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Torsional oscillations of neutron stars with highly tangled magnetic fields

To determine the frequencies of magnetic oscillations in neutron stars with highly tangled magnetic fields, we derive the perturbation equations. We assume that the field strength of the global magnetic structure is so small that such fields are negligible compared with tangled fields, which may still be far from a realistic configuration. Then, we systematically examine the spectra of the magnetic oscillations, as varying the magnetic field strength and stellar mass. The frequencies without crust elasticity are completely proportional to the strength of the magnetic field, whose proportionality constant depends strongly on the stellar mass. On the other hand, the oscillation spectra with crust elasticity become more complicated, where the frequencies even for weak magnetic fields are different from the crustal torsional oscillations without magnetic fields. For discussing spectra, the critical field strength can play an important role, and it is determined in such a way that the shear velocity is equivalent to the Alfvén velocity at the crust basis. Additionally, we find that the effect of the crust elasticity can be seen strongly in the fundamental oscillations with a lower harmonic index, $\ell$. Unlike the stellar models with a pure dipole magnetic field, we also find that the spectra with highly tangled magnetic fields become discrete, where one can expect many of the eigenfrequencies. Maybe these frequencies could be detected after the violent phenomena breaking the global magnetic field structure.