  The quest for an unambiguous detection of magnetorotational instability (MRI)
in experiments is still ongoing despite recent promising results. To
conclusively identify MRI in the laboratory, a large cylindrical Taylor-Couette
experiment with liquid sodium is under construction within the DRESDYN project.
In this paper, we investigate the linear and nonlinear dynamics of
non-axisymmetric MRI in the magnetized Taylor-Couette flow of liquid sodium,
which is a model flow in this experiment. We show that the achievable highest
Lundquist $Lu = 10$ and magnetic Reynolds $Rm = 40$ numbers in this experiment
are large enough for the linear instability of non-axisymmetric modes with
azimuthal wavenumber $|m|=1$, although the corresponding critical values of
these numbers are usually higher than those for the axisymmetric mode. The
structure of the ensuing nonlinear saturated state and its scaling properties
with respect to Reynolds number $Re$ are analyzed, which are important for the
DRESDYN-MRI experiment having very high $Re \gtrsim 10^6$. It is shown that for
$Re \lesssim 4\times 10^4$, the non-axisymmetric MRI modes eventually decay,
since the modified shear profile of the mean azimuthal velocity due to the
nonlinear axisymmetric MRI appears to be stable against non-axisymmetric
instabilities. By contrast, for larger $Re \gtrsim 4\times 10^4$, a rapid
growth and saturation of the non-axisymmetric modes of nonmagnetic origin
occurs, which are radially localized near the inner cylinder wall, forming a
turbulent boundary layer. However, for all the parameters considered, the
saturation amplitude of these non-axisymmetric modes is always a few orders
smaller than that of the axisymmetric MRI mode. Therefore, the results of our
previous axisymmetric study on the scaling properties of nonlinear MRI states
also hold when non-axisymmetric modes are included.
