Past exam of the short options paper shared by Parts A and B of the physics course of the University of Oxford / 2019 / S25 / 2

$F$ is the imposed radiative forcing; $-\lambda T_s$ is the restoring change in outgoing radiation; and $-\gamma (T_s-T_d)$ transfers heat from the surface layer to the deep ocean, whose equal and opposite gain appears in its own energy balance. The additional $-{\lambda }^{\prime }(T_s-T_d)$ is a radiative adjustment correlated with the surface–deep-ocean temperature contrast, representing a pattern-dependent effective feedback rather than another conserved interlayer heat flux.

For constant forcing, final equilibrium has $T_s=T_d=F/\lambda$. During transient warming $T_d<T_s$, so the radiative response inferred from a recent global energy budget is ${\lambda }_{\mathrm{eff}}=\lambda +{\lambda }^{\prime }(1-T_d/T_s)>\lambda$. If that response were mistaken for the final feedback, the inferred equilibrium climate sensitivity would be too small. Summing the two equations makes the distinction explicit: $d(C_s{T}_s+C_d{T}_d)/dt=F-\lambda T_s-{\lambda }^{\prime }(T_s-T_d)$.

Set ${\lambda }^{\prime }=0$. A decaying normal mode $e^{-kt}$ obeysFor $C_d\gg C_s$, its sum and product of roots giveInitially $T_s=T_d=T_0$, so $a_1+a_2=1$ and $a_1{k}_1+a_2{k}_2=\lambda /C_s$. To leading order,On the slow normal mode, the first equation gives $T_d/T_s=(\lambda +\gamma -C_s{k}_2)/\gamma \simeq (\lambda +\gamma )/\gamma$, verifying proportionality after the fast transient has decayed. The small parameter must also make $k_2/k_1$ small; the supplied positive coefficients and $C_d\gg C_s$ satisfy this.

A linearly increasing radiative forcing produces a fast adjustment over about four years superposed on a slow deep-ocean response over about four hundred years. Once forcing stops increasing at year 70, surface warming continues and approaches the final $3\mathrm{K}$ equilibrium. The right panel of solution gives a model response normalized to the supplied $2\mathrm{K}$ endpoint and $3\mathrm{K}$ equilibrium.

Write $u=t-70$ after stabilization and $3-T_s=A_f{e}^{-u/4}+A_s{e}^{-u/400}$, with $A_f+A_s=1\mathrm{K}$. For a 70-year forcing ramp, the normalized step response is $1-b_f{e}^{-t/4}-b_s{e}^{-t/400}$, $b_f+b_s=1$. The ramp endpoint impliesThis gives $b_s\simeq 0.322$, $A_f\simeq 0.116\mathrm{K}$ and $A_s\simeq 0.884\mathrm{K}$. Thusat $u=10,15,20$ years respectively. The usual order estimate after the fast tail has largely disappeared is $\boxed{{\dot{T}}_s\sim (3-2)/400\simeq 0.0025\mathrm{K}\mathrm{y}{\mathrm{r}}^{-1}}$, with a still visible fast contribution near year 10.

To hold $T_s=T_{\ast }$ fixed, the deep layer obeys $T_s-T_d\propto e^{-\gamma u/C_d}$ and the surface equation requires $F=\lambda T_{\ast }+\gamma (T_{\ast }-T_d)$. Hence $\dot{F}=-{\gamma }^2(T_{\ast }-T_d)/C_d$. Using $b_f\simeq \lambda /(\lambda +\gamma )$ gives $\gamma /\lambda \simeq 0.475$ and $\gamma /C_d\simeq 1/(400b_f)\simeq 0.00369\mathrm{y}{\mathrm{r}}^{-1}$. With $T_{\ast }=2\mathrm{K}$ and initial forcing $F_0=3\lambda$, the initial fractional decline isThe rate declines slowly as the ocean contrast disappears. A coarser estimate treating the post-fast warming as an effective equilibrium change gives $0.0025/3\simeq 0.08\%$ per year; distinguishing the instantaneous surface feedback from the equilibrium feedback explains that difference.

Wording qualification: the printed text says a “1% increase … over 70 years”. The conventional transient climate response experiment uses $1\%$ per year, which nearly doubles concentration in 70 years. A literal total $1\%$ increase would not define the usual equilibrium climate sensitivity or transient climate response. The supplied $3\mathrm{K}$ and $2\mathrm{K}$ are used here as the endpoint normalizations, and a logarithmically uniform concentration growth gives the assumed linear forcing ramp.