of the Rossby number in the saturated state at $\Lambda=0.03, 0.05, 0.1$ calculated from the total angular velocity, $\Omega+u_{\phi}/r$, averaged in the azimuthal $\phi$ and axial $z$ directions. The initial radial profile of the Rossby number corresponding to the equilibrium TC flow is shown as the black dashed line. The Rayleigh line (dotted) corresponds to $Ro =
-1$.](fig3.eps)

![Parallel evolution of the energy of small perturbations at $\Lambda=0.05$. The solid line corresponds to the time-history, starting with the initially imposed TC profile (4), (i.e., for the same run shown in figure 2), while the dashed line to that starting with the modified azimuthal velocity profile, which is established in the saturated state at this $\Lambda$, having the radial distribution of $Ro$ shown in figure 3 (green line). Calculating the inclinations of these two linear curves, we infer that the latter velocity profile indeed results in an 3.77 times reduced growth rate of HMRI than the former.](fig4.eps)

Results
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We move to the analysis of the HMRI, from its linear growth to nonlinear saturation. To have an idea about the regimes of the onset and operation of HMRI in the given TC flow configuration, in figure 1 we show the growth 