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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
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: ajackson@ethz.ch

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 $\unicode[STIX]{x03C0}$.

Type
JFM Papers
Copyright
© 2018 Cambridge University Press 

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References

Alexakis, A. 2011 Searching for the fastest dynamo: laminar ABC flows. Phys. Rev. E 84, 026321.Google ScholarPubMed
Backus, G. 1958 A class of self-sustaining dissipative spherical dynamos. Ann. Phys. 4 (4), 372447.Google Scholar
Bullard, E. C. & Gellman, H. 1954 Homogeneous dynamos and terrestrial magnetism. Phil. Trans. R. Soc. Lond. 247 (928), 213278.Google Scholar
Bullard, E. C. & Gubbins, D. 1977 Generation of magnetic fields by fluid motions of global scale. Geophys. Astrophys. Fluid Dyn. 8 (1), 4356.Google Scholar
Busse, F. 1975 A necessary condition for the geodynamo. J. Geophys. Res. 80, 278280.Google Scholar
Chen, L., Herreman, W. & Jackson, A. 2015 Optimal dynamo action by steady flows confined in a cube. J. Fluid Mech. 783, 2345.Google Scholar
Childress, S.1969 Thèorie magnétohydrodynamique de l’effet dynamo. Technical Report, Department Méchanique de la Faculité des Sciences, Université de Paris.Google Scholar
Christensen, U. R. & Aubert, J. 2006 Scaling properties of convection-driven dynamos in rotating spherical shells and application to planetary magnetic fields. Geophys. J. Intl 166, 97D114.Google Scholar
Dudley, M. L. & James, R. W. 1989 Time-dependent kinematic dynamos with stationary flows. Proc. R. Soc. Lond. A 425, 407429.Google Scholar
Duguet, Y., Monokrousos, A., Brandt, L. & Henningson, D. S. 2013 Minimal transition thresholds in plane Couette flow. Phys. Fluids 25 (8), 084103.Google Scholar
Farrell, B. & Ioannou, P. 1996 Generalized stability theory. Part 1. Autonomous operators. J. Atmos. Sci. 53 (14), 20252040.Google Scholar
Gubbins, D. 1973 Numerical solutions of the kinematic dynamo problem. Phil. Trans. R. Soc. Lond. A 274, 493521.Google Scholar
Gubbins, D. 2008 Implication of kinematic dynamo studies for the geodynamo. Geophys. J. Intl 173 (1), 7991.Google Scholar
Gubbins, D., Barber, C. N., Gibbons, S. & Love, J. J. 2000a Kinematic dynamo action in a sphere. I. Effects of differential rotation and meridional circulation on solutions with axial dipole symmetry. Proc. R. Soc. Lond. A 456, 13331353.Google Scholar
Gubbins, D., Barber, C. N., Gibbons, S. & Love, J. J. 2000b Kinematic dynamo action in a sphere. II. Symmetry selection. Proc. R. Soc. Lond. A 456, 16691683.Google Scholar
Herreman, W. 2016 Minimal flow perturbations that trigger kinematic dynamo in shear flows. J. Fluid Mech. 795, R1.Google Scholar
Holme, R. 1997 Three-dimensional kinematic dynamos with equatorial symmetry: application to the magnetic fields of Uranus and Neptune. Phys. Earth Planet. Inter. 102 (1), 105122.Google Scholar
Holme, R. 2003 Optimised axially-symmetric kinematic dynamos. Phys. Earth Planet. Inter. 140 (1), 311.Google Scholar
Kerswell, R. R., Pringle, C. C. T. & Willis, A. P. 2014 An optimization approach for analysing nonlinear stability with transition to turbulence in fluids as an exemplar. Rep. Progr. Phys. 77 (8), 085901.CrossRefGoogle ScholarPubMed
Khalzov, I. V., Brown, B. P., Cooper, C. M., Weisberg, D. B. & Forest, C. B. 2012 Optimized boundary driven flows for dynamos in a sphere. Phys. Plasmas 19 (11), 112106.Google Scholar
Kumar, S. & Roberts, P. H. 1975 A spectral solution of the magneto-convection equations in spherical geometry. Proc. R. Soc. Lond.  A 344, 235238.Google Scholar
Li, K., Jackson, A. & Livermore, P. W. 2011 Variational data assimilation for the initial value dynamo problem. Phys. Rev. E 84, 056321.Google Scholar
Li, K., Livermore, P. W. & Jackson, A. 2010 An optimal Galerkin scheme to solve the kinematic dynamo eigenvalue problem in a full sphere. J. Comput. Phys. 229, 86668683.Google Scholar
Livermore, P. W.2009 A compendium of Galerkin orthogonal polynomials.http://escholarship.org/uc/item/9vk1c6cm.Google Scholar
Livermore, P. W. 2010 Galerkin orthogonal polynomials. J. Comput. Phys. 229, 20462060.Google Scholar
Livermore, P. W., Hughes, D. W. & Tobias, S. M. 2007 The role of helicity and stretching in forced kinematic dynamos in a spherical shell. Phys. Fluids 19 (5), 057101.Google Scholar
Livermore, P. W. & Ierley, G. 2010 Quasi-L p norm orthogonal Galerkin expansions in sums of Jacobi polynomials. Numer. Algorithms 54 (333), 533569.CrossRefGoogle Scholar
Livermore, P. W. & Jackson, A. 2004 On magnetic energy instability in spherical stationary flows. Proc. R. Soc. Lond. A 460 (2045), 14531476.CrossRefGoogle Scholar
Livermore, P. W. & Jackson, A. 2005 A comparison of numerical schemes to solve the magnetic induction eigenvalue problem in a spherical geometry. Geophys. Astrophys. Fluid Dyn. 99 (6), 467480.Google Scholar
Livermore, P. W. & Jackson, A. 2006 Transient magnetic energy growth in spherical stationary flows. Proc. R. Soc. Lond. A 462 (2072), 24572479.Google Scholar
Love, J. J. & Gubbins, D. 1996a Dynamos driven by poloidal flow exist. Geophys. Res. Lett. 23 (8), 857860.Google Scholar
Love, J. J. & Gubbins, D. 1996b Optimized kinematic dynamos. Geophys. J. Intl. 124 (3), 787800.Google Scholar
Marie, L., Burguete, J., Daviaud, F. & Léorat, J. 2003 Numerical study of homogeneous dynamo based on experimental von Karman type flows. Eur. Phys. J. B 33, 469485.Google Scholar
Moffatt, H. K. 1983 Magnetic Field Generation in Electrically Conducting Fluids. Cambridge University Press.Google Scholar
Namikawa, T. & Matsushita, S. 1970 Kinematic dynamo problem. Geophys. J. R. Astron. Soc. 19 (4), 395415.CrossRefGoogle Scholar
Pekeris, C., Accad, Y. & Shkoller, B. 1973 Kinematic dynamos and the Earth’s magnetic field. Phil. Trans. R. Soc. Lond. A 275 (1251), 425461.Google Scholar
Pringle, C. C. T., Willis, A. P. & Kerswell, R. R. 2012 Minimal seeds for shear flow turbulence: using nonlinear transient growth to touch the edge of chaos. J. Fluid Mech. 702, 415443.CrossRefGoogle Scholar
Proctor, M. R. E. 1977 On Backus’ necessary condition for dynamo action in a conducting sphere. Geophys. Astrophys. Fluid Dyn. 9, 8993.Google Scholar
Proctor, M. R. E. 1979 Necessary conditions for the magnetohydrodynamic dynamo. Geophys. Astrophys. Fluid Dyn. 14 (1), 127145.Google Scholar
Proctor, M. R. E. 2015 Energy requirement for a working dynamo. Geophys. Astrophys. Fluid Dyn. 109 (6), 611614.Google Scholar
Ravelet, F., Chiffaudel, A., Daviaud, F. & Léorat., J. 2005 Towards an experimental von Karman dynamo: numerical studies for an optimized design. Phys. Fluids 17, 117104.Google Scholar
Sadek, M., Alexakis, A. & Fauve, S. 2016 Optimal length scale for a turbulent dynamo. Phys. Rev. Lett. 116, 074501.Google Scholar
Sarson, G. R. 2003 Kinematic dynamos driven by thermal wind flows. Proc. Math. Phys. Engng. Sci. 459 (2033), 12411259.Google Scholar
Stefani, F., Gerbeth, G. & Gailitis, A. 1999 Velocity profile optimization for the Riga dynamo experiment. In Transfer Phenomena in Magnetohydrodynamic and Electroconducting Flows, pp. 3144. Springer.CrossRefGoogle Scholar
Willis, A. P. 2012 Optimization of the magnetic dynamo. Phys. Rev. Lett. 109 (25), 251101.Google Scholar
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