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Stokes drift dynamos

Published online by Cambridge University Press:  19 April 2011

W. HERREMAN  and
P. LESAFFRE
W. HERREMAN*
Affiliation:
LIMSI, CNRS, 91403 Orsay CEDEX, France
P. LESAFFRE
Affiliation:
LERMA/LRA, CNRS/UMR8112, Ecole Normale Supérieure, 24, rue Lhomond, 75231 Paris, CEDEX 05, France
*
†Email address for correspondence: [email protected]
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Abstract

Fluid particles can have a mean motion in time, even when the Eulerian mean flow disappears everywhere in space. In the present article, we demonstrate that this phenomenon, known as the Stokes drift, plays an essential role in the problem of magnetic field generation by fluctuation flows (kinematic dynamo) at high Rm. At leading order, the dynamo is generated by the Stokes drift that acts as if it were a mean flow. This result is derived from a mean-field dynamo theory in terms of time averages, which reveals how classical expressions for alpha and beta tensors actually recombine into a single Stokes drift contribution. In a test case, we find fluctuation flows that have a G. O. Roberts flow as Stokes drift and this allows to confront our model to direct integration of the induction equation. We find an excellent quantitative agreement between the prediction of our model and the results of our simulations. We finally apply our Stokes drift model to prove that a broad class of inertial waves in rapidly rotating flows cannot drive a dynamo.

JFM classification

MHD and Electrohydrodynamics: Dynamo theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 679 , 25 July 2011 , pp. 32 - 57
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Aldridge, K. D. & Toomre, A. 1969 Axisymmetric inertial oscillations of a fluid in a rotating spherical container. J. Fluid Mech. 37, 307–323.CrossRefGoogle Scholar
Backus, G. E. 1958 A class of self-sustaining dissipative spherical dynamos. Ann. Phys. 4, 372–447.CrossRefGoogle Scholar
Braginsky, S. I. 1964 Self excitation of a magnetic field during the motion of a highly conducting fluid. Sov. Phys. JETP 20, 726–735.Google Scholar
Brandenburg, A. 2009 Advances in theory and simulation of large-scale dynamos. Space Sci. Rev. 1–4, 87–104.CrossRefGoogle Scholar
Bryan, G. H. 1889 The waves on a rotating liquid spheroid of finite ellipticity. Phil. Trans. R. Soc. Lond. A180, 187–219.Google Scholar
Bullard, E. & Gellman, H. 1954 Homogeneous dynamos and terrestrial magnetism. R. Soc. Lond. Phil. Trans. Ser. A 247, 213–278.Google Scholar
Busse, F. H. 2002 Convective flows in rapidly rotating sphere and their dynamo action. Phys. Fluids 14, 1301.CrossRefGoogle Scholar
Childress, S. & Gilbert, A. D. 1995 Stretch, Twist, Fold: The Fast Dynamo. Springer.Google Scholar
Christensen, U. R. 2008 Earth science: a sheet-metal geodynamo. Nature 454, 1058–1059.CrossRefGoogle ScholarPubMed
Courvoisier, A., Hughes, D. W. & Tobias, S. M. 2006 Alpha effect in a family of chaotic flows. Phys. Rev. Lett. 96 (3), 034503.CrossRefGoogle Scholar
Falkovich, G. 2009 Could waves mix the ocean. J. Fluid Mech. 638, 1–4.CrossRefGoogle Scholar
Glatzmaier, G. A. & Roberts, P. H. 1995 A three-dimensional self-consistent computer simulation of a geomagnetic field reversal. Nature 377, 203–209.CrossRefGoogle Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.Google Scholar
Kelvin, Lord 1880 Vibrations of a columnar vortex. Phil. Mag. 10, 155–168.Google Scholar
Kerswell, R. R. 2002 Elliptical instability. Annu. Rev. Fluid Mech. 34, 83–113.CrossRefGoogle Scholar
Krause, F. & Rädler, K.-H. 1980 Mean-Field Magnetohydrodynamics and Dynamo. Berlin and Pergamon Press.Google Scholar
Lacaze, L., Herreman, W., Le Bars, M., Le Dizès, S. & Le Gal, P. 2006 Magnetic field induced by elliptical instability in a rotating spheroid. Geophys. Astrophys. Fluid Dyn. 100, 299–317.CrossRefGoogle Scholar
Malkus, W. V. R. 1968 Precession of the Earth as the cause of geomagnetism. Science 160 (3825), 259–264.CrossRefGoogle ScholarPubMed
Moffatt, H. K. 1975 Dynamo action associated with random inertial waves in a rotating conducting fluid. J. Fluid Mech. 44, 705–719.CrossRefGoogle Scholar
Moffatt, H. K. 1978 Magnetic Field Generation in Electrically Conducting Fluids. Cambridge University Press.Google Scholar
Molchanov, S. A., Ruzmaikin, A. A. & Sokolov, D. D. 1985 Kinematic dynamo in random flow. Sov. Phys. Uspekhi 28, 307–327.CrossRefGoogle Scholar
Noir, J., Hemmerlin, F., Wicht, J., Baca, S. M. & Aurnou, J. M. 2009 An experimental and numerical study of librationally driven flow in planetary cores and subsurface oceans. Phys. Earth Planet. Inter. 173 (1–2), 141–152.CrossRefGoogle Scholar
Plunian, F. & Rädler, K.-H. 2002 Harmonic and subharmonic solutions of the roberts dynamo problem. application to the karlsruhe experiment. Magnetohydrodynamics 38 (1–2), 92–103.Google Scholar
Rädler, K.-H. & Brandenburg, A. 2009 Mean-field effects in the galloway-proctor flow. Mon. Not. R. Astron. Soc. 393 (1), 113–125.CrossRefGoogle Scholar
Rädler, K.-H., Rheinhardt, M., Apstein, E. & Fuchs, H. 2002 On the mean-field theory of the karlsruhe dynamo experiment. Part I. kinematic theory. Magnetohydrodynamics 38 (1–2), 39–69.Google Scholar
Roberts, G. O. 1972 Dynamo action of fluid motions with two-dimensional periodicity. Phil. Trans. R. Soc. Lond. A271, 411–454.Google Scholar
Schrinner, M., Rädler, K. H., Schmitt, D., Rheinhardt, M. & Christensen, U. 2007 Mean-field concept and direct numerical simulations of rotating magnetoconvection and the geodynamo. Geophys. Astrophys. Fluid Dyn. 101, 81–116.CrossRefGoogle Scholar
Soward, A. M. 1987 Fast dynamo action in a steady flow. J. Fluid Mech. 180 (1), 267–295.CrossRefGoogle Scholar
Stieglitz, R. & Muller, U. 2001 Experimental demonstration of a homogeneous two-scale dynamo. Phys. Fluids 13, 561–564.CrossRefGoogle Scholar
Stokes, G. G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441–455.Google Scholar
Sur, S., Brandenburg, A. & Subramanian, K. 2008 Kinematic alpha effect in isotropic turbulence simulations. Mon. Not. R. Astron. Soc. 385, L15–L19.CrossRefGoogle Scholar
Tilgner, A. 2005 Precession driven dynamos. Phys. Fluids 17, 034104.CrossRefGoogle Scholar
Wu, C. C. & Roberts, P. H. 2008 A precessionally-driven dynamo in a plane layer. Geophys. Astrophys. Fluid Dyn. 102, 1–19.CrossRefGoogle Scholar
Wu, C. C. & Roberts, P. H. 2009 On a dynamo driven by topographic precession. Geophys. Astrophys. Fluid Dyn. 103 (6), 467–501.CrossRefGoogle Scholar
Zhang, K., Liao, X. & Earnshaw, P. 2004 On inertial waves and oscillations in a rapidly rotating spheroid. J. Fluid Mech. 504, 1–40.CrossRefGoogle Scholar