Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 332 / 1

The upward arrows at the surface are reflected short-wave flux $\alpha F_r$, emitted long-wave flux $ϵ\sigma T_s^4$ from the Stefan--Boltzmann law, and, when positive, conductive heat arriving from the planetary ice shell. The downward arrow is the incident solar flux $F_r$. The planetary surface energy balance is thereforeThe relevant planetary albedo is the Bond albedo. In a steady one-dimensional shell, Fourier's law and $k(T)=a/T$ make the upward conductive fluxwhere $z$ points downward and $T(h)=T_m$. Substitution gives the requested implicit equation

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Conservation of energy and Fourier's law giveAt steady state $k{T}_z=F_c$ is constant. Since $k=a/T$,The two temperature boundary conditions therefore giveAt the basal phase boundary, the Stefan condition isThe first term removes latent heat released by freezing, while the oceanic flux $F_w$ supplies heat to the interface. A steady shell has $\dot{h}=0$, hence $F_c=F_w$ andHere $T_s$ must simultaneously satisfy the surface balance from part a with $F_c=F_w$.

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WriteThe quasi-steady model is the coupled pairFor a thin shell, $T_s$ is close to $T_m$. A first-order asymptotic expansion givesand thereforeTo leading order $F_c\sim F_{\ast }$, so an initially freezing shell grows linearly:

For a thick shell the conductive correction in the surface balance is small. The radiative equilibrium temperature and its first correction areThus, with $A=a\log (T_m/T_e)$,When $F_w$ is negligible over an intermediate range, this ordinary differential equation gives the square-root growth lawRetaining $F_w$, its implicit solution isConsequently a sketch of $h(t)$ starts approximately linearly, crosses to square-root growth, and approaches $h_{\infty }$ with exponential decay of $h_{\infty }-h$. A shell placed above $h_{\infty }$ instead thins because the basal oceanic heat flux exceeds the conductive loss.

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Treat the warm ice shell as a very viscous layer and estimate its Rayleigh numberThe temperature-dependent ice viscosity should be evaluated carefully, because deformation is concentrated near the warm base; a useful first estimate uses a representative basal or depth-averaged viscosity. The onset time is obtained by inserting $h(t)$ and $\Delta T(t)=T_m-T_s(t)$ into this expression and finding when $\mathrm{Ra}$ first exceeds the critical value for the shell's mechanical boundary conditions, usually of order $10^3$.

Above onset, mantle convection within the ice transports heat more efficiently than thermal conduction, increases the basal heat loss for a given thickness, and generally limits further thickening. For otherwise similar bodies, a larger planetary radius normally gives larger gravitational acceleration and hence a larger Rayleigh number. Convection is therefore more likely and begins in a thinner or younger shell, although changes in pressure-dependent melting temperature, shell thickness, and viscosity can alter that comparison.

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