and differential rotation of the flow. Distinctly from the above-mentioned previous studies on nonlinear HMRI, we do not restrict ourselves to axisymmetric perturbations only and allow non-axisymmetric modes to naturally develop during dynamical evolution, so that we can quantify the degree of non-axisymmetry of the saturated state of HMRI depending on the system parameters. It is well known that HMRI is axisymmetric in the linear regime, i.e. the most unstable modes do not have any azimuthal variation [@Hollerbach_Ruediger05; @Priede_etal07; @Priede11; @Child_Hollerbach16], and it is therefore important to know if this feature is retained in the nonlinear regime, which in turn can shed light on the saturation mechanism. We show that different regimes of the saturation are realized in the flow and analyze the characteristics of these states, such as angular momentum transport and energy spectra. This study, aiming at understanding the basic nonlinear dynamics of HMRI – underlying mechanism and properties of its saturation – is intended to be guiding and preparatory for the upcoming liquid sodium TC experiments in the DRESDYN facility at HZDR, which will combine and enhance the previous experiments on HMRI [@Stefani_etal09], AMRI [@Seilmayer_etal14] and on the current-driven kink-type Tayler instability [@Seilmayer_etal12].

The paper is organized 