We collected the literatures that reported the τsoil based on measurements (Text S1 in the Supplement): (1) δ13C shifts after successive changes in C3-C4 vegetation, (2) measurements of CO2 production in laboratory SOC incubation over at least 7 months, and (3) simultaneous measurements of SOC stock and heterotrophic respiration (stock over flux). We constructed a database containing the measured τsoil from 187 sites across the globe (Fig. 1). Based on the homogenous assumption, the soil system is a time-invariant linear system at the steady state. The τsoil derived from this database is under a one-pool assumption. The information of climate (e.g., mean annual temperature and precipitation) was also collected from the literature or extracted from the WorldClim database version 1.4 (http://worldclim.org/, last access: 1 June 2019, Hijmans et al., 2005) if literature was not available. The WorldClim dataset provided a set of free global climate data for ecological modeling and Geographic Information System analysis with a spatial resolution of 0.86 km2 (Hutchinson et al., 2004). We extracted the mean temperature and precipitation by averaging the monthly climate data over 1990–2000 for those observational sites with missing climate information. The classes of biomes were processed to match the seven biome classifications adopted by the MODIS land cover product MCD12C1 (NASA LP DAAC, 2008; Friedl et al., 2010) and Todd-Brown et al. (2013) (Fig. S1): (1) tropical forest including evergreen broadleaf forest between 25∘ N and 25∘ S; (2) temperate forest including deciduous broadleaf, evergreen broadleaf outside of 25∘ N and 25∘ S, and mixed forest south of 50∘ N; (3) boreal forest including evergreen needleleaf forest, deciduous needleleaf forest, and mixed forest north of 50∘ N; (4) grassland and shrubland including woody savanna south of 50∘ N and savanna and grasslands south of 55∘ N; (5) deserts and savanna including barren or sparsely vegetated open shrubland south of 55∘ N and closed shrubland south of 50∘ N; (6) tundra; and (7) croplands. Other land cover types like permanent wetland, urban, and bare land were not included in this study.

The historical simulation outputs of 12 ESMs participating in CMIP5 from 1850 to 1860 (https://esgf-data.dkrz.de/search/cmip5-dkrz/, last access: 11 January 2016) were analyzed in this study (Table S1). For each model, the SOC, litter C, NPP, and heterotrophic respiration (Rh) were extracted from the outputs in historical simulations (cSoil, cLitter, npp, and rh, respectively, from the CMIP5 variable list). The litter and soil carbon were summed as the bulk soil carbon stock. Among the 12 models, only the inmcm4 model did not output NPP, so we calculated it as gross primary production minus autotrophic respiration. Due to the diverse spatial resolutions among the models, we aggregated the results of different models to 1∘×1∘ with the nearest interpolation method (Fig. S2). The τsoil of SOC was calculated as the ratio of carbon stock over flux (NPP or Rh): τsoil=SOCflux.

To examine whether the major findings of this data–model comparison are affected by the one-pool homogenous assumption, we fitted a three-pool model with observational data and model ensemble outputs at the biome level. In this study, a three-pool C model consisted of fast, slow, and passive pools and carbon transfers among three pools (Fig. S3a). This model shares the same framework with the CENTURY and the terrestrial ecosystem models (Bolker et al., 1998; Liang et al., 2015). The dynamics of soil carbon pools follow first-order differential kinetics. The total C stocks and CO2 efflux from observations and the CMIP5 ensemble were separated into pool-specific decomposition rates by the deconvolution analysis (Fig. S3a, Liang et al., 2015). We assumed the total soil carbon input equals total soil respiration at the steady state.

Based on the theoretical analysis, the dynamics of the three-pool model can be mathematically described by the matrix equation (Luo et al., 2003; Xia et al., 2013) as dCtdt=I(t)-AKCt, where the matrix C(t)=(C1(t), C2(t), C3(t))T is used to describe soil carbon pool sizes. A is a matrix given by A=-1f12f13f21-10f31f32-1. The elements fij are carbon transfer coefficients, indicating the fractions of the carbon entering the ith (row) pool from the jth (column) pool. K is a 3×3 diagonal matrix indicating the decomposition rates (the amounts of carbon per unit mass leaving each of the pools per year). The matrix of K is given by K=diag (k1, k2, k3).

The parameters in the three-pool model were estimated based on Bayesian probabilistic inversion (Eq. 4). The posterior probability density function Pθ|Z of model parameters (θ) can be represented by the prior probability density function (P(θ)) and a likelihood function (P(Zlθ)) (Liang et al., 2015; Xu et al., 2016). The likelihood function was calculated by the minimum error between observed and modeled values with Eq. (5). In this study, we adopted the prior ranges of model parameters from Liang et al. (2015). Pθ|Z∝P(Z|θ)⋅P(θ) PZ|θ∝exp⁡-12σi2(t)∑i=1n∑t∈obs(Z)[Zit-Xit]2 Here Zit and Xit are the observed and modeled transit times, and σi2(t) is the standard deviation of measurements. The posterior probability density function of the parameters was constructed with two steps: a proposing step and a moving step. In the first step, the dataset was generated based on the previously accepted data with a proposal distribution: θnew=θnew+d(θmax-θmin)D, where θmax and θmin are the maximum and minimum values of the given parameters, d is the random variable between -0.5 and 0.5 with uniform distribution, and D is used to control the proposing step size in this study. In the moving step, the new data θnew is tested against the Metropolis criteria to quantify whether it should be accepted or rejected. The parameters of posterior probability density function were constructed with the Metropolis–Hastings algorithm. The Metropolis–Hastings algorithm was run 50 000 times for observed data. Accepted parameter values were used in further analysis.

Based on the concepts of mean age and mean transit time published by Rasmussen et al. (2016) and Lu et al. (2018), the mean carbon age defined as the whole time carbon atoms are stored in the carbon pools and then the mean age of carbon a‾it in a certain carbon pool i could be calculated with Eq. (7): a‾it=1+∑i=13(a‾jt-a‾it)⋅fijt⋅Ci-a‾jt⋅Ii(t)Ci, where the fij(t) values are the carbon fraction transfer coefficients from jth to ith pools, and Ii(t) is the external input into the ith carbon pool. The transit time τit was defined as the mean age of carbon atoms leaving the carbon pool at a specific time: τit=∑i=1dfi(t)⋅ait, where the fit is the fraction of carbon with mean age ait.

The Community Land Model version 4.5 (CLM4.5) is the terrestrial component of the Community Earth System Model (CESM). This version mainly consists of exchanges among different carbon and nitrogen pools and other biogeochemical cycles and includes a vertical dimension of soil carbon and nitrogen transformations (Koven et al., 2013). The matrix approach was applied to extract the soil module from the original CLM4.5, which could evaluate which processes influence τsoil in the model (Huang et al., 2017). Once we obtain the total carbon pool and Rh in each pool, we can calculate the τsoil with Eq. (1). We represented the structure of SOC as seven carbon pools as (i) one coarse woody debris (CWD) pool, (ii) three litter pools (litter1, litter2, and litter3), and (iii) three soil carbon pools (soil1, soil2, and soil3). In this matrix, carbon is transferred from three litter pools and CWD to three soil pools with different transfer rates. In each layer, these transfer rates are regulated by the transfer coefficients and fractions. C inputs from litterfall were allocated into different C compartments by modifications by soil environmental factors (temperature, moisture, nitrogen, and soil oxygen) and vertical transfer process. To understand whether the incorporation of soil vertical profile affects the simulation of τsoil, we compared the results based on the matrix approach with (i.e., CLM4.5) or without (i.e., CLM4.5_noV) the soil vertical transfer process.

In CLM4.5, soil C dynamics was simulated with 10 soil layers, and the same organic matter pools among different vertical soil layers are allowed to mix mainly through diffusion and advection. The matrix approach determines the soil dynamic of each SOC pool by simulating the first-order kinetics as Eq. (9): dCtdt=BtI(t)-AξtKCtV(t)C(t), where the C(t) is the organic carbon pool size at time t. I(t) is the total organic carbon inputs while B(t) is the vector of partitioning coefficients. K is a diagonal matrix that represents the intrinsic decomposition rate of each carbon pool. The decomposition rate in the matrix approach is modified by the transfer matrix A and environmental scalars ξ. The scalar matrix ξ shown in Eq. (10) is the environmental factor to modify the SOC intrinsic decomposition rate. Each scalar matrix combines temperature (ξT), water (ξW), oxygen (ξO), depth (ξD), and nitrogen (ξN) controlled scalars for SOC decay. ξ′=ξTξWξOξDξN A is the horizontal carbon transfer matrix, which quantifies C movement among different carbon pools shown as matrix (10). The non-diagonal entries Aij shown in matrix (10) represent the fraction of carbon that moves from the jth to the ith pools. In CLM4.5 and CLM4.5_noV, transfer coefficients are the same in each soil layer. A=000000000000000000000000f440000f52f530f55f56f57000f64f65f6600000f75f76f77 V(t) is the vertical carbon transfer coefficient matrix among different soil layers, and each of the diagonal blocks is a tridiagonal matrix that describes transfer coefficients with Vij(t). In this section, CLM4.5_noV assumes no vertical transfers in all pools. Therefore, V(t) for CLM4.5_noV is a blank matrix in the simulation. In contrast, CLM4.5 was assigned by a matrix with vertical transfers in each C pool. The vertical transfer rates among different C pool categories in CLM4.5 are matrix (12). V(t)=00000000V22t0000000V33t0000000V44t0000000V55t0000000V66t0000000V77t

The median and interquartile were used for the quantification of both observational and modeling results due to the fact that the probability distribution of τsoil is not normal. To test the difference in τsoil among three approaches, we first normalized the data with the log-transformation and then applied the one-way ANOVA with a multi-comparison technique (Fig. 1b insert). The linear regression and correlation analyses were performed in R (3.2.1; R development Core team, 2015).

The Gaussian kernel density estimation was used to obtain the distributions of observed transit times (Sheather and Marron, 1990; Saoudi et al., 1997). The Gaussian kernel density estimation is a nonparametric approach to estimate the probability density function of a random variable. Let (x1x2⋯, xn) denote the observed SOC τsoil with density function f as below: f^hx=1nh∑i=1nK(x-xih), where K is the nonnegative function than integrates to one and has a mean of zero, and h>0 is a smoothing parameter called the bandwidth. The bandwidth for approaches of stable isotope 13C, stock over flux, and incubation are 48.61, 35.13, and 2.62, respectively.

The one-way ANOVA with multi-comparison analysis showed no significant difference in the log-transformed τsoil between the methods of 13C (median = 60 years; interquartile range = 8 to 29 years) and stock over flux (16; 3 to 156 years, Fig. 1b). The range of these field in situ measurements (31; 5 to 84 years) is comparable to a former estimate of mean SOC turnover time (48 with 24 to 107 years) across 20 long-term experiments in temperate ecosystems using the 13C labeling approach (Schmidt et al., 2011). However, the estimates of τsoil from laboratory studies (4; 1 to 15 year) were significantly shorter than those from the other two methods (Fig. 1b). It suggests that the τsoil could be underestimated by the measurements from the laboratory incubation studies. Thus, the τsoil values from the laboratory incubation studies were excluded in the following analyses.

We then integrated the estimates of τsoil based on the 13C and stock-over-flux approaches to examine the inter-biome difference. As shown by Fig. 2b, the longest τsoil was found in desert and shrubland (170; 58 to 508) and tundra (159; 39 to 649 years). Boreal forest (58; 25 to 170 years) has a longer τsoil than the temperate (44; 13 to 89 years) and tropical forests (15; 9 to 130 years). Grassland and savanna had short τsoil (35; 21 to 57 years) and croplands had moderate (62; 21 to 120 years) τsoil in comparison with other biomes (Fig. 2).

The longest ensemble mean τsoil values of multiple models were found in dry and cold regions (Fig. 2). In comparison with the integrated observations from 13C and stock over flux, the modeled τsoil values were significantly shorter across all biomes (Fig. 2b insert). The negative bias was larger in dry (desert, grassland, and savanna) and cold (tundra and boreal forest) regions than tropical and temperate forests. The longest modeled τsoil appeared in the tundra ecosystem with a median of 64 years. The modeled median τsoil values were also shorter than observations in tropical forest (9 years), temperate forests (13 years), boreal forest (24 years), grassland/savanna (25 years), desert and shrubland (58 years), and croplands (27 years) (Fig. 2). In comparison with the observations, the models obviously underestimated the τsoil in the cold and dry biomes (Fig. 2b). A recent global data–model comparison study at 0.5∘×0.5∘ resolution also detected a similar spatial pattern of underestimation bias in ecosystem C turnover time (Carvalhais et al., 2014), but its magnitudes of bias in the cold regions are much smaller than those found in this study.

By grouping the τsoil into different climatic categories, we found that the observed τsoil significantly covaried with MAT (y=-5.28x+156.04, r2=0.48, P<0.01) and MAP (y=-68.19x+1222.6, r2=0.60, P<0.01) (Fig. 3). These results support the previous findings of negative covariations between τsoil and temperature at both the site and global levels (Trumbore et al., 1996). Although there is no significant correlation between τsoil and MAP in the observations, the models produced negative correlations of τsoil with MAT (r2=0.24, P<0.05) and MAP (r2=0.44, P<0.05) (Fig. 3).

With the three-pool model, the total C stocks and CO2 efflux from observations and the CMIP5 ensemble were separated into pool-specific decomposition rates by the deconvolution analysis (Fig. S3a; Liang et al., 2015). Seven out of 11 parameters were constrained for tropical forest and cropland (Fig. S4, Fig. S9). Eight out of 11 parameters were constrained for temperate, boreal forest, and desert and shrubland (Figs. S5, S6, S8). Five out of 11 parameters were constrained for the tundra ecosystem (Fig. S7). For grassland and savanna, seven out of 11 parameters were constrained (Fig. S10).

The longest simulated τsoil appeared in tundra (167 years) and desert (135 years) (Fig. 4, Table S3). Temperate forest (79 years) has a longer τsoil than the boreal (66 years) and tropical forests (29 years). Grassland and savanna had short τsoil (53.8 years) and croplands had moderate (77 years) τsoil in comparison with other biomes. The τsoil calculated from the one- and three-pool models did not show a large difference across all biomes. Also, estimates based on these two model structures showed the largest underestimation of τsoil in the tundra and desert (Fig. 4).

Given that many ESMs have further developed their representations of the soil biogeochemistry in recent years, we also examined whether the τsoil estimates have been improved by one of the CMIP5 models (i.e., CESM). It is encouraging that the biases of τsoil in dry and cold regions have been substantially reduced in the new land version of CESM (i.e., version 4.5 of the Community Land Model, CLM4.5). One major improvement in CLM4.5 is the vertically resolved SOC dynamics (Koven et al., 2013). The soil organic carbon is allowed to transfer through diffusion and advection up to 3.8 m within 10 layers. In each layer, the transfer rates are regulated by the environmental scalars (i.e., temperature, soil moisture, and available oxygen). The τsoil values simulated by CLM4.5 are longer than those in CLM4 (with median value 137 years and 21 years), especially in northern high-latitudinal regions. By turning off the vertical C movements with a matrix approach (i.e., there is no vertical C transfer; thus, the vertical matrix is a zero matrix in Eq. 12), we showed a similar pattern of underestimation on τsoil by CLM4.5 (i.e., CLM4.5_noV in Fig. 5). Huang et al. (2017) also reported the longer τsoil and high carbon storage capacity in northern high latitudes. These results suggest that the vertically resolved soil biogeochemistry is promising in improving the τsoil estimates by ESMs. However, it should be noted that the spatial variation of τsoil is still largely underestimated by the CLM4.5 (Fig. 5b insert).

Higher NPP values simulated by ESMs in the cold and dry regions have been reported by previous studies (Shao et al., 2013; Smith et al., 2016; Xia et al., 2017). The models produce high NPP in cold regions largely because they overestimate the efficiency of plants transferring assimilated C to growth (Xia et al., 2017). The CMIP5 models overestimate the precipitation and underestimate the dryland expansion 4-fold during 1996–2005 (Ji et al., 2015), which could lead to high NPP and fast SOC turnover rates. These results suggest that once the NPP simulation is improved without the correction of the τsoil underestimation, the models will produce smaller SOC stock in the cold and dry ecosystems.

This study shows that adding the vertical resolved biogeochemistry is a promising approach to correct the bias of τsoil in current models. However, other processes such as microbial dynamics, SOC stabilization, and nutrient cycles could affect the estimation of τsoil but are so far fully considered by the CMIP5 models (Luo et al., 2016). For example, adding soil microbial dynamics could increase τsoil in cold regions by lowering the transfer proportion of decomposed SOC to the atmosphere (Wieder et al., 2013). By contrast, the incorporation of nitrogen cycles might shorten τsoil by increasing plant C transfers to short-lived litter pools (e.g., O-CN and CABLE models) (Gerber et al., 2010) or reducing litter C transfers to the slow soil C pools (e.g., LM3V model) (Xia et al., 2013).

Large challenges still exist in using observations derived from different methods to constrain the modeled τsoil. Laboratory incubation studies report much shorter τsoil than other methods, mainly due to the optimized soil moisture and/or temperature during the soil incubation (Stewart et al., 2008; Feng et al., 2016). This suggests that the ESM models will largely underestimate τsoil if its turnover parameters are derived from laboratory incubation studies. It should be noted that the observations from the 13C and the stock-over-flux approaches in this study are derived for the bulk soil. However, SOC is commonly represented as multiple pools with different cycling rates in most of the CMIP5 models (Luo et al., 2016; Sierra et al., 2017, 2018; Metzler and Sierra, 2018). As synthesized by Sierra et al. (2017), the observations of τsoil are useful for a specific model once its pool structure is identified. This study also detects a difference in the estimated τsoil between the one- and three-pool models (Fig. 4). Thus, model databases, such as bgc-md (https://github.com/MPIBGC-TEE/bgc-md, last access: 2 February 2019), are a useful tool to improve the integration of observations and soil C models. An enhanced transparency of C-cycle model structure in ESMs is highly recommended, especially when they participate in the future model intercomparison projects such as CMIP6 (Jones et al., 2016).