Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-21T02:16:52.352Z Has data issue: false hasContentIssue false
  • English
  • Français

Thermal and magnetically driven convection in a rapidly rotating fluid layer

Published online by Cambridge University Press:  19 April 2006

A. M. Soward
A. M. Soward
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, Los Angeles Permanent address: School of Mathematics, University of Newcastle upon Tyne, England.
Get access Rights & Permissions [Opens in a new window]

Abstract

The stability of an electrically conducting Boussinesq fluid which is confined between two horizontal planes a distance d apart is investigated. The fluid is heated from below, cooled from above and the whole system rotates rapidly with angular velocity Ωc about a vertical axis. A weak non-uniform horizontal magnetic field, whose strength is measured by the Alfvén angular velocity ΩM [[Lt ] Ωc, see (1.2)] permeates the fluid and corresponds to the flow of a uniform electric current parallel to the rotation axis. When the modified Rayleigh number R [see (2.1)] is greater than zero and q = κ/λ < 1, where κ and λ are the thermal and magnetic diffusivities respectively, instability sets in as a westward-propagating wave with a low frequency of order κ/d2.

When R = 0 and ΩM > 2(ν/λ)½ Ωc, where ν is the viscosity, Roberts & Loper (1979) have isolated an exceptional class of unstable fast inertial waves which grow on the magnetic diffusion time scale τλ = d2/λ. When R < 0 and Γ = τλΩ2M/τc, exceeds some value dependent upon q, a class of unstable slow waves also exists for a range of negative values of R. These waves propagate eastwards (westwards) when q is less (greater) than unity. In this case the fluid is stably stratified and the energy for the disturbance is taken from the magnetic field. The resulting description of the stability boundary for R < 0 in the Γ, R plane extends and clarifies the results of Roberts & Loper (1979), which are valid when both Γ and q are large.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 90 , Issue 4 , 27 February 1979 , pp. 669 - 684
Copyright
© 1979 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

Braginskii, S. I. 1964 Zh. Eksp. teor. Fiz. 47, 1084. [Trans. Sov. Phys. J. Exp. Theor. Phys. 20, 726 (1965).]
Braginskii, S. I. 1967 Geomag. Aero. (USSR) 7, 1050. [Trans. Geomag. Aero. 7, 851 (1967).]
Busse, F. H. 1970 J. Fluid Mech. 44, 441.
Busse, F. H. 1975 Geophys. J. Roy. Astr. Soc. 42, 437.
Busse, F. H. 1976 Phys. Earth Planet. Int. 12, 292.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
Childress, S. & Soward, A. M. 1972 Phys. Rev. Lett. 29, 837.
Eltayeb, I. A. 1972 Proc. Roy. Soc. A 326, 229.
Eltayeb, I. A. 1975 J. Fluid Mech. 71, 161.
Eltayeb, I. A. & Kumar, S. 1977 Proc. Roy. Soc. A 353, 145.
Furth, H. P., Killeen, J. & Rosenbluth, M. N. 1963 Phys. Fluids 6, 459.
Hide, R. 1966 Phil. Trans. Roy. Soc. A 259, 615.
Loper, D. E. 1975 Phys. Earth Planet. Int. 11, 43.
Malkus, W. V. R. 1967 J. Fluid Mech. 28, 793.
Roberts, P. H. 1968 Phil. Trans. Roy. Soc. A 263, 93.
Roberts, P. H. & Loper, D. E. 1979 J. Fluid Mech. 90, 641.
Roberts, P. H. & Stewartson, K. 1974 Phil. Trans. Roy. Soc. A 277, 287.
Rochester, M. G., Jacobs, J. A., Smylie, D. E. & Chong, K. F. 1975 Geophys. J. Roy. Astr. Soc. 43, 661.
Soward, A. M. 1974 Phil. Trans. Roy. Soc. A 275, 611.