## 2.9.4. Photosynthesis[¶](#photosynthesis "Permalink to this headline")
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Photosynthesis in C3 plants is based on the model of [Farquhar et al. (1980)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#farquharetal1980). Photosynthesis in C4 plants is based on the model of [Collatz et al. (1992)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#collatzetal1992). [Bonan et al. (2011)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#bonanetal2011) describe the implementation, modified here. In its simplest form, leaf net photosynthesis after accounting for respiration (\\(R\_{d}\\) ) is

(2.9.2)[¶](#equation-9-2 "Permalink to this equation")\\\[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)[¶](#equation-9-3 "Permalink to this equation")\\\[\\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)[¶](#equation-9-4 "Permalink to this equation")\\\[\\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)[¶](#equation-9-5 "Permalink to this equation")\\\[\\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](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Photosynthetic_Capacity/CLM50_Tech_Note_Photosynthetic_Capacity.html#rst-photosynthetic-capacity)) 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](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#collatzetal1992)). 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](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Radiative_Fluxes/CLM50_Tech_Note_Radiative_Fluxes.html#solar-fluxes)), 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)[¶](#equation-9-6 "Permalink to this equation")\\\[\\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](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Photosynthetic_Capacity/CLM50_Tech_Note_Photosynthetic_Capacity.html#rst-photosynthetic-capacity)), \\(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)[¶](#equation-9-7 "Permalink to this equation")\\\[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)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#collatzetal1991). The actual gross photosynthesis rate, \\(A\\), is given by the smaller root of the equations

(2.9.8)[¶](#equation-9-8 "Permalink to this equation")\\\[\\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](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Photosynthetic_Capacity/CLM50_Tech_Note_Photosynthetic_Capacity.html#rst-photosynthetic-capacity))

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](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Plant_Respiration/CLM50_Tech_Note_Plant_Respiration.html#rst-plant-respiration))

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](#canopy-scaling)). In C3 plants, these are adjusted for leaf temperature, \\(T\_{v}\\) (K), as:

(2.9.9)[¶](#equation-9-9 "Permalink to this equation")\\\[\\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)[¶](#equation-9-10 "Permalink to this equation")\\\[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)[¶](#equation-9-11 "Permalink to this equation")\\\[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](#table-temperature-dependence-parameters-for-c3-photosynthesis) 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)](#equation-9-16)), and \\(\\Delta S\\) is 490 J mol \-1 K \-1 for \\(R\_d\\) ([Bonan et al. 2011](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#bonanetal2011), [Lombardozzi et al. 2015](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#lombardozzietal2015)). 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)[¶](#equation-9-12 "Permalink to this equation")\\\[\\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)[¶](#equation-9-13 "Permalink to this equation")\\\[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)[¶](#equation-9-14 "Permalink to this equation")\\\[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.[¶](#id5 "Permalink to this table")   
| 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)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#kattgeknorr2007). 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)[¶](#equation-9-15 "Permalink to this equation")\\\[\\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)[¶](#equation-9-16 "Permalink to this equation")\\\[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.

