Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2012 / S25 (Physics of climate change)

Spectral irradiance $F_{\lambda }$ is radiant power crossing unit area per unit wavelength, integrated over the relevant hemisphere with the projected-area factor: $F_{\lambda }=\int I_{\lambda }\cos \theta d\Omega$. Its units are $\mathrm{W}{\mathrm{m}}^{-2}$ per unit wavelength; spectral radiance $I_{\lambda }$ additionally refers to unit solid angle. For an isotropic outgoing hemisphere with Planck function $B_{\lambda }$, the integral is $F_{\lambda }=\pi B_{\lambda }$.

The vertical optical depth above height $z$ is the dimensionless extinction integralHere $z$ increases upward, $n_a$ is absorber number density, ${\sigma }_{\lambda }$ its absorption cross section, ${\rho }_a$ its mass density, and ${\kappa }_{\lambda }$ the absorption coefficient per unit absorber mass. Equivalently, using bulk air density $\rho$ and absorber mass fraction $q_a$, $d{\tau }_{\lambda }/dz=-{\kappa }_{\lambda }{q}_a\rho$. The coefficient may depend on temperature, pressure and wavelength. For purely absorbing gas, transmission along a ray is $e^{-{\tau }_{\mathrm{ray}}}$; scattering, if present, contributes to extinction and requires its own source term.

Use the optical-depth weighting stipulated in the paper. A jump in the Planck function at ${\tau }_1$ gives $dB/d\tau =(B_1-B_0)\delta (\tau -{\tau }_1)$, soThe upper isothermal layer emits $\pi B_0(1-e^{-{\tau }_1})$ and transmits the lower region's blackbody spectral irradiance with factor $e^{-{\tau }_1}$. The lower layer and black surface have the same $B_1$, so their combined radiation entering the upper layer is $\pi B_1$, independent of how its emission is divided between the lower gas and surface.

More generally, integration by parts rewrites the supplied expression asThe terms are attenuated surface emission and the atmosphere's own emission. In the original boundary-plus-gradient form, $\pi B_0$ is the isothermal reference flux and the integral corrects for temperature variation below the top. Optical depths of order unity contribute most strongly. These formulas follow the paper's specified exponential weighting; a full angular radiative transfer calculation uses ray-angle-dependent optical paths rather than identifying every ray with a vertical path.

For the linear Planck function profile, direct integration givesin the optically thick limit. A sudden doubling of absorber abundance doubles optical depth at every physical height while initially leaving the temperature profile fixed. With $\tilde{\tau }=2\tau$, the same profile is $B=B_0+(\alpha /2)\tilde{\tau }$, not $B_0+\alpha \tilde{\tau }$. ThusIf temperature increases downward, $\alpha >0$ and outgoing spectral irradiance decreases immediately because the effective emitting region moves upward into cooler gas. Subsequent thermal adjustment is outside this instantaneous comparison.

For the quadratic Planck function profile,The optically thick limit and the instantaneous doubled-abundance result are thereforeThe new coefficient is $\beta /4$, since $B(\tilde{\tau })=B_0+\beta (\tilde{\tau }/2{)}^2$ at fixed physical temperature. For positive $\beta$, the outgoing spectral irradiance decreases. The factor differs from the linear case because the temperature-dependent source function curves with depth; doubling abundance does not mean holding its old functional coefficients fixed in the new optical-depth coordinate.

Hydrostatic equilibrium balances the vertical pressure gradient against weight, $dP/dz=-\rho g$. For a dry ideal gas parcel displaced slowly enough to maintain pressure balance but rapidly enough to exchange negligible heat, the first law givesCombining the two equations yields the dry adiabatic lapse rateusing $c_p\simeq 1004\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$. Here temperature in the thermodynamic differential is absolute, $g$ and $c_p$ are treated as constant, composition is fixed, and condensation is excluded. Hydrostatic balance alone does not force the atmosphere to be adiabatic; the parcel assumption supplies the extra relation.

Water condensation releases latent heat, reducing the cooling of a rising saturated parcel and giving a smaller moist adiabatic lapse rate. Radiation, mixing, large-scale motion and incomplete convective adjustment also shape the environmental lapse rate. Thus an observed $6$--$7\mathrm{K}\mathrm{k}{\mathrm{m}}^{-1}$ need not equal the dry parcel value and is broadly consistent with a moist, radiatively influenced atmosphere.

Write $s=T_s-273.15\mathrm{K}$ as the numerical sea-level Celsius temperature in degrees Celsius, and use the same degree increments for $\Gamma z$. The model's top temperature is $T_C=s-\Gamma z$. For positive $\alpha$ its intended cold-ice branch has $T_C<0$, so the positive melt-loss magnitude and snow gain areThe model is singular at $z=s/\Gamma$ and is not a physical melting law across that boundary. On the relevant interval $s/\Gamma <z<1/\beta$, $L$ decreases hyperbolically and $A$ decreases linearly. Assume a fixed area and positive ice density, so increasing mass increases $z$: stability has the same sign as $F^{\prime }(z)$.

The equilibrium equation iswith rootsTwo distinct physical equilibria, one stable, exist providedThis combines a positive allowed interval with a strictly positive discriminant. At the larger root $F^{\prime }<0$ and small perturbations restore the ice cap; the smaller root has $F^{\prime }>0$ and is unstable. Sincea restoring local response requiresThis is the lower height bound from the stability derivative; the stable equilibrium itself is $z_{+}$. The minimum initial height for persistence, a different notion of threshold, is the unstable separator $z_{-}$: initial heights above it approach $z_{+}$, while lower heights lose ice and leave the model's cold branch. At equality in the discriminant condition the two equilibria merge, giving a marginal fold rather than a strictly stable ice cap.

At $z=4\mathrm{km}$, the snow gain is $1-0.2(4)=0.2$ and the model's top Celsius temperature is $275-273.15-6(4)=-22.15^{\circ }\mathrm{C}$. Equilibrium givesin the normalized rate convention. Using $273\mathrm{K}$ for the Celsius zero instead would give $4.4^{\circ }\mathrm{C}$. The local slope is $-0.2+4.43(6)/(22.15{)}^2\simeq -0.1458\mathrm{k}{\mathrm{m}}^{-1}<0$, as required. Initially $z_{-}=1.3083\mathrm{km}$ and $z_{\mathrm{slope}}\simeq 2.230\mathrm{km}$, while $z_{+}=4\mathrm{km}$.

As $T_s$ slowly rises, melt loss increases at fixed $z$, the stable high branch moves downward and the unstable branch upward. They merge atBeyond the fold the model has no steady cold ice cap: its net balance is negative throughout the physical interval and ice shrinks until the assumed law ceases to apply. Quasistatic tracking presumes adjustment is fast compared with the imposed warming, an assumption that deteriorates near the fold as the restoring slope tends to zero. This singular idealized law is not a quantitative real-world melting forecast.

Complete oxidation of $10^{12}\mathrm{kg}$ of carbon adds $n_C=10^{12}/(12\times 10^{-3})=8.33\times 10^{13}\mathrm{mol}$ of carbon dioxide. With atmospheric mass about $5.1\times 10^{18}\mathrm{kg}$ and mean air molar mass $28.97\times 10^{-3}\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1}$, there are about $1.76\times 10^{20}\mathrm{mol}$ of air. The dry-air mole fraction, approximately the volume fraction, changes byIt is the carbon inventory that counts: oxidation adds oxygen to each molecule but does not change the number of carbon-containing molecules. Using $5.15\times 10^{18}\mathrm{kg}$ instead changes the estimate by only about one percent. The atmospheric constants are standard Earth estimates, not additional data read from a missing plot.^[ref]

The term $b_{1}E$ is the effective atmospheric addition from ongoing emissions. Its coefficient $0.25ppmv/GtC$ is about half the full-retention estimate $0.47ppmv/GtC$, so the model implicitly removes or partitions some emitted carbon promptly. The term $-b_0(C-C_0)$ represents slower net uptake of the excess by other reservoirs; its relaxation time is $b_0^{-1}=200$ years. Both terms have units of ppmv per year.

Thousands of years at constant $E_0$ erase the initial transient, givingThe reference $C_0$ is the zero-anthropogenic-emission equilibrium, not the equilibrium when the stated small persistent emissions are present.

Put $A=b_1{E}_0/b_0$. The proposed solution can be writtenIt gives $C(0)-C_0=A$. Differentiation and substitution give, respectively,so it satisfies both the differential equation and the initial condition. At $t=200$ years,with $E(2000)=0.02e^6\simeq 8.069\mathrm{GtC}\mathrm{y}{\mathrm{r}}^{-1}$. The earlier emissions grow on the $g^{-1}=33.3$ year timescale, much faster than the model's 200-year removal time.

Let $t=0$ now denote the year 2000 and $C_{00}=336.95\mathrm{ppmv}$. When emissions stop, the model givesThe concentration peaks at the switch, then returns exponentially to $279\mathrm{ppmv}$; in 2500 it is about $283.76\mathrm{ppmv}$. Before 2000 both scenarios share the exponential-emissions trajectory.

Using the specified logarithmic equilibrium warming relative to $C_0$,gives the fast-adjustment peak estimateAn unspecified finite thermal response time below twenty years delays and reduces that peak relative to the instantaneous-equilibrium value; its exact amount cannot be fixed without a response law and a chosen thermal time. The carbon-removal time is much longer, so the warming subsequently declines gradually in this model. Taking the 1800 concentration $280$ as the temperature reference instead changes the quoted estimate by only $0.016\mathrm{K}$.

Holding emissions at $E_{00}=8.069\mathrm{GtC}\mathrm{y}{\mathrm{r}}^{-1}$ gives equilibrium concentrationFor time $t$ since 2000,It increases monotonically on the same 200-year timescale and reaches about $654.07\mathrm{ppmv}$ in 2500. The limiting, and hence peak asymptotic, warming isIn 2500 the equilibrium warming is about $3.69\mathrm{K}$, still below that asymptote. A stable thermal adjustment much faster than the carbon timescale follows this slow approach with a short lag and has the same limiting warming. These are predictions of the specified linear reservoir model, not a claim that constant real emissions produce a finite universal atmospheric equilibrium.

The real carbon cycle has multiple reservoirs and adjustment times, rather than one universally exponential removal process. Air-sea exchange, ocean mixing and carbonate chemistry give a long-lived atmospheric tail; uptake also changes with temperature, circulation and carbon loading. For example, a warming ocean can absorb less additional carbon dioxide, so a fixed $b_0$ omits a potentially positive climate-carbon feedback. Terrestrial vegetation and soil responses provide further examples. One missing feature is the dependence of carbon uptake on the changing state of the ocean and climate.