Nonaxisymmetric instabilities of a toroidal magnetic field in a rotating sphere

It has been suggested that nonaxisymmetric instabilities of an axisymmetric toroidal magnetic field, independent of the existence of convective turbulence, provide an important dynamo effect for stars when the convective turbulence is largely suppressed by the magnetic field. We investigate analytically such three-dimensional magnetic instabilities arising from a purely toroidal magnetic field whose strength is proportional to distance from the rotation axis of a star. A three-dimensional perturbation approach is used in our stability analysis. The leading-order problem neglects dissipative effects and describes two- or three-dimensional hydromagnetic waves in spherical geometry. For the first time we are able to obtain general explicit solutions for the problem. Interesting properties of the hydromagnetic waves are revealed by the explicit solutions. We investigate the effect of ohmic dissipation on the stability of the toroidal magnetic field by first deriving the magnetic boundary layer solution and then matching it onto the interior leading-order solution. Furthermore, for the first time we are able to derive an explicit analytic expression for the growth rate of the magnetic instabilities. We find that the most unstable mode of the instabilities is always three-dimensional and characterized by small radial and latitudinal scales. Implications of these magnetic instabilities for the solar dynamo are discussed.