is normalized by the imposed axial field $B_{0z}$. The Reynolds number is defined in terms of the rotation rate of the inner cylinder $$Re=\frac{\Omega_id^2}{\nu}.$$ Its value is chosen to be $Re = 6000$ throughout the paper, which is within the range used in the related experiments [@Stefani_etal09]. In this paper, we consider a highly resistive fluid with very small magnetic Prandtl number, $Pm=\nu/\eta=1.4\times 10^{-6}$, typical to the liquid metal alloy GaInSn, so that the corresponding magnetic Reynolds number of the flow is also small, $Rm=Re\cdot Pm=8.4\times 10^{-3}$.

The strength of the imposed axial field is measured by the Hartmann number, $$Ha=\frac{B_{0z}d}{\sqrt{\mu_0\rho_0\nu\eta}},$$ whereas the strength of the azimuthal field relative to the vertical one is measured by the helicity $\beta$ parameter introduced above. Everywhere below this parameter will be fixed to $\beta=4.53$, as adopted in some of the experiments [@Stefani_etal09]. Previous linear analysis showed that HMRI is effective for relatively strong azimuthal fields, $\beta \gtrsim 1$ (e.g., [@Hollerbach_Ruediger05; @Liu_etal06; @Kirillov_Stefani10; @Kirillov_etal14]).

Consider perturbations of the velocity, pressure and magnetic field about the equilibrium, ${\bf u}'={\bf u}-{\bf U}_0$, $p'=p-p_0$, ${\bf b}'={\bf B}-{\bf B}_0$. In the limit $Rm \ll 1$, which applies here, the magnetic field perturbation induced by the perturbed flow is 