Table Of ContentGeneration of 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. P´etr´elis,2 A. Chiffaudel,1 F. Daviaud,1 B. Dubrulle,1 C. Gasquet,1 L. Mari´e,1,† and F. Ravelet1,‡
1Service de Physique de l’Etat Condens´e, Direction des Sciences de la Mati`ere,
CEA-Saclay, CNRS URA 2464, 91191 Gif-sur-Yvette cedex, France
2Laboratoire de Physique Statistique de l’Ecole Normale Sup´erieure,
CNRS UMR 8550, 24 Rue Lhomond, 75231 Paris Cedex 05, France
3Laboratoire de Physique de l’Ecole Normale Sup´erieure de Lyon,
7
CNRS UMR 5672, 46 all´ee d’Italie, 69364 Lyon Cedex 07, France
0
(Dated: February 2, 2008)
0
2
We report the observation of dynamo action in the VKS experiment, i.e., the generation of
n magnetic field by a strongly turbulent swirling flow of liquid sodium. Both mean and fluctuating
a parts of the field are studied. The dynamo threshold corresponds to a magnetic Reynolds number
J R ∼ 30. A mean magnetic field of order 40 G is observed 30% above threshold at the flow
m
6 lateralboundary. Thermsfluctuationsarelargerthanthecorrespondingmeanvaluefortwoofthe
components. The scaling of the mean square magnetic field is compared to a prediction previously
] madefor high Reynoldsnumberflows.
n
y
PACSnumbers: 47.65.-d,52.65.Kj,91.25.Cw
d
-
u
l Thegenerationofelectricityfrommechanicalworkhas Oil cooling circulation
f
. been one ofthe mainachievementsofphysics by the end
cs of the XIXth century. In 1919, Larmor proposed that a P1 P2
i similarprocesscangeneratethemagneticfieldofthesun
s
y fromthemotionofanelectricallyconductingfluid. How- 5 z 5
h ever, fluid dynamos are more complex than industrial 15 0.7
p onesanditisnoteasytofindlaminarflowconfigurations y x
[
1 tehxaatmgpelneserhaatveembaegennetfiocufinedldins [t1h].eTsewvoenstiimespl[e2]bauntdclheavveer R 206 1.00
v ledmorerecentlytosuccessfulexperiments[3]. Theseex-
5
5 periments have shown that the observed thresholds are 78 Na at rest 0.4
7
in good agreement with theoretical predictions [4] made
0
1 by considering only the mean flow, whereas the satura- 41
40 61 185 0.90 0.30
0 tion level of the magnetic field cannot be described with 16
7 a laminar flow model (without using an ad-hoc turbu-
0
lent viscosity) [5]. These observations have raised many FIG. 1: Sketch of the experimental set-up. The inner and
/
s questions: what happens for flows without geometrical outer cylinders are made of copper (in gray). The dimension
c constraints such that fluctuations are of the same order are given in millimeter (left) and normalized by R (right).
i
s of magnitude as the mean flow? Is the dynamo thresh- The 3D Hall probe is located either at point P1 in the mid-
y planeorP2. Inbothcases, theprobeisnearlyflushwiththe
oldstronglyincreaseddue tothe lackofcoherenceofthe
h innershell.
p driving flow [6, 7] or does the prediction made as if the
: mean flow were acting alone still give a reasonable order
v
of magnitude [8]? What is the nature of the dynamo bi-
i
X furcation in the presence of large velocity fluctuations? ficationsthat will be describedbelow. The flowis gener-
r Allthese questions,andothersmotivatedbygeophysical ated by rotating two disks of radius 154.5 mm, 371 mm
a or astrophysical dynamos [9], have led several teams to apart in a cylindrical vessel, 2R = 412 mm in inner di-
trytogeneratedynamosinflowswithahighleveloftur- ameter and 524 mm in length. The disks are fitted with
bulence[10,11]. Wepresentinthisletterourfirstexper- 8 curved blades of height h=41.2 mm. These impellers
imental observationof the generationof a magnetic field aredrivenatarotationfrequencyuptoΩ/2π =26Hzby
in a von K´arm´an swirling flow of liquid sodium (VKS) 300 kW available motor power. An oil circulation in the
for which velocity fluctuations and the mean flow have outercoppercylindermaintainsaregulatedtemperature
comparable kinetic energy and we discuss some of the in the range 110-160oC. The mean flow has the follow-
above issues. ing characteristics: the fluid is ejected radially outward
The experimental set-up (see Fig. 1) is similar to the by the disks; this drives an axial flow toward the disks
previousVKSexperiments [11], butinvolvesthreemodi- alongtheiraxisandarecirculationinthe oppositedirec-
tion along the cylinder lateral boundary. In addition, in
2
the case of counter-rotatingdisks studied here, the pres- from it (P1 or P2 in Fig. 1). In both cases, the probe
ence of a strong axial shear of azimuthal velocity in the is nearly flush with the inner shell, thus B~ is measured
mid-plane between the impellers generates a high level at the boundary of the turbulent flow. Fig. 2 shows
of turbulent fluctuations [12, 13]. The kinetic Reynolds the time recording of the three components of B~ when
number is Re = KR2Ω/ν, where ν is the kinematic vis- R is increased from 19 to 40. The largest component,
m
cosity and K = 0.6 is a coefficient that measures the B , is tangent to the cylinder at the measurement loca-
y
efficiency of the impellers [14]. Re can be increased up tion. It increases from a mean value comparable to the
to 5106: the corresponding magnetic Reynolds number Earth magnetic field to roughly 40 G. The mean values
is, Rm = Kµ0σR2Ω ≈ 49 (at 120oC), where µ0 is the of the other components Bx and Bz also increase (not
magnetic permeability of vacuum and σ is the electrical visible on the figure because of fluctuations). Both signs
conductivity of sodium. of the components have been observed in different runs,
A first modification with respect to earlier VKS ex- depending on the sign of the residual magnetization of
periments consists of surrounding the flow by sodium at the disks. All components display strong fluctuations as
rest in another concentric cylindrical vessel, 578 mm in couldbeexpectedinflowswithReynoldsnumberslarger
inner diameter. This has been shown to decrease the than 106.
dynamo threshold in kinematic computations based on
the mean flow velocity [14]. The total volume of liquid
20
sodium is 150 l. A second geometrical modification con- Bx
sists of attaching an annulus of inner diameter 175 mm B
0 y
and thickness 5 mm along the inner cylinder in the mid- B
z
plane between the disks. Water experiments have shown
thatitseffectonthemeanflowistomaketheshearlayer G] −20
sharper around the mid-plane. In addition, it reduces B [i
lowfrequencyturbulentfluctuations,thusthelargescale −40
flowtime-averagesfastertowardthemeanflow. However,
rmsvelocityfluctuationsarealmostunchanged(oforder −60
40−50%),thus the flowremains stronglyturbulent[15]. Hz]20 10 20 30 40
It is expectedthat reducingthe transversemotionofthe π [15
2
shearlayerdecreasesthedynamothresholdforthefollow- Ω/10
0 10 20 30 40 50
ing reasons: (i) magnetic induction due to an externally time [s]
applied field on a gallium flow strongly varies because of
the large scale flow excursions away from the time aver- FIG.2: TimerecordingatP1ofthecomponentsofthemag-
agedflow[16], (ii)the additionoflargescalenoisetothe netic field when the rotation frequency Ω/2π is increased as
displayed by theramp below (R increases from 19 to40).
Taylor-Green mean flow increases its dynamo threshold m
[7],(iii)fluctuatingmotionofeddiesincreasethedynamo
threshold of the Roberts flow [17]. Fig. 3ashowsthe meanvaluesofthecomponentshB i
i
The aboveconfigurationdoes not generate a magnetic ofthemagneticfieldandFig. 3btheirfluctuationsBirms
field up to the maximum possible rotation frequency of versusRm. The fluctuations are allin the same range(3
thedisks(Ω/2π =26Hz). Wethusmadealastmodifica- G to 8 G, at 30% above threshold) although the corre-
tion andreplaceddisks made of stainlesssteel by similar sponding mean values are very different. The time aver-
iron disks. Using boundary conditions with a high per- ageofthe squareofthe totalmagneticfield, hB~2i, is dis-
meability in order to change the dynamo threshold has played in the inset of Fig. 3a. No hysteresis is observed.
been already proposed [18]. It has been also shown that Linear fits of hByi or Birms displayed in Fig. 3 define a
in the case of a Ponomarenko or G. O. Roberts flows, criticalmagneticReynoldsnumberRc ∼31whereasthe
m
the addition ofan externalwallof high permeability can linearfitofhB~2igivesalargervalueR0 ∼35. Thelatter
m
decreasethedynamothreshold[19]. Finally,recentkine- is the one that should be considered in the case of a su-
maticsimulationsoftheVKSmeanflowhaveshownthat percritical pitchfork bifurcation. The rounding observed
different ways of taking into account the sodium behind close to threshold could then be ascribed to the imper-
the disks lead to an increase of the dynamo threshold fection due to the ambient magnetic field (Earth field,
ranging from 12% to 150% [20]. We thought that using residual magnetization of the disks and other magnetic
iron disks could screen magnetic effects in the bulk of perturbations of the set-up). The actual behavior may
the flow from the region behind the disks, although the be more complex because this bifurcation takes place on
actual behavior may be more complex. This last modi- a strongly turbulent flow, a situation for which no rig-
fication generates a dynamo above R ≃ 30. The three orous theory exists. The inset of Fig. 3b shows that
m
components of the field B~ are measured with a 3D Hall the variance B2 = h(B~ −hB~i)2i is not proportional
rms
probe, located either in the mid-plane or 109 mm away to hB2i. Below the dynamo threshold, the effect of in-
3
duction due to the ambient magnetic field is observed.
Brms/hB2i1/2 first behaves linearly at lowRm, but then 3000
increases faster as R becomes closer to the bifurcation 50 P1
m 2500
P2
threshold. We thus show that this seems to be a good
2000
quantity to look at as a precursor of a dynamo regime. 40
Inaddition,weobservethatitdisplaysadiscontinuityin 1500
<B2> [G2]
srleosppeoninsetfhuencvtiicoinnistyatopfhRamcseitnraannsiatinoanlsogoorubsifwuarcyaotifosnosmine > [G] 30 1000
Bi 500 −<B>
the presence of noise. Note however that the shape of < x
20 0 −<B>
the curves depends on the measurement point and they 20 25 30 35 40 45 y
Rm
cannotbe superimposedwithascalingfactorasdonefor <B>
z
hB2i versus R in the inset of Fig. 3a. 10
m
The above results are characteristic of bifurcations in
the presence of noise. As shown in much simpler ex- 0
10 20 30 40
periments,differentchoicesofanorderparameter(mean Rm
valueoftheamplitudeoftheunstablemodeoritshigher
moments, its most probable value, etc) can lead to qual- 14
itatively different bifurcation diagrams [21]. This illus- 12 0.5 B /<B2>1/2 BBx
trates the ambiguity in the definition of the order pa- 0.45 rms y
B
rameter for bifurcations in the presence of fluctuations 10 0.4 z
or noise. In the present experiment, fluctuations enter 0.35
banodthadmduilttiivpelliyc,adtiuveeltyo, tbheecainutseeraocftitohneotfutrhbeulveenltocvietyloficietlyd, B [G]i rms 68 00.2.35
with the ambient magnetic field. Finally, we note that
0.2
both Rc and R0 are smaller than the thresholds com- 4
m m 0.15
puted with kinematic dynamo codes taking into account 20 25 30 35 40 45
onlythemeanflow,thatareintherangeRc =43to150 2 Rm
m
depending on different boundary conditions on the disks
0
10 15 20 25 30 35 40 45
and on configurations of the flow behind them [14, 20].
Rm
The probabilitydensityfunctions (PDF)ofthe fluctu-
ations of the three components of the induced magnetic FIG. 3: a) Mean values of thethree components of the mag-
field (not displayed) are roughly gaussian. The PDFs of neticfieldrecordedatP1versusRm(T =120oC):(N)−hBxi,
fluctuations below threshold, i.e., due to the induction ((cid:4))−hByi,(•)hBzi. Theinsetshowsthetimeaverageofthe
square of the total magnetic field as a function of R , mea-
resulting from the ambient magnetic field, are similar to m
sured at P1 (•), or at P2 (⋆) after being divided by 1.8. b)
the ones observed in the self-generating regime. We do
Standarddeviationof thefluctuationsof each componentsof
notobserveanynongaussianbehaviorclosetothreshold
themagneticfieldrecordedatP1versusR . Theinsetshows
m
which would result from an on-off intermittency mecha- B /hB2i1/2. Measurements done at P1: (N) T = 120oC,
rms
nism [22]. Possible reasons are the low level of small fre- frequency increased up to 22 Hz; Measurements done at P2:
quency velocity fluctuations [23] or the imperfection of (⋆) T =120oC, frequency decreased from 22 to 16.5 Hz, ((cid:3))
the bifurcation that results from the ambient magnetic T =156oC,frequencyincreasedupto22Hz,(◦)Ω/2π =16.5
Hz,T variedfrom154to116oC,(♦)Ω/2π =22Hz,T varied
field [24].
from119to156oC.TheverticallinecorrespondstoR =32.
Fig. 4displaysboththedimensional(seeinset)anddi- m
mensionless mean squarefield as a function of R . hB2i
m
is made dimensionless using a high Reynolds number
scaling [5]: hB2i ∝ ρ/(µ (σR)2)(R −Rc )/Rc , where driving power of the order of 100 kW.
0 m m m
ρ is the fluid density. We observe that data obtained at The effect of iron disks deserves additional discussion.
different working temperature are well collapsed by this Aslighteffectofmagnetizationofironhasbeenobserved:
scaling(σdecreasesbyroughly15%from100to160oC). thedynamothresholdduringthefirstrunwasabout20%
The low Reynolds number or “weak field” scaling could larger than in the next runs for which all the measure-
also give a reasonable collapse of data obtained on this ments were then perfectly reproducible. However, no ef-
temperature rangebut the predicted order of magnitude fectofremanencethatwouldleadtoahystereticbehavior
for hB2i would be 105 too small. closetothebifurcationthresholdhasbeenobserved. De-
DissipatedpowerbyOhmiclossesisanotherimportant magnetizationofpureironoccurringforfieldamplitudes
characterization of dynamo action. Our measurements of the order of the Earth field, i.e., much smaller than
show that, 30% above threshold, it leads to an excess thefieldsgeneratedbythedynamo,theirondisksdonot
power consumption of 15−20% with respect to a flow imposeanypermanentmagnetizationbutmostlychange
4
† Present address : IFREMER, Laboratoire de Physique
des Oc´eans, CNRS UMR 5519, BP70, 29280 Plouzane,
5 103
France
‡ Presentaddress: LaboratoryforAeroandHydrodynam-
B2 [G2] vs ics, TU-Delft, The Netherlands
0.2
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[13] Induction experiments with only one disk driving liquid
uous bifurcation with a saturated magnetic field in good
sodium have been performed by B. Lehnert, Arkiv fu¨r
agreementwith ascalinglawderivedfora fluiddynamo. Fysik 13, 109 (1957). Another rapidly rotating flow ge-
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http://www.imprimerie.polytechnique.fr/Theses/Files/Ravelet.pdf.
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We gratefully aknowledge the assistance of D. Cour-
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ported by the french institutions: Direction des Sciences [24] F.P´etr´elisandS.Aumaˆıtre,E.Phys.J.B51357(2006).
de la Mati`ere and Direction de l’Energie Nucl´eaire of [25] F. J. Lowes and I. Wilkinson, Nature 198, 1158 (1963);
219, 717 (1968).
CEA, Minist`ere de la Recherche and Centre National
deRechercheScientifique(ANR05-0268-03,GDR2060).
The experiments havebeen realizedin CEA/Cadarache-
DEN/DTN.
∗ Presentaddress: LEGI,CNRSUMR5519,BP53,38041
Grenoble, France
