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 , 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, (, where and 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 is required for stagnant lid formation as 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 where is a function of E and , and is the non-dimensionalized surface temperature. Moreover, the normalized (i.e., non-dimensional) thickness of the stagnant lid, , can be obtained from a measurement of the non-dimensional surface heat flux once is determined. Stagnant-lid thicknesses increase from 10 to 30% of the shell thickness as 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 is small (less than 0.4).

中文翻译：

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具有阿累尼乌斯热粘度依赖性的流体中的球形几何对流：核心尺寸和表面温度对停滞盖厚度和内部温度缩放的影响

构成地球体内部的包核球壳的岩石和岩冰混合物具有由阿伦尼乌斯依赖性很好地描述的热确定粘度。因此，对于跨越从下地幔到地表的条件的温度范围，由表征此类天体的活化能（E）确定的隐含粘度对比可以达到超过 的值。在这项研究中，我们首先探讨了在寒冷地区实施截止以限制粘度大小的影响。使用球形环空几何结构，我们研究了核心半径、表面温度和对流活力对极端热致粘度对比导致的停滞盖形成的影响。我们证明，如果解决方案不仅在计算上可管理而且在物理上有效，则截止粘度必须随着曲率因子的减小而增加（其中 和 分别是环空的内半径和外半径）。我们发现，对于统计稳定的系统，平均温度随着核心尺寸的增加而降低，并且当降低到 0.5 以下时，形成停滞盖所需的粘度对比度至少为 1。通过对超过 80 次以静止盖为特征的计算（总共约 180 次计算）的结果进行反演，我们应用了跨热边界层热流的能量平衡模型，并发现近似等温层 (AIL) 中的无量纲温度停滞盖下的对流区域中的温度可以通过以下方式很好地预测： 其中 是 E 和 的函数， 是无量纲表面温度。此外，一旦确定了无量纲表面热通量的测量值，就可以获得停滞盖的归一化（即无量纲）厚度。随着壳厚度的减小，停滞盖厚度从壳厚度的 10% 增加到 30%，并且厚盖可以覆盖小核体中剧烈对流的底层，可能延迟长期冷却，并表明具有小核的小物体可能已经形成了厚的弹性外层。太阳系历史早期的壳层，同时保持内部强烈的对流。然而，我们还发现，对于我们检查的少量 3-D 计算，基于 2-D 计算的参数化在较小（小于 0.4）时高估了对流层的温度和导电盖的厚度。