tokens: field lines and I B / I T = O (10 −6 ) for the closed field line. Field line integrals reveal the internal structure of magnetic flux tubes. By inspecting the order of magnitude of the ratio I B / I T within the flux tube, it is possible to (i) explore the internal structure of the flux tube and (ii) verify that the field-line-averaged magnetic tension inside the tube is small. Under condition (ii), the angular velocity of the fluid enclosed in the flux tube is determined by the boundary conditions on the surface of the flux tube, which are set by the surrounding fluid. The matching to the flux tube surface occurs via viscous forces on a time-scale longer than t A . 7.3 Non-axisymmetry Non-axisymmetry arises naturally even in unmagnetized spherical Couette flow (e.g. see the phase plane in fig. 1 in Nakabayashi et al. 2002a , among many other results), and in superfluid spherical Couette flow (Melatos Peralta 2007 ). In our paper, we restrict the simulations to relatively low Reynolds number ( Re 10 3 ) and to parallel rotation and magnetic axes. We do this partly for simplicity in response to computational constraints and partly because our main application (neutron stars) is arguably a low-Reynolds-number problem with 10 −10 ≤ ΔΩ/Ω ≤ 10 −5 from measurements of rotational glitches (Espinoza et al. 2011 ). However, the low- Re assumption is certainly debatable. One can also argue that the problem is high-Re because the poorly known effective viscosity of the neutron superfluid may be lower than expected, and glitches may only relax a small fraction of the underlying shear. In that case, the flow may be fully turbulent (Melatos Peralta 2007 ), and a different style of study would be needed. Assuming low Re and parallel axes, the solution is axisymmetric, and the problem is effectively bidimensional. To test this assertion, we have performed three-dimensional runs, and verified that the solution is independent of the azimuthal coordinate φ . We have performed test runs with 180 points in the radial direction, 150 points in the θ direction, and 20 points in the φ direction for the configuration ΔΩ = 0.1 and B 0 = 1.5 × 10 −2 . Both the radial velocity and the azimuthal velocity (not plotted here) are independent of the φ variable at r = 1.04 and θ = 0.61, 0.81, and 1.21 rad. So are all the other dependent variables, e.g. magnetic field components. The system starts axisymmetric and remains so as it evolves. There are two astrophysically realistic ways to drive non-axisymmetry in the flow pattern and magnetic geometry. One way would be to tilt the magnetic axis with respect to the rotational axis, as implied by the radio polarization swings observed in pulsars. The numerical complexity of the problem depends strongly on B 0 , Re , the tilt angle α , and the grid resolution in θ and φ required to resolve turbulent meridional flows and non-axisymmetries. We perform some three-dimensional test runs with resolution 202 × 176 × 50 (in r, θ , and φ ,
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