Nonaxisymmetric modes of magnetorotational and possible hydrodynamical instabilities in the upcoming DRESDYN-MRI experiments: Linear and nonlinear dynamics

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. Recently, we have analyzed the nonlinear dynamics and scaling properties of axisymmetric standard MRI with an axial background magnetic field in the context of the DRESDYN-MRI experiment. In this sequel paper, we investigate the linear and nonlinear dynamics of nonaxisymmetric MRI in the same magnetized Taylor-Couette flow of liquid sodium. We show that the achievable highest Lundquist $\text{Lu}=10$ and magnetic Reynolds $\text{Rm}=40$ numbers in this experiment are large enough for the linear instability of nonaxisymmetric modes with azimuthal wave number $\left|m\right|=1$, 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 $\text{Re}\gtrsim {10}^6$. It is shown that for $\text{Re}\lesssim 4\times {10}^4$, the nonaxisymmetric 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 nonaxisymmetric instabilities. By contrast, for larger $\text{Re}\gtrsim 4\times {10}^4$, a rapid growth and saturation of the nonaxisymmetric 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 nonaxisymmetric 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 nonaxisymmetric modes are included.

Figure 1

A sketch of a cylindrical Taylor-Couette flow setup with an axial background magnetic field.

Figure 2

Growth rate, $\mathrm{Re}\left(\gamma \right)>0$, of the nonaxisymmetric $\left|m\right|=1$ mode of SMRI for $\text{Pm}={10}^{-5}$ and (a) $\mu =0.26$, (b) $\mu =0.27$, (c) $\mu =0.30$, (d) $\mu =0.32$, (e) $\mu =0.35$. (f) The marginal stability curves for the $\left|m\right|=1$ mode at these $\mu$.

Figure 3

Marginal stability curves for the axisymmetric $m=0$ (solid lines) and nonaxisymmetric $\left|m\right|=1$ (dashed lines) SMRI modes at different $\mu =0.27\left(\text{black}\right),0.30\left(\text{red}\right),0.35\left(\text{blue}\right)$, and $\text{Pm}={10}^{-5}$. For all $\mu , m=0$ mode has much lower critical ${\text{Lu}}_c$ and ${\text{Rm}}_c$ than the $\left|m\right|=1$ mode. Points A, B, and C, which lie, respectively, outside, near and inside the marginal stability curve of the $\left|m\right|=1$ mode for the classical TC profile with quasi-Keplerian rotation $\mu =0.35$, are used for the nonlinear analysis below.

Figure 4

Structure of the (a) radial velocity $u_r$, (b) axial velocity $u_z$, (c) radial magnetic field $b_r$, and (d) axial magnetic field $b_z$ eigenfunctions in the $(r,z)$ plane for $m=1$ mode of SMRI at $\mu =0.35,\text{Lu}=6.78,\text{Rm}=35,\text{Pm}={10}^{-5}$ (point C in Fig. 3) and $k_z=1.88$. A zoomed in axial velocity $u_z$ at the cylinder boundaries in panel (b) confirms the validity of the no-slip condition for velocity.

Figure 5

Evolution of the magnetic energy of axisymmetric $m=0$ modes, ${\mathcal{E}}_{\text{mag}}\left(0\right)$ (red), of all the nonaxisymmetric $\left|m\right|\ge 1$ modes, $E_{\text{mag},\left|m\right|\ge 1}$ (black), and their sum $E_{\text{mag}}$ (dashed) for $\text{Lu}=5, \text{Rm}=20$ (point A in Fig. 3) and (a) $\text{Re}=3\times {10}^4$, (b) $4\times {10}^4$ and (c) $6\times {10}^4$. The nonaxisymmetric modes are linearly stable and hence initially decay for all Re in the exponential growth phase of the axisymmetic SMRI mode. As seen in panels (b, c), a rapid growth of the nonaxisymmetric modes occurs at large enough Re (or small Pm) just when the dominant axisymmetric SMRI mode saturates and, as a result, the mean azimuthal flow profile changes (deviates) from the original TC flow (1). At $\text{Re}=6\times {10}^4$ in panel (c), the nonaxisymmetric modes' total magnetic energy does not decay after the growth and settles down to a constant value, which is still several orders smaller than that of the axisymmetric mode.

Figure 6

Same as in Fig. 5 but for $\text{Lu}=6.78,\text{Rm}=35$ (point C in Fig. 3) and (a) $\text{Re}={10}^4$, (b) $2\times {10}^4$, and (c) $4\times {10}^4$. Note that the point C is inside the marginal stability curve of $\left|m\right|=1$ mode obtained from the original TC profile (1) and therefore this mode grows for all Re in the initial linear regime. As in Fig. 5, much steeper amplification of the nonaxisymmetric mode energy occurs at higher Re during the saturation of the axisymmetric mode as seen in panels (b), (c), which in the latter case does not decay and saturates at a constant value. The time moments denoted by E, F, G, and H in panel (c) will be used as the reference moments in Sec. 4b.

Figure 7

Evolution of the azimuthal spectral magnetic energy density ${\mathcal{E}}_{\text{mag}}\left(m\right)$ for different $m\in [0,14]$ at the same $\text{Lu}=6.78, \text{Rm}=35$ and Re as in Fig. 6. In the initial linear regime, only $\left|m\right|=1$ mode grows exponentially, because the point C lies in the unstable regime of this mode, while other nonaxisymmetric $\left|m\right|>1$ modes are linearly stable for all considered Re. For the smallest $\text{Re}={10}^4$ (a), all the nonaxisymmetric $\left|m\right|\ge 1$ modes eventually decay in the saturated state. These modes undergo rapid growth at higher Re during the saturation of the axisymmetric mode. For all $m$ this growth is only transient at $\text{Re}=2\times {10}^4$ (b) and decays to a noise level, whereas it saturates at $\text{Re}=4\times {10}^4$ (c) at about the same level though orders of magnitude smaller than that of the axisymmetric one.

Figure 8

Axially integrated magnetic energy $b^2(r,m)$ (see text) as a function of $r$ in the saturated state at different $m\in [1,19]$ for (a) $\text{Lu}=6,\text{Rm}=30,\text{Re}=6\times {10}^4$ and (b) $\text{Lu}=6.78,\text{Rm}=35,\text{Re}=4\times {10}^4$ (points B and C in Fig. 3, respectively). From panels (a, b) it is clear that the larger-$m$ nonaxisymmetric modes are more concentrated near the inner cylinder wall, indicating that they are mainly sustained by high shear $q$ values at this boundary.

Figure 9

Radial and axial velocity, ($u_r,u_z$), and magnetic field, ($b_r,b_z$), structures in the ($r,z$) plane for $\text{Lu}=6.78, \text{Rm}=35$ and $\mathrm{Re}=4\times {10}^4$ at different evolution times of the axisymmetric and nonaxisymmetric modes corresponding to the time moments F, G, and H in Fig. 6. Panels (a)–(d) and (f)–(i) show only nonaxisymmetric modes at moments F and G, respectively. (k)–(n) show all the modes in the saturated state at moment H. Rightmost panels (e), (j), (o) show the distribution of the local shear parameter $q=-\partial \ln \mathrm{\Omega }/\partial \ln r$ in the ($r,z$) plane. Orange and green dashed lines are plotted at $z=4.16$ and $z=2.52$ and analysed in Figs. 11 and 12, respectively.

Figure 10

Evolution of the ratio of the kinetic to magnetic energies of nonaxisymmetric modes, $E_{\text{kin},\left|m\right|\ge 1}/E_{\text{mag},\left|m\right|\ge 1}$ for $\text{Lu}=5, \text{Rm}=20, \text{Re}=6\times {10}^4$ (red) and $\text{Lu}=6.78, \text{Rm}=35, \text{Re}=4\times {10}^4$ (blue) corresponding to Figs. 5 and 6, respectively. Dashed green lines mark the onset moment of the nonaxisymmetric nonmagnetic (non-MRI) modes.

Figure 11

Radial and axial velocity ($u_r,u_z$) and magnetic field ($b_r,b_z$) structures in the $(r,\varphi )$ plane in the saturated state at $z=4.16$ shown by orange dashed line in the bottom row of Fig. 9 for $\text{Lu}=6.78, \text{Rm}=35,$ and $\mathrm{Re}=4\times {10}^4$.

Figure 12

Same as Fig. 11 but at $z=2.52$ shown by the green dashed line in the bottom row of Fig. 9.

Figure 13

(a) Total magnetic energy of all $m$ and $k_z$ modes, $E_{\text{mag}}={\sum }_m{\mathcal{E}}_{\text{mag}}$, and (b) torque $G/G_{\text{lam}}-1$ in the saturated state as a function of Re at point A with $\text{Lu}=5, \text{Rm}=20$ (blue lines) and point C with $\text{Lu}=6.78, \text{Rm}=35$ (green lines). Dashed lines are the power-law fits. Vertical red dotted line indicates the change in the scaling behavior of these quantities. (c) Evolution of the magnetic energy of all the nonaxisymmetric $\left|m\right|\ge 1$ modes, $E_{\text{mag},\left|m\right|\ge 1}$, for point C and different $\text{Re}=\{1,2,3,4,6,8\}\times {10}^4$.

Figure 14

Growth rate, $\mathrm{Re}\left(\gamma \right)$, versus the axial mode number $n_z=k_z{L}_z/2\pi$ for the $\left|m\right|=1$ mode of SMRI at $\text{Lu}=6.78, \text{Rm}=35, \mu =0.35$, and different $\text{Pm}={10}^{-5},{10}^{-4}, 4\times {10}^{-4}$.

Figure 15

Radial and axial velocity ($u_r,u_z$) and magnetic ($b_r,b_z$) field structures of the nonaxisymmetric modes in the ($r,z$) plane during the early exponential growth stage [moment E in Fig. 6] obtained from the nonlinear simulations, which are dominated by the $\left|m\right|=1$ SMRI mode. The parameters are: $\mu =0.35, \text{Lu}=6.78, \text{Rm}=35$ (point C in Fig. 3), and $\text{Re}=4\times {10}^4$.

Figure 16

Azimuthal spectra of the magnetic energy density, ${\mathcal{E}}_{\text{mag}}$, versus $m$ for different resolutions $N_{\varphi }=\{10,20,40\}$ in the azimuthal and a fixed resolution $N_z=480$ in the axial directions at $\text{Lu}=6.78, \text{Rm}=35$ (point C in Fig. 3), and different (a) $\mathrm{Re}=2\times {10}^4$, (b) $4\times {10}^4$, (c) $6\times {10}^4$, and (d) $8\times {10}^4$. It is seen that the most energy-containing nonaxisymmetric modes with $\left|m\right|\lesssim 10$ are well resolved from $N_{\varphi }=20$, exhibiting the convergence of the spectra at these high Re.

Figure 17

Axial spectra of the magnetic energy density for (a), (c) all $m$ modes, ${\mathcal{E}}_{\text{mag},\left|m\right|\ge 0}$ and (b), (d) for nonaxisymmetric $\left|m\right|\ge 1$ modes, ${\mathcal{E}}_{\text{mag},\left|m\right|\ge 1}$ versus $k_z$ at different resolutions $N_z=\{240,480,800\}$ in the axial and a fixed resolution $N_{\varphi }=20$ in the azimuthal directions at the same $\text{Lu}=6.78$ and $\text{Rm}=35$ as in Fig. 16 and different (a), (b) $\mathrm{Re}=4\times {10}^4$ and (c), (d) $8\times {10}^4$. It is seen that both axial spectra well converge with resolution.

Figure 18

Same as in Fig. 5 but at $\text{Lu}=6$ and $\text{Rm}=30$ (point B in Fig. 3) at different (a) $\mathrm{Re}=2\times {10}^4$, (b) $3\times {10}^4$, (c) $4\times {10}^4$, and (d) $6\times {10}^4$.

Figure 19

Evolution of the azimuthal spectrum of the magnetic energy density, ${\mathcal{E}}_{\text{mag}}\left(m\right)$, for different $m\in [0,14]$ and $\text{Lu}=6, \text{Rm}=30$, and (a) $\text{Re}=3\times {10}^4$, (b) $4\times {10}^4$, and (c) $6\times {10}^4$.

Figure 20

Same as Figs. 13 and 13 but for higher $\text{Lu}=8.56$ and $\text{Rm}=40$. Blue dotted line shows the change in the scaling behavior for the saturated magnetic energy $E_{\text{mag}}$ and torque $G/G_{\text{lam}}-1$ due to the presence of stronger nonaxisymmetric non-MRI modes at these relatively high Lu and $\text{Rm}$.