This paper presents the analysis of axisymmetric helical magnetorotational instability (HMRI) in the inviscid limit, which is relevant for astrophysical conditions. The inductionless approximation defined by zero magnetic Prandtl number is adopted to distinguish the HMRI from the standard MRI in the cylindrical Taylor-Couette flow subject to a helical magnetic field. Using a Chebyshev collocation method convective and absolute instability thresholds are computed in terms of the Elsasser number for a fixed ratio of inner and outer radii λ=2 and various ratios of rotation rates and helicities of the magnetic field. It is found that the extension of self-sustained HMRI modes beyond the Rayleigh limit does not reach the astrophysically relevant Keplerian rotation profile not only in the narrow- but also in the finite-gap approximation. The Keppler limit can be attained only by the convective HMRI mode provided that the boundaries are perfectly conducting. However, this mode requires not only a permanent external excitation to be observable but also has a long axial wavelength, which is not compatible with limited thickness of astrophysical accretion disks.
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