VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
MHD results in a liquid sodium turbulent flow:
the VKS experiment
F. Daviaud, A. Chiffaudel, B. Dubrulle, C. Gasquet,
J. Burguete, L. Marié, F. Ravelet, R. Monchaux, V. Padilla
CEA/Saclay
S. Fauve, F. Pétrélis, N. Mordant, M.Berhanu
ENS-Paris
J.-F. Pinton, P. Odier, M. Bourgoin, R. Volk, M. Moulin
ENS-Lyon
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
The dynamo problem
• spontaneous growth of a magnetic field in a moving
conducting fluid
• instability in the presence of noise because of turbulence
• at the origin of the fields observed :
Earth
Sun
• theories and numerical simulations of models
• experiments: essential but difficult  sodium
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
MHD equations








B
 Rm (v  grad ) B  Rm ( B  grad )v  B,
t

 



v
 Rm (v  grad )v   Rm grad p  Pm v  Rm rot ( B)  B,
t
 3 control parameters:

div ( B)  0

div (v )  0
LB 2
Rm  LV , Pm   , N 
V
Typical experiments: Rm = 102, Re = 107 , Pm = 10-5
Dynamo action: instability Rm > Rmc
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Role of turbulence on dynamo action
Problem of turbulence: v = <v> + v’
• kinematic dynamo: <v> gives Rmc <v> (induction eq.)
• Leprovost-Dubrulle (EPJB 2005): Rmc= f(noise)
Rmc
Rmc <v>
Noise intensity
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Role of turbulence on dynamo action
• Ponty et al, PRL 2006
• Laval, Blaineau, Leprovost, Dubrulle, FD: PRL 2006
Rm
100
Taylor-Green flow:
Vx=sinx cosy cosz
Vy=-cosx siny cosz
Vz=0
10
1
1
26th June - 1st July, 2006
10
Re
100
Warwick, UK
1000
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Dynamo experiments
Turbulent dynamo projects : competition
*Karlsruhe
VKS
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Dynamo experiments ~ fusion?
‫ ٱ‬VKS2
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Constrained dynamos: success in 2000
Riga
Karlsruhe
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Role of turbulence on dynamo action
• Riga and Karlsruhe: «small» noise
- Rmc experiment = Rmc mean flow
- but exp. saturation Energy ≠ laminar
• Madison: « large» noise; 2006: no dynamo action?
→ VKS project since 1998: « large » noise
•VKS1 experiment (2000-2003): no dynamo action
•VKS2 experiment (2005) : Rm max > Rmc mean flow
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
VKS1 : experimental conditions
Von Karman flow in a cylindrical container (70 l)
sodium loop (250 l)
• H=2 x R = 0,4 m
• 2 motors 75 kW
• 2 propellers
• frequency : 0 à 25 Hz
(Reynolds number Re = 6.106)
• Hall probe  B
• pressure probe  P
B

R

P
H
Maximal magnetic Reynolds number: Rm=0 R2 2   40
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
VKS1 experiment in CEA/Cadarache
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
VKS1 :  effect (differential rotation)
Btt
Bi
B0
o: bx : by : bz
By
Bz
Bx
axial B0 applied  transverse by induced
Bourgoin et al. Phys. Fluids 202
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Dynamo mechanisms:  effect
V
+
B0
26th June - 1st July, 2006

Bf
=
B0
Warwick, UK
+ Bi
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
VKS1 : helicity  effect (1 propeller)
1) 1st step
Vtor
B0
Bz
o: bx : by : bz
B1
J1
Bx By
2) 2nd step
BH
JH
Vpol
B1
Bz
BH
transverse B0 applied transverse J=  B0
Petrelis et al. PRL 2003
By Bx
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Dynamo mechanisms:  effect
j ~  B
B0
B1
Bf
V1
26th June - 1st July, 2006
V2
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
VKS1 : comparison with code (1 propeller)
transverse B0
Experiment:
: bx : by : bz
Numerics:
: bx : by : bz
difference: boundary conditions, turbulence ?
Marié et al., EPJB 2003
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
VKS1 : helicity  and  effects
o: bx : by : bz
transverse B0 applied  axial bx + transverse by induced
Marié et al., Magnetohydrodynamics 2002
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
VKS1: fluctuations of B
4
14
Bx
By
Bz
12
3
-1
24 Hz
log10(spectrum)
10
8
B [G]
Bx
By
Bz
6
4
2
0
2
1
-11/3
0
-1
-2
-2
-4
-6
0
2
4
6
8
10
12
t [s]
14
16
-3
-0.5
0
0.5
1
1.5
log10(F [Hz])
Evolution of B in presence of transverse B0=3G ( = 24 Hz)
•  > 24 Hz : spectrum of Kolmogorov type
•  < 24 Hz : spectrum f-1 ~ Karlsruhe exp.
26th June - 1st July, 2006
Warwick, UK
2
2.5
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
VKS1 experiment: conclusion
• VKS 1 :
- new results on magnetic induction (mean and fluctuations)
- basic ingredients for dynamo action ( and  effects) but
no dynamo
• Limitations:
- absence of cooling system  40 secs runs at full power
- problems with seals  argon bubbles inside, reliability
- Rmc = 60 based on mean flow for TM 60 propellers:
 fc = 44 Hz (Pc =750 kW / 150 kW available)
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
VKS2 experiment (since 2005)
• VKS2: a dynamo / mean flow
- max Rm = 55 > Rmc = 43  optimisation
- max P = 300 kW > Pc = 150 kW
- cylindrical shell of sodium at rest (of thickness 0.4 R)
• Study of:
- effect of turbulence on a dynamo mean flow : dynamo?
- non linear saturation regimes; scaling of <B>
- fluctuations of Bdynamo: intermittency?
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
VKS2 experiment: optimisation
• von Karman flow inside a cylindrical copper container
• copper shell with sodium at rest (cf. Riga)
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
VKS2 experiment: optimisation

+
• Topology of the mean flow (geometry of the
impellers) : optimal impellers of radius 0.75
with curved blades
R
• Numerical variation of the ratio of the mean
poloïdal flow to the mean toroïdal flow:
an optimal value close to 0.7-0.8
P


T 
• Conducting shell thickness : strongly reduces
the threshold and changes the neutral mode
structure
w
26th June - 1st July, 2006
Warwick, UK

Vr  V z rdrdz
2
2
 V

rdrdz
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Experimental mean flow (LDV or PIV)
1
r/R
poloidal
0
toroidal
1
-0.9
0
0.9
z/R
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Kinematic dynamo code (J. Léorat)

• axially periodic flow
• cylinder of uniform
conductivity
• surrounded by infinite
insulator
FFT
- mean velocity field
- temporal integration of induction
equation
z
layer of stationary conductor of
same conductivity arround flow
0
w
r
finite differences
Marié et al., EPJB 33, 469, 2003
26th June - 1st July, 2006
FFT
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Influence of conducting shell on Rmc
180
180
w=0.4, threshold divided by 4
(power divided by 64!)
Rm
c
120
60
40
0
0
0.2
0.4
w
0.6
0.8
1
Ravelet et al.,Phys. Fluids 2005
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Comparison between codes
• Stefani et al.
EJM/B 2006
• Nore et al. 2006
max
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
max
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Neutral mode structure
Stationary in time
Azimuthal dependance m=1
w  0 Rmc  180
• concentrated near axis in 2 twisted
banana-shaped regions
• 2 folded sheets of transverse field
near the boundary : external dipole
26th June - 1st July, 2006
w  0,6 Rmc  39
• bananas are still here
• Sheets have grown in axial
and azimuthal directions
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
1
1
0.5
0.5
0
0
r
-0.5
-0.5
-1
-1
0
0.9
z
1.8
0
magnetic field B
w=0
r
B-lines are
smoothed :
weaker ohmic
dissipation
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
0
0.9
0.9
z
1.8
1.8
z
0
current density j
26th June - 1st July, 2006
w=0.6
r
r
Changes of B/j-lines topology
0.9
1.8
z
Warwick, UK
Current loops
can now close
outside the flow
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Comparison with MND
analytical flow
r/R

v r   r (1  r ) 2 (1  2r ) cos(z )
2
z
v  4r (1  r ) sin( )
2
v z  (1  r )(1  r  5r 2 ) sin(z )
1
MND
Anal.
flow
0
-1
-1
0
1
z/R
For w=0 and Γ=0.8:
- optimal experimental flow Rmc=180
- analytical flow Rmc=58
Marié, Normand and Daviaud, Phys. Fluids 2005
26th June - 1st July, 2006
r/R
1
TM73
Exp.
flow
0
1
-0.9
0
z/R
Warwick, UK
0.9
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Neutral mode structure of « MND » analytical flow
Rmc
100
w  0 Rmc  58
42.95+14.83*exp(-w / 0.20)
50
0
0
0.2
0.4
0.6
0.8
1
w
w  1 Rmc  43
• eigenmode already have developped sheets of transverse dipolar field
• layer of stationary conductor: threshold reduction of only 26%
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
VKS2 experiment
• H=2 x R = 0,6 m  170 l sodium
• 4 motors 75 kW  P = 300 kW; T = 1000 Nm
• 2 propellers : 0 à 35 Hz
• cooler
• new mechanical seals
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Experimental plateform
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
container
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Inside of container
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Response to a transverse applied B0
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Transverse applied B0 : time series of Binduced
Rm=15
26th June - 1st July, 2006
Rm=47
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Transverse applied B0 : PDF of Binduced
Rm = 40
Bx: bimodal, By: exponential tails
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
high B-state
PDF of Bx
low B-state
PDF of By
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Transverse applied B0 : spectra of Binduced
Slopes ≠ VKS1
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Transverse B0 : evolution of Binduced with Rm
Bx
By
Mean Binduced=f(Rm)
rms Binduced=f(Rm)
Bz
No saturation: ≠ VKS1
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Transverse B0 : comparison exp. / simulation
<Bx>
<By>
<Bz>
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Response to a localized B0
M: magnet
(1000 -10 G)
P: Hall probe
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Localized B0 : time series of Binduced at Rm = 25
Intermittency, no coherence between components
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Localized B0: evolution with Ω of mean and
rms Binduced
Bxrms
Byrms
Bzrms
<Bx>
<By>
<Bz>
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Localized B0: spatial decay of rms Binduced
Bxrms
Byrms
Bzrms
Decay length: 100 mm ~ integral scale
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Localized B0 : PDF of fluctuations of Binduced
Bxrms
Byrms
Bzrms
Rm = 25
Exponential tails ~ random advection of scalar field
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Localized B0 : spectra of Binduced
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Conclusion
• Response of the von karman turbulent flow to:
- transverse field : fluctuations; amplification >1; no saturation
- localized field : influence of fluctuations, intermittency
• VKS2: dynamo /mean field simulations
 influence of turbulence on threshold, dynamics, saturation?
next run in July 2006
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Comparison mean / instantaneous flow (PIV)
Mean: 5000 fields sampled at 5 Hz
26th June - 1st July, 2006
Warwick, UK
VKS collaboration: CEA - CNRS - ENS Lyon - ENS Paris
Shear layer: vortices
26th June - 1st July, 2006
Warwick, UK