Past exam of the physics of the Earth as a Planet course of the University of Cambridge / 2012 / Paper 1 / 1 / a

Write $T_s$ for the fixed surface temperature and $\theta =\overline{T}-T_s$. The heat capacity per unit volume is ${\rho }_0{c}_p=k/\kappa$. Per unit horizontal area the layer stores sensible heat $(k/\kappa )d\theta$. Its increase equals the input less the output:The same equation holds for $\theta$ because $T_s$ is constant. It assumes the supplied constant-property, fixed-thickness layer, without internal heat production, latent heat, or mechanical contributions to its heat budget. The infinite horizontal area is handled by taking the budget per unit area.