## 2 Materials and Methods
### Model Improvements  
#### 2.3.1 Soil Carbon Dynamics  
The most recent 3D-CMC FEM model version (v.5.1, [37]) lacks in representing SOC dynamics, preventing any estimation of NEE. With that perspective, we developed a simplified version of the method described in [38] to quantify, dynamically, changes of 7 different SOC pools mediated by the amount of active microbial C pool (Figure 3).  
The litter C decomposed by microbial activity is partly mineralized as CO\({}_{2}\), partly stored into microbial metabolic biomass (labile), partly in structural microbial biomass (resistant), and partly transformed in other organic compounds [38]. SOC stability is also related to its chemical recalcitrance, its accessibility, and interaction with clays [45]. We divided SOC in 7 pools to take these differences  
Figure 3: Soil Carbon dynamics in 3D-CMCC-PSM. The three macro pools are highlighted by red boxes (dead C pool) and blue box (live C pool, i.e., microbial). Blue-filled boxes represent the processes simulated by the soil model.  
Figure 2: 3D-CMCC version of PSM main flowchart modified from [36]. Red-circled boxes represent the pools and variables introduced or modified by 3D-CMCC-PSM.  
into account. Humic pool (we use the term Humads, to be consistent with [38]) was divided into a more labile (labile Humads, which stands for Humic acid), an intermediate (resistant Humads, which stands for Fulvic acid) and a more resistant sub-pool (Humus, which stands for Humine).  
We assumed that microbial C use efficiency was different for different pools, but constant within each pool [44]. Microbial activity is strictly related to micro-environmental conditions too. Given these premises we modeled C dynamics among soil pools using the following two partial differential equations:  
\[\left\{\begin{array}{l}\frac{\partial C_{p}}{\partial t}=C_{L}(t)-\beta C_{p }(t)B(t)\\ \frac{\partial B}{\partial t}=\beta C_{p}(t)B(t)-\alpha\;B(t)\end{array}\right., \tag{1}\]  
with \(\frac{\partial C_{p}}{\partial t}\) being the CO\({}_{2}\) efflux produced from a specific C pool decomposition, \(C_{L}(t)\) is the amount of new carbon entering the specific soil pool, \(C_{p}(t)\) the amount of carbon in that C pool, \(B(t)\) the microbial biomass competing for \(C(t)\). \(\alpha\) and \(\beta\) respectively represented the microbial turnover and SOM consumption factors. \(\alpha\) was treated as a constant value, as in [38].  
The consumption factor \(\beta\) was estimated as a function of both SOM stability and micro-environmental conditions:  
\[\beta\Big{(}C_{pool}\Big{)}=\mu_{T}\;\mu_{M}\mu_{A}k_{pool} \tag{2}\] \[\left\{\begin{array}{l}\mu_{T}=-0.014\;T^{2}+0.099\;T+0.02\\ \mu_{M}=-2.85\;\theta^{3}+1.49\;\theta^{2}+1.77\;\theta-0.03\\ \mu_{D}=0.6\;z^{-0.136}(-0.02\;clay_{\%}+0.03)\\ \mu_{A}=\mu_{D}\frac{\log(0.14\;clay_{\%}^{-1})}{2.3026}+1\end{array}\right.\]  
where \(\mu_{T}\) represented the temperature factor, \(\mu_{M}\) the moisture factor, \(\mu_{A}\) the accessibility, \(k_{pool}\) the recalcitrance, and \(\mu_{D}\) a clay dependent depth factor [46]. The \(k_{pool}\) factor was treated as a specific parameter depending on each compartment's biomass. \(T\) represented temperature in Celsius degrees, \(\theta\) stands for water moisture, \(z\) the mean depth of the soil, \(clay_{\%}\) is the percentage of clays in soil. Humads were decomposed using the same rationale, but had slower rates. Inert organic matter (IOM) was calculated following [47].