Magnetic instabilities in rapidly rotating spherical geometries. III. The effect of differential rotation |
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| Authors: | David R Fearn Werner S Weiglhofer |
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| Institution: | Department of Mathematics , University of Glasgow, University Gardens , Glasgow , G12 8QW , Scotland , UK |
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| Abstract: | Abstract We investigate the influence of differential rotation on magnetic instabilities for an electrically conducting fluid in the presence of a toroidal basic state of magnetic field B 0 = BMB0(r, θ)1 φ and flow U0 = UMU0 (r, θ)1φ, (r, θ, φ) are spherical polar coordinates]. The fluid is confined in a rapidly rotating, electrically insulating, rigid spherical container. In the first instance the influence of differential rotation on established magnetic instabilities is studied. These can belong to either the ideal or the resistive class, both of which have been the subject of extensive research in parts I and II of this series. It was found there, that in the absence of differential rotation, ideal modes (driven by gradients of B 0) become unstable for Ac ? 200 whereas resistive instabilities (generated by magnetic reconnection processes near critical levels, i.e. zeros of B0) require Ac ? 50. Here, Λ is the Elsasser number, a measure of the magnetic field strength and Λc is its critical value at marginal stability. Both types of instability can be stabilised by adding differential rotation into the system. For the resistive modes the exact form of the differential rotation is not important whereas for the ideal modes only a rotation rate which increases outward from the rotation axis has a stabilising effect. We found that in all cases which we investigated Λc increased rapidly and the modes disappeared when Rm ≈ O(ΛC), where the magnetic Reynolds number Rm is a measure of the strength of differential rotation. The main emphasis, however, is on instabilities which are driven by unstable gradients of the differential rotation itself, i.e. an otherwise stable fluid system is destabilised by a suitable differential rotation once the magnetic Reynolds number exceeds a certain critical value (Rm )c. Earlier work in the cylindrical geometry has shown that the differential rotation can generate an instability if Rm ) ?O(Λ). Those results, obtained for a fixed value of Λ = 100 are extended in two ways: to a spherical geometry and to an analysis of the effect of the magnetic field strength Λ on these modes of instability. Calculations confirm that modes driven by unstable gradients of the differential rotation can exist in a sphere and they are in good agreement with the local analysis and the predictions inferred from the cylindrical geometry. For Λ = O(100), the critical value of the magnetic Reynolds number (Rm )c Λ 100, depending on the choice of flow U0 . Modes corresponding to azimuthal wavenumber m = 1 are the most unstable ones. Although the magnetic field B 0 is itself a stable one, the field strength plays an important role for this instability. For all modes investigated, both for cylindrical and spherical geometries, (Rm )c reaches a minimum value for 50 ≈ Λ ≈ 100. If Λ is increased, (Rm )c ∝ Λ, whereas a decrease of Λ leads to a rapid increase of (Rm )c, i.e. a stabilisation of the system. No instability was found for Λ ≈ 10 — 30. Optimum conditions for instability driven by unstable gradients of the differential rotation are therefore achieved for ≈ Λ 50 — 100, Rm ? 100. These values lead to the conclusion that the instabilities can play an important role in the dynamics of the Earth's core. |
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| Keywords: | Earth's core hydromagnetic waves magnetic instabilities differential rotation |
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