decrease. These two torques are in balance, resulting in the stationary rotational profile (4). Below we measure the torque in the perturbed state against the laminar torque, i.e., consider the ratio $G_{i,o}/G_{lam}$ and redefine $G_{i,o}/G_{lam} \rightarrow
G_{i,o}$. Its advantage is that it does not depend on the normalization of the velocity.

![Growth rate (in units of $\Omega_i$) of the most unstable mode, which is axisymmetric ($m=0$), as a function of $\Lambda$ and $Re$ (the white area is the stable region) derived from the linear stability analysis. Note that at sufficiently high $Re\gtrsim 3000$, the region of maximum growth is practically independent of $Re$ and is determined mainly by $\Lambda$. The horizontal gray line corresponds to the range of $\Lambda$ at fixed $Re=6000$ for which our direct numerical simulations have been performed. At this value of $Re$, HMRI sets in at $\Lambda_c=0.022$, reaches a maximum at $\Lambda=0.15$ and then decreases again, disappearing at $\Lambda=1.38$.](fig1.eps)

![Evolution of the volume-averaged (a) kinetic energy and (b,c) torques at the cylinders at different $\Lambda=0.03, 0.05,
0.08, 0.1$ (i.e., $Ha=13.4, 17.3, 21.9, 24.5$). After the initial exponential growth phase, these quantities settle down into a sustained quasi-steady state, where they oscillate around constant values.](fig2.eps)

![Radial profile 