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

Assume the atmosphere is transparent to incoming shortwave radiation, as required for the stated infrared-only radiative-equilibrium model. Global energy balance givesPut $D={\tau }_{\infty }$ and $F=I_{+}-I_{-}$. Radiative equilibrium requires $dF/d\tau =0$, hence the two equations imply $I_{+}+I_{-}=2\sigma T^4$. Substitution gives $d{I}_{+}/d\tau =d{I}_{-}/d\tau =-F/2$. At the top there is no incident infrared flux, $I_{-}(D)=0$, and $I_{+}(D)=F=L_{\infty }$. ThereforeThe black ground supplies $I_{+}(0)=\sigma T_g^4$. ThusThe ground also satisfies $\sigma T_g^4=L_{\infty }+I_{-}(0)$, showing explicitly that its outgoing flux balances absorbed sunlight plus downward atmospheric radiation. In pure radiative equilibrium the air just above the ground need not have the ground's temperature: $\sigma T(0{)}^4=L_{\infty }(1+D)/2$, whereas $\sigma T_g^4=L_{\infty }(2+D)/2$. That discontinuity is relevant to surface-driven convection in part (d).