## 2.4.2. Longwave Fluxes[¶](#longwave-fluxes "Permalink to this headline")
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The net longwave radiation (W m\-2) (positive toward the atmosphere) at the surface is

(2.4.9)[¶](#equation-4-9 "Permalink to this equation")\\\[\\vec{L}=L\\, \\uparrow -L\_{atm} \\, \\downarrow\\\]

where \\(L\\, \\uparrow\\) is the upward longwave radiation from the surface and \\(L\_{atm} \\, \\downarrow\\) is the downward atmospheric longwave radiation (W m\-2). The radiative temperature \\(T\_{rad}\\) (K) is defined from the upward longwave radiation as

(2.4.10)[¶](#equation-4-10 "Permalink to this equation")\\\[T\_{rad} =\\left(\\frac{L\\, \\uparrow }{\\sigma } \\right)^{{1\\mathord{\\left/ {\\vphantom {1 4}} \\right.} 4} }\\\]

where \\(\\sigma\\) is the Stefan-Boltzmann constant (Wm\-2 K\-4) ([Table 2.2.7](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Ecosystem/CLM50_Tech_Note_Ecosystem.html#table-physical-constants)). With reference to [Figure 2.4.1](#figure-radiation-schematic), the upward longwave radiation from the surface to the atmosphere is

(2.4.11)[¶](#equation-4-11 "Permalink to this equation")\\\[\\begin{split}\\begin{array}{l} {L\\, \\uparrow =\\delta \_{veg} L\_{vg} \\, \\uparrow +\\left(1-\\delta \_{veg} \\right)\\left(1-\\varepsilon \_{g} \\right)L\_{atm} \\, \\downarrow +} \\\\ {\\qquad \\left(1-\\delta \_{veg} \\right)\\varepsilon \_{g} \\sigma \\left(T\_{g}^{n} \\right)^{4} +4\\varepsilon \_{g} \\sigma \\left(T\_{g}^{n} \\right)^{3} \\left(T\_{g}^{n+1} -T\_{g}^{n} \\right)} \\end{array}\\end{split}\\\]

where \\(L\_{vg} \\, \\uparrow\\) is the upward longwave radiation from the vegetation/soil system for exposed leaf and stem area \\(L+S\\ge 0.05\\), \\(\\delta \_{veg}\\) is a step function and is zero for \\(L+S<0.05\\) and one otherwise, \\(\\varepsilon \_{g}\\) is the ground emissivity, and \\(T\_{g}^{n+1}\\) and \\(T\_{g}^{n}\\) are the snow/soil surface temperatures at the current and previous time steps, respectively ([Soil and Snow Temperatures](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Soil_Snow_Temperatures/CLM50_Tech_Note_Soil_Snow_Temperatures.html#rst-soil-and-snow-temperatures)).

For non-vegetated surfaces, the above equation reduces to

(2.4.12)[¶](#equation-4-12 "Permalink to this equation")\\\[L\\, \\uparrow =\\left(1-\\varepsilon \_{g} \\right)L\_{atm} \\, \\downarrow +\\varepsilon \_{g} \\sigma \\left(T\_{g}^{n} \\right)^{4} +4\\varepsilon \_{g} \\sigma \\left(T\_{g}^{n} \\right)^{3} \\left(T\_{g}^{n+1} -T\_{g}^{n} \\right)\\\]

where the first term is the atmospheric longwave radiation reflected by the ground, the second term is the longwave radiation emitted by the ground, and the last term is the increase (decrease) in longwave radiation emitted by the ground due to an increase (decrease) in ground temperature.

For vegetated surfaces, the upward longwave radiation from the surface reduces to

(2.4.13)[¶](#equation-4-13 "Permalink to this equation")\\\[L\\, \\uparrow =L\_{vg} \\, \\uparrow +4\\varepsilon \_{g} \\sigma \\left(T\_{g}^{n} \\right)^{3} \\left(T\_{g}^{n+1} -T\_{g}^{n} \\right)\\\]

where

(2.4.14)[¶](#equation-4-14 "Permalink to this equation")\\\[\\begin{split}\\begin{array}{l} {L\_{vg} \\, \\uparrow =\\left(1-\\varepsilon \_{g} \\right)\\left(1-\\varepsilon \_{v} \\right)\\left(1-\\varepsilon \_{v} \\right)L\_{atm} \\, \\downarrow } \\\\ {\\qquad \\qquad +\\varepsilon \_{v} \\left\[1+\\left(1-\\varepsilon \_{g} \\right)\\left(1-\\varepsilon \_{v} \\right)\\right\]\\sigma \\left(T\_{v}^{n} \\right)^{3} \\left\[T\_{v}^{n} +4\\left(T\_{v}^{n+1} -T\_{v}^{n} \\right)\\right\]} \\\\ {\\qquad \\qquad +\\varepsilon \_{g} \\left(1-\\varepsilon \_{v} \\right)\\sigma \\left(T\_{g}^{n} \\right)^{4} } \\\\ {\\qquad =\\left(1-\\varepsilon \_{g} \\right)\\left(1-\\varepsilon \_{v} \\right)\\left(1-\\varepsilon \_{v} \\right)L\_{atm} \\, \\downarrow } \\\\ {\\qquad \\qquad +\\varepsilon \_{v} \\sigma \\left(T\_{v}^{n} \\right)^{4} } \\\\ {\\qquad \\qquad +\\varepsilon \_{v} \\left(1-\\varepsilon \_{g} \\right)\\left(1-\\varepsilon \_{v} \\right)\\sigma \\left(T\_{v}^{n} \\right)^{4} } \\\\ {\\qquad \\qquad +4\\varepsilon \_{v} \\sigma \\left(T\_{v}^{n} \\right)^{3} \\left(T\_{v}^{n+1} -T\_{v}^{n} \\right)} \\\\ {\\qquad \\qquad +4\\varepsilon \_{v} \\left(1-\\varepsilon \_{g} \\right)\\left(1-\\varepsilon \_{v} \\right)\\sigma \\left(T\_{v}^{n} \\right)^{3} \\left(T\_{v}^{n+1} -T\_{v}^{n} \\right)} \\\\ {\\qquad \\qquad +\\varepsilon \_{g} \\left(1-\\varepsilon \_{v} \\right)\\sigma \\left(T\_{g}^{n} \\right)^{4} } \\end{array}\\end{split}\\\]

where \\(\\varepsilon \_{v}\\) is the vegetation emissivity and \\(T\_{v}^{n+1}\\) and \\(T\_{v}^{n}\\) are the vegetation temperatures at the current and previous time steps, respectively ([Momentum, Sensible Heat, and Latent Heat Fluxes](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Fluxes/CLM50_Tech_Note_Fluxes.html#rst-momentum-sensible-heat-and-latent-heat-fluxes)). The first term in the equation above is the atmospheric longwave radiation that is transmitted through the canopy, reflected by the ground, and transmitted through the canopy to the atmosphere. The second term is the longwave radiation emitted by the canopy directly to the atmosphere. The third term is the longwave radiation emitted downward from the canopy, reflected by the ground, and transmitted through the canopy to the atmosphere. The fourth term is the increase (decrease) in longwave radiation due to an increase (decrease) in canopy temperature that is emitted by the canopy directly to the atmosphere. The fifth term is the increase (decrease) in longwave radiation due to an increase (decrease) in canopy temperature that is emitted downward from the canopy, reflected from the ground, and transmitted through the canopy to the atmosphere. The last term is the longwave radiation emitted by the ground and transmitted through the canopy to the atmosphere.

The upward longwave radiation from the ground is

(2.4.15)[¶](#equation-4-15 "Permalink to this equation")\\\[L\_{g} \\, \\uparrow =\\left(1-\\varepsilon \_{g} \\right)L\_{v} \\, \\downarrow +\\varepsilon \_{g} \\sigma \\left(T\_{g}^{n} \\right)^{4}\\\]

where \\(L\_{v} \\, \\downarrow\\) is the downward longwave radiation below the vegetation

(2.4.16)[¶](#equation-4-16 "Permalink to this equation")\\\[L\_{v} \\, \\downarrow =\\left(1-\\varepsilon \_{v} \\right)L\_{atm} \\, \\downarrow +\\varepsilon \_{v} \\sigma \\left(T\_{v}^{n} \\right)^{4} +4\\varepsilon \_{v} \\sigma \\left(T\_{v}^{n} \\right)^{3} \\left(T\_{v}^{n+1} -T\_{v}^{n} \\right).\\\]

The net longwave radiation flux for the ground is (positive toward the atmosphere)

(2.4.17)[¶](#equation-4-17 "Permalink to this equation")\\\[\\vec{L}\_{g} =\\varepsilon \_{g} \\sigma \\left(T\_{g}^{n} \\right)^{4} -\\delta \_{veg} \\varepsilon \_{g} L\_{v} \\, \\downarrow -\\left(1-\\delta \_{veg} \\right)\\varepsilon \_{g} L\_{atm} \\, \\downarrow .\\\]

The above expression for \\(\\vec{L}\_{g}\\) is the net longwave radiation forcing that is used in the soil temperature calculation ([Soil and Snow Temperatures](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Soil_Snow_Temperatures/CLM50_Tech_Note_Soil_Snow_Temperatures.html#rst-soil-and-snow-temperatures)). Once updated soil temperatures have been obtained, the term \\(4\\varepsilon \_{g} \\sigma \\left(T\_{g}^{n} \\right)^{3} \\left(T\_{g}^{n+1} -T\_{g}^{n} \\right)\\) is added to \\(\\vec{L}\_{g}\\) to calculate the ground heat flux (section [2.5.4](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Fluxes/CLM50_Tech_Note_Fluxes.html#update-of-ground-sensible-and-latent-heat-fluxes))

The net longwave radiation flux for vegetation is (positive toward the atmosphere)

(2.4.18)[¶](#equation-4-18 "Permalink to this equation")\\\[\\vec{L}\_{v} =\\left\[2-\\varepsilon \_{v} \\left(1-\\varepsilon \_{g} \\right)\\right\]\\varepsilon \_{v} \\sigma \\left(T\_{v} \\right)^{4} -\\varepsilon \_{v} \\varepsilon \_{g} \\sigma \\left(T\_{g}^{n} \\right)^{4} -\\varepsilon \_{v} \\left\[1+\\left(1-\\varepsilon \_{g} \\right)\\left(1-\\varepsilon \_{v} \\right)\\right\]L\_{atm} \\, \\downarrow .\\\]

These equations assume that absorptivity equals emissivity. The emissivity of the ground is

(2.4.19)[¶](#equation-4-19 "Permalink to this equation")\\\[\\varepsilon \_{g} =\\varepsilon \_{soi} \\left(1-f\_{sno} \\right)+\\varepsilon \_{sno} f\_{sno}\\\]

where \\(\\varepsilon \_{soi} =0.96\\) for soil, 0.97 for glacier, \\(\\varepsilon \_{sno} =0.97\\), and \\(f\_{sno}\\) is the fraction of ground covered by snow (section [2.8.1](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Snow_Hydrology/CLM50_Tech_Note_Snow_Hydrology.html#snow-covered-area-fraction)). The vegetation emissivity is

(2.4.20)[¶](#equation-4-20 "Permalink to this equation")\\\[\\varepsilon \_{v} =1-e^{-{\\left(L+S\\right)\\mathord{\\left/ {\\vphantom {\\left(L+S\\right) \\bar{\\mu }}} \\right.} \\bar{\\mu }} }\\\]

where \\(L\\) and \\(S\\) are the leaf and stem area indices (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)) and \\(\\bar{\\mu }=1\\) is the average inverse optical depth for longwave radiation.
