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 Benard layer with imposed magnetic field B 0 = B M s ϕ and shear U 0 = U M s Ω( 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 R t = U M d / k , where d is the depth of the layer and κ is the thermal diffusivity, and the shear becomes important when R t [ges ] O (1). It is possible to compute solutions well into the asymptotic regime R t [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 b ) Ω( z ) has a turning point at z = z T , 0 z T z T = 0, Ω″( z T ) ≠ 0). In both cases, as R t increases a critical layer centred at z = z L develops, with width proportional to ( a ) R t −1/3 , ( b ) R t −¼ . In the case where Ω( z ) has a turning point, the critical layer is located at the turning point ( z L = z T ). The critical Rayleigh number R c increases with ( a ) R c ∝ R t , ( b ) R c ∝ R R t −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.