  The helical magnetorotational instability of the magnetized Taylor-Couette
flow is studied numerically in a finite cylinder. A distant upstream insulating
boundary is shown to stabilize the convective instability entirely while
reducing the growth rate of the absolute instability. The reduction is less
severe with larger height. After modeling the boundary conditions properly, the
wave patterns observed in the experiment turn out to be a noise-sustained
convective instability. After the source of the noise resulted from unstable
Ekman and Stewartson layers is switched off, a slowly-decaying inertial
oscillation is observed in the simulation. We reach the conclusion that the
experiments completed to date have not yet reached the regime of absolute
instability.
