2.9. Stomatal Resistance and Photosynthesis

2.9.1. Summary of CLM5.0 updates relative to the CLM4.5

We describe here the complete photosynthesis and stomatal conductance parameterizations that appear in CLM5.0. Corresponding information for CLM4.5 appeared in the CLM4.5 Technical Note (Oleson et al. 2013).

CLM5 includes the following new changes to photosynthesis and stomatal conductance:

• Default stomatal conductance calculation uses the Medlyn conductance model

• \(V_{c,max}\) and \(J_{max}\) at 25 ^oC: are now prognostic, and predicted via optimality by the LUNA model (Chapter 2.10)

• Leaf N concentration and the fraction of leaf N in Rubisco used to calculate \(V_{cmax25}\) are determined by the LUNA model (Chapter 2.10)

• Water stress is applied by the hydraulic conductance model (Chapter 2.11)

2.9.2. Introduction

Leaf stomatal resistance, which is needed for the water vapor flux (Chapter 2.5), is coupled to leaf photosynthesis similar to Collatz et al. (1991, 1992). These equations are solved separately for sunlit and shaded leaves using average absorbed photosynthetically active radiation for sunlit and shaded leaves [\(\phi ^{sun}\),\(\phi ^{sha}\) W m^-2 (section 2.4.1)] to give sunlit and shaded stomatal resistance (\(r_{s}^{sun}\),\(r_{s}^{sha}\) s m^-1) and photosynthesis (\(A^{sun}\),\(A^{sha}\) µmol CO₂ m^-2 s^-1). Canopy photosynthesis is \(A^{sun} L^{sun} +A^{sha} L^{sha}\), where \(L^{sun}\) and \(L^{sha}\) are the sunlit and shaded leaf area indices (section 2.4.1). Canopy conductance is \(\frac{1}{r_{b} +r_{s}^{sun} } L^{sun} +\frac{1}{r_{b} +r_{s}^{sha} } L^{sha}\), where \(r_{b}\) is the leaf boundary layer resistance (section 2.5.3).

2.9.3. Stomatal resistance

CLM5 calculates stomatal conductance using the Medlyn stomatal conductance model (Medlyn et al. 2011). Previous versions of CLM calculated leaf stomatal resistance using the Ball-Berry conductance model as described by Collatz et al. (1991) and implemented in global climate models (Sellers et al. 1996). The Medlyn model calculates stomatal conductance (i.e., the inverse of resistance) based on net leaf photosynthesis, the leaf-to-air vapor pressure difference, and the CO₂ concentration at the leaf surface. Leaf stomatal resistance is:

(2.9.1)\[\frac{1}{r_{s} } =g_{s} = g_{o} + 1.6(1 + \frac{g_{1} }{\sqrt{D_{s}}}) \frac{A_{n} }{{c_{s} \mathord{\left/ {\vphantom {c_{s} P_{atm} }} \right.} P_{atm} } }\]

where \(r_{s}\) is leaf stomatal resistance (s m² \(\mu\)mol^-1), \(g_{o}\) is the minimum stomatal conductance (\(\mu\) mol m ^-2 s^-1), \(A_{n}\) is leaf net photosynthesis (\(\mu\)mol CO₂ m^-2 s^-1), \(c_{s}\) is the CO₂ partial pressure at the leaf surface (Pa), \(P_{atm}\) is the atmospheric pressure (Pa), and \(D_{s}=(e_{i}-e{_s})/1000\) is the leaf-to-air vapor pressure difference at the leaf surface (kPa) where \(e_{i}\) is the saturation vapor pressure (Pa) evaluated at the leaf temperature \(T_{v}\), and \(e_{s}\) is the vapor pressure at the leaf surface (Pa). \(g_{1}\) is a plant functional type dependent parameter (Table 2.9.1) and are the same as those used in the CABLE model (de Kauwe et al. 2015).

The value for \(g_{o}=100\) \(\mu\) mol m ^-2 s^-1 for C₃ and C₄ plants. Photosynthesis is calculated for sunlit (\(A^{sun}\)) and shaded (\(A^{sha}\)) leaves to give \(r_{s}^{sun}\) and \(r_{s}^{sha}\). Additionally, soil water influences stomatal resistance through plant hydraulic stress, detailed in the Plant Hydraulics chapter.

Resistance is converted from units of s m² \(\mu\) mol^-1 to s m^-1 as: 1 s m^-1 = \(1\times 10^{-9} R_{gas} \frac{\theta _{atm} }{P_{atm} }\) \(\mu\) mol^-1 m² s, where \(R_{gas}\) is the universal gas constant (J K^-1 kmol^-1) (Table 2.2.7) and \(\theta _{atm}\) is the atmospheric potential temperature (K).

Table 2.9.1 Plant functional type (PFT) stomatal conductance parameters.

PFT	g1
NET Temperate	2.35
NET Boreal	2.35
NDT Boreal	2.35
BET Tropical	4.12
BET temperate	4.12
BDT tropical	4.45
BDT temperate	4.45
BDT boreal	4.45
BES temperate	4.70
BDS temperate	4.70
BDS boreal	4.70
C3 arctic grass	2.22
C3 grass	5.25
C4 grass	1.62
Temperate Corn	1.79
Spring Wheat	5.79
Temperate Soybean	5.79
Cotton	5.79
Rice	5.79
Sugarcane	1.79
Tropical Corn	1.79
Tropical Soybean	5.79
Miscanthus	1.79
Switchgrass	1.79

2.9.4. Photosynthesis

Photosynthesis in C₃ plants is based on the model of Farquhar et al. (1980). Photosynthesis in C₄ plants is based on the model of Collatz et al. (1992). Bonan et al. (2011) describe the implementation, modified here. In its simplest form, leaf net photosynthesis after accounting for respiration (\(R_{d}\) ) is

(2.9.2)\[A_{n} =\min \left(A_{c} ,A_{j} ,A_{p} \right)-R_{d} .\]

The RuBP carboxylase (Rubisco) limited rate of carboxylation \(A_{c}\) (\(\mu\) mol CO₂ m^-2 s^-1) is

(2.9.3)\[\begin{split}A_{c} =\left\{\begin{array}{l} {\frac{V_{c\max } \left(c_{i} -\Gamma _{*} \right)}{c_{i} +K_{c} \left(1+{o_{i} \mathord{\left/ {\vphantom {o_{i} K_{o} }} \right.} K_{o} } \right)} \qquad {\rm for\; C}_{{\rm 3}} {\rm \; plants}} \\ {V_{c\max } \qquad \qquad \qquad {\rm for\; C}_{{\rm 4}} {\rm \; plants}} \end{array}\right\}\qquad \qquad c_{i} -\Gamma _{*} \ge 0.\end{split}\]

The maximum rate of carboxylation allowed by the capacity to regenerate RuBP (i.e., the light-limited rate) \(A_{j}\) (\(\mu\) mol CO₂ m^-2 s^-1) is

(2.9.4)\[\begin{split}A_{j} =\left\{\begin{array}{l} {\frac{J_{x}\left(c_{i} -\Gamma _{*} \right)}{4c_{i} +8\Gamma _{*} } \qquad \qquad {\rm for\; C}_{{\rm 3}} {\rm \; plants}} \\ {\alpha (4.6\phi )\qquad \qquad {\rm for\; C}_{{\rm 4}} {\rm \; plants}} \end{array}\right\}\qquad \qquad c_{i} -\Gamma _{*} \ge 0.\end{split}\]

The product-limited rate of carboxylation for C₃ plants and the PEP carboxylase-limited rate of carboxylation for C₄ plants \(A_{p}\) (\(\mu\) mol CO₂ m^-2 s^-1) is

(2.9.5)\[\begin{split}A_{p} =\left\{\begin{array}{l} {3T_{p\qquad } \qquad \qquad {\rm for\; C}_{{\rm 3}} {\rm \; plants}} \\ {k_{p} \frac{c_{i} }{P_{atm} } \qquad \qquad \qquad {\rm for\; C}_{{\rm 4}} {\rm \; plants}} \end{array}\right\}.\end{split}\]

In these equations, \(c_{i}\) is the internal leaf CO₂ partial pressure (Pa) and \(o_{i} =0.20P_{atm}\) is the O₂ partial pressure (Pa). \(K_{c}\) and \(K_{o}\) are the Michaelis-Menten constants (Pa) for CO₂ and O₂. \(\Gamma _{*}\) (Pa) is the CO₂ compensation point. \(V_{c\max }\) is the maximum rate of carboxylation (µmol m^-2 s^-1, Chapter 2.10) and \(J_{x}\) is the electron transport rate (µmol m^-2 s^-1). \(T_{p}\) is the triose phosphate utilization rate (µmol m^-2 s^-1), taken as \(T_{p} =0.167V_{c\max }\) so that \(A_{p} =0.5V_{c\max }\) for C₃ plants (as in Collatz et al. 1992). For C₄ plants, the light-limited rate \(A_{j}\) varies with \(\phi\) in relation to the quantum efficiency (\(\alpha =0.05\) mol CO₂ mol^-1 photon). \(\phi\) is the absorbed photosynthetically active radiation (W m^-2) (section 2.4.1), which is converted to photosynthetic photon flux assuming 4.6 \(\mu\) mol photons per joule. \(k_{p}\) is the initial slope of C₄ CO₂ response curve.

For C₃ plants, the electron transport rate depends on the photosynthetically active radiation absorbed by the leaf. A common expression is the smaller of the two roots of the equation

(2.9.6)\[\Theta _{PSII} J_{x}^{2} -\left(I_{PSII} +J_{\max } \right)J_{x}+I_{PSII} J_{\max } =0\]

where \(J_{\max }\) is the maximum potential rate of electron transport (\(\mu\)mol m^-2 s^-1, Chapter 2.10), \(I_{PSII}\) is the light utilized in electron transport by photosystem II (µmol m^-2 s^-1), and \(\Theta _{PSII}\) is a curvature parameter. For a given amount of photosynthetically active radiation absorbed by a leaf (\(\phi\), W m^-2), converted to photosynthetic photon flux density with 4.6 \(\mu\)mol J^-1, the light utilized in electron transport is

(2.9.7)\[I_{PSII} =0.5\Phi _{PSII} (4.6\phi )\]

where \(\Phi _{PSII}\) is the quantum yield of photosystem II, and the term 0.5 arises because one photon is absorbed by each of the two photosystems to move one electron. Parameter values are \(\Theta _{PSII}\) = 0.7 and \(\Phi _{PSII}\) = 0.85. In calculating \(A_{j}\) (for both C₃ and C₄ plants), \(\phi =\phi ^{sun}\) for sunlit leaves and \(\phi =\phi ^{sha}\) for shaded leaves.

The model uses co-limitation as described by Collatz et al. (1991, 1992). The actual gross photosynthesis rate, \(A\), is given by the smaller root of the equations

(2.9.8)\[\begin{split}\begin{array}{rcl} {\Theta _{cj} A_{i}^{2} -\left(A_{c} +A_{j} \right)A_{i} +A_{c} A_{j} } & {=} & {0} \\ {\Theta _{ip} A^{2} -\left(A_{i} +A_{p} \right)A+A_{i} A_{p} } & {=} & {0} \end{array} .\end{split}\]

Values are \(\Theta _{cj} =0.98\) and \(\Theta _{ip} =0.95\) for C₃ plants; and \(\Theta _{cj} =0.80\)and \(\Theta _{ip} =0.95\) for C₄ plants. \(A_{i}\) is the intermediate co-limited photosynthesis. \(A_{n} =A-R_{d}\).

The parameters \(K_{c}\), \(K_{o}\), and \(\Gamma\) depend on temperature. Values at 25 °C are \(K_{c25} ={\rm 4}0{\rm 4}.{\rm 9}\times 10^{-6} P_{atm}\), \(K_{o25} =278.4\times 10^{-3} P_{atm}\), and \(\Gamma _{25} {\rm =42}.75\times 10^{-6} P_{atm}\). \(V_{c\max }\), \(J_{\max }\), \(T_{p}\), \(k_{p}\), and \(R_{d}\) also vary with temperature.

\(J_{\max 25}\) at 25 ^oC: is calculated by the LUNA model (Chapter 2.10)

Parameter values at 25 ^oC are calculated from \(V_{c\max }\) at 25 ^oC:, including: \(T_{p25} =0.167V_{c\max 25}\), and \(R_{d25} =0.015V_{c\max 25}\) (C₃) and \(R_{d25} =0.025V_{c\max 25}\) (C₄).

For C₄ plants, \(k_{p25} =20000\; V_{c\max 25}\).

However, when the biogeochemistry is active (the default mode), \(R_{d25}\) is calculated from leaf nitrogen as described in (Chapter 2.17)

The parameters \(V_{c\max 25}\), \(J_{\max 25}\), \(T_{p25}\), \(k_{p25}\), and \(R_{d25}\) are scaled over the canopy for sunlit and shaded leaves (section 2.9.5). In C₃ plants, these are adjusted for leaf temperature, \(T_{v}\) (K), as:

(2.9.9)\[\begin{split}\begin{array}{rcl} {V_{c\max } } & {=} & {V_{c\max 25} \; f\left(T_{v} \right)f_{H} \left(T_{v} \right)} \\ {J_{\max } } & {=} & {J_{\max 25} \; f\left(T_{v} \right)f_{H} \left(T_{v} \right)} \\ {T_{p} } & {=} & {T_{p25} \; f\left(T_{v} \right)f_{H} \left(T_{v} \right)} \\ {R_{d} } & {=} & {R_{d25} \; f\left(T_{v} \right)f_{H} \left(T_{v} \right)} \\ {K_{c} } & {=} & {K_{c25} \; f\left(T_{v} \right)} \\ {K_{o} } & {=} & {K_{o25} \; f\left(T_{v} \right)} \\ {\Gamma } & {=} & {\Gamma _{25} \; f\left(T_{v} \right)} \end{array}\end{split}\]

(2.9.10)\[f\left(T_{v} \right)=\; \exp \left[\frac{\Delta H_{a} }{298.15\times 0.001R_{gas} } \left(1-\frac{298.15}{T_{v} } \right)\right]\]

and

(2.9.11)\[f_{H} \left(T_{v} \right)=\frac{1+\exp \left(\frac{298.15\Delta S-\Delta H_{d} }{298.15\times 0.001R_{gas} } \right)}{1+\exp \left(\frac{\Delta ST_{v} -\Delta H_{d} }{0.001R_{gas} T_{v} } \right)} .\]

Table 2.9.2 lists parameter values for \(\Delta H_{a}\) and \(\Delta H_{d}\). \(\Delta S\) is calculated separately for \(V_{c\max }\) and \(J_{max }\) to allow for temperature acclimation of photosynthesis (see equation (2.9.16)), and \(\Delta S\) is 490 J mol ^-1 K ^-1 for \(R_d\) (Bonan et al. 2011, Lombardozzi et al. 2015). Because \(T_{p}\) as implemented here varies with \(V_{c\max }\), \(T_{p}\) uses the same temperature parameters as \(V_{c\max}\). For C₄ plants,

(2.9.12)\[\begin{split}\begin{array}{l} {V_{c\max } =V_{c\max 25} \left[\frac{Q_{10} ^{(T_{v} -298.15)/10} }{f_{H} \left(T_{v} \right)f_{L} \left(T_{v} \right)} \right]} \\ {f_{H} \left(T_{v} \right)=1+\exp \left[s_{1} \left(T_{v} -s_{2} \right)\right]} \\ {f_{L} \left(T_{v} \right)=1+\exp \left[s_{3} \left(s_{4} -T_{v} \right)\right]} \end{array}\end{split}\]

with \(Q_{10} =2\), \(s_{1} =0.3\)K^-1 \(s_{2} =313.15\) K, \(s_{3} =0.2\)K^-1, and \(s_{4} =288.15\) K. Additionally,

(2.9.13)\[R_{d} =R_{d25} \left\{\frac{Q_{10} ^{(T_{v} -298.15)/10} }{1+\exp \left[s_{5} \left(T_{v} -s_{6} \right)\right]} \right\}\]

with \(Q_{10} =2\), \(s_{5} =1.3\) K^-1 and \(s_{6} =328.15\)K, and

(2.9.14)\[k_{p} =k_{p25} \, Q_{10} ^{(T_{v} -298.15)/10}\]

with \(Q_{10} =2\).

Table 2.9.2 Temperature dependence parameters for C3 photosynthesis.

Parameter	\(\Delta H_{a}\) (J mol-1)	\(\Delta H_{d}\) (J mol-1)
\(V_{c\max }\)	72000	200000
\(J_{\max }\)	50000	200000
\(T_{p}\)	72000	200000
\(R_{d}\)	46390	150650
\(K_{c}\)	79430	–
\(K_{o}\)	36380	–
\(\Gamma _{*}\)	37830	–

In the model, acclimation is implemented as in Kattge and Knorr (2007). In this parameterization, \(V_{c\max }\) and \(J_{\max }\) vary with the plant growth temperature. This is achieved by allowing \(\Delta S\)to vary with growth temperature according to

(2.9.15)\[\begin{split}\begin{array}{l} {\Delta S=668.39-1.07(T_{10} -T_{f} )\qquad \qquad {\rm for\; }V_{c\max } } \\ {\Delta S=659.70-0.75(T_{10} -T_{f} )\qquad \qquad {\rm for\; }J_{\max } } \end{array}\end{split}\]

The effect is to cause the temperature optimum of \(V_{c\max }\) and \(J_{\max }\) to increase with warmer temperatures. Additionally, the ratio \(J_{\max 25} /V_{c\max 25}\) at 25 °C decreases with growth temperature as

(2.9.16)\[J_{\max 25} /V_{c\max 25} =2.59-0.035(T_{10} -T_{f} ).\]

In these acclimation functions, \(T_{10}\) is the 10-day mean air temperature (K) and \(T_{f}\) is the freezing point of water (K). For lack of data, \(T_{p}\) acclimates similar to \(V_{c\max }\). Acclimation is restricted over the temperature range \(T_{10} -T_{f} \ge\) 11°C and \(T_{10} -T_{f} \le\) 35°C.

2.9.5. Canopy scaling

When LUNA is on, the \(V_{c\max 25}\) for sun leaves is scaled to the shaded leaves \(J_{\max 25}\), \(T_{p25}\), \(k_{p25}\), and \(R_{d25}\) scale similarly.

(2.9.17)\[\begin{split}\begin{array}{rcl} {V_{c\max 25 sha}} & {=} & {V_{c\max 25 sha} \frac{i_{v,sha}}{i_{v,sun}}} \\ {J_{\max 25 sha}} & {=} & {J_{\max 25 sun} \frac{i_{v,sha}}{i_{v,sun}}} \\ {T_{p sha}} & {=} & {T_{p sun} \frac{i_{v,sha}}{i_{v,sun}}} \end{array}\end{split}\]

Where \(i_{v,sun}\) and \(i_{v,sha}\) are the leaf-to-canopy scaling coefficients of the twostream radiation model, calculated as

(2.9.18)\[\begin{split}i_{v,sun} = \frac{(1 - e^{-(k_{n,ext}+k_{b,ext})*lai_e)} / (k_{n,ext}+k_{b,ext})}{f_{sun}*lai_e}\\ i_{v,sha} = \frac{(1 - e^{-(k_{n,ext}+k_{b,ext})*lai_e)} / (k_{n,ext}+k_{b,ext})}{(1 - f_{sun})*lai_e}\end{split}\]

k_{n,ext} is the extinction coefficient for N through the canopy (0.3). k_{b,ext} is the direct beam extinction coefficient calculated in the surface albedo routine, and \(f_{sun}\) is the fraction of sunlit leaves, both derived from Chapter 2.3.

When LUNA is off, scaling defaults to the mechanism used in CLM4.5.

2.9.6. Numerical implementation

The CO₂ partial pressure at the leaf surface, \(c_{s}\) (Pa), and the vapor pressure at the leaf surface, \(e_{s}\) (Pa), needed for the stomatal resistance model in equation (2.9.1), and the internal leaf CO₂ partial pressure \(c_{i}\) (Pa), needed for the photosynthesis model in equations (2.9.3)-(2.9.5), are calculated assuming there is negligible capacity to store CO₂ and water vapor at the leaf surface so that

(2.9.19)\[A_{n} =\frac{c_{a} -c_{i} }{\left(1.4r_{b} +1.6r_{s} \right)P_{atm} } =\frac{c_{a} -c_{s} }{1.4r_{b} P_{atm} } =\frac{c_{s} -c_{i} }{1.6r_{s} P_{atm} }\]

and the transpiration fluxes are related as

(2.9.20)\[\frac{e_{a} -e_{i} }{r_{b} +r_{s} } =\frac{e_{a} -e_{s} }{r_{b} } =\frac{e_{s} -e_{i} }{r_{s} }\]

where \(r_{b}\) is leaf boundary layer resistance (s m² \(\mu\) mol^-1) (section 2.5.3), the terms 1.4 and 1.6 are the ratios of diffusivity of CO₂ to H₂O for the leaf boundary layer resistance and stomatal resistance, \(c_{a} ={\rm CO}_{{\rm 2}} \left({\rm mol\; mol}^{{\rm -1}} \right)\), \(P_{atm}\) is the atmospheric pressure (Pa), \(e_{i}\) is the saturation vapor pressure (Pa) evaluated at the leaf temperature \(T_{v}\), and \(e_{a}\) is the vapor pressure of air (Pa). The vapor pressure of air in the plant canopy \(e_{a}\) (Pa) is determined from

(2.9.21)\[e_{a} =\frac{P_{atm} q_{s} }{0.622}\]

where \(q_{s}\) is the specific humidity of canopy air (kg kg^-1, section 2.5.3). Equations (2.9.19) and (2.9.20) are solved for \(c_{s}\) and \(e_{s}\)

(2.9.22)\[c_{s} =c_{a} -1.4r_{b} P_{atm} A_{n}\]

(2.9.23)\[e_{s} =\frac{e_{a} r_{s} +e_{i} r_{b} }{r_{b} +r_{s} }\]

In terms of conductance with \(g_{s} =1/r_{s}\) and \(g_{b} =1/r_{b}\)

(2.9.24)\[e_{s} =\frac{e_{a} g_{b} +e_{i} g_{s} }{g_{b} +g_{s} } .\]

Substitution of equation (2.9.24) into equation (2.9.1) gives an expression for the stomatal resistance (\(r_{s}\)) as a function of photosynthesis (\(A_{n}\) )

(2.9.25)\[ag_{s}^{2} + bg_{s} + c = 0\]

where

(2.9.26)\[\begin{split}\begin{array}{l} a = 1 \\ b = -[2(g_{o} * 10^{-6} + d) + \frac{(g_{1}d)^{2}}{g_{b}*10^{-6}D_{l}}] \\ c = (g_{o}*10^{-6})^{2} + [2g_{o}*10^{-6} + d (1-\frac{g_{1}^{2}} {D_{l}})]d \end{array}\end{split}\]

and

(2.9.27)\[ \begin{align}\begin{aligned}d = \frac {1.6 A_{n}} {c_{s} / P_{atm} * 10^{6}}\\D_{l} = \frac {max(e_{i} - e_{a},50)} {1000}\end{aligned}\end{align} \]

Stomatal conductance, as solved by equation (2.9.24) (mol m ^-2 s ^-1), is the larger of the two roots that satisfy the quadratic equation. Values for \(c_{i}\) are given by

(2.9.28)\[c_{i} =c_{a} -\left(1.4r_{b} +1.6r_{s} \right)P_{atm} A{}_{n}\]

The equations for \(c_{i}\), \(c_{s}\), \(r_{s}\), and \(A_{n}\) are solved iteratively until \(c_{i}\) converges. Sun et al. (2012) pointed out that the CLM4 numerical approach does not always converge. Therefore, the model uses a hybrid algorithm that combines the secant method and Brent’s method to solve for \(c_{i}\). The equation set is solved separately for sunlit (\(A_{n}^{sun}\), \(r_{s}^{sun}\) ) and shaded (\(A_{n}^{sha}\), \(r_{s}^{sha}\) ) leaves.