the dimensionless parameter characterizing the helicity of the field. We ignore the effects of endcaps in order to gain insight into the basic/intrinsic nonlinear evolution of HMRI. The ratio of the radii $a\equiv
R_i/R_o=0.5$ and the height of the cylinders, $L_z=10d$, where $d=R_o-R_i=R_i$ is the gap width, are chosen as in PROMISE [@Stefani_etal09]. An unperturbed flow between the cylinders, as follows from Eqs. (1)-(3), is a standard vertically uniform and axisymmetric TC flow, ${\bf U}_0=(0,r\Omega(r),0)$, with the angular velocity given by $$\Omega(r)=\frac{\Omega_oR_o^2-\Omega_iR_i^2}{R_o^2-R_i^2}+\frac{\Omega_i-\Omega_o}{R_o^2-R_i^2}\frac{R_i^2R_o^2}{r^2},$$ pressure $p_0(r)$ and constant density $\rho_0$, satisfying the hydrostatic balance equation $\rho_0r\Omega^2=dp_0/dr$. The rotation ratio of the cylinders is fixed to $\mu=\Omega_o/\Omega_i=0.27$, slightly larger than the Rayleigh line $\mu_c=0.25$ at $R_i/R_o=0.5$, ensuring that purely HD instabilities are excluded, so that the flow can become unstable solely due to the presence of the magnetic field. In the following, we introduce the nondimensional variables by using the gap width $d$ as the unit of length, $\Omega_i^{-1}$ as the unit of time, $\Omega_id$ as the unit of velocity and $\rho_0\Omega_i^2d^2$ as the unit of pressure and energy density. (Since in the present case $d=R_i$, this velocity scale is in fact the rotational velocity of the inner cylinder, $\Omega_iR_i$.) The magnetic field 