Global three-dimensional flow of a neutron superfluid in a spherical shell in a neutron star

Abstract

We integrate for the first time the hydrodynamic Hall-Vinen-Bekarevich-Khalatnikov equations of motion of a $^{1}S_{0}$-paired neutron superfluid in a rotating spherical shell, using a pseudospectral collocation algorithm coupled with a time-split fractional scheme. Numerical instabilities are smoothed by spectral filtering. Three numerical experiments are conducted, with the following results. (i) When the inner and outer spheres are put into steady differential rotation, the viscous torque exerted on the spheres oscillates quasiperiodically and persistently (after an initial transient). The fractional oscillation amplitude ($\sim 10^{-2}$) increases with the angular shear and decreases with the gap width. (ii) When the outer sphere is accelerated impulsively after an interval of steady differential rotation, the torque increases suddenly, relaxes exponentially, then oscillates persistently as in (i). The relaxation time-scale is determined principally by the angular velocity jump, whereas the oscillation amplitude is determined principally by the gap width. (iii) When the mutual friction force changes suddenly from Hall-Vinen to Gorter-Mellink form, as happens when a rectilinear array of quantized Feynman-Onsager vortices is destabilized by a counterflow to form a reconnecting vortex tangle, the relaxation time-scale is reduced by a factor of $\sim 3$ compared to (ii), and the system reaches a stationary state where the torque oscillates with fractional amplitude $\sim 10^{-3}$ about a constant mean value. Preliminary scalings are computed for observable quantities like angular velocity and acceleration as functions of Reynolds number, angular shear, and gap width. The results are applied to the timing irregularities (e.g., glitches and timing noise) observed in radio pulsars.