  We examine the MHD instabilities arising in the radiation zone of a
differentially rotating star, in which a poloidal field of fossil origin is
sheared into a toroidal field. We focus on the non-axisymmetric instability
that affects the toroidal magnetic field in a rotating star, which was first
studied by Pitts and Tayler in the non-dissipative limit. According to Spruit,
it could also drive a dynamo. The Pitts & Tayler instability is manifestly
present in our simulations, with its conspicuous m=1 dependence in azimuth. But
its analytic treatment used so far is too simplified to be applied to the real
stellar situation. Although the instability generated field reaches an energy
comparable to that of the mean poloidal field, that field seems unaffected by
the instability: it undergoes Ohmic decline, and is neither eroded nor
regenerated by the instability. The toroidal field is produced by shearing the
poloidal field and it draws its energy from the differential rotation. The
small scale motions behave as Alfven waves; they cause negligible
eddy-diffusivity and contribute little to the net transport of angular
momentum. In our simulations we observe no sign of dynamo action, of either
mean field or fluctuation type, up to a magnetic Reynolds number of 10^5.
However the Pitts & Tayler instability is sustained as long as the differential
rotation acting on the poloidal field is able to generate a toroidal field of
sufficient strength.
