Grey fixed-tropopause warming estimate ($T_s^{\prime }/T_s=2^{R_a/c_p}$)

In a prescribed dry convective column with unchanged surface pressure and upper matching temperature, doubling uniformly mixed grey absorbers halves the pressure of a fixed-optical depth matching level. The adiabat $T\propto p^{R_a/c_p}$ then gives the displayed finite change. This is conditional on the fixed matching and convection assumptions. With strictly constant mass opacity, a purely radiative Eddington grey column instead has $d\ln T/d\ln p<1/4$, so a gas with $R_a/c_p>1/4$ is not destabilized into dry convection by that radiative profile alone.