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High-speed standard magneto-rotational instability

Published online by Cambridge University Press:  20 February 2019

Kengo Deguchi Open the ORCID record for Kengo Deguchi [Opens in a new window]
Kengo Deguchi*
Affiliation:
School of Mathematical Sciences, Monash University, VIC 3800, Australia
*
†Email address for correspondence: [email protected]
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Abstract

The large Reynolds number asymptotic approximations of the neutral curve of Taylor–Couette flow subject to an axial uniform magnetic field are analysed. The flow has been extensively studied since the early 1990s as the magneto-rotational instability (MRI) occurring in the flow may explain the origin of the instability observed in some astrophysical objects. Elsewhere, the ideal approximation has been used to study high-speed flows, which sometimes produces paradoxical results. For example, ideal flows must be completely stabilised for a sufficiently strong applied magnetic field. On the other hand, the vanishing magnetic Prandtl number limit of the stability should be purely hydrodynamic, so instability must occur when Rayleigh’s stability condition is violated. Our first discovery is that this apparent contradiction can be resolved by showing the abrupt appearance of the hydrodynamic instability at a certain critical value of the magnetic Prandtl number. This is found using the asymptotically large Reynolds number limit but with a sufficiently long wavelength to retain some diffusive effects. Our second finding concerns the so-called Velikhov–Chandrasekhar paradox, namely the mismatch of the zero external magnetic field limit of the Velikhov–Chandrasekhar stability criterion and Rayleigh’s stability criterion. We show for fully wide-gap cases that the high Reynolds number asymptotic analysis of the MRI naturally yields the simple stability condition that describes smooth transition from Rayleigh to Velikhov–Chandrasekhar stability criteria with increasing Lundquist number.

JFM classification

Geophysical and Geological Flows: Waves in rotating fluids
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 865 , 25 April 2019 , pp. 492 - 522
Copyright
© 2019 Cambridge University Press 

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References

Balbus, S. & Hawley, J. F. 1991 A powerful local shear instability in weakly magnetised disks. I. Linear analysis. Astrophys. J. 376, 214–222.10.1086/170270Google Scholar
Brahme, A. 1970 On the hydromagnetic stability of a nonuniformly rotating fluid. Phys. Scr. 2, 108–112.10.1088/0031-8949/2/3/008Google Scholar
Brandenburg, A., Nordlund, A. A., Stein, R. F. & Torkelsson, U. 1995 Dynamo-generated turbulence and large scale magnetic fields in a Keplerian-shear flow. Astrophys. J. 446, 741–754.10.1086/175831Google Scholar
Chandrasekhar, S. 1953 The stability of viscous flow between rotating cylinders in the presence of a magnetic field. Proc. R. Soc. Lond. A 216, 293–309.Google Scholar
Chandrasekhar, S. 1960 The stability of non-dissipative Couette flow in hydromagnetics. Proc. Natl Acad. Sci. USA 46, 253–257.10.1073/pnas.46.2.253Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.Google Scholar
Deguchi, K. 2016 The rapid-rotation limit of the neutral curve for Taylor–Couette flow. J. Fluid Mech. 808, R2.10.1017/jfm.2016.660Google Scholar
Deguchi, K. 2017 Linear instability in Rayleigh-stable Taylor–Couette flow. Phys. Rev. E 95, 021102(R).10.1103/PhysRevE.95.021102Google Scholar
Donnelly, R. J. & Caldwell, D. R. 1964 Experiments on the stability of hydromagnetic Couette flow. J. Fluid Mech. 19, 257–263.10.1017/S0022112064000696Google Scholar
Donnelly, R. J. & Ozima, M. 1960 Hydromagnetic stability of flow between rotating cylinders. Phys. Rev. Lett. 4, 497–498.10.1103/PhysRevLett.4.497Google Scholar
Donnelly, R. J. & Ozima, M. 1962 Experiments on the stability of flow between rotating cylinders in the presence of a magnetic field. Proc. R. Soc. Lond. A 266, 272–286.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Goodman, J. & Ji, H. 2002 Magnetorotational instability of dissipative Couette flow. J. Fluid Mech. 462, 365–382.10.1017/S0022112002008704Google Scholar
Hawley, J. F., Gammie, C. F. & Balbus, S. A. 1995 Local three-dimensional magnetohydrodynamic simulations of accretion disks. Astrophys. J. 440, 742–763.10.1086/175311Google Scholar
Herron, I. & Goodman, J. 2006 The small magnetic Prandtl number approximation suppresses magneto rotational instability. Z. Angew. Math. Phys. 57, 615–622.10.1007/s00033-006-0060-yGoogle Scholar
Hollerbach, R. & Rüdiger, G. 2005 New type of magnetorotational instability in cylindrical Taylor–Couette flow. Phys. Rev. Lett. 95, 124501.10.1103/PhysRevLett.95.124501Google Scholar
Ji, H., Goodman, J. & Kageyama, A. 2001 Magnetorotational instability in a rotating liquid metal annulus. Mon. Not. R. Astron. Soc. 325, L1–L5.10.1046/j.1365-8711.2001.04647.xGoogle Scholar
Julien, K. & Knobloch, E. 2010 Magnetorotational instability: recent developments. Phil. Trans. R. Soc. A 368, 1607–1633.10.1098/rsta.2009.0251Google Scholar
Kirillov, O. N., Pelinovsky, D. E. & Schneider, G. 2011 Paradoxical transitions to instabilities in hydromagnetic Couette–Taylor flows. Phys. Rev. E 84, 065301(R).Google Scholar
Kirillov, O. N. & Stefani, F. 2011 Paradoxes of magnetorotational instability and their geometrical resolution. Phys. Rev. E 84, 036304.Google Scholar
Knobloch, E. 1992 On the stability of magnetized accretion discs. Astrophys. J. 638, 382–390.Google Scholar
Kurzweg, U. H. 1963 The stability of Couette flow in the presence of an axial magnetic field. J. Fluid Mech. 17, 52–60.10.1017/S0022112063001099Google Scholar
Lin, C. C. 1955 The theory of Hydrodynamic Stability. Cambridge University Press.Google Scholar
Liu, W., Goodman, J., Herron, I. & Ji, H. 2006 Helical magnetorotational instability in magnetized Taylor–Couette flow. Phys. Rev. E 74, 056302.Google Scholar
Priede, J. 2011 Inviscid helical magnetorotational instability in cylindrical Taylor–Couette flow. Phys. Rev. E 84, 066314.Google Scholar
Rayleigh, Lord 1917 On the dynamics of revolving fluids. Proc. R. Soc. Lond. A 93, 148–154.10.1098/rspa.1917.0010Google Scholar
Roberts, P. H. 1964 The stability of hydromagnetic Couette flow. Proc. Camb. Phil. Soc. 60, 635–651.Google Scholar
Rüdiger, G., Gellert, M., Hollerbach, R., Schultz, J. & Stefani, F. 2018a Stability and instability of hydromagnetic Taylor–Couette flows. Phys. Rep. 741, 1–89.10.1016/j.physrep.2018.02.006Google Scholar
Rüdiger, G. & Hollerbach, R. 2007 Comment on ‘Helical magnetorotational instability in magnetized Taylor–Couette flow’. Phys. Rev. E 76, 068301.Google Scholar
Rüdiger, G., Hollerbach, R., Stefani, F., Gundrum, T., Gerbeth, G. & Rosner, R. 2006 The traveling-wave MRI in cylindrical Taylor–Couette flow: comparing wavelengths and speeds in theory and experiment. Astrophys. J. 649, L145–L147.10.1086/508422Google Scholar
Rüdiger, G., Schultz, M. & Shalybkov, D. 2003 Linear magnetohydrodynamic Taylor–Couette instability for liquid sodium. Phys. Rev. E 67, 046312.Google Scholar
Rüdiger, G., Schultz, M., Stefani, F. & Hollerbach, R. 2018b Magnetorotational instability in Taylor–Couette flows between cylinders with finite electrical conductivity. Geophys. Astrophys. Fluid Dyn. 112 (4), 301–320.10.1080/03091929.2018.1508575Google Scholar
Sano, T. & Miyama, S. M. 1999 Magnetorotational instability in protoplanetary disks. I. On the global stability of weakly ionized disks with Ohmic dissipation. Proc. R. Soc. Lond. A 266, 272–286.Google Scholar
Stefani, F., Gundrum, T., Gerbeth, G., Rüdiger, G., Schultz, J., Szklarski, J. & Hollerbach, R. 2006 Experimental evidence for magnetorotational instability in a Taylor–Couette flow under the influence of a helical magnetic field. Phys. Rev. Lett. 97 (18), 184502.10.1103/PhysRevLett.97.184502Google Scholar
Stefani, F., Gundrum, T., Gerbeth, G., Rüdiger, G., Szklarski, J. & Hollerbach, R. 2007 Experiments on the magnetorotational instability in helical magnetic fields. New J. Phys. 9, 295.10.1088/1367-2630/9/8/295Google Scholar
Synge, J. L. 1938 On the stability of a viscous liquid between rotating coaxial cylinders. Proc. R. Soc. Lond. A 167, 250–256.Google Scholar
Taylor, G. I. 1923 Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. R. Soc. Lond. A 223, 289–343.10.1098/rsta.1923.0008Google Scholar
Velikhov, E. P. 1959 Stability of an ideally conducting liquid flowing between cylinders rotating in a magnetic field. Sov. Phys. JETP 36, 1389–1404.Google Scholar
Velikhov, E. P., Ivanov, A. A., Lakhin, V. P. & Serebrennikov, K. S. 2006 Magneto-rotational instability in differentially rotating liquid metals. Phys. Lett. A 356, 357–365.10.1016/j.physleta.2006.03.073Google Scholar
Willis, A. P. & Barenghi, C. F. 2002 Magnetic instability in a sheared azimuthal flow. Astron. Astrophys. 388, 688–691.10.1051/0004-6361:20020510Google Scholar