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

Boegli, Sabine; Tretter, Christiane

doi:10.1137/19m1286359

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

Boegli, Sabine; Tretter, Christiane

Authors

Dr Sabine Boegli sabine.boegli@durham.ac.uk
Associate Professor

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