Past exam of the physics of atmospheres and oceans course of the University of Oxford / 2017 / 4 / b

Assume a plane-parallel atmosphere in local thermodynamic equilibrium, negligible infrared scattering, a black surface, and surface temperature equal to the adjacent air temperature. For a specified ray, let $\mathfrak{T}$ be its transmittance from a level to space, increasing from ${\mathfrak{T}}_0$ at the surface to one at the top. The formal radiative transfer solution isUsing ${\mathfrak{T}}_0+{\int }_{{\mathfrak{T}}_0}^{1}d\mathfrak{T}=1$ yieldsThis expresses absorption and re-emission together: opacity does not simply remove surface radiation without adding atmospheric radiation. An absorber close to a thermally coupled surface has nearly the same Planck function as that surface, so $B-B_0$ is small. Its emission nearly compensates the surface radiation it absorbs, giving weak radiance sensitivity to near-surface concentration. In an exactly isothermal atmosphere at surface temperature, $L=B_0$ for every optical depth, a concrete example of complete concentration ambiguity in this thermal measurement. Finite surface emissivity, reflection, scattering or a surface-air temperature discontinuity would require additional terms.