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

Boegli, Sabine; Tretter, Christiane

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
• Public URL: https://durham-repository.worktribe.com/output/1251994