Past exam of the mathematics and the environment and mathematical physiology course of the University of Oxford / 2008 / A / 1 / Solution

The left side is atmospheric heat storage per unit surface area. Incoming solar flux $Q$ is averaged over the spherical surface by the factor $1/4$; albedo $a$ reflects a fraction of it. The outgoing term is a reduced Stefan-Boltzmann flux, with effective emissivity $\gamma$ representing atmospheric greenhouse trapping. Ice-albedo feedback usually makes $a$ decrease as temperature rises through an ice-melting range; greater carbon-dioxide pressure increases infrared opacity and therefore decreases this effective $\gamma$. The simple estimates below hold albedo fixed.

Radiative equilibrium givesDifferentiating $\gamma T^4=\text{constant}$ gives $\Delta \gamma \simeq -4{\gamma }_0\Delta T/T_0\simeq -0.02474$. A doubling from $36$ Pa changes pressure by $36$ Pa, henceReference weathering balance requires $A_L{W}_0=v_0$, so $\boxed{W_0=2.0\times 10^{-3}\mathrm{kg}{\mathrm{m}}^{-2}\mathrm{y}{\mathrm{r}}^{-1}}$.

For the requested approximate system, take $(T_0,p_0)$ as the reference radiative/weathering equilibrium, set $\theta =(T-T_0)/\Delta T_c$, $P=p/p_0$, and $\hat{t}=t/t_{\mathrm{C}{\mathrm{O}}_2}$, whereLinearize the radiative loss, retaining terms linear in $T-T_0$ and $p-p_0$. Its two terms are $4\sigma {\gamma }_0{T}_0^3(T-T_0)$ and $-\sigma {\gamma }_1{T}_0^4(p-p_0)$. The carbon balance scales exactly with $v=v_{\mathrm{C}{\mathrm{O}}_2}/v_0$. ThereforeNumerically $\boxed{t_{\mathrm{C}{\mathrm{O}}_2}\simeq 7.41\times 10^3\mathrm{yr},t_T\simeq 0.0958\mathrm{yr},ϵ\simeq 1.29\times 10^{-5},\Gamma \simeq 0.1795}$. The time conversion uses seconds for the radiative constants and years for the supplied carbon production.

Because $ϵ\ll 1$, temperature rapidly approaches $\theta =\Gamma (P-1)$, while carbon changes slowly. A sustained $v\simeq 40$ has equilibrium determined byIts left side is strictly increasing on positive pressure, giving $\boxed{P_{\ast }\simeq 16.83,\Delta T_{\mathrm{eventual}}=\Delta T_c\Gamma (P_{\ast }-1)\simeq 36.9\mathrm{K}}$ within this extrapolated toy model. Counting volcanic production as well gives $v=41$ and $37.2$ K instead. An inventory estimate for the rise is $t_{\mathrm{C}{\mathrm{O}}_2}(P_{\ast }-1)/(v-1)\simeq 3.0\times 10^3$ years. Linearization near the final equilibrium gives the e-folding timeThus the carbon/weathering response is millennial, despite the short thermal adjustment time.

There are two numerical-reference inconsistencies in the printed setup. The requested expansion treats $(T_0,p_0)$ as its reference equilibrium although earlier labels pair $T_0$ with present-day $36$ Pa rather than $p_0=28$ Pa. Also, the calibrated $\Gamma =0.1795$ gives $13^{0.3}{e}^{13\Gamma }\simeq 22.3$, not the stated hint. That hint instead corresponds to $\Gamma \simeq 3/13$, obtained by calibrating a doubling of $28$ Pa. Such a convention gives a pressure around $13$–$14$ and a similar rough warming of several tens of kelvin. The displayed results above consistently use the stated $36$ Pa doubling calibration and explicitly declared reference-model approximation. A large extrapolation of a linearized radiative law is a property of this exercise, not a quantitative climate projection.

This is a weathering feedback with fast thermal adjustment.