 Solution

Let $q$ be the mass mixing ratio of the uniformly mixed infrared absorbers, and let $k$ be their constant mass absorption coefficient. Hydrostatic integration gives downward optical depth $\tau =kqp/g$. The useful fixed-tropopause-optical-depth approximation requires a sufficiently optically thick lower column, unchanged absorbed sunlight and upper radiative profile in optical-depth coordinates, and the same convective lapse law after the perturbation. Then the matching to the radiative region occurs at the same ${\tau }_t$ and temperature $T_t$. Doubling all those absorbers givesAssume also that the surface pressure is unchanged. A dry adiabat satisfies $T_t/T_s=(p_t/p_s{)}^{\kappa }$ with $\kappa =R_a/c_p=0.287$. Consequentlyfor an initial Earth-like surface temperature $288\mathrm{K}$. Its first-order logarithmic estimate, $\Delta T_s\simeq \kappa T_s\ln 2$, is about $57\mathrm{K}$; the finite-doubling formula is preferable because the perturbation is large. This estimate is conditional on fixed ${\tau }_t,T_t$ and dry convective adjustment, not on carbon dioxide's measured spectral behavior.

There is also a consistency limitation to the most literal opacity model. A self-contained plane-parallel grey atmosphere with constant mass opacity has $T^4\propto \tau +2/3$ and $\tau \propto p$. Its radiative logarithmic gradient isIt is therefore subadiabatic for these dry heat capacities, so it does not by itself establish the assumed deep dry convective column. The preceding warming is the intended fixed-matching-level, convectively prescribed estimate. A fully self-consistent model needs opacity varying with pressure, other heating structure or another closure to supply that convection; the two statements should not be silently identified.