Livermore Philip W, Bailey Lewis M, Hollerbach Rainer
School of Earth and Environment, University of Leeds, Leeds, LS2 9JT, UK.
School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK.
Sci Rep. 2016 Mar 16;6:22812. doi: 10.1038/srep22812.
We investigate how the choice of either no-slip or stress-free boundary conditions affects numerical models of rapidly rotating flow in Earth's core by computing solutions of the weakly-viscous magnetostrophic equations within a spherical shell, driven by a prescribed body force. For non-axisymmetric solutions, we show that models with either choice of boundary condition have thin boundary layers of depth E(1/2), where E is the Ekman number, and a free-stream flow that converges to the formally inviscid solution. At Earth-like values of viscosity, the boundary layer thickness is approximately 1 m, for either choice of condition. In contrast, the axisymmetric flows depend crucially on the choice of boundary condition, in both their structure and magnitude (either E(-1/2) or E(-1)). These very large zonal flows arise from requiring viscosity to balance residual axisymmetric torques. We demonstrate that switching the mechanical boundary conditions can cause a distinct change of structure of the flow, including a sign-change close to the equator, even at asymptotically low viscosity. Thus implementation of stress-free boundary conditions, compared with no-slip conditions, may yield qualitatively different dynamics in weakly-viscous magnetostrophic models of Earth's core. We further show that convergence of the free-stream flow to its asymptotic structure requires E ≤ 10(-5).
我们通过在球壳内计算由规定体力驱动的弱粘性磁准地转方程的解，研究无滑移或无应力边界条件的选择如何影响地核中快速旋转流的数值模型。对于非轴对称解，我们表明，无论选择哪种边界条件，模型都有深度为E(1/2)的薄边界层，其中E是埃克曼数，以及收敛到形式上无粘性解的自由流。在类似地球的粘度值下，无论选择哪种条件，边界层厚度约为1米。相比之下，轴对称流在结构和量级（E(-1/2)或E(-1)）上都关键取决于边界条件的选择。这些非常大的纬向流是由于要求粘度平衡残余轴对称扭矩而产生的。我们证明，即使在渐近低粘度下，切换机械边界条件也会导致流结构的明显变化，包括在赤道附近的符号变化。因此，与无滑移条件相比，无应力边界条件的实施可能在弱粘性磁准地转地球核心模型中产生定性不同的动力学。我们进一步表明，自由流收敛到其渐近结构需要E≤10(-5)。
