  We numerically compute axisymmetric Taylor-Couette flow in the presence of
axially periodic magnetic fields, with Hartmann numbers up to $Ha^2=10^7$. The
geometry of the field singles out special field lines on which Shercliff layers
form. These are simple shear layers for insulating boundaries, versus
super-rotating or counter-rotating layers for conducting boundaries. Some field
configurations have previously studied spherical analogs, but fundamentally new
configurations also exist, having no spherical analogs. Finally, we explore the
influence of azimuthal fields $B_\phi\sim r^{-1}{\bf\hat e}_\phi$ on these
layers, and show that the flow is suppressed for conducting boundaries but
enhanced for insulating boundaries.
