feature of HMRI to survive at very small magnetic Prandtl numbers and to set in at moderate Reynolds numbers makes it an ideal playground for experimental studies with liquid metals. Indeed, shortly after its theoretical discovery, a series of specially designed TC experiments [@Stefani_etal06; @Stefani_etal09] provided the first experimental evidence of HMRI at the liquid metal facility PROMISE and reproduced the main results of the linear theory, such as the stability threshold, the wavenumber and frequency of the HMRI-wave. In the experiments, the saturation amplitude as well as the propagation speed of the HMRI wave were measured in detail as a function of the system parameters ($Re$, $Ha$, ratio of rotation frequencies of outer and inner cylinders, etc.) and some differences with the numerical results were pointed out (see e.g., figure 10 in [@Stefani_etal09]).

This prompted further theoretical studies of HMRI in TC flows both with and without endcaps, focusing more on its nonlinear development and saturation. However, even today this is still a less explored area than the linear HMRI. First axisymmetric numerical simulations explored the dependence of the established HMRI-wave amplitude and propagation speed on the Reynolds number and the rotation ratio of the inner and outer cylinders 