.. _rst_Stomatal Resistance and Photosynthesis: Stomatal Resistance and Photosynthesis ========================================= 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 (:ref:`Oleson et al. 2013 `). CLM5 includes the following new changes to photosynthesis and stomatal conductance: - Default stomatal conductance calculation uses the Medlyn conductance model - :math:`V_{c,max}` and :math:`J_{max}` at 25 :sup:`\o`\ C: are now prognostic, and predicted via optimality by the LUNA model (Chapter :numref:`rst_Photosynthetic Capacity`) - Leaf N concentration and the fraction of leaf N in Rubisco used to calculate :math:`V_{cmax25}` are determined by the LUNA model (Chapter :numref:`rst_Photosynthetic Capacity`) - Water stress is applied by the hydraulic conductance model (Chapter :numref:`rst_Plant Hydraulics`) Introduction ----------------------- Leaf stomatal resistance, which is needed for the water vapor flux (Chapter :numref:`rst_Momentum, Sensible Heat, and Latent Heat Fluxes`), is coupled to leaf photosynthesis similar to Collatz et al. (:ref:`1991 `, :ref:`1992 `). These equations are solved separately for sunlit and shaded leaves using average absorbed photosynthetically active radiation for sunlit and shaded leaves [:math:`\phi ^{sun}` ,\ :math:`\phi ^{sha}` W m\ :sup:`-2` (section :numref:`Solar Fluxes`)] to give sunlit and shaded stomatal resistance (:math:`r_{s}^{sun}` ,\ :math:`r_{s}^{sha}` s m\ :sup:`-1`) and photosynthesis (:math:`A^{sun}` ,\ :math:`A^{sha}` µmol CO\ :sub:`2` m\ :sup:`-2` s\ :sup:`-1`). Canopy photosynthesis is :math:`A^{sun} L^{sun} +A^{sha} L^{sha}` , where :math:`L^{sun}` and :math:`L^{sha}` are the sunlit and shaded leaf area indices (section :numref:`Solar Fluxes`). Canopy conductance is :math:`\frac{1}{r_{b} +r_{s}^{sun} } L^{sun} +\frac{1}{r_{b} +r_{s}^{sha} } L^{sha}` , where :math:`r_{b}` is the leaf boundary layer resistance (section :numref:`Sensible and Latent Heat Fluxes and Temperature for Vegetated Surfaces`). .. _Stomatal resistance: Stomatal resistance ----------------------- CLM5 calculates stomatal conductance using the Medlyn stomatal conductance model (:ref:`Medlyn et al. 2011`). Previous versions of CLM calculated leaf stomatal resistance is using the Ball-Berry conductance model as described by :ref:`Collatz et al. (1991)` and implemented in global climate models (:ref:`Sellers et al. 1996`). The Medlyn model calculates stomatal conductance (i.e., the inverse of resistance) based on net leaf photosynthesis, the vapor pressure deficit, and the CO\ :sub:`2` concentration at the leaf surface. Leaf stomatal resistance is: .. math:: :label: 9.1 \frac{1}{r_{s} } =g_{s} = g_{o} + 1.6(1 + \frac{g_{1} }{\sqrt{D}}) \frac{A_{n} }{{c_{s} \mathord{\left/ {\vphantom {c_{s} P_{atm} }} \right. \kern-\nulldelimiterspace} P_{atm} } } where :math:`r_{s}` is leaf stomatal resistance (s m\ :sup:`2` :math:`\mu`\ mol\ :sup:`-1`), :math:`g_{o}` is the minimum stomatal conductance (:math:`\mu` mol m :sup:`-2` s\ :sup:`-1`), :math:`A_{n}` is leaf net photosynthesis (:math:`\mu`\ mol CO\ :sub:`2` m\ :sup:`-2` s\ :sup:`-1`), :math:`c_{s}` is the CO\ :sub:`2` partial pressure at the leaf surface (Pa), :math:`P_{atm}` is the atmospheric pressure (Pa), and :math:`D` is the vapor pressure deficit at the leaf surface (kPa). :math:`g_{1}` is a plant functional type dependent parameter (:numref:`Table Plant functional type (PFT) stomatal conductance parameters`) and are the same as those used in the CABLE model (:ref:`de Kauwe et al. 2015 `). The value for :math:`g_{o}=100` :math:`\mu` mol m :sup:`-2` s\ :sup:`-1` for C\ :sub:`3` and C\ :sub:`4` plants. Photosynthesis is calculated for sunlit (:math:`A^{sun}`) and shaded (:math:`A^{sha}`) leaves to give :math:`r_{s}^{sun}` and :math:`r_{s}^{sha}`. Additionally, soil water influences stomatal resistance through plant hydraulic stress, detailed in the :ref:`rst_Plant Hydraulics` chapter. Resistance is converted from units of s m\ :sup:`2` :math:`\mu` mol\ :sup:`-1` to s m\ :sup:`-1` as: 1 s m\ :sup:`-1` = :math:`1\times 10^{-9} R_{gas} \frac{\theta _{atm} }{P_{atm} }` :math:`\mu` mol\ :sup:`-1` m\ :sup:`2` s, where :math:`R_{gas}` is the universal gas constant (J K\ :sup:`-1` kmol\ :sup:`-1`) (:numref:`Table Physical constants`) and :math:`\theta _{atm}` is the atmospheric potential temperature (K). .. _Table Plant functional type (PFT) stomatal conductance parameters: .. table:: Plant functional type (PFT) stomatal conductance parameters. +----------------------------------+-------------------+ | PFT | g\ :sub:`1` | +==================================+===================+ | 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 | +----------------------------------+-------------------+ | C\ :sub:`3` arctic grass | 2.22 | +----------------------------------+-------------------+ | C\ :sub:`3` grass | 5.25 | +----------------------------------+-------------------+ | C\ :sub:`4` 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 | +----------------------------------+-------------------+ .. _Photosynthesis: Photosynthesis ------------------ Photosynthesis in C\ :sub:`3` plants is based on the model of :ref:`Farquhar et al. (1980)`. Photosynthesis in C\ :sub:`4` plants is based on the model of :ref:`Collatz et al. (1992)`. :ref:`Bonan et al. (2011)` describe the implementation, modified here. In its simplest form, leaf net photosynthesis after accounting for respiration (:math:`R_{d}` ) is .. math:: :label: 9.3 A_{n} =\min \left(A_{c} ,A_{j} ,A_{p} \right)-R_{d} . The RuBP carboxylase (Rubisco) limited rate of carboxylation :math:`A_{c}` (:math:`\mu` \ mol CO\ :sub:`2` m\ :sup:`-2` s\ :sup:`-1`) is .. math:: :label: 9.4 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. \kern-\nulldelimiterspace} 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. The maximum rate of carboxylation allowed by the capacity to regenerate RuBP (i.e., the light-limited rate) :math:`A_{j}` (:math:`\mu` \ mol CO\ :sub:`2` m\ :sup:`-2` s\ :sup:`-1`) is .. math:: :label: 9.5 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. The product-limited rate of carboxylation for C\ :sub:`3` plants and the PEP carboxylase-limited rate of carboxylation for C\ :sub:`4` plants :math:`A_{p}` (:math:`\mu` \ mol CO\ :sub:`2` m\ :sup:`-2` s\ :sup:`-1`) is .. math:: :label: 9.6 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\}. In these equations, :math:`c_{i}` is the internal leaf CO\ :sub:`2` partial pressure (Pa) and :math:`o_{i} =0.20P_{atm}` is the O\ :sub:`2` partial pressure (Pa). :math:`K_{c}` and :math:`K_{o}` are the Michaelis-Menten constants (Pa) for CO\ :sub:`2` and O\ :sub:`2`. :math:`\Gamma _{\*}` (Pa) is the CO\ :sub:`2` compensation point. :math:`V_{c\max }` is the maximum rate of carboxylation (µmol m\ :sup:`-2` s\ :sup:`-1`, Chapter :numref:`rst_Photosynthetic Capacity`) and :math:`J_{x}` is the electron transport rate (µmol m\ :sup:`-2` s\ :sup:`-1`). :math:`T_{p}` is the triose phosphate utilization rate (µmol m\ :sup:`-2` s\ :sup:`-1`), taken as :math:`T_{p} =0.167V_{c\max }` so that :math:`A_{p} =0.5V_{c\max }` for C\ :sub:`3` plants (as in :ref:`Collatz et al. 1992 `). For C\ :sub:`4` plants, the light-limited rate :math:`A_{j}` varies with :math:`\phi` in relation to the quantum efficiency (:math:`\alpha =0.05` mol CO\ :sub:`2` mol\ :sup:`-1` photon). :math:`\phi` is the absorbed photosynthetically active radiation (W m\ :sup:`-2`) (section :numref:`Solar Fluxes`) , which is converted to photosynthetic photon flux assuming 4.6 :math:`\mu` \ mol photons per joule. :math:`k_{p}` is the initial slope of C\ :sub:`4` CO\ :sub:`2` response curve. For C\ :sub:`3` 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 .. math:: :label: 9.7 \Theta _{PSII} J_{x}^{2} -\left(I_{PSII} +J_{\max } \right)J_{x}+I_{PSII} J_{\max } =0 where :math:`J_{\max }` is the maximum potential rate of electron transport (:math:`\mu`\ mol m\ :sup:`-2` s\ :sup:`-1`, Chapter :numref:`rst_Photosynthetic Capacity`), :math:`I_{PSII}` is the light utilized in electron transport by photosystem II (µmol m\ :sup:`-2` s\ :sup:`-1`), and :math:`\Theta _{PSII}` is a curvature parameter. For a given amount of photosynthetically active radiation absorbed by a leaf (:math:`\phi`, W m\ :sup:`-2`), converted to photosynthetic photon flux density with 4.6 :math:`\mu`\ mol J\ :sup:`-1`, the light utilized in electron transport is .. math:: :label: 9.8 I_{PSII} =0.5\Phi _{PSII} (4.6\phi ) where :math:`\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 :math:`\Theta _{PSII}` \ = 0.7 and :math:`\Phi _{PSII}` \ = 0.85. In calculating :math:`A_{j}` (for both C\ :sub:`3` and C\ :sub:`4` plants), :math:`\phi =\phi ^{sun}` for sunlit leaves and :math:`\phi =\phi ^{sha}` for shaded leaves. The model uses co-limitation as described by :ref:`Collatz et al. (1991, 1992) `. The actual gross photosynthesis rate, :math:`A`, is given by the smaller root of the equations .. math:: :label: 9.9 \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} . Values are :math:`\Theta _{cj} =0.98` and :math:`\Theta _{ip} =0.95` for C\ :sub:`3` plants; and :math:`\Theta _{cj} =0.80`\ and :math:`\Theta _{ip} =0.95` for C\ :sub:`4` plants. :math:`A_{i}` is the intermediate co-limited photosynthesis. :math:`A_{n} =A-R_{d}` . The parameters :math:`K_{c}`, :math:`K_{o}`, and :math:`\Gamma` depend on temperature. Values at 25 :sup:`o` \ C are :math:`K_{c25} ={\rm 4}0{\rm 4}.{\rm 9}\times 10^{-6} P_{atm}`, :math:`K_{o25} =278.4\times 10^{-3} P_{atm}`, and :math:`\Gamma _{25} {\rm =42}.75\times 10^{-6} P_{atm}`. :math:`V_{c\max }`, :math:`J_{\max }`, :math:`T_{p}`, :math:`k_{p}`, and :math:`R_{d}` also vary with temperature. :math:`J_{\max 25}` at 25 :sup:`\o`\ C: is calculated by the LUNA model (Chapter :numref:`rst_Photosynthetic Capacity`) Parameter values at 25 :sup:`\o`\ C are calculated from :math:`V_{c\max }` \ at 25 :sup:`\o`\ C:, including: :math:`T_{p25} =0.167V_{c\max 25}`, and :math:`R_{d25} =0.015V_{c\max 25}` (C\ :sub:`3`) and :math:`R_{d25} =0.025V_{c\max 25}` (C\ :sub:`4`). For C\ :sub:`4` plants, :math:`k_{p25} =20000\; V_{c\max 25}`. However, when the biogeochemistry is active (the default mode), :math:`R_{d25}` is calculated from leaf nitrogen as described in (Chapter :numref:`rst_Plant Respiration`) The parameters :math:`V_{c\max 25}`, :math:`J_{\max 25}`, :math:`T_{p25}`, :math:`k_{p25}`, and :math:`R_{d25}` are scaled over the canopy for sunlit and shaded leaves (section :numref:`Canopy scaling`). In C\ :sub:`3` plants, these are adjusted for leaf temperature, :math:`T_{v}` (K), as: .. math:: :label: 9.10 \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} .. math:: :label: 9.11 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 .. math:: :label: 9.12 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)} . :numref:`Table Temperature dependence parameters for C3 photosynthesis` lists parameter values for :math:`\Delta H_{a}` and :math:`\Delta H_{d}` . :math:`\Delta S` is calculated separately for :math:`V_{c\max }` and :math:`J_{max }` to allow for temperature acclimation of photosynthesis (see equation :eq:`9.16`), and :math:`\Delta S` is 490 J mol :sup:`-1` K :sup:`-1` for :math:`R_d` (:ref:`Bonan et al. 2011`, :ref:`Lombardozzi et al. 2015`). Because :math:`T_{p}` as implemented here varies with :math:`V_{c\max }` , :math:`T_{p}` uses the same temperature parameters as :math:`V_{c\max}` . For C\ :sub:`4` plants, .. math:: :label: 9.13 \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} with :math:`Q_{10} =2`, :math:`s_{1} =0.3`\ K\ :sup:`-1` :math:`s_{2} =313.15` K, :math:`s_{3} =0.2`\ K\ :sup:`-1`, and :math:`s_{4} =288.15` K. Additionally, .. math:: :label: 9.14 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 :math:`Q_{10} =2`, :math:`s_{5} =1.3` K\ :sup:`-1` and :math:`s_{6} =328.15`\ K, and .. math:: :label: 9.15 k_{p} =k_{p25} \, Q_{10} ^{(T_{v} -298.15)/10} with :math:`Q_{10} =2`. .. _Table Temperature dependence parameters for C3 photosynthesis: .. table:: Temperature dependence parameters for C3 photosynthesis. +------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+ | Parameter | :math:`\Delta H_{a}` (J mol\ :sup:`-1`) | :math:`\Delta H_{d}` (J mol\ :sup:`-1`) | +========================+=================================================================+=================================================================+ | :math:`V_{c\max }` | 72000 | 200000 | +------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+ | :math:`J_{\max }` | 50000 | 200000 | +------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+ | :math:`T_{p}` | 72000 | 200000 | +------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+ | :math:`R_{d}` | 46390 | 150650 | +------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+ | :math:`K_{c}` | 79430 | – | +------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+ | :math:`K_{o}` | 36380 | – | +------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+ | :math:`\Gamma _{\*}` | 37830 | – | +------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+ In the model, acclimation is implemented as in :ref:`Kattge and Knorr (2007) `. In this parameterization, :math:`V_{c\max }` and :math:`J_{\max }` vary with the plant growth temperature. This is achieved by allowing :math:`\Delta S`\ to vary with growth temperature according to .. math:: :label: 9.16 \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} The effect is to cause the temperature optimum of :math:`V_{c\max }` and :math:`J_{\max }` to increase with warmer temperatures. Additionally, the ratio :math:`J_{\max 25} /V_{c\max 25}` at 25 :sup:`o`\ C decreases with growth temperature as .. math:: :label: 9.17 J_{\max 25} /V_{c\max 25} =2.59-0.035(T_{10} -T_{f} ). In these acclimation functions, :math:`T_{10}` is the 10-day mean air temperature (K) and :math:`T_{f}` is the freezing point of water (K). For lack of data, :math:`T_{p}` acclimates similar to :math:`V_{c\max }`. Acclimation is restricted over the temperature range :math:`T_{10} -T_{f} \ge 11`\ :sup:`o`\ C and :math:`T_{10} -T_{f} \le 35`\ :sup:`o`\ C. .. _Canopy scaling: Canopy scaling -------------------------------------------- When LUNA is on, the :math:`V_{c\max 25}` for sun leaves is scaled to the shaded leaves :math:`J_{\max 25}` , :math:`T_{p25}` , :math:`k_{p25}`, and :math:`R_{d25}` scale similarly. .. math:: :label: 9.18 \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} Where :math:`i_{v,sun}` and :math:`i_{v,sha}` are the leaf-to-canopy scaling coefficients of the twostream radiation model, calculated as .. math:: :label: 9.19 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} 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 :math:`f_{sun}` is the fraction of sunlit leaves, both derived from Chapter :numref:`rst_Surface Albedos`. When LUNA is off, scaling defaults to the mechanism used in CLM4.5. .. _Numerical implementation photosynthesis: Numerical implementation ---------------------------- The CO\ :sub:`2` partial pressure at the leaf surface, :math:`c_{s}` (Pa), and the vapor pressure at the leaf surface, :math:`e_{s}` (Pa), needed for the stomatal resistance model in equation :eq:`9.1`, and the internal leaf CO\ :sub:`2` partial pressure :math:`c_{i}` (Pa), needed for the photosynthesis model in equations :eq:`9.3`-:eq:`9.5`, are calculated assuming there is negligible capacity to store CO\ :sub:`2` and water vapor at the leaf surface so that .. math:: :label: 9.31 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 .. math:: :label: 9.32 \frac{e_{a} -e_{i} }{r_{b} +r_{s} } =\frac{e_{a} -e_{s} }{r_{b} } =\frac{e_{s} -e_{i} }{r_{s} } where :math:`r_{b}` is leaf boundary layer resistance (s m\ :sup:`2` :math:`\mu` \ mol\ :sup:`-1`) (section :numref:`Sensible and Latent Heat Fluxes and Temperature for Vegetated Surfaces`), the terms 1.4 and 1.6 are the ratios of diffusivity of CO\ :sub:`2` to H\ :sub:`2`\ O for the leaf boundary layer resistance and stomatal resistance, :math:`c_{a} ={\rm CO}_{{\rm 2}} \left({\rm mol\; mol}^{{\rm -1}} \right)`, :math:`P_{atm}` is the atmospheric CO\ :sub:`2` partial pressure (Pa) calculated from CO\ :sub:`2` concentration (ppmv), :math:`e_{i}` is the saturation vapor pressure (Pa) evaluated at the leaf temperature :math:`T_{v}` , and :math:`e_{a}` is the vapor pressure of air (Pa). The vapor pressure of air in the plant canopy :math:`e_{a}` (Pa) is determined from .. math:: :label: 9.33 e_{a} =\frac{P_{atm} q_{s} }{0.622} where :math:`q_{s}` is the specific humidity of canopy air (kg kg\ :sup:`-1`, section :numref:`Sensible and Latent Heat Fluxes and Temperature for Vegetated Surfaces`). Equations and are solved for :math:`c_{s}` and :math:`e_{s}` .. math:: :label: 9.34 c_{s} =c_{a} -1.4r_{b} P_{atm} A_{n} .. math:: :label: 9.35 e_{s} =\frac{e_{a} r_{s} +e_{i} r_{b} }{r_{b} +r_{s} } Substitution of equation :eq:`9.35` into equation :eq:`9.1` gives an expression for stomatal resistance (:math:`r_{s}` ) as a function of photosynthesis (:math:`A_{n}` ), given here in terms of conductance with :math:`g_{s} =1/r_{s}` and :math:`g_{b} =1/r_{b}` .. math:: :label: 9.36 g_{s}^{2} + bg_{s} + c = 0 where .. math:: :label: 9.37 b = 2(g_{o} * 10^{-6} + d) + \frac{(g_{1}d)^{2}}{g_{b}*10^{-6}D} c = (g_{o}*10^{-6})^{2} + [2g_{o}*10^{-6} + d \frac{1-g_{1}^{2}} {D}]d and .. math:: :label: 9.38 d = \frac {1.6 A_{n}} {c_{s} / P_{atm} * 10^{6}} D = \frac {e_{i} - e_{a}} {1000} Stomatal conductance, as solved by equation :eq:`9.36` (mol m :sup:`-2` s :sup:`-1`), is the larger of the two roots that satisfy the quadratic equation. Values for :math:`c_{i}` are given by .. math:: :label: 9.39 c_{i} =c_{a} -\left(1.4r_{b} +1.6r_{s} \right)P_{atm} A{}_{n} The equations for :math:`c_{i}` , :math:`c_{s}` , :math:`r_{s}` , and :math:`A_{n}` are solved iteratively until :math:`c_{i}` converges. :ref:`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 :math:`c_{i}` . The equation set is solved separately for sunlit (:math:`A_{n}^{sun}` , :math:`r_{s}^{sun}` ) and shaded (:math:`A_{n}^{sha}` , :math:`r_{s}^{sha}` ) leaves.