MAGNETIC INSTABILITIES IN RAPIDLY ROTATING SPHERICAL GEOMETRIES .3. THE EFFECT OF DIFFERENTIAL ROTATION

We investigate the influence of differential rotation on magnetic inst abilities for an electrically conducting fluid in the presence of a to roidal basic state of magnetic field B0 = B(M)B0(r, theta)1phi and flo w U0 = U(M)U0(r, theta)1phi, [(r, theta, phi) are spherical polar coor dinates]. The fluid is confined in a rapidly rotating, electrically in sulating, rigid spherical container. In the first instance the influen ce of differential rotation on established magnetic instabilities is s tudied. These can belong to either the ideal or the resistive class, b oth of which have been the subject of extensive research in parts I an d II of this series. It was found there, that in the absence of differ ential rotation, ideal modes (driven by gradients of B0) become unstab le for LAMBDA(c) greater than or similar to 200 whereas resistive inst abilities (generated by magnetic reconnection processes near critical levels, i.e. zeros of B0) require LAMBDA(c) greater than or similar to 50. Here, LAMBDA is the Elsasser number, a measure of the magnetic fi eld strength and LAMBDA(c) is its critical value at marginal stability . Both types of instability can be stabilised by adding differential r otation into the system. For the resistive modes the exact form of the differential rotation is not important whereas for the ideal modes on ly a rotation rate which increases outward from the rotation axis has a stabilising effect. We found that in all cases which we investigated LAMBDA(c) increased rapidly and the modes disappeared when R(m) almos t-equal-to O(LAMBDA(c)), where the magnetic Reynolds number R(m) is a measure of the strength of differential rotation. The main emphasis, h owever, is on instabilities which are driven by unstable gradients of the differential rotation itself, i.e. an otherwise stable fluid syste m is destabilised by a suitable differential rotation once the magneti c Reynolds number exceeds a certain critical value (R(m))c. Earlier wo rk in the cylindrical geometry has shown that the differential rotatio n can generate an instability if R(m) greater than or similar to O(LAM BDA). Those results, obtained for a fixed value of LAMBDA = 100 are ex tended in two ways: to a spherical geometry and to an analysis of the effect of the magnetic field strength LAMBDA on these modes of instabi lity. Calculations confirm that modes driven by unstable gradients of the differential rotation can exist in a sphere and they are in good a greement with the local analysis and the predictions inferred from the cylindrical geometry. For LAMBDA = O(100), the critical value of the magnetic Reynolds number (R(m))c greater than or similar to 100, depen ding on the choice of flow U0. Modes corresponding to azimuthal wavenu mber m = 1 are the most unstable ones. Although the magnetic field B0 is itself a stable one, the field strength plays an important role for this instability. For all modes investigated, both for cylindrical an d spherical geometries, (R(m)c reaches a minimum value for 50 less tha n or similar to LAMBDA less than or similar to 100. If LAMBDA is incre ased, (R(m)c is-proportional-to LAMBDA, whereas a decrease of A leads to a rapid increase of (R(m))c, i.e. a stabilisation of the system. No instability was found for LAMBDA less than or similar to 10 - 30. Opt imum conditions for instability driven by unstable gradients of the di fferential rotation are therefore achieved for LAMBDA almost-equal-to 50 - 100, R(m) greater than or similar to 100. These values lead to th e conclusion that the instabilities can play an important role in the dynamics of the Earth's core.