Carbon cycle
Modelling the forest carbon cycle
The calculation of the carbon cycle in FORMIND 3.0 uses a simple compartment approach consisting of the following explicit carbon stocks:
- living forest stock, which equals the amount of carbon of alive trees
- deadwood stock $S_{\text{dead}}$, which equals the amount of carbon of dead trees
- slow decomposing soil stock $S_{\text{slow}}$, which accounts for the slow decomposing share of carbon in the deadwood stock
- fast decomposing soil stock $S_{\text{fast}}$, which accounts for the fast decomposing share of carbon in the deadwood stock
Schematic visualization of the carbon cycle in FORMIND 3.0. Circles represent explicit carbon stocks and the rectangle indicates the atmosphere. Dotted arrows show carbon released to the atmosphere from the respective stock and block arrows show carbon transitions between the respective explicit carbon stocks.
The dynamics of the living forest stock (i.e. carbon storage in form of growth and carbon releases as respiration) are described earlier in section Growth of a tree. The dynamics of the remaining stocks is described by a set of differential equations:
\begin{align} \frac{\mathrm{d} S_{dead}}{\mathrm{d} t} &= S_{\text{mort}} - (t_{S_{\text{dead}}\rightarrow A} + t_{S_{\text{dead}}\rightarrow S_{\text{slow}}} + t_{S_{\text{dead}}\rightarrow S_{\text{fast}}}) \cdot S_{\text{dead}} \\ \frac{\mathrm{d} S_{\text{slow}}}{\mathrm{d} t} &= t_{S_{\text{dead}}\rightarrow S_{\text{slow}}}\cdot S_{\text{dead}}-t_{S_{\text{slow} \rightarrow A}}\cdot S_{\text{slow}} \\ \frac{\mathrm{d} S_{\text{fast}}}{\mathrm{d} t} &= t_{S_{\text{dead}} \rightarrow S_{\text{fast}}}\cdot S_{\text{dead}}-t_{S_{\text{fast} \rightarrow A}}\cdot S_{\text{fast}} \end{align}
where the parameters $t_{S_{\text{dead}}\rightarrow A}$, $t_{S_{\text{slow}}\rightarrow A}$ and $t_{S_{\text{fast}} \rightarrow A}$ denote transition rates in $\tfrac{1}{\text{yr}}$ of released carbon from the respective soil stocks to the atmosphere. The parameter $t_{S_{\text{dead}} \rightarrow S_{\text{slow}}}$ and $t_{S_{\text{dead}} \rightarrow S_{\text{fast}}}$ represent in turn decomposition rates of deadwood material in $\tfrac{1}{\text{yr}}$. The variable $S_{\text{mort}} \left[\tfrac{\text{t}_\text{C}}{\text{ha}} \right]$ represents the carbon of all trees dying within the current time step (see section Mortality).
Determining the transition rates
The transition rates depend on how fast microorganisms can decompose the fallen litter or dead trees. For describing the decomposition rates, we use an approach presented earlier by Sato et al. 2007. The annual decomposition rate $t_{s_{\text{dead}}}$ for the deadwood stock is calculated as follows:
\begin{equation} t_{S_{\text{dead}} }=\text{min}\left( 1.0,\frac{10^{-1.4553+0.0014175\cdot AET}}{12} \right) \end{equation}
where $AET$ is considered as the actual evapotranspiration in the previous year in mm. The variable $AET$ is calculated by the sum of interception $IN$ and transpiration $TR$ (cf. section Water cycle and soil water limitation).
The annual decomposition rate $t_{s_{\text{dead}} }$ is modelled as the sum of all transitions rates of the deadwood pool $S_{\text{dead}}$:
\begin{equation} t_{S_{\text{dead} }}=t_{S_{\text{dead}} \rightarrow A}+t_{S_{\text{dead} } \rightarrow S_{\text{slow}}}+t_{S_{\text{dead}} \rightarrow S_{\text{fast}}} \end{equation}
According to Sato et al. 2007 70 % of the carbon of decomposing deadwood biomass (i.e. litter) is directly released to the atmosphere, while the remaining 30 % are transferred to the slow and fast decomposing soil stocks. In detail, 98.5 % of the remaining carbon is transferred to the fast soil stock and 1.5 % to the slow soil stock. We then calculate the specific transition rates as follows:
\begin{align} t_{S_{\text{dead} }\rightarrow A} &= 0.7\cdot t_{S_{\text{dead}} } \\ t_{S_{\text{dead} }\rightarrow S_{\text{slow}}} &= 0.015\cdot 0.3\cdot t_{S_{\text{dead}} } \\ t_{S_{\text{dead} }\rightarrow S_{\text{fast}}} &= 0.985\cdot 0.3\cdot t_{S_{\text{dead}} } \end{align}
The Net Ecosystem Exchange (NEE)
The NEE is the carbon net flux of the forest. We define the $NEE \left[ \tfrac{\text{t}_\text{C}}{\text{ha yr}} \right]$ as follows:
\begin{equation} NEE = C_{GPP}-C_{R}-t_{S_{\text{dead}}\rightarrow A}\cdot S_{\text{dead}}-t_{S_{\text{slow}}\rightarrow A}\cdot S_{\text{slow}}-t_{S_{\text{fast}}\rightarrow A}\cdot S_{\text{fast}} \end{equation}
where $S_\text{dead} \left[ \tfrac{ \text{t}\text{C}}{\text{ha}} \right]$ denotes the deadwood carbon pool, $S{\text{slow}} \left[\tfrac{\text{t}\text{C}}{\text{ha}} \right]$ and $S{\text{fast}} \left[\tfrac{\text{t}\text{C}}{\text{ha}} \right]$ the soil carbon stock (i.e. slow and fast decomposing), $t{x \rightarrow A} \left[ \tfrac{1}{\text{yr}} \right]$ the corresponding transition rates of released carbon from the respective stock $x$ resulting from the microbiological respiration (cf. section Carbon cylcle) and $C_{GPP} \left[\tfrac{\text{t}\text{C}}{\text{ha yr}} \right]$ is the carbon captured in the gross primary productivity of the living forest (cf. section Gross primary production), $C_R \left[\tfrac{\text{t}\text{C}}{\text{ha yr}} \right]$ is the carbon released by the total respiration of the living forest (i.e. for maintenance and growth). We also assume here that $1 \text{g}$ organic dry matter contains 44 % carbon, which results in:
\begin{align} C_{GPP} &= 0.44\cdot \sum_{\text{all trees}}^{} GPP \\ C_{R} &= 0.44 \cdot \sum_{\text{all trees}}^{} (R_{m}+R_{g}\cdot (GPP-R_{m})) \end{align}
If the $NEE$ is positive (i.e. $NEE > 0 $), the forest is considered to be a carbon sink. If the $NEE$ is negative (i.e. $NEE < 0$), the forest is considered to be a carbon source.