Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2017 / S25 / 3

The second box represents a reservoir exchanging carbon with the atmosphere, with equal capacity in the chosen concentration units. For a unit atmospheric pulse, $C(0^{+})=1$, $C_1(0^{+})=0$ and no subsequent emissions, total $C+C_1=1$ is conserved while $d(C-C_1)/dt=-2k(C-C_1)$. ThusThe atmospheric burden falls from one to one half with time constant $(2k{)}^{-1}$, not $k^{-1}$. The model has a permanent airborne fraction because neither box contains an external sink. The carbon-response panel of the original figure in S12 shows this explicitly.

For a small additional carbon mass, logarithmic forcing has derivative inversely proportional to the background atmospheric carbon inventory. With other conversion factors fixed, $b_0$ therefore decreases as background concentration rises. Solve the temperature equation using an integrating factor and the pulse burden:At $b_1=2k$ the second term is its continuous limit $t{e}^{-b_{1}t}$. For $0<k\ll b_1$, temperature initially rises over $b_1^{-1}$ toward roughly $b_0/b_1$, then relaxes over $(2k{)}^{-1}$ toward half that value. This produces an overshoot, rather than permanent growth. The two numerical limits requested separately below also follow directly from the differential equation.

Temperature is continuous through an instantaneous carbon injection. With $T(0)=0$ and $C(0^{+})=1$, $\boxed{\dot{T}(0^{+})=b_0}$. Its initial slope is finite; the emission pulse makes the carbon burden jump but does not give this first-order thermal model an instantaneous temperature jump.

At late times $C\to 1/2$ and thermal equilibration requires $b_{0}C-b_{1}T=0$. Hence $\boxed{T(\infty )=b_0/(2b_1)}$. Both $t\gg k^{-1}$ and $t\gg b_1^{-1}$ ensure that the exchange and thermal transients have decayed.

For constant marginal annual damage $D^{\prime }$ and positive discount rate $r$, use the Laplace transform of the preceding equations:The approximation uses $r\ll b_1$ but does not require choosing an order between $r$ and $k$. At zero discount rate the integral diverges in this model because the pulse leaves permanent positive warming and damages remain constant per degree.

If economic activity is adapted to the old climate, additional warming generally pushes temperatures, water availability and extremes further from that adapted range. With convex aggregate damages, marginal annual damage per degree $D^{\prime }$ therefore tends to rise as the world warms. This is an argued expectation under the stated adaptation baseline, not a theorem: adaptation, wealth changes, regional benefits and thresholds alter both the level and curvature of damages. A constant $D^{\prime }$ in the preceding integration is a simplifying local approximation, and may understate later losses when convexity dominates.