Hostname: page-component-7b7567c8c-2rsb6 Total loading time: 0 Render date: 2026-11-12T18:38:23.721Z Has data issue: false hasContentIssue false
  • English
  • Français

Electromagnetically driven zonal flows in a rapidly rotating spherical shell

Published online by Cambridge University Press:  14 May 2013

Rainer Hollerbach,
Xing Wei,
Jérõme Noir and
Andrew Jackson
Rainer Hollerbach*
Affiliation:
Institute of Geophysics, ETH Zürich, Sonneggstrasse 5, 8092 Zürich, SwitzerlandDepartment of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
Xing Wei
Affiliation:
Institute of Geophysics, ETH Zürich, Sonneggstrasse 5, 8092 Zürich, SwitzerlandGeophysics Institute, University of Göttingen, D-37077 Göttingen, Germany
Jérõme Noir
Affiliation:
Institute of Geophysics, ETH Zürich, Sonneggstrasse 5, 8092 Zürich, Switzerland
Andrew Jackson
Affiliation:
Institute of Geophysics, ETH Zürich, Sonneggstrasse 5, 8092 Zürich, Switzerland
*
†Email address for correspondence: rh@maths.leeds.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We consider the flow of an electrically conducting fluid confined in a rotating spherical shell. The flow is driven by a directly imposed electromagnetic body force, created by the combination of an electric current flowing from the inner sphere to a ring-shaped electrode around the equator of the outer sphere and a separately imposed predominantly axial magnetic field. We begin by numerically computing the axisymmetric basic states, which consist of a strong zonal flow. We next compute the linear onset of non-axisymmetric instabilities, and fully three-dimensional solutions up to ten times supercritical. We demonstrate that an experimental liquid-sodium device 50 cm in diameter could achieve and exceed these parameter values.

JFM classification

Boundary Layers: Free shear layersMHD and Electrohydrodynamics: High-Hartmann-number flowsMHD and Electrohydrodynamics: MHD and Electrohydrodynamics

Information

Type
Papers
Information
Journal of Fluid Mechanics , Volume 725 , 25 June 2013, pp. 428 - 445
Copyright
©2013 Cambridge University Press 

References

Boisson, J., Klochko, A., Daviaud, F., Padilla, V. & Aumaître, S. 2012 Travelling waves in a cylindrical magnetohydrodynamically forced flow. Phys. Fluids 24, 044101.Google Scholar
Brito, D., Alboussière, T., Cardin, P., Gagnière, N., Jault, D., La Rizza, P., Masson, J.-P., Nataf, H.-C. & Schmitt, D. 2011 Zonal shear and super-rotation in a magnetized spherical Couette-flow experiment. Phys. Rev. E 83, 066310.CrossRefGoogle Scholar
Duck, P. W. & Foster, M. R. 2001 Spin-up of homogeneous and stratified fluids. Annu. Rev. Fluid Mech. 33, 231–263.Google Scholar
Everett, J. E. & Osemeikhian, J. E. 1966 Spherical coils for uniform magnetic fields. J. Sci. Instrum. 43, 470–474.Google Scholar
Figueroa, A., Demiaux, F., Cuevas, S. & Ramos, E. 2009 Electrically driven vortices in a weak dipolar magnetic field in a shallow electrolytic layer. J. Fluid Mech. 641, 245–261.Google Scholar
Gissinger, C., Ji, H. & Goodman, J. 2011 Instabilities in magnetized spherical Couette flow. Phys. Rev. E 84, 026308.Google Scholar
Hollerbach, R. 1994 Magnetohydrodynamic Ekman and Stewartson layers in a rotating spherical shell. Proc. R. Soc. Lond. A 444, 333–346.Google Scholar
Hollerbach, R. 2000 A spectral solution of the magneto-convection equations in spherical geometry. Intl J. Numer. Meth. Fluids 32, 773–797.Google Scholar
Hollerbach, R. 2003 Instabilities of the Stewartson layer. Part 1. The dependence on the sign of $Ro$ . J. Fluid Mech. 492, 289–302.Google Scholar
Hollerbach, R. 2009 Non-axisymmetric instabilities in magnetic spherical Couette flow. Proc. R. Soc. Lond. A 465, 2003–2013.CrossRefGoogle Scholar
Jones, C. A. 2011 Planetary magnetic fields and fluid dynamos. Annu. Rev. Fluid Mech. 43, 583–614.Google Scholar
Kelley, D. H., Triana, S. A., Zimmerman, D. S., Tilgner, A. & Lathrop, D. P. 2007 Inertial waves driven by differential rotation in a planetary geometry. Geophys. Astrophys. Fluid Dyn. 101, 469–487.Google Scholar
Lathrop, D. P. & Forest, C. B. 2011 Magnetic dynamos in the lab. Phys. Today 64, 40–45.CrossRefGoogle Scholar
Livermore, P. W. & Hollerbach, R. 2012 Successive elimination of shear layers by a hierarchy of constraints in inviscid spherical-shell flows. J. Math. Phys. 53, 073104.CrossRefGoogle Scholar
Messadek, K. & Moreau, R. 2002 An experimental investigation of MHD quasi-two-dimensional turbulent shear flows. J. Fluid Mech. 456, 137–159.Google Scholar
Moresco, P. & Alboussière, T. 2004 Experimental study of the instability of the Hartmann layer. J. Fluid Mech. 504, 167–181.CrossRefGoogle Scholar
Nataf, H.-C., Alboussière, T., Brito, D., Cardin, P., Gagnière, N., Jault, D., Masson, J.-P. & Schmitt, D. 2006 Experimental study of super-rotation in a magnetostrophic spherical Couette flow. Geophys. Astrophys. Fluid Dyn. 100, 281–298.Google Scholar
Nataf, H. C., Alboussière, T., Brito, D., Cardin, P., Gagnière, N., Jault, D. & Schmitt, D. 2008 Rapidly rotating spherical Couette flow in a dipolar magnetic field: an experimental study of the mean axisymmetric flow. Phys. Earth Planet. Inter. 170, 60–72.CrossRefGoogle Scholar
Proudman, I. 1956 The almost-rigid rotation of viscous fluid between concentric spheres. J. Fluid Mech. 1, 505–516.Google Scholar
Rossi, L., Vassilicos, J. C. & Hardalupas, Y. 2006 Electromagnetically controlled multi-scale flows. J. Fluid Mech. 558, 207–242.Google Scholar
Seilmayer, M., Stefani, F., Gundrum, T., Weier, T., Gerbeth, G., Gellert, M. & Rüdiger, G. 2012 Experimental evidence for a transient Tayler instability in a cylindrical liquid-metal column. Phys. Rev. Lett. 108, 244501.Google Scholar
Sisan, D. R., Mujica, N., Tillotson, W. A., Huang, Y. M., Dorland, W., Hassam, A. B., Antonsen, T. M. & Lathrop, D. P. 2004 Experimental observation and characterization of the magnetorotational instability. Phys. Rev. Lett. 93, 114502.Google Scholar
Soward, A. M. & Dormy, E. 2010 Shear-layers in magnetohydrodynamic spherical Couette flow with conducting walls. J. Fluid Mech. 645, 145–185.Google Scholar
Stefani, F., Gailitis, A. & Gerbeth, G. 2008 Magnetohydrodynamic experiments and cosmic magnetic fields. Z. Angew. Math. Mech. 88, 930–954.Google Scholar
Stewartson, K. 1966 On almost rigid rotations. Part 2. J. Fluid Mech. 26, 131–144.Google Scholar
Verhille, G., Plihon, N., Bourgoin, M., Odier, P. & Pinton, J. F. 2010 Laboratory dynamo experiments. Space Sci. Rev. 152, 543–564.Google Scholar
Wei, X. & Hollerbach, R. 2008 Instabilities of Shercliff and Stewartson layers in spherical Couette flow. Phys. Rev. E 78, 026309.Google Scholar