Kavli Affiliate: Robert Rosner

| First 5 Authors: Dimitrios Giannakis, Paul F. Fischer, Robert Rosner, ,

| Summary:

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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