Parameter space mapping of the Princeton magnetorotational instability experiment

Extensive simulations of the Princeton Magnetorotational Instability (MRI) Experiment with the Spectral/Finite Element code for Maxwell and Navier-Stokes Equations (SFEMaNS) have been performed to map the MRI-unstable region as a function of inner cylinder angular velocity and applied vertical magnetic field. The angular velocities of the outer cylinder and the end-cap rings follow the inner cylinder in fixed ratios optimized for MRI. We first confirm the exponential growth of the MRI linear phase using idealized conducting vertical boundaries (end caps) rotating differentially with a Taylor-Couette profile. Subsequently, we run a multitude of simulations to scan the experimental parameter space and find that the normalized volume-averaged mean-square radial magnetic field, our main instability indicator, rises significantly where MRI is expected. At various locations, the local radial components of fluid velocity and generated magnetic field are well correlated with the volume-averaged indicator. Based on this correlation, a diagnostic system that will measure the radial magnetic field at several locations on the inner cylinder is proposed as the main comparison between simulation and experiment. A detailed analysis of poloidal mode structures in the SFEMaNS code indicates that MRI, rather than Ekman circulation or Rayleigh instability, dominates the fluid behavior in the region where MRI is expected.

Figure 1

(a) Cross section of Princeton MRI experiment and (b) its mesh model for SFEMaNS simulations. The fluid domain has a mesh resolution of 100 by 200 triangular cells. A sparse vacuum mesh surrounds the system with radius $R_{\text{vacuum}}=20R_1$. Colors indicate different domains with purple as the fluid, blue as the copper end-cap rings, yellow as stainless steel inner cylinder rim, and red as vacuum.

Figure 2

(a) Evolutions of the MRI signal as a function of time for SFEMaNS simulations with ideal Taylor-Couette end-cap profile at $\text{Rm}=20$ with ${\mathrm{\Omega }}_2/{\mathrm{\Omega }}_1=0.1325$. (b) Growth rate comparison from WKB, linear, and SFEMaNS calculations with the same flow parameters. Growth rates from SFEMaNS simulations are calculated from the last $50{\mathrm{\Omega }}_1^{-1}$ of $150{\mathrm{\Omega }}_1^{-1}$ runs.

Figure 3

(a) MRI signal as a function of $\text{Rm}$ and (b) $B_0$ using the MRI configuration. (c) Contour plot of the MRI signal in the experimental parameter space of external applied magnetic field $B_z$ and inner cylinder angular velocity ${\mathrm{\Omega }}_1$. Green dots indicate the locations of changes in the slope ${(\partial A/\partial \text{Rm})}_{B_0}$. The green dashed line indicates the critical $B_0=0.35$, which is the smallest Lehnert number without any slope change, indicating the onset of magnetic stabilization. The solid black line indicates the boundary of the MRI-unstable region from the WKB approximation [Eq. (5)], while the dashed black line indicates the boundary from the linear calculation. The MRI signals are averaged over the last $200{\mathrm{\Omega }}_1^{-1}$ of each simulation.

Figure 4

(a) Radial velocity of fluid element at $(r,z)=(1.15R_1,0)$, or the jet velocity, as a function of $\text{Rm}$ and (b) $B_0$ using the MRI configuration. Gray shadows around the lines indicate their standard deviation of mean from the averaging process. (c) Contour plot of the jet velocity in the experimental parameter space of external applied magnetic field $B_z$ and inner cylinder angular velocity ${\mathrm{\Omega }}_1$. The jet velocities are averaged for the last $200{\mathrm{\Omega }}_1^{-1}$ of each simulation.

Figure 5

(a) Contour plot of the normalized point measurement of radial magnetic field in the experimental parameter space measured at $(r,z)=(R_1,1.57R_1)$ at the final time steps of simulations. (b) Relation between the value radial magnetic field at $(r,z)=(R_1,\pm 1.57R_1)$ at the final time steps to their averaged MRI signals from the last $200{\mathrm{\Omega }}_1^{-1}$.

Figure 6

Poloidal contours of $B_r/B_0$ for three different values of $B_0$: (a) 0.05, (b) 0.25, and (c) 0.5 at $\text{Rm}$ of 8 from their final time steps of simulations. The structures are in steady state for the last $200{\mathrm{\Omega }}_1^{-1}$.

Figure 7

Poloidal circulation imposed on top of the contour of radial velocity for three different values of $B_0$: (a) 0.05, (b) 0.25, and (c) 0.5 at $\text{Rm}$ of 8 from their final time steps of simulations. The structures are in steady state for the last $200{\mathrm{\Omega }}_1^{-1}$.

Figure 8

Poloidal contours of $q$ for three different values of $B_0$: (a) 0.05, (b) 0.25, and (c) 0.5 at $\text{Rm}$ of 8 from the final time steps of the simulations. The structures are in steady state for the last $200{\mathrm{\Omega }}_1^{-1}$. Critical value of Rayleigh instability at $q=2$ is denoted by white color. Flow shear of the hydrodynamic cases are nearly identical to the profile in (a).

Figure 9

(a) Poloidal contours of $B_r/B_0$, (b) poloidal circulation imposed on top of the contour of radial velocity, and (c) poloidal contours of $q$ from the final time step of simulation with $\text{Rm}$ of 4 and $B_0$ of 0.25 in the MRI-unstable region of parameter space. Identical color schemes are used as in the previous comparisons. The three contours indicate similar behaviors to the simulations from the MRI-unstable region at high $\text{Rm}$.