Farquhar approach

The Farquhar biochemical growth model (Farquhar et al., 1980) calculates photosynthesis as a function of demand and supply of CO₂. The advantage with this model is that photosynthesis is regulated not only by radiation and transpiration, but also by air humidity, leaf temperature, CO₂ availability and leaf nitrogen content, and the plant also experience radiation saturation at high levels of radiation. To function properly, driving variables need to be given as input to the simulation at least once an hour. In this module photosynthesis, P, is calculated as mole carbon per leaf area per second. Thus, P has to be converted to g carbon per unit soil area per day, C _Atm_→_a, at the end of the module:

(5.14)

where M_C is the molar mass of carbon.

Parameters and variables used in the photosynthesis model are converted in a similar manner.

There are several viewing functions that illustrate the Farquhar photosynthesis model, e.g. Farquhar model – Carbon dioxide pressure as a function of time , Farquhar model – Photosynthesis as a function of carbon dioxide pressure in the sub-stomatal cavity , Farquhar model – Photosynthesis as a function of LAI and Farquhar model – Photosynthesis as a function of radiation .

Three types of photosynthesis are calculated: Rubisco limited photosynthesis, P_V, and RuBP limited photosynthesis, P_J and TPU limited photosynthesis, P_S. Gross photosynthesis, P, (including photorespiration) will be determined by the most limiting photosynthesis process.

P_V, is the Rubisco (leaf enzyme) or carboxylation limited rate of assimilation, which is a function of light, leaf nitrogen, leaf temperature and soil moisture. Photosynthesis as a function of internal CO₂ concentration is calculated according to:

(5.15)

where V_m is a function of the maximum activity of Rubisco, c_i is the sub-stomatal cavity concentration of carbon dioxide, Γ^* is the CO₂ compensation point in the light in the absence of mitochondrial respiration, K_c is the Michaelis-Menten constant of Rubisco for CO₂, O is the oxygen concentration (partial pressure) in the atmosphere and K_o is the Michaelis-Menten constant of Rubisco for O₂. The reason for the difference between C₃ and C₄ plants, is that photorespiration occurs in C₃ plants at low levels of CO₂.

The CO₂ compensation point in the absence of mitochondrial respiration, Γ^*, is calculated as:

(5.16)

where the Q₁₀ value is calculated from the leaf temperature, T_l:

(5.17)

The Michaelis-Menten constant of Rubisco for CO₂, K_c, is calculated as:

(5.18)

and the Michaelis-Menten constant of Rubisco for O₂, K_o, is calculated as:

(5.19)

V_m, is a function of the potential maximum capacity of Rubisco, V_max and the response functions for leaf temperature, f(T_l), leaf carbon nitrogen ratio, f(CN_l) and soil moisture, f(E_ta/E_tp) described above (Eqs. (5.11) -(5.12)):

(5.20)

P_J is the RuBP regeneration limited (i.e. light-limited) rate of photosynthesis calculated as:

(5.21)

where J_m is calculated as:

(5.22)

where ε is the quantum efficiency, η is the conversion factor for biomass to carbon, R_s,pl is the absorbed short-wave radiation by the plant and J_max is the maximum electron transport rate.

Finally, the metabolism of end product limited (TPU limited) rate of photosynthesis, P_S_, is calculated as:

(5.23)

where p_atm is the atmospheric pressure at the surface.

The maximum Rubisco capacity for the bulk canopy per leaf area, V_max, can be calculated using equations similar to Beer’s law:

(5.24)

where V_cmax is the maximum Rubisco capacity per leaf area at the top the canopy respectively, k_rn is the extinction coefficient for net radiation and A_l is the leaf area index. The relationship between V_cmax and the maximum electron transportation rate a the top of the canopy, J_cmax, has been investigated by Wohlfahrt et al. (1999). They found that a the ratio between the two was relatively constant (J_cmax / V_cmax = 2.1) for a number of leaves. This relationship is used in the CoupModel to determine the maximum electron transportation rate for the bulk canopy per leaf area, J_max.

To avoid abrupt transition from one limiting rate to another, we apply two quadratic equations on the assimilation rates that are solved for their smaller roots (Collatz et al., 1991):

(5.25)

where β_vj and β_ps are empirical constants and P_S is an intermediate variable equal to the minimum of P_V and P_J.

Analogously to Fick’s law of gas diffusion, the supply of CO₂ for photosynthesis can be calculated as:

(5.26)

where c_a is the external carbon dioxide concentration, p_atm is the atmospheric pressure at the surface, and g_sc is the stomatal , g_bc is the boundary layer and g_ac is the aerodynamic conductances to CO₂, respectively. The gas diffusion from the atmosphere to the leaf is calculated step-wise, from the atmosphere, c_a, via the canopy air space, c_b, to the surface of the leaf, c_s, and finally into the sub-stomatal cavity, c_i in the following manner:

1) Carbon concentration in the atmosphere, c_a: model input.

2) Carbon concentration in the canopy air space, c_b:

(5.27)

where c_b,t-1 is the carbon concentration in the canopy air space from the previous time step, P_n is the net photosynthesis and R_soil is the sum of all respiration fluxes from the soil surface. k_CO2cap is the carbon capacity of air (mol air / m²), which is basically the mass of air under the top of the canopy, or, to be exact, from ground to displacement height. This factor, together with time, t, converts the flows (mol CO₂ / m² / s) into concentrations (mol CO₂ / mol air). The carbon capacity is calculated as:

(5.28)

where d is the displacement height, a_mol is the amount of gas in one cubic meter of air, T_f is the freezing point, T_abszero is the absolute zero temperature, p_atm is the atmospheric pressure at the soil surface given as a parameter and p_atmnorm is the normal pressure at the soil surface.

3) Carbon concentration in at the leaf surface, c_s:

(5.29)

4) Carbon concentration in the sub-stomatal cavity, c_i:

(5.30)

The functions to derive the equilibrium concentration of carbon dioxide in the sub-stomatal cavity, c_i, from the demand and the supply functions, follows the iterative procedure in the SiB2 model (Sellers et al., 1996).

The conductance from the canopy air space to the free flowing air for carbon dioxide, g_ac, is calculated from the aerodynamic resistance to water flow, r_a:

(5.31)

The boundary layer conductance for carbon dioxide, g_rc, is calculated from the boundary layer resistance for water flow, r_b, as:

(5.32)

where the boundary layer resistance, r_b, is given as an input to model simulations. 1.4 is the ratio of the diffusivities of CO₂ and H₂O in the leaf boundary layer.

The stomatal conductance for carbon dioxide, g_sc, is calculated from the resistance to water flow through stomata, r_s, as:

(5.33)

where the response function for soil water stress f(E_ta/E_tp) is multiplied with the stomatal resistance to account for stomatal closure due to plant water stress. 1.6 is the ratio of the diffusivities of CO₂ and H₂O in the stomatal pores.

The resistances to water flow are measured in s m^-1, and thus corresponding conductance is in m s^-1. To convert the conductance from m s^-1 to moles m^-2 s^-1, which is the unit used in the photosynthesis equations, the following conversion is performed:

(5.34)

Reduction of photosynthesis due to grain development is simulated in the same way as in the light use efficiency approach, by replacing ε_L with ε in Eq.(5.13).