We present a new numerical scheme for solving one-dimensional conduction problems and in the radial direction of a spherically symmetric body or shell with variable spatial domain. This numerical scheme adopts a solution of the conduction equation in each interval of the chosen discretization which is valid if the fluxes at the interval boundaries are constant in time. This ‘piece-wise steady flux’ (PWSF) numerical scheme is continuous and differentiable in the space domain. These smoothness properties are convenient for implementing the numerical scheme in an energy-conserved thermal evolution approach for a terrestrial core in which a conductive thermally stratified layer is considered that develops below the core-mantle boundary when the heat flux drops below the adiabatic heat flux. The influence of a time-variable thermally stratified region on the general evolution of the planetary body is examined, in comparison to imposing an adiabatic temperature profile for the entire core. Also, the dependence of the model's accuracy to the applied grid size of the numerical scheme is studied. We showed that a very low numerical resolution of this approach suffices for obtaining a thermal evolution of a partially conductive planetary core with high accuracy.

By considering thermal stratification in a planetary core where the heat flux is lower than the adiabatic heat flux, radial variations in the cooling rate are accounted for whereas otherwise the distribution of energy in the core is fixed by the imposed adiabat. During the growth of the thermally stratified region, the deep part of the core cools more rapidly than the outer part of the core. Therefore, the inner core grows to a larger size and the temperature and heat flux at the core-mantle boundary are higher and larger, respectively, if a thermally stratified region is considered. For the Earth, the implications are likely very minor and can be neglected in thermal evolution studies that are not specifically interested in the thermally stratified region itself. For Mercury, however, these implications are larger. For example, the age of Mercury's inner core can be underestimated by about a billion of years if thermal stratification is neglected. The consideration of thermal stratification in Mercury's core is also important for the evolution of Mercury's mantle. It increases the mantle temperature, leads to a higher Rayleigh number and therefore a larger heat flux into the lithosphere and prolongation of mantle convection.

我们提出了一种新的数值方案，用于解决具有可变空间域的球对称体或壳的径向方向上的一维传导问题。该数值方案在所选离散化的每个区间中采用传导方程的解，如果区间边界处的通量在时间上是恒定的，则该解是有效的。这种“分段稳定通量”(PWSF) 数值方案在空间域中是连续且可微的。这些平滑特性便于在陆地核心的能量守恒热演化方法中实现数值方案，其中当热通量低于绝热热通量时，在地核-地幔边界以下形成导热分层层。与对整个核心施加绝热温度分布相比，研究了随时间变化的热分层区域对行星体总体演化的影响。此外，还研究了模型精度对数值方案的应用网格大小的依赖性。我们表明，这种方法的非常低的数值分辨率足以以高精度获得部分导电行星核心的热演化。

通过考虑热通量低于绝热热通量的行星核心中的热分层，冷却速率的径向变化被考虑在内，否则核心中的能量分布由施加的绝热固定。在热分层区域的生长过程中，地核深部比地核外部冷却得更快。因此，如果考虑到热分层区域，内核会增长到更大的尺寸，并且地核-地幔边界处的温度和热通量分别更高和更大。对于地球而言，其影响可能非常小，在对热分层区域本身不特别感兴趣的热演化研究中可以忽略不计。然而，对于水星来说，这些影响更大。例如，如果忽略热分层，水星内核的年龄可能会被低估约 10 亿年。考虑水星核心的热分层对于水星地幔的演化也很重要。它增加了地幔温度，导致更高的瑞利数，从而导致更大的进入岩石圈的热通量和地幔对流的延长。