自由表面MHD耦合Orr-Sommerfeld方程和感应方程的谱Galerkin方法
A spectral Galerkin method for the coupled Orr-Sommerfeld and induction equations for free-surface MHD
摘要：
我们开发和测试谱Galerkin计划来解决耦合Orr-Sommerfeld（OS）和感应方程的并行，不可压缩的MHD在自由表面和固定边界的几何形状。该方案的离散基由Legendre内部形函数组成，并辅以节点形函数，用于弱应力和绝缘边界条件的施加。基多项式的正交性质解决了矩阵系数增长问题，并且特征值-特征函数对可以在谱阶至少为p = 3，000的情况下稳定地计算，并且具有与p无关的舍入误差。准确性反而受到由于在大流体动力学和/或磁雷诺数（Re，Rm > 4 E4）下稳定性算子的非正态性而引起的循环灵敏度的限制。在哈特曼速度和磁场分布的问题，我们采用适当的高斯求积规则来评估相关的指数加权sesquilinear形式没有错误。另一种方法，它涉及近似的形式，通过Legendre-Gauss-Lobatto（LGL）的积分在2 p- 1精度水平，被发现产生相等的特征值内roundenergyerror。作为一致性检查，我们比较模态增长率的能量增长率在非线性模拟和记录的相对差异小于$ 1E-5 $的最不稳定的模式在自由表面流在Re = 3E 4。此外，我们确认，计算的正常模式满足能量守恒定律的自由表面磁流体与误差小于1 E-6。自由表面MHD中的临界雷诺数被发现是敏感的磁普朗特数Pm，即使在Pm = O（1 E-5）制度的液体金属。
英文摘要：
We develop and test spectral Galerkin schemes to solve the coupled Orr-Sommerfeld (OS) and induction equations for parallel, incompressible MHD in free-surface and fixed-boundary geometries. The schemes' discrete bases consist of Legendre internal shape functions, supplemented with nodal shape functions for the weak imposition of the stress and insulating boundary conditions. The orthogonality properties of the basis polynomials solve the matrix-coefficient growth problem, and eigenvalue-eigenfunction pairs can be computed stably at spectral orders at least as large as p = 3,000 with p-independent roundoff error. Accuracy is limited instead by roundoff sensitivity due to non-normality of the stability operators at large hydrodynamic and/or magnetic Reynolds numbers (Re, Rm > 4E4). In problems with Hartmann velocity and magnetic-field profiles we employ suitable Gauss quadrature rules to evaluate the associated exponentially weighted sesquilinear forms without error. An alternative approach, which involves approximating the forms by means of Legendre-Gauss-Lobatto (LGL) quadrature at the 2p - 1 precision level, is found to yield equal eigenvalues within roundoff error. As a consistency check, we compare modal growth rates to energy growth rates in nonlinear simulations and record relative discrepancy smaller than $ 1E-5 $ for the least stable mode in free-surface flow at Re = 3E4. Moreover, we confirm that the computed normal modes satisfy an energy conservation law for free-surface MHD with error smaller than 1E-6. The critical Reynolds number in free-surface MHD is found to be sensitive to the magnetic Prandtl number Pm, even at the Pm = O(1E-5) regime of liquid metals.
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