We analyze numerically the magnetorotational instability of a Taylor-Couette flow in a helical magnetic field [ helical magnetorotational instability ( HMRI ) ] using the inductionless approximation defined by a zero magnetic Prandtl number ( \textrm { 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 wavelength 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 .