  We study the convective and absolute forms of azimuthal magnetorotational
instability (AMRI) in a Taylor-Couette (TC) flow with an imposed azimuthal
magnetic field. We show that the domain of the convective AMRI is wider than
that of the absolute AMRI. Actually, it is the absolute instability which is
the most relevant and important for magnetic TC flow experiments. The absolute
AMRI, unlike the convective one, stays in the device, displaying a sustained
growth that can be experimentally detected. We also study the global AMRI in a
TC flow of finite height using DNS and find that its emerging butterfly-type
structure -- a spatio-temporal variation in the form of upward and downward
traveling waves -- is in a very good agreement with the linear stability
analysis, which indicates the presence of two dominant absolute AMRI modes in
the flow giving rise to this global butterfly pattern.
