## 2.4.1. Solar Fluxes[¶](#solar-fluxes "Permalink to this headline")
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[Figure 2.4.1](#figure-radiation-schematic) illustrates the direct beam and diffuse fluxes in the canopy.

\\(I\\, \\uparrow \_{\\Lambda }^{\\mu }\\) and \\(I\\, \\uparrow \_{\\Lambda }\\) are the upward diffuse fluxes, per unit incident direct beam and diffuse flux (section [2.3.1](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Surface_Albedos/CLM50_Tech_Note_Surface_Albedos.html#canopy-radiative-transfer)). \\(I\\, \\downarrow \_{\\Lambda }^{\\mu }\\) and \\(I\\, \\downarrow \_{\\Lambda }\\) are the downward diffuse fluxes below the vegetation per unit incident direct beam and diffuse radiation (section [2.3.1](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Surface_Albedos/CLM50_Tech_Note_Surface_Albedos.html#canopy-radiative-transfer)). The direct beam flux transmitted through the canopy, per unit incident flux, is \\(e^{-K\\left(L+S\\right)}\\). \\(\\vec{I}\_{\\Lambda }^{\\mu }\\) and \\(\\vec{I}\_{\\Lambda }^{}\\) are the fluxes absorbed by the vegetation, per unit incident direct beam and diffuse radiation (section [2.3.1](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Surface_Albedos/CLM50_Tech_Note_Surface_Albedos.html#canopy-radiative-transfer)). \\(\\alpha \_{g,\\, \\Lambda }^{\\mu }\\) and \\(\\alpha \_{g,\\, \\Lambda }\\) are the direct beam and diffuse ground albedos (section [2.3.2](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Surface_Albedos/CLM50_Tech_Note_Surface_Albedos.html#ground-albedos)). \\(L\\) and \\(S\\) are the exposed leaf area index and stem area index (section [2.2.1.4](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Ecosystem/CLM50_Tech_Note_Ecosystem.html#phenology-and-vegetation-burial-by-snow)). \\(K\\) is the optical depth of direct beam per unit leaf and stem area (section [2.3.1](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Surface_Albedos/CLM50_Tech_Note_Surface_Albedos.html#canopy-radiative-transfer)).

![Image 1: ../../_images/image15.png](https://escomp.github.io/ctsm-docs/versions/master/html/_images/image15.png)

Figure 2.4.1 Schematic diagram of (a) direct beam radiation, (b) diffuse solar radiation, and (c) longwave radiation absorbed, transmitted, and reflected by vegetation and ground.[¶](#id3 "Permalink to this image")

For clarity, terms involving \\(T^{n+1} -T^{n}\\) are not shown in (c).

The total solar radiation absorbed by the vegetation and ground is

(2.4.1)[¶](#equation-4-1 "Permalink to this equation")\\\[\\vec{S}\_{v} =\\sum \_{\\Lambda }S\_{atm} \\, \\downarrow \_{\\Lambda }^{\\mu } \\overrightarrow{I}\_{\\Lambda }^{\\mu } +S\_{atm} \\, \\downarrow \_{\\Lambda } \\overrightarrow{I}\_{\\Lambda }\\\]

(2.4.2)[¶](#equation-4-2 "Permalink to this equation")\\\[\\begin{split}\\begin{array}{l} {\\vec{S}\_{g} =\\sum \_{\\Lambda }S\_{atm} \\, \\downarrow \_{\\Lambda }^{\\mu } e^{-K\\left(L+S\\right)} \\left(1-\\alpha \_{g,\\, \\Lambda }^{\\mu } \\right) +} \\\\ {\\qquad \\left(S\_{atm} \\, \\downarrow \_{\\Lambda }^{\\mu } I\\downarrow \_{\\Lambda }^{\\mu } +S\_{atm} \\downarrow \_{\\Lambda } I\\downarrow \_{\\Lambda } \\right)\\left(1-\\alpha \_{g,\\, \\Lambda } \\right)} \\end{array}\\end{split}\\\]

where \\(S\_{atm} \\, \\downarrow \_{\\Lambda }^{\\mu }\\) and \\(S\_{atm} \\, \\downarrow \_{\\Lambda }\\) are the incident direct beam and diffuse solar fluxes (W m\-2). For non-vegetated surfaces, \\(e^{-K\\left(L+S\\right)} =1\\), \\(\\overrightarrow{I}\_{\\Lambda }^{\\mu } =\\overrightarrow{I}\_{\\Lambda } =0\\), \\(I\\, \\downarrow \_{\\Lambda }^{\\mu } =0\\), and \\(I\\, \\downarrow \_{\\Lambda } =1\\), so that

(2.4.3)[¶](#equation-4-3 "Permalink to this equation")\\\[\\begin{split}\\begin{array}{l} {\\vec{S}\_{g} =\\sum \_{\\Lambda }S\_{atm} \\, \\downarrow \_{\\Lambda }^{\\mu } \\left(1-\\alpha \_{g,\\, \\Lambda }^{\\mu } \\right) +S\_{atm} \\, \\downarrow \_{\\Lambda } \\left(1-\\alpha \_{g,\\, \\Lambda } \\right)} \\\\ {\\vec{S}\_{v} =0} \\end{array}.\\end{split}\\\]

Solar radiation is conserved as

(2.4.4)[¶](#equation-4-4 "Permalink to this equation")\\\[\\sum \_{\\Lambda }\\left(S\_{atm} \\, \\downarrow \_{\\Lambda }^{\\mu } +S\_{atm} \\, \\downarrow \_{\\Lambda } \\right)=\\left(\\vec{S}\_{v} +\\vec{S}\_{g} \\right) +\\sum \_{\\Lambda }\\left(S\_{atm} \\, \\downarrow \_{\\Lambda }^{\\mu } I\\uparrow \_{\\Lambda }^{\\mu } +S\_{atm} \\, \\downarrow \_{\\Lambda } I\\uparrow \_{\\Lambda } \\right)\\\]

where the latter term in parentheses is reflected solar radiation.

Photosynthesis and transpiration depend non-linearly on solar radiation, via the light response of stomata. The canopy is treated as two leaves (sunlit and shaded) and the solar radiation in the visible waveband (\\(<\\) 0.7 µm) absorbed by the vegetation is apportioned to the sunlit and shaded leaves (section [2.3.1](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Surface_Albedos/CLM50_Tech_Note_Surface_Albedos.html#canopy-radiative-transfer)). The absorbed photosynthetically active (visible waveband) radiation averaged over the sunlit canopy (per unit plant area) is

(2.4.5)[¶](#equation-4-5 "Permalink to this equation")\\\[\\phi ^{sun} ={\\left(\\vec{I}\_{sun,vis}^{\\mu } S\_{atm} \\downarrow \_{vis}^{\\mu } +\\vec{I}\_{sun,vis}^{} S\_{atm} \\downarrow \_{vis}^{} \\right)\\mathord{\\left/ {\\vphantom {\\left(\\vec{I}\_{sun,vis}^{\\mu } S\_{atm} \\downarrow \_{vis}^{\\mu } +\\vec{I}\_{sun,vis}^{} S\_{atm} \\downarrow \_{vis}^{} \\right) L^{sun} }} \\right.} L^{sun} }\\\]

and the absorbed radiation for the average shaded leaf (per unit plant area) is

(2.4.6)[¶](#equation-4-6 "Permalink to this equation")\\\[\\phi ^{sha} ={\\left(\\vec{I}\_{sha,vis}^{\\mu } S\_{atm} \\downarrow \_{vis}^{\\mu } +\\vec{I}\_{sha,vis}^{} S\_{atm} \\downarrow \_{vis}^{} \\right)\\mathord{\\left/ {\\vphantom {\\left(\\vec{I}\_{sha,vis}^{\\mu } S\_{atm} \\downarrow \_{vis}^{\\mu } +\\vec{I}\_{sha,vis}^{} S\_{atm} \\downarrow \_{vis}^{} \\right) L^{sha} }} \\right.} L^{sha} }\\\]

with \\(L^{sun}\\) and \\(L^{sha}\\) the sunlit and shaded plant area index, respectively. The sunlit plant area index is

(2.4.7)[¶](#equation-4-7 "Permalink to this equation")\\\[L^{sun} =\\frac{1-e^{-K(L+S)} }{K}\\\]

and the shaded leaf area index is \\(L^{sha} =(L+S)-L^{sun}\\). In calculating \\(L^{sun}\\),

(2.4.8)[¶](#equation-4-8 "Permalink to this equation")\\\[K=\\frac{G\\left(\\mu \\right)}{\\mu }\\\]

where \\(G\\left(\\mu \\right)\\) and \\(\\mu\\) are parameters in the two-stream approximation (section [2.3.1](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Surface_Albedos/CLM50_Tech_Note_Surface_Albedos.html#canopy-radiative-transfer)).

The model uses the two-stream approximation to calculate radiative transfer of direct and diffuse radiation through a canopy that is differentiated into leaves that are sunlit and those that are shaded (section [2.3.1](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Surface_Albedos/CLM50_Tech_Note_Surface_Albedos.html#canopy-radiative-transfer)). The two-stream equations are integrated over all plant area (leaf and stem area) in the canopy. The model has an optional (though not supported) multi-layer canopy, as described by [Bonan et al. (2012)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#bonanetal2012). The multi-layer model is only intended to address the non-linearity of light profiles, photosynthesis, and stomatal conductance in the plant canopy.

In the multi-layer canopy, canopy-integrated radiative fluxes are calculated from the two-stream approximation. The model additionally derives the light profile with depth in the canopy by taking the derivatives of the absorbed radiative fluxes with respect to plant area index (\\(L'=L+S\\)) and evaluating them incrementally through the canopy with cumulative plant area index (\\(x\\)). The terms \\({d\\vec{I}\_{sun,\\Lambda }^{\\mu } (x)\\mathord{\\left/ {\\vphantom {d\\vec{I}\_{sun,\\Lambda }^{\\mu } (x) dL'}} \\right.} dL'}\\) and \\({d\\vec{I}\_{sun,\\Lambda }^{} (x)\\mathord{\\left/ {\\vphantom {d\\vec{I}\_{sun,\\Lambda }^{} (x) dL'}} \\right.} dL'}\\) are the direct beam and diffuse solar radiation, respectively, absorbed by the sunlit fraction of the canopy (per unit plant area) at a depth defined by the cumulative plant area index \\(x\\); \\({d\\vec{I}\_{sha,\\Lambda }^{\\mu } (x)\\mathord{\\left/ {\\vphantom {d\\vec{I}\_{sha,\\Lambda }^{\\mu } (x) dL'}} \\right.} dL'}\\) and \\({d\\vec{I}\_{sha,\\Lambda }^{} (x)\\mathord{\\left/ {\\vphantom {d\\vec{I}\_{sha,\\Lambda }^{} (x) dL'}} \\right.} dL'}\\) are the corresponding fluxes for the shaded fraction of the canopy at depth \\(x\\). These fluxes are normalized by the sunlit or shaded fraction at depth \\(x\\), defined by \\(f\_{sun} =\\exp \\left(-Kx\\right)\\), to give fluxes per unit sunlit or shaded plant area at depth \\(x\\).

