  We investigate magnetic Taylor--Couette flow in the presence of an imposed
axial magnetic field. First we calculate nonlinear steady axisymmetric
solutions and determine how their strength depends on the applied magnetic
field. Then we perturb these solutions to find the critical Reynolds numbers
for the appearance of wavy modes, and the related wavespeeds, at increasing
magnetic field strength. We find that values of imposed magnetic field which
alter only slightly the transition from circular--Couette flow to
Taylor--vortex flow, can shift the transition from Taylor--vortex flow to wavy
modes by a substantial amount. The results are compared against onset in the
absence of a magnetic field.
