influence of turbulence on the dynamo threshold
DESCRIPTION
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. Maxwell equations. Navier-Stokes equations. Field strectching. Field diffusion. Competition characterized by magnetic Reynolds number. - PowerPoint PPT PresentationTRANSCRIPT
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
vRBBcurlRpRvvRv
BBvcurlRB
mmmmt
mt
Re
Maxwell equations
Navier-Stokes equationsField 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
Rm Pm=1Pm=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 50vv0.3
“TM60”, no dynamo Field “TM28”, dynamo
No dynamo
Dynamo
VKS ExperimentSodium
MeasureOptimisation
Kinematic code
…Failure!
1
10
100
1 10 100 1000R
mRe
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 2Who 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
vRBBcurlRpRvvRv
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) 1Z
B / 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 et a 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 dynamowindows
Results for noisy delta-correlatedForcing 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
= <V
2 >/ <
<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