Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-26T05:46:08.252Z Has data issue: false hasContentIssue false
  • English
  • Français

The magnetoelliptic instability of rotating systems

Published online by Cambridge University Press:  27 July 2009

K. A. MIZERSKI  and
K. BAJER
K. A. MIZERSKI*
Affiliation:
Department of Applied Mathematics, University of Leeds, Woodhouse Lane, Leeds LS2 9JT, UK Institute of Geophysics, University of Warsaw, Pasteura 7, 02-093 Warsaw, Poland
K. BAJER
Affiliation:
Institute of Geophysics, University of Warsaw, Pasteura 7, 02-093 Warsaw, Poland
*
†Email address for correspondence: krzystof.mizerski@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We address the question of stability of the Euler flow with elliptical streamlines in a rotating frame, interacting with uniform external magnetic field perpendicular to the plane of the flow. Our motivation for this study is of astrophysical nature, since many astrophysical objects, such as stars, planets and accretion discs, are tidally deformed through gravitational interaction with other bodies. Therefore, the ellipticity of the flow models the tidal deformations in the simplest way. The joint effect of the magnetic field and the Coriolis force is studied here numerically and analytically in the limit of small elliptical (tidal) deformations (ζ ≪ 1), using the analytical technique developed by Lebovitz & Zweibel (Astrophys. J., vol. 609, 2004, pp. 301–312). We find that the effect of background rotation and external magnetic field is quite complex. Both factors are responsible for new destabilizing resonances as the vortex departs from axial symmetry (ζ ≪ 1); however, just like in the non-rotating case, there are three principal resonances causing instability in the leading order. The presence of the magnetic field is very likely to destabilize the system with respect to perturbations propagating in the direction of the magnetic field if the basic vorticity and the background rotation have opposite signs (i.e. for anticyclonic background rotation). We present the dependence of the growth rates of the modes on various parameters describing the system, such as the strength of the magnetic field (h), the inverse of the Rossby number (ℛv), the ellipticity of the basic flow (ϵ) and the direction of propagation of modes (ϑ). Our analytical predictions agree well with the numerical calculations.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 632 , 10 August 2009 , pp. 401 - 430
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Aldridge, K. D. & Lumb, L. I. 1987 Inertial waves identified in the Earth's fluid outer core. Nature 325, 421–423.CrossRefGoogle Scholar
Bajer, K. & Mizerski, K. A. 2009 Elliptical flow instability triggered by a magnetic field. Phys. Rev. Lett. Submitted.Google Scholar
Balbus, S. A. & Hawley, J. F. 1991 A powerful local shear instability in weakly magnetized disks. Part 1. Linear analysis. Astrophys. J. 376, 214–222.CrossRefGoogle Scholar
Bayly, B. J. 1986 Three-Dimensional instability of elliptical flow. Phys. Rev. Lett. 57 (17), 2160–2163.CrossRefGoogle ScholarPubMed
Braginsky, S. I. 1991 Towards a realistic theory of the geodynamo. Geophys. Astrophys. Fluid Dyn. 60, 89–134.CrossRefGoogle Scholar
Braginsky, S. I. 1999 Dynamics of the stably stratified ocean at the top of the core. Phys. Earth Planet. Inter. 111, 21–34.CrossRefGoogle Scholar
Cadot, O., Douady, S. & Couder, Y. 1995 Characterization of the low pressure filaments in three-dimensional turbulent shear flow. Phys. Fluids 7, 630.CrossRefGoogle Scholar
Chandrasekhar, S. 1969 Ellipsoidal Figures of Equilibrium, Yale University Press.Google Scholar
Craik, A. D. D. 1986 Exact solutions of non-conservative equations for three-wave and second-harmonic resonance. Proc. R. Soc. Lond. A 406, 1–12.Google Scholar
Craik, A. D. D. 1988 A class of exact solutions in viscous incompressible magnetohydrodynamics. Proc. R. Soc. Lond. A 417, 235–244.Google Scholar
Craik, A. D. D. 1989 The stability of unbounded two- and three-dimensional flows subject to body forces: some exact solutions. J. Fluid Mech. 198, 275–292.CrossRefGoogle Scholar
Craik, A. D. D. & Criminale, W. O. 1986 Evolution of wave-like disturbances in shear flows. A class of exact-solutions of the Navier–Stokes equations. Proc. R. Soc. Lond. A 406, 13–26.Google Scholar
Crossley, D. J., Hinderer, J. & Legros, H. 1991 On the excitation, detection and damping of core modes. Phys. Earth Plant. Inter. 68, 97–116.CrossRefGoogle Scholar
Eloy, C h., Le Gal, P. & Le Dizès, S. 2000 Experimental study of the multipolar vortex instability. Phys. Rev. Lett. 85, 3400–3403.CrossRefGoogle ScholarPubMed
Fabre, D. & Jacquin, L. 2004 Short-wave cooperative instabilities in representative aircraft vortices. Phys. Fluids 16, 1366.CrossRefGoogle Scholar
Goodman, J. 1993 The local instability of tidally distorted accretion disks. Astrophys. J. 406, 596–613.CrossRefGoogle Scholar
Hasegawa, H. 2005 Structure of the magnetopause boundary layers discovered by Cluster multipoint observations. In Proceedings of the Cluster and Double Star Symposium: Fifth Anniversary of Cluster in Space, Noordwijk, Denmark.Google Scholar
Hille, E. 1962 Analytic Function Theory, vol. 2. Ginn.CrossRefGoogle Scholar
Ji, H., Burin, M., Schartman, E. & Goodman, J. 2006 Hydrodynamic turbulence cannot transport angular momentum effectively in astrophysical disks. Nature 444, 343–346. Doi:10.1038/nature05323.CrossRefGoogle ScholarPubMed
Kelvin, Lord W. Thomson 1887 Stability of fluid motion: rectilinear motion of viscous fluid between two parallel plates. Philos. Mag. 24, 188–196.Google Scholar
Kerswell, R. R. 1994 Tidal excitation of hydromagnetic waves and their damping in the Earth. J. Fluid Mech. 274, 219–241.CrossRefGoogle Scholar
Kerswell, R. R. 2002 Elliptical instability. Annu. Rev. Fluid Mech. 34, 83–113.CrossRefGoogle Scholar
Kerswell, R. R. & Malkus, W. V. R. 1998 Tidal instability as the source for Io's magnetic signature. Geophys. Res. Lett. 25 (5), 603–606.CrossRefGoogle 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 (4–5), 299–317.CrossRefGoogle Scholar
Landman, M. J. & Saffman, P. G. 1987 The three dimensional instability of strained vortices in a viscous fluid. Phys. Fluids 30 (8), 2339–2342.CrossRefGoogle Scholar
Le Bars, M. & Le Dizès, S. 2006 Thermo-elliptical instability in a rotating cylindrical shell. J. Fluid. Mech. 563, 189–198.CrossRefGoogle Scholar
Lebovitz, N. R. & Lifschitz, A. 1996 Short wavelength instabilities of Riemann ellipsoids. Phil. Trans. R. Soc. Lond. A 354, 927–950.Google Scholar
Lebovitz, N. R. & Saldanha, K. I. 1999 On the weakly nonlinear development of the elliptic instability. Phys. Fluids 11, 3374–3379.CrossRefGoogle Scholar
Lebovitz, N. R. & Zweibel, E. 2004 Magnetoelliptic instabilities. Astrophys. J. 609, 301–312.CrossRefGoogle Scholar
Le Dizès, S. 2003 Modal growth and non-modal growth in a stretched shear layer. Eur. J. Mech. B 22, 411–430.CrossRefGoogle Scholar
Le Dizès, S. & Lacaze, L. 2005 An asymptotic description of vortex Kelvin modes. J. Fluid Mech. 542, 69–96.CrossRefGoogle Scholar
Le Dizès, S. & Laporte, F. 2002 Theoretical predictions for the elliptic instability in a two-vortex flow. J. Fluid Mech. 471, 169.CrossRefGoogle Scholar
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, 30–34.CrossRefGoogle Scholar
Leweke, T. & Williamson, C. H. K. 1998 Cooperative elliptic instability of a vortex pair. J. Fluid Mech. 360, 85.CrossRefGoogle Scholar
Melchior, P. W., Crossley, D. J., Dehant, V. P. & Ducarme, B. 1988 Have inertial waves been identified from the Earth's core?. In Structure and Dynamics of the Earth's Deep Interior (ed. Smylie, D. E. & Hide, R.), pp. 1–12. American Geophysical Union.Google Scholar
Melchior, P. W. & Ducarme, B. 1986 Detection of inertial gravity oscillations in the Earth's core with a superconducting gravimeter at Brussels. Phys. Earth Planet. Inter. 42, 129.CrossRefGoogle Scholar
Meunier, P., Le Dizès, S. & Leweke, T. 2005 Physics of vortex merging. C. R. Phys. 6, 431–450.CrossRefGoogle Scholar
Meunier, P. & Leweke, T. 2001 Three-dimensional instability during vortex merging. Phys. Fluids 13, 2747.CrossRefGoogle Scholar
Miyazaki, T. 1993 Elliptical instability in a stably stratified rotating fluid. Phys. Fluids A 5 (11), 2702–2709.CrossRefGoogle Scholar
Miyazaki, T. & Fukumoto, Y. 1992 Three-dimensional instability of strained vortices in a stably stratified fluid. Phys. Fluids A 4 (11), 2515–2522.CrossRefGoogle Scholar
Mizerski, K. A. & Bajer, K. 2007 The magnetoelliptic instability in the presence of inetrtial forces. In Advances in Turbulence XI, Springer Proceedings in Physics, vol. 117, pp. 121–123. Springer. Also in Proceedings of the 11th EUROMECH European Turbulence Conference, Porto, Portugal.CrossRefGoogle Scholar
Mizerski, K. A., Bajer, K. & Moffatt, H. K. 2009 The α-effect generated by elliptical instability. J. Fluid Mech. Submitted.Google Scholar
Mozer, F. S., Phan, T. D. & Bale, S. D. 2003 The complex structure of the reconnecting magnetopause. Phys. Plasmas 10 (6), 2480–2485.CrossRefGoogle Scholar
Noir, J., Brito, D., Aldridge, K. & Cardin, P. 2001 Experimental evidence of inertial waves in a precessing spheroidal cavity. Geophys. Res. Lett. 19, 3785–3788.CrossRefGoogle Scholar
Orszag, S. A. & Patera, A. T. 1983 Secondary instability of wall-bounded shear flows. J. Fluid Mech. 128, 347–385.CrossRefGoogle Scholar
Pierrehumbert, R. T. 1986 Universal short-wave instability of two-dimensional eddies in an inviscid fluid. Phys. Rev. Lett. 57 (17), 2157–2159.CrossRefGoogle Scholar
Roberts, P. H. & Soward, A. M. 1992 Dynamo theory. Annu. Rev. Fluid Mech. 24, 459–512.CrossRefGoogle Scholar
Roth, M., De Keyser, J., Darrouzet, F. & Čadež, V. 2001 Structure and dynamics of the Earth's magnetopause. In Space Scientific Research in Belgium vol. 2, part 2, pp. 81, Federal Office for Scientific, Technical and Cultural Affairs (Brussels).Google Scholar
Sipp, D. & Jacquin, L. 2003 Widnall instabilities in vortex pairs. Phys. Fluids 15, 1861.CrossRefGoogle Scholar
Soward, A. M. 1991 The Earth's dynamo. Geophys. Astrophys. Fluid Dyn. 62, 191–209.CrossRefGoogle Scholar
Suess, S. T. 1970 Some effects of gravitational tides on a model Earth's core. J. Geophys. Res. 75, 6650–6661.CrossRefGoogle Scholar
Thess, A. & Zikanov, O. 2007 Transition from two-dimensional to three-dimensional magnetohydrodynamic turbulence. J. Fluid Mech. 579, 383–412.CrossRefGoogle Scholar
Tilgner, A. 2005 Precession driven dynamos. Phys. Fluids 17, 034104.CrossRefGoogle Scholar
Vanyo, J., Wilde, P., Cardin, P. & Olson, P. 1995 Experiments on precessing flows in the Earth's liquid core. Geophys. J. Intl 121, 136–142.CrossRefGoogle Scholar
Waleffe, F. 1990 On the three-dimensional instability of strained vortices. Phys. Fluids A 2 (1), 76–80.CrossRefGoogle Scholar
Wienbruch, U. & Spohn, T. 1995 A self-sustained magnetic field on Io?. Planet. Space. Sci. 43, 1045–1057.CrossRefGoogle Scholar