Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T03:30:48.663Z Has data issue: false hasContentIssue false
  • English
  • Français

On the hydromagnetic stability of a rotating fluid annulus

Published online by Cambridge University Press:  29 March 2006

D. J. Acheson
D. J. Acheson
Affiliation:
School of Mathematics and Physics, University of East Anglia, Norwich Present address: Geophysical Fluid Dynamics Laboratory, Meteorological Office, Bracknell, Berkshire.
Get access Rights & Permissions [Opens in a new window]

Abstract

A non-dissipative fluid rotates uniformly in the annular region between two infinitely long cylinders and is permeated by a magnetic field varying with distance from the axis of rotation. The hydromagnetic stability of this system is examined theoretically. When the magnetic field is azimuthal the system can always be rendered stable to axisymmetric disturbances by sufficiently rapid rotation (Michael 1954). Unless the magnetic field everywhere decreases with radius, however, the system may be unstable to non-axisymmetric disturbances even when the rotation speed exceeds a typical Alfvén speed by many orders of magnitude. ‘Slow’ hydromagnetic waves, akin to those invoked in a recent theory of the geomagnetic secular variation (Hide 1966), may then be generated by the spatial variations of the magnetic field. All unstable waves so generated propagate against the basic rotation, i.e. ‘westward’, when the field is azimuthal, and this property is in fact remarkably insensitive to variations in both magnitude and direction of the imposed field.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 52 , Issue 3 , 11 April 1972 , pp. 529 - 541
Copyright
© 1972 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

Acheson, D. J. 1971 The magnetohydrodynamics of rotating fluids. Ph.D. Thesis, University of East Anglia.
Benton, E. R. & Loper, D. E. 1969 On the spin-up of an electrically conducting fluid. Part 1. The unsteady hydromagnetic Ekman-Hartmann boundary layer problem. J. Fluid Mech. 39, 561.Google Scholar
Braginsky, S. I. 1967 Magnetic waves in the Earth's core. Geomagnetism & Aeronomy, 6, 851.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
Gilman, P. A. 1967 Stability of baroclinic flows in a zonal magnetic field. Part I. J. Atmos. Sci. 24, 101.Google Scholar
Gilman, P. A. & Benton, E. R. 1968 Influence of an axial magnetic field on the steady linear Ekman boundary layer. Phys. Fluids, 14, 7.Google Scholar
Hide, R. 1966 Free hydromagnetic oscillations of the Earth's core and the theory of the geomagnetic secular variation. Phil. Trans. A 259, 615.Google Scholar
Hide, R. 1969a On hydromagnetic waves in a stratified rotating fluid. J. Fluid Mech. 39, 283.Google Scholar
Hide, R. 1969b Dynamics of the atmospheres of the major planets - with an appendix on the viscous boundary layer at the rigid bounding surface of an electrically-conducting rotating fluid in the presence of a magnetic field. J. Atmos. Sci. 26, 841.Google Scholar
Joseph, D. D. & Munson, B. R. 1970 Global stability of spiral flow. J. Fluid Mech. 43, 545.Google Scholar
Lehnert, B. 1954 Magnetohydrodynamic waves under the action of the Coriolis force I. Astrophys. J. 119, 647.Google Scholar
Loper, D. E. & Benton, E. R. 1970 On the spin-up of an electrically conducting fluid. Part 2. Hydromagnetic spin-up between infinite flat insulating plates. J. Fluid Mech. 43, 785.Google Scholar
Malkus, W. V. R. 1967a Hydromagnetic planetary waves. J. Fluid Mech. 28, 793.Google Scholar
Malkus, W. V. R. 1967b Hydromagnetic instability of the sub-Alfvén equations. Proc. Symp. Appl. Math. 18, 63.Google Scholar
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, 45.Google Scholar
Rickard, J. A. 1970 Planetary waves. Ph.D. Thesis, University of London.
Roberts, P. H. 1967 A n Introduction to Magnetohydrodynamics. Longmans.
Rossby, C. G. 1939 Relation between variations in the intensity of the zonal circulation of the atmosphere and the displacements of the semi-permanent centres of action. J. Mar. Res. 2, 38.Google Scholar
Schubert, G. 1968 The stability of toroidal magnetic fields in stellar interiors. Astrophys. J. 151, 1099.Google Scholar
Shercliff, J. A. 1965 A Textbook of Magnetohydrodynamics. Pergamon Press.
Stewartson, K. 1967 Slow oscillations of fluid in a rotating cavity in the presence of a toroidal magnetic field. Proc. Roy. Soc. A 299, 173.Google Scholar
Velikhov, E. P. 1959 Stability of an ideally conducting fluid flowing between cylinders rotating in a magnetic field. J. Exp. Theor. Phys. (U.S.S.R.) 36, 1398.Google Scholar