Spherical geometry convection in a fluid with an Arrhenius thermal viscosity dependence: The impact of core size and surface temperature on the scaling of stagnant-lid thickness and internal temperature

The rock and rock-ice mixtures of the core-enveloping spherical shells comprising terrestrial body interiors have thermally determined viscosities well described by an Arrhenius dependence. Accordingly, the implied viscosity contrasts determined from the activation energies (E) characterizing such bodies can reach values exceeding ${10}^{40}$, for a temperature range that spans the conditions found from the lower mantle to the surface. In this study, we first explore the impact of implementing a cut-off to limit viscosity magnitude in cold regions. Using a spherical annulus geometry, we investigate the influence of core radius, surface temperature, and convective vigour on stagnant lid formation resulting from the extreme thermally induced viscosity contrasts. We demonstrate that the cut-off viscosity must be increased with decreasing curvature factor, $f$ ($=r_{\text{in}}/r_{\text{out}}$, where $r_{\text{in}}$ and $r_{\text{out}}$ are the inner and outer radii of the annulus, respectively), if the solutions are to be not only computationally manageable but physically valid. We find that for statistically-steady systems, the mean temperature decreases with core size, and that a viscosity contrast of at least ${10}^7$ is required for stagnant lid formation as $f$ decreases below 0.5. Inverting the results from over 80 calculations featuring stagnant lids (from a total of approximately 180 calculations), we apply an energy balance model for heat flow across the thermal boundary layers and find that the non-dimensionalized temperature in the Approximately Isothermal Layer (AIL) in the convecting region under a stagnant lid is well predicted by $T_{\mathrm{AIL}}^{\prime }=\frac{1}{2}\left(-\left(2T_{\text{out}}^{\prime }+\gamma \right)+\sqrt{{\gamma }^2+4\gamma \left(1+T_{\text{out}}^{\prime }\right)}\right)$ where $\gamma$ is a function of E and $f$, and $T_{\text{out}}^{\prime }$ is the non-dimensionalized surface temperature. Moreover, the normalized (i.e., non-dimensional) thickness of the stagnant lid, $L^{\prime }$, can be obtained from a measurement of the non-dimensional surface heat flux once $T_{\mathrm{AIL}}^{\prime }$ is determined. Stagnant-lid thicknesses increase from 10 to 30% of the shell thickness as $f$ is decreased, and thick lids can overlie vigorously convecting underlying layers in small core bodies, potentially delaying secular cooling and suggesting that small objects with small cores may have developed thick elastic outer shells early in the solar system's history while maintaining vigorously convecting interiors. However, we also find that for the small number of 3-D calculations that we examined, parametrizations based on 2-D calculations overestimate the temperature of the convecting layer and the thickness of the conductive lid when $f$ is small (less than 0.4).