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 B 0 ( r , θ)ϕ and azimuthal shear flow U 0 ( 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 R c is found to reach a minimum here and the qualitative behaviour of the thermally driven instabilities in the absence of a shear flow ( U 0 = 0) is similar to that found by earlier workers (e.g. Fearn 1979 b ) for the simpler basic state B 0 = r sin θ. The effect of a shear flow is followed as its strength (measured by the magnetic Reynolds number R m ) 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 R m = O(q) . For R m > q three features are evident as R m is increased: the perturbation in the temperature field (but not the other variables when R m < O (1)) becomes increasingly localized at some point ( r L , θ L ); the phase speed of the instability tends towards the fluid velocity at that point; and R c increases with R m with the suggestion that R c ∝ R m / q for R m [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.