Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2015 / S25 / 1

The tropopause is the transition between the lower, often convecting troposphere and the more stably stratified atmosphere above, where the decrease of temperature with altitude weakens or reverses. Its definition concerns the thermal profile, not a fixed universal altitude. An optical depth is a dimensionless path-integrated extinction, ${\tau }_{\nu }=\int {\kappa }_{\nu }\rho ds$ for mass extinction coefficient ${\kappa }_{\nu }$; without emission into the ray its direct intensity is attenuated by $e^{-{\tau }_{\nu }}$.

In hydrostatic equilibrium, the upward pressure force balances weight:The second equality uses the ideal gas law with specific gas constant $R_a$. For constant $T$ and $g$,For variable temperature the corresponding result is $p(z)=p(z_0)\exp [-{\int }_{z_0}^{z}g/(R_{a}T(z^{\prime }))d{z}^{\prime }]$. The fixed-temperature derivative does not require the entire column to be isothermal.

Assume a well-mixed ideal gas with constant $g,c_p$, no latent heat, and sufficiently strong surface heating to maintain dry convection near the surface. An adiabatically displaced unit mass has specific volume $V=1/\rho$, so the stated relation and hydrostatic equilibrium giveThis is the dry neutral lapse rate; infrared absorption alone does not guarantee that a radiative profile is steep enough to trigger convection. Combining the same equations with the ideal gas law yields $T\propto p^{R_a/c_p}$ along that adiabat.

High above the main infrared-emitting region, optical depth to space becomes small. A transparent-to-sunlight atmosphere has no substantial direct solar heating there; a thin grey layer absorbs the upwelling planetary infrared flux $Q=(1-\alpha )S_0/4$ and emits from both faces. In the optically thin radiative-equilibrium limit,The factor four in $Q$ is projected-area versus whole-sphere averaging; the additional factor two counts emission upward and downward by the tenuous layer. For Earth-like $S_0\simeq 1360\mathrm{W}{\mathrm{m}}^{-2}$ and $\alpha \simeq 0.30$, this is about $214\mathrm{K}$. Thus temperature approaches a height-independent skin value as $\tau \to 0$, rather than continuing the dry adiabat to zero. This is an asymptotic grey-atmosphere approximation, not a proof that an entire finite stratosphere is exactly isothermal.

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.

The actual Earth response to doubling carbon dioxide is expected to be much smaller than this roughly $60\mathrm{K}$ all-grey-absorbers estimate. Carbon dioxide absorbs in selected spectral bands rather than doubling opacity across the entire thermal spectrum, and doubling it does not double the other infrared absorbers. Saturated band centres and overlap with water absorption further distinguish it from uniform opacity multiplication. Changes in water vapour, clouds and the lapse rate introduce important climate feedbacks, but do not justify treating the real atmosphere as the stipulated dry, wavelength-independent absorber. The idealized number is therefore not an estimate of real-world equilibrium climate sensitivity.