Carbonyl sulfide, like any other trace gas, can be present in the soil matrix in three forms: (1) vaporised in the air-filled pore space, (2) dissolved in the water-filled pore space or (3) adsorbed on the surface of the soil matrix (mineral and organic matter solid particles). The total OCS concentration Ctot (mol m-3 soil) is thus the sum of the OCS concentration in each phase weighted by their volumetric content: Ctot=εaC+θCl+ρbCs where εa (m3 air m-3 soil) is the volumetric air content, θ (m3 water m-3 soil) is the volumetric water content, ρb (kg m-3) is soil bulk density, C (mol m-3 air) and Cl (mol m-3 water) denote OCS concentration in soil air and liquid water respectively and Cs (mol kg-1 soil) denotes the OCS concentration adsorbed on the soil matrix.

In the following we will assume full equilibrium between the three phases. We will also assume linear sorption/desorption behaviour (a fair assumption at ambient OCS concentrations), so that Cl and Cs can be linearly related to C: Cl=BC where B (m3 water m-3 air) is the solubility of OCS in water and Cs=(Ksg+BKsw)C where Ksg (m3 air kg-1 soil) and Ksw (m3 water kg-1 soil) are the solid/vapour and solid/liquid partitioning coefficients respectively (Olesen et al., 2001). The solubility B is related to Henry's law constant KH (mol m-3 Pa-1): B=KH RT where R= 8.31446 J mol-1 K-1 is the ideal gas constant and T (K) is soil water temperature. It has been shown that KH is fairly independent of pH (at least for pH below 9, see De Bruyn et al. (1995); Elliott et al., 1989) but decreased with temperature and salinity (De Bruyn et al., 1995; Elliott et al., 1989). In the following we will use the parameterisation of Wilhelm et al. (1977) assuming low salinity levels in the soil: KH=0.021exp⁡[24900/R(1/T-1/298.15)]. We preferred this expression rather than the more recent expression proposed by De Bruyn et al. (1995) that was based on one single data set rather than a compilation of multiple data sets. The difference between the two expressions is shown in Fig. 1a.

Expressions of Ksg and Ksw for OCS are currently not available. For organic vapours it has been shown that Ksw is highly correlated with soil characteristics such as C content (Petersen et al., 1995), specific surface area or clay content (Yamaguchi et al., 1999), and that Ksg is usually significant at soil water contents corresponding to less than five molecular layers of water coverage (Petersen et al., 1995). In this range of soil moisture, direct chemical adsorption onto dry mineral surfaces dominates and can increase the adsorption capacity of soils by several orders of magnitude. For these organic vapours the relationship of Ksg with soil moisture can be related to soil specific surface area (Petersen et al., 1995) or clay content (Yamaguchi et al., 1999). However these relationships obtained for organic vapours are unlikely to be applicable for OCS because the adsorption mechanisms may be completely different. Liu and colleagues have estimated OCS adsorption capacities of several mineral oxides and found that quartz (SiO2) and anatase (TiO2) did not adsorb OCS but other oxides with higher basicity adsorbed, reversibly or not, rather large quantities of OCS (Liu et al., 2008, 2009, 2010a). They also recognised that these estimates of the adsorption capacity of the minerals were an upper limit owing to the competitive adsorption of other gases such as CO2, H2O and NOx that occur in the real Earth's atmosphere (Liu et al., 2009, 2010a) and the somewhat lower OCS partial pressure in ambient air compared to that used in their experimental setup. Also, at steady state, adsorption should have little influence on the soil–air OCS exchange rate, unless heterogeneous (surface) reactions occur and continuously remove OCS from the adsorbed phase (Liu et al., 2010a). In the following we will neglect adsorption of OCS on solid surfaces, but we recognise that this assumption might be an oversimplification.

The transport of OCS through the soil matrix occurs by either pressure-driven (advective-dispersive) or concentration-driven (diffusive) fluxes. Carbonyl sulfide can also be destroyed or emitted, owing to abiotic and/or biotic processes. The general mass balance equation for OCS in a small soil volume can then be written as follows: ∂εtotC∂t=-∇Fdiff-∇Fadv+P-S, where εtot=εa+θB+ρb (Ksg+ BKsw)≈ εa+θB (m3 air m-3 soil) is total OCS soil porosity, Fdiff (mol m-2 s-1) represents the diffusional flux of OCS through the soil matrix, Fadv (mol m-2 s-1) is the advective flux of OCS, P (mol m-3 s-1) the OCS production rate, S (mol m-3 s-1) the OCS consumption rate and ∇=∂/∂x+∂/∂y+∂/∂z denotes the differential operator, i.e. the spatial gradient in all three directions x, y and z.

If the soil is horizontally homogeneous (that is, the soil properties are independent of x and y) and the soil lateral dimensions are much larger than its total depth (minimal edge effects), the OCS concentration is only a function of soil depth z and time t. Eq. (1) may be simplified: ∂εtotC∂t=-∂Fdiff∂z-∂Fadv∂z+P-S,

Diffusion in the gas phase is commonly described by Fick's first law (Bird et al., 2002; Scanlon et al., 2002): Fdiff,a=-Deff,a∂C∂z, where Fdiff,a (mol m-2 s-1) is the diffusive flux of gaseous OCS and Deff,a (m3 air m-1 soil s-1) is the effective diffusivity of gaseous OCS through the soil matrix. The latter is commonly expressed relative to the binary diffusivity of OCS in free air D0,a (m2 air s-1): Deff,a/D0,a=τaεa where τa is the so-called air tortuosity factor that accounts for the tortuosity of the air-filled pores, as well as their constrictivity and water-induced disconnectivity (e.g. Moldrup et al., 2003). The air-filled porosity (εa) appears in this equation to account for the reduced cross-sectional area in the soil matrix relative to free air, although the effective porosity for diffusion could be smaller if the soil contains small pores that do not contribute to the overall transport such as dead end or blind pores. Expressions for τa differ depending on whether the soil is repacked or undisturbed (Moldrup et al., 2003). For undisturbed soils the most commonly used equations are those of Penman (1940); τa= 0.66, hereafter referenced as Pen40, and Millington and Quirk (1961); τa=εa7/3/φ2, where φ is total soil porosity, hereafter referred to as MQ61. For repacked soils, equations proposed by Moldrup et al. (2003); τa=εa3/2/φ, hereafter referred to as Mol03r are preferred. For undisturbed soils with high porosity such as volcanic ash, the expression proposed by Moldrup et al. (2003; τa=εa1+3/b/φ3/b, where b is the pore-size distribution parameter) seems a better predictor (Moldrup et al., 2003). Recently a new density-corrected expression for undisturbed soils has also been proposed by Deepagoda et al. ((2011); τa=[0.2(εa/φ)2+0.004]/φ) that seems to be superior to previous formulations and has the advantage of not requiring knowledge of the pore-size distribution parameter b. A summary of these different formulations of the tortuosity factor and their range of application is given in Table 1.

Diffusion in the liquid phase is described in a similar fashion to the gas phase (Olesen et al., 2001): Fdiff,l=-Deff,l∂Cl∂z=-Deff,lB∂C∂z+CdBdT∂T∂z, where Fdiff,l (mol m-2 s-1) is the diffusive flux of dissolved OCS in soil water and Deff,l (m3 water m-1 soil s-1) is the effective diffusivity of dissolved OCS through the soil matrix. As for gaseous diffusion Deff,l is commonly expressed relative to the binary diffusivity of OCS in free water D0,l (m2 water s-1): Deff,l/D0,l=τlθ where τl is the tortuosity factor for solute diffusion. Different expressions for τl can also be found in the literature (Table 1).

Diffusion of OCS in the adsorbed phase can theoretically occur and can be described in a similar fashion to other trace gases (e.g. see Choi et al. (2001) for ozone). However we will neglect such a diffusion flux in the adsorbed phase because it is expected to be orders of magnitude smaller than in the two other phases. Also the binary diffusivity of any trace gas is several orders of magnitude higher in the air than it is for its dissolved counterpart in liquid water so that, in unsaturated (oxic) soils, Fdiff=Fdiff,a+Fdiff,l is dominated by the gas-phase OCS diffusion flux Fdiff,a. The role of Fdiff,l in the OCS transport equations becomes significant only when the soil is waterlogged.

The binary diffusivity D0,a depends on pressure and temperature and is assumed here to follow the Chapman–Enskog theory for ideal gases (i.e. Bird et al., 2002): D0,a(T,p)=D0,a(T0,p0) (T/T0)1.5 (p0/p). A value for D0,a (25 ∘C, 1 atm) of 1.27 × 10-5 m2 s-1 is used and derived from the value for the diffusivity of water vapour in air at 25 ∘C (2.54 × 10-5 m2 s-1, see Massman, 1998) and the CO2 / OCS diffusivity ratio of 2.0 ± 0.2 derived from the Chapman–Enskog theory and the difference in molar masses of OCS and CO2 (Seibt et al., 2010). The binary diffusivity D0,l also depends on temperature (Ulshöfer et al., 1996). Because the Stokes–Einstein equation only applies to spherical suspended particles, we preferred to use an empirical equation that works well for both the self-diffusivity of water and the diffusivity of dissolved CO2 in liquid water (Zeebe, 2011): D0,l(T)=D0,l(T0) (T/T0-1)2, with D0,l (25 ∘C) = 1.94 × 10-9 m2 s-1 (Ulshöfer et al., 1996) and T0= 216 K. This value of T0 was chosen to be intermediate between the value used for water (215.05 K) and dissolved CO2 (217.2 K) (Zeebe, 2011), and results in a temperature dependency of D0,l for OCS in water in very good agreement with relationships found in other studies (Fig. 1b).

Advection of OCS can occur in both the liquid and gas phases when the carrier fluid (water or air) moves relative to the soil matrix: Fadv,l=qlCl=qlBC,Fadv,a=qaC, where ql (m s-1) and qa (m s-1) are the velocity fields for liquid water and air respectively. If the flow in the porous soil is laminar these velocity fields are given by Darcy's law (Massman et al., 1997; Scanlon et al., 2002): ql=-klμl∂Ψl∂z=-Kl∂hl∂z+1,qa=-kaμa∂pa∂z+ρag. In Eqs. (6a) and (6b) kl and ka (m2) denote soil permeabilities for liquid water and air respectively, μl and μa (kg m-1 s-1) are water and air dynamic viscosities, Ψl=ρlg(hl+z) is total soil water potential (Pa), ρl is water density (1000 kg m-3), hl (m) is matric potential height, g is gravitational acceleration (9.81 m s-2), ρa is air density (ca. 1.2 kg m-3) and pa (Pa) is air pressure. We also defined the soil hydraulic conductivity Kl (m s-1): Kl=klρlg/μl. In practice pa can be expressed as the sum of the hydrostatic pressure (pah=-ρagz) and a fluctuating (non-hydrostatic) part: pa=-ρagz+pa′ so that Eq. (6b) can be replaced by: qa=-kaμa∂pa′∂z. From Eq. (6c) we can see that advection in the gas phase can result from pressure fluctuations, caused by, e.g. venting the soil surface (according to Bernouilli's equation) or turbulence above the soil surface. Typical air pressure fluctuations are of the order of 10 Pa (Maier et al., 2012; Massman et al., 1997). Pressure fluctuations can also result from non-hydrostatic density fluctuations caused by a change in the air composition with gas species of different molar mass as air or by temperature gradients, but the resulting flux is significant only in highly permeable (i.e. fractured) soils.

When averaged over a long enough timescale (> 1 h) the advective flux starts to become negligible compared to the diffusive flux (e.g. Massman et al., 1997). Integration timescales of a few minutes were already assumed to allow liquid–vapour equilibration in Eq. (5a). In the following we will thus neglect advective fluxes in the OCS budget equation, keeping in mind that such an assumption is valid only for time scales of about 1h or longer.

Even when advective fluxes are negligible, advection through porous media generates a diffusive-like flux called mechanical dispersion that reflects the fact that not everything in the porous medium travels at the average water or gas flow speed. Some paths are faster, some slower, some longer and some shorter, leading to a net spreading of the gas or solute plume that looks very much like diffusive behaviour. Since mechanical dispersion depends on the flow, it is expected to increase with increasing flow speed: Fdisp,l=-Ddisp,l∂Cl∂z=-αlql∂BC∂z,Fdisp,a=-Ddisp,a∂C∂z=-αaqa∂C∂z, where αl (m) and αa (m) are the longitudinal dynamic dispersivity of liquid water and air flow respectively and Ddisp,l (m2 s-1) and Ddisp,a (m2 s-1) are the corresponding dispersive diffusivities. Transverse dispersion (i.e. in a plane perpendicular to the flow) can also occur but will be neglected here.

In practice, because of advective–dispersive fluxes, we must know the liquid water and air velocity fields ql and qa in order to solve the trace gas OCS mass budget Eq. (2). This requires solving the total mass balance equations for liquid water and air separately. However, except during rain infiltration and immediate redistribution, ql rarely exceeds a few millimetres per day while the drift velocity, defined as the ratio Fdiff,a/C, is typically of the order of a few millimetres per minute. For this reason, advection fluxes are generally neglected in soil gas transport models. Dispersive fluxes can still be accounted for as a correction factor to true diffusion, provided we have parameterisations of the dispersion diffusivities that are independent of the advective flux (e.g. expressions for Ddisp,a independent of qa). For example Maier et al. (2012) proposed expressions of Ddisp,a/D0,a that rely on the air-filled porosity (εa) and permeability (μa) of the soil and the degree of turbulence above the soil surface (characterised by the friction velocity u*).

The processes of consumption or production of OCS in a soil are not fully understood. Carbonyl sulfide can be consumed through hydrolysis in the bulk soil water at an uncatalysed rate kuncat (s-1) that depends mostly on temperature T and pH (Elliott et al., 1989). In the following we will use the expression proposed by Elliott et al. (1989) because it covers the widest range of temperature and pH: kuncat=2.15⋅10-5exp⁡-104501T-1298+12.7⋅10-pKw+pHexp⁡-60401T-1298, where pKw is the dissociation constant of water. Other expressions are available in the literature and compared to Eq. (8) for both temperature (Fig. 1c) and pH (Fig. 2a) responses. Using Eq. (8) the uncatalysed OCS uptake rate is then computed as Suncat=kuncatBθC. The volumetric soil water content θ appears in this equation to convert the hydration rate from mol m-3 water s-1 to mol m-3 soil s-1.

This uncatalysed rate is rather small and cannot explain the large OCS uptake rates observed in oxic soils (Kesselmeier et al., 1999; J. Liu et al., 2010; Van Diest and Kesselmeier, 2008). The main consumption of OCS is thought to be enzymatic and governed by soil microorganism CA activity (Kesselmeier et al., 1999; J. Liu et al., 2010; Van Diest and Kesselmeier, 2008). We will assume that such a catalysed reaction by CA-containing organisms can be described by Michaelis–Menten kinetics, as was observed for OCS in several marine algae species (Blezinger et al., 2000; Protoschill-Krebs et al., 1995) and one flour beetle (Haritos and Dojchinov, 2005). Because of the low concentrations of OCS in ambient air (500 ppt) and the comparatively high values of the Michaelis–Menten coefficient for OCS (Km, see Ogawa et al. (2013); Protoschill-Krebs et al., 1995, 1996) the catalysed uptake rate Scat (mol m-3 s-1) can be approximated: Scat=θkcatCABCKm+BC≈kcatKmCABθC, where kcat (s-1) and Km (mol m-3) are the turnover rate and the Michaelis–Menten constant of the enzymatic reaction respectively and [CA] (mol m-3) is the total CA concentration in soil water. We recognise that Eq. (9) is an oversimplification of the reality in the sense that kcat and Km are not true kinetic parameters but rather volume-averaged parameters for the entire soil microbial community. Also Eq. (9) neglects the competition for CA by CO2 molecules and the co-limitation of the uptake by diffusional constraints. Given the Michaelis–Menten constant of CA for CO2 (Km,CO2, of the order of 3 mM at 25 ∘C and pH 8–9) and the range of CO2 mixing ratios encountered in soil surfaces (300–5000 ppm or 0.01–0.15 mM at 25 ∘C and 1 atm), we can conclude that the competition with CO2 is negligible (i.e. the denominator in Eq. (9) would need to be multiplied by a factor 1 + [CO2]/Km,CO2 which would deviate from unity by less than 5 %). We recognise that the CO2 concentration inside microbial cells (i.e. at the CA sites) must be somewhat larger than in the surrounding soil water but certainly not to an extent to justify accounting for competition between the two substrates. Also, using typical values of transfer conductance across cell wall and plasma membrane (Evans et al., 2009), we can show that the limitation of OCS uptake by diffusion into the microbial cells is negligible for calculating the OCS uptake rate (see Appendix A for a derivation). In the following we will therefore assume Eq. (9) to be valid.

As found for any enzymatic reaction, kcat and Km depend on temperature and internal pH (pHin). In the following we will assume that the ratio kcat/Km has a temperature dependency that can be approximated: kcatKm∝xCA(T)=exp⁡(-ΔHa/RT)1+exp⁡(-ΔHd/RT+ΔSd/R), where ΔHa, ΔHd and ΔSd are thermodynamic parameters. In the following we will take ΔHa= 40 kJ mol-1, ΔHd= 200 kJ mol-1 and ΔSd= 660 J mol-1 K-1, that leads to a temperature optima Topt,CA= 25 ∘C and reproduces well the temperature response of β-CA found on maize leaf extracts observed in the range 0–17 ∘C by Burnell and Hatch (1988) (Fig. 1d). To our knowledge this is the only study that reports the temperature response of β-CA, the dominant CA class expected in soils (Smith et al., 1999). Interestingly our parameterisation of xCA(T), based on direct measurements on β-CA from Burnell and Hatch (1988), is very different from the one used by Sun et al. (2015), especially at temperatures above 20 ∘C (Fig. 1d).

The pH response of CA activity for OCS hydrolysis was described by a monotonically decreasing function towards more acidic pHin, as observed in plant β-CA for both OCS (Protoschill-Krebs et al., 1996) and CO2 (Rowlett et al., 2002). In the following we will use the expression proposed by Rowlett et al. (2002) for CO2: kcatKm∝yCA(pHin)=11+10-pHin+pKCA. A value of pKCA= 7.2 was used that corresponds to the CA response of the wild-type Arabidopsis thaliana (Rowlett et al., 2002). The shape of the function yCA is shown in Fig. 2b.

A β-CA KM value for OCS (39 µM at 20 ∘C and pH 8.2) was estimated on pea (Pisum sativum) by Protoschill-Krebs et al. (1996). From a reanalysis of the same data set we also estimated a kcat of 93 s-1 at the same temperature and pH, leading to a kcat/Km value of 2.39 s-1 µM-1. To our knowledge this is the only report of kcat and Km values for OCS in β-CA.

The breaking of water film continuity that occurs at low soil water content leads to a reduction in microbial activity owing to the spatial separation of the microbes and their respiratory substrates (Manzoni and Katul, 2014). In our case soil water discontinuity should not affect OCS supply as gaseous OCS should be equally available in all soil pores. On the other hand different organisms may have different kcat/Km values so that the spatially-averaged kcat/Km could vary with drought-induced changes in microbial diversity. However our knowledge of how kcat/Km for OCS varies amongst different life forms is too scarce to know if it should increase or decrease during drought stress. We will therefore assume that soil water discontinuity does not affect kcat/Km directly. CA concentration ([CA]) could also vary during drought stress, although it is not clear in which direction. During water stress, microbial activity such as respiration or growth is usually reduced, but slow growth rates and heat stress have been shown to cause an up-regulation of CA-gene expression in Escherichia coli (Merlin et al., 2003), probably because of a need of bicarbonate for lipid synthesis. For this study we thus make the simplifying assumption that CA concentration does not vary with soil water content. The catalysed OCS uptake rate Scat is then simply proportional to soil water content (Eq. 9).

Destruction of OCS can also occur in the solid phase and was observed on pure mineral oxides with high basicity (Liu et al., 2008, 2009, 2010a). However, such catalytic reaction should be significant only in very dry soils (with only a few molecular layers of water) and in the absence of other competitive adsorbents such as CO2 (Liu et al., 2008, 2010b) and is therefore neglected in our model. The total soil OCS uptake rate is thus computed as S= kB θC: k=kuncat(T,pH)+xCA(T)xCA(20∘C)yCA(pHin)yCA(8.2)2390CA. Following common practice in the CO2 literature, we will also express k with respect to the uncatalysed rate at 25 ∘C and pH 4.5: k=fCAkuncat(25∘C,pH=4.5)xCA(T)/xCA(25∘C), where fCA is the so-called soil CA enhancement factor. We can see from Eqs. (11a)–(b) that fCA is not an intrinsic property of the soil and will vary with temperature and pH, even at constant CA concentration. In the case where the catalysed rate dominates k in Eq. (11a) and the internal pH is close to 8.2 we have: fCA≈ 111 [CA], where [CA] is in nM.

In some situations the OCS uptake rates can be overridden by OCS production. This is the case when soil temperature rises above 25 ∘C (Maseyk et al., 2014; Whelan and Rhew, 2015) or soil redox potential falls below -100 mV (Devai and Delaune, 1995). Light has also been proposed as an important trigger of OCS production, assuming photoproduction processes similar to those observed in ocean waters can occur (Whelan and Rhew, 2015). However the literature and data on this possible mechanism is still too scarce and not quantitative enough to be accounted for in our model.

The soil redox potential (Eh) is a very dynamic variable that is not easily measured in the field, especially in unsaturated soils (e.g. van Bochove et al., 2002). Although Eh and pH are linked, their relationship is not unique and depends on the set of oxidants and reductants present in the soil solution (e.g. Delaune and Reddy, 2005). Furthermore the soil redox potential is probably a more direct trigger for OCS production, as it defines when sulfate ions start to become limiting for the plants or the soil microbes (Husson, 2012). For this study we thus consider that, for anoxic soils at least, Eh is the primary driver of OCS production, independently of pH: P=PrefyP(Eh)Q10(T-Tref)/10, where Pref (mol m-3 s-1) is the production rate at temperature Tref (K) and low Eh (typically -200 mV) and Q10 is the multiplicative factor of the production rate for a 10 ∘C temperature rise. Because soil OCS emission, when observed in oxic soils, usually occurs at temperature around 25 ∘C or higher, we will set Tref as 25 ∘C and thus Pref=P25. According to results from Devai and DeLaune (1995), the function yP(Eh) may be expressed in the following manner: yP(Eh)=11+exp⁡-(Eh-100mV)/20mV, For oxic soils, Eq. (12a) would probably need to be modified to incorporate the effect of light on the OCS production rate (Whelan and Rhew, 2015) and the function yP(Eh) given by Eq. (12b) may not hold. In any case it would be difficult to evaluate. Whether we should use UV light only or total solar radiation could also be debated. For all these reasons we decided in this study to only look at the effect of temperature on the OCS production rate and its consequences on the total OCS deposition rate.

The one-dimensional mass balance equation (Eq. 2) can be rewritten: ∂εtC∂t=∂∂zDeff,a+αaqa∂C∂z+Deff,l+αlql∂BC∂z+P-kBθC. Assuming steady-state conditions, isothermal and uniform soil moisture and porosity through the soil column, this simplifies to the following: Dd2Cdz2-kBθC=-P, with: D=Deff,a+αaqa+Deff,l+αlqlB. Boundary conditions are C(z= 0) =Ca, the OCS concentration in the air above the soil column and dC/dz (z=zmax)=0, i.e. zero flux at the bottom of the soil column, located at depth zmax (the case for laboratory measurements). With such boundary conditions, the solution of Eq. (14) is the following: C(z)=z12P+(Ca-z12P)e-z/z1+ξ2e+z/z11+ξ2, with z12=D/kBθ and ξ=e-zmax/z1. This leads to an OCS efflux at the soil surface: F=kBθDCa-z12PD1-ξ21+ξ2, from which we can deduce the deposition velocity Vd=-F/Ca.

For field data sets, the condition at the lower boundary should be modified to dC/dz (z→∞) = 0 and the production rate P should be positive and uniform only over a certain depth zP below the surface. In this case the steady-state solution becomes the following: F=kBθD⋅Ca-z12PD(1-exp⁡(-zP/z1). We can verify that both equations give the same results if zmax→∞ and zP→∞ and also that Eq. (17) leads to F→-P×zP when k→0.

The steady-state OCS deposition model presented here (Eq. 16) was evaluated against measurements performed on different soils in the laboratory. For this purpose we revisited the data set presented in Van Diest and Kesselmeier (2008). Volumetric soil moisture content (θ, in m3 (H2O) m-3(soil)) was converted from gravimetric soil water content data (Mw,soil, in g(H2O) g(soil)-1) by means of the bulk density of the soil inside the chamber (ρb, in g cm-3): θ=Mw,soilρb/ρw, where ρw=1 g cm-3 is the density of liquid water. The soil bulk density was itself estimated from the maximum soil moisture content after saturation (θmax=Mw,soil,maxρb/ρw), assuming the latter corresponded to soil porosity (φ=1-ρb/2.66), i.e. (ρb=1/(Mw,soil,max/ρw+1/2.66). Soil thickness (zmax) was further estimated using ρb, soil dry weight (200 g for the German soil, 80 g for the other soils) and soil surface area (165.1 cm2) assuming soil density was uniform. Air porosity was calculated as εa=φ-θ. These estimates of θ, φ and εa were then used to compute D (Eq. (15), assuming |qa|=|ql| = 0) and F (Eq. (16b), with P= 0 and k estimated using Eq. (11b), with different fCA values for each soil temperature incubation). Note that in these experiments, the air in the chamber headspace was stirred with fans above the soil surface so that dispersion fluxes may be large (i.e. |qa| may not be zero). Without any more information about turbulence intensity at the soil surface in these experiments, we had to neglect this possible complication. We will discuss below how this simplification may affect the results of our simulations of these experiments.

Given the large diversity of expressions for the air tortuosity factor (τa) used to compute the effective diffusivity of OCS through the soil matrix, we felt it important to perform a sensitivity analysis of the model to different formulations available in the literature for τa. In Fig. 3 we show how the steady-state soil OCS deposition velocity model (Eq. 16b) responds to soil moisture or soil temperature for three different formulations of τa: Pen40 (τa= 0.66), MQ61 (τa=εa7/3/φ2) and Mol03r (τa=εa3/2/φ). We also indicate the optimal soil moisture (θopt) and temperature (Topt,Vd) for each formulation.

We found that the optimal temperature and the general shape of the response to temperature were not affected by the choice of the diffusivity model (Fig. 3, right panel). On the other hand the optimal soil moisture and the general shape of the response to soil moisture strongly depended on the choice made for τa (Fig. 3, left panel). In particular the model of Penman (1940) gives a perfectly symmetric response to soil moisture with an optimal value at θopt= 0.50φ, unlike other formulations: θopt≈0.23φ for Millington and Quirk (1961) and θopt≈0.29φ for Moldrup et al. (2003).

It is also noticeable on the right panel of Fig. 3 that the optimal temperature for Vd (Topt,Vd) is actually lower than the prescribed optimal temperature for the catalysed OCS hydrolysis rate (Topt,CA= 25 ∘C in this case), even in the absence of an OCS source term. This is because Topt,Vd integrates other temperature responses from the total effective diffusivity (D) and the OCS solubility (B). Although these variables do not exhibit a temperature optimum, their temperature responses affect the overall value of Topt,Vd. It can be shown analytically that this leads to Topt,Vd < Topt,CA.

Laboratory-based measurements of soil–air OCS fluxes are generally performed on small soil samples with a thickness of no more than a few centimetres. In contrast flux measurements performed in the field account for the entire soil column beneath the chamber enclosure. In order to see whether results from laboratory measurements could be directly applied to field conditions we performed a sensitivity analysis of the model to soil thickness (Fig. 4). We found that the responses to both soil moisture and soil temperature were affected by maximum soil depth (zmax), at least when zmax was below a few centimetres. Thin soils lead to lower maximum deposition rates but higher values of θopt and Topt,Vd. In Fig. 4 this is true mostly for zmax= 1 cm, and as soon as zmax reaches values above or equal to 3 cm, the response curve becomes almost indistinguishable from that obtained with zmax= 100 cm.

However this threshold on zmax also depends on soil CA activity. Results shown in Fig. 4 were obtained with an enhancement factor for OCS hydrolysis fCA of only 10 000. An even smaller enhancement factor would have led to a deeper transition zone (e.g. about 10 cm with fCA of 1000). This is because in Eq. (16b), the steady-state model of OCS deposition is proportional to tanh (zmax/z1). Given the shape of the hyperbolic tangent function, we expect our steady-state OCS deposition velocity model to become insensitive to zmax as soon as zmax/z1≥2. With z1=D/kBθ and because k is proportional to fCA we can see that this condition on zmax/z1 will depend on fCA. At fCA= 1000, we have z1(θopt)∼ 5 cm while at fCA= 10 000 we have z1(θopt) ∼ 1.5 cm.

This response to soil depth was already observed by Kesselmeier et al. (1999), who reported measurements of OCS deposition velocity that increased linearly with the quantity of soil in their soil chamber enclosure up to 200 g of soil and then reached a plateau at around 400 g. Because their soil samples were evenly spread inside the soil chamber, an increase in the quantity of soil directly translates into an increase in soil thickness. Using an enhancement factor fCA of 27 000 we were able to reproduce their saturation curve with soil weight using our steady-state model (Fig. 5). A lower fCA value would have reduced the curvature of the model but would have also lowered the maximum Vd (not shown, but see Fig. 6). A value for fCA of 27 000 was the best compromise to match the observed saturation curve. Because different soil weights were measured at different times with new soil material each time, it is possible that they would correspond to slightly different fCA values and this could explain the slight mismatch between the model and the fitted curve on the observations.

Our model has two main parameters that need to be constrained by observations: these are the CA concentration (or conversely the CA enhancement factor fCA) and the OCS production rate at 25 ∘C (P25). A sensitivity analysis of our steady-state OCS deposition model to these two parameters is shown in Figs. 6 and 7. Both parameters affect the maximum deposition rates but in opposite directions, with high fCA values leading to higher Vd and high P25 values leading to lower Vd. This was expected from Eq. (16b) as Vd is proportional to fCA and is linearly and negatively related to P25.

Interestingly, the optimal soil moisture is not modified by changes in fCA (Fig. 6, left panel) and only slightly by P25 (Fig. 7, left panel). This means that, provided that zmax is known precisely (or larger than 2z1, see Sect. 3.2), the overall shape of the response to soil moisture (as typically measured during a drying cycle) and the exact value of θopt are indicative solely of the diffusivity model to be used (Fig. 3). This result is important and should help us to at least decide whether the Pen40 formulation for τa must be used instead of a more asymmetrical one (the Mol03r and MQ61 formulations are harder to distinguish, see Fig. 3).

The value of Topt,Vd is also insensitive to changes in fCA (Fig. 6, right panel), but diminishes when P25 increases (Fig. 7, right panel). This means that very low optimal temperature values Topt,Vd (i.e. unusually low compared to expected values for enzymatic activities and Topt,CA) should be indicative of an OCS emission term, even if the values of Vd remain positive (i.e. the soil acts as a sink) in the temperature range explored. Of course at higher temperatures, and because in our model the OCS source term responds exponentially with temperature (Q10 response) while k exhibits an optimal temperature (Topt,CA), the Vd should reach negative values if the value of P25 is large and fCA is low. In some extreme cases where P25 fully dominates over fCA, our model could even predict OCS fluxes close to zero at temperatures below ∼ 10 ∘C that would increase exponentially at warmer temperatures, as it has been observed in some agricultural soils (J. Liu et al., 2010; Maseyk et al., 2014; Whelan and Rhew, 2015).

The sensitivity of our model to different soil pH was also tested. Because the effect of soil pH is mostly to modify the hydration rate k, we could not set a constant value of fCA. Instead we fixed the CA concentration in the soil (330 nM) and also adjusted the internal pH, assuming partial homeostasis with changes in soil pH, as observed in bacteria (Krulwich et al., 2011): pHin= 6 + 0.25 pH (Fig. 8). By assuming pHin to vary with changes in soil pH, we changed kcat (Eq. 11a) and this was equivalent to changing fCA. Indeed results shown in Fig. 8 are very similar to those shown in Fig. 6 where low pH (and pHin) correspond to low fCA values. If we had assumed that pHin was not modified by soil pH (and fixed at 8.2) no change in kcat would have been observed and the change in k would have only been caused by the effect of soil pH on kuncat (Eq. 11a). Unless the soil contains very little CA or the soil pH moves to very alkaline values (Fig. 2), this change in kuncat would have been too small to significantly affect Vd. Indeed at a CA concentration of 330 nM and with a pHin maintained at 8.2, our model Eq. (16b) gives exactly the same values for soil pH ranging from 4 to 9. In summary, within the range of soil pH found in nature, the response of Vd to this environmental factor is only happening through its influence on pHin and hence on kcat (Eq. 10b and Fig. 2b).

Our steady-state OCS deposition model was further evaluated against experimental data from Van Diest and Kesselmeier (2008) and results are shown in Figs. 9–12 for different soils. Because OCS deposition values observed by Van Diest and Kesselmeier (2008) were all positive we set the source term to zero (P25= 0), although we recognise that this may be an oversimplification. We also set the optimum temperature for the catalysed OCS hydration rate to 25 ∘C. A value for fCA was then manually adjusted for each soil and each temperature, between 21 600 and 336 000, depending on the soil origin and temperature (Figs. 9–12). Once this adjustment on fCA was done, our model, with the diffusivity formulation of Moldrup et al. (2003), was able to reproduce most observed response curves to soil drying (Figs. 9–12, left and middle panels). The model was also able to reproduce, within the measurement uncertainties, the temperature dependency of Vd at a soil moisture level of 0.12 m3 m-3 (far right panels in Figs. 9–12).

Many studies have clearly demonstrated that soil moisture strongly modulates OCS uptake by soils, with an optimal soil moisture content usually around 12 % of soil weight (Kesselmeier et al., 1999; J. Liu et al., 2010; Van Diest and Kesselmeier, 2008). As noted in some of these studies, such a bell-shape response is indicative of reactional and diffusional limitations at low and high soil moisture contents respectively. Using our steady-state formulation for shallow soils (Eq. 16b) we were able to reproduce the soil moisture response observed experimentally (Figs. 9–12). We also found that the observed asymmetric response to soil moisture was best captured by the soil diffusivity model of Moldrup et al. (2003) or Millington and Quirk (1961) and showed that the optimum soil moisture could be related to soil porosity: θopt= 0.3φ/1.3 for MQ61 and θopt= 2φ/7 for Mol03r. Using our model we were also able to explain the response of OCS uptake to soil weight (i.e. soil thickness) observed by Kesselmeier et al. (1999) (Fig. 5).

We also tested our model against observations of the temperature response of Vd. Empirical studies showed that, for a given soil, the maximum OCS uptake rate was modulated by incubation temperature, with an optimal temperature ranging from 15 to 35 ∘C (Kesselmeier et al., 1999; J. Liu et al., 2010; Van Diest and Kesselmeier, 2008). This temperature response was interpreted as an enzymatically catalysed process, governed by soil microorganism CA activity (Kesselmeier et al., 1999; J. Liu et al., 2010; Van Diest and Kesselmeier, 2008). To reproduce this response of Vd to incubation temperature using our steady-state model, we had to manually adjust fCA for each incubation temperature. We will argue here that using different fCA values on the same soil is justified given the way measurements were performed. Van Diest and Kesselmeier (2008) wanted to characterise the Vd response to soil drying at a set temperature and for this, they saturated a soil sample with water and acclimated it to a given temperature (between 5 and 35 ∘C), they then recorded the OCS exchange immediately and continued to measure until the soil was completely dry, which usually lasted one to two days. The same soil sample, or a different one from the same geographical location, was then rewatered and reacclimated to a different temperature and another cycle of measurements started. Sometimes several months separated measurements at two different temperatures but storage time (at 5 ∘C) did not seem to affect the soil CA activity (measurements on the same soil and incubation temperature were reproducible). On the other hand incubation temperature clearly differ and, at least for the German soil, samples were not all collected in the same season. This means that, for a given soil origin, the microbial community was experiencing different environmental conditions and history between each drying curve. Thus, the size and diversity of the microbial population were likely different for each incubation temperature, thus justifying the use of different enhancement factors at each temperature. Interestingly fCA tends to increase with incubation temperature, as we would expect for the microbial biomass. Only the German soil has a higher fCA at low temperature (15 ∘C) and this corresponds to a soil sampled at a different period (March) than the other two incubation temperatures (June).

Following this argument it seems that the optimum temperatures observed by Van Diest and Kesselmeier (2008) for different soil types are not a good proxy for the optimal temperature of CA activity (Topt,CA). Using our model we already showed that the optimum temperature for Vd (Topt,Vd) was different from Topt,CA, at least for deep soils (Fig. 4). A closer inspection of the results shown in Figs. 9–12 also show that the adjusted fCA values closely follow the patterns of the maximum Vd at θopt (see right panels in Figs. 9–12). This means that the optimum temperature observed by Van Diest and Kesselmeier (2008) is a better indicator of maximum fCA or equivalently maximum CA concentration (assuming all the CAs in the soil have similar kcat/KM as the pea extracts measured by Protoschill-Krebs et al. (1996)). This could explain why the optimum for the German soil was so low (around 15 ∘C), i.e. lower than expected for Topt,CA. The presence of a competing enzymatic process, such as OCS emission, could have explained this low Topt,Vd value (Fig. 7) but it is more likely that the soil sample studied at 15 ∘C contained more CA than those used for other incubation temperatures. Measurements on microbial biomass could have helped confirm this hypothesis but were unfortunately not made.

Because fCA is a fitting parameter in our model, it is important to see if the values that we derived for the different soils are realistic. There are two ways to do so. First, we have a relatively good idea of how much CA is needed inside the cytosol of leaf mesophyll cells or in unicellular algae, which is of the order of 100 µM (Tholen and Zhu, 2011). Assuming this CA concentration value is also applicable to microbial cells and using estimates of the soil microbial population size, we can convert this physiological CA concentration ([CA]in) into a CA concentration in the soil matrix ([CA]): [CA]θ= [CA]inρmic, where ρmic (m3 microbes m-3 soil) denotes the volumetric microbial content of the soil. Using a typical microbial population size of 3 × 109 cm-3 and an average cell size of 1 µm3 (Wingate et al., 2009), we obtain a microbial content of ρmic=0.003 m3 m-3 and a soil CA content of about 1000 nM (we used θ= 0.3). Using this value of [CA] and the kcat/KM value for OCS (2.39 s-1 µM-1 at 20 ∘C and pHin 8.2) this leads to an fCA value of about 127 000 for OCS, which is in the same order of magnitude as those found for the different soils in this study (between 21 600 and 336 000, with a median value at 66 000). From this crude calculation we can conclude that our fCA estimates are physiologically meaningful.

Another way of checking if our fCA estimates are meaningful is to convert them into fCA equivalents for soil CO2 isotope fluxes, for which we have a better idea of what the expected values should be (Seibt et al., 2006; Wingate et al., 2008, 2009, 2010). The kcat/KM value for CO2 in pea extracts has been measured for a pH range of 6–9 and at 25 ∘C (Bjorkbacka et al., 1999). The pH response described a similar pattern as the one found for Arabidopsis by Rowlett et al. (2002) (Fig. 2) with a pKa of 7.1. Using xCA(T) and yCA (pHin) to convert those values to pHin 8.2 and 20 ∘C, we obtain a kcat/KM value for CO2 of 50 s-1 µM-1, i.e. about 20 times greater than the kcat/KM for OCS. Given the difference in uncatalysed hydration rates between the two gas species (12 000 µs-1 for CO2 and 21.5 µs-1 for OCS at 25 ∘C and pH = 4.5) this means that at equal soil CA concentration, the fCA for CO2 should be about 30 times smaller than that derived for OCS. This corresponds to a median fCA value of 2200 for CO2, i.e. at the higher end of values observed in different soils (Wingate et al., 2009).

The calculation above considers only β-CA kinetic parameters to relate the soil CA enhancement factor for OCS to the fCA for CO2. However other enzymes can catalyse OCS hydrolysis and not have a strong affinity to CO2. For example Smeulders et al. found a carbon disulfide hydrolase from an acido-thermophilic archaeon that was very efficient at catalysing OCS hydrolysis but did not have CO2 as one of its substrates (Smeulders et al., 2012). More recently, Ogawa et al. (2013) found in Thiobacillus thioparus, a sulfur-oxidising bacterium widely distributed in soils and freshwaters, an enzyme that shared a high similarity with β-CAs and was able to catalyse OCS hydrolysis with a similar efficiency (KM= 60 µM, kcat= 58 s-1 at pH 8.5 and 30 ∘C) but whose CO2 hydration activity was 3–4 orders of magnitude smaller than that of β-CAs. For this reason they called this enzyme carbonyl sulfide hydrolase (COSase). The carbon disulfide hydrolase identified by Smeulders et al. (2012) may only be present in extremely acidic environments such as volcanic solfataras, but the COSase found in T. thioparus may be more ubiquitous in soils. If this was the case this would imply that the fCA ratio of OCS to CO2 is not unique and could, in some soils, be higher than the same ratio derived from β-CA kinetic parameters only. This could partly explain the highest fCA values obtained here for OCS.

Higher than expected values of fCA could also be explained by the fact that we neglected dispersion fluxes when we compared the model against observations. Indeed dispersion fluxes would enhance OCS diffusion (Eq. 15) and result in larger deposition velocities (Eq. 16b) for the same level of CA concentration. Results from Maier et al. (2012) show that the diffusivity D could be easily doubled by the presence of turbulence above the soil surface, which would be equivalent to a doubling of k (D and k appear as a product in the sink term of Eq. 16b). This means that if dispersion occurred in the experiments (a possibility that we cannot rule out), the fCA values that we derived from them may be overestimated by a factor of two, bringing them closer to values compatible with CO2 studies.

To conclude, the fCA values derived here for OCS seem compatible with physiological CA contents and also compatible with fCA values reported in CO2 isotope studies, given possible affinity differences of some CAs towards OCS and CO2 and possible artefacts of mechanical dispersion caused by fans in some laboratory experiments.

Response curves of OCS deposition rates to soil moisture and temperature have been derived from laboratory experiments similar to those presented here (Kesselmeier et al., 1999) and the derived equations have been used to estimate the OCS uptake by soils at the global scale (Kettle et al., 2002). In addition Van Diest and Kesselmeier have proposed that the optimum (gravimetric) soil moisture content for OCS deposition was around 0.12 g g-1, independently of soil type (Van Diest and Kesselmeier, 2008). Our model allows us to verify if such simplification or extrapolation is justified, on a theoretical point of view at least. For semi-infinite soil columns we showed that θopt varied with soil porosity from 0.23φ to 0.5φ, depending on the soil diffusivity model used. Assuming soil bulk density is 2.66(1-φ), this leads to gravimetric soil moisture contents of between 0.61φ(1-φ) and 1.33φ(1-φ), which is clearly dependent on soil type. Also from Fig. 4 we can see that the general shape of the soil moisture response and θopt strongly depend on the exact soil depth used during the experiment, at least for soil less than 3 cm thick (or more if the CA activity is lower). For thicker soils the deepest soil layers do not contribute to the exchange and we reach the saturation point with soil weight shown in Fig. 5. However in both aforementioned studies (Kesselmeier et al., 1999; Van Diest and Kesselmeier, 2008), care was taken not to reach the saturation point (using soil weights of about 80 g). From our model results we can see that this would lead to an overestimation of θopt and an overall underestimation of Vd (Fig. 4). Thus based on this observation we would recommend to use soil depths of at least 5–6 cm in future studies so that the results can be more readily extrapolated to field conditions.

Another difficulty when we want to extrapolate laboratory data to the natural environment is that soil disturbance prior to the experiment (sieving, repacking …) strongly modifies the gas diffusivity properties of the soil. Our results show that OCS deposition rates can be extremely sensitive to the choice of the diffusivity model used (Fig. 3). In highly compacted, highly aggregated soils the gas diffusivity response to soil moisture content can even become bimodal (Deepagoda et al., 2011), which would certainly have a strong impact on the Vd-θ relationship. Even without such a complication, our results suggest that deposition rate measurements on repacked soils may not be representative of field conditions because the soil treatment would modify the diffusivity properties of the soil and alter the soil moisture response of the OCS deposition rate. On the other hand, our model (Eq. 17, for semi-infinite soil column) with a soil diffusivity formulation for undisturbed soils (i.e. Mol03u or Deepa11, see Table 1) could be used for interpreting field measurements.