an essential participant in the MRI process) must be produced from the axial field by induction effects, which are proportional to the magnetic Reynolds number $Rm$ of the flow. $Rm$, in turn, is proportional to the hydrodynamic (HD) Reynolds number $Re$ according to $Rm = Pm\cdot Re$. Therefore, in order to achieve $Rm \gtrsim 1$ necessary for SMRI to operate (see e.g., [@Sano_Miyama99; @Ji_etal01]), taking into account the very small values of $Pm$, $Re \sim 10^5-10^6$ is needed. This makes SMRI experiments with TC flows extremely challenging, although evidence for slow magneto-Coriolis waves [@Nornberg_etal10] and for a free-Shercliff layer instability [@Roach_etal12] have already been obtained. Within the DRESDYN project at HZDR [@Stefani_etal12], it is planned to set-up a large liquid sodium TC experiment, which will allow to broaden the parameter range in such a way as to make SMRI reachable.

On the theoretical side, SMRI in TC flow has been extensively studied both in the linear and nonlinear regimes, with more focus on the low-$Pm$ regime as it is typical in experiments. Still, the typical Reynolds numbers $Re\sim 10^3$ used in these analyses are orders of magnitude smaller than experimental values $Re\gtrsim
10^6$, which are extremely demanding from the computational 