Past exam of the physics of atmospheres and oceans course of the University of Oxford / 2018 / 3 / a / Solution

$I_{+}$ and $I_{-}$ are the positive upward and downward hemispherically integrated infrared fluxes, in $\mathrm{W}{\mathrm{m}}^{-2}$, rather than directional specific intensities. The source is the blackbody hemispheric flux $\sigma T^4$. Absorption removes a fraction proportional to the flux along a ray, and thermal emission replaces it according to Kirchhoff's law. Local thermodynamic equilibrium sets that emission source to the local Planck function.

A plane-parallel, non-scattering, gray atmosphere with a two-stream/angular closure reduces the angle- and frequency-dependent transfer equation to these two flux equations. The angular path factors are absorbed into the effective optical depth. Since the supplied coordinate increases upwards, the upward flux loses $I_{+}d\tau$ while gaining $\sigma T^{4}d\tau$ in moving upward; the downward ray traverses the coordinate in the opposite direction, reversing the differential signs in its equation. The equations therefore encode emission as well as absorption and are consistent with an isothermal source approaching thermal radiation in an optically thick medium. The source's use of “radiance” for these integrated quantities should not introduce an extra factor of $\pi$.