Photosynthesis stomatal conductance model

Governing equations

The photosynthesis model has two equations (first and second) that involves three unknowns ($c_i$, $A_n$, and $g_{sw}$). Thus, an additional equation is needed for stomatal conductance to close the system of equations. In the literature, multiple stomatal conductance models (SCMs) have been developed that are based on empirical, semi-empirical, or optimization approach. Furthermore, SMCs can exclude or include plant hydraulics.

Semi-empirical models without accounting for plant hydraulics

Ball-Berry model

The semi-empirical Ball-Berry (BB) SCM¹ is given as

\[\begin{equation} \label{eqn_bb} g_{sw} = g_0 + g_1\frac{A_n}{c_s}h_s \end{equation}\]

where $g_0$ [mol H$_2$O m$^{-2}$ s$^{-1}$] is the minimum stomatal conductance, $g_1$ [mol H$_2$O m$^{-2}$ s$^{-1}$] is the slope of the relationship, and $h_s $ [-] is the fractional humidity at the leaf surface. The fractional humidity at the leaf surface is $h_s = e_s/e_{sat}(T_\ell)$ where $e_s$ and $e_{sat}(T_\ell)$ are the vapor pressure at the leaf surface and saturated vapor pressure at leaf temperature, $T_\ell$, respectively. The vapor pressure at leaf surface can be given as

\[\begin{equation} \label{eqn_vp_leaf} e_s = \frac{g_{bw}e_a + g_{sw} e_{sat}(T_\ell) }{g_{bw} + g_{sw}} \end{equation}\]

Substituting equation \eqref{eqn_vp_leaf} in equation \eqref{eqn_bb} leads the following quadratic equation in which $g_{sw}$ is the larger root of the equation.

\[\begin{equation} \alpha g_{sw}^2 + \beta g_{sw} + \gamma = 0 \end{equation}\]

where

\[\begin{eqnarray} \alpha &=& 1 \\ \beta &=& g_{gw} - g_0 - \frac{g_1 A_n}{c_s} \\ \gamma &=& - g_{bw} \left[ g_0 + \frac{g_1 A_n e_a}{c_s e_{sat}(T_\ell)} \right] \end{eqnarray}\]

Medlyn model

The semi-empirical Medlyn SCM² is given as

\[\begin{equation} \label{eqn_medlyn} g_{sw} = g_0 + 1.6\frac{A_n}{c_s} \left( 1 + \frac{g_1}{\sqrt D_s}\right) \end{equation}\]

where $g_0$ [mol H$_2$O m$^{-2}$ s$^{-1}$] is the minimum stomatal conductance, $g_1$ [mol H$_2$O m$^{-2}$ s$^{-1}$] is the slope of the relationship, and $D_s = (e_{sat}(T_\ell) - e_s)$ [KPa] is the vapor pressure deficit. Similar to BB model, substituting equation \eqref{eqn_vp_leaf} in equation \eqref{eqn_medlyn} leads to following quadratic equations, whose larger root is $g_{sw}$.

\[\begin{equation} \alpha g_{sw}^2 + \beta g_{sw} + \gamma = 0 \end{equation}\]

where

\[\begin{eqnarray} \alpha &=& 1 \\ \beta &=& - 2 \left( g_{0} - 1.6\frac{g_1 A_n}{c_s}\right) - \left( \frac{1.6 A_n g_1}{c_s} \right)^2 \frac{1}{g_{bw} D_\ell} \\ \gamma &=& - \left[ 2g_0 + \frac{1.6A_n}{c_s} \left( 1 - \frac{g_\ell^2}{D_\ell} \right) \right] \frac{1.6 A_n g_1}{c_s} \end{eqnarray}\]

with $D_\ell = e_{sat}(T_\ell) - e_a$.

Optimization-based models

Optimization-based SCMs maximize carbon update, $A_n$, while minimizing the cost associated with carbon update, $\Theta$, related a measure of stomatal opening, $\chi$³. Such models can be formulated as

\[\begin{equation} \label{eqn_opt_obj_fn} \begin{aligned} \max_{\chi} \quad & (A_n - \Theta) \end{aligned} \end{equation}\]

and the solution of equation \eqref{eqn_opt_obj_fn} is obtained by finding $\chi$ that satisfies the following equation

\[\begin{equation} \label{eqn_opt_soln} \frac{\partial A_n}{\partial \chi} - \frac{\partial \Theta}{\partial \chi} = 0 \end{equation}\]

Marginal water-use efficiency (WUE) model without accounting for plant hydraulics

In this model, $\chi = E$ and $\Theta = \xi E$, where $\xi$ is a constant model parameter. Thus, equation \eqref{eqn_opt_soln} reduces to

\[\begin{equation} \label{eqn_wue} \frac{\partial A_n}{\partial E} = \xi \end{equation}\]

The LHS term of equation \eqref{eqn_wue} can be derived⁴ as

\begin{equation} \frac{\partial A_n}{\partial E} = \left( \frac{c_a - c_i}{w_l} \right) \left( \frac{ \partial A_n/\partial c_i}{\partial A_n/\partial c_i + g_{lc}} \right) 1.6 \frac{g_{lc}^2}{g_{\ell w}^2} \end{equation} where $w_\ell = \left[ e_{sat}(T_\ell) - e_a \right]/P_{ref}$ [mol mol$^{-1}$] is the vapor pressure deficit.

Intrinsic WUE (iWUE) model without accounting for plant hydraulics

In this model, $\chi = g_{sw}$ and the cost function is similar to that of the WUE model (i.e. $\Theta = \xi E$). The equation \eqref{eqn_opt_soln} then reduces to

\begin{equation} \label{eqn_iwue} \begin{split} \frac{\partial A_n}{\partial g_{sw}} &= \xi \frac{\partial E}{\partial g_{sw}} \ &= \xi \frac{\partial}{\partial g_{sw}} \left( \frac{(e_{sat}(T_\ell) - e_s ) g_{sw} }{P_{ref}} \right) \ &\approx \xi w_s \end{split} \end{equation} where $w_s (= [e_{sat}(T_\ell) - e_s]/P_{ref})$ [mol mol$^{-1}$] is the water vapor deficit at the leaf surface. In equation \eqref{eqn_iwue}, the term $\partial e_s/\partial g_{sw}$ is neglected.

WUE model including plant hydraulics

The marginal WUE model can be modified to account for loss of xylem conductivity by including dependence of $\xi$ in equation \eqref{eqn_wue} on leaf water potential⁵ as

\[\begin{equation} \xi(\psi_\ell) = \exp(\beta \psi_\ell) \end{equation}\]

where $\beta$ is a model parameter.

Co-optimization model of Bonan2014

Bonan2014⁶ is a SCM that maximizes $g_{sw}$ while satisfying two constraints: (1) WUE or iWUE is greater than a threshold (i.e. $\partial A_n/\partial E \ge \xi$ or $(\partial A_n/\partial g_{sw})/w_s \ge \xi$, and (2) leaf water potential is greater than a threshold (i.e. $\psi_\ell \ge \psi_{\ell, min})$. The plant hydraulics model assumes leaves (sunlit or shaded) at any height are directly connected to multiple soil layers via a root system. The leaf water storage is given by

\[\begin{equation} \label{eqn_bonan14_phm} \frac{d\psi_\ell}{dt} = \frac{K_L (\psi_s - \psi_\ell - \rho_wgh) - 10^3 \times E}{C_p} \end{equation}\]

where $\psi_\ell$ [MPa] is the leaf water potential, $\psi_s$ [MPa] is the soil water potential, $\rho_wgh$ [MPa] is the gravitational head, $C_p$ [$\mu$mol H$_2$O { }m$^{-2}_\ell$ MPa$^{-1}$] is the leaf capacitance, and $K_L$ [$\mu$mol H$_2$O { }m$^{-2}_\ell$ s$^{-1}$ MPa$^{-1}$] is the whole plant hydraulic conductance, and $E$ [mmol H$_2$O { }m$^{-2}_\ell$ s$^{-1}$] is the transpiration flux. The $K_L$ is independent of $\psi_l$ and thus integrating equation \ref{eqn_bonan14_phm} provides an analytical expression for the change of leaf water potential, $\Delta \psi_\ell^{t+\Delta t}$, for time step, $\Delta t$, as

\[\begin{equation} \Delta\psi_\ell^{t+\Delta t} = \left[ \psi_s - \psi_\ell^{t} - \rho_w g h - \frac{10^3 \times E}{K_L}\right] \left( 1 - e^{-K_L\Delta t/C_p}\right) \end{equation}\]

The second constraint of the co-optimization approach leads to

\[\begin{equation} \psi_{\ell}^t + \Delta\psi_\ell^{t+\Delta t} \ge \psi_{\ell, min} \end{equation}\]

The whole plant hydraulic conductance depends on soil-to-stem conductance, $K_{L,s2s}$ [$\mu$mol H$_2$O { }m$^{-2}_\ell$ s$^{-1}$ MPa$^{-1}$], and stem-to-leave conductance, $K_{L,s2\ell}$ [$\mu$mol H$_2$O { }m$^{-2}_\ell$ s$^{-1}$ MPa$^{-1}$] as

\[\begin{equation} \frac{1}{K_L} = \frac{1}{K_{L,s2s}} + \frac{1}{K_{L,s2\ell}} \end{equation}\]

Modified Bonan2014 model

In this study, we have developed a modified plant hydraulic model of Bonan2014 by including dependence of leaf water potential on stem-to-soil conductance, which is modeled by a Weibull function as

\begin{equation} K_{L,s2\ell}(\psi_\ell) = \psi_{L,s2\ell}^{max} \exp \left[ \left(\frac{-\psi_{\ell}}{b}\right)^c \right] \end{equation} where $b$ [MPa] and $c$ [-] are parameters. The modified equation \eqref{eqn_bonan14_phm} is solved using the forward Euler time-integration scheme.

Wang2020 model

In Wang2020 model³, $\chi = E$ and the cost function is given as

\[\begin{equation} \Theta = A_n\frac{E_{\ell}}{E_{critical}} \end{equation}\]

Semi-empirical model with downregulation due to plant hydraulics

Empirical models have been proposed for reducing $g_{sw}$ to account for loss of xylem hydraulic conductivity with water potential. Examples of such empirical stomatal downregulation models include equation \eqref{eqn_chris}⁷ and equation \eqref{eqn_bohrer}⁸.

\[\begin{eqnarray} g_{sw} &=& g_{sw,max} \left[ 1 + \left(\frac{-\psi_{\ell}}{\psi_{50}}\right)^a \right]^{-1} \label{eqn_chris}\\ g_{sw} &=& g_{sw,max} \exp \left[ -\left( \frac{\psi_{\ell}}{\psi_{50}}\right)^a \right] \label{eqn_bohrer} \end{eqnarray}\]

where $a$ [-] and $\psi_{50}$ [KPa] are model parameters. In these empirical stomatal downregulation models, $g_{sw,max}$ is obtained from equation \ref{eqn_bb} or \ref{eqn_medlyn} or \ref{eqn_wue} or \ref{eqn_iwue}.

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