as follows. The basic equations are introduced in section 2. Direct numerical simulations of the nonlinear saturation and evolution of HMRI as well as the spectral characteristics of the nonlinear state are presented in section 3. Summary and conclusions are given in section 4.

Main equations
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The basic equations of non-ideal MHD governing the motion of an incompressible conductive fluid with constant kinematic viscosity $\nu$ and Ohmic resistivity $\eta$ are $$\frac{\partial {\bf u}}{\partial t}+({\bf u}\cdot \nabla) {\bf
u}=-\frac{1}{\rho}\nabla p+\frac{1}{\mu_0\rho}(\nabla \times {\bf
B})\times {\bf B} + \nu\nabla^2 {\bf u},$$ $$\frac{\partial {\bf B}}{\partial t}=\nabla\times \left( {\bf
u}\times {\bf B}\right)+\eta\nabla^2{\bf B},$$ $$\nabla\cdot {\bf u}=0,~~~\nabla\cdot {\bf B}=0,$$ where $\rho$ is the constant density, $p$ is the thermal pressure, ${\bf u}$ is the velocity, ${\bf B}$ is the magnetic field and $\mu_0$ is the magnetic permeability of vacuum.

The equilibrium state is an axisymmetric cylindrical magnetized TC flow between two coaxial cylinders with the inner radius $R_i$ and the outer radius $R_o$, rotating with the angular velocities $\Omega_i$ and $\Omega_o$, respectively. The flow is threaded by an externally imposed helical magnetic field ${\bf
B}_0=(0,B_{0\phi}(r),B_{0z})$ consisting of constant axial, $B_{0z}$, and radially varying current-free azimuthal, $B_{0\phi}(r)=\beta B_{0z}R_i/r$, components in cylindrical coordinates $(r,\phi,z)$, where $\beta$ is 