We use a face-centered finite-volume grid for discretizing the soil column, with 26 computational nodes down to 1 m depth. The vertical grid is constructed using an equation similar to that in the Community Land Model 4.5 , which is more densely spaced at the upper boundary. This has advantages of resolving litter layers at the surface, and reducing the computational demand. The depths of the computational nodes are defined as zi=exp⁡0.2i-5,i=0,…,N, where N=25. The thicknesses of control volumes are Δzi=z0+z12,i=0zi+1-zi-12,i=1,…,N-1zN-zN-1,i=N.

The depths of the layer interfaces are defined as zi+1/2=zi+zi+12,i=0,…,N-1zN+zN-zN-12,i=N.

The use of face-centered rather than node-centered control volumes generally gives better evaluation of diffusive fluxes across interfaces.

We use the Crank–Nicolson method to discretize the diffusion–reaction equation, which ensures numerical stability when using large time steps . The method is implemented with second-order accuracy in space and in time. The discretized finite difference equations are a sparse linear system that can be easily solved.

We first discretize spatially Eq. () on the defined grid by integrating in each control volume , ηiΔzidCidt=Ja→0-J0→1+S0Δz0,i=0Ji-1→i-Ji→i+1+SiΔzi,1≤i≤N-1JN-1→N+SNΔzN,i=N, where ηi=(kHθw+θa) at z=zi is a simplifying notation, Ja→0 represents diffusive flux from the atmosphere to soil layer 0, and Ji-1→i represents diffusive flux from layer i-1 to layer i. The flux term J can be evaluated with a central difference scheme, Ji-1→i=Ji-1/2=-D∂C∂zi-1/2=-Di-1/2Ci-Ci-1zi-zi-1.

The diffusivity at the interface, Di-1/2, can be interpolated linearly, Di-1/2=Di+Di-1/2, since the interface locates at the center of two computational nodes. For the ease of denotation, we define Gi-1/2=Di-1/2/zi-zi-1 as the conductance at the interface.

The flux at the topmost layer is thus Ja→0=-Da→0C0-Caz0-0=-Ga→0(C0-Ca).

Again, Da→0 is the diffusivity at the soil–atmosphere boundary, and Ga→0 is the conductance. We use the harmonic mean of soil diffusivity D0 and air diffusivity Da for Da→0, following the conductance model in , Da→0=21D0+1Da.

By rewriting Eq. () in a matrix form, we obtain an ordinary differential equation (ODE) system for time, AddtC=BC+S, where C=C0,C1,…,CNT,A=diagη0Δz0,η1Δz1,…,ηNΔzN,B=-Ga→0-G1/2G1/20⋯000G1/2-G1/2-G3/2G3/2⋯000⋮⋮⋮⋱⋮⋮⋮000⋯GN-3/2-GN-3/2-GN-1/2GN-1/2000⋯0GN-1/2-GN-1/2,S=S0Δz0+Da→0z0Ca,S1Δz1,…,SNΔzNT.

The above linear ODE system is then discretized at time step t=(n+1/2)Δt: ACn+1-CnΔt=BCn+1+Cn2+S.

Therefore, the evolution in time is, Cn+1=(2A-ΔtB)-1(2A+ΔtB)Cn+2ΔtS.

At each time step, the concentration profile is evolved to the next time step with the diffusion–reaction operator. Thus, the model can simulate transient conditions in real time.

The diffusivity of COS in soil media (D, m2 s-1) is calculated following , DDm=θa2θaθsat3/bTTrefn, where θsat (m3 m-3) is the soil total porosity (θsat=θa+θw), Dm (m2 s-1) is the COS-air diffusivity at Tref=25 ∘C and 1 atm, n=1.5 , and b is parameterized with a water retention function . We assume that the same function holds for litter layer diffusivity since litter is also a porous medium like the soil.

For COS, Dm is calculated as 1.337×10-5 m2 s-1 at 25 ∘C and 1 atm, based on the theoretical CO2 / COS diffusivity ratio of 1.21 derived in from the Chapman–Enskog kinetic theory, and the molecular diffusivity of CO2 (1.618×10-5 m2 s-1, 25 ∘C and 1 atm) is calculated from .

In solubility equilibrium, the chemical potentials of COS in gas phase (μg) and in aqueous solution (μaq) are equal: μg=μg⊖(T)+RTln⁡pp⊖+RTln⁡ϕ=μaq*(T,p)+RTln⁡x+RTln⁡γ=μaq, where p is the partial pressure in gas phase, x is the molar fraction in aqueous phase, ϕ is the fugacity coefficient, γ is the activity coefficient, and p⊖ is the standard pressure. Here, μg⊖ and μaq* are both chemical potentials under chosen standard conditions. Note that μg⊖ depends only on temperature, whereas μaq* depends on both temperature and pressure.

For a dilute solution, as x→0 and p/p⊖→0, we have ϕ→1 and γ→1. This is valid for COS in natural environments because its molar fraction is only ∼10-10 in the atmosphere and ∼10-12 in seawater . Therefore, μaq*-μg⊖=-RTln⁡xp/p⊖≡-RTln⁡kH,xp, where kH,xp is Henry's law constant defined as the ratio of aqueous-phase molar fraction to gas-phase partial pressure. According to the van't Hoff equation, ∂ln⁡kH,xp∂T=ΔsolHm⊖RT2, where ΔsolHm⊖ is the standard partial molar enthalpy of dissolution. We assume that it is constant within the temperature range of ambient conditions. With respect to a reference state T0, integrating yields ln⁡kH,xp=-ΔsolHm⊖R1T-1T0.

Temperature dependence of kH,xp can thus be described by the following function, with A and B constants: kH,xp=exp⁡AR+BRT.

The dimensionless Henry law constant, kH,cc, is defined as the ratio of aqueous molar concentration to gaseous molar concentration. kH,cc≡caqcg=ρwx/Mwp/(RT)=ρwRTMwp⊖⋅xp/p⊖=ρwRTMwp⊖kH,xp=Texp⁡AR+ln⁡ρwRTMwp⊖+BRT.

Thus, we can build a nonlinear regression model for temperature dependence of the dimensionless Henry law constant: kH,cc(T)=(T/K)exp⁡α+βT/K.

Using data for the regression (Fig. ), we obtained that α=-20.00-1.46+1.48, and β=4050-430+424 (95 % confidence interval).

Because both COS uptake and production activities exist in soils and the net COS flux exhibits a linear response to COS concentration , we assume that uptake and production are separable terms.

Soil COS uptake (U) is formulated based on Michaelis–Menten kinetics, with dependence on soil temperature and moisture: U=-VSU,max⋅kHCKm+kHC⋅f(T)⋅g(w), where Km=1.9 mol m-3 is the Michaelis constant for COS hydrolysis by CA , VSU,max (mol m-3 s-1) is the soil COS uptake capacity, and f(T) and g(w) are temperature and moisture limitation functions.

We assume that the temperature limitation function for soil COS uptake reflects the temperature dependence of enzyme activity, which is described as f(T)=ATTexp⁡-Δ‡GcatRT1+exp⁡-ΔHeqR1T-1Teq, where Δ‡Gcat (J mol-1) is the activation free energy of the transition state of the CA enzyme in terms of COS reactions, ΔHeq (J mol-1) is the enthalpy change during conversion from activated to inactivated form of the enzyme, Teq (K) is the temperature at which the concentrations of the activated enzyme and the inactivated enzyme are equal, AT is the pre-exponential factor to normalize the equation so that max⁡f(T)=1. The function has a temperature optimum, Topt, defined at which the function reaches its maximum value. This temperature optimum is close to the parameter Teq and always smaller than it .

Other enzymes in soil microbes may also contribute to COS uptake, for example, COSase, nitrogenase and CS2 hydrolase . These enzymes are not considered here due to lack of information on their kinetics and distribution in the microbial community. They are also unlikely to be as ubiquitous as CA.

Typical fit values of Δ‡Gcat for a range of enzymes are between 51 and 88 kJ mol-1, whereas ΔHeq varies from 86 to 826 kJ mol-1 . We assume that Δ‡Gcat=84.10 kJ mol-1 (20.1 kcal mol-1) for CA-catalyzed COS hydrolysis, based on calculations of CA–COS nucleophilic attack, the rate-determining step of COS hydrolysis, which used [(H3N)3Zn(OH)]+ as a structural analog of the active site of CA .

The temperature dependence (Eq. ) has a similar mathematical form as that in , but the temperature limitation of COS uptake is now linked to enzyme kinetics that can be determined in the laboratory. Using Δ‡Gcat=84.10 kJ mol-1, ΔHeq=358.9 kJ mol-1, and Teq=293.14 K, we can approximate the temperature function φ(T) in (Fig. a). In this study, Teq will be an adjustable parameter to fit temperature dependence for different soils (see Sect. 2.4.2 and Table ).

For the parameterization of moisture dependence, we use a simple bell-shaped function, described by the form of the Rayleigh distribution function: g(w)=Aw⋅wwopt2⋅exp⁡-w2wopt2, where wopt (m3 m-3) is the optimal water content for COS uptake, and AW is the normalization factor such that max⁡g(w)=1. The parameter wopt can be adjusted to fit observed data (see Sect. 2.4.2 and Table ). The function has a wider range than that in (Fig. b), and does not approach zero at high soil moisture. It represents the dependence of microbial uptake per se instead of a combination of diffusion and microbial uptake, because diffusion has already been accounted for. We found no empirical reason to assume that microbial uptake of COS be zero at high soil water content if it is not diffusion-limited.

Soil COS production is represented as an exponential function of temperature, P=VSP,max⋅exp⁡kT(T-Tref), where VSP,max (mol m-3 s-1) is the soil COS production capacity, kT (K-1) is the temperature dependence factor that is equivalent to Q10=1.9, and Tref=25∘C. The Q10 value here is set as the average value of the Q10 associated with the two regression lines of COS flux vs. temperature in . Lab incubation of various soils has found that soil COS emissions generally exhibit exponential responses with temperature . Although COS may be produced by both soil microbes and roots , we use a single production term in Eq. () due to a lack of data on the individual temperature responses of the different components.

Litter was present at one of the model evaluation sites (Stunt Ranch). We obtained an equation for litter COS fluxes based on an incubation experiment with litter at the Stunt Ranch site , FSL=-VLU,max⋅kHCKm+kHC⋅sinh⁡kLwL+VLP,max⋅exp⁡kT(T-Tref), where VLU,max (mol m-3 s-1) and VLP,max (mol m-3 s-1) are the litter COS uptake and production capacities, kT (K-1) is the temperature dependence factor for COS production equivalent to Q10=1.9, and kL (dimensionless) is the moisture limitation factor for COS uptake. The hyperbolic sine function is chosen to describe the exponential dependence on litter water content.

Litter porosity is usually much larger than that of soils. In the model, the porosity at the grid point near the litter–soil interface is interpolated so as to prevent numerical instability caused by discontinuity (Fig. ).

Temperatures of the litter layers were interpolated between soil and chamber air temperature by assuming a logarithmic temperature profile in the surface layer, according to mixing length theory . The roughness length was assumed to be 0.01 m .

Southern Great Plains, OK (SGP): soil COS fluxes were measured from 1 April to 31 May 2012 in a wheat field at the ARM (Atmospheric Radiation Measurement) Southern Great Plains Central Facility (36.61∘ N, 97.49∘ W) near Billings, OK. The soils are mainly silt loam to clay loam, with 33 % sand, 22 % silt, and 45 % clay . Further details on the site and data are provided in .

Stunt Ranch, CA (SR): surface COS fluxes were measured from 1 April to 15 April 2013 in a Mediterranean oak woodland at the University of California Stunt Ranch Santa Monica Mountains Reserve (34∘05′38′′ N, 118∘39′26′′ W) in southern California. The soil is a gravelly sandy loam, covered with 2 cm of leaf litter from a coast live oak tree (Quercus agrifolia). The site received 2 mm of precipitation the day before the measurements started. The experimental setup was largely identical to that used at SGP see. Further details on the site and data are provided in .

Site-specific parameters are summarized in Table . We used site-specific values for soil porosity that were constant with depth except in the top few centimeters when there is loose topsoil. Soil porosity is estimated as 0.50 m3 m-3 at the SGP site from the maximum observed soil moisture , but that in the top 2 cm is assumed to be 0.60 m3 m-3 as the topsoil is looser due to agricultural activity. At the SR site, soil porosity was measured as 0.35  m3 m-3. We assume the field capacity (θFC) is 0.20 m3 m-3 at the SR site based on soil textures . For the parameter b in Eq. (), we use 4.9 for SR (sandy loam) and 5.3 for SGP (silt loam) based on .

Soil temperature profiles are modeled with the observations at 5 cm depth. The temperature signals are considered as a superposition of fast varying diurnal signals (TF) and slowly varying synoptic-scale signals (TS) , T(z,t)=TS+TF⋅exp⁡-z/zTsin⁡(ωt+ψ-z/zT), where zT is the damping depth for diurnal temperature waves, ω=2π day-1 is the angular frequency, and ψ is the phase constant. zT is determined from zT=2αT/ω where αT (m2 s-1) is the soil thermal diffusivity, calculated from soil mineral composition and water content using empirical formulae in and . Soil thermal diffusivities are 2.5–5.0×10-7 m2 s-1 at the SGP site and 6.8–8.1×10-7 m2 s-1 at the SR site, assuming typical mineral compositions. The thermal diffusivities are translated to average damping depths of 0.11 m (SGP) and 0.14 m (SR), respectively.

For the SR data, we observed that the temperature optimum for COS uptake is Topt≈13 ∘C . This is reproduced by setting Teq at 15 ∘C in Eq. (). For the SGP soil, because COS flux did not show a temperature optimum for uptake during the observed temperature range (7–47 ∘C) in , we set Teq=10 ∘C (equivalent to Topt≈7 ∘C) to get a monotonic relationship with temperature.

Soil moisture was measured at 5 cm depth at both sites, and additionally at 30 cm at SGP. We generate soil moisture profiles for the SGP site by interpolating between the 5 and 30 cm data and assuming constant soil moisture below 30 cm (cf. 1-D simulations with gravity drainage boundary condition in ). For the SR site, we assume soil moisture at 1 m depth is at field capacity (θFC, m3 m-3), and interpolated smoothly between the values at the surface and 1 m depth (Fig. ). The conditions below 10 cm do not significantly affect surface COS flux according to Fig.  (see also Sect. 4.2). The optimum water content for soil COS uptake, wopt, is assumed to be 0.20 m3 m-3 for SGP soils, corresponding to a threshold for the COS flux-temperature response . For SR soils, we set wopt=0.14 m3 m-3, estimated from lab incubation data of the same soils .

Since litter was present at the SR site, litter layers are included in model simulations when validating the model with the SR data set. The 2 cm thick litter layer is represented in the top six grid points. Litter porosity was measured to be 0.94 m3 m-3. Litter water content (LWC) is an important variable controlling litter COS fluxes. Since LWC was not measured at SR, we test two scenarios: (1) fast decreasing LWC, and (2) slowly decreasing LWC following the precipitation event on the day before the measurements started. We also evaluate the surface COS flux in the absence of litter fluxes.