Liquid metal dynamo experiments

Transcription

1 Liquid metal dynamo experiments Sébastien Aumaître CEA-Saclay and VKS team Dynamics and turbulent transport in plasmas and conducting fluids Les Houche-2011

2 Bibliography H. K. Moffatt : Magnetic field generation in electrically conducting fluids, 1978 Cambridge university Press P.H. Robert T.H. Jensen, Phys. Fluids B 5-7 (1993) A.Gailitis et al, Rev. Mod. Phys. 74 (2002) F.H. Busse, Annu. Rev. Fluid Mech 32 (2000) S. Fauve & D. Lathtrop, Laboratory experiments on liquid metal dynamos and liquid metal MHD turbulence in Fluid dynamics and dynamo in astrophysics and geophysics (2005)

3 Outlook Background & Motivation Constrained lab dynamos Riga & Karlsruhe exp. Unconstrained lab dynamo: the VKS experiment Structure and dynamics of the Magnetic field Elements for dynamo action in VKS.

4 Outlook Background & Motivation Constrained lab dynamos Riga & Karlsruhe exp. Unconstrained lab dynamo: the VKS experiment Structure and dynamics of the Magnetic field Elements for dynamo action in VKS.

5 Dynamo action Def : Instability that transfers mechanical energy into magnetic energy Example : the Bullard dynamo L di dt = ( Mω R)I L the self inductance of the spire M the mutual inductance spire-disk R the circuit resistivity More realistic solid dynamo:

6 Tools for Liquid dynamos r r r r r r r 1 r tb + u B = B u + B µ oσ r r r r r r r r r r ρ + u u = P + J B + η u ; B = [ ] µ J t u o NB : Pm = << 1 2 UL σb L Rm = σµ oul ; Re = ; N = ν ρu Rm = µ o σν Re Pm=10-5 for Na => The dynamo grows on a turbulent flow 2 3 P P ρl U Rm = µ oσ ρl Rm 2 Rm P 8 P 1/3 r

7 Tools for Liquid dynamos. Transfer mechanism in the homogeneous dynamos : Ω-effect U α-effect U B B // B J B // α 2, α-ω Dynamo

8 The oldest experienced Dynamo: The Earth. 1 st compass ~ 1040 in China 1600 in Europe Magnetic dipole almost aligned with rotation axis Since 1979 measurement with satellite Oersted 1 ( ) From CHAMP ( )

9 Characteristics of the Earth dynamo Larmor 1919 : Earth magnetic field= dynamo due to the convective motion of the liquid core. Magnetic Field B~10-4 T Magnetic Energy B 2 L 3 /2µ o ~ J (L~ ) Diffusion time µ o σ L 2 ~ 10 4 years Reynold Number ~10 9 Magnetic Reynolds Number~500 Rossby number :U/LΩ~10-6 [from Zeldovich,. et al., Magnetic Fields in Astrophysics, 1983 (Gordon & Breach: New York)]

10 Paleomagnetisme and Earth reversal Inversion of the earth polarity T~10 6 [Glatzmaier Roberts Nature 377 (1999)]

11 Other example : the sun Magnetic Field B~10-4 T Magnetic Energy B 2 L 3 /2µ o ~ J (L~ ) Diffusion time µ o σ L 2 ~ 10 6 years Reynold Number ~10 14 Magnetic Reynolds Number~10 8 Dynamics with a period of 11 years

12 Motivations for controlled Dynamo experiments Dynamo with small Pm (<< simulation) closer to geo & astrophysical object Instability on turbulent flow Role on dynamo onset? Instability with multiplicative noise Intermittency? Scaling for saturation B Energy partition Dynamical regime β B ( Rm Rmc) ; β?

13 Outlook Background & Motivation Constrained lab dynamos Riga & Karlsruhe exp. Unconstrained lab dynamo: the VKS experiment Structure and dynamics of the Magnetic field Elements for dynamo action in VKS.

14 First dynamos generation Constrained flows reproducing some theoretical prediction The Riga Dynamo and the Ponomarenko flow The Karlsruhe dynamo and the Robert flow

15 The Ponomarenko flow and the Riga Experiment u θ u z = rω = Cte r = u 0 r Ponomarenko flow : helicoidal flow + fluid at rest Rm c =18 oscillating (propagating) dynamo [Ponomarenko J. Appl. Mech Tech. Phys. 14 (1972)] Ω<40 Hz; P<300kW

16 The Riga Dynamo [Gailitis et al PRL (2001); PoP 11-5 (2004)] Growth of an oscillating magnetic field at almost the expected value Ohmic lost about 10% β B ( Ω Ωc ) ; β = f ( z)

17 The Robert flow and the Karlsruhe experiment r o Re C Re The Robert s flow Prediction [Busse et al Magnetohydrodynamics 32 (1996)] H > 4π { 1+ [( 3.83/ π ) ( d / r )]} o Sketch of the Karlsruhe exp. =>Electromagnetic Pump with a flow rate up to 150 m 3 /h

18 The Karlsruhe dynamo [Stieglitz & Mőller PoF 13-3 (2001)] Stationary imperfect bifurcation

19 Outlook Background & Motivation Constrained lab dynamos Riga & Karlsruhe exp. Unconstrained lab dynamo: the VKS experiment Structure and dynamics of the Magnetic field Elements for dynamo action in VKS.

20 The von Karman flow Poloidal & Toroidal (P/T)

21 The VKS experiment Involve : LPS-ENS, Lab. Phys. ENS-Lyon & CEA-Saclay. Device in CEA-Cadarache Diameter 500 mm, 150 l of liquid sodium (120º C) Pm = motors 300 kw Sensors : 40 Hall probes (3 components), 2 torquemeters, 1 potentiel velocity probe (2 components) Rm= µ 0 σ R cell.r disk (2 π F) 62; Re 10 6 Fluctuation rate : u 2 u u %

22 VKS In its 1st Dynamo configuartion 10 years of Optimization Aspect ratio +Blades curvatures + Steel imp. +Sodium at rest; No Dyna. +Ring in the equatorial plane +Soft-Iron impellers Dyna.. Dynamo!!! (september 2006) Copper Soft iron Stainless steel

23 At F 1 =F 2 : Stationnary dynamo [Monchaux et al PRL ] B axe B θ B r Ohmic lost of few % compared to stainless steel impellers

24 Dynamo onset Imperfect bifurcation due to iron impellers magnetization Earth magnetic field B β ( Rm Rmc) ; 1/ 2 β < 1 Demagnetized impellers Magnetized impellers

25 Scaling for the saturated field [Petrelis Mordant Fauve GAFD 101 (2007)] Turbulent flow => balance : r r r ρu u 1 µ o r r r ( B) B u ~ u c B o 2 2 B ρ 2 σ µ L 2 ρµ u u o Rm c c Rm VKS Riga Karlsruhe

26 Correlation length Instantaneous correlations on scales of order R [monchaux et al PoF 21 (2009) ]

27 The magnetic mode (F 1 =F 2 ) Mainly an axial dipole Numerical simulations (mean flow) Equatorial dipole Cowling Theorem : Axisymmetrical flow CANNOT generates an axial dipole. No-axisymmetrical fluctuations are necessary to get this dynamo

28 The magnetic mode (F 1 =F 2 ) Rm=34 Rm=36 Rm=41 The shape of the mode does not depend on Rm

29 Dynamical regimes for F 1 F 2 Breaks the R π symmetry ~Adding a global rotation π No dynamo Stat. Dyn. Reversal II Bursts Reversal I extinction

30 Reversal (Earth) F 1 /F 2 =16/22 Earth experiment τ τ T exp exp τ = T Earth Earth T [Berhanu et al EPL (2007)]

31 Similarity with Earth reversal VKS VKS Earth Earth VKS Earth [From F. Petrelis, S. Fauve, E. Dormy, J. P. Valet PRL 102 (2009) ]

32 Limit cycle F 1 /F 2 =18.5/22 axe r [Ravelet et al PRL 101, (2008)]

33 Bursts / B θ (t) (G) time (s) 15/22 [Ravelet et al PRL 101, (2008)]

34 Oscillations F1/F2 = 12/28 11/28 10/28 9/28 8/28 8/25 8/20 B axe (G) B θ (G) B r (G) Sun? time (s)

35 Dynamo with one impeller : Bistability [Berhanu et al JFM 641 (2009)] Two kinds of dynamo in the same range but not by the same path

36 F 2 =26 Hz F 2 =26.5 Hz F 1 (Hz)

37 Low dimensional model [F. Petrelis and S. Fauve, J.Phys. Cond. Mat. 20 (2008) ; F. Petrelis, S. Fauve, E. Dormy, J. P. Valet PRL 102(2009) ] Neart the onset only 2 interacting modes D(t).d(r) symmetric and Q(t).q(r) anti-symmetric Time evolution for this 2 modes D(t) and Q(t)? A(t)=D(t)+iQ(t) Symmetry considerations : B B D D; Q Q and A A A & = µ A + νa + β1a + β2 A A + β3 A A + β4 A + o( A 5 ) For F 1 =F 2 R π : D D; Q Q and A A All coefficient real Multiplicative noise ξ

38 Low dimensional model ( ) ( ) ( ) ( ) ( ) ( ) ' ' ' ' D d Q D c D bq Q a Q D Q Q dq D cq Q bd ad D Q D D i i r r i i r r = = ξ µ ν ν µ ξ µ ν ν µ & & 2 1 for 0 F F r > ν At F 1 =F 2 : D and Q decoupled Dipole grows first : µ r and ν r even function of F 1 -F 2 µ i and ν i odd function of F 1 -F 2 r r r r ν µ ν µ + < 0 < For : Q(t) is solution but instable via a dipolar perturbation

39 A = D + iq = R exp( iθ ) Q Unstable Stable X Q X D D X X

40 Path in phase space for limit cycle θ r x 60 Q X X D Br [G] B θ [G] 22 17

41 Path in phase space for limit cycle r r θ θ x x X Q X D Br [G] Q X X D B θ [G]

42 Reversal Noise strong enough to cross the instable mode and trigger reversal Q Unstable (Q) Stable (P) X D X Model can describe bi-stability by a codimension-2 bifurcation point

43 Outlook Background & Motivation Constrained lab dynamos Riga & Karlsruhe exp. Unconstrained lab dynamo: the VKS experiment Structure and dynamics of the Magnetic field Elements for dynamo action in VKS.

44 Ingredients for dynamo Action Reminder : Dynamo reaches with A ring in the mid-plane A optimized curvature of the blade (ratio P/T) A given aspect ratio Continuous electrical BC Soft iron impellers Question : Which ingredients are really necessary and why??

45 Role of the ring in the equatorial plane Copper Soft iron Stainless steel X No dynamo Stationary Dynamical

46 Curvature of the blades The threshold is increased + in the reverse sense _ Without ring in the equatorial plane, no dynamo P T + r + r < P T r r < P T wr wr The sense of rotation not essential but pumping and shear act on the threshold

47 Soduim at rest and Aspect Ratio Without the inner cylinder dynamo at samerm= μ 0 σr cell.r disk (2 πf) semaine F2 [Hz] F 1 [Hz] Conclusion : Magnetic boundary conditions on the impellers is the main element that improves dynamo action

48 Possible mechanism [Petrelis et al GAFD. 101 (2007)] Requirement : axial magnetic field necessitates non axis-symmetric fluctuations of the flow B θ B axe J vortices at the edge of the blades : B θ B axe by α-effect Action of the Soft-Iron impellers? Shear : B axe B θ by Ω effect

49 Kinematic simulation of non-axis-symmetric flow [CJP Gissinger EPL ] Synthetic mean flow + 8 rotating vortices Axial magnetic field at the threshold Axial dipole and quadrupole threshold similar => dynamical regime if R π symmetry is broken Suggest that :Soft-Iron boundary conditions (at rest) decrease the threshold,

50 Localized alpha-effect and periodic boundary condition [G. Gieseck, F. Stefani, G.Gerbert PRL 104 (2010) ; FH Busse J Wicht GAFD 64 (1992)] Local α-effect axial dipole [Laguerre et al PRL 101 (2008)] Realistic impellers realistic α-effect Periodic boundary conditions help to decrease the intensity of the α-effect by Busse & Wicht mechanism U Suggest that the periodic permittivity due to blades are important

51 Omega effect the moving disk [G. Verhille N Plihon M Bourgoin Ph Odier J-F Pinton NJP 12 (2010) Herzenberg et Lowes Phil. Trans. R. Soc. A 249 (1957)] Omega effect due to the jump of the magnetic permeability of the moving disks B θ B ax Rm disk V r disk 3 Role of the soft iron blades?

52 Distance to the threshold 6 Decay time Below the threshold of an applied B (axial or equatorial) 3 Bax B 2 c τ 16 = 0.41 s τ 26 = 0.52 s time (s) 1 lim 0 τ ( Rm ) Rm Rmc Bapp lim 2 B( Rm) Form induction 0 2 Rm Rmc time (s)

53 Copper impellers Role of electrical conductivity of the impeller? σ > σ > σ Cu Iron Steel Copper Soft iron Stainless steel

54 Copper impellers Role of electrical conductivity of the impeller? σ > σ > σ Cu Iron Steel Higher conductivity dynamo action Copper Soft iron Stainless steel

55 No flow behind the impellers Idea : soft-iron impellers decouple the magnetic field from the unfavorable flow behind them [Stefani et al Euro. J. Mech. B 25 (2006)] Copper Soft iron Stainless steel

56 No flow behind the impellers Idea : soft-iron impellers decouple the magnetic field from the unfavorable flow behind them [Stefani et al Euro. J. Mech. B 25 (2006)] Copper Soft iron Stainless steel : No dynamo but the induction is improved and decay time =>Rmc~70-80 (which mode?)

57 Motionless soft-iron disks Motion of the iron necessary? Copper Soft iron Stainless steel

58 Motionless soft-iron disks Motion of the iron necessary? Copper Soft iron Stainless steel Answer : YES

59 High permittivity BC Full soft iron the most favorable (following C. Gissinger EPL)? Copper Soft iron Stainless steel

60 High permittivity BC Full soft iron the most favorable (following C. Gissinger EPL) Copper Soft iron Stainless steel Answer : No but the threshold them closer than without (by study of the decay time)

61 Mixed configuration Question : rotation of the iron impeller necessary? Copper Soft iron Stainless steel

62 Mixed configuration Question : rotation of the iron impeller necessary? Answer : Yes, if the soft-iron impeller does not move : no dynamo

63 Mixed impellers I Question : Improvement due to the soft-iron blades? Gieseck et al suggestion ; Constrain on field-line (S. Fauve)? B Copper Soft iron Stainless steel

64 Mixed impellers I Question : Improvement due to the soft-iron blades? Gieseck et al suggestion ; Constrain on field-line (perpendicular to the blades at µ )? Copper Soft iron Stainless steel Answer : NO decay time and induction gives a threshold above Rm~100

65 Mixed impellers II Question : Dynamo due to the disk? Verhille et al Suggestion Most of the iron masse Copper Soft iron Stainless steel

66 Mixed impellers II Question : Dynamo due to the disk? Verhille et al Suggestion Most of the iron masse Copper Answer : NO!!! Soft iron Stainless steel

67 Conclusions Generation of the dynamo : no-axisymmetrical fluctuations axial magnetic dipole No exact understanding of the role of BC on field generation Dynamical Regimes :Few modes involved in the dynamics, fluctuations act as noise triggering the reversals Share similarities with Earth reversal Accurate velocity and power measurements (feedback u B) Unconstrained dynamo without soft iron