Past exam of the short options paper shared by Parts A and B of the physics course of the University of Oxford / 2019 / S25 / 1 / b / Solution

The usual estimate adds a grey, infrared-absorbing radiative equilibrium upper atmosphere to the convective lower atmosphere. Consider a very optically thin upper layer: it receives upwelling infrared flux approximately $F_0$ from below, absorbs a small fraction $d\tau$, and emits $\sigma T^{4}d\tau$ into each hemisphere. Radiative balance givesfor $F_0=300\mathrm{W}{\mathrm{m}}^{-2}$. The corresponding effective emission temperature is $(F_0/\sigma {)}^{1/4}\simeq 270\mathrm{K}$. Without shortwave absorption, the model has no solar-heated stratospheric inversion.

This skin estimate requires the grey absorption/emission and angular approximations just stated. Horizontal mixing and no shortwave absorption alone do not determine an exact minimum for a spectrally selective nitrogen–carbon-dioxide atmosphere: absorption coefficients and spectral windows can change its upper-atmosphere radiative equilibrium.