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 / 2 / 2

Write $s=T_s-273.15\mathrm{K}$ as the numerical sea-level Celsius temperature in degrees Celsius, and use the same degree increments for $\Gamma z$. The model's top temperature is $T_C=s-\Gamma z$. For positive $\alpha$ its intended cold-ice branch has $T_C<0$, so the positive melt-loss magnitude and snow gain areThe model is singular at $z=s/\Gamma$ and is not a physical melting law across that boundary. On the relevant interval $s/\Gamma <z<1/\beta$, $L$ decreases hyperbolically and $A$ decreases linearly. Assume a fixed area and positive ice density, so increasing mass increases $z$: stability has the same sign as $F^{\prime }(z)$.

The equilibrium equation iswith rootsTwo distinct physical equilibria, one stable, exist providedThis combines a positive allowed interval with a strictly positive discriminant. At the larger root $F^{\prime }<0$ and small perturbations restore the ice cap; the smaller root has $F^{\prime }>0$ and is unstable. Sincea restoring local response requiresThis is the lower height bound from the stability derivative; the stable equilibrium itself is $z_{+}$. The minimum initial height for persistence, a different notion of threshold, is the unstable separator $z_{-}$: initial heights above it approach $z_{+}$, while lower heights lose ice and leave the model's cold branch. At equality in the discriminant condition the two equilibria merge, giving a marginal fold rather than a strictly stable ice cap.