Shortwave radiation

Governing equations

The downwelling, \(I_i^{\downarrow}\) [Wm\(^{-2}\)], and upwelling, \(I_{i+1}^{\uparrow}\) [Wm\(^{-2}\)], scattered radiation fluxes at \(i\)-th and \((i+1)\)-th level are given as

\[\begin{eqnarray} I_{i }^{\downarrow} & = & I_{i+1 }^{\downarrow} \left[ \mathcal{\tau}_{d,i+1} + (1 - \tau_{d,i+1})\tau_{\ell,i+1}\right] + I_{i }^{\uparrow} (1 - \tau_{d,i+1})\rho_{\ell,i+1} \nonumber \\ \label{eqn_sw_dn} & & + I_{sky,b}^{\downarrow} T_{b,i+1} (1 - \tau_{b,i+1})\tau_{\ell,i+1} \\ I_{i+1}^{\uparrow} & = & I_{i }^{\uparrow} \left[ \tau_{d,i+1} + (1 - \tau_{d,i+1})\tau_{\ell,i+1}\right] + I_{i+1 }^{\downarrow} (1 - \tau_{d,i+1})\rho_{\ell,i+1} \nonumber \\ & & + I_{sky,b}^{\downarrow} T_{b,i+1} (1 - \tau_{b,i+1})\rho_{\ell,i+1} \label{eqn_sw_up} \end{eqnarray}\]

where \(\tau_{d,i+1}\) [-] and \(\tau_{b,i+1}\) [-] are the diffuse and direct beam transmittances through \((i+1)\)-th layer, \(\rho_{\ell,i+1}\) [-] is the leaf reflectance of \((i+1)\)-th layer, \(I_{sky,b}\) [Wm\(^{-2}\)] is the direct beam radiation incident on the top of the canopy, and \(T_{b,i+1}\) [-] is the fraction of direct beam radiation that is not intercepted through the cumulative leaf area above the \((i+1)\)-th layer.

Transmittance

The direct beam transmittance through the \((i+1)\)-th layer with leaf area index \(\Delta L_{i+1}\) is

\[\begin{equation} \tau_{d,i+1} = 2 \int_{0}^{\pi/2} \exp \left[ - \frac{G(Z_i)\Omega}{\cos (Z_i)} \Delta L_{i+1} \right] \sin (Z_i) \cos (Z_i) dZ \end{equation}\]

where \(Z\) is the sky zenith angle and \(\Omega\) is the leaf clumping factor. The Ross-Goudriann function, \(G(Z)\), is given by

\begin{equation} \label{eqn_rg_function} G(Z) = \phi_1 + \phi_2 \cos(Z) \end{equation} where \(\phi_1 = 0.5 -0.633\chi_\ell - 0.33\chi_\ell^2\) and \(\phi_2 = 0.877 (1 - 2\phi_1\)). The leaf departure angle from from spherical orientation, \(\chi_\ell\), in restricted to \(-0.4 \le \chi_\ell \le 0.6\). The equation \eqref{eqn_rg_function} is numerically approximation for nine sky zones as

\begin{equation} \tau_{d,i+1} = 2 \sum_{i=1}^{9} \exp \left[ - \frac{G(Z_i)\Omega}{\cos (Z_i)} \Delta L_{i+1} \right] \sin (Z_i) \cos (Z_i) \Delta Z_i \end{equation} with \(\Delta Z_i = \pi/18\).

The diffuse beam transmittance through the \((i+1)\)-th layer is

\begin{equation} \tau_{b,i+1} = \exp \left( -K_{b,i+1} \Omega \Delta L_{i+1} \right) \end{equation} % where \(K_{b,i+1} = G(Z)/\cos(Z)\) is the extinction coefficient for the direct beam. The fraction of direct been that is not intercepted through the cumulative leaf area above the \((i+1)\)-th layer is computed as

\[\begin{equation} T_{b,i+1} = \prod_{j=i+1}^N \exp \left( -K_{b,j} \Omega_j \Delta L_j \right) \end{equation}\]

Linear system

The equation \eqref{eqn_sw_dn}-\eqref{eqn_sw_up} can be written as a system of linear equations

\[\begin{eqnarray} \label{eqn_sw_linear_system_up} -a_i I_i^{\uparrow} + I_i^\downarrow - b_iI_{i+1}^\uparrow &=& d_i \\ \label{eqn_sw_linear_system_dn} -e_{i+1}I_i^\downarrow + I_{i+1}^\uparrow - f_{i+1}I_{i+1}^\downarrow &=& c_{i+1} \end{eqnarray}\]

where

\[\begin{eqnarray} a_i &=& f_{i+1} = (1-\tau_{d,i+1})\rho_{\ell,i+1} - \frac{[\tau_{d,i+1} + (1 - \tau_{d,i+1}) \tau_{\ell,i+1} ]^2} {(1 - \tau_{d,i+1})/\rho_{\ell,i+1}} \\ b_i &=& e_{i+1} = \frac{\tau_{d,i+1} + (1 - \tau_{d,i+1})\tau_{\ell,i+1}} {(1 - \tau_{d,i+1}) \rho_{\ell,i+1}} \\ c_i &=& I_{sky,b}^\downarrow T_{b,i } (1 - \tau_{b,i }) (\rho_{\ell,i } - \tau_{\ell,i. } e_i)\\ d_i &=& I_{sky,b}^\downarrow T_{b,i+1} (1 - \tau_{b,i+1}) (\tau_{\ell,i+1} - \rho_{\ell,i+1} b_i) \end{eqnarray}\]

The boundary conditions for the downwelling radiation at the bottom layer, \(i=0\), and the upwelling radiation at the top layer, \(i=N\), are given by

\[\begin{eqnarray} c_0 &=& \rho_{gb} I_{sky,b}^\downarrow T_{b,0} \\ f_0 &=& \rho_{gd} \\ d_N &=& I_{sky,d}^\downarrow \end{eqnarray}\]

where \(\rho_{gb}\) [-] is the surface albedo for beam radiation, \(\rho_{gd}\) [-] is the surface albedo for diffuse radiation, and \(I_{sky,d}\) [Wm\(^{-2}\)] is the diffuse radiation incident on the top of the canopy.

The linear system of equations given in equation \eqref{eqn_sw_linear_system_up}-\eqref{eqn_sw_linear_system_dn} can be written as

\[\begin{align} \begin{bmatrix} 1 & -\rho_{gd} & & & & & & \\[.6em] -a_0 & 1 & -b_0 & & & & & \\[.6em] & -e_1 & 1 & -f_1 & & & & \\[.6em] & & -a_1 & 1 & -b_1 & & & \\[.6em] & & & \ddots & \ddots & \ddots & & \\[.6em] & & & & \ddots & \ddots & \ddots & \\[.6em] & & & & & -e_N & 1 & -f_N \\[.6em] & & & & & & 0 & 1 \end{bmatrix} \begin{bmatrix} \hspace{5pt} I_0^\uparrow \hspace{5pt} \\[.5em] I_0^\downarrow \\[.5em] I_1^\uparrow \\[.5em] I_1^\downarrow \\[.5em] \vdots \\[.5em] \vdots \\[.5em] I_N^\uparrow \\[.5em] I_N^\downarrow \end{bmatrix} = \begin{bmatrix} \hspace{5pt} c_0 \hspace{5pt} \\[.6em] d_0 \\[.6em] c_1 \\[.6em] d_1 \\[.6em] \vdots \\[.6em] \vdots \\[.6em] c_N \\[.6em] d_N \end{bmatrix} \end{align}\]

Absorbed radiative fluxes

The diffuse, \(\overrightarrow{I}_{cd,i}\) [Wm\(_{ground}^{-2}\)], and direct beam, \(\overrightarrow{I}_{cb,i}\) [Wm\(_{ground}^{-2}\)], radiation absorbed by the \(i\)-th canopy layer is

\[\begin{eqnarray} \overrightarrow{I}_{cd,i} &=& \left( I_i^\downarrow + I_{i-1}^\uparrow \right) (1 - \tau_{d,i})(1 - \omega_{\ell,i}) \\ \overrightarrow{I}_{cb,i} &=& I_{sky,b}^\downarrow T_{b,i}(1 - \tau_{b,i})(1 - \omega_{\ell,i}) \end{eqnarray}\]

The radiation absorbed at the ground, \(\overrightarrow{I}_{g}\) [Wm\(_{ground}^{-2}\)], is

\[\begin{equation} \overrightarrow{I}_g = \left( 1 - \rho_{gd} \right) I_0^\downarrow + \left( 1 - \rho_{gb} \right) I_{sky,b}^\downarrow T_{b,0} \end{equation}\]

It is assumed that the shaded leaves only absorb diffuse radiation, while sunlit leaves receive direct and diffuse radiations. The absorbed radiation fluxes by shaded, \(\overrightarrow{I}_{\ell sha, i}\) [Wm\(_{leaf}^{-2}\)], and sunlit, \(\overrightarrow{I}_{\ell sun, i}\) [Wm\(_{leaf}^{-2}\)], is given as

\[\begin{eqnarray} \overrightarrow{I}_{\ell sha, i} &=& \frac{I_{cd,i}}{\Delta L_i} \\[0.5em] \overrightarrow{I}_{\ell sun, i} &=& \overrightarrow{I}_{\ell sha, i} + \frac{\overrightarrow{I}_{cb,i}}{f_{sun,i} \Delta L_i} \end{eqnarray}\]

where \(f_{sun}\) [-] is the fraction of sunlit leaves at \(i\)-th layer.

The absorbed canopy fluxes for shaded, \(\overrightarrow{I}_{cSha}\) [Wm\(_{ground}^{-2}\)], and sunlit, \(\overrightarrow{I}_{cSun}\) [Wm\(_{ground}^{-2}\)], are

\[\begin{eqnarray} \overrightarrow{I}_{cSha} &=& \sum_{i=1}^N \overrightarrow{I}_{\ell sha, i} ( 1 - f_{sun}) \Delta L_i = \sum_{i=1}^N I_{cd,i} (1 - f_{sun})\\ \overrightarrow{I}_{cSun} &=& \sum_{i=1}^N \overrightarrow{I}_{\ell sun, i} f_{sun} \Delta L_i = \sum_{i=1}^N \left( \overrightarrow{I}_{cd, i} f_{sun} + \overrightarrow{I}_{cb,i}\right) \end{eqnarray}\]