  Dynamical instabilities in protoneutron stars may produce gravitational waves
whose observation could shed light on the physics of core-collapse supernovae.
When born with sufficient differential rotation, these stars are susceptible to
a shear instability (the "low-T/|W| instability"), but such rotation can also
amplify magnetic fields to strengths where they have a considerable impact on
the dynamics of the stellar matter. Using a new magnetohydrodynamics module for
the Spectral Einstein Code, we have simulated a differentially-rotating neutron
star in full 3D to study the effects of magnetic fields on this instability.
Though strong toroidal fields were predicted to suppress the low-T/|W|
instability, we find that they do so only in a small range of field strengths.
Below 4e13 G, poloidal seed fields do not wind up fast enough to have an effect
before the instability saturates, while above 5e14 G, magnetic instabilities
can actually amplify a global quadrupole mode (this threshold may be even lower
in reality, as small-scale magnetic instabilities remain difficult to resolve
numerically). Thus, the prospects for observing gravitational waves from such
systems are not in fact diminished over most of the magnetic parameter space.
  Additionally, we report that the detailed development of the low-T/|W|
instability, including its growth rate, depends strongly on the particular
numerical methods used. The high-order methods we employ suggest that growth
might be considerably slower than found in some previous simulations.
