Supplement to "Torsional oscillations of a magnetar with a tangled magnetic field" (2016, ApJL, 823, L1)

We introduce a mean-field formalism with which to study the oscillations of a neutron star or other stellar object with a dipolar magnetic field plus a topologically distinct isotropic magnetic tangle that stabilizes the field configuration. In terms of the ratio of the energy density of the tangled field to that in the dipolar component b _t ², we obtain separable equations for the eigenfunctions and eigenfrequencies for a star of uniform density. We show that finite b _t ² breaks the Alfvén continuum that is supported by the dipolar component of the field into discrete normal modes, and we quantify the splitting. Assuming the estimated dipole fields of $7\times {10}^{14}$ G in SGR 1900+14 and $2\times {10}^{15}$ G in SGR 1806–20, and tuning b _t ² to match the lowest-frequency quasi-periodic oscillations observed to accompany flares in these objects, we infer $b_t^2=13.3$ and 0.17, respectively. The predicted spectrum is about three times denser than that observed in SGR 1900+14, and much denser in SGR 1806–20. If the dipole field of SGR 1806–20 has been overestimated, the spectrum of normal modes will be less dense. We show that for $b_t^2\gg 1$, as stability considerations suggest, the crust is nearly dynamically irrelevant. Moreover, density stratification generally shifts the eigenfrequencies up by about 20%, a change that can be mostly subsumed in a uniform density model with a slightly different choice of stellar radius or mass.