Moisture function

Function development: The moisture function, f_m, was developed based on the primary physicochemical and biological processes controlling HR in soils. Organic carbon (C) is assumed to initially adsorb onto soil mineral surfaces and is consumed by microorganisms after two steps: the SOC converts to DOC after desorption, and the DOC is diffused to regions where microorganisms inhabit.

The flux of DOC released from SOC can be estimated by²⁴

where θ is water content [m³ m⁻³], K_θ is a moisture constant reflecting the impact of moisture content on C desorption [m³ m⁻³]²⁴, α is the mass transfer coefficient between SOC and DOC [s⁻¹]⁴⁶, and m_SOC is organic C content per unit area of soils [kg m⁻²]. The released DOC is biologically degraded after diffusing into regions containing microorganisms, thus the turnover rate of SOC is related to the degree of collocation between SOC and microorganisms.

In soils where SOC and microorganisms are completely separated, the flux of bioavailable DOC for HR can be described by⁴⁵

where ϕ is soil porosity [−], m_s and n_s are cementation and saturation exponents [−], accounting for the effects of pore structure and water connectivity on DOC diffusion⁴⁵.

In soils where SOC and microorganisms are completely collocated, the released DOC can be degraded locally without diffusion. Therefore, the flux of bioavailable DOC for HR is the same as the flux of total available DOC

For most soils, microorganisms are partly separated from SOC: released DOC is degraded either locally or after diffusion. We introduce a parameter a, the SOC–microorganism collocation factor, to represent the degree of collocation between SOC and microorganisms. Consequently, the flux of bioavailable DOC for soil HR can be described by

where a increases as the degree of collocation between SOC and microorganisms decreases. Given that a = 0 when SOC and microorganisms are completely collocated (Eq. ⁴) and a = 1 when they are completely separated (Eq. ³), 0 < a < 1 is presumed when they are partly collocated. Therefore, Eq. ⁵ with 0 ≤ a ≤ 1 uniformly describes the relationship between HR rates and water content for soils with full degrees of collocation between SOC and microorganisms.

When soil HR is limited by organic C bioavailability, the HR rate is determined by the flux of bioavailable DOC and its response to water content should be the same as for the flux of bioavailable DOC. Therefore, we hypothesize that, when organic C is limiting, the relationship between soil HR rates and water content can be described by the SOC–microorganism collocation, which is represented by a as in Eq. (⁵).

Soil HR becomes O₂ limited when water content is above the optimal value, θ > θ_op. Considered that O₂ diffusion through liquid can be ignored compared with that through air⁶⁵, the supply rate of O₂ from the atmosphere to soils can be estimated using the gaseous O₂ diffusion at the soil–atmosphere interface,

where D_GO is the effective diffusion coefficient of gaseous O₂ at the soil–atmosphere interface [m² s⁻¹], and can be estimated by⁶⁶

where m_g and n_g are cementation and saturation exponents accounting for the effects of pore structure and air connectivity on O₂ diffusion in soils⁴⁵, respectively, and D_GO,0 is the diffusion coefficient of O₂ in pure air [m² s⁻¹]. ∇_O2 is the gradient of gaseous O₂ concentrations between the top soil surface and the atmosphere [g l⁻¹ m⁻¹], and can be expressed by¹⁸

where k_GO is a coefficient representing the degree of oxygen depletion in soils, and ω reflects the impact of soil pore connectivity on O₂ transport. Substituting Eqs. (⁷) and (⁸) into Eq. (⁶), we have

where b, b = 1 + n_g + ω, is a parameter reflecting the effects of soil characteristics on O₂ supply at the soil–atmosphere interface, called the O₂ supply restriction factor.

The supplied O₂ diffuses into soils and enters water to form dissolved oxygen (DO), which is eventually consumed by microorganisms. Regardless of O₂ delivery from plant roots, the flux of bioavailable DO for soil HR should be the same as the flux of O₂ supply from the atmosphere

When soil HR is limited by O₂ bioavailability, its rate response to water content should be the same as for the flux of bioavailable DO. Therefore, we hypothesize that, when O₂ is limiting, the relationship between soil HR rates and water content can be described by the O₂ supply restriction factor, b, as shown in Eq. (⁹).

Theoretically, the soil HR rate maximizes when bioavailable DOC and DO are both limiting¹⁸, i.e., the supplied DO is stoichiometrically enough to react with the bioavailable DOC, F_DO = ν_DOF_DOC, where ν_DO is the stoichiometric coefficient of DO with respect to DOC [g g⁻¹]. Correspondingly, the water content was regarded as optimum water content, θ_op, that can be calculated by

Considered only water content related terms and normalized to the maximum HR rate, the process-based moisture function, f_m, can be expressed by

Function parameterization and evaluation: Parameter and initial values used in the simulations are presented in Supplementary Table ¹. For the evaluation of analytical θ_op (Fig. 4), a = 1 and b = 1.7 were used to calculate the values of analytical θ_op because homogenous soils were utilized, and m_SOC was calculated using ρ_s(1 − ϕ)HC_SOC where ρ_s is the density of soil mineral³⁹. k_GO was estimated by the intercepts of log(∇_O2) − log(ϕ − θ) curves with y-coordinate, whose value equals log(k_GO) + ω log(ϕ − θ) (Fig. 3a). The simulation results showed that k_GO primarily changed with SOC content and its value could be estimated using kGO=0.7465CSOC0.512 (Fig. 3a, Supplementary Fig. ²).

For the applications of f_m to laboratory and field observations (Fig. 7), the values of α and ν_DO were not given in the literature and were estimated using Eq. (¹¹) in Methods section, in which a, b, and θ_op were fitted using the measured data. Note b = 0.75 was used for the sandy loam in the laboratory incubation¹⁰, where the measured HR rates were available only for θ < θ_op. Consequently, α × ν_DO = 3.56 × 10⁻⁸ s⁻¹ for the sandy loam¹⁰ and 2.9 × 10⁻⁷ s⁻¹ for the loam²⁵.

Function application:The application of f_m requires to determine six parameters. The SOC–microorganism collocation factor, a, can be estimated using clay content c_c, a = 2.8c_c − 0.046. For soils with low clay content (c_c < 0.016 g g⁻¹), we assume a = 0; for soils with high clay content (c_c > 0.37 g g⁻¹), we assume a = 1. The O₂ supply restriction factor, b, is assumed as constant, b = 0.75, in practical applications. The optimum water content, θ_op, can be calculated implicitly using Eq. (¹¹) in Methods section, and its value depends on soil properties, as well as the values of a and b. If these properties are not available, we assume θ_op/ϕ = 0.65, a value widely observed in laboratory and fields^4,18,26. The soil porosity, ϕ, can be estimated using bulk density ρ_b⁶⁷, ϕ=1-ρbρs. The saturation exponent, n_s, is relatively invariable, and can be assumed to be constant⁴⁵, n_s = 2. The moisture constant, K_θ, depends on organo-mineral associations, and its value can be assumed to be constant, K_θ = 0.1, when unavailable²⁴. The determination of parameter values in the application of f_m were summarized in Table ¹.