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The quest for an unambiguous detection of magnetorotational instability (MRI)in experiments is still ongoing despite recent promising results. Toconclusively identify MRI in the laboratory, a large cylindrical Taylor-Couetteexperiment with liquid sodium is under construction within the DRESDYN project.In this paper, we investigate the linear and nonlinear dynamics ofnon-axisymmetric MRI in the magnetized Taylor-Couette flow of liquid sodium,which is a model flow in this experiment. We show that the achievable highestLundquist $Lu = 10$ and magnetic Reynolds $Rm = 40$ numbers in this experimentare large enough for the linear instability of non-axisymmetric modes withazimuthal wavenumber $|m|=1$, although the corresponding critical values ofthese numbers are usually higher than those for the axisymmetric mode. Thestructure of the ensuing nonlinear saturated state and its scaling propertieswith respect to Reynolds number $Re$ are analyzed, which are important for theDRESDYN-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 thenonlinear axisymmetric MRI appears to be stable against non-axisymmetricinstabilities. By contrast, for larger $Re \gtrsim 4\times 10^4$, a rapidgrowth and saturation of the non-axisymmetric modes of nonmagnetic originoccurs, which are radially localized near the inner cylinder wall, forming aturbulent boundary layer. However, for all the parameters considered, thesaturation amplitude of these non-axisymmetric modes is always a few orderssmaller than that of the axisymmetric MRI mode. Therefore, the results of ourprevious axisymmetric study on the scaling properties of nonlinear MRI statesalso hold when non-axisymmetric modes are included.

Non-axisymmetric modes of magnetorotational and possible hydrodynamical  instabilities in the upcoming DRESDYN-MRI experiments -- linear and nonlinear  dynamics