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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.

A thermally driven fluid layer rotating about the vertical axis experiences a Coriolis force that couples vertical velocity to vertical vorticity. Its onset depends on the Rayleigh number, Prandtl number and Taylor number, as well as thermal and mechanical boundary conditions. The force does no mechanical work, but changes the linear mode balance and nonlinear convection roll competition.

Substitution of zero growth rate into the rotating-convection cubic gives the displayed Rayleigh number. For the first vertical mode, setting $t=k^2$, $m={\pi }^2$, differentiating with respect to $t$ gives $(t+m{)}^2(2t-m)=\mathrm{Ta}m$. At rapid rotation, $k_c\sim (\mathrm{Ta}{\pi }^2/2{)}^{1/6}$ and $R_c\sim 3(\mathrm{Ta}{\pi }^2/2{)}^{2/3}$. Discrete allowed wavenumbers or different plate conditions require a different minimization.

For a fixed-temperature layer with stress-free boundary conditions, a vertical Fourier sine series has $a^2={\lambda }^2+n^2{\pi }^2$, where $\lambda$ is horizontal wavenumber. The coupled vertical velocity and temperature amplitudes satisfy the displayed quadratic for their growth rate $\sigma$. Its roots are real for positive Rayleigh number, while both roots have negative real parts for negative Rayleigh number. Setting $\sigma =0$ gives the free-slip convection neutral curve.

A thermal boundary layer is a thin region in which temperature changes rapidly and thermal conduction remains important. Its thickness can be estimated by balancing thermal diffusion against advection or by requiring a local Rayleigh number to be near its critical value.

Using layer depth $d$ and thermal diffusivity $\kappa$ gives time scale $d^2/\kappa$, velocity scale $\kappa /d$, and perturbation pressure scale ${\rho }_0{\kappa }^2/d^2$. With temperature scale $\Delta T$, the Linearized Boussinesq equations have viscous coefficient Prandtl number $\mathrm{Pr}=\nu /\kappa$ and buoyancy coefficient $\mathrm{Ra}\mathrm{Pr}$, where Rayleigh number $\mathrm{Ra}=g{\alpha }_T\Delta T{d}^3/(\nu \kappa )$.

For the free-slip plane layer, onset selects the smallest Rayleigh number across all admissible stationary and oscillatory normal modes. The first vertical mode wins because increasing vertical index raises viscous cost and reduces oscillatory admissibility. Oscillatory minimization replaces the right-hand side of the stationary wave-number equation by $P^2\mathrm{Ta}{\pi }^2/(1+P{)}^2$. For the oscillatory minimum also require positive angular frequency squared. Where both selected curves are admissible, at rapid rotation the ratio of the minimized thresholds is $R_{o,c}/R_{s,c}\sim 2P^{4/3}/(1+P{)}^{1/3}$. Equality gives $8P^4-P-1=0$, with positive root about $0.67660$; below it sufficiently rapid rotation can select oscillatory primary onset.