Optimal energy bounds in spherically symmetric -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 | References | Similar Articles | Additional Information

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 this number may be compared with the critical Reynolds number indicating the onset of dynamo action for a given velocity field . 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 -mean-field dynamos, where the single scalar field 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* -profile (measured in a suitable norm), which, in fact, yields an improved energy bound compared to the best hitherto known bound . This bound is close to the best hitherto known critical Reynolds number , which belongs to a constant -profile, and is, moreover, optimal since it is connected to an -profile whose critical Reynolds number exceeds by less than .

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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