IDEAL INSTABILITIES IN RAPIDLY ROTATING MHD SYSTEMS THAT HAVE CRITICAL LAYERS

The study of MHD instabilities in Earth's fluid core is simplified by making the usual assumptions that the fluid is uniform, inviscid, inco mpressible, and electrically-conducting (conductivity, sigma). It is a lso supposed that it is in a state of rapid, and nearly solid-body rot ation about the polar axis, Oz; in that frame, the prevailing velocity , V0, is slow and, like the magnetic field, B0, is nearly azimuthal. I n an ideal fluid (defined as one in which sigma is infinite), an impor tant class of instabilities arise, namely Acheson's field gradient mod es. In the case considered here, in which V0 = 0, these arise when the s-gradient of (B0/s)2, where s is the distance from the axis of rotat ion, is sufficiently large. Commonly (and here), these instabilities a re studied using a simple model in which B0 is a function of s alone, and in which the fluid fills a cylindrical annulus s(i) less-than-or-e qual-to s less-than-or-equal-to s(o). When such cylindrical models hav e been studied in the past, it has usually been supposed that the fiel d has no ''critical levels'', i.e. that B0 has no zeros in s(i) < s < s(o). Here, however, the explicit aim is to understand better what hap pens when critical levels are present. Acheson's analysis is extended to this case, and a procedure is developed that generalizes his stabil ity criterion. To the same qualitative accuracy as his method, the gro wth rate, p, of the instability is estimated. A ''double-turning point '' technique is presented that provides quantitatively accurate p. Thi s technique also explains why and how the eigenfunctions are highly lo calized in s. The results of numerical integrations for finite value s igma are also presented.