### Abstract

We develop and test spectral Galerkin schemes to solve the coupled Orr-Sommerfeld 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 = 3000 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 ≳ 4 × 10^{4}). 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 quadrature at the 2 p - 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 10^{- 5} for the least stable mode in free-surface flow at Re = 3 × 10^{4}. Moreover, we confirm that the computed normal modes satisfy an energy conservation law for free-surface MHD with error smaller than 10^{- 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 (10^{- 5}) regime of liquid metals.

Original language | English (US) |
---|---|

Pages (from-to) | 1188-1233 |

Number of pages | 46 |

Journal | Journal of Computational Physics |

Volume | 228 |

Issue number | 4 |

DOIs | |

State | Published - Mar 1 2009 |

### Fingerprint

### Keywords

- Eigenvalue problems
- Free-surface MHD
- Hydrodynamic stability
- Orr-Sommerfeld equations
- Spectral Galerkin method

### ASJC Scopus subject areas

- Computer Science Applications
- Physics and Astronomy (miscellaneous)

### Cite this

*Journal of Computational Physics*,

*228*(4), 1188-1233. https://doi.org/10.1016/j.jcp.2008.10.016

**A spectral Galerkin method for the coupled Orr-Sommerfeld and induction equations for free-surface MHD.** / Giannakis, Dimitrios; Fischer, Paul F.; Rosner, Robert.

Research output: Contribution to journal › Article

*Journal of Computational Physics*, vol. 228, no. 4, pp. 1188-1233. https://doi.org/10.1016/j.jcp.2008.10.016

}

TY - JOUR

T1 - A spectral Galerkin method for the coupled Orr-Sommerfeld and induction equations for free-surface MHD

AU - Giannakis, Dimitrios

AU - Fischer, Paul F.

AU - Rosner, Robert

PY - 2009/3/1

Y1 - 2009/3/1

N2 - We develop and test spectral Galerkin schemes to solve the coupled Orr-Sommerfeld 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 = 3000 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 ≳ 4 × 104). 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 quadrature at the 2 p - 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 10- 5 for the least stable mode in free-surface flow at Re = 3 × 104. Moreover, we confirm that the computed normal modes satisfy an energy conservation law for free-surface MHD with error smaller than 10- 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 (10- 5) regime of liquid metals.

AB - We develop and test spectral Galerkin schemes to solve the coupled Orr-Sommerfeld 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 = 3000 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 ≳ 4 × 104). 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 quadrature at the 2 p - 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 10- 5 for the least stable mode in free-surface flow at Re = 3 × 104. Moreover, we confirm that the computed normal modes satisfy an energy conservation law for free-surface MHD with error smaller than 10- 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 (10- 5) regime of liquid metals.

KW - Eigenvalue problems

KW - Free-surface MHD

KW - Hydrodynamic stability

KW - Orr-Sommerfeld equations

KW - Spectral Galerkin method

UR - http://www.scopus.com/inward/record.url?scp=58149170559&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=58149170559&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2008.10.016

DO - 10.1016/j.jcp.2008.10.016

M3 - Article

VL - 228

SP - 1188

EP - 1233

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 4

ER -