  Hydrodynamic instabilities and turbulence in neutron stars have been
suggested to be related to observable spin variations in pulsars, such as spin
glitches, timing noise, and precession (nutation). Accounting for the
stabilizing effects of the stellar magnetic field, we revisit the issue of
whether the inertial modes of a neutron star can become unstable when the
neutron and proton condensates flow with respect to one another. The neutron
and proton condensates are coupled through the motion of imperfectly pinned
vorticity (vortex slippage) and vortex-mediated scattering (mutual friction).
Two-stream instabilities that occur when the two condensates rotate with
respect to one another in the outer core are stabilized by the toroidal
component of the magnetic field. This stabilization occurs when the Alfv\'en
speed of the toroidal component of the magnetic field becomes larger than the
relative rotational velocity of the condensates, corresponding to toroidal
field strengths in excess of $\simeq 10^{10}\,{\rm G}$. In contrast with
previous studies, we find that spin down of a neutron star under a steady
torque is stable. The Donnelly--Glaberson instability is not stabilized by the
magnetic field, and could play an important role if neutron stars undergo
precession.
