 Solution

ID: 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 τ

, and emits

σ T^{4} d τ

into each hemisphere. Radiative balance gives

2 σ T_{skin}^{4} = F_{0}, T_{m i n} ≃ T_{skin} = (\frac{F _{0}}{2 σ})^{1/4} ≃ 227 K

for

F_{0} = 300 W m^{- 2}

. The corresponding effective emission temperature is

(F_{0} / σ)^{1/4} ≃ 270 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.

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