This report is the result of the use of the python package bgc_md, as means to translate published models to a common language. The underlying yaml file was created by Verónika Ceballos-Núñez (Orcid ID: 0000-0002-0046-1160) on 16/3/2016.
The model depicted in this document considers carbon allocation with a process based approach. It was originally described by Murty & McMurtrie (2000).
global
| Name | Description | Unit |
|---|---|---|
| \(C_{f}\) | Foliar carbon mass | \(kgC\cdot m^{-2}\) |
| \(C_{r}\) | Root carbon | \(kgC\cdot m^{-2}\) |
| \(C_{w}\) | Carbon in woody tissue | \(kgC\cdot m^{-2}\) |
| Name | Description | Expression | Unit |
|---|---|---|---|
| \(C_{sw}\) | Sapwood carbon content | \(C_{sw}=1.11\cdot C_{w}^{0.77}\) | \(kgC\cdot m^{-2}\) |
| \(N_{f}\) | Nitrogen content of foliage | - | \(kgN\cdot m^{-2}\) |
| \(N_{r}\) | Nitrogen content of fine roots | - | - |
| \(n_{f}\) | Foliar N:C ratio | - | - |
| \(n_{crit}\) | Foliar N:C ratio below which production is N-limited | - | - |
| \(T_{a}\) | Mean air temperature | - | - |
| \(Q_{10}\) | - | - | |
| \(Q_{010}\) | - | - |
| Name | Description | Expression | Unit |
|---|---|---|---|
| \(I_{0}\) | Incident PAR | - | \(GJ\cdot m^{-2}\) |
| \(\sigma\) | Leaf area per unit carbon | - | \(m^{2}\cdot kgC^{-1}\) |
| \(k\) | Light extinction coefficient | - | \(kgC\cdot m^{-2}\) |
| \(APAR\) | Absorbed photosynthetically active radiation | \(APAR=I_{0}\cdot\left(1 - e^{- C_{f}\cdot k\cdot\sigma}\right)\) | - |
| \(E_{nf}\) | Function that represents the dependence of NPP on foliar N:C ratio (n_f) | \(E_{nf}=\begin{cases}\frac{\left(n_{crit} + 0.017\right)\cdot\left(1.84\cdot n_{f} - 0.01\right)}{\left(1.84\cdot n_{crit} - 0.01\right)\cdot\left(n_{f} + 0.017\right)} &\text{for}\: n_{crit} > n_{f}\\1 &\text{for}\: n_{crit} < n_{f}\end{cases}\) | - |
| \(\epsilon_{young}\) | Maximum gross PAR utilization efficiency of young stands | - | \(gC\cdot MJ^{-1}\) |
| \(\epsilon_{old}\) | Maximum gross PAR utilization efficiency of old stands | - | \(gC\cdot MJ^{-1}\) |
| \(\epsilon_{0}\) | Maximum gross PAR utilization efficiency | \(\epsilon_{0}=\begin{cases}\epsilon_{young} &\text{for}\: t{\leq} t_{1}\\\begin{cases}\epsilon_{young} -\frac{\left(-\epsilon_{old} +\epsilon_{young}\right)\cdot\left(t - t_{1}\right)}{- t_{1} + t_{2}} &\text{for}\: t > t_{1}\\\begin{cases}\epsilon_{young} -\frac{\left(-\epsilon_{old} +\epsilon_{young}\right)\cdot\left(t - t_{1}\right)}{- t_{1} + t_{2}} &\text{for}\: t < t_{2}\\\epsilon_{old} &\text{otherwise}\end{cases} &\text{otherwise}\end{cases} &\text{otherwise}\end{cases}\) | \(gC\cdot MJ^{-1}\) |
| \(GPP\) | Gross primary production | \(GPP=APAR\cdot E_{nf}\cdot\epsilon_{0}\) | - |
| \(NPP\) | Annual net primary production | \(NPP=GPP - R_{c} - R_{m}\) | \(kgC\cdot m^{-2}\cdot year^{-1}\) |
| Name | Description | Expression | Unit |
|---|---|---|---|
| \(R_{c}\) | Total construction respiration | - | - |
| \(R_{0}\) | Respiration rate per unit nitrogen content corresponding to a temperature of 0°C | - | \(kgC\cdot kgN^{-1}\cdot year^{-1}\) |
| \(R_{mf}\) | Annual maintenance respiration rate of foliage (dark period only) | \(R_{mf}=0.5\cdot N_{f}\cdot Q_{10}^{\frac{T_{a}}{10}}\cdot R_{0}\) | - |
| \(R_{mr}\) | Annual maintenance respiration rate of fine roots (dark period only) | \(R_{mr}=N_{r}\cdot Q_{10}^{\frac{T_{a}}{10}}\cdot R_{0}\) | - |
| \(R_{msw}\) | Annual maintenance respiration rate of sapwood (dark period only) | \(R_{msw}=0.00876\cdot C_{sw}\cdot Q_{010}^{\frac{T_{a}}{10}}\) | - |
| \(R_{m}\) | Total maintenance respiration | \(R_{m}=R_{mf} + R_{mr} + R_{msw}\) | - |
| Name | Description | Expression |
|---|---|---|
| \(a_{f}\) | Allocation fraction to foliar biomass | - |
| \(a_{r}\) | Allocation fraction to roots biomass | - |
| \(a_{w}\) | Allocation fraction to wood (in stem, branches and large structurl roots) biomass | \(a_{w}=- a_{f} - a_{r} + 1\) |
| Name | Description | Unit |
|---|---|---|
| \(\gamma_{f}\) | Foliage senescence rate | \(yr^{-1}\) |
| \(\gamma_{r}\) | Roots senescence rate | \(yr^{-1}\) |
| \(\gamma_{w}\) | Wood senescence rate | \(yr^{-1}\) |
| Name | Description | Expression |
|---|---|---|
| \(x\) | vector of states for vegetation | \(x=\left[\begin{matrix}C_{f}\\C_{r}\\C_{w}\end{matrix}\right]\) |
| \(u\) | scalar function of photosynthetic inputs | \(u=NPP\) |
| \(b\) | vector of partitioning coefficients of photosynthetically fixed carbon | \(b=\left[\begin{matrix}a_{f}\\a_{r}\\a_{w}\end{matrix}\right]\) |
| \(A\) | matrix of senescence (cycling) rates | \(A=\left[\begin{matrix}-\gamma_{f} & 0 & 0\\0 & -\gamma_{r} & 0\\0 & 0 & -\gamma_{w}\end{matrix}\right]\) |
| \(f_{v}\) | the righthandside of the ode | \(f_{v}=u b + A x\) |
\(C_{f}: a_{f}\cdot\left(- 0.0097236\cdot C_{w}^{0.77}\cdot Q_{010}^{\frac{T_{a}}{10}} + I_{0}\cdot\left(1 - e^{- C_{f}\cdot k\cdot\sigma}\right)\cdot\left(\begin{cases}\epsilon_{young} &\text{for}\: t{\leq} t_{1}\\\begin{cases}\epsilon_{young} -\frac{\left(-\epsilon_{old} +\epsilon_{young}\right)\cdot\left(t - t_{1}\right)}{- t_{1} + t_{2}} &\text{for}\: t > t_{1}\\\begin{cases}\epsilon_{young} -\frac{\left(-\epsilon_{old} +\epsilon_{young}\right)\cdot\left(t - t_{1}\right)}{- t_{1} + t_{2}} &\text{for}\: t < t_{2}\\\epsilon_{old} &\text{otherwise}\end{cases} &\text{otherwise}\end{cases} &\text{otherwise}\end{cases}\right)\cdot\left(\begin{cases}\frac{\left(n_{crit} + 0.017\right)\cdot\left(1.84\cdot n_{f} - 0.01\right)}{\left(1.84\cdot n_{crit} - 0.01\right)\cdot\left(n_{f} + 0.017\right)} &\text{for}\: n_{crit} > n_{f}\\1 &\text{for}\: n_{crit} < n_{f}\end{cases}\right) - 0.5\cdot N_{f}\cdot Q_{10}^{\frac{T_{a}}{10}}\cdot R_{0} - N_{r}\cdot Q_{10}^{\frac{T_{a}}{10}}\cdot R_{0} - R_{c}\right)\)
\(C_{r}: a_{r}\cdot\left(- 0.0097236\cdot C_{w}^{0.77}\cdot Q_{010}^{\frac{T_{a}}{10}} + I_{0}\cdot\left(1 - e^{- C_{f}\cdot k\cdot\sigma}\right)\cdot\left(\begin{cases}\epsilon_{young} &\text{for}\: t{\leq} t_{1}\\\begin{cases}\epsilon_{young} -\frac{\left(-\epsilon_{old} +\epsilon_{young}\right)\cdot\left(t - t_{1}\right)}{- t_{1} + t_{2}} &\text{for}\: t > t_{1}\\\begin{cases}\epsilon_{young} -\frac{\left(-\epsilon_{old} +\epsilon_{young}\right)\cdot\left(t - t_{1}\right)}{- t_{1} + t_{2}} &\text{for}\: t < t_{2}\\\epsilon_{old} &\text{otherwise}\end{cases} &\text{otherwise}\end{cases} &\text{otherwise}\end{cases}\right)\cdot\left(\begin{cases}\frac{\left(n_{crit} + 0.017\right)\cdot\left(1.84\cdot n_{f} - 0.01\right)}{\left(1.84\cdot n_{crit} - 0.01\right)\cdot\left(n_{f} + 0.017\right)} &\text{for}\: n_{crit} > n_{f}\\1 &\text{for}\: n_{crit} < n_{f}\end{cases}\right) - 0.5\cdot N_{f}\cdot Q_{10}^{\frac{T_{a}}{10}}\cdot R_{0} - N_{r}\cdot Q_{10}^{\frac{T_{a}}{10}}\cdot R_{0} - R_{c}\right)\)
\(C_{w}: \left(- a_{f} - a_{r} + 1\right)\cdot\left(- 0.0097236\cdot C_{w}^{0.77}\cdot Q_{010}^{\frac{T_{a}}{10}} + I_{0}\cdot\left(1 - e^{- C_{f}\cdot k\cdot\sigma}\right)\cdot\left(\begin{cases}\epsilon_{young} &\text{for}\: t{\leq} t_{1}\\\begin{cases}\epsilon_{young} -\frac{\left(-\epsilon_{old} +\epsilon_{young}\right)\cdot\left(t - t_{1}\right)}{- t_{1} + t_{2}} &\text{for}\: t > t_{1}\\\begin{cases}\epsilon_{young} -\frac{\left(-\epsilon_{old} +\epsilon_{young}\right)\cdot\left(t - t_{1}\right)}{- t_{1} + t_{2}} &\text{for}\: t < t_{2}\\\epsilon_{old} &\text{otherwise}\end{cases} &\text{otherwise}\end{cases} &\text{otherwise}\end{cases}\right)\cdot\left(\begin{cases}\frac{\left(n_{crit} + 0.017\right)\cdot\left(1.84\cdot n_{f} - 0.01\right)}{\left(1.84\cdot n_{crit} - 0.01\right)\cdot\left(n_{f} + 0.017\right)} &\text{for}\: n_{crit} > n_{f}\\1 &\text{for}\: n_{crit} < n_{f}\end{cases}\right) - 0.5\cdot N_{f}\cdot Q_{10}^{\frac{T_{a}}{10}}\cdot R_{0} - N_{r}\cdot Q_{10}^{\frac{T_{a}}{10}}\cdot R_{0} - R_{c}\right)\)
\(C_{f}: C_{f}\cdot\gamma_{f}\)
\(C_{r}: C_{r}\cdot\gamma_{r}\)
\(C_{w}: C_{w}\cdot\gamma_{w}\)
Murty, D., & McMurtrie, R. E. (2000). The decline of forest productivity as stands age: A model-based method for analysing causes for the decline. Ecological Modelling, 134(2-3), 185–205. http://doi.org/10.1016/s0304-3800(00)00345-8