Photosynthesis numerical solution

The set of nonlinear equations for photosynthesis model include:

1. biological demand (see here),
2. diffusion (see here), and
3. stomatal conductance model.

The set of nonlinear equations are numerically solved when the residual equation, \(R(x)\), where \(x\) is the unknown variable. The solution, \(x^*\), of the nonlinear is obtained when \(R(x^*) = 0\). The residual equation for photosynthesis model with various SCMs is provided below.

BB and Medlyn model

The unknown variable is \(c_i\) and the residual equation is

\[\begin{equation} \label{eqn_residual_semiempirical} R(c_i) \equiv A_n - g_{\ell c} (c_a - c_i) = 0 \end{equation}\]

Marginal WUE

The unknown variable is \(g_\ell\) and with the residual equation for model without plant hydraulics (i.e. \(\xi\) is independent of \(\psi_\ell\)) is

\[\begin{equation} \label{eqn_residual_wue} R(g_\ell) \equiv \frac{\partial A_n}{\partial g_{sw}} - \xi = 0 \end{equation}\]

For models that include plant hydraulics, equation \eqref{eqn_residual_wue} is modified by making \(\xi\) depend on \(\psi_\ell\).

iWUE

The unknown variable is \(g_\ell\) and with the residual equation for model without plant hydraulics (i.e. \(\xi\) is independent of \(\psi_\ell\)) is

\[\begin{equation} \label{eqn_residual_iwue} R(g_\ell) \equiv \frac{\partial A_n}{\partial g_{sw}} - \xi w_s = 0 \end{equation}\]

For models that include plant hydraulics, equation \eqref{eqn_residual_iwue} is modified by make \(\xi\) depend on \(\psi_\ell\).

SCM with stomatal dowregulation

Depending on the choice of \(g_{sw,max}\) (described here), the residual equation is given by equation \eqref{eqn_residual_semiempirical} or \eqref{eqn_residual_wue} or \eqref{eqn_residual_iwue}.

Original and modified Bonan2014 co-optimization model

The unknown variable is \(g_\ell\) and the residual equation for the first constraint is given by equation \eqref{eqn_residual_wue} or \eqref{eqn_residual_iwue}. The residual equation for the second constraint is given as

\[\begin{equation} R(g_\ell) \equiv \psi_{\ell}^t + \Delta\psi_\ell^{t+\Delta t} - \psi_{\ell, min} = 0 \end{equation}\]

Wang2020

The unknown variable is \(g_\ell\) and the residual equation is

\[\begin{equation} R(g_\ell) \equiv \left( 1 - \frac{E_\ell}{E_{critical}} \right) \frac{\partial \Theta}{\partial E} - \frac{A_n}{E_{critical}} \frac{\partial E_\ell}{E} = 0 \end{equation}\]