Longwave radiation
Governing equations
The downwelling, \(L_{i}^\downarrow\) [Wm\(^{-2}\)], and upwelling \(L_{i+1}^\uparrow\) [Wm\(^{-2}\)], longwave radiation can
be similarly described as the shortwave radiation model by replacing the direct beam
scattering term with a thermal radiation source term as
\[\begin{eqnarray}
L_{i }^{\downarrow} & = & L_{i+1 }^{\downarrow} \left[ \mathcal{\tau}_{d,i+1} + (1 - \tau_{d,i+1})\tau_{\ell,i+1}\right]
+ L_{i }^{\uparrow} (1 - \tau_{d,i+1})\rho_{\ell,i+1} \nonumber \\
& & + \varepsilon_\ell \sigma T_{\ell,i+1}^4 (1 - \tau_{d,i+1}) \\
L_{i+1}^{\uparrow} & = & L_{i }^{\uparrow} \left[ \tau_{d,i+1} + (1 - \tau_{d,i+1})\tau_{\ell,i+1}\right]
+ L_{i+1 }^{\downarrow} (1 - \tau_{d,i+1})\rho_{\ell,i+1} \nonumber \\
& & + \varepsilon_\ell \sigma T_{\ell,i+1}^4 (1 - \tau_{d,i+1})
\end{eqnarray}\]
where \(\varepsilon_{\ell}\) [-] is the leaf emissivity and
\(T_\ell\) [K] is the leaf temperature.
If sunlit and shaded leaves are modeled explicitly,
an effective leaf temperature is defined based on the
sunlit and shaded leaf fraction.
Linear system
The linear system of equations for the longwave model can be written as
\[\begin{align}
\begin{bmatrix}
1 & -(1-\varepsilon_{g}) & & & & & & \\[.6em]
-a_0 & 1 & -b_0 & & & & & \\[.6em]
& -e_1 & 1 & -f_1 & & & & \\[.6em]
& & -a_1 & 1 & -b_1 & & & \\[.6em]
& & & \ddots & \ddots & \ddots & & \\[.6em]
& & & & \ddots & \ddots & \ddots & \\[.6em]
& & & & & -e_N & 1 & -f_N \\[.6em]
& & & & & & 0 & 1
\end{bmatrix}
\begin{bmatrix}
L_0^\uparrow \hspace{5pt} \\[.5em]
L_0^\downarrow \\[.5em]
L_1^\uparrow \\[.5em]
L_1^\downarrow \\[.5em]
\vdots \\[.5em]
\vdots \\[.5em]
L_N^\uparrow \\[.5em]
L_N^\downarrow
\end{bmatrix}
=
\begin{bmatrix}
c_0 \hspace{5pt} \\[.6em]
d_0 \\[.6em]
c_1 \\[.6em]
d_1 \\[.6em]
\vdots \\[.6em]
\vdots \\[.6em]
c_N \\[.6em]
d_N
\end{bmatrix}
\end{align}\]
where
\[\begin{eqnarray}
a_i &=& f_{i+1} = (1-\tau_{d,i+1})\rho_{\ell,i+1} -
\frac{[\tau_{d,i+1} + (1 - \tau_{d,i+1}) \tau_{\ell,i+1} ]^2}
{(1 - \tau_{d,i+1})/\rho_{\ell,i+1}} \\[1em]
b_i &=& e_{i+1} = \frac{\tau_{d,i+1} + (1 - \tau_{d,i+1})\tau_{\ell,i+1}}
{(1 - \tau_{d,i+1}) \rho_{\ell,i+1}} \\[1em]
c_i &=& (1 - e_i)(1 - \tau_{d,i }) \varepsilon_\ell \sigma T_{\ell,i }^4 \\[1em]
d_i &=& (1 - b_i)(1 - \tau_{d,i+1}) \varepsilon_\ell \sigma T_{\ell,i+1}^4
\end{eqnarray}\]
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 as
\[\begin{eqnarray}
c_0 &=& \varepsilon_{g} \sigma T_g^4 \\
d_N &=& L_{sky}^\downarrow
\end{eqnarray}\]
where
\(\varepsilon_g\) [-] is ground surface emissivity,
\(T_g\) [K] is the ground surface temperature, and
\(L_{sky}^\downarrow\) [Wm\(^{-2}\)] is incident longwave radiation at the top of the canopy.
Absorbed fluxes
The net longwave flux absorbed per unit ground area, \(\overrightarrow{L}_i\) [Wm\(_{ground}^{-2}\)], and
per unit leaf area, \(\overrightarrow{L}_{\ell,i}\) [Wm\(_{leaf}^{-2}\)], by the \(i\)-th layer are
\[\begin{eqnarray}
\overrightarrow{L}_i &=& \varepsilon_{\ell} \left( L_i^\downarrow + L_{i-1}^\uparrow \right) (1 - \tau_{d,i})
- 2 \varepsilon_\ell \sigma T_{\ell,i}^4 (1 - \tau_{d,i}) \\[.2em]
\overrightarrow{L}_{\ell,i} &=& \frac{\overrightarrow{L}_i}{\Delta L_i}
\end{eqnarray}\]
The radiation absorbed by the canopy, \(\overrightarrow{L}_{c}\) [Wm\(_{ground}^{-2}\)], and
the ground, \(\overrightarrow{L}_{g}\) [Wm\(_{ground}^{-2}\)], is given by
\[\begin{eqnarray}
\overrightarrow{L}_c &=& \sum_{i=1}^{N} \overrightarrow{L}_c \\[.2em]
\overrightarrow{L}_{g} &=& \varepsilon_g L_0^\downarrow - \varepsilon_g \sigma T_g^4
\end{eqnarray}\]