Nonaxisymmetric simulations of the Princeton magnetorotational instability experiment with insulating and conducting axial boundaries

Stability and nonlinear evolution of rotating magnetohydrodynamic flows in the Princeton magnetorotational instability (MRI) experiment are examined using three-dimensional non-axisymmetric simulations. In particular, the effect of axial boundary conductivity on a free Stewartson-Shercliff layer (SSL) is numerically investigated using the spectral finite-element Maxwell and Navier Stokes (SFEMaNS) code. The free SSL is established by a sufficiently strong magnetic field imposed axially across the differentially rotating fluid with two rotating rings enforcing the boundary conditions. Numerical simulations show that the response of the bulk fluid flow is vastly different in the two different cases of insulating and conducting end caps. We find that, for the insulating end caps, there is a transition from stability to instability of a Kelvin-Helmholtz-like mode that saturates at an azimuthal mode number $m=1$, whereas for the conducting end caps, the reinforced coupling between the magnetic field and the bulk fluid generates a strong radially localized shear in the azimuthal velocity resulting in axisymmetric Rayleigh-like modes even at reduced thresholds for the axial magnetic field. For reference, three-dimensional nonaxisymmetric simulations have also been performed in the MRI unstable regime to compare the modal structures.

Figure 1

Initialization of the simulation domain. The inner cylinder is composed of an insulating shell with stainless steel ends, and the outer cylinder is composed of stainless steel in the experiments and an insulator in the simulations. The inner and outer rings are composed of copper. The working fluid is GaInSn.

Figure 2

Visualization of the parameter space involved in the simulations and experiments with the curve representing marginal stability and the shaded area representing the MRI unstable regime. The red and blue diamonds, respectively, represent the split-stable experiment and simulations that were conducted in the slow-rotation regime. The black dot represents the MRI simulation in the fast-rotation regime with the background field $B=6500\mathrm{G}$ and the differential rotation rate $\mathrm{\Delta }\mathrm{\Omega }={\mathrm{\Omega }}_1-{\mathrm{\Omega }}_2=4250\mathrm{rpm}$.

Figure 3

Plots of the fluid rotation rate $\mathrm{\Omega }$ as a function of radius taken at the midplane ($z=0$) of the system. (a) Initialization of the hydrodynamically stable initial state is performed by relaxing a piecewise uniform state matched at the rotation rates of the boundaries to more precisely emulate real experimental fluid flows. The simulation flow for the split-stable case is indicated by a dashed blue line, the experimental flow by a dotted red line, and the ideal Taylor-Couette solution by a solid black line. The experimental flow is collected using ultrasound Doppler velocimetry from a run with the same ratio of inner and outer components but at a slightly higher rotation rate. (b) Reference MRI unstable case with piecewise solid body (PSB) initial state and the relaxed rotation profile. The relaxed rotation profile is the effective initial state, which could trigger MRI for this reference case.

Figure 4

Calculated volumetrically averaged kinetic energy contributions for the S-S insulating boundary simulations from azimuthal modes $m=1$ to $m=4$. Higher azimuthal modes are not active.

Figure 5

The mode structures of the split-stable insulating simulation calculated for the normalized fluctuating azimuthal velocity $V_{\theta ,\mathrm{fl}}=(V_{\theta }-V_{\mathrm{mean}})/V_{\mathrm{mean}}$. (a) shows the azimuthal mode structure at the midplane ($z=0$), and (b) shows the axial ($r-z$) mode structure and streamlines at the azimuthal cross section $\theta =\pi /2$, (c) shows poloidal streamlines. The computational data were taken at $t=2.1\mathrm{s}$ when the $m=1$ contribution was dominant.

Figure 6

Calculated volumetrically averaged kinetic energy contributions for conducting boundary S-S simulations. (a) Azimuthal modes $m=1$ to $m=4$, and (b) $m=0$, and the inset shows the flow profile before initialization and after the $m=0$ mode growth. Higher azimuthal modes are not active.

Figure 7

The mode structures of the S-S conducting simulation calculated for the fluctuating radial and axial velocity $V_r,V_z$. (a) shows the azimuthal mode structure of $V_r$ at the midplane ($z=0$) and halfway to the axial boundaries ($z=1$), and (b) shows the axial mode structures of $V_r,V_z$ and streamlines at the azimuthal cross section $\theta =\pi /2$. The data were taken at $t=0.1\mathrm{s}$ when the $m=0$ contribution was growing rapidly.

Figure 8

The current responses $J_r$ to the background magnetic field for the S-S conducting and insulating simulations. The current response and resulting Lorentz forces are strong and immediate for conducting case whereas the current response is weak for the insulating case.

Figure 9

$q=-\partial ln\mathrm{\Omega }/\partial lnr$ plots at times $t=0.1\mathrm{s}$ colored in black and $t=0.7\mathrm{s}$ colored in red, taken at the midplane for the S-S simulations. The hydrodynamically stable initial state is colored in green. Early time behavior suggests that the response of the shear is rapid in the conducting case, whereas it is slower for the insulating case. Later time behavior has the shear above the Rayleigh threshold $q=2$ for both the conducting and the insulating cases, but the increased shear is maintained in the conducting case whereas it is flattened for the insulating case as the K-H mode develops.

Figure 10

$q=-\partial ln\mathrm{\Omega }/\partial lnr$ plots versus radius at late times, taken at the midplane for the S-S simulations and experiments. $q$ values above 2 are linearly unstable to Rayleigh's centrifugal instability. The black line indicates the $q$ profile of the conducting end caps, whereas the red line indicates the $q$ profile of the insulating end caps. The blue line indicates the $q$ profile of the MRI unstable configuration with conducting end caps.

Figure 11

The evolution of volumetrically averaged $B_r$ (MRI signal) for the MRI unstable and S-S simulations. The MRI signal saturates at a much higher threshold in the MRI unstable configuration compared to the split-stable configuration. Slight differences in the MRI signal evolution for the 2D and 3D cases are due to computational mesh size; the 3D simulations are conducted with a mesh that is four times finer, so the initial perturbation is smaller than the 2D simulations.

Figure 12

Axial ($r-z$) mode structures of (a) $V_r$, (b) $V_z$, (c) $B_r$, and (d) $B_{\theta }$ for the reference MRI unstable simulation. Structures remain axisymmetric even in the presence of nonaxisymmetric modes in the simulations.

Figure 13

Azimuthal mode structures of $B_r$ and $V_r$ of the S-S and MRI unstable simulations. Mode structures of $V_r$ are similar in both cases with differences in radial distributions. The azimuthal mode structures of $B_r$ of the MRI unstable configuration and the split-stable configuration are vastly different from each other with a strong dominant axisymmetric $m=0$ component in the MRI unstable configuration compared to a weak fluctuation in the split-stable configuration.