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.. _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 <Olesonetal2013>`).
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 <Collatzetal1991>`, :ref:`1992 <Collatzetal1992>`). 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<Medlynetal2011>`).
Previous versions of CLM calculated leaf stomatal resistance is using the Ball-Berry conductance
model as described by :ref:`Collatz et al. (1991)<Collatzetal1991>` and implemented in global
climate models (:ref:`Sellers et al. 1996<Sellersetal1996>`). 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 <deKauwe2015>`).
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)<Farquharetal1980>`. Photosynthesis in C\ :sub:`4` plants is
based on the model of :ref:`Collatz et al. (1992)<Collatzetal1992>`. :ref:`Bonan et al. (2011)<Bonanetal2011>`
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 <Collatzetal1992>`). 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)
<Collatzetal1991>`. 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<Bonanetal2011>`, :ref:`Lombardozzi et al. 2015<Lombardozzietal2015>`).
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) <KattgeKnorr2007>`. 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)<Sunetal2012>` 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.
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