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The mean electromotive force generated by elliptic instability

  • K. A. Mizerski (a1), K. Bajer (a2) (a3) and H. K. Moffatt (a4)


The mean electromotive force (EMF) associated with exponentially growing perturbations of an Euler flow with elliptic streamlines in a rotating frame of reference is studied. We are motivated by the possibility of dynamo action triggered by tidal deformation of astrophysical objects such as accretion discs, stars or planets. Ellipticity of the flow models such tidal deformations in the simplest way. Using analytical techniques developed by Lebovitz & Zweibel (Astrophys. J., vol. 609, 2004, pp. 301–312) in the limit of small elliptic (tidal) deformations, we find the EMF associated with each resonant instability described by Mizerski & Bajer (J. Fluid Mech., vol. 632, 2009, pp. 401–430), and for arbitrary ellipticity the EMF associated with unstable horizontal modes. Mixed resonance between unstable hydrodynamic and magnetic modes and resonance between unstable and oscillatory horizontal modes both lead to a non-vanishing mean EMF which grows exponentially in time. The essential conclusion is that interactions between unstable eigenmodes with the same wave-vector can lead to a non-vanishing mean EMF, without any need for viscous or magnetic dissipation. This applies generally (and not only to the elliptic instabilities considered here).


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1. Aldridge, K. D., Lumb, L. I. & Henderson, G. A. 1989 A Poincaré model for the earth’s fluid core. Geophys. Astrophys. Fluid Dyn. 48, 523.
2. Bajer, K. & Mizerski, K. A. 2012 Elliptical flow instability triggered by a magnetic field. Phys. Rev. Lett. (submitted).
3. Bender, C. M. & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill.
4. Braginsky, S. I. 1964a Self excitation of a magnetic field during the motion of a highly conducting fluid. Sov. Phys. JETP 20, 726735.
5. Braginsky, S. I. 1964b Theory of the hydromagnetic dynamo. Sov. Phys. JETP 20, 14621471.
6. Braginsky, S. I. 1975 An almost axially symmetric model of the hydromagnetic dynamo of the earth. Part I. Geomagn. Aeron. 15, 149156.
7. Braginsky, S. I. 1976 On the nearly axially-symmetrical model of the hydromagnetic dynamo of the earth. Phys. Earth Planet. Inter. 11, 191199.
8. Braginsky, S. I. 1978 An almost axially symmetric model of the hydromagnetic dynamo of the earth. Part II. Geomagn. Aeron. 18, 240351.
9. Bushby, P. J. & Proctor, M. R. E. 2010 The influence of -effect fluctuations and the shear-current effect upon the behaviour of solar mean-field dynamo models. Mon. Not. R. Astron. Soc. 409 (4), 16111618.
10. Cambon, C., Benoit, J. P., Shao, L. & Jacquin, L. 1994 Stability analysis and large eddy simulation of rotating turbulence with organized eddies. J. Fluid Mech. 278, 175200.
11. Chandrasekhar, S. 1969 Ellipsoidal Figures of Equilibrium. Yale University Press.
12. Courvoisier, A., Hughes, D. W. & Tobias, S. M. 2006 -effect in a family of chaotic flows. Phys. Rev. Lett. 96, 034503.
13. Craik, A. D. D. & Criminale, W. O. 1986 Evolution of wavelike disturbances in shear flows: a class of exact solutions of the Navier–Stokes equations. Proc. R. Soc. A 406 (1830), 1326.
14. Gilbert, A. 2003 Dynamo theory. In Handbook of Mathematical Fluid Dynamics (ed. Friedlander, S. & Serre, D. ), vol. 2, pp. 355441. Elsevier.
15. Hawley, J. F., Gammie, C. F. & Balbus, S. A. 1995 Local three-dimensional magnetohydrodynamic simulations of accretion disks. Astrophys. J. 440, 742763.
16. Kerswell, R. R. 1993 The instability of precessing flow. Geophys. Astrophys. Fluid Dyn. 72 (1), 107144.
17. Kerswell, R. R. 1994 Tidal excitation of hydromagnetic waves and their damping in the earth. J. Fluid Mech. 274, 219241.
18. Kerswell, R. R. & Malkus, W. V. R. 1998 Tidal instability as the source for Io’s magnetic signature. Geophys. Res. Lett. 25 (5), 603606.
19. Lacaze, L., Le Gal, P. & Le Dizès, S. 2004 Elliptical instability in a rotating spheroid. J. Fluid Mech. 505, 122.
20. Lacaze, L., Le Gal, P. & Le Dizès, S. 2005 Elliptical instability of the flow in a rotating shell. Phys. Earth Planet. Inter. 151 (3/4), 194205.
21. 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 (4/5), 299317.
22. Landman, M. J. & Saffman, P. G. 1987 The three-dimensional instability of strained vortices in a viscous fluid. Phys. Fluids 30 (8), 23392342.
23. Leblanc, S. & Cambon, C. 1997 On the three-dimensional instabilities of plane flows subjected to Coriolis force. Phys. Fluids 9 (5), 13071316.
24. Leblanc, S. 1997 Stability of stagnation points in rotating flows. Phys. Fluids 9 (11), 35663569.
25. Lebovitz, N. R. & Lifschitz, A. 1996 Short wavelength instabilities of Riemann ellipsoids. Phil. Trans. R. Soc. Lond. A 354, 927950.
26. Lebovitz, N. R. & Zweibel, E. 2004 Magnetoelliptic instabilities. Astrophys. J. 609, 301312.
27. Le Gal, P., Lacaze, L. & Le Dizès, S. 2005 Magnetic field induced by elliptical instability in a rotating tidally-distorted sphere. J. Phys. Conf. Ser. 14, 3034.
28. Lesur, G. & Papaloizou, J. C. B. 2009 On the stability of elliptical vortices in accretion discs. Astron. Astrophys. 498, 112.
29. Malkus, W. V. R. 1968 Precession of the earth as the cause of geomagnetism: experiments lend support to the proposal that precessional torques drive the earth’s dynamo. Science 160 (3825), 259264.
30. Malkus, W. V. R. 1989 An experimental study of global instabilities due to the tidal (elliptical) distortion of a rotating elastic cylinder. Geophys. Astrophys. Fluid Dyn. 48 (1), 123134.
31. Mizerski, K. A. & Bajer, K. 2009 The magnetoelliptic instability of rotating systems. J. Fluid Mech. 632 (1), 401430.
32. Mizerski, K. A. & Bajer, K. 2011 The influence of magnetic field on short-wavelength instability of Riemann ellipsoids. Physica D 240, 16291635.
33. Moffatt, H. K. 1970 Dynamo action associated with random inertial waves in a rotating conducting fluid. J. Fluid Mech. 44, 705719, available at
34. Moffatt, H. K. 1974 The mean electromotive force generated by turbulence in the limit of perfect conductivity. J. Fluid Mech. 65, 110, available at
35. Moffatt, H. K. 1976 Generation of magnetic fields by fluid motion. Adv. Appl. Mech. 16, 119181, available at
36. Moffatt, H. K. 1978 Magnetic Field Generation in Electrically Conducting Fluids. Cambridge University Press, available at
37. Moffatt, H. K. 1983 Induction in turbulent conductors. In Stellar and Planetary Magnetism (ed. Soward, A. M. ). pp. 316. Gordon and Breach, available at
38. Noir, J., Brito, D., Aldridge, K. & Cardin, P. 2001 Experimental evidence of inertial waves in a precessing spheroidal cavity. Geophys. Res. Lett. 28 (19), 37853788.
39. Proctor, M. R. E. 2007 Effects of fluctuation on alpha–omega dynamo models. Mon. Not. R. Astron. Soc. 382 (1), L39L42.
40. Rädler, K. H. & Brandenburg, A. 2009 Mean-field effects in the Galloway–Proctor flow. Mon. Not. R. Astron. Soc. 393 (1), 113125.
41. Richardson, K. J. & Proctor, M. R. E. 2010 Effects of -effect fluctuations on simple nonlinear dynamo models. Geophys. Astrophys. Fluid Dyn. 104 (5), 601618.
42. Roberts, G. O. 1972 Dynamo action of fluid motions with two-dimensional periodicity. Phil. Trans. R. Soc. A 271 (1216), 411454.
43. Rüdiger, G. O. & Hollerbach, R. 2004 The Magnetic Universe: Geophysical and Astrophysical Dynamo Theory. Wiley.
44. Seehafer, N. 1995 The turbulent electromotive force in the high-conductivity limit. Astron. Astrophys. 301, 290292.
45. Soward, A. M. 1972 A kinematic theory of large magnetic Reynolds number dynamos. Phil. Trans. R. Soc. A 272 (1227), 431462.
46. Suess, S. T. 1970 Some effects of gravitational tides on a model earth’s core. J. Geophys. Res. 75, 66506661.
47. Tilgner, A. 2005 Precession driven dynamos. Phys. Fluids 17, 034104.
48. Vanyo, J., Wilde, P., Cardin, P. & Olson, P. 1995 Experiments on precessing flows in the Earth’s liquid core. Geophys. J. Intl 121 (1), 136142.
49. Wienbruch, U. & Spohn, T. 1995 A self sustained magnetic field on Io?. Planet. Space Sci. 43 (9), 10451057.
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The mean electromotive force generated by elliptic instability

  • K. A. Mizerski (a1), K. Bajer (a2) (a3) and H. K. Moffatt (a4)


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