Summary of the Article:

## Longwave Fluxes

### Net Longwave Radiation at the Surface
The net longwave radiation (positive toward the atmosphere) at the surface is given by the 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.

### Radiative Temperature
The radiative temperature $T_{rad}$ is defined from the upward longwave radiation as:
$T_{rad} = \left(\frac{L\, \uparrow}{\sigma}\right)^{1/4}$
where $\sigma$ is the Stefan-Boltzmann constant.

### Upward Longwave Radiation
The upward longwave radiation from the surface to the atmosphere is given by the equation:
$L\, \uparrow = \delta_{veg} L_{vg} \, \uparrow + (1-\delta_{veg})(1-\varepsilon_{g})L_{atm} \, \downarrow + (1-\delta_{veg})\varepsilon_{g}\sigma(T_{g}^{n})^{4} + 4\varepsilon_{g}\sigma(T_{g}^{n})^{3}(T_{g}^{n+1}-T_{g}^{n})$
This equation is reduced for non-vegetated and vegetated surfaces.

### Net Longwave Radiation Flux
The net longwave radiation flux for the ground is given by:
$\vec{L}_{g} = \varepsilon_{g}\sigma(T_{g}^{n})^{4} - \delta_{veg}\varepsilon_{g}L_{v} \, \downarrow - (1-\delta_{veg})\varepsilon_{g}L_{atm} \, \downarrow$
The net longwave radiation flux for vegetation is given by:
$\vec{L}_{v} = \left[2-\varepsilon_{v}(1-\varepsilon_{g})\right]\varepsilon_{v}\sigma(T_{v})^{4} - \varepsilon_{v}\varepsilon_{g}\sigma(T_{g}^{n})^{4} - \varepsilon_{v}\left[1+(1-\varepsilon_{g})(1-\varepsilon_{v})\right]L_{atm} \, \downarrow$

### Emissivity
The emissivity of the ground and vegetation are defined by equations involving soil, snow, leaf, and stem area indices.