Farquhar Model

This section breaks down the Farquhar model that describes the biochemical process of photosynthesis in plants as environmental conditions change.

The biochemical processes within a leaf determine the rate of photosynthesis, particularly the diffusion of CO$_2$ into the leaf, the assimilation of CO$_2$ during photosynthesis, and the transpiration of water vapor. It takes into account factors such as light intensity, temperature, and CO$_2$ concentration to estimate the rate at which plants convert light energy into chemical energy through photosynthesis.

The net assimilation by a leaf (An) is calculated based on the biochemistry of C3 and C4 photosynthesis to determine potential (unstressed by water availability) leaf-level photosynthesis. This is calculated in terms of two potentially-limiting rates:

An vs. air Temperature (T, °C) and Photosynthetically Active Radiation (PAR, μmol m⁻² s⁻¹)

An vs. air Temperature (T, °C) and intra-cellular CO2 (ci, ppm)

Rubisco limited rate

\[\begin{equation} a_1(T, c_a, VPD) = \begin{cases} V_{cmax}(T) \frac{(c_i(T, c_a, VPD) - \Gamma^*(T))}{(c_i(T, c_a, VPD) + K_c(T)*(1+o_i/K_o(T)))} & \text{for C3}\\ V_{cmax}(T) & \text{for C4} \end{cases} \end{equation}\]

The dependence on the atmospheric CO$_2$ concentration $c_a$ (mol/mol) and vapor pressure deficit $VPD$ arise in the expression for $c_i$,

\[\begin{align} c_i(T, c_a, VPD) = \max{(c_a(1-1/m(VPD)), \Gamma^*(T)}), \end{align}\]

where and $m$ is the Medlyn factor (see Stomatal Conductance).

We also have

\[ \Gamma^*(T) = \Gamma^*_{25}\exp\left(\Delta H_{\Gamma^*}\frac{T - T_o}{T_o R T}\right),\]

where $\Delta H_{\Gamma^*}$ is the activation energy per mol for $\Gamma^*$.

Light limited rate

\[\begin{equation} a_2 = \begin{cases} J(T, PAR) (c_i - \Gamma^*)/4(c_i + 2 \Gamma^*) & \text{for C3}\\ J(T, PAR) & \text{for C4} \end{cases} \end{equation}\]

where J is the rate of electron transport, which has units of mol photon per m$^2$ per s. It depends on $PAR$ via $APAR$, as described below, and on $T$ via the dependence on $J_{max}$.

J is given by the root of the equation

\[\begin{align} \theta_j J^2 - (I + J_{max}) J + I J_{max} &= 0 \nonumber \\ I &= \frac{\phi}{2} (APAR) \nonumber \\ J_{max}(T) &= V_{cmax}(T)\times e \exp\left(\Delta H_{J_{max}}\frac{T - T_o}{T_o R T}\right),\nonumber \\ J(T, PAR) &= \frac{(I + J_{max} - \sqrt{(I + J_{max})^2 - 4\theta_j I \times J_{max}}}{2\theta_j}, \end{align}\]

where $\phi = 0.6$ and $\theta_j = 0.9$ are the quantum yield of photosystem II and a curvature function (Bonan's book), and $\Delta H_{J_{max}}$ is the energy of activation of $J_{max}$.

The total net carbon assimilation (A$_n$, mol CO$_2$ m$^{-2}$ s$^{-1}$) is given by the weighted sum of C3 and C4 net carbon assimilation fractions following:

\[\begin{align} A_n(T, PAR, VPD, c_a) = \text{max}(0, \text{min}(a_1 \beta, a_2) - R_d) \end{align}\]

where $\beta$ is the moisture stress factor which is related to the mean soil moisture concentration in the root zone and R$_d$ is the leaf dark respiration calculated as

\[\begin{align} R_{d,25}(\psi_l) &= f V_{cmax,25}\beta(\psi_l), \nonumber \\ R_d (T, \psi_l) & = R_{d,25}(\psi_l)\exp\left(\Delta H_{R_{d}}\frac{T - T_o}{T_o R T}\right), \end{align}\]

where $f = 0.015$ is a constant, $\Delta H_{R_d}$ is the energy of activation for $R_d$, and finally Vcmax is calculated as

\[\begin{equation} V_{cmax}(T) = V_{cmax,25} \exp\left(\Delta H_{Vcmax}\frac{T - T_o}{T_o R T}\right)\\ \end{equation}\]

with $V_{cmax,25}$ is a parameter (Vcmax at the reference temperature 25 C), and $\Delta H_{Vcmax} = 65,330 J/mol$.

The moisture stress factor is related to the leaf water potential $\psi_l$ as

\[\begin{align} \beta = \frac{1+ \exp{(s_c \psi_c)}}{1+ \exp{(s_c(\psi_c - \psi_l))}}, \end{align}\]

where $s_c = 4$MPa$^{-1}$, $\psi_c = -2$MPa, and $\psi_l$ is the leaf water potential computed by the plant hydraulics model.

GPP is the total canopy photosynthesis calculated as the integral of leaf-level photosynthesis over the entire canopy leaf area index:

\[\begin{align} GPP(T, PAR, c_a, VPD, \theta_s) = A_n (1 - \exp(-K LAI \Omega))/K. \end{align}\]

This is not currently needed by other components, but is used for offline validation of the model.

We need to supply the following parameters and “drivers"

• $K_{c,25}$ and $K_{o,25}$, $V_{cmax, 25}$, $\Gamma^*_{25},\phi$, $\theta_j$, $o_i$, $s_c$, $\psi_c$
• $\psi_l$, to compute $\beta$
• Temperature $T$, $PAR$, $c_a$, VPD, $\theta_s$.