rate of the most unstable mode (which is axisymmetric) as a function of $\Lambda$ and $Re$, obtained by solving the linear stability problem with the same parameters and boundary conditions. From this figure we see that HMRI operates in a certain interval of $\Lambda$ provided the Reynolds number is larger than a critical value, $Re>Re_c=1244$. The instability interval with respect to the interaction parameter broadens with the increase of $Re$, but converges in the inviscid limit (not shown in this figure). It is also seen in figure 1 that at sufficiently high $Re\gtrsim 3000$, the maximum growth of HMRI is mainly determined by $\Lambda$ and attained at its nearly constant value. For the chosen $Re=6000$, the instability first appears in the flow at the critical value $\Lambda_c=0.022$, reaches a maximum at $\Lambda=0.15$ and then decreases again, disappearing at $\Lambda=1.38$. Having outlined the main linear features of HMRI, we can now move to nonlinear evolution.

We initialize the simulations by imposing random noise perturbations of the velocity and magnetic field on the equilibrium flow. The subsequent evolution of the total energy $E$ and torques $G_i,G_o$ are shown in figure 2 at different interaction parameters, which are all above the critical value 