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Eigenvalues of magnetohydrodynamic mean-field dynamo models: bounds and reliable computation. (English) Zbl 1452.35214

The authors analyze the linear stability of three magnetohydrodynamic (MHD) mean-field dynamo models used in astrophysics. After some computations, they end with the problem written in spherical coordinates as: \[\partial _{t}\left( \begin{array}{c} S \\ T \end{array} \right) =\left( \begin{array}{cc} \Delta & \alpha \\ -\alpha \Delta -\frac{1}{r}\alpha ^{\prime }\partial _{r}r+\omega ^{\prime }\sin \theta \partial _{\theta } & \Delta \end{array} \right) \left( \begin{array}{c} S \\ T \end{array} \right) -\omega \partial _{\varphi }\left( \begin{array}{c} S \\ T \end{array} \right) ,\] for scalar-valued functions \(S\) and \(T\) which satisfy the normalizations \(\int_{0}^{\pi }\int_{0}^{2\pi }S(r,\theta ,\varphi ;t)\sin \theta d\varphi d\theta =0=\int_{0}^{\pi }\int_{0}^{2\pi }T(r,\theta ,\varphi ;t)\sin \theta d\varphi d\theta \) for every \(r\in \lbrack 0,1]\). Then the authors expand the functions \(S\) and \(T\) in spherical harmonics \( Y_{l}^{m}\): \(S(r,\theta ,\varphi ;t)=\sum_{m=-\infty }^{\infty }\sum_{l=\left\vert m\right\vert }^{\infty }\frac{x_{l,m}(r)}{r} Y_{l}^{m}(\theta ,\varphi )e^{\lambda _{l,m}t}\) and a similar expression for \(T\) with coefficients \(y_{l,m}(r)\) and they prove that \(\lambda _{l,m}\) are independent of \(l\) and that \(y_{l,m}(r)\) and \(x_{l,m}(r)\) satisfy an infinite and coupled system of ordinary equations with coefficients \(\alpha\), \(\alpha ^{\prime }\) and \(\omega ^{\prime }\). They consider three different dynamo models: the \(\alpha ^{2}\)-model, in which the \(\alpha \)-effect is assumed to dominate and the term with \(\omega ^{\prime }\) is neglected, the \( \alpha \omega \)-model, in which the \(\omega \)-effect is assumed to dominate and the terms with \(\alpha \) and \(\alpha ^{\prime }\) are neglected, and the \( \alpha ^{2}\omega \)-model, for which both effects are kept and no term is neglected. In each case, the authors prove antidynamo theorems, as they 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. They establish upper bounds for the real part of the spectrum and they prove resolvent estimates. They use interval truncation and finite section methods to regularize the singular differential expressions and the infinite number of coupled equations, and they prove that these methods are spectrally exact. The paper ends with the presentation of numerical examples.

MSC:

35Q86 PDEs in connection with geophysics
35Q35 PDEs in connection with fluid mechanics
47A10 Spectrum, resolvent
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
76W05 Magnetohydrodynamics and electrohydrodynamics
35P15 Estimates of eigenvalues in context of PDEs
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