From helical to standard magnetorotational instability: predictions for upcoming liquid sodium experiments

We conduct a linear analysis of axisymmetric magnetorotational instability (MRI) in a magnetized
cylindrical Taylor-Couette (TC) flow for its standard version (SMRI) with a purely axial background
magnetic field and two further types – helically modified SMRI (H-SMRI) and helical MRI (HMRI)
– in the presence of combined axial and azimuthal magnetic fields. This study is intended as
preparatory for upcoming large-scale liquid sodium MRI experiments planned within the DRESDYN
project at Helmholtz-Zentrum Dresden-Rossendorf, so we explore these instability types for typical
values of the main parameters: the magnetic Reynolds number, the Lundquist number and the ratio
of the angular velocities of the cylinders, which are attainable in these experiments. In contrast to
previous attempts of detecting MRI in the lab, our results demonstrate that SMRI and its helically
modified version can in principle be detectable in the DRESDYN-TC device for the range of the
above parameters, including the astrophysically most important Keplerian rotation, despite the
extremely small magnetic Prandtl number of liquid sodium. Since in the experiments we plan to
approach (H-)SMRI from the previously studied HMRI regime, we characterise the continuous and
monotonous transition between the both regimes. We show that H-SMRI, like HMRI, represents
an overstability (travelling wave) with non-zero frequency linearly increasing with azimuthal field.
Because of its relevance to finite size flow systems in experiments, we also analyse the absolute form
of H-SMRI and compare its growth rate and onset criterion with the convective one.