^{1}, Michael Berhanu

^{2}, Sébastien Aumaître

^{1}, Arnaud Chiffaudel

^{1}, François Daviaud

^{1}, Bérengère Dubrulle

^{1}, Florent Ravelet

^{1}, Stephan Fauve

^{2}, Nicolas Mordant

^{2}, François Pétrélis

^{2}, Mickael Bourgoin

^{3}, Philippe Odier

^{3}, Jean-François Pinton

^{3}, Nicolas Plihon

^{3}and Romain Volk

^{3}

### Abstract

The von Kármán Sodium (VKS) experiment studies dynamo action in the flow generated inside a cylinder filled with liquid sodium by the rotation of coaxial impellers (the von Kármán geometry). We first report observations related to the self-generation of a stationary dynamo when the flow forcing is -symmetric, i.e., when the impellers rotate in opposite directions at equal angular velocities. The bifurcation is found to be supercritical with a neutral mode whose geometry is predominantly axisymmetric. We then report the different dynamical dynamo regimes observed when the flow forcing is not symmetric, including magnetic field reversals. We finally show that these dynamics display characteristic features of low dimensional dynamical systems despite the high degree of turbulence in the flow.

We thank M. Moulin, C. Gasquet, J.-B Luciani, A. Skiara, D. Courtiade, J.-F. Point, P. Metz, and V. Padilla for their technical assistance. This work was supported by Grant No. ANR05-0268-03, Direction des Sciences de la Matière et Direction de l’Énergie Nucléaire of CEA, Ministère de la Recherche and CNRS. The experiment was operated at CEA/Cadarache DEN/DTN.

I. INTRODUCTION

A. Experimental dynamos

B. The VKS experiment

C. Setup details

II. DYNAMO GENERATION, SYMMETRIC DRIVING

A. Self-generation

B. Bifurcation

C. Geometry of the dynamo field

D. Saturation

E. Power consumption issues

F. Fluctuations

III. DYNAMICAL REGIMES WITH ASYMMETRIC FORCING

A. Asymmetric flow forcing

B. A variety of regimes

C. Stationary dynamo regimes,

D. Reversals

E. Bursts

F. Extinction regimes

G. Oscillatory regimes

H. Transition and dynamics

IV. CONCLUDING REMARKS

### Key Topics

- Magnetic fields
- 73.0
- Turbulent flows
- 58.0
- Bifurcations
- 31.0
- Sodium
- 28.0
- Reynolds stress modeling
- 27.0

## Figures

Experimental setup. Note the curved impellers, the inner copper cylinder which separates the flow volume from the blanket of surrounding sodium, and the thin annulus in the midplane. Also shown are the holes through which the 3D Hall probes are inserted into the copper vessel for magnetic measurements. Location in coordinates of available measurement points: , , , , and (see Table I for details concerning available measurements in different experimental runs). When referring to the coordinates of magnetic field vector measured in the experiment at these different points, we will use either the Cartesian projection on the frame or the cylindrical projection on the frame ( and for measurements at points P1, P2, and P3). We will use in figures the following color code for the magnetic field components: axial in blue, azimuthal in red, and radial in green.

Experimental setup. Note the curved impellers, the inner copper cylinder which separates the flow volume from the blanket of surrounding sodium, and the thin annulus in the midplane. Also shown are the holes through which the 3D Hall probes are inserted into the copper vessel for magnetic measurements. Location in coordinates of available measurement points: , , , , and (see Table I for details concerning available measurements in different experimental runs). When referring to the coordinates of magnetic field vector measured in the experiment at these different points, we will use either the Cartesian projection on the frame or the cylindrical projection on the frame ( and for measurements at points P1, P2, and P3). We will use in figures the following color code for the magnetic field components: axial in blue, azimuthal in red, and radial in green.

Three components of the magnetic field generated by dynamo action at measured at point P1 in the experiment VKS2g. (a) Growth of the magnetic field as the impellers’ rotation rate is increased from 10 to 22 Hz. (b) Two independent realizations at same frequency above threshold showing opposite field polarities.

Three components of the magnetic field generated by dynamo action at measured at point P1 in the experiment VKS2g. (a) Growth of the magnetic field as the impellers’ rotation rate is increased from 10 to 22 Hz. (b) Two independent realizations at same frequency above threshold showing opposite field polarities.

Bifurcation curves. (a) Azimuthal field measured at P1 for VKS2h, growing with either polarity (only measurements with initially demagnetized impellers are shown here). The solid lines correspond to a best fit with a scaling behavior above threshold. (b) Magnetic field amplitude at P3 for VKS2i. Impellers are counter-rotating at equal rotation rates in the positive direction shown in Fig. 1 (closed blue circles) or in the opposite direction, i.e., with the blades on the impellers moving in a scooping or negative direction (open red squares). Changes in the efficiency of the stirring are taken into account in the definition of ; with in the normal, positive direction of rotation and in the opposite direction.

Bifurcation curves. (a) Azimuthal field measured at P1 for VKS2h, growing with either polarity (only measurements with initially demagnetized impellers are shown here). The solid lines correspond to a best fit with a scaling behavior above threshold. (b) Magnetic field amplitude at P3 for VKS2i. Impellers are counter-rotating at equal rotation rates in the positive direction shown in Fig. 1 (closed blue circles) or in the opposite direction, i.e., with the blades on the impellers moving in a scooping or negative direction (open red squares). Changes in the efficiency of the stirring are taken into account in the definition of ; with in the normal, positive direction of rotation and in the opposite direction.

Bifurcation diagrams measured for the azimuthal component at P1 in VKS2h. The curve with closed blue circles is built when increasing with demagnetized impellers and crossing the dynamo threshold for the first time. Successive cycles in across the bifurcation threshold all lie on the curve with open red square symbols.

Bifurcation diagrams measured for the azimuthal component at P1 in VKS2h. The curve with closed blue circles is built when increasing with demagnetized impellers and crossing the dynamo threshold for the first time. Successive cycles in across the bifurcation threshold all lie on the curve with open red square symbols.

Magnetic field measured at points P1 and P5 in VKS2h: (a) Time traces of the azimuthal magnetic field components. (b) Cross correlation function of the two signals for increasing magnetic Reynolds numbers in the range of 28–45.

Magnetic field measured at points P1 and P5 in VKS2h: (a) Time traces of the azimuthal magnetic field components. (b) Cross correlation function of the two signals for increasing magnetic Reynolds numbers in the range of 28–45.

Cross correlation function of the axial component and azimuthal component of the field measured inside the flow for increasing magnetic Reynolds numbers in the range of 28–45. Measurements are done at point P1 in VKS2h.

Cross correlation function of the axial component and azimuthal component of the field measured inside the flow for increasing magnetic Reynolds numbers in the range of 28–45. Measurements are done at point P1 in VKS2h.

Radial profiles of magnetic field. VKS2i measurements with the probe array inserted at P3. Measurements for flows at , above the dynamo onset. The magnetic field components are normalized by the largest value of the -component (azimuthal direction): axial is blue, azimuthal is red, and radial is green. The thick vertical line around the seventh sensor indicates the position of the inner copper shell; the tenth sensor at is located into the outer copper wall of the vessel.

Radial profiles of magnetic field. VKS2i measurements with the probe array inserted at P3. Measurements for flows at , above the dynamo onset. The magnetic field components are normalized by the largest value of the -component (azimuthal direction): axial is blue, azimuthal is red, and radial is green. The thick vertical line around the seventh sensor indicates the position of the inner copper shell; the tenth sensor at is located into the outer copper wall of the vessel.

Geometry of the dynamo mode. The magnetic field amplitudes measured along the probe array in Fig. 7 is represented by arrows: (a) toroidal component, (b) poloidal component, and (c) proposed dipole structure for the neutral mode.

Geometry of the dynamo mode. The magnetic field amplitudes measured along the probe array in Fig. 7 is represented by arrows: (a) toroidal component, (b) poloidal component, and (c) proposed dipole structure for the neutral mode.

Laminar and turbulent scalings for the magnetic field intensity at saturation. Measurements are shown here for several rotation rates of the impellers and for different operating temperatures ; the magnetic Reynolds number is rescaled accordingly. Measurement at P1 in VKS2h, : (a) laminar scaling [Eq. (5) ] and (b) turbulent scaling [Eq. (6) ]. Closed red triangles: , ; closed cyan squares: , ; open blue circles: , ; and open green stars: , .

Laminar and turbulent scalings for the magnetic field intensity at saturation. Measurements are shown here for several rotation rates of the impellers and for different operating temperatures ; the magnetic Reynolds number is rescaled accordingly. Measurement at P1 in VKS2h, : (a) laminar scaling [Eq. (5) ] and (b) turbulent scaling [Eq. (6) ]. Closed red triangles: , ; closed cyan squares: , ; open blue circles: , ; and open green stars: , .

Evolution with of the power number defined in Eq. (7) and measured from the drives of the motors. The blue circles (and dashed line as an eye guide) are from nondynamo runs (VKS2f) with stainless steel impellers. The red stars (VKS2g) and black squares (VKS2h) come from two dynamo runs, with same geometry but with soft-iron impellers and also without any magnetic probes in the flow bulk.

Evolution with of the power number defined in Eq. (7) and measured from the drives of the motors. The blue circles (and dashed line as an eye guide) are from nondynamo runs (VKS2f) with stainless steel impellers. The red stars (VKS2g) and black squares (VKS2h) come from two dynamo runs, with same geometry but with soft-iron impellers and also without any magnetic probes in the flow bulk.

Evolution with of rms values of the magnetic field at saturation. Measurements at point P1 in VKS2h. (a) Azimuthal component . (b) Magnetic field amplitude . The closed blue circles corresponds to a first run with almost demagnetized impellers (see text) and the open red squares to the subsequent runs.

Evolution with of rms values of the magnetic field at saturation. Measurements at point P1 in VKS2h. (a) Azimuthal component . (b) Magnetic field amplitude . The closed blue circles corresponds to a first run with almost demagnetized impellers (see text) and the open red squares to the subsequent runs.

Evolution of the ratio of rms fluctuations to second moment of the magnetic field at point P1.

Evolution of the ratio of rms fluctuations to second moment of the magnetic field at point P1.

PDFs of local fluctuations of . Measurement at point P1, VKS2h. Magnetic Reynolds numbers are given in the legend and profiles below onset cannot be distinguished. Inset: PDFs for centered and normalized variables.

PDFs of local fluctuations of . Measurement at point P1, VKS2h. Magnetic Reynolds numbers are given in the legend and profiles below onset cannot be distinguished. Inset: PDFs for centered and normalized variables.

Time spectra of magnetic field fluctuations measured with the probe array at point P3 (location of the innermost sensor) in VKS2i. Different curves on each plot correspond to different sensor depth, labeled in the legend by their distance to the rotation axis. The rotation frequency of the impellers is 20 Hz: (a) Axial component, (b) azimuthal component, and (c) radial component.

Time spectra of magnetic field fluctuations measured with the probe array at point P3 (location of the innermost sensor) in VKS2i. Different curves on each plot correspond to different sensor depth, labeled in the legend by their distance to the rotation axis. The rotation frequency of the impellers is 20 Hz: (a) Axial component, (b) azimuthal component, and (c) radial component.

Time spectra of magnetic field fluctuations measured for different values of at point P4 during VKS2i: (a) axial component and (b) radial component.

Time spectra of magnetic field fluctuations measured for different values of at point P4 during VKS2i: (a) axial component and (b) radial component.

Time spectra of magnetic field fluctuations at point P1. (a) Three components, ; [(b) and (c)] log-log plot and semilog plots of spectra for increasing values.

Time spectra of magnetic field fluctuations at point P1. (a) Three components, ; [(b) and (c)] log-log plot and semilog plots of spectra for increasing values.

(a) Measurement of the reduced dimensionless torque vs . Data are from VKS2g (open circles) and VKS2h (closed triangles), i.e., without probes in the flow bulk which strongly affect in the one-cell regimes. Data have been symmetrized. (b) Schematic configuration of the mean flow in the two-cell regime for small , as measured in water. (c) Schematic configuration of the mean flow in the one-cell regime for large .

(a) Measurement of the reduced dimensionless torque vs . Data are from VKS2g (open circles) and VKS2h (closed triangles), i.e., without probes in the flow bulk which strongly affect in the one-cell regimes. Data have been symmetrized. (b) Schematic configuration of the mean flow in the two-cell regime for small , as measured in water. (c) Schematic configuration of the mean flow in the one-cell regime for large .

Parameter space and dynamo regimes for VKS2g, VKS2h, and VKS2i. (a) In the plane. (b) In the plane, i.e., symmetrized data.

Parameter space and dynamo regimes for VKS2g, VKS2h, and VKS2i. (a) In the plane. (b) In the plane, i.e., symmetrized data.

Examples of observed dynamo regimes for increasing values of . See text for details. Color code [see caption of Fig. 1 and legend of Fig. 2(a) ] for the magnetic field components: axial in blue, azimuthal in red, and radial in green.

(a) Power spectrum density. Frequency is normalized by the average frequency of the impellers . The dashed line shows the −1 slope as a guide for the eye. (b) Centered PDF, normalized by its rms value. Quantities are computed from the azimuthal component in the time signal in Fig. 19 for the regimes Stat1, Stat2, and Stat3.

(a) Power spectrum density. Frequency is normalized by the average frequency of the impellers . The dashed line shows the −1 slope as a guide for the eye. (b) Centered PDF, normalized by its rms value. Quantities are computed from the azimuthal component in the time signal in Fig. 19 for the regimes Stat1, Stat2, and Stat3.

Reversals of the magnetic field generated by driving the flow with counter-rotating impellers at frequencies and ( , VKS2g). (a) Time recording of the three magnetic field components at P2: axial in blue, azimuthal in red, and radial in green [see legend of Fig. 2(a) ]. (b) superimposition of the azimuthal component for successive reversals from negative to positive polarity together with successive reversals from positive to negative polarity with the transformation . For each of them the origin of time has been shifted such that it corresponds to .

Reversals of the magnetic field generated by driving the flow with counter-rotating impellers at frequencies and ( , VKS2g). (a) Time recording of the three magnetic field components at P2: axial in blue, azimuthal in red, and radial in green [see legend of Fig. 2(a) ]. (b) superimposition of the azimuthal component for successive reversals from negative to positive polarity together with successive reversals from positive to negative polarity with the transformation . For each of them the origin of time has been shifted such that it corresponds to .

Nearly periodic reversals of the azimuthal magnetic field generated at and , VKS2h. Temperature of the outer copper cylinder compared to about for the regime with irregular reversals shown in Fig. 21 .

Nearly periodic reversals of the azimuthal magnetic field generated at and , VKS2h. Temperature of the outer copper cylinder compared to about for the regime with irregular reversals shown in Fig. 21 .

(a) Bursting regimes, VKS2h; , , ; three components of the magnetic field measured at point P1 [blue: axial component; red: minus azimuthal component; and green: radial component; see legend of Fig. 2(a) ]. (b) and (c) are zoomed on, respectively, quiet periods and active bursts.

(a) Bursting regimes, VKS2h; , , ; three components of the magnetic field measured at point P1 [blue: axial component; red: minus azimuthal component; and green: radial component; see legend of Fig. 2(a) ]. (b) and (c) are zoomed on, respectively, quiet periods and active bursts.

Bursting regimes, VKS2h; , , [cf. Fig. 23 ]. (a) Time correlations of the magnetic field components. (b) log-linear plot of the autocorrelation of the largest field component . (c) Corresponding time-power spectra—the black solid lines shows an scaling. (d) PDF for .

Bursting regimes, VKS2h; , , [cf. Fig. 23 ]. (a) Time correlations of the magnetic field components. (b) log-linear plot of the autocorrelation of the largest field component . (c) Corresponding time-power spectra—the black solid lines shows an scaling. (d) PDF for .

Bursting regimes, VKS2h; , , and . (a) Three components of the magnetic field measured at point P1 [blue: axial component; red: azimuthal component; and green: radial component; see legend of Fig. 2(a) ]. (b) and (c) are further zoomed on the oscillations (note the change in polarity of the low state mode before and after the oscillations). (d) Zoom including the ending extinction around .

Bursting regimes, VKS2h; , , and . (a) Three components of the magnetic field measured at point P1 [blue: axial component; red: azimuthal component; and green: radial component; see legend of Fig. 2(a) ]. (b) and (c) are further zoomed on the oscillations (note the change in polarity of the low state mode before and after the oscillations). (d) Zoom including the ending extinction around .

Bursting regimes, VKS2h; , , [cf. Fig. 25 ]. (a) Time correlations of the magnetic field components. (b) Zoom around zero time lags. (c) Corresponding time-power spectra. (d) PDFs of the three magnetic field components. Color code for (c) and (d) as in Fig. 25 .

(a) Oscillatory time signals for in the range of 0.44–0.56. The vertical dashed lines denote impeller frequency changes. Starting with kept constant, we observe successively: stationary dynamo and oscillatory dynamo . Then is reduced below dynamo onset by lowering to 25 Hz (damped oscillations) and 20 Hz (no dynamo). (b): Evolution of the magnetic field amplitude and the dimensionless oscillation frequency with for Osc oscillatory dynamo regimes (closed circles) and Stat3 stationary dynamo regimes (open squares). Amplitude of oscillations is a peak value (see text). Quadratic fits are described in text and the vertical line stands for the oscillatory mode threshold .

(a) Oscillatory time signals for in the range of 0.44–0.56. The vertical dashed lines denote impeller frequency changes. Starting with kept constant, we observe successively: stationary dynamo and oscillatory dynamo . Then is reduced below dynamo onset by lowering to 25 Hz (damped oscillations) and 20 Hz (no dynamo). (b): Evolution of the magnetic field amplitude and the dimensionless oscillation frequency with for Osc oscillatory dynamo regimes (closed circles) and Stat3 stationary dynamo regimes (open squares). Amplitude of oscillations is a peak value (see text). Quadratic fits are described in text and the vertical line stands for the oscillatory mode threshold .

Sketch of the axial dipolar (a) and quadrupolar (b) magnetic modes. (c) Location of the different states in the plane: fixed points corresponding to the stationary regimes for frequencies ; limit cycle (LC) observed for impellers counter-rotating at different frequencies (22,18.5) Hz (red cycle). The magnetic field is time averaged over 1 s to remove high frequency fluctuations caused by the turbulent velocity fluctuations. (d) Time recording of the components of the magnetic field for frequencies (22,18.5) Hz .

Sketch of the axial dipolar (a) and quadrupolar (b) magnetic modes. (c) Location of the different states in the plane: fixed points corresponding to the stationary regimes for frequencies ; limit cycle (LC) observed for impellers counter-rotating at different frequencies (22,18.5) Hz (red cycle). The magnetic field is time averaged over 1 s to remove high frequency fluctuations caused by the turbulent velocity fluctuations. (d) Time recording of the components of the magnetic field for frequencies (22,18.5) Hz .

Plot of a cut in a phase space reconstruction with for three regimes: thick dashed black line for field reversals reported in Fig. 21 , the magnetic field being rescaled by an ad hoc factor accounting for the fact that the probe location is not in the midplane. The medium blue line is for symmetric bursts and the thin red line for asymmetric bursts . In these last two plots the magnetic field is time averaged over 0.25 s to remove high frequency fluctuations.

Plot of a cut in a phase space reconstruction with for three regimes: thick dashed black line for field reversals reported in Fig. 21 , the magnetic field being rescaled by an ad hoc factor accounting for the fact that the probe location is not in the midplane. The medium blue line is for symmetric bursts and the thin red line for asymmetric bursts . In these last two plots the magnetic field is time averaged over 0.25 s to remove high frequency fluctuations.

## Tables

Experimental configuration of the four successive runs discussed here. VKS2g (September 2006) is the first run with dynamo action. Labels MP or G indicate whether a 3D probe array of Hall sensors or a single 3D gaussmeter was used. P# (cf. Fig. 1 ) is the location of the measurement—the innermost sensor for the probe array.

Experimental configuration of the four successive runs discussed here. VKS2g (September 2006) is the first run with dynamo action. Labels MP or G indicate whether a 3D probe array of Hall sensors or a single 3D gaussmeter was used. P# (cf. Fig. 1 ) is the location of the measurement—the innermost sensor for the probe array.

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