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 / 2

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.