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Eigenvalues of Magnetohydrodynamic Mean-Field Dynamo Models: Bounds and Reliable Computation

Boegli, Sabine; Tretter, Christiane

Eigenvalues of Magnetohydrodynamic Mean-Field Dynamo Models: Bounds and Reliable Computation Thumbnail


Authors

Christiane Tretter



Abstract

This paper provides the first comprehensive study of the linear stability of three important magnetohydrodynamic (MHD) mean-field dynamo models in astrophysics, the spherically symmetric $\alpha^2$-model, the $\alpha^2\omega$-model, and the $\alpha\omega$-model. For each of these highly nonnormal problems, we establish upper bounds for the real part of the spectrum, prove resolvent estimates, and derive thresholds for the helical turbulence function $\alpha$ and the rotational shear function $\omega$ below which no MHD dynamo action can occur for the linear models (antidynamo or bounding theorems). In addition, we prove that interval truncation and finite section method, which are employed to regularize the singular differential expressions and the infinite number of coupled equations, are spectrally exact. This means that all spectral points are approximated and no spectral pollution occurs, thus confirming, for the first time, that numerical eigenvalue approximations for the highly nonnormal MHD dynamo problems are reliable.

Citation

Boegli, S., & Tretter, C. (2020). Eigenvalues of Magnetohydrodynamic Mean-Field Dynamo Models: Bounds and Reliable Computation. SIAM Journal on Applied Mathematics, 80(5), 2194-2225. https://doi.org/10.1137/19m1286359

Journal Article Type Article
Acceptance Date May 20, 2020
Online Publication Date Sep 30, 2020
Publication Date 2020
Deposit Date Nov 4, 2020
Publicly Available Date Nov 4, 2020
Journal SIAM Journal on Applied Mathematics
Print ISSN 0036-1399
Electronic ISSN 1095-712X
Publisher Society for Industrial and Applied Mathematics
Peer Reviewed Peer Reviewed
Volume 80
Issue 5
Pages 2194-2225
DOI https://doi.org/10.1137/19m1286359

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Copyright Statement
© 2020, Society for Industrial and Applied Mathematics.





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