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The optimal kinematic dynamo driven by steady flows in a sphere

Published online by Cambridge University Press:  25 January 2018

L. Chen Open the ORCID record for L. Chen [Opens in a new window] ,
W. Herreman ,
K. Li ,
P. W. Livermore ,
J. W. Luo  and
A. Jackson Open the ORCID record for A. Jackson [Opens in a new window]
L. Chen
Affiliation:
Institute of Geophysics, ETH Zurich, Sonneggstrasse 5, Zurich 8092, Switzerland
W. Herreman
Affiliation:
LIMSI, CNRS, Université Paris-Sud, Orsay 91405, France
K. Li
Affiliation:
Institute of Geophysics, ETH Zurich, Sonneggstrasse 5, Zurich 8092, Switzerland
P. W. Livermore
Affiliation:
School of Earth and Environment, University of Leeds, Leeds LS2 9JT, UK
J. W. Luo
Affiliation:
Institute of Geophysics, ETH Zurich, Sonneggstrasse 5, Zurich 8092, Switzerland
A. Jackson*
Affiliation:
Institute of Geophysics, ETH Zurich, Sonneggstrasse 5, Zurich 8092, Switzerland
*
†Email address for correspondence: [email protected]
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Abstract

We present a variational optimization method that can identify the most efficient kinematic dynamo in a sphere, where efficiency is based on the value of a magnetic Reynolds number that uses enstrophy to characterize the inductive effects of the fluid flow. In this large-scale optimization, we restrict the flow to be steady and incompressible, and the boundary of the sphere to be no-slip and electrically insulating. We impose these boundary conditions using a Galerkin method in terms of specifically designed vector field bases. We solve iteratively for the flow field and the accompanying magnetic eigenfunction in order to find the minimal critical magnetic Reynolds number $Rm_{c,min}$ for the onset of a dynamo. Although nonlinear, this iteration procedure converges to a single solution and there is no evidence that this is not a global optimum. We find that $Rm_{c,min}=64.45$ is at least three times lower than that of any published example of a spherical kinematic dynamo generated by steady flows, and our optimal dynamo clearly operates above the theoretical lower bounds for dynamo action. The corresponding optimal flow has a spatially localized helical structure in the centre of the sphere, and the dominant components are invariant under rotation by $\unicode[STIX]{x03C0}$.

JFM classification

MHD and Electrohydrodynamics: Dynamo theory Mathematical Foundations: Variational methods
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 839 , 25 March 2018 , pp. 1 - 32
Copyright
© 2018 Cambridge University Press 

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