation was highest around midnight, the 
15N values were lowest and when when N2O saturation was lowest during mid-day, the 
15N values were greatest. The diel pattern in 
18O values was more complex at Blair than at Bridgeport suggesting that daytime and nighttime were associated with different 
18O values of N2O production. These patterns were similar to those of scenario #5 (variable N2O production and variable 
values of the source N2O coinciding when maximum production rates coincided with highest source 
values) and the result of a five-fold variability in day-to-night N2O production and variability in 
15N and 
18O of the N2O produced in the river.
For both Bridgeport and Blair data, the cause of poorer fits for 
18O than 
15N deserve further research. Adding concomitant measurements of 
15N and 
18O values of 
may provide clues about N cycling and help explain some of the observed variability in N2O [53]. Predicting 
18O-N2O values from nitrification [11] and denitrification [54] is difficult because of the complex relationship between 
18O-H2O values and 
18O-N2O values. Additionally, diel variability in N2O reduction to N2
[45], [46], may also manifest itself in 
18O-N2O values because of the strong O isotope fractionation factor during denitrification [55].
Discussion
Calculating the 
values of emitted or source N2O is critical for regional and global N2O isotopic budgets and also provides information about the source of N2O and thus N cycling processes. However, SIDNO can simulate the relationships between the 
values of dissolved, source, and N2O emitted from aquatic ecosystems to the atmosphere.In systems with N2O production at dynamic steady state, the 
values of dissolved N2O will not always be directly indicative of the 
values of the source N2O. The difference between dissolved and source 
values increases as N2O saturation decreases (as demonstrated in Figures 2 and 3). Even above 1000% saturation (from high production rates and/or low 
), the 
values of dissolved N2O will only approach 
values of the source but offset by 0.7‰ for 
15N and 1.9‰ for 
18O, a result of the 
values (Figure 3). At constant N2O production rates and 
values, the source and emitted 
values can be quantified since the 
values of emitted N2O must be identical to those of the source and can be calculated from dissolved values (Figure 3; equations 5 and 6).
Our modelling results identified the limitations associated with simple interpretation of dissolved N2O isotope data since the 
values of dissolved and emitted data are synchronous but rarely offset by a constant value. If N2O saturation changes with time, the N2O production rate must also have changed with time, provided 
had been constant (compare model scenarios #1 and #2 in Figures 4 and 5). In contrast, changes in 
values of dissolved N2O do not require a change in the source 
values (model scenario #1 and Figure 4), while constant 
values of dissolved N2O do not require constant source 
values (for example model scenario #5 in Figure 8).
When 
values of the source N2O are variable, the relationship between emitted and source N2O becomes complicated. The 
values of emitted N2O will lag behind those of the source and the amplitude of the diel range of 
values will be dampened relative to the source. The amount of lag and dampening is a function of 
, N2O production rate and timing, and the proximity of the source 
values to those of the atmosphere (compare Figures 2 with 3 and Figures 4 with 6). Qualitatively, the 
values of emitted N2O will be similar to the source if the equilibration time of dissolved N2O is small relative to the period of source variability (e.g., 24 h period due to diel changes in N cycling [36], [45]). Assuming homogeneous N2O release upstream, the equilibration time can be approximated from a decay curve as 
, where 
is mean depth [49]. If 
is small and/or 
is high, the equilibration time will be short and the 
values of the emitted N2O will be close to the source. With decreasing 
(or increasing equilibration time), the 
values of emitted N2O will lag farther behind and will always have a smaller range of 
values than the source. At the most extreme case, the variability in the 
values of emitted N2O will be reduced to nearly zero and 
values of the emitted N2O would be equal to the average production-weighted source 
values. At very long equilibration times, the probability of N2O consumption increases, a process not explicitly included in SIDNO where the 
value of the source N2O is simply that which is released to the water column.
Separating N2O production into nitrification and denitrification requires independent knowledge about the 
values of the source N and O in aquatic ecosystems. It is therefore not possible to state a single 
15N value for nitrification–N2O and one for denitrification–N2O applicable to all aquatic ecosystems. The 
15N value of the N2O precursors 
and 
vary across ecosystem as a result of human impact and N loading (agricultural and WWTP) as well as the source of N, and additional N transformations in the aquatic ecosystem. For example, along the length of the Grand River, 
15N values of 
and 
exhibit systematic trends resulting from the confluence of agricultural tributaries and large urban waste-water treatment plants (Schiff et al., unpublished results, [53]). Nevertheless, these values can be measured and biogeochemical relationships between N species, redox, and N2O can be used as supporting information for process separation (e.g., [5], [12], [36], [45]). The 
18O value of N2O will also vary across ecosystems as a result of its close relationship with 
18O-H2O and to a lesser extent 
18O-O2
[10], [11], [54], [56]. Fortunately, 
18O-H2O values can be easily predicted and measured [57]. Thus, once 
values of N2O precursors have been identified, biogechemical data can provide an indication about the diel pattern of N2O production processes, and ranges of potential end-member 
values can be calculated (e.g., [5] summarize isotopic fractionation for 
15N and [10], [11], [54], [56] for 
18O) and the model used to fit the field data.
Conclusions
In aquatic ecosystems, the instantaneous 
values of N2O emitted to the atmosphere are easily calculated if the water temperature and dissolved N2O concentration and 
values are known. Our modelling efforts illustrate that complex relationships exist between dissolved and source N2O and that the 
values of dissolved N2O are not always representative of either the N2O produced or emitted to the atmosphere. Thus, calculated 
values of the emitted N2O are the values that should be used to draw conclusions about N2O emission from aquatic systems and the global N2O cycle rather than the more commonly used instantaneous values (Table 4). The flux-weighted 
values of emitted N2O can provide average production-weighted 
values of the N2O source under dynamic steady-state in aquatic ecosystems.
Table 4. Summary of the results of the SIDNO modelling as the predictive relationship between observations and implications.
| Observed Parameter | Implications | Examples |
| Dissolved N2O concentration is constant with time | The N2O production rate is constant with time (if k and temperature are also constant). The N2O flux to the atmosphere is equal to the production rate. This may not be true if the concentration is close to atmospheric equilibrium. | Scenarios #2 and #3 |
| Dissolved N2O concentration is variable with time in a sinusoidal pattern | The N2O production rate is variable with time (if concentration change cannot be explained by change in k or temperature). The average N2O flux to the atmosphere is equal to the average production rate. | Scenario #1 |
15N and 
18O of dissolved N2O is constant with time |
The observation is inconclusive. At concentrations near atmospheric equilibrium, isotopic composition of dissolved N2O will approximate tropospheric N2O, egardless of source values. A constant isotopic signature of dissolved N2O that is different from tropospheric N2O can indicate either a constant source (if production rate is constant), or a variable source. | Scenario #5 |
15N and 
18O of dissolved N2O is variable with time (slope of data on a 
18O–
15N cross-plot tends toward tropospheric N2O) |
The change in 
15N and 
18O of dissolved N2O is likely a result of a change in concentration, but it is possible that the source is variable with time, if the 
15N and 
18O values of the source also trend through the value for tropospheric N2O. |
Scenarios #1, #2, #4, and #6 |
Calculated 
15N and 
18O of emitted N2O is constant with time |
The isotopic composition of the source is constant with time, and equal to the calculated value for emitted N2O | Scenario #1 |
Calculated 
15N and 
18O of emitted N2O is variable with time |
The isotopic composition of the source is variable with time. The range in 
15N and 
18O of emitted N2O is the minimum for the range in that of the source. The flux weighted average 
15N and 
18O of emitted N2O is equal to the production weighted average source values. |
Scenarios #2–#6 |
Long residence time relative to variability of source (need to independently determine  ) |
The changes in N2O concentration and 
15N and 
18O of emitted N2O will be dampened relative to, and lag behind, that of the source |
Scenarios #3 and #6 |
Short residence time relative to variability of source (need to independently determine  ) |
The changes in N2O concentration and 
15N and 
18O of emitted N2O will be indicative of changes in the source |
Scenarios #1, #2, #3 and #4 |
If the 
values of emitted N2O are constant with time, either the 
values of the source must also be constant or the N2O equilibration time is very long. However, if the calculated 
values of emitted N2O vary with time then the 
values of the source must also vary with time producing a diagnostic pattern. These findings are more robust than using dissolved 
values alone since dissolved 
values can change simply with a change in N2O production rate, changes in source 
values, and changes in 
. N2O residence time, dependent on production rate, 
, and 
, will determine the lag time between the 
values of emitted and source N2O. The difference in timing between maxima and minima 
values of emitted N2O and the maxima and minima of dissolved N2O is indicative of how the 
values of the source change. For all these reasons, we urge caution when using single samples of N2O concentration and 
to calculate fluxes of N2O to the atmosphere and inferring N2O production pathways.
Ultimately, the dynamic model SIDNO may be used to estimate 
, N2O production rate and 
values of the N2O source, an indication of the production pathway and N cycling, in aquatic ecosystems via inverse modelling. If physical properties, such as depth and temperature are known, SIDNO may be used to fit the measured field data (N2O concentration and 
values) by adjusting the N2O source parameters. SIDNO can also be used to explore the dynamics between dissolved, source, and emitted N2O to query production scenarios and design field campaigns for studies of N cycling processes.
Acknowledgments
We thank MS Rosamond, HM Baulch, the reviewers, and the academic editor for their helpful comments.
Funding Statement
Funding was provided by a Natural Sciences and Engineering Research Council (NSERC) and BIOCAP grant 336807-06 (SLS), an NSERC scholarship (SJT), and Environment Canada's Science Horizons Youth Internship Program. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
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