as well as the role of endcaps [@Szklarski_Ruediger06; @Liu_etal07; @Szklarski07]. However, the parameter range adopted in these studies was much narrower than that in the above experiments, being near the linear instability threshold of HMRI, when its dynamics is weakly nonlinear (see also [@Clark_Oishi16b]). More recent axisymmetric simulations [@Child_Hollerbach16; @Hollerbach_etal17] looked into the intrinsic nonlinear dynamics of HMRI in an axially infinite TC domain, avoiding complications because of the endcaps [@Avila12]. Different types of nonlinear regimes were analyzed and the applicability of the generalized quasi-linear approximation was tested by tracing the dynamics of large-scale Fourier modes. Despite these efforts, which undoubtedly contributed to a better understanding of the nonlinear dynamics of HMRI, detailed physics of its saturation and sustenance is still missing, especially when comparison with experiments is concerned. Evidently, it is this nonlinear saturated state which is observed in experiments, while the initial transient phase is much harder to identify due to the limited sensitivity of the applied ultrasonic flow measurement techniques.

In some respect, more progress has been made recently on the nonlinear dynamics of a “sibling” of HMRI – the azimuthal magnetorotational instability (AMRI). This non-axisymmetric instability, which emerges in the presence of an imposed purely azimuthal 