  Pinning of superfluid vortices to magnetic flux tubes in the outer core of a
neutron star supports a velocity difference of $\sim 10^5$ \cms\ between the
neutron superfluid and the proton-electron fluid as the star spins down. Under
the Magnus force that arises on the vortex array, vortices undergo {\em vortex
creep} through thermal activation or quantum tunneling. We examine the
hydrodynamic stability of this situation. Vortex creep introduces two
low-frequency modes, one of which is unstable above a critical wavenumber for
any non-zero flow velocity of the neutron superfluid with respect to the
charged fluid. For typical pinning parameters of the outer core, the superfluid
flow is unstable over wavelengths $\lambda\lap 10$ m and over timescales of
$\sim (\lambda/{1 m})^{1/2}$ yr down to $\sim 1$ d. The vortex lattice could
degenerate into a tangle, and the superfluid flow would become turbulent. We
suggest that superfluid turbulence could be responsible for the red timing
noise seen in many neutron stars, and find a predicted spectrum that is
generally consistent with observations.
