Influence of turbulence on the dynamo threshold

B. Dubrulle, GIT/SPEC

N. Leprovost, R. Dolganov, P. Blaineau

J-P. Laval and F. Daviaud

Basic equations

v

RBBcurlRpRvvRv

BBvcurlRB

mmmmt

mt

Re

Maxwell equations

Navier-Stokes equations

Field strectching

Field diffusion

Competition characterized by magnetic Reynolds number

Dynamo if Rm > Rmc (Instability)

« Classical dynamo » paradigm

Pas dynamo Dynamo

Em

t

Em

t

Rmc Rm

No dynamo

Dynamo

2 ln B2

tIndicator:

Dynamos in the Universe

Def: Magnetic field generation through movement of a conductor

In the universe….

stars, galaxies

planets

Control Parameters: Re UL /Rm UL /Pr Rm / Re /

Re 108

Rm 106

vv0.1 0.2

Problem

Turbulent flow:

v

V

v '

t

B Rmcurl

V

B B Rmcurl

v '

B

Multiplicative noiseClassical linear instability 

Mean Flow Fluctuation

« Classical turbulence » paradigm

v '

B 2

B curl

B Mean Field argument:

t

B 2

B ( )

B

k ( )k 2

km

2( )

m 2

4( )

Mean Field equation:

Mean Field dispersion:

Mean Field instability:

Turbulence creates dynamo « most of the time »« Helical turbulence is good for dynamo »

Numerical test ?

1

10

100

1 10 100 1000

Rm

Re

Schekochihin et al, 2004 Ponty et al, 2005, 2006Laval et al, 2006

Re

RmPm=1

Pm=1

Whithout mean velocity With mean-velocity

*Karlsruhe

Experimental test ?

Dynamos with low “unstationarity”: success

Riga

Karlsruhe

R Stieglitz, U. Müller, Phys.of Fluids,2001A. Gailitis et al., Phys. Rev. Lett., 2001

Experiments with unstationarity…

Re 107

Rm 50

vv0.3

“TM60”, no dynamo Field “TM28”, dynamo

No dynamo

Dynamo

VKS ExperimentSodium

MeasureOptimisation

Kinematic code

…Failure!

1

10

100

1 10 100 1000

Rm

Re

Turbulence increases thresholdWith respect to time-averaged!

Explanation: numerics

Simulation with time averaged velocity

Simulation with real velocity

Explanation: theory

Kraichnan model

t B

B B

Mean Field Theorie

Perturbative computation (Petrelis, Fauve)

Rmc C v'B(1)

C V B(0)

CB(0)

CB(0)

(<V>=0)

(with mean velocity fiel)

(<V>=0)

v'(x, t)v '(x r, t ) (1 r )()

Dynamo only for

1 2

Who is right ??????

Importance of the order parameter

MFT, Petrelis, Fauve: transition over <B>

KM : transition over <B2>

<B>=0… No dynamo (MFT, Petrelis)

<B2> non zero…Dynamo (KM)

Vote: What is good order parameter?

Problem

Turbulent flow:

v

V

v '

t

B Rmcurl

V

B B Rmcurl

v '

B

Multiplicative noiseClassical linear instability 

Mean Flow Fluctuation

Troubles

B B BModel equation:

Problem: how to define threshold?

B

(D ) B

B2

2(2D ) B2

Instability threshold depends on moment order!!! Etc, etc...

Solution: work with PDF and Lyapunov exponent

2 ln B2

t

2 ln B2

t

Stochastic approach

v

RBBcurlRpRvvRv

BBvcurlRB

mmmmt

mt

Re

Basic Equations

Approximation 1

Approximation 2

v

V

v '

Noise delta-correlated in timeMean Flow

BKBBvcurlBt

�2

Fokker-Planck Equation

tP VkkP kVi BiBkP KBi

B2BiP k kl lP 2Bi

BkliklP ijklBiB jBk

BlP with

Equation for P(B,x,t)

kl vk' vl

'

ijk vi'kv j

'

ijkl jvi'lvk

'

Mean-Field Equation

t Bi Vkk Bi kVi Bk K B2Bi

k kll Bi 2kilk Bl

beta effect(turbulent diffusion) Alpha effect

Helicity if isotropy

Mean Field Theory EquationStability governed by alpha et beta….!!!???

Isotropic case

t

B 2

B ( )

B

t B2 B2 ( ) B2

Mean Field Magnetic energy

k ( )k 2

( )k 2

km

2( )

m 2

4( )

km 0

m

Stationary solutions

P B Always solution!

Other solution: P P(B)G ei ,x j Bi Bei

P(B) 1

ZB / a D exp KB2 /a

a ijkleie jekel G

kVieiek G ijkl ikeiek kjeiel

G

Z: normalizationD: space dimensiona et : coefficients

Lyapunov exponent!

Bifurcations

Non-zero Solution (normalisation) Most probable value

a 0 eta 0

aD

0 aDNo dynamo IntermittentDynamo

TurbulentDynamo

New theoretical turbulent paradigm

RmRm1 Rm2No dynamo IntermittentDynamo

TurbulentDynamo

Pas dynamo Dynamo

Em

t

Em

t

Rmc

Turbulent

Laminar

Rm

The Lyapunov exponent…

a ijkleie jekel G 0

kVieiek G ijkl ikeiek kjeiel

G

Orientation (<0)(zero if <V>=0)

>0 and proportional to noise( KM effect)

Unstable Direction

Rmc

Expected result

StableDirection

Noise intensity

Rmc <V>

Leprovost, Dubrulle, EPJB 2005

Illustration: Bullard

Homopolar Dynamo

Noise intensity

Intermittent Dynamo

No Dynamo

Leprovost, PhD thesis

Discussion

Noise influences threshold through mu AND vector orientation

Influence of alpha and beta through vector orientation

Threshold different from Mean Field Theory prediction

Dangerous to optimize dynamo experiments from mean field!

Turbulent threshold can be very different from « laminar » ones

Simulations

t

B Rmcurl

V

B B Rmcurl

v '

B

V Time-average of velocity field computed through Navier-Stokes

v 'Type of simulation

MHD-DNSKinematicNoisy

Computed through NS0

Markovian noise (F,tc, ki)

Numerical code

Spectral methodIntegration scheme: Adams-BrashfordResolution: 64*64*64 to 256*256*256Forcing with Taylor-Green vortexConstant velocity forcing

Cf Ponty et al, 2004, 2005

Time-averaged vs real dynamo

• Laval, Blaineau, Leprovost, Dubrulle, FD: PRL 2006

1

10

100

1 10 100 1000

Rm

Re

2 dynamo

windows

Results for noisy delta-correlated

Forcing at ki=1

Forcing at ki=16

0

10

20

30

40

50

60

1 10

Rm

a)

0

10

20

30

40

50

60

1 10

Rm

c)

Linear in (-1)(Fauve-Petrelis)

v

V

v '

Results for noisy tc=0.1

Forcing at ki=1

Forcing at ki=16

0

10

20

30

40

50

60

1 10

Rm

0

10

20

30

40

50

60

1 10

Rm

d)

Results for noisy tc=1 s

Forcing at ki=1

0

10

20

30

40

50

60

1 10

Rm

b)

Summary of noisy

ki=1

ki=160

10

20

30

40

50

1 10

Rm

a)

0

5

10

15

20

25

30

35

40

1 2 3 4

Rm

DNS

Tc=1

Compa DNSStochastic noise k=1Tauc=1 s

Summary of noisysimulations

00.11

50

8

Interpretation

0

5

10

15

20

0 0,2 0,4 0,6 0,8 1

Rm

*

b)

Kinetic energy of of theVelocity Fluid

Rm*

Rm

Universal curve

In VKS =30=0.97

Definition of a universal « control parameter »

1

2

3

4

0 20 40 60 80 100 120

= <

V2>

/ <<

V>

2>

Re

V 2

X/ V 2

X

Comparison stochastic/DNS

0

10

20

30

40

50

1 10

Rm

a)

Compa DNSStochastic noise k=1Tauc=1 s

Summary of noisysimulations

Tauc

ki=1

ki=16

10

100

10 100

Rm

Re3

2

Comparison DNS and mean flow

Laval, Blaineau, Leprovost, Dubrulle, Daviaud (2005)

Dynamo CM

No dynamo

Intermittent Dynamo

1

10

100

1 10 100 1000

Rm

Re

ConclusionsIn Taylor-Green, turbulence is not favourable to dynamo

Large scale turbulence (unstationarity) increases dynamo threshold-> desorientation effectSmall scale turbulence decreases dynamo threshold-> « friction »

Turbulence looks like a large scale noiseBad influence through desorientation effect

Possible transition to dynamo via intermittent scenario

In natural objects: importance of Coriolis force (kills large-scale)

Possibility of stochastic simulations to replace DNS