Magnetic induction and diffusion mechanisms in a liquid sodium spherical Couette experiment

We present a reconstruction of the mean axisymmetric azimuthal and meridional flows in the Derviche Tourneur Sodium installation in Grenoble liquid sodium experiment. The experimental device sets a spherical Couette flow enclosed between two concentric spherical shells where the inner sphere holds a strong dipolar magnet, which acts as a magnetic propeller when rotated. Measurements of the mean velocity, mean induced magnetic field, and mean electric potentials have been acquired inside and outside the fluid for an inner sphere rotation rate of 9 Hz $\left(\text{Rm}\simeq 28\right)$. Using the induction equation to relate all measured quantities to the mean flow, we develop a nonlinear least-squares inversion procedure to reconstruct a fully coherent solution of the mean velocity field. We also include in our inversion the response of the fluid layer to the nonaxisymmetric time-dependent magnetic field that results from deviations of the imposed magnetic field from an axial dipole. The mean azimuthal velocity field we obtain shows superrotation in an inner region close to the inner sphere where the Lorentz force dominates, which contrasts with an outer geostrophic region governed by the Coriolis force, but where the magnetic torque remains the driver. The meridional circulation is strongly hindered by the presence of both the Lorentz and the Coriolis forces. Nevertheless, it contributes to a significant part of the induced magnetic energy. Our approach sets the scene for evaluating the contribution of velocity and magnetic fluctuations to the mean magnetic field, a key question for dynamo mechanisms.

Figure 1

Meridional maps of data coverage. Black lines displaying four spherical shells mark successively the copper, the fluid, and the stainless steel shells. The azimuthal data are shown on the left. The colored lines (crossing fluid shell) are the projections in the meridional plane of the ultrasound beams from which Doppler angular velocity profiles are obtained. Colored (gray) dots mark the position of the magnetic probes measuring the azimuthal component of the magnetic field $B_{\phi }$. Electric potentials are measured in the outer shell at positions displayed by the black squares. The meridional data are shown on the right. The colored lines (crossing fluid shell) are the projections in the meridional plane of the ultrasound beams from which Doppler meridional velocity profiles are obtained. Colored (gray) arrows indicate the position of the magnetic probes measuring the radial and orthoradial components of the magnetic field inside and outside the fluid. The magnetic probes provide both the mean and the time-dependent signals used in this study.

Figure 2

Plot of the $2\pi$ longitudinal motifs of the nonaxisymmetric magnetic field recorded by a Hall magnetometer measuring $B_{\phi }$ for $r=0.79$ at a latitude of $-{20}^{\circ }$, for increasing rotation rates of the inner sphere $f=-9.4,-18.4$, and $-30$ Hz. The magnetic field is normalized by $B_o$ as indicated in Table 4. The motif for $f=0$ is computed from the scalar magnetic potential of the magnet.

Figure 3

Radial profiles of velocity and magnetic modes for all harmonic degrees. The top plots show the toroidal $u_l^T\left(r\right)$ (left) and poloidal $u_l^P\left(r\right)$ (right) velocity modes. The bottom plots show the axisymmetric toroidal $b_{l,m=0}^T\left(r\right)$ (left) and poloidal $b_{l,m=0}^P\left(r\right)$ (right) magnetic field modes. All fields are normalized as given in Appendix pp1. The radius axis of the magnetic plots extends from $r_i-{\delta }_{Cu}$ to $r_o+{\delta }_{ss}$ and horizontal lines indicate the fluid-solid interfaces at $r_i$ and $r_o$.

Figure 4

Radial profiles of the induced axisymmetric poloidal magnetic modes for all harmonic degrees. Each latitudinal degree from $l=1$ to $l=6$ is represented using colored lines (grayscale) depicted in the legend. The field is made dimensionless as given in Appendix pp1. The radius axis extends from $r_i-{\delta }_{Cu}$ to $r_o+{\delta }_{ss}$ and horizontal lines indicate the fluid-solid interfaces at $r_i$ and $r_o$.

Figure 5

Maps of the velocity and magnetic fields in a meridional $(r,\theta )$ section. Black spherical lines mark successively the copper, the fluid, and the stainless steel shells. (a) Contour map of the angular velocity. (b) Stream lines of the meridional flow. The flow is centrifugal along the equator and values of the stream function range from $-4\times {10}^{-4}$ to $4\times {10}^{-4}$. (c) Contour map of the azimuthal magnetic field $B_{\phi }$ at longitude $\phi =0$. (d) Field lines of the induced meridional magnetic field at $\phi =0$.

Figure 6

Fits of the angular velocity predictions to ultrasound velocimetry Doppler profiles. Measured data are thin lines presented with their error bars. Thick lines are predictions. All profiles are function of the distance $d$ along the ray of the ultrasound beam. Colors (grayscale) refer to ultrasound beam path at different latitudes as displayed on the data coverage map in Fig. 1. All data are dimensionless as given in Appendix pp1.

Figure 7

Fits of velocity predictions to ultrasound velocimetry Doppler profiles. These plots compare the measured data with their error bars to our kinematic model predictions of the meridional (left) and radial (right) velocities. Both profiles are a function of the distance $d$ along the ray of the ultrasound beam. All data are dimensionless as given in Appendix pp1.

Figure 8

Azimuthal predictions and measurements. Shown on the left is the azimuthal magnetic field: azimuthal magnetic field predictions inside the fluid with the measured data and their error bars. The triangles with error bars are our measurements at three latitudes: ${40}^{\circ }$ (bottom red markers), ${10}^{\circ }$ (middle blue markers), and $-{20}^{\circ }$ (top green markers). The solid lines are our model predictions at the same latitudes. The velocity model contains both the azimuthal and the meridional flows. The dashed lines are the predictions for the same flow without meridional circulation. The radius axis extends from $r_i-{\delta }_{cu}$ and $r_o+{\delta }_{ss}$ and vertical dash-dotted lines indicate the fluid-solid interfaces at $r_i$ and $r_o$. Shown on the right is the surface electric potential. The black triangles with error bars are the differences in electric potential between electrodes ${10}^{\circ }$ apart that we measure at the surface of the outer shell. The red squares are our model predictions. All fields are dimensionless as given in Appendix pp1.

Figure 9

Meridional predictions and measurements. These plots represent the measured data using marker symbols with their error bars and numerical predictions from our kinematic model using solid lines. The radial description of the magnetic field at three different latitudes $[{40}^{\circ }$ (squares and solid lines), ${10}^{\circ }$ (triangles and light gray dashed lines), and $-{20}^{\circ }$ (triangles and dark gray dashed lines)] is shown: the orthoradial component (left) and the radial component (center). Radial (light gray) and orthoradial (dark gray) components of the induced magnetic field measured at the surface of the outer shell are also shown (right). The radius ranges from $r_i-{\delta }_{\text{Cu}}$ to $r_o+{\delta }_{ss}$ and the vertical dotted lines indicate the fluid-solid interfaces at $r_i$ and $r_o$. All fields are dimensionless (see Appendix pp1).

Figure 10

Plot of the radial evolution of the nonaxisymmetric magnetic data for experimental measurements and predictions at latitude $-{10}^{\circ }$. For experimental data, we plot the imaginary (vertical axis) and real (horizontal axis) parts of the Fourier coefficients of the azimuthal motifs presented in Sec. 3a. Predictions display a full radial description for three different models, presenting the real and imaginary parts of $B_{\phi }^m(r,\theta )$. We present on this graphic the azimuthal mode $m=2$ as an example. Lines represent the radial evolution of $B_{\phi }^m$ from the inner boundary to the outer sphere surface (where the magnetic field is numerically solved). The thick solid blue (dark gray) line account for our kinematic model predictions, the red dotted (bottom) line shows the static case, and the black dotted (top) line is the skin effect case (see Sec. 5d for details). Colored (grayscale) markers labeled in the legend account for experimental data collected in DTS from magnetic probes in the sleeve. Markers radially increase from P6 to P1 (see Fig. 1). Nonaxisymmetric data are all dimensionless (see Appendix pp1).

Figure 11

Plot of nonaxisymmetric magnetic data of five azimuthal modes from $m=1$ to 5, for two magnetic components at two different latitudes. The azimuthal component $B_{\phi }^m(r,\theta )$ is presented in the five top graphics for latitude $-{20}^{\circ }$. The radial component $B_r^m(r,\theta )$ is presented in the five bottom graphics for latitude ${10}^{\circ }$. Colored (grayscale) symbols labeled in the legend mark experimental data collected in DTS from magnetic probes in the sleeve. The radial position of the probes increases from P6 to P1 for the azimuthal component and from R3 to R1 for the radial component (see Fig. 1). Nonaxisymmetric data are dimensionless as given in Appendix pp1. See the caption of Fig. 10 for more details.

Figure 12

Location of the magnetic measurements used to infer the scalar magnetic potential of the inner sphere's magnet. In this meridional cross section, the arrows indicate which component $(B_r$ in blue and $B_{\theta }$ in red) is measured, with the small black circles corresponding to the $B_{\phi }$ data. The measurements are repeated every $5^{\circ }$ or ${10}^{\circ }$ in longitude. Black spherical lines mark successively the copper, the fluid, and the stainless steel shells.

Figure 13

Contour map of the scalar magnetic potential of the magnet reconstructed at the surface of the inner sphere. The potential is normalized by $r_o^{*}{B}_o$ as indicated in Table 4. The nonaxisymmetric part is shown on the top and the nondipole axisymmetric part on the bottom.