Experimental Observation and Characterization of the Magnetorotational Instability

Differential rotation occurs in conducting flows in accretion disks and planetary cores. In such systems, the magnetorotational instability can arise from coupling Lorentz and centrifugal forces to cause large radial angular momentum fluxes. We present the first experimental observation of the magnetorotational instability. Our system consists of liquid sodium between differentially rotating spheres, with an imposed coaxial magnetic field. We characterize the observed patterns, dynamics, and torque increases, and establish that this instability can occur from a hydrodynamic turbulent background.

Figure 1

The spherical apparatus and characteristic induced magnetic field instability patterns. The device (a) consists of a thin stainless spherical outer vessel, a rotating inner copper sphere, and liquid sodium in between. The line adjacent to $\mathbf{v}$ indicates the location of the velocity measurements. The resulting dynamics change character under the influence of an externally applied magnetic field coaxial with the rotation. As the external field increases, one sees states, in an equal area projection of the induced magnetic field just outside the outer sphere, dominated by (b) a rotating odd pattern with azimuthal wave number $m=1$ (${\mathrm{O}}_1$) at $R_m=26$ (50 Hz), $S=4.19$ ($84.6\mathrm{m}\mathrm{T}$), (c) an even $m=0$ pattern (${\mathrm{E}}_0$) at $R_m=10.4$ ($20\mathrm{H}\mathrm{z}$), $S=4.37$ ($88.1\mathrm{m}\mathrm{T}$), and (d) a rotating $m=1$ even pattern (${\mathrm{E}}_1$) at $R_m=10.4$ ($20\mathrm{H}\mathrm{z}$), $S=7.08$ ($14.29\mathrm{m}\mathrm{T}$). Red and blue indicate outward and inward radial (cylindrical) induced magnetic field, and green indicates the null values.

Figure 2

The zero field angular momentum and rotation rate profiles for ${\Omega }_o/2\pi =30\mathrm{H}\mathrm{z}$ ($R_m=16$). From ultrasound velocimetry measurements, the rising mean angular momentum density (a) shows the system is stable to centrifugal instabilities except for thin boundary layers. The rotation curve (b) decreases with cylindrical radial distance from the center, a necessary condition for the base state to be unstable to the magnetorotational instability. The smooth curve is a Keplerian profile for comparison. The inset (c) shows the velocity exponent $\zeta$ with the Keplerian value $\zeta =-3/2$ indicated by a dashed line.

Figure 3

Coupled fluctuations in the velocity and magnetic fields. Here the amplitude of the induced field for the two Gauss modes ${\mathrm{O}}_1$ (top curves) and the amplitude of a velocity component (lower black curve) below the outer sphere substantiate that Lorentz forces are key to this instability. These data were taken at ${\Omega }_o/2\pi =7.5\mathrm{H}\mathrm{z}$ ($R_m=4$); the velocity was measured $0.115\mathrm{m}$ from the outer wall. Each mode with nonzero azimuthal wave number has two components: when they oscillate $\pi /2$ out of phase the pattern precesses. The precession rate depends (nonlinearly) on both applied field strength and rotation rate.

Figure 4

The torque increase and variance of induced field coefficients as the applied field is varied. We characterize the saturated state by the mode with the largest variance. These measurements were taken for a fixed rotation rate of $30\mathrm{H}\mathrm{z}$ ($R_m=16$). The field data reflect fluctuations in the coefficients from their means, except the ${\mathrm{E}}_0$ state which shows significant mean amplitude (squared average shown in green, variance of fluctuations in blue), and does so at the same applied field values where it shows ample fluctuations. At zero field, the base state torque (including confounding errors due to seals) here is $1.14$ N-m, the black curve shows the increase in torque above this base value.

Figure 5

Phase diagram of saturated states. Regions are defined by the mode with the largest variance of fluctuations (see Fig. 4). Some secondary instabilities show hysteresis; these data are for increasing $S$, for fixed $R_m$. The states have regions associated with background turbulence (tb), mode ${\mathrm{O}}_1$ dominated, ${\mathrm{E}}_0$ dominated, followed by ${\mathrm{O}}_1$, ${\mathrm{E}}_1$, ${\mathrm{O}}_2$, and ${\mathrm{E}}_2$ modes. The lowest $R_m$ and $S$ numbers for these states (□) are obtained by extrapolating the maximum saturated state amplitude versus $R_m$ (a linear trend) to zero amplitude. Also shown are theoretical stability boundaries for the longest wavelength (red) and second longest wavelength (blue) instabilities, calculated from the magnetorotational dispersion relation.