fields in TC liquid metal flows, had been first identified after HMRI [@Seilmayer_etal14; @Hollerbach_etal10] (see also [@Kirillov_etal14] and references therein). Using numerical simulations, Guseva et al. [@Guseva_etal15; @Guseva_etal16; @Guseva_etal17] probed much broader ranges of Reynolds and Hartmann numbers than those done for HMRI so far and identified different regimes of nonlinear saturation, from supercritical Hopf bifurcation near the linear instability threshold up to a catastrophic transition to spatio-temporal defects, which are mediated by a subcritical subharmonic Hopf bifurcation, and ultimately to turbulence. The scaling of the angular momentum transport of AMRI in the nonlinear regime with respect to Reynolds, Hartmann and Prandtl numbers was also explored in these papers.

Motivated by the experimental results of [@Stefani_etal09] and by the recent progress on understanding the nonlinear dynamics of AMRI [@Guseva_etal15], in this paper we investigate the evolution of HMRI, from its linear exponential growth to nonlinear saturation in liquid metal TC flows at small magnetic Prandtl numbers using numerical simulations. We consider infinite cylinders with periodic boundary conditions in the axial direction in order to avoid complications because of the endcaps (Ekman pumping) and concentrate on the intrinsic dynamics of HMRI driven by the combination of an imposed helical magnetic field 