  We consider superfluid helium inside a container which rotates at constant
angular velocity and investigate numerically the stability of the array of
quantized vortices in the presence of an imposed axial counterflow. This
problem was studied experimentally by Swanson {\it et al.}, who reported
evidence of instabilities at increasing axial flow but were not able to explain
their nature. We find that Kelvin waves on individual vortices become unstable
and grow in amplitude, until the amplitude of the waves becomes large enough
that vortex reconnections take place and the vortex array is destabilized. The
eventual nonlinear saturation of the instability consists of a turbulent tangle
of quantized vortices which is strongly polarized. The computed results compare
well with the experiments. Finally we suggest a theoretical explanation for the
second instability which was observed at higher values of the axial flow.
