The stabilizing role of differential rotation on hydromagnetic waves

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 = BM sϕ and shear U0 = UM sΩ(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 = UM d/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.