  The problem of the stability of magnetic fields in stars has a long history
and has been investigated in detail in perturbation theory. Here we consider
the nonlinear evolution of a non-rotating neutron star with a purely poloidal
magnetic field, in general relativity. We find that an instability develops in
the region of the closed magnetic field lines and over an Alfven timescale, as
predicted by perturbation theory. After the initial unstable growth, our
evolutions show that a toroidal magnetic field component is generated, which
increases until it is locally comparable in strength with the poloidal one. On
longer timescales the system relaxes to a new non-axisymmetric configuration
with a reorganization of the stellar structure and large-amplitude
oscillations, mostly in the fundamental mode. We discuss the energies involved
in the instability and the impact they may have on the phenomenology of
magnetar flares and on their detectability through gravitational-wave emission.
