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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.