We investigate the global transition from a turbulent state of superfluid vorticity (quasi-isotropic vortex tangle) to a laminar state (rectilinear vortex array), and vice versa, in the outer core of a neutron star. By solving numerically the hydrodynamic Hall-Vinen-Bekarevich-Khalatnikov equations for a rotating superfluid in a differentially rotating spherical shell, we find that the meridional counterflow driven by Ekman pumping exceeds the Donnelly-Glaberson threshold throughout most of the outer core, exciting unstable Kelvin waves which disrupt the rectilinear vortex array, creating a vortex tangle. In the turbulent state, the torque exerted on the crust oscillates, and the crust-core coupling is weaker than in the laminar state. This leads to a new scenario for the rotational glitches observed in radio pulsars: a vortex tangle is sustained in the differentially rotating outer core by the meridional counterflow, a sudden spin-up event (triggered by an unknown process) brings the crust and core into corotation, the vortex tangle relaxes back to a rectilinear vortex array (in < ∼ 10 5 s), then the crust spins down electromagnetically until enough meridional counterflow builds up (after < ∼ 1 yr) to reform a vortex tangle. The turbulent-laminar transition can occur uniformly or in patches; the associated time-scales are estimated from vortex filament theory. We calculate numerically the global structure of the flow with and without an inviscid superfluid component, for Hall-Vinen (laminar) and Gorter-Mellink (turbulent) forms of the mutual friction. We also calculate the post-glitch evolution of the angular velocity of the crust and its time derivative, and compare the results with radio pulse timing data, predicting a correlation between glitch activity and Reynolds number. Terrestrial laboratory experiments are proposed to test some of these ideas.

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 pseudo-spectral 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. (1) When the inner and outer spheres are put into steady differential rotation, the viscous torque exerted on the spheres oscillates quasi-periodically and persistently (after an initial transient). The fractional oscillation amplitude ($10 À2 ) increases with the angular shear and decreases with the gap width.(2) 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 (1). The relaxation timescale is determined principally by the angular velocity jump, whereas the oscillation amplitude is determined principally by the gap width. (3) 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 timescale is reduced by a factor of $3 compared to (2), and the system reaches a stationary state in which the torque oscillates with fractional amplitude $10 À3 about a constant mean value. Preliminary scalings are computed for observable quantities such as angular velocity and acceleration as functions of the 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.

The gravitational-wave signal generated by global, nonaxisymmetric shear flows in a neutron star is calculated numerically by integrating the incompressible Navier-Stokes equation in a spherical, differentially rotating shell. At Reynolds numbers Re տ 3 #10 3 the laminar Stokes flow is unstable, and helical, oscillating Taylor-Görtler vortices develop. The gravitational-wave strain generated by the resulting kinetic energy fluctuations is computed in both plus and cross polarizations as a function of time. It is found that the signal-to-noise ratio for a coherent, 10 8 s integration with LIGO II scales as 6.5[Q * /(10 4 rad s Ϫ1 )] 7/2 for a star at 1 kpc with angular velocity Q * . This should be regarded as a lower limit: it excludes pressure fluctuations, herringbone flows, Stuart vortices, and fully developed turbulence (for Re տ 10 6 ).

We solve numerically for the first time the two-fluid Hall-Vinen-BekarevichKhalatnikov (HVBK) equations for an He-II-like superfluid contained in a differentially rotating spherical shell, generalizing previous simulations of viscous spherical Couette flow (SCF) and superfluid Taylor-Couette flow. The simulations are conducted for Reynolds numbers in the range 1 × 10 2 6 Re 6 3 × 10 4 , rotational shear 0.1 6 Ω/Ω 6 0.3, and dimensionless gap widths 0.2 6 δ 6 0.5. The system tends towards a stationary but unsteady state, where the torque oscillates persistently, with amplitude and period determined by δ and Ω/Ω. In axisymmetric superfluid SCF, the number of meridional circulation cells multiplies as Re increases, and their shapes become more complex, especially in the superfluid component, with multiple secondary cells arising for Re > 10 3 . The torque exerted by the normal component is approximately three times greater in a superfluid with anisotropic Hall-Vinen (HV) mutual friction than in a classical viscous fluid or a superfluid with isotropic Gorter-Mellink (GM) mutual friction. HV mutual friction also tends to 'pinch' meridional circulation cells more than GM mutual friction. The boundary condition on the superfluid component, whether no slip or perfect slip, does not affect the large-scale structure of the flow appreciably, but it does alter the cores of the circulation cells, especially at lower Re. As Re increases, and after initial transients die away, the mutual friction force dominates the vortex tension, and the streamlines of the superfluid and normal fluid components increasingly resemble each other. In non-axisymmetric superfluid SCF, three-dimensional vortex structures are classified according to topological invariants. For misaligned spheres, the flow is focal throughout most of its volume, except for thread-like zones where it is strain-dominated near the equator (inviscid component) and poles (viscous component). A wedge-shaped isosurface of vorticity rotates around the equator at roughly the rotation period. For a freely precessing outer sphere, the flow is equally strain-and vorticity-dominated throughout its volume. Unstable focus/contracting points are slightly more common than stable node/saddle/saddle points in the viscous component, but not in the inviscid component. Isosurfaces of positive and negative vorticity form interlocking poloidal ribbons (viscous component) or toroidal tongues (inviscid component) which attach and detach at roughly the rotation period. 222

The uniform flow past a sphere undergoing steady rotation about an axis transverse to the free stream flow was investigated numerically. The objective was to reveal the effect of sphere rotation on the characteristics of the vortical wake structure and on the forces exerted on the sphere. This was achieved by solving the time-dependent, incompressible Navier–Stokes equations, using an accurate Fourier–Chebyshev spectral collocation method. Reynolds numbers Re of 100, 250 and 300 were considered, which for a stationary sphere cover the axisymmetric steady, non-axisymmetric steady and vortex shedding regimes. The study identified wake transitions that occur over the range of non-dimensional rotational speeds Ω* = 0 to 1.00, where Ω* is the maximum velocity on the sphere surface normalized by the free stream velocity. At Re = 100, sphere rotation triggers a transition to a steady double-threaded structure. At Re = 250, the wake undergoes a transition to vortex shedding for Ω* ≥ 0.08. With an increasing rotation rate, the recirculating region is progressively reduced until a further transition to a steady double-threaded wake structure for Ω* ≥ 0.30. At Re = 300, wake shedding is suppressed for Ω* ≥ 0.50 via the same mechanism found at Re = 250. For Ω* ≥ 0.80, the wake undergoes a further transition to vortex shedding, through what appears to be a shear layer instability of the Kelvin–Helmholtz type.

The flow past a transversely rotating sphere at Reynolds numbers of $\mathit{Re}=500{-}1000$ is directly simulated using an unstructured finite volume collocated code. The effect of rotation rate on the flow is studied by increasing the dimensionless rotation rate, ${\it\Omega}^{\ast }$, from 0 to 1.20, where ${\it\Omega}^{\ast }$ is the maximum sphere surface velocity normalised by the free stream velocity. This study investigates the marked unsteadiness of the flow structures at $\mathit{Re}=500{-}1000$. Comparison with previous numerical data (Giacobello et al., J. Fluid Mech., vol. 621, 2009, pp. 103–130; Kim, J. Mech. Sci. Technol., vol. 23, 2009, pp. 578–589) reveals a new flow regime, namely a ‘shear layer–stable foci’ regime, besides the widely reported ‘vortex shedding’ and ‘shear layer instability’ regimes. The ‘shear layer–stable foci’ regime is observed at $\mathit{Re}=500$ and ${\it\Omega}^{\ast }=1.00$; $\mathit{Re}=640{-}1000$ and ${\it\Omega}^{\ast }\geqslant 0.80$. In this flow regime, the shear layer on the advancing side of the sphere (where the sphere surface velocity vector opposes the free stream velocity) shortens significantly while fluid from the retreating side (opposite to the advancing side) is drawn towards the mid-plane normal to the peripheral velocity. This results in the formation of a stable focus near the onset of the shear layer instability. This stable focus becomes more pronounced with increasing $\mathit{Re}$ and ${\it\Omega}^{\ast }$. It increases the oscillation magnitude and decreases the oscillation frequency of the hydrodynamic forces.

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