Hydromagnetic waves in a differentially rotating sphere

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.