Because rocks and ices viscosities strongly depend on temperature, planetary mantles and ice shells are often thought to be animated by stagnant-lid convection. Their dynamics is further impacted by the release of internal heat, either through radioactive isotopes decay or tidal dissipation. Here, we quantify the impact of internal heating on stagnant-lid convection. We perform numerical simulations of convection combining strongly temperature-dependent viscosity and mixed (basal and internal) heating in 3D-Cartesian and spherical geometries, and use these simulations to build scaling laws relating surface heat flux, Φ_surf, interior temperature, T_m, and stagnant lid thickness, d_lid, to the Rayleigh number, heating rate, H, and top-to-bottom viscosity ratio, Δη. These relationships show that T_m increases with H but decreases with Δη, while Φ_surf increases with H and Δη. Importantly, they also describe heterogeneously heated systems well, provided that the maximum dissipation occurs in hottest regions. For H larger than a value H_crit, the bottom heat flux turns negative and the system cools down both at its top and bottom. Two additional interesting results are that (a) while the rigid lid stiffens (its mobility decreases) with increasing H, it also thins; and (b) H_crit increases with increasing Δη. We then use our scaling laws to assess the impact of tidal heating on Europa's ice shell properties and evolution. Our calculations suggest a shell thickness in the range 20–80 km, depending on viscosity and dissipated power, and show that internal heating has a stronger influence than the presence of impurities in the sub-surface ocean.

中文翻译：

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混合加热静止盖对流的缩放定律及其在木卫二中的应用

由于岩石和冰的粘度强烈依赖于温度，因此行星地幔和冰壳通常被认为是由停滞盖对流激活的。它们的动力学进一步受到内部热量释放的影响，无论是通过放射性同位素衰变还是潮汐消散。在这里，我们量化了内部加热对停滞盖对流的影响。我们在 3D 笛卡尔几何和球面几何中结合强烈依赖温度的粘度和混合（基础和内部）加热对对流进行数值模拟，并使用这些模拟来建立与表面热通量、Φ _surf、内部温度、T _m相关的比例定律，和停滞盖厚度，d_盖，瑞利数，加热速率，H和上下粘度比 Δ η。这些关系表明T _m随H增加但随 Δ η减小，而 Φ _surf随H和 Δ η增加。重要的是，他们还很好地描述了异质加热系统，前提是最大耗散发生在最热的区域。对于ħ比某值大ħ_爆击，底部热在其顶部和底部磁通匝负和系统冷却两者。另外两个有趣的结果是 (a) 虽然刚性盖子会随着H 的增加而变硬（其流动性降低），它也变薄了；(b) H _crit随着 Δ η 的增加而增加。然后，我们使用缩放定律来评估潮汐加热对木卫二冰壳特性和演化的影响。我们的计算表明壳层厚度在 20-80 公里范围内，具体取决于粘度和耗散功率，并表明内部加热的影响比次表层海洋中杂质的存在更大。