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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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