Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Optimal energy bounds in spherically symmetric $ \alpha^2$-dynamos


Authors: Ralf Kaiser and Andreas Tilgner
Journal: Quart. Appl. Math. 76 (2018), 437-461
MSC (2010): Primary 49R05; Secondary 76W05, 85A30
DOI: https://doi.org/10.1090/qam/1501
Published electronically: February 21, 2018
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Abstract: In kinematic dynamo theory energy bounds provide necessary conditions for dynamo action valid for every velocity field. When expressed by the magnetic Reynolds number $ R$ this number $ R_E$ may be compared with the critical Reynolds number $ R_c = R_c (\mathbf {v})$ indicating the onset of dynamo action for a given velocity field $ \mathbf {v}$. Typically, there is an (often large) gap between both numbers, which suggests the question: are there better (energy) bounds or are the most critical velocity fields not yet known (or are both conjectures false)?

Here we answer this question in a simplified setting, viz. for spherically symmetric $ \alpha ^2$-mean-field dynamos, where the single scalar field $ \alpha $ takes the role of the velocity field and where spherical symmetry allows the reliable numerical solution of a non-linear variational problem. The non-linear problem arises from the simultaneous variation of magnetic field and $ \alpha $-profile (measured in a suitable norm), which, in fact, yields an improved energy bound $ R_E^{opt} = 4.4717$ compared to the best hitherto known bound $ R_E = 3.0596$. This bound is close to the best hitherto known critical Reynolds number $ R_c = 4.4934$, which belongs to a constant $ \alpha $-profile, and is, moreover, optimal since it is connected to an $ \alpha $-profile whose critical Reynolds number exceeds $ R_E^{opt}$ by less than $ 10^{-4}$.


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Additional Information

Ralf Kaiser
Affiliation: Fakultät für Mathematik und Physik, Universität Bayreuth, D-95440 Bayreuth, Germany
Email: ralf.kaiser@uni-bayreuth.de

Andreas Tilgner
Affiliation: Institut für Geophysik, Universität Göttingen, D-37077 Göttingen, Germany
Email: andreas.tilgner@geo.physik.uni-goettingen.de

DOI: https://doi.org/10.1090/qam/1501
Keywords: Dynamo theory, $\alpha^2$-dynamo, energy bound, variational method
Received by editor(s): July 26, 2017
Published electronically: February 21, 2018
Article copyright: © Copyright 2018 Brown University

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