We simulate GPP (\(P\)) using a light use efficiency model developed from first principles . The equations in Section are also described by and . Total canopy \(P\) is \[ P = \psi_P \; I_{\mathrm{abs}} \label{eq:pmod0} \] where \(\psi_P\) is the light use efficiency and \(I_{\mathrm{abs}}\) is the light absorbed by the canopy, modelled by the Beer-Lambert Law for light extinction. \[ I_{\mathrm{abs}} = I_0(1-e^{-kL}) \label{eq:iabs} \] \(I_0\) is the light intensity at the top of the canopy, \(k\) is the light extinction coefficient in the canopy, and \(L\) is the leaf area index. \(\psi_P\) is predicted by assuming the coordination hypothesis, which states that under typical daytime conditions, photosynthetic C assimilation operates at the point of co-limitation by the capacity of Rubisco for carboxylation of RuBP (\(A_c\)) and the electron transport for RuBP regeneration (\(A_J\)). Their functional forms are given by the Farquhar model Farquhar et al., 1980: \[ A_c = V_{\mathrm{cmax}} \; \frac{\chi - \gamma}{\chi + \kappa} \]
\[ A_J = I_{\mathrm{abs}}\; \varphi_0 \frac{\chi - \gamma}{\chi + 2\gamma} \] with \(\chi=c_i/c_a\), \(\gamma = \Gamma^{\ast}/c_a\), and \(\kappa = K/c_a\), where \(\varphi_0\) is the quantum yield, \(c_i\) and \(c_a\) are the leaf-internal and ambient concentrations, and \(\Gamma^{\ast}\) is the compensation point in the absence of dark respiration, calculated after (see Appendix AXXX), and \(K\) is the Michaelis-Menten coefficient of Rubisco. is the maximum rate of Rubisco activity. Following the electron transport-limited case, \(I_{\mathrm{abs}}\) can be interpreted as the total light absorbed by the canopy and the light use efficiency of photosynthesis is thus defined as: \[ \psi_P = \varphi_0 \frac{\chi - \gamma}{\chi + 2\gamma} \label{eq:lue} \] The acclimation of photosynthetic capacity to co-limitation operates at time scales of Rubisco turnover. Here, we assume acclimation to monthly mean climate conditions. %>>> %(xxx the following sentences until <<< are copied from KeenanXX) % %\(\Gamma^{\ast}\) depends on temperature, as estimated through a biochemical rate parameter \(x\) as described in : \[ \Gamma^{\ast} = x_{25} \; \exp( \frac{\Delta H(T-298.15)}{298.15\;R\;T} ) \] The ratio of leaf-internal to ambient (\(\chi\)) is predicted by accounting for an optimisation of costs associated with the carboxylation and transpiration capacities following . The relative costs depend on ambient , air temperature \(T\) and vapour pressure deficit \(D\) and the optimal \(\chi\) is derived by as \[ \chi = \frac{\Gamma^{\ast}}{c_a} + \left( 1 - \frac{\Gamma^{\ast}}{c_a}\right)\;\frac{\xi}{\xi + \sqrt{D}} \] With \(\xi = \sqrt{\frac{\beta\;(K+\Gamma^{\ast})}{1.6\;\eta^{\ast}}}\). \(D\) is estimated as the difference between saturated and actual vapour pressure. \(\beta\) is a parameter for the ratio of the unit costs of carboxylation versus transpiration capacity and is chosen based on observational data (Wang Han subm.). \(\eta^{\ast}=\eta/\eta_{25}\) where \(\eta\) is the viscosity of water and \(\eta_{25}\) is viscosity at standard conditions (25\(^{\circ}\)C and 1013.25 Pa) (see Appendix AXXX-AXXX for \(K\), \(D\), \(\eta\) and \(\Gamma^{\ast}\) as a function of environmental conditions). Equation can thus be written as \[ \psi_P = \varphi_0 \; \frac{c_a - \Gamma^{\ast}}{c_a + 2\;\Gamma^{\ast} + 3\;\Gamma^{\ast}\;\sqrt{\frac{1.6\;D\;\eta^{\ast}}{\beta\left(K+\Gamma^{\ast}\right)}}} \]
Accounting for an upper limit of \(A_J\) introduces a modification of \(m\) in Eq.: \[ m' = m\;\sqrt{1 - \frac{c^{2/3}}{m^{2/3}}} \] with \(c = 0.41\).
Consistent with the coordination hypothesis (\(A_c=A_J\)), is \[ V_{\mathrm{cmax}} = \varphi_0 \; I_{\mathrm{abs}} \; \frac{\chi + \kappa}{\chi + 2\gamma} \label{eq:vcmax} \] This prediction of holds only at time scales of multiple days to months. Therefore, \(I_{\mathrm{abs}}\) represents typical daytime light conditions, implemented as the photon flux density, averaged over daylight seconds (PPFD, mol m\(^{-2}\) s). We model total leaf respiration to be proportional to canopy-level . \[ R_l = a_{\mathrm{R}} \; V_{\mathrm{cmax}}\;. \] We can thus define a net light use efficiency \(\psi_{Pn}\) so that \(P-R_l = I_{\mathrm{abs}} \psi_{Pn}\) with \[ \psi_{Pn} = \frac{\varphi_0}{\chi + 2\gamma}\left( \chi - \gamma - a_{\mathrm{R}} (\chi + \kappa) \right)\;. \label{eq:luenet} \]
In brief, N in foliage is modelled as the sum of a metabolic fraction (N in Rubisco, \(N_v\)) and a structural fraction (\(N_s\)). \(N_v\) follows from the prediction of \(V_{cmax}\) (see Sect.) at standard temperature (25 K), termed \(V_{cmax25}\). \(N_s\) is modelled as a linear function of \(N_v\) at the leaf-level. To distinguish quantities expressed at the leaf versus canopy level, we henceforth use superscripts \(l\), and \(v\), respectively, with \(N^l=N^c/L\). LMA is calulated by multiplying \(N_s\) with a PFT (species-) specific factor that relates canopy structural N to structural C. The equations are as follows. The total canopy N content per unit ground area (mol N m\(^{-2}\)) is \[ N^c = L\;(N_v^l + N_s^l)\;. \label{eq:canopynassum} \] The N content per unit leaf area is therefore \(N_{\mathrm{area}} = N^c/L = N^l = N_v^l + N_s^l\).
The metabolic component of leaf nitrogen is proportional to \(V_{c max 25}^c\) at the canopy level. We assume a monthly time-scale of acclimation and use the maximum monthly \(V_{c max 25}\) for a given year with \(I_{\mathrm{abs}}\) in units of mol m\(^{-2}\) s\(^{-1}\), averaged over the monthly daylight period. The canopy-level metabolic nitrogen content is \[ N_v^c = n_v\; V_{cmax25}^c \label{eq:metabolicn} \] \(N_v^c\) is expressed in units of mol N m\(^{-2}\)-ground area. \(V_{cmax25}\) is the maximum rate of Rubisco activity at standard temperature (25 K). This accounts for the effect that \(V_{cmax}\) per unit metabolic leaf N is higher (lower) at temperatures above (below) 25 K, with: \[ V_{\mathrm{cmax25}} = f_{25} V_{\mathrm{cmax}} \\ f_{25} = \exp( -\frac{\Delta H_V}{R} \left( \frac{1}{298} - \frac{1}{T_K} \right) ) \label{eq:vcmax25} \] where \(\Delta H_V =\) 65330 J mol\(^{-1}\) and \(R=\) 8.3145 J mol\(^{-1}\) K\(^{-1}\). \(n_v\) is in mol N s (mol CO\(_2\))\(^{-1}\) and specifies the amount of metabolic leaf N per unit \(V_{\mathrm{cmax25}}\). \(T_K\) is air temperature in Kelvin. Follwoing , \(n_v\) is 336.4 mol N s (mol CO\(_2\))\(^{-1}\) and can be decomposed into \[ n_v = \frac{M_R\;[N_R]}{k_{\mathrm{cat}}\;n_R}. \] \(M_R\) is the molecular weight of Rubisco (\(5.5\cdot 10^5\) gR (mol R\(^{-1}\)), \([N_R]\) is the N concentration in Rubisco (\(1.14 \cdot 10^{-2}\) mol N (gR)\(^{-1}\)), \(k_{\mathrm{cat}}\) is the catalytic turnover rate per site (3.5 mol CO\(_2\) (mol R sites)\(^{-1}\) s\(^{-1}\)). Following Harrison et al., 2009, we use a value of \(k_{\mathrm{cat}}=2.33\) accounting for non-active Rubisco. \(n_R\) is the number of catalytic sites per mol Rubisco, 8 mol R sites (mol R)\(^{-1}\).
We assume a linear relationship between metabolic and structural N at the leaf-level. \[ N_s^l = a_{\mathrm{fol}} + b_{\mathrm{fol}} \; N_v^l \label{eq:structuraln} \] The parameter \(b_{\mathrm{fol}}\) determines the minimum investment into structures as \(N_v^l\) tends toward zero. This relationship is based on data of Hikosaka et al., 2009 (xxx update with better data!xxx).
We assume a constant ratio of leaf C to structural N which may vary between PFTs. At the leaf-level, the C content per unit leaf area is \[ C^l = c_{\mathrm{fol}}\;N_s^l \label{eq:leafc} \] Taken together, these relationships imply a declining LMA (taken as \(2.0 \cdot C^l\)) with increasing LAI and an increasing leaf C:N ratio with decreasing metabolic N. While the total metabolic N component is proportional to \(V_{c max 25}\) at the canopy-level and thus reaches an asymptote with increasing LAI, the minimum amount of N needed for foliage tissue always implies a non-zero cost for adding more leaves. The N content per unit leaf mass is therefore \(N_{\mathrm{mass}} = N^c/C^c = N^l/C^l = (N_s^l + N_v^l)/c_{\mathrm{fol}} N_s^l.\)
%This is illustrated in Figure . These relationships are consistent with observations that suggest that within the canopy, LMA varies with depth, more-or-less paralleling \(V_{c max 25}\). As a result, leaf C:N ratio is almost invariant between the upper and lower leaves.
The following equations are derived from relationships described above and are implemented in the model to simulate the non-linear relationships between foliage C, N and \(L\). This is illustrated in Fig.. By combining Equations , , and , total canopy metabolic N can be written as a function of absorbed light \(N_v^c = n_v^{\ast}\;I_{\mathrm{abs}}\) with \[ n_v^{\ast} = f_{25}\;\varphi_0\;\frac{\chi+\kappa}{\chi+2\gamma}\;n_v \label{eq:nvstar} \] This quantity follows from the predictions of photosynthesis, given VPD, temperature, , and light conditions, and is calculated at the beginning of each simulation year for each month’s average conditions, given as model input. We further assume that the metabolic leaf N predicted here represents a potential maximum, achievable by re-activation of temporally deactivated Rubisco. Thus we use the annual maximum of monthly \(n_v^{\ast}\) values for the allocation algorithm. By combining equations , , , and , total canopy N is thus \[ N^c = M_N \left( I_0\;n_v^{\ast}\;(1+b_{\mathrm{fol}})(1-e^{-kL}) + a_{\mathrm{fol}}L \right)\;. \label{eq:canopyn} \] \(M_N\) is the molecular weight of nitrogen (14.0067 gN (mol N)\(^{-1}\)). Using Eq., the total foliage C can be calculated analogously: \[ C^c = c_{\mathrm{fol}} M_c \left( I_0\;n_v^{\ast}\;b_{\mathrm{fol}}\;(1-e^{-kL}) + a_{\mathrm{fol}}\;L \right) \label{eq:canopyc} \] When a plant starts growing, i.e. for small \(L\) and \((1-e^{-kL})\simeq k\), the initial foliage C:N ratio can be derived by a Taylor approximation of Eq. divided by Eq. : \[ r_{\mathrm{C:N}}^0 = \frac{c_{\mathrm{fol}}\;M_C(I_0\;n_v^{\ast}\;k\;b_{\mathrm{fol}} + a_{\mathrm{fol}})}{M_N(I_0 \;n_v^{\ast} \;k (b_{\mathrm{fol}}+1) + a_{\mathrm{fol}} )} \] Furthermore, in the model, \(L\) needs to be calculated as a function of \(C^c\). However, solving Eq. for \(L\) leads to a transcendental equation of the form \[ \alpha(1-e^{-kL}) + \beta L - \gamma = 0 \label{eq:form} \] with \(\alpha = I_0 \; n_v^{\ast} \; b_{\mathrm{fol}}\), \(\beta = a_{\mathrm{fol}}\), and \(\gamma = c_{\mathrm{fol}}^{-1} M_C^{-1} C^c\). The solution of this equation requires the Lambert-W function \(W\): \[ L = \frac{1}{\beta k} \left( -\alpha k + \gamma k + \beta \; W \left[ \frac{\alpha k}{\beta} \exp( \frac{(\alpha - \gamma)k}{\beta} ) \right] \right) \label{eq:lambertlai} \] We apply the TOMS743 algorithm xxx
We assume that N uptake, , comes at a C cost , i.e. an additional respiration term that reduces the amount of C allocatable for new growth. generally represents C delivered to and consumed by energy-consuming processes responsible for the acquisition and processing of N from the soil (reduction of , C export to mycorrhizae, C exudation into the rhizosphere). The instantaneous N uptake efficiency \(\psi_N\) is proportional to N availability : \[ \frac{\partial N_{\mathrm{up}}}{\partial C_{\mathrm{ex}}} = \psi_N \; N_{\mathrm{av}} \] We further assume that \(\partial\)\(=-\partial\) and that is independent of . The model is solved for discrete time steps (daily), with net mineralisation and N uptake being calculated sequentially (see Sec.). Therefore, for a given initial amount of available N, , the total daily is \[ N_{\mathrm{up}} = N_{0}\;(1-\exp(-\psi_N\;C_{\mathrm{ex}})) \] is a function of root mass, \(C_r\). Here, we assume a proportional relationship between \(C_r\) and : \[ C_{\mathrm{ex}} = \varphi \; C_r \] Taken together, is a saturating function of \(C_r\) with \(\lim_{C_r \to \infty} N_{\mathrm{up}} = N_0\), whereby the uptake efficiency parameter \(\psi\) determines the depletion rate of for a given and the slope of the curve \((C_r)\) for \(C_r \to 0\) (see Fig.). The instantaneous uptake efficiency in turn declines with increasing as gets progressively depleted and it attains 0 for $ $. The formulation chosen here differs from other formulations (xxx refs) in that the of is limited by root mass, but its absolute magnitude is not. The daily amount of scales proportionally with . Using root mass for determining and not root surface area is a simplification and avoids the requirement of an additional parameter representing root mass-specific surface area.
The amount of N acquired through symbiotic biological N fixation (BNF), (gN d\(^{-1}\)), is simulated for respective PFTs by accounting for the relative efficiency of N uptake from the soil versus the efficiency of BNF. As for N uptake from the soil, N acquired by BNF comes at a C cost (). In contrast, the efficiency of N acquisition by BNF (\(\psi_{\mathrm{fix}}\)) is independent of in the soil (infinite atmospheric N\(_2\) pool) but is simulated as a function of soil temperature. \[ \psi_{\mathrm{fix}} = \frac{\partial N_{\mathrm{fix}}}{\partial C_{\mathrm{ex}}^{\mathrm{fix}}} = \psi_{\mathrm{fix}}^{\mathrm{max}}\cdot 1.25 \cdot\exp(a_{\mathrm{fix}} + b_{\mathrm{fix}}\;T_{\mathrm{soil}}\;(1 - 0.5\;\frac{T_{\mathrm{soil}}}{T_{\mathrm{fix}}^{\mathrm{opt}}} )) \] This is a bell-shaped function with a maximum at \(T_{\mathrm{fix}}^{\mathrm{opt}}=25.15\) and asymptotically declines to zero for low and high temperatures. We assume here that the functional form of \(\psi_{\mathrm{fix}}\) is identical to the fitted relationship between observed nitrogenase activity and soil temperature given by . The factor 1.25 is introduced to normalise the function to vary between zero and \(\psi_{\mathrm{fix}}^{\mathrm{max}}\). The latter is given by the inverse of the minimum cost of N fixation (4.8 gC/gN), found by . In contrast to other models (ref Houlton08, Wang 09, …), we introduce a temperature dependence of N fixation efficiency here to account for the apparently strong inhibition of nitrogenase activity at low temperatures.
In each time step, can be spent to acquire N by soil uptake and BNF. Their relative shares is found by finding the maximum N return on investment, i.e. is spent on soil uptake until a point is reached where soil uptake efficiency falls below the efficiency of BNF at the given soil temperature. This threshold efficiency is given by \[ \psi^{\ast} = \frac{\partial N_{\mathrm{up}}}{\partial C_{\mathrm{ex}}} \Bigr|_{C_{\mathrm{ex}}=C_{\mathrm{ex}}^{\ast}} = \psi_{\mathrm{fix}} \] The amount of C expended for soil uptake can then be calculated as \[ C_{\mathrm{ex}}^{\ast} = - 1 / \psi\;\ln( \frac{\psi_{\mathrm{fix}}}{\psi} )\;, \] and total amount of N taken up from the soil plus BNF is \[ N_{\mathrm{up}} = N_{0}\;(1-\exp(-\psi\;C_{\mathrm{ex}}^{\ast})) + \psi_{\mathrm{fix}} ( C_{\mathrm{ex}} - C_{\mathrm{ex}}^{\ast} )\;. \label{eq:nuptot} \]