Prediction of the vertical scaling of soil organic carbon in temperate forest soils using percolation theory

. Forest soil stores a large portion of soil organic carbon (SOC), making it one of the essential components of global carbon cycling. There is apparent spatial variability of SOC in forest soils, but the mechanism that regulates the vertical pattern of SOC is still not clear. Understanding the vertical distribution as well as the transport process of SOC can be of importance in developing comprehensive SOC models in forest soils, as well as in better estimating terrestrial carbon cycling. 10 We propose a theoretical scaling derived from percolation theory to predict the vertical scaling of SOC with soil depth in temperate forest soils, with the hypothesis that the content of SOC along soil profile is limited by the transport of solute. The powers of the vertical scaling of 5 published datasets across different regions of the world are -0.920, -1.097, -1.196, - 1.062, and -1.038, comparing with the theoretical value of -1.149. Field data from Changbai Mountain region, Jilin, China, with spatial variation of SOC correlating strongly to temperature, precipitation, and sampling slope is constrained well by 15 theoretical boundaries predicted from percolation theory, indicating that the vertical transport so as the content of SOC along soil profile is limited by solute transport, which can be described by percolation theory in both small and large scales. Prediction of SOC content in Changbai Mountain region based on an estimated SOC content at 0.15 m from available data demonstrates a good agreement with field observation, suggesting the potential of collaborating the presented model with other surface soil models to predict SOC storage and carbon cycling in temperate forest soils.

transport of SOC in temperate forest soils as 3D saturated conditions, where D b = 1.87 (Sheppard et al., 1999), since the percolation of SOC in soil is a wetting process. There also could be possibilities of 2D condition, for example, if along a fracture plane, which might be less common in forest soils. Table 1 summarizes the values of D b in different scenarios. 85 Table 1. Values of fractal dimensionality of percolation backbone (D b ) (from Hunt, 2015a) Dimension and saturated conditions D b 2-D Saturated (random) 1.64 3-D Saturated or wetting (random) 1.87 2-D Unsaturated (invasion) 1.22 3-D Unsaturated or drying (invasion) 1.46 Thus, in the context of the transport of SOC in temperate forest soil, we have, where t is the transport time of solute, and h is soil depth. By taking the time derivative of h, one can obtain the vertical scaling of delivery rate of SOC, R soc, with soil depth, If we only consider the downward delivery of SOC from the soil surface as the source of SOC input to the subsurface layer, and neglect the input of carbon from roots and the deformation of the soil matrix, one can derive the scaling of SOC, C soc 95 with soil depth h as, Eq.
(3) can be rewritten in a more useful form for predicting SOC at any given h, with C s as a known SOC content (in most cases, the SOC content in the surface layer of soil), and h s as the corresponding 100 soil depth. Since SOC in the surface soil depends on the balance between carbon input and output, C s can be different across sites.
Theoretical scaling of SOC with soil depth from Eq. (3) and (4) were compared with our field data as well as data from published papers, to evaluate the application of percolation theory in predicting the vertical pattern of SOC in temperate 105 forest soils in small and large scales. Field data were sampled from 6 temperate forests in Changbai Mountain region, Jilin, China, and published data were referenced from 5 datasets collected from temperate forest soils across different regions in the world.  Five sampling points were randomly set at each plot. Soil samples were collected using a stainless steel corer down to 1m or to C layer (if C layer is shallower than 1m), which were further divided into 5 soil intervals, 0-0.1 m, 0.1-0.2 m, 0.2-0.3 m, 0.3-0.5 m, and 0.5-1 m. Samples at the same soil layer from the 5 sampling points in each plot were mixed as one, and naturally air-dried with any plant residue removed before analyzing. Dry samples were then milled and sieved through 10mesh screen. SOC of each soil layer was determined using potassium dichromate oxidation method with external heating 125 following the standard procedure (LY/T 1237-1999). The average depth of each soil interval was taken as the corresponding soil depth.

Sources of external data from published papers
Sources of the published data (Table 3) (2000), which cover 60 samples for the temperate deciduous forest, and 123 for the temperate evergreen forest across the world, and 3 field studies done in Hailun, China (Hao et al., 2015), Qinling Mountains, China (Wang et al., 2015), and Hainich, Germany, (Braakhekke et al., 2011).
Soil depths were taken as the averaged depth of each soil interval, and the unit of SOC was converted from g kg -1 to g 100g -1 , 135 except data from Jobbágy and Jackson (2000), which was presented in relative proportion content throughout the first meter of soil in the original article.

Results
Sampling results from 6 temperate forest soils in Changbai Mountain region is summarized in Table 4, with surface SOC ranging from 2% to 8%. The SOC content demonstrates spatial variability that strongly correlates to temperature, precipitation, and slope ( Fig. 1 and 2). In general, cooler sites retained more SOC than warmer ones (JL-P1 vs. HS-P1 and JL-P2 vs. HS-P2 in Fig. 1(a)), and precipitation favours the production of SOC (CS-P1 vs. LX-P1 in Fig.1(b)). Effects of 145 temperature and precipitation on SOC are in accordance with previous studies (e.g. Jenny, 1941;Jobbágy and Jackson, 2000;Wang et al., 2002;Liu et al., 2016). We examined the effect of slope, and sites with identical climatic conditions but steeper slopes show lower SOC contents (JL-P1 vs. JL-P2, HS-P1 vs. HS-P2, CS-P1 vs. TS-P1, and LX-P1 vs. LX-P2, in Fig. 1).    Figure 2 shows the effects of temperature and precipitation on SOC when coupled with slope. Temperature in CS-P2 and LX-P2 are close, but the higher precipitation and lower slope in CS-P2 tend to retain more water, resulting in higher SOC.
One the contrary, JL-P1 has similar temperature with HG-P1, but it is wetter and steeper. The SOC contents in these two 165 subsites are close, indicating that the feedbacks from higher precipitation and steeper slope seem to be neutralized here.
When precipitation is close, SOC in LX-P2 is lower than that in HG-P1. Cooler climate in LX-P2 favours the accumulation of SOC, however, the much more steeper (17 degrees vs. 5 degrees) topography has a more negative effect such that SOC is lower in LX-P2. In Fig. 3, we plot our field observations from the 6 study sites comparing with theoretical scalings predicted from Eq. (4).
Minimum and maximum predictions were calculated with setting C s(max) = 12.78%, C s(min) = 0.49%, and h s = 0.1485m. Dai et 175 al. (2009)   As shown in Fig. 3, all observed data are within the predicted boundaries from percolation theory, with variation of SOC in 180 the same soil depth among sites, which is affected by climate and topography. Variation of the environmental conditions causes the spatial variability of SOC in the surface layer of soil ( Fig.1 and 2, and the intercept on y-axis), but has no significant effect on the vertical scaling of SOC (shown as the slopes on the log-log plot), indicating that the vertical distribution of SOC is limited by the vertical transport of solute in temperate forest soils regardless of the carbon input from soil surface. 185  (2000), and 3 field studies done in Hailun (Hao et al., 2015), and Qinling Mountains, China (Wang et al., 2015), and Hainich, Germany, (Braakhekke et al., 2011) 190 Figure 4 demonstrates the vertical scaling of SOC content with soil depth from published data (Wang et al., 2015, Hao et al., 2015, Jobbágy & Jackson, 2000, and Braakhekke et al., 2011. The scaling powers are -0.920, -1.097, -1.196, -1.062, and -1.038 from the 5 datasets (average is -1.063), comparing with the theoretical scaling, -1.149 in Eq. (3), demonstrating a good agreement between prediction and observation. Data referenced from Jobbágy and Jackson (2000) with scaling power of -1.062 is quite convincing for it include a total of 183 soil profiles in temperature forest soils across different regions of the 195 world, indicating that the presented theoretical scaling can be applied in large-scale of prediction.