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

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.

Solved by gpt-5.6-sol high.