Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-06-04T03:02:37.224Z Has data issue: false hasContentIssue false
  • English
  • Français

Hydromagnetic waves in a differentially rotating sphere

Published online by Cambridge University Press:  20 April 2006

D. R. Fearn  and
M. R. E. Proctor
D. R. Fearn
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge CB3 9EW
M. R. E. Proctor
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge CB3 9EW
Get access Rights & Permissions [Opens in a new window]

Abstract

The linear stability of a uniformly internally heated, self-gravitating, rapidly rotating fluid sphere is investigated in the presence of an azimuthal magnetic field B0(r, θ)ϕ and azimuthal shear flow U0(r, θ)ϕ (where (r, θ, ϕ) are spherical polar coordinates). Solutions are calculated numerically for magnetic field strengths that produce a Lorentz force comparable in magnitude to that of the Coriolis force. The critical Rayleigh number Rc is found to reach a minimum here and the qualitative behaviour of the thermally driven instabilities in the absence of a shear flow (U0 = 0) is similar to that found by earlier workers (e.g. Fearn 1979b) for the simpler basic state B0 = r sin θ. The effect of a shear flow is followed as its strength (measured by the magnetic Reynolds number Rm) is increased from zero. In the case where the ratio q of thermal to magnetic diffusivities is small (q [Lt ] 1) the effect of the flow becomes significant when Rm = O(q). For Rm > q three features are evident as Rm is increased: the perturbation in the temperature field (but not the other variables when Rm < O(1)) becomes increasingly localized at some point (rL, θL); the phase speed of the instability tends towards the fluid velocity at that point; and Rc increases with Rm with the suggestion that Rc ∝ Rm/q for Rm [Gt ] q although the numerical resolution is insufficient to verify this. Greater resolution is achieved for a simpler problem which retains the essential physics and is described in the accompanying paper (Fearn & Proctor 1983). The possible significance of these results to the geomagnetic secular variation is discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 128 , March 1983 , pp. 1 - 20
Copyright
© 1983 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. 1973 Hydromagnetic wavelike instabilities in a rapidly rotating stratified fluid J. Fluid Mech. 61, 609–624.Google Scholar
Acheson, D. J. 1978a Magnetohydrodynamic waves and instabilities in rotating fluids. In Rotating Fluids in Geophysics (ed. P. H. Roberts & A. M. Soward), pp. 315–349. Academic.
Acheson, D. J. 1978b On the instability of toroidal magnetic fields and differential rotation in stars. Phil. Trans. R. Soc. Lond A 289, 459–500.Google Scholar
Acheson, D. J. 1980 Stable density stratification as a catalyst for instability J. Fluid Mech. 96, 723–733.Google Scholar
Acheson, D. J. 1982 Thermally convective and magnetohydrodynamic instabilities of a rotating fluid – I. Unpublished manuscript.
Acheson, D. J. & Hide, R. 1973 Hydromagnetics of rotating fluids Rep. Prog. Phys. 36, 159–221.Google Scholar
Braginsky, S. I. 1980 Magnetic waves in the core of the Earth – II Geophys. Astrophys. Fluid Dyn. 14, 189–208.Google Scholar
Busse, F. H. & Hood, L. L. 1982 Differential rotation driven by convection in a rapidly rotating annulus Geophys. Astrophys. Fluid Dyn. 21, 59–74.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.
Eltayeb, I. A. 1972 Hydromagnetic convection in a rapidly rotating fluid layer. Proc. R. Soc. Lond A 326, 229–254.Google Scholar
Eltayeb, I. A. 1975 Overstable hydromagnetic convection in a rotating fluid layer J. Fluid Mech. 71, 161–179.Google Scholar
Eltayeb, I. A. 1981 Propagation and stability of wave motions in rotating magnetic systems Phys. Earth Planet. Interiors 24, 259–271.Google Scholar
Eltayeb, I. A. & KUMAR 1977 Hydromagnetic convective instability of a rotating, self-gravitating fluid sphere containing a uniform distribution of heat sources. Proc. R. Soc. Lond A 353, 145–162.Google Scholar
Eltayeb, I. A. & Roberts, P. H. 1970 On the hydromagnetics of rotating fluids Astrophys. J. 162, 699–701.Google Scholar
Fearn, D. R. 1979a Thermally driven hydromagnetic convection in a rapidly rotating sphere. Proc. R. Soc. Lond A 369, 227–242.Google Scholar
Fearn, D. R. 1979b Thermal and magnetic instabilities in a rapidly rotating fluid sphere Geophys. Astrophys. Fluid Dyn. 14, 103–126.Google Scholar
Fearn, D. R. & Loper, D. E. 1981 Compositional convection and stratification of Earth's core Nature 289, 393–394.Google Scholar
Fearn, D. R. & Proctor, M. R. E. 1983 The stabilizing role of differential rotation on hydromagnetic waves J. Fluid Mech. 128, 21–36.Google Scholar
Gubbins, D., Masters, T. G. & Jacobs, J. A. 1979 Thermal evolution of the Earth's core Geophys. J. R. Astr. Soc. 59, 57–99.Google Scholar
Jepps, S. A. 1975 Numerical models of hydromagnetic dynamos J. Fluid Mech. 67, 625–646.Google Scholar
Loper, D. E. & Roberts, P. H. 1983 Compositional convection and the gravitationally powered dynamo. In Stellar and Planetary Magnetism (ed. A. M. Soward). Gordon & Breach.
Malkus, W. V. R. 1967 Hydromagnetic planetary waves. J. Fluid Mech. 28. 793–802.Google Scholar
Moffatt, H. K. 1978 Magnetic Field Generation in Electrically Conducting Fluids. Cambridge University Press.
Peters, G. & Wilkinson, J. H. 1971a Eigenvectors of real and complex matrices by LR and QR triangularisations. In Handbook for Automatic Computation, vol. 2: Linear Algebra (ed. J. H. Wilkinson & C. Reinsch), pp. 370–395. Springer.
Peters, G. & Wilkinson, J. H. 1971b The calculation of specified eigenvectors by inverse iteration. In Handbook for Automatic Computation, vol. 2: Linear Algebra (ed. J. H. Wilkinson & C. Reinsch), pp. 418–439. Springer.
Proctor, M. R. E. 1975 Non-linear mean field dynamo models and related topics. Ph.D. thesis, University of Cambridge.
Roberts, P. H. 1968 On the thermal instability of a rotating-fluid sphere containing heat sources. Phil. Trans. R. Soc. London. A 263, 93–117.Google Scholar
Roberts, P. H. 1972 Kinematic dynamo models. Phil. Trans. R. Soc. Lond A 272, 663–703.Google Scholar
Roberts, P. H. 1978 Magneto-convection in a rapidly rotating fluid. In Rotating Fluids in Geophysics (ed. P. H. Roberts & A. M. Soward), pp. 421–435. Academic.
Roberts, P. H. & Loper, D. E. 1979 On the diffusive instability of some simple steady magnetohydrodynamic flows J. Fluid Mech. 90, 641–668.Google Scholar
Soward, A. M. 1977 On the finite amplitude thermal instability of a rapidly rotating fluid sphere. Geophys. Astrophys. Fluid Dyn. 9, 19–74.Google Scholar
Soward, A. M. 1979a Convection driven dynamos Phys. Earth Planet. Interiors 20, 134–151.Google Scholar
Soward, A. M. 1979b Thermal and magnetically driven convection in a rapidly rotating fluid layer J. Fluid Mech. 90, 669–684.Google Scholar
Whaler, K. A. 1980 Does the whole of the Earth's core convect? Nature 287, 528–530.Google Scholar