= Solution
Let $t=0$ now denote the year 2000 and $C_{00}=336.95\,\mathrm{ppmv}$. When emissions stop, the model gives
$$
\boxed{C(t)=C_0+(C_{00}-C_0)e^{-t/(200\,\mathrm{yr})}.}
$$
The 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 $C_0$,
$$
\Delta T_{\rm eq}(C)=3\log_2(C/C_0)\,\mathrm K,
$$
gives the fast-adjustment peak estimate
$$
\boxed{\Delta T_{\rm peak}\approx3\log_2(336.95/279)\simeq0.82\,\mathrm K.}
$$
An 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 $0.016\,\mathrm K$.
\Image[/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-carbon-emission-scenarios.png]
{title=The two specified carbon-emission scenarios from 1800 to 2500}
{description=Both curves share the prescribed emissions growth up to 2000. Zero later emissions give exponential return to 279 ppmv; holding emissions constant gives approach to 682.43 ppmv. The inset-free lower panel shows equilibrium warming rather than an invented thermal adjustment history.}
Back to article page
