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‎components/elm/docs/figures/longwave_radiation.png‎

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

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- [Shortwave Radiation](shortwave_radiation.md):
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Shortwave radiation model
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- [Longwave Radiation](longwave_radiation.md): Longwave radiation model
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- [TOP Parameterization](top_solar_parameterization.md):
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Parameterization of sub-grid topographical effects on solar radiation.

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

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

‎docs/refs/elm.bib‎

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

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