  The magnetorotational instability (MRI) in cylindrical Taylor-Couette flow
with external helical magnetic field is simulated for infinite and finite
aspect ratios. We solve the MHD equations in their small Prandtl number limit
and confirm with time-dependent nonlinear simulations that the additional
toroidal component of the magnetic field reduces the critical Reynolds number
from $O(10^6)$ (axial field only) to $O(10^3)$ for liquid metals with their
small magnetic Prandtl number. Computing the saturated state we obtain velocity
amplitudes which help designing proper experimental setups. Experiments with
liquid gallium require axial field $\sim 50$ Gauss and axial current $\sim 4$
kA for the toroidal field. It is sufficient that the vertical velocity $u_z$ of
the flow can be measured with a precision of 0.1mm/s.
  We also show that the endplates enclosing the cylinders do not destroy the
traveling wave instability which can be observed as presented in earlier
studies. For TC containers without and with endplates the angular momentum
transport of the MRI instability is shown as to be outwards.
