  The magnetorotational instability (MRI) is considered to be one of the most
powerful sources of turbulence in hydrodynamically stable quasi-Keplerian
flows, such as those governing accretion disk flows. Although the linear
stability of these flows with applied external magnetic field has been studied
for decades, the influence of the instability on the outward angular momentum
transport, necessary for the accretion of the disk, is still not well known. In
this work we model Keplerian rotation with Taylor-Couette flow and imposed
azimuthal magnetic field using both linear and nonlinear approaches. We present
scalings of instability with Hartmann and Reynolds numbers via linear analysis
and direct numerical simulations (DNS) for the two magnetic Prandtl numbers of
$1.4 \cdot 10^{-6}$ and $1$. Inside of the instability domains modes with
different axial wavenumbers dominate, resulting in sub-domains of
instabilities, which appear different for each $Pm$. The DNS show the emergence
of 1- and 2-frequency spatio-temporally oscillating structures for $Pm=1$ close
the onset of instability, as well as significant enhancement of angular
momentum transport for $Pm=1$ as compared to $Pm=1.4 \cdot 10^{-6}$.
