We analyze numerically the magnetorotational instability of a Taylor-Couette
flow in a helical magnetic field (HMRI) using the inductionless approximation
defined by a zero magnetic Prandtl number (Pm=0). The Chebyshev collocation
method is used to calculate the eigenvalue spectrum for small amplitude
perturbations. First, we carry out a detailed conventional linear stability
analysis with respect to perturbations in the form of Fourier modes that
corresponds to the convective instability which is not in general
self-sustained. The helical magnetic field is found to extend the instability
to a relatively narrow range beyond its purely hydrodynamic limit defined by
the Rayleigh line. There is not only a lower critical threshold at which HMRI
appears but also an upper one at which it disappears again. The latter
distinguishes the HMRI from a magnetically-modified Taylor vortex flow. Second,
we find an absolute instability threshold as well. In the hydrodynamically
unstable regime before the Rayleigh line, the threshold of absolute instability
is just slightly above the convective one although the critical wave length of
the former is noticeably shorter than that of the latter. Beyond the Rayleigh
line the lower threshold of absolute instability rises significantly above the
corresponding convective one while the upper one descends significantly below
its convective counterpart. As a result, the extension of the absolute HMRI
beyond the Rayleigh line is considerably shorter than that of the convective
instability. The absolute HMRI is supposed to be self-sustained and, thus,
experimentally observable without any external excitation in a system of
sufficiently large axial extension.Comment: 16 pages, 15 figures; minor revision, Phys. Rev. E (in press
