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The stabilizing role of differential rotation on hydromagnetic waves

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
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Abstract

The role that differential rotation plays in the hydromagnetic stability of rapidly rotating fluids has recently been investigated by Fearn & Proctor (1983) (hereinafter referred to as I) as part of a wider study related to the geodynamo problem. Starting with a uniformly rotating fluid sphere, the strength of the differential rotation was gradually increased from zero and several interesting features were observed. These included the development of a critical region whose size decreased as the strength of the shear increased. The resolution of the two-dimensional numerical scheme used in I is limited, and consequently it was only possible to consider small shear strengths. This is unfortunate because differential rotation is probably an important effect in the Earth's core and a more detailed study at higher shear strengths is desirable. Here we are able to achieve this by studying a rapidly rotating Bénard layer with imposed magnetic field B0 = BMsϕ and shear U0 = UMsΩ(z)ϕ, where (s, ϕ, z) are cylindrical polar coordinates. In the limit where the ratio q of the thermal to magnetic diffusivities vanishes (q = 0), the governing equations are separable in two space dimensions and the problem reduces to a one-dimensional boundary-value problem. This can be solved numerically with greater accuracy than was possible in the spherical geometry of I. The strength of the shear is measured by a modified Reynolds number Rt = UMd/k, where d is the depth of the layer and κ is the thermal diffusivity, and the shear becomes important when Rt [ges ] O(1). It is possible to compute solutions well into the asymptotic regime Rt [Gt ] 1, and details of the behaviour observed are dependent on the nature of Ω(z). Specifically, two cases were considered: (a) Ω(z) has no turning point in 0 < z < 1, and (b) Ω(z) has a turning point at z = zT, 0 < zT < 1 (Ω′(zT = 0, Ω″(zT) ≠ 0). In both cases, as Rt increases a critical layer centred at z = zL develops, with width proportional to (a) Rt−1/3, (b) Rt−¼. In the case where Ω(z) has a turning point, the critical layer is located at the turning point (zL = zT). The critical Rayleigh number Rc increases with (a) Rc ∝ Rt, (b) Rc ∝ RRt−¼, and the instability is carried around with the fluid velocity at the critical layer. The relevance of these results to the geomagnetic secular variation is discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 128 , March 1983 , pp. 21 - 36
Copyright
© 1983 Cambridge University Press

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References

Acheson, D. J. 1972 On the hydromagnetic stability of a rotating fluid annulus J. Fluid Mech. 52, 529–541.Google Scholar
Acheson, D. J. 1973 Hydromagnetic wavelike instabilities in a rapidly rotating stratified fluid J. Fluid Mech. 61, 609–624.Google Scholar
Acheson, D. J. 1978 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. 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. 1967 Magnetic waves in the Earth's core Geomag. Aeron. 7, 851–859.Google Scholar
Braginsky, S. I. 1980 Magnetic waves in the core of the Earth. II Geophys. Astrophys. Fluid Dyn. 14, 189–208.Google Scholar
Braginsky, S. I. & Roberts, P. H. 1975 Magnetic field generation by baroclinic waves. Proc. R. Soc. Lond A 347, 125–140.Google Scholar
Davey, A. 1973 A simple numerical method for solving Orr-Sommerfeld problems Q. J. Mech. Appl. Maths 26, 401–411.Google Scholar
Davey, A. 1978 Numerical methods for solution of linear differential eigenvalue problems. In Rotating Fluids in Geophysics (ed. P. H. Roberts & A. M. Soward), pp. 485–498. Academic.
Eltayeb, I. A. 1981 Propagation and stability of wave motions in rotating magnetic systems Phys. Earth Planet. Interiors 24, 259–271.Google Scholar
Fearn, D. R. 1979 Thermal and magnetic instabilities in a rapidly rotating fluid sphere Geophys. Astrophys. Fluid Dyn. 14, 103–126.Google Scholar
Fearn, D. R. & Proctor, M. R. E. 1983 [I] Hydromagnetic waves in a differentially rotating sphere. J. Fluid Mech. 128, 1–20.Google Scholar
Peters, G. & Wilkinson, J. H. 1971a 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.
Peters, G. & Wilkinson, J. H. 1971b Eigenvalues 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.
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
Roberts, P. H. & Soward, A. M. 1972 Magnetohydrodynamics of the Earth's core Ann. Rev. Fluid Mech. 4, 117–154.Google Scholar
Soward, A. M. 1979 [S79] Thermal and magnetically driven convection in a rapidly rotating fluid layer. J. Fluid Mech. 90, 669–684.Google Scholar