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 / c

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.