Gas exchange

The exchange module computes the rates of net carbon assimilation of C3 plants (\(A_n\)), stomatal conductance to CO2 (\(g_{s, \ CO_2}\)) and water vapor (\(g_{s, \ H_2O}\)), and transpiration (\(E\)) per unit leaf surface area as a function of micrometeorological conditions and leaf water status.

\(A_n\) and \(g_{s, \ CO_2}\) are coupled based on the analytical solution proposed by Yin et al. (2009) which also considers mesophyll conductance to CO2 diffusion \(g_m\). The solution is based on the following equations (following Evers et al. 2010 supporting information):

\[ \begin{align}\begin{aligned}\begin{array}\\\begin{split}A_n = \frac{(C_c - \Gamma) \cdot x_1}{C_c + x_2} - R_d \\\end{split}\\\begin{split}C_c = C_i - \frac{A_n}{g_m} \\\end{split}\\\begin{split}g_{s, \ CO_2} = g_{s0, \ CO_2} + m_0 \cdot \frac{A_n + R_d}{C_i - \Gamma} \cdot f_w \\\end{split}\\g_{s, \ CO_2} = \frac{A_n}{C_a - C_i - A_n \cdot r_{tb}}\\\end{array}\end{aligned}\end{align} \]

where \(A_n \ [\mu mol \ m^{-2} \ s^{-1}]\) is net carbon assimilation rate, \(R_d \ [\mu mol \ m^{-2} \ s^{-1}]\) is mitochondrial respiration in the light, \(\Gamma \ [\mu bar]\) is \(CO_2\) compensation point in the absence of mitochondrial respiration, \(x_1 \ [\mu mol \ m^{-2} \ s^{-1}]\) and \(x_2 \ [\mu bar]\) are intermediate parameters, \(g_m \ [\mu mol \ m^{-2} \ s^{-1} \ {\mu bar}^{-1}]\) is mesophyll conductance for \(CO_2\) diffusion, \(g_{s, \ CO_2} \ [mol \ m^{-2} \ s^{-1} \ {\mu bar}^{-1}]\) is stomatal conductance to \(CO_2\), \(g_{s0, \ CO_2} \ [mol \ m^{-2} \ s^{-1} \ {\mu bar}^{-1}]\) is the residual stomatal conductance to \(CO_2\), \(m_0 \ [-]\) is a shape parameter regulating the slope between \(A_n\) and \(g_{s, \ CO_2}\), \(f_w \ [-]\) is a dimensionless function representing the response of stomatal conductance to soil or plant water status, \(r_{tb} \ [m^2 \ s \ \mu bar \ {\mu mol}^{-1}]\) is the combined turbulence and boundary layer resistance for \(CO_2\), \(C_a \ [\mu bar]\) is air \(CO_2\) partial pressure, \(C_i \ [\mu bar]\) is intercellular \(CO_2\) partial pressure, and \(C_c \ [\mu bar]\) is chloroplast \(CO_2\) partial pressure.

Net carbon assimilation rate per unit leaf area

\(A_n\) is given as:

\[A_n = V_c \cdot \left(1 - \frac{\Gamma}{C_c} \right) - R_d\]

where \(V_c \ [\mu mol \ m^{-2} \ s^{-1}]\) is carboxylation rate, \(\Gamma \ [\mu {mol}_{CO_2} \ {mol}_{CO_2}^{-1}]\) \(CO_2\) compensation point in the absence of mitochondrial respiration, \(C_c \ [\mu {mol}_{CO_2} \ {mol}^{-1}]\) chloroplast \(CO_2\) concentration, and \(R_d \ [\mu mol \ m^{-2} \ s^{-1}]\) is mitochondrial respiration rate in the light.

\(V_c\) is given as:

\[V_c = \min \left(W_c, W_j, W_p \right)\]

where \(W_c\), \(W_j\), and \(W_p \ [\mu mol \ m^{-2} \ s^{-1}]\) are respectively Rubisco-limited carboxylation rate, RuBP-limited carboxylation rate, and Triose phosphates-limited carboxylation rate, given as:

\[ \begin{align}\begin{aligned}\begin{array}\\\begin{split}W_c = \frac{C_c \cdot V_{c, \ max}}{C_c + K_c \cdot \left(1 + \frac{O}{K_o} \right)} \\\end{split}\\\begin{split}W_j = \frac{J}{4 + 8 \cdot \frac{\Gamma}{C_c}} \\\end{split}\\W_p = \frac{3 \cdot TPU}{\left(1 - \frac{\Gamma}{C_c} \right)}\\\end{array}\end{aligned}\end{align} \]

where \(V_{c, \ max} \ [\mu mol \ m^{-2} \ s^{-1}]\) is the maximum carboxylation rate, \(J \ [\mu mol \ m^{-2} \ s^{-1}]\) electron transport rate, \(TPU \ [\mu mol \ m^{-2} \ s^{-1}]\) Triose phosphates transport rate, \(K_c \ [\mu mol \ {mol}^{-1}]\) Michaelis-Menten constant for the carboxylase, \(K_o \ [mmol \ {mol}^{-1}]\) Michaelis-Menten constant for the oxygenase, and \(O \ [mmol \ {mol}^{-1}]\) oxygen concentration.

Finally, \(J\) is given as:

\[J = \frac{\alpha \cdot {PPFD}}{\sqrt{1 + \frac{\alpha^2 \cdot {PPFD}^2}{J_{max}^2}}}\]

where \(J_{max} \ [\mu mol \ m^{-2} \ s^{-1}]\) is maximum electron transport rate, and \(PPFD \ [\mu mol \ m^{-2} \ s{-1}]\) is photosynthetic photon flux density.

The impact of leaf temperature on the photosynthesis parameters is accounted for using Arrhenius functions. For \(V_{c, \ max}\), \(J_{max}\), \(TPU\), and \(R_d\) temperature dependency writes:

\[P = P^{25} \cdot \ \frac {\exp \left(c - \frac{\Delta H_a}{R \cdot T_{leaf}} \right)} {1 + \exp \left(\frac {\Delta S \cdot T_{leaf} - \Delta H_d} {R \cdot T_{leaf}} \right)}\]

where \(P\) denotes any of \(V_{c, \ max}\), \(J_{max}\), \(TPU\), and \(R_d\) parameters, \(P^{25}\) is the value of \(P\) at 25 \(^\circ C\), \(c \ [-]\) is a shape parameter, \(\Delta H_a \ [kJ \ {mol}_{CO_2}^{-1}]\) is activation energy of the Arrhenius functions, \(\Delta H_d \ [kJ \ {mol}_{CO_2}^{-1}]\) is deactivation energy of the Arrhenius functions, \(\Delta S \ [kJ \ K^{-1} \ {mol}_{CO_2}^{-1}]\) is entropy term, \(R \ [kJ \ K^{-1} \ {mol}^{-1}]\) is the ideal gas constant, and \(T_{leaf} \ [K]\) is leaf temperature.

Finally, for \(\Gamma\), \(K_c\), and \(K_o\) temperature dependency writes:

\[P = \exp \left( c - \frac{\Delta H_a}{R \cdot T_{leaf}} \right)\]

Variable intra-canopy photosynthetic capacities

Leaf photosynthetic traits ( \(V_{cmax}\), \(J_{max}\), \(TPU\) and \(R_d\); cf. Appendix I in Albasha et al., 2019) are set to vary as a function leaf nitrogen content per unit leaf surface area (\(N_a, \ g_N \ m^{-2}\)) following Prieto et al. (2012):

\[P^{25} = S_{N_a} \ N_a - b_{N_a}\]

where \(P^{25} \ [\mu mol \ m^{-2} \ s^{-1}]\) is the value at 25 \(^\circ C\) for any of the rates of \(V_{cmax}\), \(J_{max}\), \(TPU\) or \(R_d\), \(S_{N_a} \ [\mu mol \ g_N^{-1} \ s^{-1}]\) and \(b_{N_a} \ [\mu mol \ m^{-2} \ s^{-1}]\) are the slope and the intercept of the linear relationship with \(N_a\) specific to each rate.

\(N_a\) is calculated as the product of nitrogen content per unit leaf dry mass ( \(N_m, \ g_N \ g_{drymatter}^{-1}\)) and leaf dry mass per area (\(LMA, \ g_{drymatter} \ m^{-2}\)). \(N_m\) linearly varies with plant age, expressed as the thermal time cumulated since budburst, and \(LMA\) is determined by leaf exposure to light during the last past days (Prieto et al., 2012), as expressed respectively in the two following equations:

\[ \begin{align}\begin{aligned}\begin{array}\\\begin{split}N_m = a_N \cdot \sum_{i=budburst}^d {\left( \max{\left( 0, T_{air, \ i} - T_b \right)} \right)} + b_N \\\end{split}\\LMA = a_M \cdot \ \ln{(PPFD_{10})} + b_M\\\end{array}\end{aligned}\end{align} \]

where \(T_{air, \ i} \ [^\circ C]\) is the mean temperature of the day \(i\), \(T_b \ [^\circ C]\) is the base temperature (minimum required for growth), set to 10 \(\ ^\circ C\) for grapevine and used for the calculation of thermal time since budburst, \(a_N \ [g_N \ g_{drymatter}^{-1} \ ^\circ C \ d^{-1}]\) and \(b_N \ [g_N \ g_{drymatter}^{-1}]\) are the slope and intercept of the linear relationship between \(N_m\) and accumulated thermal time since budburst, \(PPFD_{10} \ [mol_{photon} \ m^{-2} \ d^{-1}]\) is the cumulative photosynthetic photon flux density irradiance intercepted by the leaf (output of the energy module) averaged over the past 10 days, \(a_M \ [g_{drymatter} \ mol_{photon}^{-1} \ d^{-1}]\) and \(b_M \ [g_{drymatter} \ m^{-2}]\) are the slope and intercept of the linear relationship between \(LMA\) and the logarithm of \(PPFD_{10}\).

Photoinhibition

HydroShoot is provided with an empirical photoinhibition model which assumes that combined heat and water stresses inhibit photosynthesis by reducing the electron transport rate (\(J\)):

\[ \begin{align}\begin{aligned}\begin{array}\\\begin{split}\Delta H_d = \Delta H_{d, \ max} - \max \left( 0, \ \left( \Delta H_{d, \ max - \Delta H_{d, \ T}} \right) \cdot \min \left( 1, \ \frac{\Psi_{leaf} - \Psi_{leaf, \ max}}{\Psi_{leaf, \ min} - \Psi_{leaf, \ max}} \right) \right) \\\end{split}\\\Delta H_{d, \ t} = \Delta H_{d, \ t1} - \left( \Delta H_{d, \ t1} - \Delta H_{d, \ t2} \right) \cdot \min \left( 1, \ \max \left( 0, \ \frac{T_{leaf} - T_{leaf1}}{T_{leaf2} - T_{leaf1}} \right) \right)\\\end{array}\end{aligned}\end{align} \]

where \(\Delta H_d \ [kJ \ mol^{-1}]\) is calculated after accounting for the joint effects of leaf water potential \(\Psi_{leaf} \ MPa\) and temperature \(T_{leaf} \ [K]\), \(\Delta H_{d, \ max} \ [kJ \ mol^{-1}]\) is the value of \(\Delta H_d\) without accounting for photoinhibition, \(\Delta H_{d, \ T} \ [kJ mol^{-1}]\) is the value of \(\Delta H_d\) after accounting for the effect of \(T_{leaf}\), \(\Psi_{leaf, \ max}\) and \(\Psi_{leaf, \ min} \ [MPa]\) are leaf water potential values at which photoinhibition starts and reaches its maximum effect, respectively, finally, \(\Delta H_{d, \ T1}\) and \(\Delta H_{d, \ T2} \ [kJ mol^{-1}]\) are empirical thresholds corresponding to leaf temperatures \(T_{leaf1}\) and \(T_{leaf2}\) which are temperatures at which photoinhibition starts and reaches its maximum effect, respectively.

Transpiration rate per unit leaf area

The transpiration rate (\(E, \ mol \ m^{-2} s^{-1}\)) is calculated as:

\[E = \frac{1}{\frac{1}{g_{b, \ H_2O}} + \frac{1}{1.6 \cdot g_{s, \ CO_2}}} \left( \frac{VPD}{P_a} \right)\]

where \(P_a \ [kPa]\) is the atmospheric pressure and \(g_{b, \ H_2O} \ [mol \ m^{-2} s^{-1}]\) is the boundary layer conductance to water vapor derived from Nobel (2005) as:

\[g_{b, \ H_2O} = \frac{D_{H_2O} \cdot P_v}{R \cdot T_{leaf} \cdot \Delta x}\]

with

\[D_{H_2O} = D_{H_2O, 0} \cdot \frac{P_a}{P_v} \cdot \left( \frac{T_{leaf}}{273} \right)^{1.8}\]

where \(D_{H_2O}\) is the diffusion coefficient of H2O in the air at 0 \(^\circ C\) (\(2.13 \cdot {10}^{-5} \ m^2 s^{-1}\)), \(P_a \ [MPa]\) is the ambient air pressure at 0 \(^\circ C\), \(P_v \ [MPa]\) is water vapor partial pressure, and \(\Delta x \ [m]\) is the thickness of the boundary layer defined following Nobel (2005) as:

\[\Delta x = 0.004 \sqrt{\frac{l}{v}}\]

where \(l \ [m]\) is the mean length of the leaf in the downwind direction (set to 70% of blade length), and \(v \ [m \ s^{-1}]\) is wind speed in the vicinity of the leaf.

Finally, the impact of water stress on stomatal conductance (i.e. via the \(f_w\) function) is calculated using one of the following options:

\[\begin{split}f_w = \left \{ \begin{array}{11} \frac{1}{1+\left( \frac{VPD}{D_0} \right)} & (a) \\ \frac{1}{1+\left( \frac{\Psi_{leaf}}{\Psi_{crit, \ leaf}} \right)^n} & (b) \\ \frac{1}{1+\left( \frac{\Psi_{soil}}{\Psi_{crit, \ leaf}} \right)^n} & (c) \\ \end{array} \right.\end{split}\]

where \(VPD \ [kPa]\) is vapor pressure deficit (between the leaf and the air), \(D_0 \ [kPa]\) shape parameter, \(\Psi_{leaf} \ [MPa]\) leaf bulk xylem potential, \(\Psi_{soil} \ [MPa]\) soil bulk water potential (assumed equal to xylem potential at the base of the shoot), and \(\Psi_{crit, leaf} \ [MPa]\) leaf water potential at which stomatal conductance reduces to half its maximum value.

In case the option \(a\) is used, stomatal conductance reduction is considered independent from the soil water status (i.e. following Leuning, 1995). In contrast, Both options \(b\) and c allow simulating stomatal conductance as a function either of leaf water potential (i.e. regarding shoot hydraulic structure) or soil water potential (i.e. disregarding the hydraulic structure of the shoot).