Adds tech note about longwave radiation

Author: bishtgautam

components/elm/docs/figures/longwave_radiation.png

@@ -6,3 +6,4 @@
 Shortwave radiation model
 - [TOP Parameterization](top_solar_parameterization.md):
 Parameterization of sub-grid topographical effects on solar radiation.
+- [Longwave Radiation](longwave_radiation.md): Longwave radiation model

components/elm/docs/tech-guide/index.md

@@ -0,0 +1,283 @@
+# Overview
+
+The longwave radiation in ELM solves the amount of longwave
+radiation absorbed by the ground and the vegetation, and the
+amount of outgoing radiation to the atmosphere (Figure 1).
+The model represents the ground surface as a mixture of snow,
+soil, and standing surface water. The shaded and sunlit leaves
+are combined as a single leaf within the model. The incoming
+longwave atmosphere radiation, $L^\downarrow_{atm}$, is a
+boundary condition for the model.
+
+![Image title](../figures/longwave_radiation.png)
+
+Fig 1. Two-stream longwave radiation model for
+(a) non-vegetated and (b) vegetated surfaces.
+
+## Governing Equations For Non-vegetated Surfaces
+
+The emitted longwave radation from ground, $L^{\uparrow}_g$, is
+
+$$
+\begin{equation}
+L^{\uparrow}_{g} = (1 - \epsilon_g)L^\downarrow_{atm} + \epsilon_g \sigma T_{g}^4
+\label{eq:lg_up_nonveg}
+\end{equation}
+$$
+
+where $\epsilon_g$ is the emissivity of the ground,
+$\sigma$ is the Stefan-Boltzmann constant, and
+$T_g$ is the ground temperature. The first term on the
+right-hand side represents the reflected atmospheric longwave
+radiation, while the second term represents the emitted longwave
+radiation by the ground. The emitted longwave radiation is computed as
+
+$$
+\begin{equation}
+\epsilon_g \sigma T_g^4 = \epsilon_g \sigma \left[ f_{sno} T^4_{sno} + \left( 1 - f_{sno} - f_{h2osfc}\right) T^4_{soi,1} + f_{h2osfc} T^4_{h2osfc} \right]
+\label{eqn:tg}
+\end{equation}
+$$
+
+where $T_{sno}$, $T_{soi,1}$ and $T_{h2osfc}$
+the temperature of the top snow layer, the first soil layer, and
+the standing surface water, respectively, and
+$f_{sno}$ and $f_{h2osfc}$ are fraction of snow and
+standing surface water.
+
+The radiation absorbed by the ground, $\overrightarrow{L}_g$, is
+
+$$
+\begin{equation}
+\overrightarrow{L}_g = \epsilon_g \sigma T_g^4 + \epsilon_g L_{atm}^\downarrow
+\label{eqn:lg_net_nonveg}
+\end{equation}
+$$
+
+## Governing Equations For Vegetated Surfaces
+
+The longwave radiation below the canopy, $L_v\downarrow$, is
+
+$$
+\begin{equation}
+L_v^\downarrow = (1 - \epsilon_v)L_{atm}^\downarrow + \epsilon_v \sigma T_v^4
+\end{equation}
+$$
+
+where $\epsilon_v$ is the emissivity of the vegetation
+and $T_v$ is the temperature of the canopy. The model assumes
+the sunlit and shaded leaves are at the same temperature.
+The first term on the right-hand side of the equation represents
+the transmitted atmospheric longwave radiation through the
+canopy and the second term represents the emitted longwave
+radiation by the canopy.
+
+The upwelling longwave radiation from the ground is
+
+$$
+\begin{eqnarray}
+L_g^\uparrow &=& (1 - \epsilon_g) L_v^\downarrow + \epsilon_g \sigma T_g^4 \nonumber\\[0.5em]
+&=& (1 - \epsilon_g)(1 - \epsilon_v)L_{atm}^\downarrow + (1 - \epsilon_g)\epsilon_v \sigma T_v^4 + \epsilon_g \sigma T_g^4
+\end{eqnarray}
+$$
+
+where the $T_g$ is given by equation \eqref{eqn:tg}.
+
+Lastly, the upwelling radiation from the top of the canopy to
+the atmosphere is
+
+$$
+\begin{eqnarray}
+L_{vg}^\uparrow &=& (1 - \epsilon_v) L_g^\uparrow + \epsilon_v \sigma T_v^4 \nonumber \\[0.5em]
+&=& (1 - \epsilon_v) \left[ (1 - \epsilon_g) L_v^\downarrow + \epsilon_g \sigma T_g^4 \right] + \epsilon_v \sigma T_v^4 \nonumber \\[0.5em]
+&=& (1 - \epsilon_v) (1 - \epsilon_g) L_v^\downarrow
++ (1 - \epsilon_v) \epsilon_g \sigma T_g^4 + \epsilon_v \sigma T_v^4 \nonumber \\[0.5em]
+&=& (1 - \epsilon_v) (1 - \epsilon_g) \left[ (1 - \epsilon_v)L_{atm}^\downarrow + \epsilon_v \sigma T_v^4 \right] \nonumber\\[0.5em]
+& & + \epsilon_v \sigma T_v^4 + (1 - \epsilon_v) \epsilon_g \sigma T_g^4 \nonumber\\[0.5em]
+&=& (1 - \epsilon_v) (1 - \epsilon_g) (1 - \epsilon_v)L_{atm}^\downarrow \nonumber\\[0.5em]
+& & +(1 - \epsilon_v) (1 - \epsilon_g) \epsilon_v \sigma T_v^4 \nonumber\\[0.5em]
+& & + \epsilon_v \sigma T_v^4 + (1 - \epsilon_v) \epsilon_g \sigma T_g^4
+\end{eqnarray}
+$$
+
+The radiation absorbed by the vegetation, $\overrightarrow{L}_{v}$, with
+positive value towards the atmosphere, is
+
+$$
+\begin{eqnarray}
+\overrightarrow{L}_v & = &  2 \epsilon_v T_v^4 - \epsilon_v L_g^\uparrow - \epsilon_v L_{atm}^\downarrow \nonumber\\[0.5em]
+& = & 2 \epsilon_v T_v^4  - \epsilon_v \left[ (1 - \epsilon_g)(1 - \epsilon_v)L_{atm}^\downarrow + (1 - \epsilon_g)\epsilon_v \sigma T_v^4 + \epsilon_g \sigma T_g^4 \right] \nonumber\\[0.5em]
+& & - \epsilon_v L_{atm}^\downarrow \nonumber\\[0.5em]
+& = & \left[ 2 - \epsilon_v (1 - \epsilon_g)\right] \epsilon_v \sigma T_v^4 - \epsilon_v \epsilon_g \sigma T_g^4 - \epsilon_v \left[ 1 + (1 - \epsilon_g)(1 - \epsilon_v) \right] L_{atm}^\downarrow \label{eq:net_lv}
+\end{eqnarray}
+$$
+
+The radiation absorbed by the ground with a positive value towards the atmosphere is
+
+$$
+\begin{equation}
+\overrightarrow{L_g} = \epsilon_g \sigma T_g^4 - \epsilon_g L_{v}^\downarrow
+\label{eq:net_lg_veg1}
+\end{equation}
+$$
+
+## Temporal Discretization of Ground Temperature
+
+The three components of ground temperature
+(i.e. $T_{sno,1}$, $T_{soi,1}$ and $T_{h2osoi}$) that contribute
+to the upward longwave radiation at the ground are coupled to the
+temperature of deeper snow and soil layers. This *coupling* of
+the temporally discretized equation of the surface energy
+balance (that includes absorbed shortwave radiation, $\overrightarrow{S}_{soi}$,
+absorbed longwave radiation, sensible heat flux, $H_{soi}$,
+latent heat flux, $\lambda E_{soi}$, and ground heat flux, $G$)
+with the spatio-temporally discretized equations
+of the vertical heat diffusion model within the snow and soil layers
+leads to complexity.
+This complexity is discussed in Section 7.3 of Bonan (2019)[@bonan2019climate]
+and briefly summarized below.
+
+### Non-vegetated Surface
+
+For simplicity, let's consider the non-vegetated case in which
+snow and standing are absent. In such a case, the ground temperature is the
+temperature of the first soil layer i.e. $T_g = T_{soi,1}$.
+At the current time step $n+1$, the absorbed
+longwave radiation given by equation \eqref{eqn:lg_net_nonveg} is a function
+of soil temperature at time $n+1$, $T_{sol,1}^{n+1}$, which is unknown.
+The ground heat flux at the top of the soil is given as
+
+$$
+\begin{eqnarray}
+G_{soi} &=& \overrightarrow{S}_{soi} - \overrightarrow{L}_g - H_{soi} - \lambda E_{soi} \nonumber\\[0.5em]
+& = & \overrightarrow{S}_{soi} - \epsilon_{soi} \sigma T_{soi,1}^4 - \epsilon_{soi}L_{atm}^\downarrow - H_{soi} - \lambda E_{soi}
+\end{eqnarray}
+$$
+
+The vertical heat diffusion model in ELM uses the Crank-Nicholson temporal discretization
+method in which the fluxes between cells are computed as an average of the flux
+at $n$-th and $(n+1)$-th time step. The top boundary heat flux (i.e. $G$) in
+ELM is computed only at $(n+1)$-th time step and is linearized as
+
+$$
+\begin{eqnarray}
+G_{soi}^{n+1} &=& G_{soi}^{n} + \dfrac{\partial G}{\partial T_{soi,1}}
+\left(T_{soi,1}^{n+1} - T_{soi,1}^n\right) \nonumber \\[0.5em]
+& = & \overrightarrow{S}_{soi} - \left[ \epsilon_{soi} \sigma (T_{soi,1}^{n})^4 + 4 \epsilon_{soi} \sigma (T_{soi,1}^{n})^3 \Delta T_{soi}^{n+1} + \epsilon_{soi}L_{atm}^\downarrow \right] \nonumber \\[0.5em]
+& & - \left[ H_{soi}^n + \dfrac{\partial H}{\partial T_{soi,1}} \Delta T_{soi}^{n+1} \right] \nonumber\\[0.5em]
+& & - \left[ \lambda E_{soi}^n + \dfrac{\partial \lambda E}{\partial T_{soi,1}} \Delta T_{soi}^{n+1} \right] \nonumber \\[0.5em]
+& = & \overrightarrow{S}_{soi} - \overrightarrow{L}_g^{n+1} - H_{soi}^{n+1} - \lambda E_{soi}^{n+1} \label{eqn:G_bc_discretized}
+\end{eqnarray}
+$$
+
+Thus, the temporally discretized net absorbed longwave radiation in Equation \eqref{eqn:G_bc_discretized} is
+
+$$
+\begin{equation}
+\overrightarrow{L}_g^{n+1} = \left[ \epsilon_{soi} \sigma (T_{soi,1}^{n})^4 + 4 \epsilon_{soi} \sigma (T_{soi,1}^{n})^3 \Delta T_{soi}^{n+1} + \epsilon_{soi}L_{atm}^\downarrow \right]
+\label{eqn:lg_net_nonveg2}
+\end{equation}
+$$
+
+Comparing Equation \eqref{eqn:lg_net_nonveg} and \eqref{eqn:lg_net_nonveg2}, one can
+interpret the second term on the right hand side, $4\epsilon_{soi}\sigma$ ($T_{soi}^n)^3 \Delta T^{n+1}$,
+as an additional source of emitted longwave radiation. However, this term can only
+be computed after the vertical heat diffusion model is solved, i.e, after $\Delta T_{soi}^{n+1}$
+is known. Furthermore, this additional longwave radiation source term is added to
+the upward longwave radiation to the atmosphere given by $L_g^\uparrow$ in
+equation \eqref{eq:lg_up_nonveg).
+
+For the more general case, when snow and standing surface water are
+present on the ground, the temporally discretized net absorbed longwave radiation is
+
+$$
+\begin{equation}
+\overrightarrow{L}_g^{n+1} = \epsilon_{g} \sigma (T_{g}^{n})^4 + 4 \epsilon_{g} \sigma (T_{g}^{n})^3 \Delta T_{g}^{n+1} + \epsilon_{soi}L_{atm}^\downarrow
+\label{eqn:lg_net_nonveg3}
+\end{equation}
+$$
+
+### Vegetated Surface
+
+The same coupling of the surface ground energy flux equations and vertical
+heat diffusion model leads to a similar model complexity and the temporally
+discretized net longwave radiation for vegetated given by equation \eqref{eq:net_lg_veg1}
+is
+
+$$
+\begin{equation}
+\overrightarrow{L_g}^{n+1} = \epsilon_g \sigma (T_g^n)^4 + 4 \epsilon_{g} \sigma (T_{g}^{n})^3 \Delta T_{g}^{n+1} - \epsilon_g L_{v}^\downarrow
+\end{equation}
+$$
+
+The additional *apparent* emitted longwave radiation represented by the second term on the right
+hand side of the above equation is not absorbed by the canopy and directly sent upwards to
+the atmosphere by adding it in $L_{vg}^\uparrow$.
+
+## Temporally Discretized Governing Equations
+
+For the sake of completeness and clarity, we list below the temporally discretized
+longwave equations used in ELM.
+
+### Non-vegetated Surfaces
+
+The temporally discretized upwelling longwave radiation to the atmosphere and
+absorbed longwave radiation by the ground are
+
+$$
+\begin{eqnarray}
+L^{\uparrow n+1}_{g} &=& (1 - \epsilon_g)L^{\downarrow n+1}_{atm} + \epsilon_g \sigma (T_{g}^{n})^4 + 4 \epsilon_{g} \sigma (T_{g}^{n})^3 \Delta T_{g}^{n+1} \\[0.5em]
+\overrightarrow{L}_g^{n+1} &=& \epsilon_g \sigma (T_g^n)^4 + \epsilon_g L_{atm}^{\downarrow n+1} + 4 \epsilon_{g} \sigma (T_{g}^{n})^3 \Delta T_{g}^{n+1}
+\end{eqnarray}
+$$
+
+### Vegetated Surfaces
+
+When solving for $T_v^{n+1}$, ELM uses a diagnostic heat model in which leaves
+have no heat capacity and the sum of net absorbed solar and longwave radiation
+must balance the latent and sensible heat energy. This leads to a nonlinear
+equation for the vegetation canopy temperature, which is solved iteratively.
+When the solution for vegetation temperature has been found after $k$-th iteration,
+$T_v^{n+1,k}$, ELM uses a linear approximation of the non-linear term related
+to canopy temperature in the canopy emitted upward and downward longwave radiation
+equations. The linear approximation is as follows.
+
+$$
+\begin{equation}
+\epsilon_v \sigma_v (T_v^{n+1,k+1})^4 = \epsilon_v \sigma_v (T_v^{n+1,k})^4 + 4\epsilon_v \sigma_v (T_v^{n+1,k})^3 \Delta T_v^{n+1,k}
+\end{equation}
+$$
+
+The temporally discretized downward longwave radiation by leaves is
+
+$$
+\begin{equation}
+L_v^{\downarrow n+1} = (1 - \epsilon_v)L_{atm}^{\downarrow n+1} + \epsilon_v \sigma (T_v^{n+1,k})^4
++ 4 \epsilon_v \sigma (T_v^{n+1,k})^3 (\Delta T_v^{n+1,k+1})
+\end{equation}
+$$
+
+The temporally discretized upward longwave from the canopy
+
+$$
+\begin{eqnarray}
+L_{vg}^{\uparrow n+1}
+&=& (1 - \epsilon_v) (1 - \epsilon_g) (1 - \epsilon_v)L_{atm}^{\downarrow n+1} \nonumber \\[0.5em]
+& & + (1 - \epsilon_v) (1 - \epsilon_g) \epsilon_v \sigma (T_v^{n+1,k})^4 \nonumber\\[0.5em]
+& & + 4 (1 - \epsilon_v) (1 - \epsilon_g) \epsilon_v \sigma (T_v^{n+1,k})^3 (\Delta T_v^{n+1,k+1}) \nonumber\\[0.5em]
+& & + \epsilon_v \sigma (T_v^{n+1,k})^4 + 4 \epsilon_v \sigma (T_v^{n+1,k})^3 \Delta T_v^{n+1,k+1} \nonumber\\[0.5em]
+& & + (1 - \epsilon_v) \epsilon_g \sigma (T_g^n)^4
+\end{eqnarray}
+$$
+
+The temporally discretized upward longwave from the canopy and ground to the atmosphere is
+
+$$
+\begin{equation}
+L^{\uparrow n + 1} = L_{vg}^{\uparrow n+1}  + 4 \epsilon_g \sigma (T_g^n)^3 (\Delta T_g^{n+1})
+\end{equation}
+$$
+
+Note the $T_v$ used in computing the net longwave radiation absorbed by
+the leaf given by equation \eqref{eq:net_lv} is $T_v^{n+1,k}$ and it is not
+adjusted after the solution for vegetation temperature is found.

components/elm/docs/tech-guide/longwave_radiation.md

@@ -117,3 +117,9 @@ @book{goudriaan1977crop
   publisher={Wageningen University and Research}
 }
 
+@book{bonan2019climate,
+  title={Climate change and terrestrial ecosystem modeling},
+  author={Bonan, Gordon},
+  year={2019},
+  publisher={Cambridge University Press}
+}