Title: Influence of Boundary Conditions and Heating Modes on the Onset of Columnar Convection in Rotating Spherical Shells Show Chinese title

Title: 边界条件和加热模式对旋转球壳中柱状对流起始的影响

Abstract: We investigate the linear onset of thermal convection in rotating spherical shells with a focus on the influence of mechanical boundary conditions and thermal driving modes. Using a spectral method, we determine critical Rayleigh numbers, azimuthal wavenumbers, and oscillation frequencies over a wide range of Prandtl numbers and shell aspect ratios at moderate Ekman numbers. We show that the preferred boundary condition for convective onset depends systematically on both aspect ratio and Prandtl number: for sufficiently thick shells or for large $\text{Pr}$, the Ekman boundary layer at the outer boundary becomes destabilising, so that no-slip boundaries yield a lower $\text{Ra}_c$ than stress-free boundaries. Comparing differential and internal heating, we find that internal heating generally raises $\text{Ra}_c$, shifts the onset to larger wavenumbers and frequencies, and relocates the critical column away from the tangent cylinder. Mixed boundary conditions with no-slip on the inner boundary behave similarly to purely stress-free boundaries, confirming the dominant influence of the outer surface. These results demonstrate that boundary conditions and heating mechanisms play a central role in controlling the onset of convection and should be carefully considered in models of planetary and stellar interiors. Show Chinese abstract

Abstract: 我们研究了旋转球壳中热对流的线性起始，重点在于机械边界条件和热驱动模式的影响。 使用谱方法，在中等埃克曼数下，我们确定了广泛范围的普朗特数和壳层纵横比下的临界瑞利数、方位波数和振荡频率。 我们表明，对流起始的首选边界条件系统地取决于纵横比和普朗特数：对于足够厚的壳层或较大的$\text{Pr}$，外边界处的埃克曼边界层变得不稳定，因此无滑移边界条件产生的$\text{Ra}_c$比无应力边界条件更低。 比较差异加热和内部加热，我们发现内部加热通常会提高$\text{Ra}_c$，将起始转移到更大的波数和频率，并将临界柱体从切线圆柱体重新定位。 内边界为无滑移的混合边界条件的行为类似于纯无应力边界条件，证实了外表面的主导影响。 这些结果表明，边界条件和加热机制在控制对流起始方面起着核心作用，在行星和恒星内部模型中应仔细考虑。