  Magnetorotational instability (MRI) is the most likely mechanism driving
angular momentum transport in astrophysical disks. However, despite many
efforts, a conclusive experimental evidence of MRI is still missing. Recently,
performing 1D linear analysis of the standard MRI (SMRI) in a cylindrical
Taylor-Couette (TC) flow with an axial magnetic field, we showed that SMRI can
be detected in the upcoming DRESDYN-MRI experiment based on a magnetized TC
flow of liquid sodium. In this study, also related to DRESDYN-MRI experiments,
we focused on the nonlinear evolution and saturation properties of SMRI and
analyzed its scaling behavior with respect to the main parameters of the TC
flow. We did a detailed analysis over the extensive ranges of magnetic Reynolds
number $Rm\in [8.5, 37.1]$, Lundquist number $Lu\in[1.5, 15.5]$ and Reynolds
number, $Re\in[10^3, 10^5]$. We considered small magnetic Prandtl numbers, $Pm
\ll 1$, down to $Pm\sim 10^{-4}$, aiming at values typical of liquid sodium in
the experiments. In the saturated state, the magnetic energy of SMRI and torque
due to perturbations on the cylinders, which characterizes angular momentum
transport, both increase with $Rm$ for fixed $(Lu, Re)$, while for fixed $(Lu,
Rm)$, the magnetic energy decreases and torque increases with increasing $Re$.
We studied the scaling of the magnetic energy and torque in the saturated state
as a function of $Re$ and find a power law dependence $Re^{-0.6...-0.5}$ for
the magnetic energy and $Re^{0.4...0.5}$ for the torque at all $(Lu, Rm)$ and
high $Re\geq 4000$. We also explored the dependence on Lundquist number and
angular velocity of the cylinders. These scaling laws will be instrumental in
the subsequent analysis of more realistic finite-length TC flows and comparison
of numerical results with those obtained from the DRESDYN-MRI experiments to
unambiguously identify SMRI in laboratory.
