A spectral Galerkin method for the coupled Orr-Sommerfeld and induction equations for free-surface MHD

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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