  Gravitational-wave diagnostics are developed for discriminating between
varieties of mixed poloidal-toroidal magnetic fields in neutron stars, with
particular emphasis on differentially rotating protoneutron stars. It is shown
that tilted torus magnetic fields, defined as the sum of an internal/external
poloidal component, whose axis of symmetry is tilted with respect to the
rotation axis, and an internal toroidal component, whose axis of symmetry is
aligned with the rotation axis, deform the star triaxially, unlike twisted
torus fields, which deform the star biaxially. Utilizing an analytic tilted
torus example, we show that these two topologies can be distinguished by their
gravitational wave spectrum and polarization phase portraits. For example, the
relative amplitudes and frequencies of the spectral peaks allows one to infer
the relative strengths of the toroidal and poloidal components of the field,
and the magnetic inclination angle. Finally, we show how a tilted torus field
arises naturally from magnetohydrodynamic simulations of differentially
rotating neutron stars, and how the gravitational wave spectrum evolves as the
internal toroidal field winds up. These results point to the sorts of
experiments that may become possible once gravitational wave interferometers
detect core-collapse supernovae routinely.
