Conception and parameterization of field-scale models for simulating ammonia loss from fertilized lands: a review

Abstract

Ammonia is a crucial component in the biogeochemical cycle of nitrogen, with various harmful environmental effects. The primary source of NH₃ is agriculture, particularly the application of fertilizers in crop cultivation. A significant portion of the nitrogen content from fertilizers, when applied without utilization, is released into the environment, becoming a source of loss and pollution. Emissions occur both from the soil and through stomata. However, if the compensation point concentration of the apoplast is lower than the nearby concentration of NH₃, stomatal absorption occurs. Additionally, cuticular deposition processes and bidirectional exchange of droplets on foliage (rain, dew, guttation) contribute to the ammonia cycle within the canopy. Depending on the conditions, a considerable amount of the ammonia emitted by the soil can be recaptured by the canopy. This recapture helps reduce both nitrogen loss from fertilizers and environmental pollution. This article presents a general review of models simulating the bi-directional exchange of ammonia in the soil—plant—atmosphere system, focusing on determining ammonia loss and amounts recycled by the canopy. The review covers concepts and parameterization of various model inputs.

Introduction

The accumulation of nitrogen compounds in Earth’s ecosystems is one of the most serious environmental problems today. In the biogeochemical cycle of nitrogen, called the nitrogen cascade (Galloway et al. 2003), reactive nitrogen (N_r) has a variety of negative consequences in different forms and at different timescales. N_r harms human health, visibility (PM_2.5 particles), and ecosystems (biodiversity, eutrophication). It also takes part in the decomposition of the stratospheric ozone layer (as NO from N₂O), has a greenhouse effect (N₂O), is responsible for soil acidification, and increases nitrate levels in groundwater. Since the beginning of the twentieth century, more and more inert atmospheric nitrogen (N₂) has been converted into ammonia with the Haber–Bosch synthesis, primarily for fertilizer production. Currently, ammonia synthesis capacity has reached 236 million tonnes per annum (Statista 2023). After industrial use and transformation (e.g., into ammonium, nitrate, or urea), all the converted nitrogen enters the biosphere and begins its biogeochemical cycle. Only a negligible part of N_r is converted back into inert di-nitrogen by denitrification processes. Among N_r, ammonia (NH₃) plays an important role in the N-cycle as the main N-supplier to ecosystems. The main sources of ammonia are animal husbandry, manure treatment, and crop production.

According to Xu et al. (2018, 2019), the global ammonia emission from fertilizers has increased from 1.9 ± 0.03 Tg N yr^–1 to 16.7 ± 0.5 Tg N yr^–1 between 1961 and 2010. Considering the trends in the rate of ammonia synthesis and fertilizer use, as well as the impact of global climate change on the increase in ammonia emissions, global emissions may double by the end of the century (Sutton et al. 2013). For this reason, it is crucial to increase the efficiency of fertilizers used.

There are two main sources of ammonia from the soil–plant-atmosphere system: volatilization from the ground and emission through the stomata of leaves. Both can be significant sources, and their ratio strongly depends on physical, chemical, and plant physiological conditions, especially on the application technology and the amount and chemical and physical form of fertilizers.

Synthesized nitrogen fertilizers are applied in a variety of forms, such as urea (in majority), ammonium nitrate, ammonium sulphate, calcium ammonium nitrate, urea ammonium nitrate (UAN), ammonium hydrogen carbonate, di-ammonium phosphate (DAP), or monoammonium phosphate (MAP) (Amhamed et al. 2022). After the application of N fertilizers, one part of ammonium is taken up by vegetation by roots or oxidizes by bacterial nitrification, and the other part is emitted into the atmosphere.

The emission of ammonia from the soil layer reaches its maximum rate following the application of fertilizers, especially in the case of urea (Génermont and Celier 1997; Søgaard et al. 2002; Meyers et al. 2006; Sintermann et al. 2012; Flechard et al. 2015). According to estimates by Baligar and Bennett (1986a, b); Baligar et al. (2001); Linquist et al. (2012); Coskun et al. (2017); Bindraban et al. (2020); Dimkpa et al. (2020) the overall efficiency of applied nitrogen (N) fertilizer is less than 50% on global level. The volatilization rate, and consequently the emission factor of ammonia, strongly depends on soil characteristics, climate circumstances, and management practices (Sommer et al. 2004). Fu et al. (2015) reported a dependency of ammonia volatilization rate on the timing and amount of fertilization, as well as local circumstances such as temperature and soil acidity (Roelle and Aneja 2002; Corstanje et al. 2008).

Nitrogen loss from fertilizers occurs partly due to soil gas emissions (NH₃, NO, N₂O, N₂) and partly due to nitrate leaching in the soil. There is significant variability among crop-specific gaseous emission factors, particularly based on region, fertilizer type, and application technology. For synthetic N fertilizer, the NH₃ emission factor is highest in southeastern Asia (19.5%) and lowest in Europe (6%). This discrepancy may be linked to regional differences in N fertilizer use and management practices in agricultural production (Ma et al. 2021). Zhan et al. (2020) estimated a globally averaged volatilization rate of 12.6% ± 2.1%, consistent with previous data-driven studies (13.3% ± 3.1%). However, this result is approximately one-quarter lower than process-based models (16.5% ± 3.1%). Regarding N₂O losses after synthetic fertilizer application, reported values for the emission factor vary widely, ranging from 0.03 to 12.9, as documented by Walling and Vaneeckhaute (2020). Moreover, the relationship between the fertilizer dose and the emitted ammonia is not linear. Jiang et al. (2017) showed with a meta-analysis of 70 experiments that the emission varies exponentially rather than linearly with the dose.

Nelson et al. (2017) quantified a significant NH₃ loss from fertilizer above an intensively managed maize field. They found that volatilization of ammonia accounted for 79% of the total nitrogen loss during the entire vegetation period, particularly in the first three weeks after fertilization. The rate of volatilization is influenced by the type of fertilizer used. He et al. (1999) discovered that the potential maximum NH₃ volatilization, as predicted by the Langmuir equation under experimental conditions, decreased in the following order: NH₄HCO₃ (23.2% of applied NH₄-N) > (NH₄)₂SO₄ (21.7%) > CO(NH₂)₂ (21.4%) > NH₄NO₃ (17.6%). Moreover, an increase in NH₄ application rate led to a significant increase in NH₃ volatilization for (NH₄)₂SO₄, CO(NH₂)₂, and NH₄HCO₃ but a decrease for NH₄NO₃.

The use of manure represents another significant source of ammonia loss. On a European scale, the application of organic fertilizers is estimated to contribute to 30–40% of total NH₃ emissions (Sintermann et al. 2012). Soil emissions typically experience an increase by one or more orders of magnitude following slurry application (Flechard et al. 2010).

Mitigation measures offer an opportunity to increase nitrogen use efficiency while maintaining crop yields, as demonstrated by Sanz-Cobena et al. (2014). According to a comprehensive synthesis by Pan et al. (2016), based on 824 observations investigating the impacts of farming practices (irrigation, residue retention, amendments), and enhanced efficiency fertilizers (fertilizers with urease inhibitors, nitrification inhibitors, or controlled release coatings), they found that globally, an average of 18% of applied nitrogen was lost as NH₃. The use of non-urea-based fertilizers, deep placement of fertilizers, irrigation, and mixing with amendments (pyrite, zeolite, and organic acids) significantly decreased NH₃ volatilization by 75%, 55%, 35%, and 35%, respectively. Furthermore, urease inhibitors and controlled-release fertilizers decreased NH₃ volatilization by 54% and 68%, respectively. These results highlight that ammonia volatilization represents a substantial nitrogen loss from agricultural systems, and this loss can be mitigated through the adoption of appropriate fertilizer products and/or improved management practices.

In recent decades, intensive measurement and modelling projects have been launched to measure and model the atmosphere–surface exchange of ammonia in croplands. Various field-scale, soil–plant-atmosphere, bidirectional exchange models have been designed and applied, as detailed in Section 2. Measurements were carried out over non-fertilized managed fields (e.g., Sutton et al. 1993, 1995, 1998, 2000, 2009a; Flechard and Fowler 1998; Nemitz et al. 2000a, b; Milford et al. 2001a, b; Wichink Kruit et al. 2007; Whitehead et al. 2008; Loubet et al. 2012) and after the application of fertilizers (Sutton et al. 1993; Milford et al. 2001a; 2009; Bash et al. 2010; Flechard et al 2010).

Several review articles discuss different model conceptions, including those by Zhang et al. (2010); Wen et al. (2014) with some focusing on conceptual aspects (Massad and Loubet 2015) and parameterization affairs (Flechard et al. 2013, 2015).

Despite the progress, many open questions remain regarding the study of ammonia exchange in fertilized crops. To what extent do stomata and soil emissions contribute to these emissions? What is the role of absorption on the wet leaf surface in the bi-directional exchange processes? Is a specific model suitable for modelling dry–wet and day-night conditions, or should a different model concept be used for these different conditions? What is the amount of ammonia released by the soil and absorbed by the foliage? Is the parameterization of input parameters appropriate?

This review represents a step forward in different ways. Firstly, it places special focus on modelling ammonia release resulting from fertilization, in contrast to broader-topic reviews. Secondly, it seeks connections between different models and proposes the application of varying model concepts based on changing meteorological conditions. Lastly, this study compiles parameterizations for model input data found in the literature, making it a unique initiative.

The purpose of this review article is to describe the existing field-scale models and examine their application possibilities in different environmental conditions, especially regarding the extent of fertilizer loss and loss retention (recapture). In the following section the model concepts are outlined followed by a detailed overview of parameterization of input parameters.

Model conceptions

Conceptual principles

Model structures are based on the electrotechnical analogy (Fig. 1). Fluxes are defined as the ratio of a concentration difference between two points and a resistance term, expressed as F =—Δχ/R. In the bi-directional exchange models, upward fluxes are positive, and vice versa. Within the soil–plant-atmosphere system, various pathways exist for ammonia exchange. The leaf apoplast, via stomata, as well as the liquid phase on the leaf surface (such as drops, dew, or guttation, if present), determine compensation point concentrations denoted by χ_s and χ_d. The compensation point concentration is determined by the emission potential, which is governed by the ratio of ammonium and hydrogen ion concentrations, denoted as Г = [NH₄⁺]/[H⁺], in the liquid phase. This applies to various contexts such as the leaf apoplast, wet leaf surface, and soil moisture. In the absence of a liquid phase on the leaf surface, uni-directional dry deposition occurs on the wax layer of the cuticle. These three exchange pathways are characterized by resistance terms R_s and R_d, which ultimately determine the canopy compensation point concentration, χ_c while R_w controls the uni-direction adsorption on leaves. During the exchange processes, the resistance of the quasi-laminar boundary layer is represented by R_b. Finally, the direction and magnitude of the flux are influenced by the difference between the concentration of ammonia of the ambient air (χ_a) and the concentration between the aerodynamic layer and the boundary layer (χ_z0). The resistance of the turbulent layer is characterized by the term R_a. Additionally, since soil can act as an important sink or source, it also plays a role in the bi-directional exchange of ammonia. The flux between the air and soil traverses the soil boundary layer and the in-canopy turbulent layer (represented by R_bg and R_ac), contributing to the overall concentrations χ_z0 or χ_c as well.

Fig. 1

Some possible pathways for ammonia flux in soil/plant/air systems in bidirectional exchange models. Pathways: black: all cases; brown: soil exchange; purple: recapture of soil emission by foliage allowed; red: foliage recapture excluded; blue: exchange with wet leaves; yellow: adsorption on dry cuticula; green: stomatal exchange. Concentrations: χ_g, χ_d, χ_s, χ_c: soil/litter, wet leaf, stomatal/mesophilic, canopy compensation point potentials, χ_a, χ_z0: concentrations of the ambient air and at z₀ height, respectively. Resistances: R_bg, R_ac: quasi-laminar and turbulent layer resistances below the canopy and R_b, R_a: above the canopy; R_d, R_w, R_s: resistances for dynamic ammonia exchange, cuticular deposition, stomatal/mesophilic exchange, respectively. Fluxes: (F = Δχ/R) marked by F, accordingly

Ecosystem modelling presents several challenges. On one hand, it is crucial to determine the quantity of soil and plant exchange process separately. On the other hand, understanding their contribution to various emission pathways is equally important. The literature offers a diverse range of models, which can be categorized based on the review by Butterbach-Bahl et al. (2013) into three groups: simplified, conceptual, and complex models. Advantages of simple models include low computational demands, minimal input requirements, and easy validation. However, they have limitations in their application to multi-ecosystems due to their low temporal resolution. Complex models, on the other hand, offer high temporal resolution and are applicable in climate change scenarios. However, they come with high computational demands and require extensive input.

Regarding model conceptions and parameterizations, several challenging factors persist, although some have been partially resolved. For instance, soil emission estimation is simplified by considering ammonium concentration and acidity in soil, independent of the entire nitrogen cycle in the soil. This approach was discussed by Grosz et al. (2023) in the context of modelling soil nitrogen effluxes. The role of litter in the emission process is critical as well, especially after fertilization. In some cases, litter emissions may even surpass those from the soil itself, as demonstrated by Sutton et al. (2009b).

According to Lathière et al. (2015), some knowledge has been acquired regarding the chemistry effects on NH₃ deposition. These effects include SO₂-NH₃ co-deposition and the leaf wetness effect, which involves aqueous chemistry. However, only a few models incorporate leaf surface or in-canopy chemistry effects.

Another frequently neglected factor is the air column chemistry within and above the canopy. The reversible thermodynamic equilibria of ammonium salts and ammonia, as well as nitric acid and hydrochloric acid gases, significantly impact the surface-atmosphere exchange rates of ammonia (as discussed by Brost et al. 1988; Nemitz et al. 2004a, b; Nemitz et al. 2009a).

Numerous other challenges exist in ecosystem models. As Massad and Loubet (2015) pointed out, the ideal surface-atmosphere NH₃ exchange model should dynamically address all ecosystem NH_x-related processes, fluxes, and pools. These include fertilizer volatilization and recapture, soil biogeochemistry, plant biochemistry and physiology, air and surface chemistry, and atmosphere exchange. All these factors should be considered within a multiple-layer canopy framework, which encompasses in-canopy profiles of turbulence, radiation, temperature, humidity, green vs. senescent leaves, and soil layers.

Evaluation of model conceptions

Big-leaf (single-layer) models simulate plant-atmosphere exchange involving stomatal and cuticular flux. This approach has been demonstrated by Sutton et al. (1995; 1998; 2000; 2001); Massad et al. (2010); Flechard et al. (2013); and Massad and Loubet (2015). Big-leaf approach does not distinguish multiple layers vertically in the canopy and does not simulate in-canopy turbulent transfer. In this model vegetation is assumed to behave as a single leaf. Ammonia exchange can be represented by the black, yellow, and green pathways in Fig. 1. Additionally, in Wichink Kruit et al. (2010), a single-layer model was coupled with a chemistry transport model.

The two-layer models, employed by Nemitz et al. (2000a, 2001); Sutton et al. (2000, 2001); Zhang et al. (2003, 2010); Burkhardt et al. (2009); Cooter et al. (2010); Massad et al. (2010); Wichink Kruit et al. (2010), Pleim and Ran (2011); Flechard et al. (2013), Pleim et al. (2013; 2019); Massad and Loubet (2015) involve soil/litter interactions alongside plant-air exchange as well (brown pathway in Fig. 1). Bidirectional ammonia exchange models can be coupled with agroecosystem models (Bash et al. 2013). More sophisticated process-based models involve the numerical simulation of chemical transformations in the soil -canopy-air exchange procedure (e.g., Flechard et al. 1999; Móring et al. 2016, 2017).

The three-layer models distinguish between foliage and siliques (Nemitz et al. 2000a) or upper and lower vegetative parts (Sutton et al. 2001).

Some of the models detailed in this paper are based on the bidirectional two-layer model of Nemitz et al. (2000a, 2001), see Fig. 2a and c, case 1. In this model flux is influenced by three key parameters χ_s, χ_g, and R_w. Beside bidirectional fluxes between stomata-air and soil-air, the uni-directional flux onto the thin liquid layer on the cuticula of leaves is represented by R_w. The Nemitz model can be applied during the daytime when photosynthetically active radiation I_p > 0 and leaf wetness LW < 1 (Fig. 2a, case 1). Night-time when stomatal uptake is practically zero the model of Fig. 2c, case 1 is applied. However, during wet conditions (i.e., high leaf wetness or in the presence of water drops on the leaf surface), LW ≈ 1, and the emission potential in the liquid phase (Γ = [NH₄⁺]/[H⁺]) determines a compensation point potential χ_d, making possible the re-absorption of ammonia. To parameterize this effect, Burkhardt et al. (2009) modified the Nemitz model, introducing a new term of χ_d (see Fig. 2b, d, case 1). When surface (soil, litter, fertilizer) emitted ammonia does not appear at z₀ height, the pathway of soil flux is connected to the canopy budget χ_c (Fig. 2, case 2).

Fig. 2

Various approaches to simulate ammonia fluxes in different conditions. a, c, case 1: soil emission is not connected to the canopy compensation point concentration (χ_c); (model of Nemitz et al. 2000a). a, c, case 2: soil emission contributes to the canopy compensation point concentration (χ_c); (model of Zhang et al. 2010). a, c, case 1 or 2: cuticular uptake is uni-directional (model of Nemitz et al. 2000a; and Zhang et al. 2010). b, d, case 1: bi-directional exchange is assumed between the wet leaves and the air (model of Burkhardt et al. 2009)

The uni-directional approach in dry cuticular exchange tends to underestimate fluxes due to the underestimation of the R_w parameter (Schrader et al. 2016). In the quasi-bidirectional nonstomatal compensation-point model proposed by Wichink Kruit et al. (2010) the compensation point concentration of cuticle (χ_w) was introduced. However, χ_w model exhibits a more complex response. It tends to underestimate R_w during warm conditions and moderately high ambient NH₃ concentrations, while overestimating R_w during colder conditions. As an interim solution for improving flux simulations, Schrader et al. (2016) recommend reducing the minimum allowed R_w and adjusting the temperature response parameter in the unidirectional model. Additionally, they suggest revisiting the temperature-dependent cuticular compensation point potential (Г_w) parameterization of the bidirectional model. Despite the efforts, the complexity of cuticular exchange still poses numerous unanswered questions that require further investigation.

Emission of ammonia after fertilization

The measurement and modelling of ammonia loss after application of N-fertilizers provide valuable information for budget calculations and mitigation purposes. Research questions have arisen from previous flux studies examining this process. Contradictory results exist in the literature regarding the amount of ammonia emission after fertilization, both from the soil and vegetation, and their ratio. Measuring and partitioning the strength of the two main emission sources is complicated, as it requires the simultaneous determination of soil and stomatal emissions separately. Since the surface emission potential (Γ_g) of fertilized soil is higher by orders of magnitude compared to that of leaf stomata (e.g., in Zhang et al. 2010; Walker et al. 2013), it seems evident that the soil source for ammonia is stronger, even when considering that stomatal and soil fluxes are represented by different conductivities (reciprocal resistances).

Soil vs. stomatal emission

As Loubet et al. (2012) state, it is difficult to determine whether the soil or the stomata are the main sources. Additional measurements of NH₄⁺ and pH of the soil surface, the leaves, and the leaf surfaces should be performed in the future to help partition NH₃ fluxes.

Milford et al. (2009) documented the diurnal pattern of ammonia emission following the fertilization of 108 kg N ha^–1 with CaNH₄(NO₃)₃, noting the highest fluxes during the day. Substantial emissions occurred only for two nights immediately after fertilizer application. However, nocturnal emissions ceased thereafter, reaching a minimum of 1.4 ng m^–2 s^–1 compared to the daytime maximum of 3821 ng m^–2 s^–1. They emphasized the necessity of process models and additional measurements, such as the soil chamber method, to interpret the contributions of different sources and sinks.

Sutton et al. (1998) concluded that over wheat, the main processes regulating exchange are the capture and release of NH₃ by plant cuticles, along with exchange with leaf tissues. Additionally, emission may occur from the soil surface after nitrogen fertilization. The relative magnitudes of different component fluxes on annual scales (e.g., stomatal emissions, cuticular desorption emissions, cuticular recapture of stomatal emissions) and the fraction of NH₃ deposited onto cuticles assimilated vs. emitted to the atmosphere remain open questions. For N-rich agricultural vegetation in summer, emissions are likely dominated by stomatal exchange, as well as by soil and litter emissions.

The ‘missing’ night emission

While high daytime emission rates are frequently observed, with a peak during the day, nighttime emissions are practically zero (Horváth et al. 2005; Whitehead et al. 2008; Milford et al. 2009; Sutton et al. 2009a). In the study by Horváth et al. (2005), high daytime ammonia emission was observed for two weeks after fertilization (100 kg N ha^–1). In contrast, emissions were practically zero at night, according to measurements and modelling. The substantial difference between measured daytime and nighttime emissions (262 and 9.1 ng m² s^–1, respectively) suggests that even if emissions were solely from the soil with a constant emission potential, fluxes would be smaller during the night and highest during the daytime due to limited turbulence at night (resulting in high R_a values). Whitehead et al. (2008) presented nighttime flux minima in two experiments conducted during the spring and autumn seasons. After the application of 70 and 150 k N ha⁻¹ fertilizer, high daytime peaks and practically zero nighttime flux were observed.

Although some researchers describe incompletely closed stomata for many C3 and C4 plant species at night, very little is understood about this phenomenon (Caird et al. 2007). Generally, stomata are considered closed at night (when I_p = 0). In this condition, when canopy emission is practically zero (F_c ≈ 0), solely the soil/litter emission should have appeared in the nighttime ammonia flux, even though slightly lower emission is expected due to the decrease in temperature but most likely due to the limited turbulence at night-time stable stratification. There may be several reasons for the contradiction, i.e., for the lack of soil emission measured at night:

• i) Parameterization of soil emission potential: If observations do not align reasonably with soil emission potential modelling, it may be due to an overestimation of the soil emission potential. The ratio of ammonium/hydrogen ions (gamma), determining the compensation point potential of the soil, is derived from the bulk ammonium content of the soil. However, a portion of ammonium is adsorbed to the solid phase (Neftel et al. 1998), leading to a lower equilibrium concentration by Henry's law, which overestimates the soil emission potential and soil flux (χ_g, F_g in Fig. 1).

• ii) Parameterization of stomatal emission potential: In the absence of detailed bioassay measurements, stomatal emission potential is often empirically estimated using the bulk ammonium concentration of leaf tissue. This is a potential source of underestimation for stomatal compensation point concentration, stomata flux, and stomatal/soil flux ratio as well (χ_s, F_s in Fig. 1).

• iii) Chemistry of thin liquid phase (dew, guttation, or rain droplets) on cuticula: Soil-derived ammonia can be absorbed by water droplets (dew) on the surface of leaves. Wentworth et al. (2016) demonstrated, based on laboratory experiments, that dew on the surface of leaves can act as a night-time reservoir and morning source for atmospheric NH₃ (see F_d in Fig. 1). This was confirmed by the model of Burkhardt et al. (2009), focusing on the chemical interactions occurring on leaf surfaces, indicating the importance of chemical interactions on wet leaf surfaces as a potential ammonia sink at night. They suggested further work to develop their model approach to deal with desorption from concentrated solutions and dry surfaces. Neirynck and Ceulemans (2008) found that empirical descriptions for cuticular resistances adsorption in static models (see R_w, F_w in Fig. 1), developed as a function of micrometeorological conditions and co-deposition effects, cannot reproduce the measured fluxes. To parameterize the bidirectional flux of adsorbed ammonia on cuticula, Sutton et al. (1998); Nemitz et al. (2001); Cooter et al. (2010); Flechard et al. (2013) applied a capacitor term in their models. A better agreement with measurements was obtained using dynamic models, which simulate net emission during the daytime. Wichink Kruit et al. (2007) observed deposition fluxes in the evening, night, and early morning due to leaf surface wetness, while in the afternoon, emission fluxes were observed due to high canopy compensation points in the summer season. Their observations above non-fertilized managed grassland also demonstrate the role of a wet canopy in controlling even the direction and magnitude of fluxes.

• iv) Interaction among gas-phase NH₃, HNO₃, and particle NH₄NO₃ below the canopy: The interaction among gas-phase NH₃, HNO₃, and particle NH₄NO₃ below the canopy can modify ammonia fluxes. Measurements of surface exchange fluxes of ammonia, nitric acid, and ammonium nitrate are often strongly affected by phase changes between gaseous and particulate compounds of the triad, making measurements of the individual species necessary for a correct interpretation of the measured concentration differences (Wolff et al. 2010). Nemitz et al. (2004a, b, Nemitz et al. 2009a) also agree that gas-particle interactions below the canopy may influence the concentration regime through chemical and physical processes like the dissociation equilibrium of NH₄NO₃ ↔ NH₃ + HNO₃ particle/gas. As the low temperature and high relative humidity favour condensation of ammonium nitrate particles where nitric acid vapour concentration is comparable with that of ammonia.

• v) Recapture of soil derived ammonia by the foliage.

Foliar recapture of soil emission

The canopy can prohibit the exchange with the atmosphere. Consequently, the contribution of soil emissions to the atmospheric flux may be less significant than that of the stomatal and cuticular pathways. Surprisingly, this fact often goes unnoticed in most measurement observations after applying fertilizers. There is evidence that the net total emission (F_t) measured above the canopy layer at z₀ height is different compared to the sum of fluxes from the ground. Part of ammonia emitted from the soil and from granulates of fertilizer in soil or on the surface (F_g) and by soil-covered litter (F_l) can be adsorbed on the cuticle of leaves or recaptured by stomata; see the two different pathways for soil emission on Fig. 1. (Harper et al. 2000; Nemitz et al. 2000a; Meyers et al. 2006; Denmead et al. 2008; Nemitz et al. 2009b; Bash et al. 2010; Walker et al. 2013; Flechard et al. 2015; Schoninger et al. 2018). According to Flechard et al. (2015), in the case of agricultural plants, the soil/litter emission during the growing season is largely recovered by the leaves, either by stomatal uptake or uptake on a wet surface (Nemitz et al. 2000b; Meyers et al. 2006). In practice, the recoverable fraction of NH₃ varies greatly. Meyers et al. (2006) showed, by modelling air-surface exchange of ammonia, that nearly all the ammonia emitted from the soil was taken up by the canopy, with some additional removal of ammonia from the atmosphere. In-canopy source/sink modelling by Walker et al. (2013) showed that the canopy has a large capacity to recapture emissions from the soil during the day once the canopy has fully developed.

According to the inverse source/sink and resistance modelling by Walker et al. (2013), the canopy recaptured approximately 76% of soil emissions near peak LAI in a maize canopy. Stomatal uptake may account for 12–34% of the total uptake by foliage during the day, compared to 66–88% deposited onto the cuticle. Schoninger et al. (2018), using the ¹⁵N tracer technique, showed that 23–68% of applied urea was volatilized, and a significant amount, up to 15% of volatilized N was recaptured by corn plants. They verified a direct and positive relationship between the leaf area index and the percentage of NH₃ uptake. The complexity of exchange processes proved by Meyers et al. (2006) as air-surface exchange of NH₃ is complicated by the fact that both the vegetation and soil can act as a sink or source of NH₃. Emissions can occur at the soil surface and through leaf stomata while at the same time deposition to the plant is ongoing. In the absence of a strong soil surface emission, the soil does not play a significant role as either a source or sink of NH₃.

According to Bash et al. (2010), the fertilized soil surface acted as a source of NH₃ for one month following fertilization, while the canopy typically served as a net sink, with the lower canopy being a constant sink. The estimated rate for soil-emitted and recaptured ammonia was 70%. They concluded that the parametrization of within-canopy processes in air quality models is necessary. In-canopy source/sink estimates for stable night-time conditions indicate a net NH₃ deposition to the canopy.

Although measurements, modelling, and the exponential decrease of concentration above the soil suggest a strong emission from the surface during the daytime (unstable stratification), the measured near-zero flux above the canopy at night—due to limited turbulence—still raise questions about soil emission and the relationship between the flux-gradient and stability. Theobald et al. (2015) established that current in-canopy turbulence transport models are insufficient to properly explain net exchange processes. Many phenomena identified in the field cannot be simulated using existing simple parameterizations, and the importance of these phenomena for net exchange processes is also unclear.

All the findings and observations listed above underscore the fact that a significant part of the ammonia emitted by the surface (soil, fertilizer, litter) does not reach the boundary layer between the surface and the atmosphere in the case of mature vegetation after emission.

Parameterisations

Calculation of χ_c and χ_z0

The direction and magnitude of the bi-directional ammonia flux within the soil – plant – atmosphere system can be calculated using either the compensation point concentration over the canopy (denoted as χ_z0) or the canopy compensation point (denoted as χ_c) depending on the applied model. Knowledge of ambient ammonia concentration (χ_a) is essential for these calculations.

In Nemitz et al. (2000a) model (case 1 in Fig. 2a and c, or the yellow and red pathways in Fig. 1), the total (F_t) fluxes can be calculated as follows:

$$\begin{array}{cc}{F}_{{\text{t}}}=-\frac{{\chi }_{{\text{a}}}-{\chi }_{{\text{z}}0}}{{R}_{{\text{a}}}},& {F}_{{\text{c}}}=-\frac{{\chi }_{{\text{a}}}-{\chi }_{{\text{c}}}}{{R}_{{\text{a}}}{+R}_{{\text{b}}}}.\end{array}$$

(1)

The parameterization of R_a and R_b is described in subsections 3.2, 3.3, and 3.4. The calculation represents the case when ammonia recapture is not relevant. Furthermore, all resistance terms and compensation point concentrations χ_s, χ_g can be calculated, while χ_a is measured. The compensation point concentration at z₀ height is given by:

$${\chi }_{{\text{z}}0}={\chi }_{{\text{c}}}\cdot \left(1+\frac{{R}_{{\text{b}}}}{{R}_{{\text{s}}}}+\frac{{R}_{{\text{b}}}}{{R}_{{\text{w}}}}\right)-\frac{{\chi }_{{\text{s}}}{\cdot R}_{{\text{b}}}}{{R}_{{\text{s}}}},$$

(2)

and the expression for canopy compensation point concentration can be calculated by:

$${\chi }_{{\text{c}}}=\frac{{\chi }_{{\text{a}}}\cdot {\left({R}_{{\text{a}}}\cdot {R}_{{\text{b}}}\right)}^{-1}+{\chi }_{{\text{s}}}\cdot \left[{\left({R}_{{\text{a}}}\cdot {R}_{{\text{s}}}\right)}^{-1}+{\left({R}_{{\text{b}}}\cdot {R}_{{\text{s}}}\right)}^{-1}+{\left({R}_{{\text{g}}}\cdot {R}_{{\text{s}}}\right)}^{-1}\right]+{\chi }_{{\text{g}}}\cdot {\left({R}_{{\text{b}}}\cdot {R}_{{\text{g}}}\right)}^{-1}}{{\left({R}_{{\text{a}}}\cdot {R}_{{\text{b}}}\right)}^{-1}+{\left({R}_{{\text{a}}}\cdot {R}_{{\text{s}}}\right)}^{-1}+{\left({R}_{{\text{a}}}\cdot {R}_{{\text{w}}}\right)}^{-1}+{\left({R}_{{\text{b}}}\cdot {R}_{{\text{g}}}\right)}^{-1}+{\left({R}_{{\text{b}}}\cdot {R}_{{\text{s}}}\right)}^{-1}+{\left({R}_{{\text{b}}}\cdot {R}_{{\text{w}}}\right)}^{-1}+{\left({R}_{{\text{g}}}\cdot {R}_{{\text{s}}}\right)}^{-1}+{\left({R}_{{\text{g}}}\cdot {R}_{{\text{w}}}\right)}^{-1}},$$

(3)

where R_g = R_ac + R_bg. The parameterization of R_ac, R_bg, R_s, R_w, and χ_s, χ_g is detailed in subsections 3.5, 3.6, 3.7, 3.8, 3.9, and 3.10, respectively. If F_g is measured parallel with total flux F_t, and because of F_t = F_g + F_c, the difference between the two measured fluxes gives the canopy uptake or emission, F_c. Furthermore, as F_c and F_t can also be calculated from Eqs. (1) hence ${\chi }_{{\text{c}}}={-F}_{{\text{g}}}{\cdot R}_{{\text{b}}}+{\chi }_{{\text{a}}}+{F}_{{\text{t}}}\left({R}_{{\text{a}}}+{R}_{{\text{b}}}\right)$ the canopy compensation point concentration can directly be calculated during dry periods when R_w ≈ ∞ and checked against the χ_c calculated from bioassay measurements.

In Burkhardt et al. (2009), a new term R_d, appears (see in 3.8) representing the bidirectional flux of ammonia of the drops/liquid layer at cuticula (case 1 in Fig. 2b and d, or by the blue and red pathways in Fig. 1).

$${\chi }_{{\text{c}}}=\frac{{\chi }_{{\text{a}}}\cdot {\left({R}_{{\text{a}}}\cdot {R}_{{\text{b}}}\right)}^{-1}+{\chi }_{{\text{s}}}\cdot \left[{\left({R}_{{\text{a}}}\cdot {R}_{{\text{s}}}\right)}^{-1}+{\left({R}_{{\text{b}}}\cdot {R}_{{\text{s}}}\right)}^{-1}+{\left({R}_{{\text{g}}}\cdot {R}_{{\text{s}}}\right)}^{-1}\right]+{\chi }_{{\text{g}}}\cdot {\left({R}_{{\text{b}}}\cdot {R}_{{\text{g}}}\right)}^{-1}+{\chi }_{{\text{d}}}\cdot \left[{\left({R}_{{\text{a}}}\cdot {R}_{{\text{d}}}\right)}^{-1}+{\left({R}_{{\text{b}}}\cdot {R}_{{\text{d}}}\right)}^{-1}+{\left({R}_{{\text{g}}}\cdot {R}_{{\text{d}}}\right)}^{-1}\right]+{\chi }_{{\text{g}}}\cdot {\left({R}_{{\text{b}}}\cdot {R}_{{\text{g}}}\right)}^{-1}}{{\left({R}_{{\text{a}}}\cdot {R}_{{\text{b}}}\right)}^{-1}+{\left({R}_{{\text{a}}}\cdot {R}_{{\text{s}}}\right)}^{-1}+{\left({R}_{{\text{a}}}\cdot {R}_{{\text{d}}}\right)}^{-1}+{\left({R}_{{\text{b}}}\cdot {R}_{{\text{g}}}\right)}^{-1}+{\left({R}_{{\text{b}}}\cdot {R}_{{\text{s}}}\right)}^{-1}+{\left({R}_{{\text{b}}}\cdot {R}_{{\text{d}}}\right)}^{-1}+{\left({R}_{{\text{g}}}\cdot {R}_{{\text{s}}}\right)}^{-1}+{\left({R}_{{\text{g}}}\cdot {R}_{{\text{d}}}\right)}^{-1}}.$$

(4)

In the model where soil and canopy fluxes are not separated, we use the model described by Zhang et al. (2010) and applied in a transport model by Wen et al. (2014) (case 2 in Fig. 2a and c, or the yellow and purple pathways in Fig. 1):

$$\begin{array}{cc}{F}_{{\text{t}}}= -\frac{\left({\chi }_{{\text{a}}}-{\chi }_{{\text{c}}}\right)}{\left({R}_{{\text{a}}}+{R}_{{\text{b}}}\right)}& {\text{and}}\end{array}$$

(5)

$${\chi }_{{\text{c}}}= \frac{\frac{{\chi }_{{\text{a}}}}{{R}_{{\text{a}}}+{R}_{{\text{b}}} }+ \frac{{\chi }_{{\text{s}}}}{{R}_{{\text{s}}}} + \frac{{\chi }_{{\text{g}}}}{{R}_{{\text{g}}}}}{\frac{1}{{R}_{{\text{a}}}+{R}_{{\text{b}}}} + \frac{1}{{R}_{{\text{s}}}} + \frac{1}{{R}_{{\text{g}}}} + \frac{1}{{R}_{{\text{w}}}}}.$$

(6)

It should be noted that the three different model concepts (Burkhardt, Nemitz, and Zhang) may lead to different results, even if the parameterization is similar. For this reason, it can be useful to perform the simulation with all three models and validate it with the measured fluxes.

Resistance of the turbulent layer above the canopy R_a

The parameterisation of the resistance of the aerodynamic layer above the canopy as well as the resistance of the quasi-laminar layer both control the overall exchange of ammonia between the ambient air and the plant-soil system.

The calculations are based on the Monin–Obukhov similarity theory. However, different approaches exist in the literature. The basic equations and the general form of the universal functions are detailed in Appendix 1. It should be noted that there are many universal functions in the literature (Weidinger et al. 2000; Foken 2017). The most frequently used and indirectly verified formulae are presented here.

According to Garland (1977); Monteith and Unsworth (1990); Flechard and Fowler (1998); Flechard et al. (2010), the resistance of the turbulent layer above the canopy is as follows:

$${R}_{{\text{a}}({\text{z}}-{\text{d}})}=\frac{1}{k{u}_{*}}\left[{\text{ln}}\left(\frac{z-d}{{z}_{0}}\right)-{\widehat{\psi }}_{{\text{H}}}\left(\frac{z-d}{L}\right)+{\widehat{\psi }}_{{\text{H}}}\left(\frac{{z}_{0}}{L}\right)\right],$$

(7)

while according to Massad et al. (2010):

$${R}_{{\text{a}}({\text{z}}-{\text{d}})}=\frac{{u}_{({\text{z}}-{\text{d}})}}{{u}_{*}^{2}}-\frac{{\widehat{\psi }}_{{\text{H}}}\left(\frac{z-d}{L}\right)-{\widehat{\psi }}_{{\text{M}}}\cdot \left(\frac{z-d}{L}\right)}{k\cdot {u}_{*}},$$

(8)

where ${\widehat{\psi }}_{{\text{H}}}$ and ${\widehat{\psi }}_{{\text{M}}}$ are integrated stability functions for momentum and sensible heat flux, supposing that the roughness length for these parameters is equal. The universal functions were selected according to Monteith and Unsworth (1990).

The definition for aerodynamic resistance of Sommer et al. (2004) is:

$${R}_{{\text{a}}}=\frac{{\text{ln}}\left[(z-d)/{z}_{0}\right]-{\widehat{\psi }}_{{\text{H}}}\left(\frac{z-d}{L}\right)}{k\cdot {u}_{*}},$$

(9)

where u_* the friction velocity can be derived from:

$${u}_{*}=\frac{k\cdot u}{{\text{ln}}\left[(z-d)/{z}_{0}\right]-{\widehat{\psi }}_{{\text{M}}}\left(\frac{z-d}{L}\right)},$$

(10)

where ${\widehat{\psi }}_{{\text{M}}}$ and ${\widehat{\psi }}_{{\text{H}}}$ are integrated stability functions, z₀ ≈ 0.1· h_c, d = 0.6· h_c. In neutral conditions: ${\widehat{\psi }}_{{\text{M}}}={\widehat{\psi }}_{{\text{H}}}=0$, in stable cases: ${\widehat{\psi }}_{{\text{M}}}={\widehat{\psi }}_{{\text{H}}}=-5\cdot \left(z-d\right)/L$, and for unstable stratifications: ${\widehat{\psi }}_{{\text{M}}}=2{\text{ln}}\left[\left(1+x\right)/2\right]+{\text{ln}}\left[\left(1+{x}^{2}\right)/2\right]-2 {{\text{tg}}}^{-1}x+\pi /2$, and ${\widehat{\psi }}_{{\text{H}}}=2{\text{ln}}\left[\left(1+x\right)/2\right]$, $x={\left(1-16\left(z-d\right)/L\right)}^{1/4}$ based on Dyer and Hicks (1970), where L is the Monin–Obukhov length:

$$L= -\frac{{u}_{*}^{3}\cdot T\cdot \rho \cdot {c}_{p}}{k\cdot g\cdot H}.$$

(11)

Personne et al. (2009) and Simpson et al. (2012) gave similar definition with the same universal functions as used by Sommer et al. (2004).

All the parameterizations presented here are based on the classical Monin–Obukhov similarity theory. There may be variations in the choice of the universal function. During application, we recommend considering site-specific optimization. Equation (8) includes the difference between integrated universal functions for sensible heat ${\widehat{\psi }}_{{\text{H}}}\left(\frac{z-d}{L}\right)$ and momentum ${\widehat{\psi }}_{{\text{M}}}\left(\frac{z-d}{L}\right)$ transport, but the author did not analyze its physical background. For calculations, we suggest using the classical Eq. (7).

Resistance of the quasi-laminar layer above the canopy R_b

Similar to R_a, there are several parameterizations for the quasi-laminar layer resistance (R_b) found in the literature.

According to Monteith and Unsworth (1990) and Garland (1977):

$${R}_{{\text{b}}}=\frac{1.45\cdot {\left(\frac{{z}_{0}{\cdot u}_{*}}{\nu }\right)}^{0.24}\cdot {\left(\frac{\nu }{{D}_{{\text{NH}}3}}\right)}^{0.8}}{{u}_{*}},$$

(12)

or: R_b = 1.45∙Re_*^0.24 ∙ Sc^0.8∙u_*^–1, where Re_* the turbulent Reynolds-number: Re_* = z₀ ∙u_*∙ ν^–1, ν is the kinematic viscosity of air, Sc = ν_a∙ D_NH3^–1 is the Schmidt-number, D_NH3 is the molecular diffusion coefficient of ammonia 1.978∙10^–5 m² s^–1, (Massman 1998).

Other empirical parameterizations are given by Sommer et al. (2004) and Thom (1972): R_b = 6.2∙u_*^–0.67, and Personne et al. (2009):

$${R}_{{\text{b}}}= {\left(\frac{{D}_{{{\text{NH}}}_{3}}}{{D}_{{\text{w}}}}\right)}^{-2/3}\cdot \frac{{\alpha }_{{\text{u}}}}{2\cdot a\cdot LAI}\cdot {\left(\frac{{L}_{w}}{{u}_{{{\text{h}}}_{{\text{c}}}}}\right)}^\frac{1}{2}\cdot {\left(1-{{\text{e}}}^{-\frac{{\alpha }_{{\text{u}}}}{2}}\right)}^{-1},$$

(13)

where α = 0.01 s m^–1/2, L_W is the characteristic width of leaves (m), D_w is the diffusion coefficient of water. Choudhury and Montheith (1988) gave an expression for the calculation of α_u:

$${u}_{{\text{z}}}={u}_{{{\text{h}}}_{c}}\cdot {{\text{e}}}^{{\alpha }_{{\text{u}}}\left(\frac{z}{{h}_{{\text{c}}}}-1\right)},$$

(14)

where ${u}_{{\text{z}}}$ is the wind velocity at the given height (z).

The expression for R_b by Garland (1977) and Flechard and Fowler (1998) is given by: R_b = B·u_*^–1, where B the Stanton-number, a function of Schmidt-number and roughness Reynolds-number: ${Re}_{*}= {z}_{0} \cdot {u}_{*}\cdot {\nu }^{-1}$ (Garland 1977). Other parameterizations are given by Simpson et al. (2012):

$${R}_{{\text{b}}}=\frac{2}{k\cdot {u}_{*}}{\left(\frac{Sc}{Pr}\right)}^{2/3},$$

(15)

and by Pleim and Ran (2011):

$${R}_{{\text{b}}}=\frac{{B}^{-1}}{{k}_{*}}{\left(\frac{Sc}{Pr}\right)}^{2/3}.$$

(16)

The formulae for determining R_b vary in complexity. Since we are directly above the canopy surface, we can assume an indifferent near-layer stratification. The mechanical turbulence effect is characterized by the neutral layer wind profile (u_* friction velocity). The simplest approach is the classical (15) (16) parameterization. Apart from specifying u_*, it does not require any other input data. In the parametrization methods of Monteith and Unsworth (1990) and Garland (1977), the surface type is already included via the roughness height (z₀). If LAI values and the structure of leaf characteristics (the characteristic width of leaves, Lw) are available, we recommend using the vegetation-specific (13) parametrization.

Sum of aerodynamic and quasi laminar layer resistances R_a + R_b

A simple method exists for determining the sum of the resistances R_a + R_b from the horizontal wind velocity (u) and the friction velocity (u*) as proposed by Baldocchi and Meyers (1991) and Lamaud et al. (2002).

$${R}_{{\text{a}}}+{R}_{{\text{b}}}=\frac{u}{{u}_{* }^{2}}+\frac{2}{{\text{k}}{u}_{*}}{\left(\frac{Sc}{Pr}\right)}^\frac{2}{3},$$

(17)

where ${u}_{*}$ can be derived from momentum flux (τ) calculated using ultrasonic anemometer data:

$$\tau =\rho \cdot {u}_{*}^{2}=\sqrt{{(\overline{{u }{'}{w}{'}})}^{2}+{(\overline{{v }{'}{w}{'}})}^{2}},$$

(18)

where $\stackrel{-}{u{'}w{'}}$ and $\stackrel{-}{v{'}w{'}}$ denotes the covariances of the two horizontal (u, v) and the vertical (w) components of the wind speed and prime means the turbulent fluctuating component around the mean.

The calculation of the sum of aerodynamic and quasi-laminar layer resistances using Eq. (17) is a classical and straightforward parametrization that remains in use today for assessing boundary layer resistance to water vapor transfer. This approach incorporates the “omega” theory proposed by Jarvis and McNaughton (1986) (cited from Baldocchi and Meyers 1991). The advantage of this calculation lies in its simplicity compared to separate estimations. Additionally, stability is considered through the measurement of friction velocity. The characteristic width of leaves (LW) and the leaf area index (LAI) play a role in the R_b parameterizations used in contemporary modelling. Furthermore, the shape of universal functions is considered during the parameterization of R_a. The parameterizations for R_a and R_b in Sections 3.2 and 3.3 incorporate more information, resulting in more accurate results.

Resistance of the turbulent layer above soil, inside canopy R_ac

The resistance term R_g, represents the soil-atmosphere exchange and is the sum of the aerodynamic resistance (R_ac) and boundary layer resistance (R_bs) above the soil. They are indicated by the brown line in Fig. 1.

The calculation of R_ac the requires knowledge of the friction velocity and vegetation characteristics are. A simple method to calculate R_ac is used in the model of Zhang et al. (2002, 2003) and Wen et al. (2014), where the R_ac is given by:

$${R}_{{\text{ac}}}= \frac{{R}_{{\text{ac}},0}\cdot {LAI}^\frac{1}{4}}{{u}_{*}^{2}},$$

(19)

where R_ac,0 = 10–40 s m^–1 and 10–50 s m^–1 for crops and maize, respectively. R_ac,0 is the reference value for in-canopy aerodynamic resistance, which varies with the change in canopy structure at different times of the growing season. The minimum values, R_{ac,0 (min)}, correspond to the earlier growing periods for agricultural lands. The maximum values, R_{ac,0 (max)}, correspond to the full-leaf period for agricultural lands. A simple formula is given for R_ac,0 values for any day of the year based on minimum and maximum LAI values. This approach requires only the knowledge of LAI and u_*. However, the estimation of empirical constants carries some uncertainty.

Another approach applied in the model of Pleim et al. (2013) where R_ac calculated as suggested by Erisman et al. (1994):

$${R}_{{\text{ac}}}={\text{b}}\cdot LAI\cdot \frac{{h}_{{\text{c}}}}{{u}_{*}},$$

(20)

where b = 14 m^–1 is an empirical constant. This calculation method has the advantage of simplicity but it does not distinguish among different plant species.

Nemitz et al. (2000a) and Massad et al. (2010) suggest a more sophisticated approach for calculation of R_ac:

$${R}_{{\text{ac}}(d+{z}_{0})}={\alpha }_{\left(d+{z}_{0}\right)}{\cdot u}_{*}^{-1},$$

(21)

where α is a proportionality factor, z₀ = 0.13∙h_c, h_c is the canopy height, and α can be calculated by Shuttleworth and Wallace (1985):

$${\alpha }_{\left(d+{z}_{0}\right)}= \frac{{h}_{{\text{c}}}}{k\cdot {\text{n}}\cdot ({h}_{{\text{c}}}-d)}\cdot \left[{{\text{e}}}^{{\text{n}}}-{{\text{e}}}^{{\text{n}}\cdot \left(1-\frac{d+{z}_{0}}{{h}_{{\text{c}}}}\right)}\right],$$

(22)

where n is an exponential decay constant. Lafleur and Rouse (1990) proposed an expression for n as: n = 2.6 ∙ (LAI)^0.36, (3.62 > n > 1.87). Burkhardt et al. (2009) used same calculation for R_ac as:

$${R}_{{\text{ac}}(d+{z}_{0})}={\alpha }_{\left(d+{z}_{0}\right)}\cdot {u}_{*}^{-1}=\frac{40\cdot {h}_{{\text{c}}}}{0.45\cdot {u}_{*}},$$

(23)

where α = 40 was estimated for a canopy height of 0.45 m.

A more complicated calculation is provided by Nemitz et al. (2000a, 2001), based on Raupach (1989). In this calculation, resistance of the turbulent layer, between z₁ and z₂ height, below the canopy is the integral of the reciprocal of turbulent diffusion constant:

$$\begin{array}{ccc}{R}_{{\text{ac}}, ({{\text{z}}}_{1}{,\mathrm{ z}}_{2})}=\underset{{z}_{1}}{\overset{{z}_{2}}{\int }}{K}_{{\text{H}}}^{-1}{\text{d}}z,& {\text{and}}& {R}_{{\text{ac}}({\text{d}}+{z}_{0})}=\underset{0}{\overset{ {d+z}_{0}}{\int }}{K}_{{\text{H}}}^{-1} \left(z\right){\text{d}}z.\end{array}$$

(24)

For dense canopy d ≈ 0.7 ∙ h_c and z₀ = 0.13 ∙ h_c. K_H can be calculated from the standard deviation of vertical wind component (σ_w) and the Lagrangian integral timescale:

$${K}_{{\text{H}}}\left(z\right)={\tau }_{{\text{L}},{\text{z}}}\cdot {\sigma }_{w,z}^{2}.$$

(25)

Parameterization of τ_L is given by Raupach (1989):

$$\frac{{\tau }_{{\text{L}},{\text{z}}}{\cdot u}_{*}}{{h}_{{\text{c}}}}={\text{max}}\left\{0.3;\frac{k\cdot (z-d)}{1.25\cdot {h}_{{\text{c}}}}\right\},$$

(26)

where τ_L,z is the Lagrangian integral timescale at the height (z) above ground. Because of both σ_w, and τ_L are generally a function of u_*, R_ac can be derived by Eq. (21).

Resistance of the boundary layer above soil inside canopy R_bg

Similarly to R_ac, R_bg can be calculated from the friction velocity and plant characteristics. In the following three estimations the soil layer friction velocity (u_*g) is considered by different approaches: According to Personne et al. (2009) following Hicks et al. (1987):

$${R}_{{\text{bg}}}=\frac{2}{k\cdot {u}_{*{\text{g}}}}\cdot {\left(\frac{{Sc}_{{{\text{NH}}}_{3}}}{Pr}\right)}^\frac{2}{3},$$

(27)

where ${Sc}_{{{\text{NH}}}_{3}}$= ν/${D}_{{{\text{NH}}}_{3}}$, Pr = 0.72, the Prantdl number. The calculation for the friction velocity near the soil surface (u_*g) is provided by Loubet et al. (2006) and is given by the following equation:

$${u}_{*{\text{g}}}= {\left[{u}_{*}^{2}\cdot {{\text{e}}}^{1.2\cdot LAI\cdot \left(\frac{{z}_{0}}{{h}_{{\text{c}}}}-1\right)}\right]}^{1/2}.$$

(28)

In calculation by Pleim et al. (2013) the process is simpler and does not require knowledge of the canopy height:

$${R}_{{\text{bg}}}=\frac{Sc-{\text{ln}}\frac{{\delta }_{0}}{{z}_{{\text{r}}}}}{k\cdot {u}_{*{\text{g}}}}.$$

(29)

Soil layer friction velocity is defined as: u_*g = u_*∙e^–LAI, where z_r = 0.1 m is the top of the logarithmic wind profile layer, δ₀ the thickness of the laminar layer above ground:

$${\delta }_{0}=\frac{\nu }{k\cdot {u}_{*{\text{g}}}}.$$

(30)

Nemitz et al. (2000a, 2001) parameterized the R_bg following Schuepp (1977) with the consideration of a constant canopy height:

$${R}_{{\text{bg}}}^{-1}= \frac{{k\cdot u}_{*g}}{Sc-{\text{ln}}({\delta }_{0}/{z}_{1})},$$

(31)

where k is the von Kármán constant (0.41) and Sc the Schmitt number Sc = ν/${D}_{{{\text{NH}}}_{3}}$, δ₀ the distance above ground where molecular diffusivity (Dχ) equals the eddy diffusivity (Dχ ≈ k⋅u_∗g⋅ δ₀), and z₁ is the upper height of the logarithmic wind profile. In this calculation, the Leaf Area Index (LAI) and the canopy height and (h_c)are not considered.

Stomatal resistance R_s

There are various approaches for calculation of the stomatal resistance (R_s is indicated by the green color in Fig. 1), such as using the evaporation term or the global or photosynthetically active radiations.

According to Flechard et al. (2010) the stomatal resistance to the transpiration of water vapour during daytime, for fertilized grass, when stomatal transpiration (E) is dominant, when cuticular, and soil evaporation are negligible, then R_s can be calculated from the water vapour deficit (D) at z₀ height and E (Thom 1975; Monteith and Unsworth 1990):

$${R}_{{\text{s}}, {{\text{H}}}_{2}{\text{O}}}=\frac{\rho \cdot \varepsilon \cdot D}{p\cdot E},$$

(32)

where: ρ is the density of the moist air, p is the pressure, ε is the molar ratio of water and air (18/29), D is the water vapour pressure deficit, and E is the stomatal transpiration. The resistance of water vapour and ammonia differs with their molecular diffusivities.

There are three calculations that use radiation data to calculate stomatal resistance. Nemitz et al. (2000a, 2001) calculated R_s as:

$${R}_{{\text{s}}}={\text{min}}[{R}_{{\text{s}},\mathrm{ max}};{R}_{{\text{s}},{\text{min}}}(1+ \frac{{\alpha }_{1}}{{R}_{{\text{g}}}})],$$

(33)

where R_{s max} = 5000 s m^–1, R_s,min = 35 s m^–1, α₁ = 180 W m^–2, R_g is the global radiation in W m^–2 and

$${R}_{{\text{s}},{\text{min}}}\cdot \frac{1+ \frac{b}{{I}_{p}}}{{f}_{e}{\times f}_{w}{\times f}_{T}{\times f}_{s}},$$

(34)

where R_s,min and b (W m^–2) species and LAI dependent constants, I_p the photosynthetically active radiation (W m^–2). The f factors, which represent the effects of humidity, dry stress and difference in diffusion, while f_s modifies the stomatal resistance for the differences in molecular diffusivity between the trace gas and water vapour. These factors are provided by Hicks et al. (1987).

Personne et al. (2009) calculated stomatal conductivity in flux dimension after Emberson et al. (2000):

$${g}_{{\text{s}}}=\frac{{D}_{{{\text{NH}}}_{3}}}{{D}_{{\text{w}}}}\left\lfloor{g}_{{\text{max}}}\cdot {\text{max}}\left({g}_{{\text{min}}};{g}_{{\text{t}}}\cdot {g}_{{\text{Ip}}}\cdot {g}_{{\text{D}}}\cdot {g}_{{\text{SWP}}}\right)\right\rfloor,$$

(35)

where g_max = 0.0115 m s^–1, g_min the observed daily minimum conductivities, further factors are representing the effect of other physical drivers as: g_Ip = 1 – e^–0.009×Ip (μmol m^–2 s^–1); g_D = 0 if D > 3 kPa, g_D = 1, if D < 1.3 kPa; g_D = – (D/1.7) + 1.76 kPa, when 1.3 kPa < D < 3 kPa; g_SWP = 1, when SWP > –0.49, g_SWP = 0 when SWP < –1.5, g_SWP = (SWP/1.01) + 1.49, if –1.5 < SWP < –0.49, where SWP the soil water potential (MPa), and ${g}_{{\text{s}},{\text{t}}}=1-{\left(\frac{{t}_{{\text{z}}0}^{\mathrm{^{'}}}-{t}_{{\text{opt}}}}{{t}_{{\text{opt}}}-{t}_{{\text{min}}}}\right)}^{2}$. The conductivity is at the maximum when t_opt = 26 °C, below t_min = 12 °C and above t_max = 40 °C is zero.

Zhang et al. (2003) described a parameterization for R_s:

$${R}_{{\text{s}}}=\frac{1}{{g}_{{\text{s}},{{\text{I}}}_{{\text{p}}}}\cdot {{\text{f}}}_{{\text{T}}}\cdot {{\text{f}}}_{{\text{D}}}\cdot {{\text{f}}}_{\uppsi }\cdot \frac{{D}_{{\text{i}}}}{{D}_{{\text{v}}}}},$$

(36)

where ${g}_{{\text{s}},{{\text{I}}}_{{\text{p}}}}$ a stress-free conductance in the function I_p. D_v and D_i are the molecular diffusivities for water vapour and the ammonia, respectively. Calculation of ${g}_{{\text{s}},{{\text{I}}}_{{\text{p}}}}$ is given by Zhang et al. (2002). The dimensionless functions f_T, f_D, and f_ψ represent the effect of different factors (Brook et al. 1999).

For special dry cases Nemitz et al. (2004a) provided an expression for dry conditions using water pressure and transpiration data:

$${R}_{{\text{s}}}=\frac{{D}_{{{\text{H}}}_{2}{\text{O}}}}{{D}_{{{\text{NH}}}_{3}}}{R}_{{\text{s}},{{\text{H}}}_{2}\mathrm{O }}=\frac{{D}_{{{\text{H}}}_{2}{\text{O}}}}{{D}_{{{\text{NH}}}_{3}}}\cdot \frac{{e}_{{\text{w}},{\text{z}}{0}{'}}-{e}_{{\text{sat}},\mathrm{ T},{\text{z}}{0}{'}}}{E},$$

(37)

where e_w,z0’ the water vapour pressure at z_0’heigt,${e}_{{\text{sat}},\mathrm{ T},{\text{z}}{0}{'}}$ the saturated water pressure at T temperature.

Cuticular resistance R_w and R_d

Cuticular resistance has been defined in various ways. R_w represents the uni-directional deposition of substances on leaves, as indicated by the yellow lines in Fig. 1). This concept focuses on the adsorption of substances onto the leaf surface and describes how the cuticle (the wax layer on the leaf surface) affects the exchange of materials between the leaf and air.

This parameterization poses challenges due to its complexity. In the context of R_w, it is assumed that the cuticular compensation point concentration (χ_w) at the leaf surface is zero. Consequently, the cuticle is considered a consistent sink for ammonia. There are no consensual parameterizations for this term, partly due to technical and methodological difficulties, and large uncertainties (Flechard et al. 2011). Since resistance is an intensive quantity, it obviously depends on the quantity of the surface where exchange processes take place, i.e., on the leaf area index. Nonetheless, most of the studies in the literature derive the cuticular resistance solely from relative humidity, as it affects the moisture of the leaf surface and thus the adsorption capacity. Some of them are available in Sutton et al. (1993, 1998); Erisman et al. (1994); Jakobsen et al. (1997); Nemitz et al. (2000a, 2001); Milford et al. (2001a); Zhang et al. (2003); Bleeker et al. (2004); van Jaarsveld (2004); Wichink Kruit et al. (2007); Flechard et al. (2010); Massad et al. (2010); Loubet et al. (2012); Móring et al. (2016) proposed rates of R_w = 20, 50, and 100 s m^–1 depending on the period, season, and relative humidity of RH ≥ 87% or RH < 87%, for grass and crop. Later, the threshold of RH was modified to 71% (Wichink Kruit et al. 2007). In the listed papers, the R_w is frequently determined indirectly from nighttime flux measurements when stomata are closed, and soil interactions are negligible.

However, besides humidity, other factors can also control the cuticular adsorption process, like the effect of acid gases, too high ammonia concentration, temperature, leaf surface area, and properties (Massad et al. 2010).

In the presence of acid gases on cuticula, it favours the adsorption of ammonia (Erisman and Wyers 1993; Flechard et al. 1999, 2011; Massad et al. 2010). A similar phenomenon was pointed out in several studies, in which the co-deposition of ammonia and sulphur dioxide was supposed or discussed (McLeod et al. 1990; Bobbink et al. 1992; Bobbink and Heil 1993; Cape et al. 1995, 1998; Horvath et al. 2009).

Massad et al. (2010) compiled literature data for cuticular resistance applied in various models. They suggested a combined formula for correction for R_w reported in the literature based on Flechard et al. (2010) and Zhang et al. (2003) considering the joint effect of temperature and LAI. In this study, important scatter in the dataset and no clear relationship either between R_w and RH or R_w and the AR (acid–base molar ratio) index were observed. For oilseed rape and wheat, they reported R_w = 0.93–5 s m^–1 depending on RH, LAI, T, χ_a, and total acid/ammonia ratio, which is calculated as the molar ratio of acidic gases and ammonia in the air: AR = {2 [SO₂] + [HNO₃] + [HCl]}/[NH₃]. Considering the leaf area index, the corrected R_w can be expressed by the following formula (Flechard et al. 2010; Zhang et al. 2003):

$${R}_{{\text{w}},\mathrm{ korr}}=\frac{{R}_{{\text{w}}}\cdot {{\text{e}}}^{0.15\cdot T}}{{LAI}^{\mathrm{0.5}}}.$$

(38)

Above RH > 95% the equation is: ${R}_{{\text{w}},{\text{korr}}(95\mathrm{\%})}=29.9\cdot {AR}^{-1}$. Supposing a linear relationship between RH and R_w,korr in the range of 95–100%, then:

$${R}_{{\text{w}},\mathrm{ korr},\mathrm{ min}}={R}_{{\text{w}},\mathrm{ korr}(100\mathrm{\%})}=31.5\cdot {AR}^{-1}.$$

(39)

Combining Eqs. (38, 39), we receive:

$${R}_{{\text{w}},{\text{korr}}}=31.5\cdot {AR}^{-1}\cdot {{\text{e}}}^{a\cdot \left(100-RH\right)},$$

(40)

where the supposed rate for α = 0.148 ± 0.113 for crops.

In the χ_g/χ_s/R_w models (e.g., Nemitz et al. 2000a, 2001) the term R_w controls the deposition/adsorption of ammonia in the wax layer of the cuticula covered by a thin liquid layer. According to Nemitz et al. (2001) the R_w depends on molar ratio of sulphur dioxide and ammonia because of the co-deposition of these two gases, with strong dependence on RH. They report some R_w values from the literature. E.g., for oilseed rape and wheat, the R_w ranges 1–500,000 s m^–1 and 2–1500 s m^–1, respectively in the RH interval of 60% and 100% with a rapid decrease for higher humidity.

Simpson et al. (2012) in a continental-scale chemical transport model, proposed the equations:

$${R}_{w}=\gamma \cdot {F}_{1\left({t}_{{\text{s}}}, {\text{RH}}\right)}\cdot {F}_{2\left({\alpha }_{{\text{SN}}}\right)},$$

(41)

where:

$${F}_{1\left({t}_{{\text{s}}}, RH\right)}=10\cdot {\text{log}}\left({t}_{{\text{s}}}+2\right)\cdot {e}^{\frac{100-RH}{7}},$$

(42)

$${F}_{2\left({\alpha }_{{\text{SN}}}\right)}={10}^{-1.1099\cdot {\alpha }_{{\text{SN}}}+1.6769},$$

(43)

$${\alpha }_{{\text{SN}}}=0.6\cdot \frac{\left[{{\text{NH}}}_{3}\right]}{\left[{{\text{SO}}}_{2}\right]},$$

(44)

and t_s is the temperature measured at 2 m height (°C). Considering that base/acid ratio is a governing factor for cuticular ammonia adsorption. The term γ = 0.0455 is a normalising factor, 0.6 is a correction factor for simulated ammonia concentration. With direct ammonia measurements, the simple [NH₃]/[SO₂] ratio is used for calculation of α_SN (Wichink Kruit et al. 2007).

According to Smith et al. (2000), cited by Flechard et al. (2011):

$${R}_{{\text{w}}}={t}_{{\text{w}}}\cdot {e}^{\left(\frac{100-RH}{7}\right)}.$$

(45)

In frosty conditions below t_w = –5 °C R_w = 1000 s m^–1, while between t_w = 0 °C and –5 °C R_w = 200 s m^–1.

Parameterization based on Benner et al. (1992) and van Hove et al. (1989) cited by Wichink Kruit et al. (2010):

$${R}_{{\text{w}}}=\alpha \cdot {{\text{e}}}^{\left(\frac{100-RH}{\beta }\right)*},$$

(46)

where $\alpha$ =2, $\beta$ =12, for wheat at hight SO₂ concentrations. According to Milford et al. (2001b) $\alpha$ = 0.5, $\beta$ = 12 for moorland, with low NH₃ and SO₂ concentration, while for intensively managed, fertilized croplands $\alpha$ = 30, $\beta$ = 7, with low SO₂ concentrations.

Pleim et al. (2013) reported an estimation for cuticular resistance considering the acid/base circumstances on the cuticula:

$${R}_{{\text{w}}}=\frac{{R}_{{\text{w}},{\text{o}}}}{LAI\cdot {H}_{{\text{eff}}}}\cdot \left(\frac{{\chi }_{{\text{c}}}}{{\chi }_{{\text{ref}}}}+1\right)+{a}_{{\text{h}}}\cdot \left(1-{f}_{{{\text{RH}}}_{{\text{s}}}}\right),$$

(47)

where H_eff = K_H,NH3 ∙ (1.0 + K_a/[OH^–]), dimensionless effective Henry-constant for ammonia, ${f}_{{{\text{RH}}}_{{\text{s}}}}$ a fractional (0 to 1) RH on leaves (i.e., the compensation point of water), ${R}_{{\text{w}},{\text{o}}}$= 125,000 s m^–1, a_h = 100 s m^–1, χ_ref = 1 μg m^–3, K_a is the dissociation constant of ammonium ion, and K_H,NH3 is the Henry’s low constant for ammonia. For wet foliage, during precipitation of dew events the Eq. (48) becomes simple and gives an expression for R_d, (originally marked with R_w), the resistance term for bidirectional exchange of gases between liquid and gaseous phases:

$${R}_{{\text{d}}}=\frac{{R}_{{\text{w}},0}}{LAI\cdot {H}_{{\text{eff}}}}.$$

(48)

Flechard et al. (2011); Zhang et al. (2002; 2003) refer to cuticular resistances either for dry (R_w) and wet leaf (R_d) (note: R_w = dry and R_d = wet indexing may cause misunderstanding; however, we retained the traditional use in the literature):

$${R}_{{\text{w}}}= \frac{{R}_{{\text{w}},0}}{{{{\text{e}}}^{0.03\cdot RH}\cdot u}_{*}\cdot {LAI}^{0.25}},$$

(49)

$${R}_{{\text{d}}}= \frac{{R}_{{\text{d}},0}}{{u}_{*}\cdot {LAI}^{0.5}},$$

(50)

where ${R}_{{\text{w}},0}$ and ${R}_{{\text{d}},0}$ are land-use specific reference values given in Zhang et al. (2003).

There is a model conception where bi-directional cuticular exchange of ammonia is allowed and controlled by the compensation point concentration (χ_d) of the liquid phase, such as drops on the surface of leaves, i.e., in case of the wet cuticle. Flechard et al. (1999) compiled a dynamic model considering the acid–base interactions of different inorganic compounds in the liquid drops on the cuticula. In this model, the cuticular resistance is not a fixed value but varies dynamically based on chemical interactions. It considers factors such as surface chemistry, composition, and reactivity of the leaf cuticle. The resistance term accounts for both deposition and emission processes, making it more comprehensive. The model includes a bidirectional exchange between gaseous and liquid phases according to Henry’s law (as indicated by the blue line in Fig. 1). A new resistance term R_d is introduced to represent the resistance to transfer across the air–water interface and is parameterized as a function of ionic strength (I) in the liquid phase:

$${R}_{{\text{d}}}={\text{exp}}\left(100 I\right)$$

(51)

In the model of Burkhardt et al. (2009) the bidirectional exchange is allowed by introducing a non-zero cuticular water film equilibrium concentration χ_d, coupled with an exchange resistance R_d, however, parameterization of these is not reported there.

The bi-directional exchange of dry cuticle is assumed in the model proposed by Wichink Kruit et al. (2010) which provides a different perspective on R_w. In this conception R_w specifically refers to cuticular resistance of dry leaves in the context of bi-directional exchange. The cuticle plays a crucial role in regulating this bidirectional exchange, affecting both the uptake and emission of materials. Wichink Kruit et al. (2010) parameterized R_w night-time when stomatal exchange is zero as:

$${R}_{{\text{w}}} \approx \frac{{\chi }_{{\text{w}}}-{\chi }_{{\text{a}}}}{{F}_{{\text{t}}}}-\left({R}_{{\text{a}}}+{R}_{{\text{b}}}\right)$$

(52)

In a first approximation, Wichink Kruit et al. (2010) assumed that the atmospheric ammonia concentration at the external leaf surface is zero (χ_w = 0) to derive R_w, similar to the approach taken by other authors (e.g., Sutton et al. 1993; Nemitz et al. 2004a).

In summary, there are cuticular resistance parametrizations for dry leaves (Eqs. 38–47 and 49, model 2a and 2c) and wet leaves (Equations. 48, 50, and 51, model 2b and 2d). If we assume that a compensation point concentration also develops on the dry leaves, then Eq. 52 is applicable although this approach is not completely exact due to the assumption of χ = 0.

Stomatal compensation point concentration χ_s

The compensation point concentration through stomata is controlled by the ratio of ammonium and hydrogen ions in the apoplast Γ_s = [NH₄⁺]/[H⁺] based on the following relationship, considering the thermodynamic equilibrium of ammonia between gas and liquid phases, and the acid–base equilibrium between ammonia-ammonium ion (after Schjoerring et al. 1998; Nemitz et al. 2001; Personne et al. 2009):

$${\chi }_{{\text{s}}}\left({\text{mol}}{L}^{-1}\right)=K\cdot {\text{exp}}\left(\frac{\sum G}{{\text{R}}}\cdot \left(\frac{1}{298.15}+\frac{1}{T}\right)\right)\cdot {\Gamma }_{{\text{s}}}=\frac{\mathrm{161,500}}{T}{{\text{e}}}^{-}\frac{\mathrm{10,380}}{T}\cdot \frac{\left[{{\text{NH}}}_{4}^{+}\right]}{{{\text{[H]}}}^{+}},$$

(53)

where K is the product of Henry's constant and the dissociation constant of ammonia for a given temperature (T), and ΣG is the sum of the free enthalpy of the two processes, R is the universal gas constant. Γ_s can be determined directly by the measurement of [NH₄⁺] and [H⁺] in the apoplast (Schjoerring et al. 1993, 1998, 2000; Husted and Schjoerring 1995; Husted et al. 2000a, b; Mattsson and Schjoerring 2003).

The [NH₄⁺] and [H⁺] concentrations in apoplast can be determined by the bioassay analysis in the extracellular liquid of plant tissues, which controls the stomatal emission potential. Several papers have appeared in the recent decades describing the applied technics (Husted et al. 2000b; Husted and Schjoerring 1995; Mattsson and Schjoerring 2003; Schjoerring et al. 1993, 1998, 2000). Later on, O'Leary et al. (2014) used the infiltration-centrifugation technique for extraction of apoplastic fluid.

Instead of complicated bioassay measurements Γ_s is often estimated by empirically, e.g., from ammonium content of the bulk tissue. According to Massad et al. (2010) Γ_s (N_in, kg ha yr^–1) can be calculated from the average N-intake, which is mainly based on the annual fertilizer intake, including the atmospheric deposition, excluding the period immediately after fertilization: Γ_s = 66.4 + 0.0853 $\cdot$ N_in^1.59. Mattsson et al. (2009) found a correlation between Γ_s and bulk NH₄⁺. For grassland and crops, Massad et al. (2010) provided an exponential function for this relationship:

$${\Gamma }_{{\text{s}}}=19.3\cdot {{\text{e}}}^{0.0506\cdot {{[{\text{NH}}}_{4}^{+}]}_{{\text{bulk}}}},$$

(54)

where [NH₄⁺]_bulk is the mass ratio of ammonium and the fresh tissue (µg g^–1). Directly after fertilization:

$$\begin{array}{ccc}{\Gamma }_{{\text{s}}}= {\Gamma }_{{\text{s}}({\text{max}})}\cdot {{\text{e}}}^{-t/\tau },& {\text{where}}& {\Gamma }_{{\text{s}}({\text{max}})}=12.3\cdot {N}_{{\text{app}}}+20.3,\end{array}$$

(55)

where t is the time from fertilization and τ = 2.88 (days). The dynamics of Γ_s after fertilization mainly depends on the amount of fertilizer (N_app, kg N ha^–1), the application technique, soil, and weather conditions. After about 10 days, the conditions like before fertilization exist.

Compensation point concentration of ground (soil, litter, fertilizer) χ_g

Similarly, to the formula for stomata, the compensation point concentration of the ground surface is controlled by the ratio of ammonium and hydrogen ions in the liquid phase of soil/litter, and is given by:

$${\chi }_{{\text{g}}}= \frac{\mathrm{161,500}}{T} {{\text{e}}}^{-\frac{\mathrm{10,380}}{T}}\cdot \frac{{[{\text{NH}}}_{4}^{+}]}{\left[{{\text{H}}}^{+}\right]}.$$

(56)

For background soil (no fertilizer), the emission potential can be calculated from the molar ratio of ammonium/hydrogen ions. However, the litter should definitely be involved in the sampling, because it can be a significant ammonia source rather than soil. Moreover, ammonia flux from background soils is partly regarded as negligible by Nemitz et al. (2000a). On the other hand, as it mentioned in Section 2. i. a part of ammonium is adsorbed in the solid phase of soil (Neftel et al. 1998), hence emission potential of soil can be overestimated when bulk ammonium content of soil used in calculation.

For fertilized soil, according to the calculation of Sommer et al. (2004) χ_g is determined by total ammoniacal nitrogen, TAN = [NH₃] + [NH₄⁺] and by the equilibrium processes in the liquid phase:

$$\begin{array}{ccc}{\left[{{\text{NH}}}_{3}\right]}_{{\text{L}}}=\frac{TAN}{{1+10}^{\left(0.09018+\frac{2729.92}{T}-pH\right)}}& {\text{and}}& {\chi }_{{\text{g}}}=\end{array}{\left[{{\text{NH}}}_{3}\right]}_{{\text{L}}}\cdot {10}^{\frac{1477.7}{T}-1.69},$$

(57)

where [NH₃]_L and pH are the concentration of ammonia and the pH in soil liquid, respectively.

Loubet et al. (2012) determined the emission potential of the soil layer empirically to get the best fit between the measured and the modelled fluxes. After application of fertilizer the emission potential is exponentially decreasing (Massad et al. 2010): Γ_g = Γ_g(max) $\cdot {{\text{e}}}^{{-}_{\tau }^{t}}$, where τ = 2.88 days. Γ_g(max) can be derived from:)

$${\Gamma }_{{\text{g}}({\text{max}})}=\frac{\frac{{N}_{{\text{app}}}}{{\theta }_{{\text{g}}}}\cdot {M}_{{\text{N}}}\cdot {l}_{{\text{g}}}\cdot {h}_{{\text{m}}}}{{10}^{-{\text{pH}}}},$$

(58)

where is N_app the mass of nitrogen applied (kg N ha^–1), ${\theta }_{{\text{s}}}$ is the soil wetness (%), M_N is a conversion factor (1/14 mol g^–1), l_g is the depth of soil layer where a fertilizer is applied (typically 0.05 m), h_m is another conversion factor (10^–4 ha m^–2), pH is the pH of soil solution. Fu et al. (2015) and Xu et al. (2018) give a similar expression:

$${\Gamma }_{{\text{g}}}=\frac{\frac{{N}_{{\text{app}}}}{{\theta }_{{\text{s}}}{\cdot M}_{N}\cdot {l}_{s}}}{{10}^{-{\text{pH}}}},$$

(59)

where N_app is the mass of fertilizer applied (g N m^–1), is the ${\theta }_{{\text{s}}}$ soil water content (m³ m^–3).

Compensation point concentration of drops on leaves: χ_d

Theoretically, the compensation point concentration of the liquid (dew, rain drops) on cuticula can be derived from Eq. (53), similarly for the equations for the stomata and soil/litter:

$${\chi }_{{\text{d}}}\left({\text{mol}}{L}^{-1}\right)=\frac{\mathrm{161,500}}{T}{{\text{e}}}^{-\frac{\mathrm{10,380}}{T}}\cdot \frac{\left[{{\text{NH}}}_{4}^{+}\right]}{{{\text{[H]}}}^{+}}.$$

(60)

In practice, the determination of Γ_d = [NH₄⁺]/[H⁺] faces difficulties due to the lack of knowledge of the concentrations. Nevertheless, attempts have been made to calculate them in the literature. Burkhardt et al. (2009) measured the chemical composition of dew, guttation, and rain droplets on the leaves and used them in a dynamic chemical model. The results are encouraging, but widespread application is difficult as it would require continuous sampling. Walker et al. (2013) analysed the dew and guttation on the cuticle and incorporated them into their model. As stated there, they achieved "a significant advancement over previous approaches, but the evaluation and improvement of such modelling systems require process-level field measurements over extended periods," pointing out again the limit of applicability of continuous leaf chemistry observations.

Wentworth et al. (2016) demonstrated the role of dew as a nighttime sink and daytime source of ammonia. After the evaporation of dew, they observed a morning increase in ammonia concentration. They also mentioned the role of organic acids, which makes the calculations more complicated.

χ_d can be determined indirectly at night (I_p = 0, closed stomata) by the parallel measurement of F_t and F_g according to the equation: F_d = F_t – F_g = χ_d/R_d. However, the exact parameterization of R_d is questionable.

Gas/particle phase ammonium nitrate, ammonia, and nitric acid equilibrium

The reversible equilibrium of NH₃/HNO₃/NH₄NO₃ is temperature and humidity-dependent (Mozurkewich 1993) and influences the exchange of ammonia between the surface and atmosphere (Brost et al. 1988). Depending on the mixing ratios of NH₃, HNO₃, and NH₄NO₃, on temperature, and RH in the air column within and above vegetation, gas/particle conversion may alter the net NH₃ flux, as exchange velocities for gas-phase NH₃ and aerosol-phase NH₄⁺ are different (Flechard et al. 2015; Nemitz et al. 2004a). Partly, this process can cause the apparent disappearance of ammonia and the subsequent daytime release, morning peaks when latent ammonia appears again. Moreover, this equilibrium process affects the AR ratio because of the transformation of nitric acid vapor into quasi-neutral ammonium nitrate particles and vice versa.

The equilibrium ratio of gas-phase ammonia and nitric acid to ammonium nitrate is determined by the dissociation constant (K_d, ppb²) of NH₄NO₃ below the deliquescence relative humidity (Lin and Cheng 2007):

$${K}_{{\text{d}}}=\left[{{\text{NH}}}_{3}\right]\cdot \left[{{\text{HNO}}}_{3}\right]=118.87- \frac{\mathrm{24,084}}{T}-6.025\cdot {\text{ln}}\left(T\right).$$

(61)

The deliquescence relative humidity (DRH) is at 60–66% (Peng et al. 2022). More recent studies have clarified the DRH of ammonium nitrate to 61.5% (cited by Lightstone et al. 2000). Above this RH the dissociation constant is described by (Seinfeld and Pandis 1998):

$${K}_{{\text{d}}}^{\mathrm{^{'}}}=\left[{P}_{1}-{P}_{2}\left(1-{a}_{{\text{w}}}\right)+{P}_{3}{\left(1-{a}_{{\text{w}}}\right)}^{2}\right]\times \left(1-{a}_{{\text{w}}}\right)+{\left(1-{a}_{{\text{w}}}\right)}^{1.75}\times {K}_{{\text{d}}},$$

(62)

where ${a}_{w}=\frac{RH}{100}$, and values for P₁, P₂ and P₃ are:

$${\text{ln}}\left({P}_{1}\right)=-135.94+\frac{8763}{T}+19.12\mathrm{ ln}\left(T\right),$$

(63)

$${\text{ln}}\left({P}_{2}\right)=-122.65+\frac{9969}{T}+16.22\mathrm{ ln}\left(T\right),$$

(64)

$${\text{ln}}\left({P}_{3}\right)=-182.61+\frac{13875}{T}+24.46\mathrm{ ln}\left(T\right).$$

(65)

By using Eq. (61) and (62) the nitric acid concentration can be calculated either for RH < 61.5% or RH > 61.5% in the knowledge of the measured ammonia concentration (χ_a).

In our calculations Kelvin effect was not taken into consideration. The mean diameter of ammonium nitrate particles is around the 400 nm (Larson and Taylor 1967; Bergin et al. 1987) where the vapour pressure of curved surface is practically equal with that of flat plain.

It should be noted that the dissociation of ammonium chloride has same effect as ammonium nitrate but considering the concentration ratio of nitrate/chloride > 20, in the regional background air, it is practically negligible.

Summary conclusions

There are several different models for field-scale modelling of ammonia emissions from fertilized agricultural areas, which share similar basic concepts. Only a portion of the applied fertilizers is taken up by plants, while the remaining portion is transformed or emitted from the soil. A portion of the ammonia released by the soil-litter is emitted into the atmosphere, while another portion is adsorbed on the dry leaf surface or absorbed in the droplets on the leaf surface. Stomata can be a sink or a source of ammonia depending on the conditions. A portion of the recaptured ammonia can be used by the plant as a substrate, reducing the loss and environmental pollution. However, there are discrepancies in the literature regarding the estimation of ammonia recaptured by the canopy. Not only the magnitude but also the direction (emission, deposition) of the various fluxes is questionable.

Since current in-canopy turbulence transport models are insufficient to properly explain net exchange processes, more work is therefore needed to develop multi-layer canopy models that can simulate these processes.

Most of the discrepancies can be attributed to the parameterization of the models. These can be as follows:

• Soil emissions are generally calculated empirically from the ammoniacal content and pH of the soil or the amount of applied fertilizers. However, it is obvious that the simplified approximations carry errors. This is partly because only a portion of the soil ammonium content is active in terms of emissions. On the other hand, the nitrogen cycle of the soil, including the transformations of ammonia, is complex enough to be parameterized with sufficient accuracy using empirical formulas. Doubts could be eliminated by direct soil emission chamber measurements, which could establish a relationship between emissions and various physical and chemical soil parameters.

• The stomatal compensation point concentration is usually calculated empirically from the bulk leaf ammonia/ammonium concentrations and pH. However, this approach may underestimate the emission potential. This can be eliminated by bioassay measurements, i.e., by measuring the apoplastic ammoniacal concentration and pH.

• The role of acidic gases in cuticular adsorption and absorption processes is often not adequately considered. On the other hand, the equilibrium processes of NH₄NO₃, NH₃, and HNO₃ also play a role in the ammonia cycle within the canopy, especially considering the daily course of RH and T. Estimating these processes would improve the accuracy of the simulation.

• Most of the discrepancies can be eliminated by more accurately determining the input parameters. In this case, the empirical approximation could be replaced by specific measurements parallel to modelling. However, parallel measurements with numerical simulation pose a great challenge. Finally, as Walker et al. (2013) stated, future studies in fertilized cropping systems should focus on the temporal dynamics of net emission to the atmosphere from fertilization to peak leaf area index (LAI) and improvement of soil and cuticular resistance parameterizations.

Appendix

The general expression of aerodynamic resistance (${R}_{a}$) can be delivered according to the basic equations of Monin–Obukhov similarity theory as follows.

Based on the turbulent characteristics, the momentum flux ($\tau$), the sensible heat flux (H), and the flux of a given trace substance (${F}_{t}$) can be expressed as:

$$\tau =\sqrt{{(\overline{{u }{'}{w}{'}})}^{2}+{(\overline{{v }{'}{w}{'}})}^{2}}=\rho \cdot {u}_{*}^{2},$$

(66)

$$H=\rho {\cdot c}_{{\text{pm}}}\left(\overline{{{\Theta }_{v}}{'}{w}{'}}\right)=-\rho {\cdot c}_{{\text{pm}}}\cdot {u}_{*}{\cdot T}_{*},$$

(67)

$${F}_{t}=\left(\overline{{\chi }{'}{w}{'}}\right)=-{\chi }_{*}{u}_{*}$$

(68)

where $\rho$ and ${c}_{{\text{pm}}}$ are the density and specific heat capacity of the wet air at constant pressure, ${u}_{*}$, ${T}_{*}$, ${\chi }_{*}$ are the friction velocity, the characteristic dynamical temperature, and the characteristic dynamical concentration, respectively. Terms marked by prime are the fluctuations according to the rules of Reynolds averaging.

In the eddy covariance calculation of turbulent fluxes several correction procedures must be applied (Foken 2017). In the case of positive values, the transport of heat and material is directed from the surface to the atmosphere. The mean gradients of the wind velocity ($V$), the virtual potential temperature $({\Theta }_{v}$), and the concentration (χ) can be expressed as:

$$\begin{array}{ccc}\frac{\partial V}{\partial z}=\frac{{u}_{*}}{k}{\varphi }_{M}\left(\zeta \right),& \frac{\partial {\Theta }_{v}}{\partial z}=\frac{{T}_{*}}{k}{\varphi }_{H}\left(\zeta \right),& \frac{\partial \chi }{\partial z}=\frac{{c}_{*}}{k}{\varphi }_{c}\left(\zeta \right),\end{array}$$

(69)

where ${\varphi }_{M}$, ${\varphi }_{H}$, and ${\varphi }_{c}$ are the universal functions of the transfer of momentum, sensible heat, and trace material in the function of dimensionless stability $(\zeta =(z-d)/L)$, where $d$ is the displacement height, and $L$ is the Monin–Obukhov length:

$$L=- \frac{1}{\beta k}\frac{{\left(\tau /\rho \right)}^{3/2}}{H/\rho {c}_{{\text{pm}}}}=\frac{{u}_{*}^{2}}{\beta k{T}_{*}},$$

(70)

where $\beta$ is the stability parameter, expressed by:

$$\beta =\frac{g}{{\overline{\Theta } }_{v}},$$

(71)

The virtual potential temperature is:

$${\Theta }_{v}={T}_{v}{\left(\frac{{p}_{00}}{p}\right)}^{\frac{{R}_{{\text{m}}}}{{c}_{pm}}},$$

(72)

where ${T}_{v}$ is the virtual temperature, ${R}_{m}$ is the specific gas constant of wet air, ${p}_{00}=1000 {\text{hPa}}$, the reference pressure. The virtual temperature is given in the function of specific humidity ($q):$

$${T}_{v}=T\left(1-\mathrm{0,608}\cdot q\right).$$

(73)

The specific humidity, the ratio of densities of the water vapour and the wet air can be given in the knowledge of temperature and relative humidity. When $p\cong {p}_{00}$, then ${\Theta }_{v}\cong {T}_{v}\cong T$, this approximation is frequently used in trace materials’ modelling. In practice we suppose an equality between the universal functions for heat and trace material fluxes:

$${\varphi }_{c}={\varphi }_{H}.$$

(74)

For detailed information of shape of universal functions depending on stability and on the integral formulae of profile equation refer to e.g., Högström (1988); Arya (2001); Weidinger et al. (2000); Foken (2017). The equations are integrated between ${(z}_{0}+d$) and ($z-d$) heights. The general form of aerodynamic resistance is:

$${R}_{a}=-\frac{{\chi }_{{\text{a}}}-{\chi }_{{\text{z}}0}}{{F}_{t}}=-\frac{\chi \left(z-d\right)-\chi \left({z}_{0}\right)}{{F}_{t}}=\frac{\Delta \chi }{{u}_{*}{c}_{*}}=\frac{1}{{ku}_{*}}\left[{\text{ln}}\left(\frac{z-d}{{z}_{0}}\right)-{\psi }_{{\text{H}}}\left(\frac{z-d}{L}\right)+{\psi }_{{\text{H}}}\left(\frac{{z}_{0}}{L}\right)\right].$$

(75)

The ${u}_{*}$ can be determined from wind velocity (V) measurements at z height as:

$$\frac{1}{{u}_{*}}=\frac{1}{kV}\left[{\text{ln}}\left(\frac{z-d}{{z}_{0}}\right)-{\psi }_{{\text{M}}}\left(\frac{z-d}{L}\right)+{\psi }_{{\text{M}}}\left(\frac{{z}_{0}}{L}\right)\right].$$

(76)

The integral formulae of universal functions, ${\psi }_{{\text{M}}}$, ${\psi }_{{\text{H}}}$ express the deviance of profiles from logarithmic shape:

$$\begin{array}{cc}{\psi }_{{\text{M}}}=\int (1-{\varphi }_{{\text{M}}})\mathrm{dln\zeta },& {\psi }_{{\text{H}}}=\int (1-{\varphi }_{{\text{H}}})\mathrm{dln\zeta }.\end{array}$$

(77)