  Mature neutron stars are thought to be sufficiently cold that nuclei in the
outer layers freeze, solidifying a crust. Crustal elasticity allows the star to
support a set of seismic modes, such as torsional oscillations. These
axial-parity modes can couple to the polar sector in a number of ways, for
example via rotation or a magnetic field. Even in a static,
spherically-symmetric star however these modes can couple at non-linear order.
In this study, such couplings in the crust are examined for the first time: we
derive the axisymmetric perturbation equations for second-order axial
eigenfunctions, which are sourced by axial-polar couplings at first-order, and
solve the resulting equations in the time domain. Through our studies, we find
that the second-order spectrum contains additional oscillation modes, not
predicted by the linear analysis with either axial or polar perturbations,
which can be excited to relatively large amplitudes.
