energy spectra calculations in section 3.3 below. The radial boundary conditions at the cylinders are standard no-slip for the velocity perturbation, ${\bf u}|_{r=R_i,R_o}=0$, and insulating for the magnetic field perturbation, meaning that current does not penetrate into cylinders, that is, $\nabla \times {\bf
b}=0$ at $r<R_i$ and $r>R_o$. These radial boundary conditions are implemented in the code (see details in [@Willis_Barenghi02; @Willis_Barenghi02a; @Guseva_etal15]), while periodic conditions are imposed for all variables in the axial $z$-direction, with a period $L_z$, and in the azimuthal $\phi$-direction. In this first effort to gain a deeper insight into the process of the nonlinear saturation of HMRI, we vary the $\Lambda$ number at fixed $Re$ and other parameters of the flow (which is equivalent to varying $Ha$).

Energy and torque
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The perturbation energy density, composed of kinetic and magnetic parts, $e=({\bf u}^2+Ha^2Pm\cdot {\bf b}^2)/2$ (here ${\bf b}$ is the normalized magnetic field perturbation as defined in Eqs. 5-7), and torques at the cylinders, governing the evolution of angular momentum $l=ru_{\phi}$ (in nondimensional units), are among the important characteristics/diagnostics of the flow. We see that at very small $Pm$, the magnetic energy is much smaller than the kinetic one. The evolution equations for these quantities, 