these small terms for completeness. In the chosen nondimensional units, the rotational frequency (4) of the background velocity becomes $$\Omega(r)=\frac{1}{1-a^2}\left(\mu-a^2+\frac{a^2}{(1-a)^2}\frac{1-\mu}{r^2}\right)$$ and the imposed field is ${\bf B}_0=(0,\beta/r,1)$. Since $\eta,
\mu, \beta$ and $Re$ are fixed, only $Ha$ is a free parameter, entering Eqs. (5)-(7) through the Elsasser number, or the interaction parameter: $$\Lambda\equiv \frac{Ha^2}{Re}=\frac{B_{0z}^2}{\mu_0\rho_0\eta
\Omega_i},$$ which is composed of the imposed magnetic field, rotation and resistivity and is independent of viscosity. The interaction parameter has been shown to play a special role in the linear dynamics of the HMRI, as it determines its modal growth rate, the corresponding critical wavenumber and other characteristics [@Priede11; @Kirillov_etal14; @Mamatsashvili_Stefani16]. Specifically, in the inviscid case ($Re\rightarrow \infty$), the HMRI can exist in a certain range of $\Lambda$ and disappears at sufficiently large or small values of this parameter. In the first limit, the magnetic field is too strong and stabilizes the flow, whereas in the second limit it is too weak to support the instability. At finite $Re$, this implies that for a fixed $Ha$, HMRI operates in a range of $Re$ and for a fixed $Re$, in a range of $Ha$ [@Priede_etal07; @Priede_Gerbeth09; @Ruediger_etal06], which was also confirmed experimentally [@Stefani_etal07; @Stefani_etal09]. Since $\Lambda$ 