From helical to standard magnetorotational instability: Predictions for upcoming 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 additional 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 new cutting-edge 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 at detecting MRI in the laboratory, our results demonstrate that SMRI and its helically modified version can in principle be detected 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 characterize the continuous and monotonous transition between these two regimes. We show that H-SMRI, like HMRI, represents an overstability (traveling wave) with nonzero frequency linearly increasing with azimuthal field. Because of its relevance to finite-size flow systems in experiments, we also analyze the absolute form of H-SMRI and compare its growth rate and onset criterion with the convective one.