Skip to main content Accessibility help
×
Home

Local instabilities in magnetized rotational flows: a short-wavelength approach

  • O. N. Kirillov (a1), F. Stefani (a1) and Y. Fukumoto (a2)

Abstract

We perform a local stability analysis of rotational flows in the presence of a constant vertical magnetic field and an azimuthal magnetic field with a general radial dependence. Employing the short-wavelength approximation we develop a unified framework for the investigation of the standard, helical and azimuthal version of the magnetorotational instability (MRI), as well as of current-driven kink-type instabilities. Considering the viscous and resistive setup, our main focus is on the case of small magnetic Prandtl numbers which applies e.g. to liquid-metal experiments but also to the colder parts of accretion disks. We show that the inductionless versions of MRI that were previously thought to be restricted to comparatively steep rotation profiles extend well to the Keplerian case if only the azimuthal field slightly deviates from its current-free (in the fluid) profile. We find an explicit criterion separating the pure azimuthal inductionless MRI from the regime where this instability is mixed with the Tayler instability. We further demonstrate that for particular parameter configurations the azimuthal MRI originates as a result of a dissipation-induced instability of Chandrasekhar’s equipartition solution of ideal magnetohydrodynamics.

Copyright

Corresponding author

Email address for correspondence: o.kirillov@hzdr.de

References

Hide All
Acheson, D. J. 1978 On the instability of toroidal magnetic fields and differential rotation in stars. Phil. Trans. R. Soc. Lond. A 289 (1363), 459500.
Acheson, D. J. & Hide, R. 1973 Hydromagnetics of rotating fluids. Rep. Prog. Phys. 36, 159221.
Altmeyer, S., Hoffmann, C. & Lücke, M. 2011 Islands of instability for growth of spiral vortices in the Taylor–Couette system with and without axial through flow. Phys. Rev. E 84, 046308.
Armitage, P. J. 2011 Protoplanetary disks and their evolution. Annu. Rev. Astron. Astrophys. 49, 67117.
Balbus, S. A. & Hawley, J. F. 1991 A powerful local shear instability in weakly magnetized disks 1. Linear analysis. Astrophys. J. 376, 214222.
Balbus, S. A. & Hawley, J. F. 1992 A powerful local shear instability in weakly magnetized disks 4. Nonaxisymmetric perturbations. Astrophys. J. 400, 610621.
Bayly, B. J. 1988 Three-dimensional centrifugal-type instabilities in inviscid two-dimensional flows. Phys. Fluids 31, 5664.
Bilharz, H. 1944 Bemerkung zu einem Satze von Hurwitz. Z. Angew. Math. Mech. 24, 7782.
Bogoyavlenskij, O. I. 2004 Unsteady equipartition MHD solutions. J. Math. Phys. 45, 381390.
Brandenburg, A., Nordlund, A., Stein, R. F. & Torkelsson, U. 1995 Dynamo-generated turbulence and large-scale magnetic fields in a Keplerian shear flow. Astrophys. J. 446, 741754.
Chandrasekhar, S. 1956 On the stability of the simplest solution of the equations of hydromagnetics. Proc. Natl Acad. Sci. USA 42, 273276.
Chandrasekhar, S. 1960 The stability of non-dissipative Couette flow in hydromagnetics. Proc. Natl Acad. Sci. USA 46, 253257.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
Dobrokhotov, S. & Shafarevich, A. 1992 Parametrix and the asymptotics of localized solutions of the Navier–Stokes equations in $R^{3}$ , linearized on a smooth flow. Math. Notes 51, 4754.
Done, C., Gierlinski, M. & Kubota, A. 2007 Modelling the behaviour of accretion flows in X-ray binaries. Astron. Astrophys. Rev. 15, 166.
Eckhardt, B. & Yao, D. 1995 Local stability analysis along Lagrangian paths. Chaos, Solitons Fractals 5 (11), 20732088.
Eckhoff, K. S. 1981 On stability for symmetric hyperbolic systems, I. J. Differ. Equ. 40, 94115.
Eckhoff, K. S. 1987 Linear waves and stability in ideal magnetohydrodynamics. Phys. Fluids 30, 36733685.
Fleming, T. P., Stone, J. M. & Hawley, J. F. 2010 The effect of resistivity on the nonlinear stage of the magnetorotational instability in accretion disks. Astrophys. J. 530, 464477.
Friedlander, S. & Lipton-Lifschitz, A. 2003 Localized instabilities in fluids. In Handbook of Mathematical Fluid Dynamics, vol. II (ed. Friedlander, S. J. & Serre, D.), pp. 289353. Elsevier.
Friedlander, S. & Vishik, M. M. 1995 On stability and instability criteria for magnetohydrodynamics. Chaos 5, 416423.
Fromang, S. & Papaloizou, J. 2007 MHD simulations of the magnetorotational instability in a shearing box with zero net flux. Astron. Astrophys. 476, 11131122.
Fuchs, H., Rädler, K.-H. & Rheinhardt, M. 1999 On self-killing and self-creating dynamos. Astron. Nachr. 320, 127131.
Gailitis, A., Lielausis, O., Dement’ev, S., Platacis, E., Cifersons, A., Gerbeth, G., Gundrum, T., Stefani, F., Christen, M., Hänel, H. & Will, G. 2000 Detection of a flow induced magnetic field eigenmode in the Riga dynamo facility. Phys. Rev. Lett. 84, 43654368.
Goedbloed, H., Keppens, R. & Poedts, S. 2010 Advanced Magnetohydrodynamics. Cambridge University Press.
Golovin, S. V. & Krutikov, M. K. 2012 Complete classification of stationary flows with constant total pressure of ideal incompressible infinitely conducting fluid. J. Phys. A: Math. Theor. 45, 235501.
Hattori, Y. & Fukumoto, Y. 2003 Short-wavelength stability analysis of thin vortex rings. Phys. Fluids 15 (10), 31513163.
Herault, J., Rincon, F., Cossu, C., Lesur, G., Ogilvie, G. I. & Longaretti, P.-Y. 2011 Periodic magnetorotational dynamo action as a prototype of nonlinear magnetic-field generation in shear flows. Phys. Rev. E 84 (10), 036321.
Hollerbach, R. & Rüdiger, G. 2005 New type of magnetorotational instability in cylindrical Taylor–Couette flow. Phys. Rev. Lett. 95 (12), 124501.
Hollerbach, R., Teeluck, V. & Rüdiger, G. 2010 Nonaxisymmetric magnetorotational instabilities in cylindrical Taylor–Couette flow. Phys. Rev. Lett. 104, 044502.
Howard, L. N. & Gupta, A. S. 1962 On the hydrodynamic and hydromagnetic stability of swirling flows. J. Fluid Mech. 14, 463476.
Ji, H. & Balbus, S. 2013 Angular momentum transport in astrophysics and in the lab. Phys. Today 66 (8), 2733.
Ji, H., Goodman, J. & Kageyama, A. 2001 Magnetorotational instability in a rotating liquid metal annulus. Mon. Not. R. Astron. Soc. 325 (2), L1L5.
Kagan, D. & Wheeler, J. C. 2014 The role of the magnetorotational instability in the Sun. Astrophys. J. 787 (1), 21.
Käpylä, P. J. & Korpi, M. J. 2011 Magnetorotational instability driven dynamos at low magnetic Prandtl numbers. Mon. Not. R. Astron. Soc. 413, 901907.
Kirillov, O. N. 2009 Campbell diagrams of weakly anisotropic flexible rotors. Proc. R. Soc. Lond. A 465, 27032723.
Kirillov, O. N. 2013 Nonconservative Stability Problems of Modern Physics, De Gruyter Studies in Mathematical Physics, vol. 14. De Gruyter.
Kirillov, O. N. & Stefani, F. 2010 On the relation of standard and helical magnetorotational instability. Astrophys. J. 712, 5268.
Kirillov, O. N. & Stefani, F. 2011 Paradoxes of magnetorotational instability and their geometrical resolution. Phys. Rev. E 84 (3), 036304.
Kirillov, O. N. & Stefani, F. 2012 Standard and helical magnetorotational instability: How singularities create paradoxal phenomena in MHD. Acta Appl. Math. 120, 177198.
Kirillov, O. N. & Stefani, F. 2013 Extending the range of the inductionless magnetorotational instability. Phys. Rev. Lett. 111, 061103.
Kirillov, O. N., Stefani, F. & Fukumoto, Y. 2012 A unifying picture of helical and azimuthal MRI, and the universal significance of the Liu limit. Astrophys. J. 756, 83.
Kirillov, O. N., Stefani, F. & Fukumoto, Y. 2014 Instabilities of rotational flows in azimuthal magnetic fields of arbitrary radial dependence. Fluid Dyn. Res. 46, 031403.
Kirillov, O. N. & Verhulst, F. 2010 Paradoxes of dissipation-induced destabilization or who opened Whitney’s umbrella? Z. Angew. Math. Mech. 90 (6), 462488.
Krolik, J. H. 1998 Active Galactic Nuclei. Princeton University Press.
Krueger, E. R., Gross, A. & Di Prima, R. C. 1966 On relative importance of Taylor-vortex and non-axisymmetric modes in flow between rotating cylinders. J. Fluid Mech. 24 (3), 521538.
Landman, M. J. & Saffman, P. G. 1987 The three-dimensional instability of strained vortices in a viscous fluid. Phys. Fluids 30, 23392342.
Lesur, G. & Longaretti, P.-Y. 2007 Impact of dimensionless numbers on the efficiency of magnetorotational instability induced turbulent transport. Mon. Not. R. Astron. Soc. 378 (8), 14711480.
Lifschitz, A. 1989 Magnetohydrodynamics and Spectral Theory. Kluwer.
Lifschitz, A. 1991 Short wavelength instabilities of incompressible three-dimensional flows and generation of vorticity. Phys. Lett. A 157, 481486.
Lifschitz, A. & Hameiri, E. 1991 Local stability conditions in fluid dynamics. Phys. Fluids A 3, 26442651.
Liu, W., Goodman, J., Herron, I. & Ji, H. 2006 Helical magnetorotational instability in magnetized Taylor–Couette flow. Phys. Rev. E 74 (5), 056302.
Marcus, P. S., Pei, S., Jiang, C.-H. & Hassanzadeh, P. 2013 Three-dimensional vortices generated by self-replication in stably stratified rotating shear flows. Phys. Rev. Lett. 111 (8), 084501.
Michael, D. H. 1954 The stability of an incompressible electrically conducting fluid rotating about an axis when current flows parallel to the axis. Mathematika 1, 4550.
Monchaux, R., Berhanu, M., Bourgoin, M., Moulin, M., Odier, Ph., Pinton, J.-F., Volk, R., Fauve, S., Mordant, N., Petrelis, F., Chiffaudel, A., Daviaud, F., Dubrulle, B., Gasquet, C., Mari, L. & Ravelet, F. 2007 Generation of a magnetic field by dynamo action in a turbulent flow of liquid sodium. Phys. Rev. Lett. 98, 044502.
Montgomery, D. 1993 Hartmann, Lundquist, and Reynolds: the role of dimensionless numbers in nonlinear magnetofluid behavior. Plasma Phys. Control. Fusion 35, B105B113.
Müller, U. & Stieglitz, R. 2000 Can the Earth’s magnetic field be simulated in the laboratory? Naturwissenschaften 87, 381390.
Nornberg, M. D., Ji, H., Schartman, E & Roach, A. 2010 Observation of magnetocoriolis waves in a liquid metal Taylor–Couette experiment. Phys. Rev. Lett. 104, 074501.
Ogilvie, G. I. & Pringle, J. E. 1996 The non-axisymmetric instability of a cylindrical shear flow containing an azimuthal magnetic field. Mon. Not. R. Astron. Soc. 279, 152164.
Oishi, J. S. & Mac Low, M.-M. 2011 Magnetorotational turbulence transports angular momentum in stratified disks with low magnetic Prandtl number but magnetic Reynolds number above a critical value. Astrophys. J. 740, 18.
Petitdemange, L., Dormy, E. & Balbus, S. A. 2008 Magnetostrophic MRI in the Earth’s outer core. Geophys. Res. Lett. 35, L15305.
Priede, J. 2011 Inviscid helical magnetorotational instability in cylindrical Taylor–Couette flow. Phys. Rev. E 84, 066314.
Rüdiger, G., Gellert, M., Schultz, M. & Hollerbach, R. 2010 Dissipative Taylor–Couette flows under the influence of helical magnetic fields. Phys. Rev. E 82, 016319.
Rüdiger, G. & Hollerbach, R. 2007 Comment on ‘Helical magnetorotational instability in magnetized Taylor–Couette flow’. Phys. Rev. E 76, 068301.
Rüdiger, G., Kitchatinov, L. & Hollerbach, R. 2013 Magnetic Processes in Astrophysics. Wiley-VCH.
Rüdiger, G. & Schultz, M. 2010 Tayler instability of toroidal magnetic fields in MHD Taylor–Couette flows. Astron. Nachr. 331, 121129.
Seilmayer, M., Galindo, V., Gerbeth, G., Gundrum, T., Stefani, F., Gellert, M., Rüdiger, G., Schultz, M. & Hollerbach, R. 2014 Experimental evidence for non-axisymmetric magnetorotational instability in an azimuthal magnetic field. Phys. Rev. Lett. 113, 024505.
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.
Shi, J., Krolik, J. H. & Hirose, S. 2010 What is the numerically converged amplitude of magnetohydrodynamics turbulence in stratified shearing boxes? Astrophys. J. 708, 17161727.
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.
Spruit, H. C. Dynamo action by differential rotation in a stably stratified stellar interior. Astron. Astrophys. 381, 923932.
Squire, J. & Bhattacharjee, A. 2014 Nonmodal growth of the magnetorotational instability. Phys. Rev. Lett. 113, 025006.
Stefani, F., Eckert, S., Gerbeth, G., Giesecke, A., Gundrum, T., Steglich, C., Weier, T. & Wustmann, B. 2012 DRESDYN—a new facility for MHD experiments with liquid sodium. Magnetohydrodynamics 48, 103113.
Stefani, F., Gailitis, A. & Gerbeth, G. 2008 Magnetohydrodynamic experiments on cosmic magnetic fields. Z. Angew. Math. Mech. 88, 930954.
Stefani, F., Gerbeth, G., Gundrum, T., Hollerbach, R., Priede, J., Rüdiger, G. & Szklarski, J. 2009 Helical magnetorotational instability in a Taylor–Couette flow with strongly reduced Ekman pumping. Phys. Rev. E 80, 066303.
Stefani, F., Gundrum, T., Gerbeth, G., Rüdiger, G., Schultz, M., Szklarski, J. & Hollerbach, R. 2006a Experimental evidence for magnetorotational instability in a Taylor–Couette flow under the influence of a helical magnetic field. Phys. Rev. Lett. 97, 184502.
Stefani, F., Gundrum, T., Gerbeth, G., Rüdiger, G., Szklarski, J. & Hollerbach, R. 2006b Experiments on the magnetorotational instability in helical magnetic fields. New J. Phys. 9, 295.
Tayler, R. 1973 The adiabatic stability of stars containing magnetic fields. Mon. Not. R. Astron. Soc. 161, 365380.
Umurhan, O. M. 2010 Low magnetic-Prandtl number flow configurations for cold astrophysical disk models: speculation and analysis. Astron. Astrophys. 513, A47.
Velikhov, E. P. 1959 Stability of an ideally conducting liquid flowing between cylinders rotating in a magnetic field. Sov. Phys. JETP-USSR 9, 995998.
Vishik, M. & Friedlander, S. 1998 Asymptotic methods for magnetohydrodynamic instability. Q. Appl. Maths 56, 377398.
Weber, N., Galindo, V., Stefani, F., Weier, T. & Wondrak, T. 2013 Numerical simulation of the Tayler instability in liquid metals. New J. Phys. 15, 043034.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed