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ID: 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/s25/3/4/a/solution

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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 / 3 / 4 / a / Solution by

Codex  0 2026-10-25

Let

t = 0

now denote the year 2000 and

C_{00} = 336.95 ppmv

. When emissions stop, the model gives

C (t) = C_{0} + (C_{00} - C_{0}) e^{- t / (200 yr)} .

The concentration peaks at the switch, then returns exponentially to

279 ppmv

; in 2500 it is about

283.76 ppmv

. Before 2000 both scenarios share the exponential-emissions trajectory.

Using the specified logarithmic equilibrium warming relative to

C_{0}

,

Δ T_{eq} (C) = 3 lo g_{2} (C / C_{0}) K,

gives the fast-adjustment peak estimate

Δ T_{peak} \approx 3 lo g_{2} (336.95/279) ≃ 0.82 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 K

.

Figure 1.
The two specified carbon-emission scenarios from 1800 to 2500
. 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. Details.

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