Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-29T13:51:15.566Z Has data issue: false hasContentIssue false
  • English
  • Français

Magneto-gravity-elliptic instability

Published online by Cambridge University Press:  15 May 2023

Abdelaziz Salhi Open the ORCID record for Abdelaziz Salhi [Opens in a new window]  and
Claude Cambon Open the ORCID record for Claude Cambon [Opens in a new window]
Abdelaziz Salhi*
Affiliation:
Département de Physique, Faculté des Sciences de Tunis, Université Tunis El Manar, 2092 El Manar, Tunis, Tunisia Université de Lyon, Laboratoire de Mécanique des Fluides et d'Acoustique, UMR 5509, Ecole Centrale de Lyon, CNRS, UCBL, INSA, F-69134 Ecully CEDEX, France
Claude Cambon
Affiliation:
Université de Lyon, Laboratoire de Mécanique des Fluides et d'Acoustique, UMR 5509, Ecole Centrale de Lyon, CNRS, UCBL, INSA, F-69134 Ecully CEDEX, France
*
†Email address for correspondence: lazizsalhi@yahoo.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

Magneto-gravity-elliptic instability is addressed here considering an unbounded strained vortex (with constant vorticity $2\varOmega$ and with ellipticity parameter $\varepsilon$) of a perfectly conducting fluid subjected to a uniform axial magnetic field (with Alfvén velocity scaled from the basic magnetic field $B)$ and an axial stratification (with a constant Brunt–Väisälä frequency $N$). Such a simple model allows us to formulate the stability problem as a system of equations for disturbances in terms of Lagrangian Fourier (or Kelvin) modes which is universal for wavelengths of the perturbation sufficiently small with respect to the scale of variation of the basic velocity gradients. It can model localised patches of elliptic streamlines which often appear in some astrophysical flows (stars, planets and accretion discs) that are tidally deformed through gravitational interaction with other bodies. In the limit case where the flow streamlines are exactly circular ($\varepsilon =0),$ there are fast and slow magneto-inertia-gravity waves with frequencies $\omega _{1,2}$ and $\omega _{3,4},$ respectively. Under the effect of finite ellipticity, the resonant cases of these waves, $\omega _i-\omega _j=n\varOmega$ $(i\ne j)$ ($n$ being an integer), can become destabilising. The maximal growth rate of the subharmonic instability (related to the resonance of order $n=2)$ is determined by extending the asymptotic method by Lebovitz & Zweibel (Astrophys. J., vol. 609, 2004, pp. 301–312). The domains of the $(k_0B/\varOmega, N/\varOmega )$ plane for which this instability operates are identified ($1/k_0$ being a characteristic length scale). We demonstrate that the $N\rightarrow 0$ limit is, in fact, singular (discontinuous). The axial stable stratification enhances the subharmonic instability related to the resonance between two slow modes because, at large magnetic field strengths, its maximal growth rate is twice that found in the case without stratification.

JFM classification

Instability: Parametric instability Geophysical and Geological Flows: Stratified flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 963 , 25 May 2023 , A9
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press

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

Aravind, H.M., Dubos, T. & Mathur, M. 2022 Local stability analysis of homogeneous and stratified Kelvin–Helmholtz vortices. J. Fluid Mech. 943, A18.CrossRefGoogle Scholar
Bajer, K. & Mizerski, K.A. 2013 Elliptical flow instability in a conducting fluid triggered by an external magnetic field. Phys. Rev. Lett. 110, 104503.CrossRefGoogle Scholar
Balbus, S.A. & Hawley, J.F. 1991 A powerful local shear instability in weakly magnetized disks. I. Linear analysis. Astrophys. J. 376, 214–222.CrossRefGoogle Scholar
Barker, A.J. & Lithwick, Y. 2013 Non-linear evolution of the tidal elliptical instability in gaseous planets and stars. Mon. Not. R. Astron. Soc. 435 (4), 3614–3626.CrossRefGoogle Scholar
Barker, A.J., Braviner, H.J. & Ogilvie, G.I. 2016 Non-linear tides in a homogeneous rotating planet or star: global modes and elliptical instability. Mon. Not. R. Astron. Soc. 459 (1), 924–938.CrossRefGoogle Scholar
Barker, A.J. 2016 Non-linear tides in a homogeneous rotating planet or star: global simulations of the elliptical instability. Mon. Not. R. Astron. Soc. 459 (1), 939–956.CrossRefGoogle Scholar
Barker, A.J. & Lithwick, Y. 2014 Non-linear evolution of the elliptical instability in the presence of weak magnetic fields. Mon. Not. R. Astron Soc. 437 (1), 305–315.CrossRefGoogle Scholar
Bayly, B.J. 1986 Three-dimensional instability of elliptical flow. Phys. Rev. Lett. 57 (17), 2160–2163.CrossRefGoogle ScholarPubMed
Benkacem, N., Salhi, A., Khlifi, A., Nasraoui, S. & Cambon, C. 2022 Destabilizing resonances of precessing inertia-gravity waves. Phys. Rev. E 105 (3), 035107.CrossRefGoogle ScholarPubMed
Cambon, C. 1982 Etude spectrale d'un champ turbulent incompressible, soumis à des effets couplés de déformation et de rotation, imposés extérieurement. Doctoral dissertation, Université Claude Bernard-Lyon I, Villeurbanne.Google Scholar
Cambon, C., Teissedre, C. & Jeandel, D. 1985 Etude d'effets couplés de déformation et de rotation sur une turbulence homogène. J. Méc. 4 (5), 629–657.Google Scholar
Cébron, D., Le Bars, M., Le Gal, P., Moutou, C., Leconte, J. & Sauret, A. 2013 Elliptical instability in hot Jupiter systems. Icarus 226 (2), 1642–1653.CrossRefGoogle Scholar
Chang, C. & Smith, S.G.L. 2021 Density and surface tension effects on vortex stability. Part 2. Moore–Saffman–Tsai–Widnall instability. J. Fluid Mech. 913, A15.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.Google Scholar
Chagelishvili, G.D., Zahn, J.P., Tevzadze, A.G. & Lominadze, J.G. 2003 On hydrodynamic shear turbulence in Keplerian discs: via transient growth to bypass transition. Astron. Astrophys. 402, 401–407.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
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
Crow, S.C. 1970 Stability theory for a pair of trailing vortices. AIAA J. 8 (12), 2172–2179.CrossRefGoogle Scholar
Davidson, P.A. 2013 Turbulence in Rotating, Stratified and Electrically Conducting Fluids. Cambridge University Press.CrossRefGoogle Scholar
Éloy, C. & Le Dizes, S. 2001 Stability of the Rankine vortex in a multipolar strain field. Phys. Fluids 13 (3), 660–676.CrossRefGoogle Scholar
Feys, J. & Maslowe, S.A. 2016 Elliptical instability of the Moore–Saffman model for a trailing wingtip vortex. J. Fluid Mech. 803, 556–590.CrossRefGoogle Scholar
Fukumoto, Y. 2003 The three-dimensional instability of a strained vortex tube revisited. J. Fluid Mech. 493, 287–318.CrossRefGoogle Scholar
Gledzer, E.B., Dolzhansky, F.V., Obukhov, A.M. & Pononmarev, V.M. 1975 An experimental and theoretical study of the stability of a liquid in an elliptical cylinder. Isv. Atmos. Ocean. Phys. 11, 617–622.Google Scholar
Godeferd, F.S., Cambon, C. & Leblanc, S. 2001 Zonal approach to centrifugal, elliptic and hyperbolic instabilities in Stuart vortices with external rotation. J. Fluid Mech. 449, 1–37.CrossRefGoogle Scholar
Guimbard, D., Le Dizès, S., Le Bars, M., Le Gal, P. & Leblanc, S. 2010 Elliptic instability of a stratified fluid in a rotating cylinder. J. Fluid Mech. 660, 240–257.CrossRefGoogle Scholar
Herreman, W., Cébron, D., Le Dizès, S. & Le Gal, P. 2010 Elliptical instability in rotating cylinders: liquid metal experiments under imposed magnetic field. J. Fluid Mech. 661, 130–158.CrossRefGoogle Scholar
Kelvin, L. 1887 Stability of fluid motion: rectilinear motion of viscous fluid between two parallel plates. Phil. Mag. 24 (5), 188–196.Google Scholar
Kerswell, R.R. 1993 a The instability of precessing flow. Geophys. Astrophys. Fluid Dyn. 72 (1–4), 107–144.CrossRefGoogle Scholar
Kerswell, R.R. 1993 b Elliptical instabilities of stratified, hydromagnetic waves. Geophys. Astrophys. Fluid Dyn. 71 (1–4), 105–143.CrossRefGoogle 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
Kuchment, P.A. 1993 Floquet Theory for Partial Differential Equations, vol. 60. Springer.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. & Zweibel, E. 2004 Magnetoelliptic instabilities. Astrophys. J. 609, 301–312.CrossRefGoogle Scholar
Le Reun, T., Favier, B. & Le Bars, M. 2019 Experimental study of the nonlinear saturation of the elliptical instability: inertial wave turbulence versus geostrophic turbulence. J. Fluid Mech. 879, 296–326.CrossRefGoogle Scholar
Lesur, G. & Papaloizou, J.C.B. 2009 On the stability of elliptical vortices in accretion discs. Astron. Astrophys. 498, 1–12.CrossRefGoogle Scholar
Leweke, T., Le Dizes, S. & Williamson, C.H. 2016 Dynamics and instabilities of vortex pairs. Annu. Rev. Fluid Mech. 48, 507–541.CrossRefGoogle Scholar
Lifschitz, A. & Hameiri, E. 1991 Local stability conditions in fluid dynamics. Phys. Fluids A 3 (11), 2644–2651.CrossRefGoogle Scholar
Lifschitz, A. 1994 On the stability of certain motions of an ideal incompressible fluid. Adv. Appl. Maths 15, 404–436.CrossRefGoogle Scholar
Malkus, W.V.R. 1989 An experimental study of the global instabilities due to the tidal (elliptical) distortion of a rotating elastic cylinder. Geophys. Astrophys. Fluid Dyn. 48, 123–134.CrossRefGoogle Scholar
McKeown, R., Ostilla-Mónico, R., Pumir, A., Brenner, M.P. & Rubinstein, S.M. 2020 Turbulence generation through an iterative cascade of the elliptical instability. Sci. Adv. 6 (9), eaaz2717.CrossRefGoogle ScholarPubMed
Mizerski, K.A. & Bajer, K. 2009 The magnetoelliptic instability of rotating systems. J. Fluid Mech. 632 (1), 401–430.CrossRefGoogle Scholar
Mizerski, K.A. & Lyra, W. 2012 On the connection between the magneto-elliptic and magneto-rotational instabilities. J. Fluid Mech. 698, 358–373.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
Moffatt, H.K. 2010 Note on the suppression of transient shear-flow instability by a spanwise magnetic field. J. Engng Maths 68, 263–268.CrossRefGoogle Scholar
Moore, D.W. & Saffman, P.G. 1975 The instability of a straight vortex filament in a strain field. Proc. R. Soc. Lond. A 346, 413–425.Google Scholar
Nornberg, M.D., Ji, H., Schartman, E., Roach, A. & Goodman, J. 2010 Observation of magnetocoriolis waves in a liquid metal Taylor–Couette experiment. Phys. Rev. Lett. 104 (7), 074501.CrossRefGoogle Scholar
Ogilvie, G.I. 2014 Tidal dissipation in stars and giant planets. Annu. Rev. Astron. Astrophys. 52, 171–210.CrossRefGoogle Scholar
Otheguy, P., Chomaz, J.M. & Billant, P. 2006 Elliptic and zigzag instabilities on co-rotating vertical vortices in a stratified fluid. J. Fluid Mech. 553, 253–272.CrossRefGoogle Scholar
Pedlosky, J. 2013 Geophysical Fluid Dynamics. Springer.Google 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
Sagaut, P. & Cambon, C. 2008 Homogeneous Turbulence Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Schecter, D.A., Boyd, J.F. & Gilman, P.A. 2001 “Shallow-water” magnetohydrodynamic waves in the Solar tachocline. Astrophys. J. 551 (2), L185.CrossRefGoogle Scholar
Salhi, A. & Cambon, C. 1997 An analysis of rotating shear flow using linear theory and DNS and LES results. J. Fluid Mech. 347, 171–195.CrossRefGoogle Scholar
Salhi, A. & Cambon, C. 2009 Precessing rotating flows with additional shear: stability analysis. Phys. Rev. E 79 (3), 036303.CrossRefGoogle ScholarPubMed
Salhi, A., Lehner, T. & Cambon, C. 2010 Magnetohydrodynamic instabilities in rotating and precessing sheared flows: an asymptotic analysis. Phys. Rev. E 82 (1), 016315.CrossRefGoogle ScholarPubMed
Salhi, A., Lehner, T., Godeferd, F. & Cambon, C. 2012 Magnetized stratified rotating shear waves. Phys. Rev. E 85 (2), 026301.CrossRefGoogle ScholarPubMed
Salhi, A., Baklouti, F.S., Godeferd, F., Lehner, T. & Cambon, C. 2017 Energy partition, scale by scale, in magnetic Archimedes Coriolis weak wave turbulence. Phys. Rev. E 95 (2), 023112.CrossRefGoogle ScholarPubMed
Salhi, A., Khlifi, A. & Cambon, C. 2020 Nonlinear effects on the precessional instability in magnetized turbulence. Atmosphere 11 (1), 14.CrossRefGoogle Scholar
Singh, S. & Mathur, M. 2019 Effects of Schmidt number on the short-wavelength instabilities in stratified vortices. J. Fluid Mech. 867, 765–803.CrossRefGoogle Scholar
Sipp, D., Lauga, E. & Jacquin, L. 1999 Vortices in rotating systems: centrifugal, elliptic and hyperbolic type instabilities. Phys. Fluids 11 (12), 3716–3728.CrossRefGoogle Scholar
Sipp, D. & Jacquin, L. 2003 Widnall instabilities in vortex pairs. Phys. Fluids 15, 1861–1874.CrossRefGoogle Scholar
Slane, J. & Tragesser, S. 2011 Analysis of periodic nonautonomous inhomogeneous systems. Nonlinear Dyn. Syst. Theory 11 (2), 183–198.Google Scholar
Suzuki, S., Hirota, M. & Hattori, Y. 2018 Strato-hyperbolic instability: a new mechanism of instability in stably stratified vortices. J. Fluid Mech. 854, 293–323.CrossRefGoogle Scholar
Tsai, C.Y. & Widnall, S.E. 1976 The stability of short waves on a straight vortex filament in a weak externally imposed strain field. J. Fluid Mech. 73 (4), 721–733.CrossRefGoogle Scholar
Waleffe, F. 1989 The 3-D instability of a strained vortex and its relation to turblence. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA.Google Scholar
Waleffe, F. 1990 On the three-dimensional instability of strained vortices. Phys. Fluids A 2 (1), 76–80.CrossRefGoogle Scholar
Waleffe, F. 1992 The nature of triad interactions in homogeneous turbulence. Phys. Fluids A 4 (2), 350–363.CrossRefGoogle Scholar
Wang, Y., Gilson, E.P., Ebrahimi, F., Goodman, J. & Ji, H. 2022 Observation of axisymmetric standard magnetorotational instability in the laboratory. Phys. Rev. Lett. 129 (11), 115001.CrossRefGoogle ScholarPubMed
Wilczyński, F., Hughes, D.W. & Kersalé, E. 2022 Magnetic buoyancy instability and the anelastic approximation: regime of validity and relationship with compressible and Boussinesq descriptions. J. Fluid Mech. 942, A46.CrossRefGoogle Scholar
Zwirner, L., Tilgner, A. & Shishkina, O. 2020 Elliptical instability and multiple-roll flow modes of the large-scale circulation in confined turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 125 (5), 054502.CrossRefGoogle ScholarPubMed