Dynamic domains of the Derviche Tourneur sodium experiment: Simulations of a spherical magnetized Couette flow

The Derviche Tourneur sodium experiment, a spherical Couette magnetohydrodynamics experiment with liquid sodium as the medium and a dipole magnetic field imposed from the inner sphere, recently underwent upgrades to its diagnostics to better characterize the flow and induced magnetic fields with global rotation. In tandem with the upgrades, a set of direct numerical simulations were run to give a more complete view of the fluid and magnetic dynamics at various rotation rates of the inner and outer spheres. These simulations reveal several dynamic regimes, determined by the Rossby number. At positive differential rotation there is a regime of quasigeostrophic flow, with low levels of fluctuations near the outer sphere. Negative differential rotation shows a regime of what appear to be saturated hydrodynamic instabilities at low negative differential rotation, followed by a regime where filamentary structures develop at low latitudes and persist over five to ten differential rotation periods as they drift poleward. We emphasize that all these coherent structures emerge from turbulent flows. At least some of them seem to be related to linear instabilities of the mean flow. The simulated flows can produce the same measurements as those that the physical experiment can take, with signatures akin to those found in the experiment. This paper discusses the relation between the internal velocity structures of the flow and their magnetic signatures at the surface.

Figure 1

Time evolution of the induced magnetic field measured along a meridian at the surface of the DTS-$\mathrm{\Omega }$ experiment spinning at 10 rotations per second. The induced magnetic field is expressed as a percentage of $B_o$, the intensity of the imposed magnetic field at the outer sphere's equator. (a) $B_r$ component for $\mathrm{\Delta }f=-10$ Hz. (b) $B_{\varphi }$ component for $\mathrm{\Delta }f=-20$ Hz.

Figure 2

Dynamic regimes of simulations of the DTS device. The heights of the dark (light) markers scale linearly with the kinetic (magnetic) fluctuation levels. The fluctuation levels are calculated by integrating the maps, calculated by Eq. (7), in $r$ and $\theta$. The velocities and magnetic fields are both in nondimensional units $(1/U_o$ for velocity, $V_A/U_o$ for magnetic fields). The $\text{Em}$ and $\mathrm{\Lambda }$ of the simulations are chosen to match those of the machine at the various outer rotation rates (see Table 1). The rotation frequency for each ${\mathrm{\Lambda }}_b$ value is indicated on the right axis. The $\text{Ek}$ of the simulations are set to ${10}^{-6}$.

Figure 3

Colormaps showing kinetic energy in each $m>0$ azimuthal wave number as a function of spherical radius for (a) $\text{Ro}=-2.0,{\mathrm{\Lambda }}_b=0.095$ and (b) $\text{Ro}=-1.0,{\mathrm{\Lambda }}_b=0.095$. The $m\in [1,5]$ are colored, the others have their opacity set as a function of the largest value in the bulk flow. A zoom of the last 0.01 is shown on the right.

Figure 4

Colormaps showing fluctuation levels of the magnetic and velocity fields. The first row (a, b, and c) shows the kinetic fluctuations, the second row (d, e, and f) shows the magnetic fluctuations. The numbers on the lower right indicate the maximum value of the energy density (excluding the Ekman layer). The contour lines in the first row represent isocontours of angular momentum of the bulk fluid. Inside the dashed line the bulk fluid is rotating faster than the inner sphere in the reference frame of the outer sphere. The contour lines in the second row are field lines of the mean poloidal magnetic field. Each column presents a set of parameters firmly in the quasigeostrophic (a and d), modal (b and e), and filamentary (c and f) regimes. All of the fluctuation levels are calculated for ${\mathrm{\Lambda }}_b=0.095$.

Figure 5

As described in the caption of Fig. 4, but only for modal flows. The contour lines represent streamlines of poloidal circulation. Solid lines are counterclockwise circulation; dashed lines are clockwise circulation. All of the energies are calculated for ${\mathrm{\Lambda }}_b=0.095$.

Figure 6

Kinetic fluctuation levels at the equator (solid lines) for Ro = $-1.2$ and Ro = $-1.0$ and ${\mathrm{\Lambda }}_b=0.095$ in log-scale (on the left) as a function of $r$. The (a) bulk flows' cylindrically radial velocity, (b) bulk flows' angular velocities, and (c) imposed vertical magnetic field in the equatorial plane are plotted as dashed lines with the vertical scales on the right. The radii of the peak fluctuation levels are noted on the $x$ axis.

Figure 7

(a–c) Colormaps showing azimuthally averaged kinetic energy density of the first three singular modes of the $\text{Ro}=-1.0,{\mathrm{\Lambda }}_b=0.095$ flow. The upper hemisphere shows the equatorially antisymmetric component of the mode; the lower hemisphere shows the equatorially symmetric component. (d) The energy in each mode (a–c) as a function of time, scaled against the mean energy in the kinetic fluctuations of the raw signal. Only the region $r\in [0.425,0.975]$ is included in the PCA.

Figure 8

(a–c) Colormaps showing azimuthally averaged kinetic energy density of the first three singular modes of the $\text{Ro}=-1.2,{\mathrm{\Lambda }}_b=0.095$ flow. The upper hemisphere shows the equatorially antisymmetric component of the mode; the lower hemisphere shows the equatorially symmetric component. (d) The energy in each mode (a–c) as a function of time, scaled against the mean energy in the kinetic fluctuations of the raw signal. Only the region $r\in [0.425,0.975]$ is included in the PCA. See also Supplemental Materials [22] and [23].

Figure 9

Paraview rendering of the velocity component (a) and the surface magnetic field (c) of the first principle component at $\text{Ro}=-1.2,{\mathrm{\Lambda }}_b=0.095$. Panels (b) and (d) are the same for $\text{Ro}=-1.0,{\mathrm{\Lambda }}_b=0.095$. The surfaces in (a) and (b) are isocontours of $\left|\mathbf{u}\right|$, at $0.15U_0$ and $0.10U_0$, respectively, with the azimuthal velocity painted on the surface. The azimuthal magnetic field of the components (scaled against $B_o)$ at the surface of the outer sphere are shown in (c) and (d). The inner sphere is visible in (a); the green cylinders represent the rotation axis. See also Supplemental Material movie [22].

Figure 10

Paraview [29] rendering of two individual snapshots of the flow at $\text{Ro}=-2.0$ and ${\mathrm{\Lambda }}_b=0.095$. The renderings show (a) an isosurface of $\left|\tilde{\mathbf{v}}\right|$ corresponding to 0.$025U_0$ and (b) the same contour $16\mathrm{\Delta }{f}^{-1}$ later. The isosurfaces are painted with the azimuthal velocity. The surface magnetic fields, scaled against $B_o$ of these snapshots are shown in (c) and (d), respectively, along with the magnetic field lines originating at the isocontours. The latitudinal field is shown in (c), the azimuthal field in (d). The green cylinders represent the rotation axis. See also Supplemental Material movie [27].

Figure 11

(a) Cylindrically Integrated kinetic energy vs. latitude and time. (b) Azimuthally integrated ${\widetilde{b_{\theta }}}^2$ at the surface (scaled against $\left.B_o\right)$ vs. latitude. Filamentary regime at $\text{Ro}=-2.0$ and ${\mathrm{\Lambda }}_b=0.095$. The vertical lines indicate the snapshots the renderings of Fig. 10 are taken from. See also Supplemental Material movie [27].

Figure 12

Colormaps of the intensity of the most unstable mode (white is zero). Top row (a, b, c): $\text{Ro}=-1.00$ and ${\mathrm{\Lambda }}_b=0.095$. Bottom row (d, e, f): $\text{Ro}=-1.20$ and ${\mathrm{\Lambda }}_b=0.095$. Left (a, d) shows the hydrodynamic instability while middle (b, e) and right (c, f) represent, respectively, the velocity and magnetic intensity of the magnetohydrodynamic instability. The contours in (a, b, d, e) are isolines of the angular velocity of the mean flow ${\overline{\mathbf{U}}}_{\varphi }/s$ with magenta for super-rotation. The contours in (c, f) are isolines of the the toroidal mean magnetic field ${\overline{\mathbf{B}}}_{\varphi }$ (dashed for negative). All velocity eigenmodes are equatorially symmetric while the magnetic ones are antisymmetric.

Figure 13

Radial component of the surface magnetic field (as a percentage of $B_o$ at a single meridian on the outer sphere vs time at (a) Ro = $-1.2$ and (b) Ro = $-1.0$ at ${\mathrm{\Lambda }}_b=0.095$. The signal from the dominant principle component is shown below in (c) and (d), respectively.

Figure 14

The surface magnetic field, as a percentage of $B_o$ at a single meridian on the outer sphere vs. time at $\text{Ro}=-2.0$ and ${\mathrm{\Lambda }}_b=0.095$, broken up into (a) latitudinal and (b) longitudinal components. The second row (c and d, respectively) shows the power spectral density of the magnetic signal at each latitude on that meridian. The color scale is truncated at one half the maximum spectral level to help highlight the higher harmonics.

Figure 15

(a–c) Colormaps showing azimuthally averaged kinetic energy density of the first three singular modes of the $\text{Ro}=-1.0,{\mathrm{\Lambda }}_b=0.190$ flow. The upper hemisphere shows the equatorially antisymmetric component of the mode; the lower hemisphere shows the equatorially symmetric component. (d) The energy in each mode (a–c) as a function of time, scaled against the mean energy in the kinetic fluctuations of the raw signal. Only the region $r\in [0.425,0.975]$ is included in the PCA.

Figure 16

(a–c) Colormaps showing azimuthally averaged kinetic energy density of the first three singular modes of the $\text{Ro}=-1.2,{\mathrm{\Lambda }}_b=0.190$ flow. The upper hemisphere shows the equatorially antisymmetric component of the mode; the lower hemisphere shows the equatorially symmetric component. (d) The energy in each mode (a–c) as a function of time, scaled against the mean energy in the kinetic fluctuations of the raw signal. Only the region $r\in [0.425,0.975]$ is included in the PCA.