Modelling nitrous oxide emissions: comparing algorithms in six widely used agro-ecological models

Keywords: agriculture, agro-ecological models, algorithm comparison, denitrification, N₂O emissions, nitrification, soil, soil-atmosphere flux.

Nitrous oxide is a long-lived greenhouse gas and the major source of stratospheric nitric oxides (NO_x; Cicerone 1989; Williams et al. 1992), contributing approximately 6% to global warming (Dalal et al. 2003; Smith et al. 2008). Atmospheric N₂O concentration has risen from the pre-industrial value (1850–1990) of 270 to 332 ppb (v/v) in 2019, with 70% of the increase occurring since 1970 (IPCC 2014, 2021). Soils are estimated to contribute between 15 and 30% of the anthropogenic N₂O at the global scale (Bouwman 1990; Tubiello et al. 2013), and agriculture is cited as being the single largest contributor (Mosier et al. 1996, 1998; IPCC 2014). Since 1980, global agricultural expansion and intensification (increased nitrogen fertiliser and manure use) has accounted for 30% of the total increased N₂O concentration (Tian et al. 2020; IPCC 2021). Globally, about 3–5% of the synthetic N fertiliser used in agriculture is estimated to be returned to the atmosphere as N₂O (Crutzen et al. 2016). The predicted warmer and wetter conditions in response to climate change are expected to further increase N₂O emissions (Griffis et al. 2017). The IPCC synthesis reports (IPCC 2014, 2021) and other studies (Bouwman 1989; Mosier et al. 1998; FAO/IFA 2001) suggest that more than 60% of the annual global N₂O emissions from soils are from arable land.

Although N₂O production relates to three or four microbial processes (Baggs 2011), about 70% of global N₂O emissions are estimated to originate from microbial nitrification and denitrification (Braker and Conrad 2011; Syakila and Kroeze 2011). However, most models only simulate emissions from nitrification and denitrification, which are regulated by the oxygen (O₂) status in soil profiles, mainly controlled by soil moisture (SW), temperature (T_s), carbon (C) and nitrogen (N) substrates, soil pH and the microbial community (Nömmik 1956; Focht 1974; Weier et al. 1993; Smith et al. 1998; Weier 1999; Saggar et al. 2013). The complex microbial production and consumption processes, and the links to biotic (species competition, plant–microbe interactions) and abiotic (climate, soil chemistry and physics) factors are not fully understood and quantified (Baggs 2011; Butterbach-Bahl et al. 2013). Emissions are highly variable and difficult to predict because soil N₂O production and consumption are controlled by multiple biotic and abiotic factors within soil micro-sites (Butterbach-Bahl et al. 2013; Griffis et al. 2017). Recently, mechanistic approaches have been proposed to represent microsites in soil aggregates in models (Ebrahimi and Or 2018; Sihi et al. 2020). An ‘anaerobic balloon’, as used in the model DNDC, can be considered an early representation of non-uniformity in soil aeration status within any soil layer. Given that none of the models reviewed include microsites within soil aggregates, this was not considered in this review.

N₂O emissions estimates from agricultural soils range from a simple emission factor (Bouwman 1996; IPCC 2014; Del Grosso et al. 2020 and reference within) to more complex process-based models such as APSIM (Keating et al. 2003; Holzworth et al. 2014), DayCent (Parton et al. 1996, 2001; Del Grosso et al. 2000), WNMM (Li et al. 2007) or DNDC (Li et al. 1992, 2000; Li 2000). Although emission factors are generally used for regional-scale inventories, process-based models capable of predicting the effect of environmental conditions and management practices play an important role in modifying emission factors (Xing et al. 2013; Shen et al. 2018), and may possibly replace them. Bouwman (1996) highlighted the inadequacy of emission factors for estimating emissions at the local scale. The recent trend is to use process-based models, such as DayCent and DNDC, to calculate field-scale N₂O emissions, or to develop a series of emission factors for different land uses and management to estimate the larger scale fluxes (Li 2000; Del Grosso et al. 2005, 2006, 2020; Giltrap et al. 2010, 2011; Leip et al. 2011). Irrespective of the process-based model used, it is important to understand how processes are represented, the environmental rate modifiers and subsequent impact on model predictions.

Models play an essential role in estimating soil N₂O emissions, understanding management impacts on N₂O fluxes, simulating emissions over variable time scales, and extrapolating results to regional or national scales. Despite increasing applications, questions remain as to whether models can accurately predict the fluxes from soils in response to climate variability and change and agronomic management modifications (including fertiliser use). Giltrap et al. (2020) discussed how inaccuracies with model parameters can propagate through the model leading to uncertainty in predictions. When comparing a number of simplified denitrification process models, Heinen (2006) found that changes in SW were critical to estimating N₂O emissions and must be accurately determined. Pan et al. (2022), using a global data set, found that denitrification rate was positively related to soil water-filled pore space (WFPS), NO₃⁻ concentration and soil temperature (P < 0.01), whereas the rate decreased with higher soil oxygen content (P < 0.01).

The most commonly used process-based models for predicting N₂O production from soils include APSIM (Thorburn et al. 2010; Xing et al. 2011), DayCent (Del Grosso et al. 2000; Parton et al. 2001), DNDC (Li 2000; Li et al. 2000), FASSET (Chatskikh et al. 2005), NOE (Hénault et al. 2005) and WNMM (Li et al. 2007). Chen et al. (2008) reviewed and summarised the strengths, limitations and applications of commonly used field scale N₂O emissions models (i.e. DayCent, DNDC, NLOSS, ecosyss, Expert-N, FASSET, WNMM, and CERES-NOE). They concluded that significant gaps existed in the accurate partitioning of denitrification products and the diffusion and assimilation of N₂O from depth. More recently, Giltrap et al. (2020) reviewed the concepts of APSIM, DayCent and DNDC. Other studies have compared the performance of process-based models at predicting N₂O emissions (e. g., Li et al. 2007; Xing et al. 2011; Ehrhardt et al. 2018). There are two ways to compare model performance: (1) extracting the specific algorithms from the different models and recoding them in the same platform (model), then comparing N₂O emissions prediction performance after calibrating and validating against measurements (e.g. Li et al. 2007; Xing et al. 2011; Inatomi et al. 2019); and (2) comparing the predicted productivity and N₂O emissions by applying default parameters (benchmark simulations) to multiple process-based models using experimental data from multiple sites (e.g. Ehrhardt et al. 2018). Interestingly in the latter study, the prediction errors for N₂O emissions only marginally improved after model calibration. Evaluating model structure and performance helps to explain model differences. The consensus from these studies is that the processes controlling N₂O emissions involve multiple complex interactions.

Uncertainty still exists regarding the spatial and temporal variability of N₂O emission as a function of soil properties and nitrogen fertiliser applications. Furthermore, not all the relevant environmental and biogeochemical drivers and processes are currently represented in the most widely used models. Del Grosso et al. (2020) suggested that a systematic evaluation of N₂O and N₂ production from nitrification and denitrification in relation to key drivers and comparison to measurements is needed to highlight the algorithms and parameter values most responsible for inaccuracies in predicted N₂O emissions.

As models become more mechanistic, they should, in principle, be better suited to simulate the impacts of site-specific climate, soil drivers, and management on N₂O emissions (Giltrap et al. 2020). However, increasing the complexity in a model is only useful if, for example, the measured total N₂O emission can be attributed to specific N transformation rates, microbial, and physical processes (Zhang et al. 2015a). Most process-based models have been evaluated against a range of field experiments using net N₂O emissions (see for example Frolking et al. 1998; Li et al. 2005; Vogeler et al. 2011). Li et al. (2005) compiled the algorithms for predicting N₂O emissions from WNMM, DayCent and DNDC into the WNMM framework, and compared the performance of the different modules. Vogeler et al. (2011) compared APSIM and DNDC predictions for nitrification, denitrification and N₂O emissions from pastoral systems. They found the SW threshold value, T_s, and potential denitrification rate were important for predicting N₂O emissions from agricultural soils. Thorburn et al. (2010) and Xing et al. (2011) evalutated APSIM against field and laboratory data sets on net N₂O emissions and found that APSIM underestimated N₂O emissions from denitrification. Xing et al. (2011) highlighted the importance of soil depth when simulating emissions as the atmospheric emission includes variable contributions of N₂O produced in deeper layers. Del Grosso et al. (2010) showed that in DayCent, spatial variblity was mainly a result of differences in fertiliser or organic N additions, whereas temporal variablity in the annual N₂O emission was correlated with N mineralisation rates and climatic varialibity. Such comparisons indicate a need to better quantify the controlling variables on nitrification, denitrification and N₂O emissions. Accurate simulation of SW dynamics and its linkage to denitrification and N₂O flux is a key component of each model (Frolking et al. 1998). Predicting N₂O emissions is further challenged by a range of measurement artefacts, primarily SW dynamics and T_s, caused by the use of field chambers and laboratory incubations in calibration and validation data sets (Davidson et al. 2002; Livingston et al. 2006; Debouk et al. 2018). The temporal frequency of these measurements is also critical thus ensuring relatively large episodic events are captured (Grace et al. 2020).

Model comparisons, across a range of scales, indicate that model structure and parameterisation lead to different spatial and temporal patterns of the simulated N₂O emissions (Chen et al. 2008; Wu et al. 2015; Tian et al. 2018). At global and regional scales, uncertainty in agronomic management (cultivation, nitrogen fertiliser applications and species) results in significant variations in simulated N₂O emissions (Wu et al. 2015; Tian et al. 2018). Simulations of annual cumulative N₂O emissions by DNDC, LandscapeDNDC and IAP-N-GAS aligned well with annual observations, but there were discrepancies between simulated and observed daily fluxes related to the date, duration, and the magnitude of the peak fluxes (Zhang et al. 2015a). The error in daily predicted N₂O fluxes was hypothesised to be due to inaccurate SW prediction.

Specific comparisons between model components to understand the effect of process modifiers on the simulated N₂O fluxes are rarely done. This study evaluated the algorithms in six process-based models (APSIM, DayCent, DNDC, FASSET, NOE, and WNMM) for nitrification and denitrification, and N₂O emissions from these processes, including the effect of environmental rate modifiers. We show the difference in the algorithms for modelling nitrification, denitrification and N₂O fluxes and identify the measurements needed to improve model process-understanding and predictability for soil N₂O emissions.

The biochemical processes that lead to N₂O production during nitrification and denitrification are well known (Sahrawat and Keeney 1986; Firestone and Davidson 1989; Davidson 1993; Braker and Conrad 2011; Syakila and Kroeze 2011). Most of the research related to N₂O emissions explores emission response to environment, soil and land management in different agro-ecological zones (see for example, Li 2000; Del Grosso et al. 2010; Tian et al. 2018; Yu and Zhuang 2019).

All of the models reviewed in this study predict N₂O emissions from nitrification and denitrification and except for DNDC, ignore chemical pathways of N₂O production (Chalk and Smith 1983, 2020; Howden and O’Leary 1997). Additionally, N₂O emission during the NO₃⁻ reducing processes of nitrate ammonification (dissimilatory NO₃⁻ reduction to NH₄⁺), and co-denitrification are not represented in current models (Baggs 2011).

Nitrification is regarded as the main source of N₂O emissions during the oxidation of NH₄⁺ or heterotrophic nitrification of organic N to nitrate (NO₃⁻) in aerobic soil (Bremner and Blackmer 1978; Butterbach-Bahl et al. 2013; Zhang et al. 2015b; Inatomi et al. 2019). In contrast, denitrification is the main source of N₂O emissions under anaerobic conditions (Bremner and Shaw 1958; Focht 1974; Bremner and Blackmer 1980; Tiedje et al. 1983; Butterbach-Bahl et al. 2013) where NO₃⁻ is reduced to dinitrogen (N₂) and oxygen (O₂) and the NO₃⁻ is used as the terminal electron acceptor. Although denitrification is an anaerobic process, in soils it can occur at high O₂ pressures because of anaerobic microsites within aggregates in otherwise aerobic soil (Khalil et al. 2004; Sihi et al. 2020).

The models reviewed in this study use two different approaches to model N₂O production as a function of aeration and the availability of oxygen, which is commonly indexed using SW or more precisely the WFPS (the equations used to calculate soil moisture content (SW) and soil water-filled porosity (WFPS) are given in Cresswell and Hamilton (2002)). The first approach assumes that denitrification occurs only if WFPS exceeds a ‘threshold’ value that would indicate anaerobiosis. The WFPS threshold value for the various models range from 0.55 to 0.62. Nitrification is also a function of WFPS and nitrification and denitrification can occur simultaneously in the same soil layer. DayCent, WNMM, NOE, FASSET and APSIM adopt this approach. Field or laboratory measurements generally give total N₂O emissions and cannot ascribe the fluxes to either nitrification or denitrification. The use of ¹⁵N isotopes, added to different mineral N pools, is required to attribute N₂O emissions to their source processes (Zhang et al. 2015b; Denk et al. 2019).

The second approach assumes that the soil is partitioned into aerobic and anaerobic fractions within each soil layer. Nitrification occurs in the aerobic fraction while denitrification happens in the anaerobic fraction. The size of the aerobic or anaerobic fraction is calculated as a function of the ‘bulk’ soil redox potential (Eh). This approach is used by DNDC to simulate N₂O emission from both nitrification and denitrification in the same soil layer at the same time (Li et al. 2000). The size of the anaerobic fraction is defined by the simulated O₂ partial pressure calculated from O₂ diffusion and consumption rates in the soil (Table 1, Li et al. 2000).

Table 1.  The faction of N loss as N₂O during nitrification (K₁) used in different models.

Each of the reviewed models simulate net nitrification rates (see Supplementary Table S1). The algorithms therein do not distinguish the distinct phases of nitrification, namely NH₄⁺ oxidation to NO₂⁻ and NO₂⁻ oxidation to NO₃⁻. Nitrite is generally not considered because of its low concentration relative to NH₄⁺ and/or NO₃⁻. However, high concentrations of NO₂⁻ can occur following banded applications of urea or anhydrous ammonia (NH₃) that uncouples the two phases of nitrification (Chalk et al. 1975; Venterea et al. 2020). In the models reviewed, NH₄⁺ concentration, SW, T_s, and pH regulate nitrification, although not all factors are included in the different algorithms (Hanson et al. 2000; Li et al. 2000, 2007; Parton et al. 2001; Thorburn et al. 2010). Nitrifier biomass is included in DNDC (Li et al. 2000). Nitrous oxide is produced as an intermediate gaseous product during the microbial oxidation of NH₄⁺ to NO₃⁻. Since there is insufficient data to quantify the intermediary steps and regulatory mechanisms for N₂O production during nitrification, the process is represented as a ‘hole-in-the-pipe’, or ‘leaky pipe’ first proposed by Firestone and Davidson (1989). In most models, a proportion of nitrified NH₄⁺ is emitted as N₂O (Fig. 1; Li et al. 2000, 2007; Parton et al. 2001; Khalil et al. 2004; Chatskikh et al. 2005). Recently, Zhang et al. (2015b) reported significant gross heterotrophic nitrification of organic N and N₂O emissions from acidic soil with high C/N ratio. They proposed introducing N₂O production via heterotrophic nitrification of organic N into the conceptual ‘hole-in-the-pipe’ model of N₂O emission. Currently, the models reviewed in this study are calibrated to represent net N₂O emissions without distinguishing between autotrophic and heterotrophic nitrification.

Fig. 1.  Conceptual diagram of N₂O production during nitrification and denitrification and the leaky pipe concept. WFPS, T_s and pH modify the fraction of N₂O produced during nitrification and denitrification.

The models reviewed in this study simulate different nitrification rates under a given NH₄⁺ concentration even when other soil conditions (temperature, water, and pH) are at the optimal level. Nitrification is therefore not limited by any factor except NH₄⁺ concentration (Fig. 2a). The relationship between nitrification rates and NH₄⁺ concentration is as a linear response in DNDC, FASSET and WNMM. Michaelis–Menten kinetics is used in APSIM, and a Langmuir isotherm equation in used NOE. The detailed equations for nitrification and the modifiers in the models are given in Table S1.

Fig. 2.  Effect of (a) NH₄⁺-N concentration, (b) WFPS, (c) soil temperature (T_s) and (d) pH on the nitrification rate in APSIM (black solid line; APSIM_max – long dash), DNDC (dark red dash-dot line), DayCent (red square-dotted line) FASSET (black round-dotted line), NOE (blue dash line) and WNMM (black long dash-dot-dot line). The response to each factor (modifier) are dimensionless and range from 0 to 1. The nitrification rate in APSIM was either divided by V_max (APSIM) or by the maximum nitrification rate (32.65 mg N kg⁻¹ day⁻¹: APSIMmax). WFPS was calculated from volumetric water content assuming a bulk density (BD) of 1480 kg m⁻³.

Potential nitrification in APSIM follows Michaelis–Menten kinetics (Meier et al. 2006) with a maximum reaction rate (V_max) of 40 mg NH₄⁺-N kg⁻¹ soil day⁻¹ and a NH₄-N concentration of 90 mg NH₄⁺-N kg⁻¹ soil at V_max/2 (K_m) as default. Potential nitrification rate is also modified by SW, and T_s. Meier et al. (2006) have suggested that the values of V_max and K_m should be reduced to 12 mg NH₄⁺-N kg⁻¹ soil day⁻¹ and 30 mg NH₄⁺-N kg⁻¹ soil, respectively. Similarly, Smith et al. (2020) suggested that the parameter values in APSIM need to be reduced and a minimum NH₄⁺ concentration introduced into the model. Meier et al. (2006) cite Reuss and Innis (1977) as the source of APSIM’s default values. However, there is some confusion regarding the units as Reuss and Innis (1977) list the units for V_max and K_m as being g N m⁻³ day⁻¹ and g m⁻³, respectively. They do have the same value when the soil bulk density is 1.

In DayCent, nitrification follows exponential decay (1 – EXP form) that asymptotes at the maximum rate of 1 (Fig. 2a; Table S1) with a base flux equivalent to 0.1 g N ha⁻¹ day⁻¹. For comparison purposes we convert all the model outputs to the dimension ha⁻¹ day⁻¹ and volumetric soil water content, assuming a bulk of 1480 kg m⁻³ in the surface 0.15 m. A minimum NH₄⁺ concentration is set to allow nitrification to occur. The effect of NH₄⁺ on nitrification is very similar in DayCent and APSIM_max (scaled using the maximum value), but if the APSIM rate is scaled using V_max, the maximum value is 83% of what is given in DayCent.

Although the effect of NH₄⁺ concentration in DayCent and APSIM appear to be similar (Fig. 2a), there is a maximum fraction of NH₄⁺ nitrified in DayCent (Parton et al. 1996, 2001). The equations are (Parton et al. 1996):

where is the fraction of NH₄⁺ nitrified and is the ammonium concentration in mg kg⁻¹ soil. Eqn (1a) was revised to include net daily mineralisation (Eqn 1b; Parton et al. 2001).

R_{n_Max} is the maximum fraction of NH₄⁺ that goes to NO₃⁻˙ (gN m⁻² day⁻¹), N_min is the daily net N mineralisation from the soil organic matter decomposition, K₁ is the fraction of N_min that is assumed to be nitrified each day (K₁ = 0.2), is the model-derived soil ammonium concentration (gN m⁻²), K_max is the maximum fraction of NH₄⁺ nitrified (K_max = 0.10 day⁻¹), f_n(ST) is the effect of soil temperature on nitrification, f_n(SW) is the effect of soil water content and soil texture, and f_n(pH) is the effect of soil pH on nitrification. The absolute maximum rate (R_{p_n} = 0.4; it has a similar meaning to the potential nitrification rate used in other models) used for the comparison (Fig. 3), was taken from the nitrification code (SJ Del Grosso, pers. comm.).

Fig. 3.  Effect of WFPS (water filled pore space) and NH₄⁺-N on nitrification rate (mg N kg⁻¹ soil day⁻¹) from APSIM, DayCent, NOE, WNMM and FASSET. All models have been converted to the same unit using a soil depth of 0.15 m, and a bulk density of 1480 kg m⁻³. The WFPS scale only shows values between the lower limit (LL = 0.15) and drained upper limit (DUL = 0.31).

In NOE, the relationship between nitrification and NH₄⁺ follows a Langmuir isotherm-type equation (Fig. 2a) where NH₄⁺ only effects the nitrification rate at low concentrations and achieves 80% of the maximum rate at 10 mg NH₄⁺-N kg⁻¹ soil.

Whilst all the models recognise O₂ as a key controlling factor of nitrification, soil O₂ dynamics are difficult to measure and predict. Soil moisture content is often used as a proxy for O₂ status (Heinen 2006). DNDC simulates O₂ partial pressure, which is calculated based on O₂ diffusion and consumption rates in the soil and partitions the soil into aerobic and anaerobic fractions (Li et al. 1992, 2000; Li 2000). As stated previously, nitrification in DNDC is only allowed to occur in the aerobic fraction. APSIM and WNMM use volumetric water content, DayCent and NOE use WFPS, and FASSET uses soil water potential (ψ) to describe the impacts of O₂ on nitrification. For a given soil type, volumetric water content and soil water potential are related to WFPS, which enables the response to nitrification rate to changes in soil water conditions to be compared. For the comparison in this review, SW was converted to ψ using the equation in van Genuchten (1980) and the soil water characteristic, [θ(ψ)], measured on surface layer of a Solodic soil (Smith et al. 2020). The comparison revealed significant differences between the models in depicting the effect of SW on nitrification (Fig. 2b). The differences are reflected not only in the threshold values (i.e. the range of WFPS within which nitrification occurs), but also in the type of response within the ranges (Fig. 2b). The response to SW was the same in WNMM and APSIM until about 0.65 WFPS. The modifier increases around 0.4 WFPS and decreases around 0.8 WFPS. The response in FASSET, which uses water potential, was different. Decreasing the ψ from −310 MPa to −31 MPa (Petersen et al. 2005a) means that the modifier remains at 1 over a larger range of WFPS before declining to 0.6 as ψ approaches 0. As outlined in Cresswell (2002), the shape of the water characteristic curves varies with soil texture. The use of ψ requires knowledge of the θ(ψ) characteristics of the soil layers.

Whilst DNDC uses a simple equation to describe the effect of moisture when WFPS > 0.05, (and 0 when WFPS is <0.05), the water factor modifies the relative growth rate of the nitrifiers (Li 2000). In turn, the nitrification rate is a function of the net increase in nitrifier biomass. DNDC also allocates NH₄⁺ substrate between the aerobic and anaerobic soil fractions and only the substrate in the aerobic fraction is available for nitrification. The detailed equations for DNDC are given in the Appendix of Li (2000).

APSIM and WNMM have a narrow range for the effect of SW (and WPFS) on nitrification (Fig. 2b). Pan et al. (2021) reported that APSIM and WNMM estimated no nitrification when the WFPS was <0.3. However, their global analysis (Pan et al. 2022) indicates nitrification can occur when WFPS is <0.3. Based on studies in sandy soils of Western Australia, Asseng et al. (1998) reduced the lower limit for nitrification in APSIM to 5% of the potential rate at the lower limit of plant available water (LL). The default potential mineralisation rate and the production of NH₄⁺ substrate for nitrification in APSIM was developed for medium- and fine-textured soils and higher mineralisation rates are more appropriate for coarser-textured soils (Asseng et al. 1998; Mielenz et al. 2016a).

Large variations exist in the models in the values used to represent the dimensionless modifier of nitrification to T_s (Fig. 2c). There are contrasting temperature thresholds (minimum, optimum and maximum temperatures), and different response curves employed between models. For example, DayCent uses an optimal temperature for nitrification that is a function of the average maximum monthly air temperature for the warmest month of the year (Parton et al. 2001).

Both DNDC and DayCent assume nitrification occurs in pH range 0–14. DNDC increases the nitrification rate linearly with soil pH, while DayCent uses a logistic curve and simulates the maximum nitrification rate at pH 9.0 (Fig. 2d). Nitrification is assumed to only occur between pH values of 4–9 in APSIM, and 4.5 to 10 in WNMM, with both models using a trapezoid response curve (Fig. 2d). FASSET and NOE do not consider the effect of pH on nitrification. However, there is compelling evidence that under alkaline conditions, NO₂⁻ accumulates due to the inhibition of NO₂⁻ oxidising organisms and is a precursor to biogenic and abiotic formation of N₂O (Chalk and Smith 1983, 2020).

Of all the models reviewed, only DNDC considers the impact of soluble organic carbon on nitrification. It assumes that the population of nitrifiers (U_g) increases with dissolved organic carbon (C_DOC; kg C ha⁻¹) according to the equation:

where U_max is the maximum growth rate for nitrifiers (4.87 day⁻¹) and f_n(SW) is a moisture factor (Li et al. 2000). The relative growth rate is calculated with double-Monod kinetics, which has the same form as the Michaelis–Menten equation (Li et al. 1992, 2000).

Ammonium concentration and SW content are the two most important factors that influence the rate of nitrification. 3D response surfaces (Fig. 3) demonstrate the combined effects of these two factors on the daily nitrification rate in the different models with the WFPS scaled between the lower limit (LL; volumetric water content) and drained upper limit (DUL). The other rate modifiers (soil pH and T_s) were set to 1, equivalent to the most favourable conditions for nitrification.

Changes in NH₄⁺ concentrations and SW (and the derived WFPS) produce distinctly different daily nitrification rates for each of the models (Fig. 3). WNMM produces the highest nitrification rate, although the effect of SW on nitrification rate is similar to that observed in APSIM (Fig. 2b). At SW below LL, the water factor for nitrification is zero, and this increases linearly to 1 at a soil water content of LL + 0.25 × (DUL-LL). The lowest daily nitrification rates were predicted by the NOE and DayCent models. The nitrification rate in DayCent reached a maximum value at a NH₄⁺ concentration of 12 mg N kg⁻¹ soil but increased in response to increases in WFPS. In the NOE model, there is a large initial increase in the daily nitrification rate with increasing NH₄⁺-N concentration.

The regulating mechanisms for quantifying the fraction of N₂O produced in the nitrification process are important but poorly understood (Inatomi et al. 2019). Only a few studies have investigated the responses of to T_s and moisture content (Stark and Firestone 1996; Bateman and Baggs 2005). Farquharson (2016) found no strong relationship between the fraction of N₂O emitted from nitrification and environmental factors that included soil moisture. Between 0.03% and 1% of N was released as N₂O accompanying nitrification in soils. Inatomi et al. (2019) also found no consistent relationship between and environmental factors. When summarised, the results from these studies indicate ranged from 0.006 to 29.4% with a median value of 0.19%. After removing outliers from the data, the mean was 0.43% and the median was 0.14%.

The generalised equations used to predict N₂O production from nitrification (N₂O_n) are given in Eqns 5–7. The equations used in each model are given in Table S1.

where R_n and R_P__n are the actual and potential nitrification rates (kg N ha⁻¹ day⁻¹), respectively; and , f_n(SW), f_n(ST) and f_n(pH) are the rate modifiers for the effects of NH₄⁺ concentration, soil moisture, T_s and pH conditions on nitrification, respectively. N₂O_n is the N₂O flux produced from nitrification; K₁, is the potential fraction of nitrified N lost as N₂O flux, and ranges from 0.002 to 0.047 in different models (Table 1); and are the rate modifiers for the effects of soil moisture and T_s on K₁, respectively. Additionally, soil moisture and T_s influence N₂O production and the diffusion of N₂O to the atmosphere (Davidson and Swank 1986; Clough et al. 2005). In APSIM and DayCent, once the N₂O is produced, it is directly released to the atmosphere.

Although DNDC uses the same ‘leaky pipe’ concept, it is represented as (Li et al. 2000):

The equation to predict N₂O from nitrification reported in Li (2000) is:

where K₁ is 0.0006 (or 0.0024 in Eqn 8b), R_n is the nitrification rate, R_{p_n} is maximum nitrification rate (1/h), which is similar to the definition of potential nitrification rate used in other models, is the rate modifiers for the effects of NH₄⁺ concentration (kg N ha⁻¹) on nitrification and B_n is the biomass of nitrifiers (kg C ha⁻¹), is temperature factor and WFPS is water-filled porosity. The main difference between Eqn 7 and Eqn 4 is that the biomass of nitrifiers (B_n) is included in Eqn 7, which is driven by dissolved organic carbon, SW and T_s (Table S1). In summary, the proportion of K₁ is a constant in APSIM, and DayCent (Parton et al. 2001; Thorburn et al. 2010), whereas it is modified by changes in soil moisture content in NOE (Khalil et al. 2004), and changed with T_s and SW in DNDC, FASSET and WNMM (Li et al. 2000, 2007; Chatskikh et al. 2005).

The fraction of N₂O produced by nitrification increases linearly with WFPS in FASSET to a maximum of 1% (Fig. 4a). In WNMM, no N₂O is produced when WFPS is <0.25, a linear increase to 1% at a WFPS of 0.4, plateauing between 0.4 and 0.8 WFPS and then a linear decline to 0% at SW saturation. In NOE, the WFPS modifier increases slowly from 0.06 to 0.12 within the range of WFPS that nitrification is considered to occur (Fig. 4a).

Fig. 4.  The impacts of (a) WFPS and (b) soil temperature on the fraction of N₂O from nitrification in FASSET (black round-dotted line), NOE (blue dash line) and WNMM (black long dash-dot-dot line). WFPS was calculated from volumetric water content assuming a bulk density of 1480 kg m⁻³.

FASSET and WNMM both assume that there is an optimum T_s (35°C) for the fraction of N₂O emitted during nitrification (Fig. 4b). This fraction decreases when T_s is higher than the optimum temperature in FASSET, whereas in WNMM T_s above the optimum temperature has limited effect.

Except for DNDC, the models reviewed use the ‘leaky pipe’ concept to simulate denitrification and associated N₂O production (Fig. 1). The general equations for calculating daily denitrification rate and N₂O emission from denitrification are given in Eqns 9–11. The equations used in each model are given in Tables S2 and S3. Denitrification is calculated as a function of NO₃⁻, SW, T_s, pH, and soil organic carbon (SOC) (Eqn 9). The amount of N₂O produced from denitrification is a function of the total denitrification (N₂O plus N₂). The fraction of N₂O to total denitrification (Eqn 11) is positively related to SOC, SW, and T_s, and inversely related to NO₃⁻ concentration (Parton et al. 2001; Khalil et al. 2004; Chatskikh et al. 2005; Li et al. 2007). The algorithms to calculate N₂O from denitrification in APSIM came from DayCent (Thorburn et al. 2010). The models reviewed use different OC components to modify the denitrification rate and associated N₂O emissions. To develop a generic set of equations, we used OC as a surrogate for the different C pools in Eqn 9–11. The specific component of OC used is given in Tables S2 and S3.

where R_d and R_P__d are the actual and potential rates of denitrification (kg N ha⁻¹ day⁻¹); , f_(OC), f_{(SW_d)}, f_{(ST_d)} and f_{(pH_d)} are dimensionless rate modifiers for the effects of NO₃⁻, OC, SW, T_s and pH on denitrification, respectively. N₂O_d is the N₂O flux produced from denitrification (kg N ha⁻¹ day⁻¹); K₂ is the potential fraction of N₂O production to the total denitrification gases (NO, N₂O and N₂). K_con is the N₂O consumption rate during diffusion in the soil and is estimated from clay content and the depth at which the N₂O was produced in the soil profile. The terms , , and are rate modifiers for the effects of NO₃⁻ concentration, OC, SW and T_s on K₂.

In DNDC (Li 2000; Li et al. 2000), denitrification is simulated using a stepwise approach according to the biochemical process. The relative intensity of each step is related to the activity of the generic denitrifying microorganisms and is controlled by Eh, SW, T_s, pH, concentration of nitrogen oxides (NO₃⁻, NO₂⁻, NO and N₂O) and dissolved organic carbon (DOC). The general equation for denitrification (Eqn 9) can be modified to Eqn 12 to estimate the activity of all denitrifier groups. The stepwise equations are listed in Table S3. Denitrifier activity for each step is a function of the activity of all the denitrifying populations and is modified by soil moisture (f_{(SW_d)}), temperature (f_{(ST_d)}), pH (f_{(pH_d)}) and the substrates (DOC and NO₃⁻, or NO₂⁻, or NO, or N₂O; see Eqn 13). Net N₂O production (N₂O_{d_E}) is estimated as the difference in denitrifier activity between N₂O production (NO₂⁻ → N₂O and NO → N₂O) and N₂O reduction (N₂O → N₂), and N₂O consumption (reduction of N₂O to N₂) as the gas diffuses through the soil to the atmosphere (Eqn 14).

where G_d is the activity of all denitrifying microorganisms; G_P__d is the potential denitrifier activity; , f_(DOC), f_{(SW_d)}, f_{(ST_d)}, and f_{(pH_d)} are the dimensionless rate modifiers for the effects of nitrogen oxides (that includes NO₃⁻, NO₂⁻, NO, N₂O) concentrations, DOC, SW, T_s and pH on denitrifier activity, respectively; NO_{x_d} is the production of NO_x (NO₂⁻, NO, N₂O and N₂); K_x is the potential fraction of NO_x (NO₂⁻, NO, N₂O) production; The terms , , and are the rate modifiers based on the concentrations of the substrates (NO₃⁻, NO₂⁻, NO, N₂O and DOC), soil temperature and pH on G_d; K_com is the N₂O consumption rate during diffusion in the soil.

If other factors are non-limiting, the effect of NO₃⁻ concentration on denitrification, or denitrifier activity is handled differently in the various models. APSIM and WNMM assume that the denitrification rate increases linearly with NO₃⁻ concentration. In contrast, DayCent adopts a power function to describe the impact of NO₃⁻, and NOE and FASSET use a Langmuir-type equation (Fig. 5a; Tables S2 and S3).

Fig. 5.  The rate modifiers of denitrification rate to (a) nitrate concentration, (b) soil carbon and respiration, (c) soil volumetric water content (θ) and WFPS, (d) temperature, (e) pH and (f) clay content in APSIM (black solid line), DayCent (red square-dotted line), DNDC (dark red dash-dot line), FASSET (black round-dotted line), NOE (blue dash line) and WNMM (black long dash-dot-dot line). WFPS was calculated from soil volumetric water content assuming a bulk density of 1480 kg m⁻³.

Assuming other factors are non-limiting, the effect of SOC, or its surrogate (e.g. soil heterotrophic CO₂ respiration as used in DayCent) has either no impact (NOE) or a significant impact (APSIM, DayCent, DNDC, FASSET and WNMM) on denitrification. DNDC assumes that denitrifier activity continues to increase in response to increasing DOC. Denitrification potential (g N m⁻² day⁻¹) in FASSET is assumed to be proportional to the SOC mineralisation rate (g C m⁻² day⁻¹) and to clay content (Chatskikh et al. 2005). DayCent and WNMM assume that denitrification rate asymptotes to the maximum with increasing soil heterotrophic CO₂ respiration (DayCent) or organic carbon content (Fig. 5b). DayCent and FASSET assume a combined effect of NO₃⁻ and SOC on denitrification. Liebig’s Law of the Minimum is used in DayCent to describe the combined effect, thus, at any given time either NO₃⁻ or CO₂ limits the denitrification rate to a value permitted by the most limiting substrate (Parton et al. 1996; Del Grosso et al. 2000). APSIM uses active carbon defined by the organic C concentration in soil humus (HUM) and fresh organic (FOM) carbon pools (Thorburn et al. 2010).

Apart from DNDC, the models assume that there is a lower limit to WFPS at which denitrification occurs. The dimensionless SW modifier (f_{(SW_d)}) increases from 0 to 1 as SW or WFPS increase from the lower limit to the upper limit, following different curves (Fig. 5c). The lower and upper limits of WFPS vary among the models, reflecting slightly different representations of the same generalised equation. In DNDC, SW content determines the size of the anaerobic zone that controls the size of the denitrifier population, and this will modify the activity and capacity of each stage during the transformation of NO₃⁻ to N₂ (Li 2000). Denitrification was found to be very sensitive to WFPS or porosity at which denitrification commences in APSIM (dnit_lim; Mielenz et al. 2016a, 2016b). Mielenz et al. (2016a) hypothesised that the relationship between dnit_lim and WFPS_DUL is a soil specific variable that may be predicted from DUL.

Denitrification rate is modified by T_s in all the models except DayCent. Although DayCent assumes T_s has no direct effect on denitrification, T_s it does modify nitrification and NO₃⁻ production (Fig. 5d). The other models assume that denitrifier activity increases in response to T_s between a lower and upper limit. The lower limit, optimum T_s and the response function vary in each of the models.

Except for DNDC, all the models assume that the pH has no effect on denitrification. In DNDC, denitrification does not occur if the soil pH is <4.2 or >9.5 and occurs at the maximum rate when the soil pH is between 6.0 and 7.0 (Fig. 5e).

Soil texture (clay content) is directly considered in FASSET (Fig. 5f) whereas, APSIM, DayCent, DNDC and WNMM consider it to have an indirect effect. APSIM and DNDC consider soil texture effects denitrification through soil water retention curves. For example, WFPS at DUL is the threshold for denitrification in APSIM, and DUL varies with soil texture and depth. Mielenz et al. (2016a) reported variability in WFPS or porosity at which denitrification commences for different soils. Consistent with the suggestion of Mielenz et al. (2016a), DayCent and WNMM assume that denitrification response curves to SW change with texture. Three soil textural classes (coarse, medium, and fine) are used in DayCent to predict denitrification response to WFPS, whereas six soil textural classes are considered in WNMM.

Soil water content or WFPS may not adequately represent the O₂ partial pressure that regulates nitrifier and denitrifier activity in different soils or within soil aggregates (Li et al. 2000). Del Grosso et al. (2000) points out that WFPS interacts with soil physical properties and O₂ demand, to determine the O₂ status of the soil. Modellers have addressed this by using ‘soil texture’ to characterise SW or WFPS effect on denitrification. WNMM (Li et al. 2005) adopts six response curves, related to soil texture, to describe the effect of WFPS on denitrification rate. However, the ‘soil texture’ classes are not directly related to measurable soil texture. Experience is needed to judge the selection of the arbitrary ‘soil textural class’, which introduces uncertainty into N₂O emission predictions. Using measured soil particle size data would be an improvement on using the arbitrary textural classes but will require modellers to develop equations that use measured soil particle-size distribution to modify the denitrification–WFPS function.

The combined effect of WFPS and NO₃⁻ concentration on denitrification rate for the different models is shown in Fig. 6. The WFPS scale is limited to the range of values where denitrification occurs. The other rate modifiers (soil pH, and T_s) were set to 1, equivalent to the most favourable condition for denitrification. A potential denitrification rate (R_{p_d}, gN m⁻² day⁻¹) was required in order to make comparisons between FASSET and the other models (Hansen et al. 1991; Chatskikh et al. 2005; Petersen et al. 2005a, 2005b). This was estimated using the algorithm in FASSAT, which is proportional to mineralisation of organic C and depends on clay content (Eqn 15).

where a_d and b_d (a_d = 0.151 gN gC⁻¹, b_d = 0.015 gN gC⁻¹) are constants and CLAY is clay content (%) and N_min (g C m⁻² day⁻¹) is the soil organic matter mineralisation rate (Chatskikh et al. 2005). For the comparison in this study, the mineralisation rate was set to 30 kg C ha⁻¹ day⁻¹ (equivalent to 3 g C m⁻² day⁻¹), which is equivalent to the non-limiting soil respiration used in DayCent (Fig. 3, Parton et al. 1996). The actual denitrification rate (R_d, g N m⁻² day⁻¹) is the product of R_{p_d}, which is modified by the T_s function (f_{(ST_d)}), water filled porosity (F_Q(Qwfp)) and NO₃⁻ concentration (, mg N kg⁻¹). The equation (Chatskikh et al. 2005) is:

Fig. 6.  Effect of WFPS (water filled pore space) and NO₃⁻-N on denitrification (mg N kg⁻¹ soil day⁻¹) from APSIM, DayCent, NOE and FASSET. All models have been converted to the same unit using a soil depth of 0.2 m and a bulk density of 1480 kg m⁻³.

There was more than a 1000-fold difference in total denitrification rates between the models using the default parameters set for each model (Fig. 6). The values shown in Fig. 6 have been converted to the same soil depth (0.2 m) because the DayCent, NOE and WNMM models only calculate denitrification from this soil layer. The lowest rate was observed with DayCent, whilst the highest rate was with FASSET. The denitrification rate plateaued when the NO₃⁻ concentration was >200 mg N kg⁻¹ soil in FASSET and NOE (not shown). By contrast the denitrification rate continued to increase with NO₃⁻ in APSIM and DayCent (Figs 5, 6).

APSIM, DayCent, FASSET and WNMM assume the fraction of N₂O to the total denitrification fluxes (NO, N₂O and N₂) declines as WFPS increases (Fig. 7a). Values for the threshold and the optimal WPFS, similar to the denitrification rate modifier, vary amongst the models. The other two major differences with respect to WFPS are the minimum and maximum values of the dimensionless WFPS modifier, and the response functions (Fig. 7a).

Fig. 7.  The effect of (a) WFPS, (b) soil temperature, (c) NO₃⁻, and (d) active carbon or soil heterotrophic CO₂ respiration on the fraction of N₂O produced during denitrification in APSIM (black solid line), DayCent (red square-dotted line), DNDC (dark red dash-dot line), FASSET (black round-dotted line) and WNMM (black long dash-dot-dot line). WFPS is converted from soil volumetric water content assuming a bulk density of 1480 kg m⁻³. DayCent/APSIM solid line (part a, c and d) indicates that the two lines fall on top of each other.

The effect of soil temperature on the proportion of N₂O to the total denitrification fluxes (NO, N₂O and N₂) is only considered in DNDC and FASSET (Fig. 7b). DNDC assumes that microbe activity for NO₂ → N₂O, NO → N₂O and N₂O → N₂ increases with soil temperature when it is below the optimal value (22.5°C). The temperature factor is a standard exponential function equal to 1.0 at 22.5°C (Q₁₀ = 2; Li et al. 1996, 2000). The same temperature factor is applied to the activities of NO₃⁻, NO₂⁻ and N₂O denitrifiers. In addition, DOC and N oxides are the main substrates controlling the growth of denitrifier growth (Li et al. 2000). The fraction of N₂O produced during denitrification decreases with increasing temperature in FASSET (Fig. 7b; Fang et al. 2015).

The effect of soil pH on denitrification and N₂O production is only considered in DNDC through pH factor (F_PH2; Li et al. 2000), which modifies the relative growth rate of NO₂⁻ and NO denitrifiers, and F_PH3, which is a pH factor for N₂O denitrifiers (see table 4 in Li et al. 2000). The production and reduction of N₂O both increase with increasing soil pH (Fig. 8).

Fig. 8.  Effect of pH on the relative growth rate of total denitrifiers. F_PH1 = pH factor for NO₃⁻ on denitrifiers relative growth rate (solid line); F_PH2 = pH factor for nitrite (NO₂⁻) and NO on denitrifiers (dash line); F_PH3 = pH factor for N₂O on denitrifiers (dot line).

The direct impact of NO₃⁻ on the fraction of N₂O produced is considered in APSIM and DayCent (Fig. 7d). DNDC is not included in Fig. 7d because NO₃⁻ modifies the relative growth of denitrifiers and, through a series of interactions, modifies the N₂O fraction produced (see Appendix: equation and parameters; Li 2000). APSIM and DayCent assume that the proportion of N₂O increases as NO₃⁻ concentration increases from 0 to optimal and does not increase above the optimal NO₃⁻ concentration at a given soil heterotrophic respiration level (Del Grosso et al. 2000). In DNDC, N₂O and N₂ production increases in response to increases in substrate concentration following Monod type kinetics (Li et al. 1992) with the same value for the half-saturated nitrogen oxides (Fig. 8). Net N₂O production is estimated by the difference between N₂O produced from NO and NO₂⁻, and reduction of N₂O to N₂.

APSIM, DayCent and DNDC consider the effect of SOC on the relative N₂O production from denitrification (Fig. 7d). APSIM uses ‘active’ carbon which is a function of SOC (defined as the carbon concentration in the HUM and FOM soil C pools; Thorburn et al. 2010), DayCent and APSIM assume that the proportion of N₂O declines to a minimum as soil heterotrophic CO₂ respiration (which is a surrogate for labile carbon) increases from 0 to the optimal level for any given NO₃⁻ concentration. In DNDC, denitrifier activity for N₂O production and reduction is directly linked with DOC and follows double-Monod kinetics (nutrient-dependent Michaelis–Menten-type growth; Li et al. 1996). The proportion of N₂O emitted relative to total denitrification declined with increasing DOC, which is comparable to the assumption used in DayCent and APSIM. Increasing DOC allows the denitrifiers to complete the denitrification process to N₂ (Li et al. 1996).

Nitrous oxide consumption is only considered in DNDC, FASSET, WNMM and in the APSIM version modified by Xing et al. (2011) (Fig. 9). The NOE and DayCent models assume that there is no N₂O consumption, and once N₂O is produced it is released to the atmosphere, irrespective of depth in the soil profile. The N₂O consumption rate increases with clay content in DNDC and FASSET. The N₂O consumption algorithm in the modified APSIM (Xing et al. 2011) was adopted from FASSET. This algorithm assumes that N₂O consumption increases with soil depth and all N₂O produced below 0.55 m is reduced to N₂. The WNMM model has a similar assumption, but the depth limitation does not exist (Fig. 9b). DNDC assumes that N₂O produced in deeper soil layers diffuses into the aerobic and anaerobic zones in the above layer (Li et al. 2000). The fraction that diffused into the aerobic zone passes directly into the next aerobic or anaerobic zones or is emitted from the soil surface. The portion that diffuses into the anaerobic zone in the upper layers is reduced to N₂, depending on the substrate (DOC) and electron acceptors (NO, N₂O) that control denitrifier growth (Li et al. 2000). The proportion of N₂O produced that diffuses into the aerobic and anaerobic zones in the upper soil layers depends on the fraction of aerobic and anaerobic zones in the soil layer, which is controlled by the soil O₂ condition.

Fig. 9.  N₂O consumption rate (%) in response to (a) clay content and (b) soil depth in APSIM (solid line) modified by Xing et al. (2011), FASSET (round-dot line), DNDC (dash-dot line) and WNMM (long dash-dot-dot line). FASSET and APSIM are on the same line.

Although N₂O diffusion and consumption are both known to be influenced by soil variables (Li et al. 2000; Warneke et al. 2015), the two major reasons limiting the incorporation of the N₂O consumption and diffusion processes into models are: (1) the lack of data on N₂O movement through soil profiles; and (2) the calibration and validation of these processes are limited, because only net fluxes to the air are reported from the soil surface. Moldrup et al. (2000) established a predictive relationship between gaseous diffusion coefficient and air-filled porosity which is an improvement on the widely used soil type-independent gas diffusivity models. Whilst consumption terms can be approximated, accurate measurements are needed to improve the modelling of N₂O consumption, net N₂O emission and N balance in the soil profiles.

While there are differences in the algorithms used to predict N₂O emissions from nitrification and denitrification in the models, conceptually the models use a similar approach. For example, different kinetics are used to represent the biogeochemical processes. These differences arise from the availability of data on gaseous (NO_x, N₂O, N₂) production and emissions from the nitrification and denitrification pathways and the simplified representations of complex mechanisms needed to describe the impact of soil type, climate, land use, and vegetation. Although N₂ is a major product of denitrification, modelling studies rarely evaluate simulated N₂ emissions because few observations are available (see Del Grosso et al. 2020). Measuring N₂, the final product of denitrification, remains an analytical challenge and as is the partitioning of N₂O and N₂ ratio. Progress in this area will be most effective if experimental scientists and modellers work together to design and implement the appropriate experiments to cover the diverse range of soil and climatic conditions. There are a larger number of historical laboratory and field experiments that use inhibitor (e.g. acetylene), which have proven to be unreliable.

The use of ¹⁵N isotopes and a stable isotope model (e.g. SIMONE), are promising tools to obtain information on rates of nitrification and denitrification and address weaknesses in N cycling of ecosystem models (Denk et al. 2019). The differences in the modelling approaches demonstrate: (1) the importance of calibrating and validating models before they are used to predict N₂O emissions and analyse responses to environment and management at any scale; and (2) the need for detailed data sets, including diffusion and consumption that can be used to calibrate and validate the different processes to reduce uncertainty. The scarcity of accessible measurements as compared with the extent of simulated processes is a general problem for the calibration and validation of complex biogeochemical models (Del Grosso et al. 2020). There is renewed interest in using helium/O₂ chambers to measure N₂, in addition to N₂O. Although the technique has the potential to improve the quantitative understanding of denitrification, the methodology needs to ensure: (1) stringent quality control to ensure there are no air leaks; (2) an N₂-free soil atmosphere is established; and (3) flushing with helium/O₂ does not modify the anaerobic zones or micropores within the soil column, before reliable denitrification data sets can be obtained (Friedl et al. 2020).

A potential improvement is to quantify nitrification and denitrification responses to O₂ pressure in the soil atmosphere, analogous to that used in DNDC (Li et al. 1992, 2000). Adopting this approach will require prediction capability for soil O₂ concentration to be incorporated into the models (e.g. Cook 1995) and measurement of O₂ concentrations in the soil profile using the techniques given by Hanslin et al. (2005). This will increase the model complexity and require more parameters, but these are measurable and physically based.

Previous laboratory and field studies focused on one or a limited number of variables that influence N₂O production and ignore the other controlling factors. For example, soil texture is not always reported, or disturbed soil samples are used in most experiments making it difficult to relate the finding to field conditions. This leads to difficulties in quantifying the reasons why response curves are different when the same treatments are used in the different experiments. Several processes that control N₂O emissions are missing from the models (e.g. chemodenitrification, the two steps in nitrification, and heterotrophic nitrification of organic N), although the literature shows that they are important and contribute to total N₂O emissions. These processes and their response to environmental factors need to be quantified and understood before it is possible to incorporate them into models. Measurements that apportion N₂O to the different biogeochemical process are also needed, rather than measurements of net emissions at the soil-air surface.

Finally, future model development needs to balance complexity and precision. Models may potentially increase in response to process understanding but the number of parameters and how they are estimated must be rigorously evaluated to reduce model uncertainty.

Supplementary material is available online.

Data sharing is not applicable as no new data generated or analysed during this study.

The authors declare no conflicts of interest.

This study was supported in part by funding from the Department of Agriculture and the Grains Research and Development Corporation, Australia.