Generation of a Magnetic Field by Dynamo Action in a Turbulent Flow of Liquid Sodium
R. Monchaux,
1
M. Berhanu,
2
M. Bourgoin,
3,
*
M. Moulin,
3
Ph. Odier,
3
J.-F. Pinton,
3
R. Volk,
3
S. Fauve,
2
N. Mordant,
2
F. Pe
´
tre
´
lis,
2
A. Chiffaudel,
1
F. Daviaud,
1
B. Dubrulle,
1
C. Gasquet,
1
L. Marie
´
,
1,†
and F. Ravelet
1,‡
1
Service de Physique de l’Etat Condense
´
, Direction des Sciences de la Matie
`
re, CEA-Saclay,
CNRS URA 2464, 91191 Gif-sur-Yvette cedex, France
2
Laboratoire de Physique Statistique de l’Ecole Normale Supe
´
rieure, CNRS UMR 8550,
24 Rue Lhomond, 75231 Paris Cedex 05, France
3
Laboratoire de Physique de l’Ecole Normale Supe
´
rieure de Lyon, CNRS UMR 5672, 46 alle
´
e d’Italie, 69364 Lyon Cedex 07, France
(Received 25 October 2006; published 25 January 2007)
We report the observation of dynamo action in the von Ka
´
rma
´
n sodium experiment, i.e., the generation
of a magnetic field by a strongly turbulent swirling flow of liquid sodium. Both mean and fluctuating parts
of the field are studied. The dynamo threshold corresponds to a magnetic Reynolds number R
m
30.A
mean magnetic field of the order of 40 G is observed 30% above threshold at the flow lateral boundary.
The rms fluctuations are larger than the corresponding mean value for two of the components. The scaling
of the mean square magnetic field is compared to a prediction previously made for high Reynolds number
flows.
DOI: 10.1103/PhysRevLett.98.044502 PACS numbers: 47.65.d, 52.65.Kj, 91.25.Cw
The generation of electricity from mechanical work was
one of the main achievements of physics at the end of the
19th century. In 1919, Larmor proposed that a similar
process could generate the magnetic field of the Sun
from the motion of an electrically conducting fluid.
However, fluid dynamos are more complex than industrial
ones and it is not easy to find laminar flow configurations
that generate magnetic fields [1]. Two simple but clever
examples were found in the 1970s [2] and have led more
recently to successful experiments [3]. These experiments
have shown that the observed thresholds are in good agree-
ment with theoretical predictions [4] made by considering
only the mean flow, whereas the saturation level of the
magnetic field cannot be described with a laminar flow
model (without using an ad hoc turbulent viscosity) [5].
These observations have raised many questions: what hap-
pens for flows without geometrical constraints such that
fluctuations are of the same order of magnitude as the mean
flow? Is the dynamo threshold strongly increased due to the
lack of coherence of the driving flow [6,7] or does the
prediction made as if the mean flow were acting alone still
give a reasonable order of magnitude [8]? What is the
nature of the dynamo bifurcation in the presence of large
velocity fluctuations? All of these questions, and others
motivated by geophysical or astrophysical dynamos [9],
have led several teams to try to generate dynamos in flows
with a high level of turbulence [10,11]. We present in this
Letter our first experimental observation of the generation
of a magnetic field in a von Ka
´
rma
´
n swirling flow of liquid
sodium (VKS) for which velocity fluctuations and the
mean flow have comparable kinetic energy and we discuss
some of the above issues. The experimental setup (see
Fig. 1) is similar to the previous VKS experiments [11],
but involves three modifications that will be described
below. The flow is generated by rotating two disks of radius
154.5 mm, 371 mm apart in a cylindrical vessel, 2R
412 mm in inner diameter and 524 mm in length. The disks
are fitted with 8 curved blades of height h 41:2mm.
These impellers are driven at a rotation frequency up to
=2 26 Hz by 300 kW available motor power. An oil
circulation in the outer copper cylinder maintains a regu-
lated temperature in the range 110–160
C. The mean flow
has the following characteristics: the fluid is ejected radi-
ally outward by the disks; this drives an axial flow toward
the disks along their axis and a recirculation in the opposite
direction along the cylinder lateral boundary. In addition,
in the case of counterrotating disks studied here, the pres-
ence of a strong axial shear of azimuthal velocity in the
midplane between the impellers generates a high level of
turbulent fluctuations [12,13]. The kinetic Reynolds num-
0.751.00
0.4
0.30185
16
206
5
61
78
41
155
Oil cooling circulation
Na at rest
P2
P1
x
z
y
0.90
40
R
FIG. 1. Sketch of the experimental setup. The inner and outer
cylinders are made of copper (in gray). The dimension are given
in millimeter (left) and normalized by R (right). The 3D Hall
probe is located either at point P1 in the midplane or P2. In both
cases, the probe is nearly flush with the inner shell.
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0031-9007=07=98(4)=044502(4) 044502-1 © 2007 The American Physical Society
ber is Re KR
2
=, where is the kinematic viscosity
and K 0:6 is a coefficient that measures the efficiency of
the impellers [14]. Re can be increased up to 5 10
6
: the
corresponding magnetic Reynolds number is R
m
K
0
R
2
49 (at 120
C), where
0
is the magnetic
permeability of vacuum and is the electrical conductivity
of sodium.
A first modification with respect to earlier VKS experi-
ments consists of surrounding the flow by sodium at rest in
another concentric cylindrical vessel, 578 mm in inner
diameter. This has been shown to decrease the dynamo
threshold in kinematic computations based on the mean
flow velocity [14]. The total volume of liquid sodium is
150 l. A second geometrical modification consists of at-
taching an annulus of inner diameter 175 mm and thickness
5 mm along the inner cylinder in the midplane between the
disks. Water experiments have shown that its effect on the
mean flow is to make the shear layer sharper around the
midplane. In addition, it reduces low frequency turbulent
fluctuations, thus the large scale flow time-averages faster
toward the mean flow. However, rms velocity fluctuations
are almost unchanged (of order 40%–50%), thus the flow
remains strongly turbulent [15]. It is expected that reducing
the transverse motion of the shear layer decreases the
dynamo threshold for the following reasons: (i) magnetic
induction due to an externally applied field on a gallium
flow strongly varies because of the large scale flow excur-
sions away from the time averaged flow [16], (ii) the
addition of large scale noise to the Taylor-Green mean
flow increases its dynamo threshold [7], (iii) fluctuating
motion of eddies increase the dynamo threshold of the
Roberts flow [17].
The above configuration does not generate a magnetic
field up to the maximum possible rotation frequency of the
disks (=2 26 Hz). We thus made a last modification
and replaced disks made of stainless steel by similar iron
disks. Using boundary conditions with a high permeability
in order to change the dynamo threshold has been already
proposed [18]. It has been also shown that in the case of a
Ponomarenko or G. O. Roberts flows, the addition of an
external wall of high permeability can decrease the dy-
namo threshold [19]. Finally, recent kinematic simulations
of the VKS mean flow have shown that different ways of
taking into account the sodium behind the disks lead to an
increase of the dynamo threshold ranging from 12% to
150% [20]. We thought that using iron disks could screen
magnetic effects in the bulk of the flow from the region
behind the disks, although the actual behavior may be more
complex. This last modification generates a dynamo above
R
m
’ 30. The three components of the field
~
B are measured
with a 3D Hall probe, located either in the midplane or
109 mm away from it (P1 or P2 in Fig. 1). In both cases,
the probe is nearly flush with the inner shell, thus
~
B is
measured at the boundary of the turbulent flow. Figure 2
shows the time recording of the three components of
~
B
when R
m
is increased from 19 to 40. The largest component
B
y
is tangent to the cylinder at the measurement location. It
increases from a mean value comparable to the Earth
magnetic field to roughly 40 G. The mean values of the
other components B
x
and B
z
also increase (not visible on
the figure because of fluctuations). Both signs of the com-
ponents have been observed in different runs, depending on
the sign of the residual magnetization of the disks. All
components display strong fluctuations as could be ex-
pected in flows with Reynolds numbers larger than 10
6
.
Figure 3(a) shows the mean values of the components
hB
i
i of the magnetic field and Fig. 3(b) their fluctuations
B
irms
versus R
m
. The fluctuations are all in the same range
(3 to 8 G, at 30% above threshold) although the corre-
sponding mean values are very different. The time average
of the square of the total magnetic field, h
~
B
2
i, is displayed
in the inset of Fig. 3(a). No hysteresis is observed. Linear
fits of hB
y
i or B
irms
displayed in Fig. 3 define a critical
magnetic Reynolds number R
c
m
31 whereas the linear fit
of h
~
B
2
i gives a larger value R
0
m
35. The latter is the one
that should be considered in the case of a supercritical
pitchfork bifurcation. The rounding observed close to
threshold could then be ascribed to the imperfection due
to the ambient magnetic field (Earth field, residual magne-
tization of the disks and other magnetic perturbations of the
setup). The actual behavior may be more complex because
this bifurcation takes place on a strongly turbulent flow, a
situation for which no rigorous theory exists. The inset of
Fig. 3(b) shows that the variance B
2
rms
h
~
B h
~
Bi
2
i is
not proportional to hB
2
i. Below the dynamo threshold, the
effect of induction due to the ambient magnetic field is
observed. B
rms
=hB
2
i
1=2
first behaves linearly at low R
m
, but
then increases faster as R
m
becomes closer to the bifurca-
tion threshold. We thus show that this seems to be a good
quantity to look at as a precursor of a dynamo regime. In
addition, we observe that it displays a discontinuity in
slope in the vicinity of R
c
m
in an analogous way of some
10 20 30 40
−60
−40
−20
0
20
B
i
[G]
B
x
B
y
B
z
0 1020304050
10
15
20
Ω
/2π [Hz]
time
[
s
]
FIG. 2 (color online). Time recording at P1 of the components
of the magnetic field when the rotation frequency =2 is
increased as displayed by the ramp below (R
m
increases from
19 to 40).
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044502-2
response functions at phase transitions or bifurcations in
the presence of noise. Note, however, that the shape of the
curves depends on the measurement point and they cannot
be superimposed with a scaling factor as done for hB
2
i
versus R
m
in the inset of Fig. 3(a).
The above results are characteristic of bifurcations in the
presence of noise. As shown in much simpler experiments,
different choices of an order parameter (mean value of the
amplitude of the unstable mode or its higher moments, its
most probable value, etc.) can lead to qualitatively differ-
ent bifurcation diagrams [21]. This illustrates the ambigu-
ity in the definition of the order parameter for bifurcations
in the presence of fluctuations or noise. In the present
experiment, fluctuations enter both multiplicatively, be-
cause of the turbulent velocity, and additively, due to the
interaction of the velocity field with the ambient magnetic
field. Finally, we note that both R
c
m
and R
0
m
are smaller than
the thresholds computed with kinematic dynamo codes
taking into account only the mean flow, that are in the
range R
c
m
43 to 150 depending on different boundary
conditions on the disks and on configurations of the flow
behind them [14,20].
The probability density functions (PDF) of the fluctua-
tions of the three components of the induced magnetic field
(not displayed) are roughly Gaussian. The PDFs of fluctu-
ations below threshold, i.e., due to the induction resulting
from the ambient magnetic field, are similar to the ones
observed in the self-generating regime. We do not observe
any non-Gaussian behavior close to threshold which would
result from an on-off intermittency mechanism [22].
Possible reasons are the low level of small frequency
velocity fluctuations [23] or the imperfection of the bifur-
cation that results from the ambient magnetic field [24].
Figure 4 displays both the dimensional (see inset) and
dimensionless mean square field as a function of R
m
. hB
2
i
is made dimensionless using a high Reynolds number
scaling [5]: hB
2
i/=
0
R
2
R
m
R
c
m
=R
c
m
, where
is the fluid density. We observe that data obtained at differ-
ent working temperature are well collapsed by this scaling
( decreases by roughly 15% from 100 to 160
C). The
low Reynolds number or ‘‘weak field’’ scaling could also
give a reasonable collapse of data obtained on this tem-
10 15 20 25 30 35 40 45
0
0.1
0.2
<B
2
> µ
0
σ
2
R
2
/ρ
Rm
16 20 22
0
B
2
[G
2
] vs
Ω/(2 π) [Hz]
5 10
3
FIG. 4 (color online). The dimensionless quantity,
hB
2
i
0
R
2
= is displayed as a function of R
m
for different
working temperatures and frequencies [measurements done at
P2 and identical symbols as in the inset of Fig. 3(b)]. The inset
shows the same data in dimensional form B
2
versus rotation
frequency for different temperatures.
10 20 30 40
0
10
20
30
40
50
Rm
<B
i
> [G]
−<B
x
>
−<B
y
>
<B
z
>
20 25 30 35 40 45
0
500
1000
1500
2000
2500
3000
Rm
<B
2
> [G
2
]
P1
P2
10 15 20 25 30 35 40 45
0
2
4
6
8
10
12
14
Rm
B
i rms
[G]
B
x
B
y
B
z
20 25 30 35 40 45
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
B
rms
/<B
2
>
1/2
Rm
FIG. 3 (color online). (a) Mean values of the three components
of the magnetic field recorded at P1 versus R
m
(T 120
C):
(䉱) hB
x
i,(䊏) hB
y
i,(䊉) hB
z
i. The inset shows the time
average of the square of the total magnetic field as a function
of R
m
, measured at P1 (䊉), or at P2 (夹) after being divided by
1.8. (b) Standard deviation of the fluctuations of each compo-
nents of the magnetic field recorded at P1 versus R
m
. The inset
shows B
rms
=hB
2
i
1=2
. Measurements done at P1:(䉱) T 120
C,
frequency increased up to 22 Hz; Measurements done at P2:(夹)
T 120
C, frequency decreased from 22 to 16.5 Hz, (䊐) T
156
C, frequency increased up to 22 Hz, () =2 16:5Hz,
T varied from 154 to 116
C,(䉫) =2 22 Hz, T varied from
119 to 156
C. The vertical line corresponds to R
m
32.
PRL 98, 044502 (2007)
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044502-3
perature range but the predicted order of magnitude for
hB
2
i would be 10
5
too small.
Dissipated power by Ohmic losses is another important
characterization of dynamo action. Our measurements
show that, 30% above threshold, it leads to an excess power
consumption of 15%–20% with respect to a flow driving
power of the order of 100 kW.
The effect of iron disks deserves additional discussion.
A slight effect of magnetization of iron has been observed:
the dynamo threshold during the first run was about 20%
larger than in the next runs for which all the measurements
were then perfectly reproducible. However, no effect of
remanence that would lead to a hysteretic behavior close to
the bifurcation threshold has been observed. Demagne-
tization of pure iron occurring for field amplitudes of the
order of the Earth field, i.e., much smaller than the fields
generated by the dynamo, the iron disks do not impose any
permanent magnetization but mostly change the boundary
condition for the magnetic field generated in the bulk of the
flow. This changes the dynamo threshold and the near
critical behavior for amplitudes below the coercitive field
of pure iron. It should be also emphasized that the axisym-
metry of the setup cannot lead to Herzenberg-type dyna-
mos [25]. In addition, these rotor dynamos display a sharp
increase of the field at threshold and their saturation is
mostly limited by the available motor power [25]. On the
contrary, we observe a continuous bifurcation with a satu-
rated magnetic field in good agreement with a scaling law
derived for a fluid dynamo.
The different mechanisms at work, effect of magnetic
boundary conditions, effect of mean flow with respect to
turbulent fluctuations, etc., will obviously motivate further
studies of the VKS dynamo. A preliminary scan of the
parameter space has shown that when the disks are rotated
at different frequencies, other dynamical dynamo regimes
are observed including random inversions of the field
polarity. Their detailed description together with experi-
ments on the relative effect of the mean flow and the
turbulent fluctuations on these dynamics are currently in
progress.
We gratefully acknowledge the assistance of D.
Courtiade, J.-B. Luciani, P. Metz, V. Padilla, J.-F. Point,
and A. Skiara and the participation of J. Burguete to the
early stage of VKS experiment. This research is supported
by the French institutions: Direction des Sciences de la
Matie
`
re and Direction de l’Energie Nucle
´
aire of CEA,
Ministe
`
re de la Recherche and Centre National de
Recherche Scientifique (No. ANR NT05-1 42110, GDR
2060). The experiments have been realized in CEA/
Cadarache-DEN/DTN.
*Present address: LEGI, CNRS UMR 5519, BP53, 38041
Grenoble, France.
†
Present address: IFREMER, Laboratoire de Physique des
Oce
´
ans, CNRS UMR 5519, BP70, 29280 Plouzane,
France.
‡
Present address: Laboratory for Aero and Hydrodynamics,
TU-Delft, The Netherlands.
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044502-4
![FIG. 4 (color online). The dimensionless quantity, hB2i 0 R 2= is displayed as a function of Rm for different working temperatures and frequencies [measurements done at P2 and identical symbols as in the inset of Fig. 3(b)]. The inset shows the same data in dimensional form B2 versus rotation frequency for different temperatures.](/figures/fig-4-color-online-the-dimensionless-quantity-hb2i-0-r-2-is-2n2eblp4.webp)
