Examples of non-stationarity and slow dynamics in turbulence (experimental observations) JFP, laboratoire de physique de l’Ecole normale supérieure de Lyon Alain Arnéodo, Laurent Chevillard, Jean Delour, Peter Holdsworth, Emmanuel Lévêque, Nicolas Mordant, Philippe Odier, Romain Volk VKS Collaboration: Mickael Berhanu, Mickael Bourgoin, Arnaud Chiffaudel, François Daviaud, Bérengère Dubrulle, Stephan Fauve, Romain Monchaux, Nicolas Plihon, François Pétrélis, Florent Ravelet
Fully developped turbulence
Flow forcing at large scale (non linear interactions) (energy transfer : ) Dissipation at small scale
Fully developed turbulence
Fully developed turbulence
Inertial range
Fully developed turbulence long time uncorrelated fluct short-time memory
Questions: • universality of small scale motion (independence from large scale forcing) • large scale / long time dynamics ?
Question: emergence of long time dynamics in turbulence ?
Exemple 1: motion of fluid particles
Exemple 2: dynamics of velocity gradients
Exemple 3: the dynamo instability
More questions …
The Kolmogorov picture (1 particle) • Random walks
• White acceleration, spectrum :
The Kolmogorov picture (1 particle) • Random walks
• Link with K41 theory in Eulerian framework
Lagrangian measurements Ott, Mann, J. Fluid Mech., 422, 207, (2000) Voth, Satyanarayan, Bodenschatz, Phys. Fluids, 10, 2268, (2000) Mordant, Pinton, Michel, J. Acoust. Soc. Am., 112, 108, (2002)
Particles ! 10 "m Rates ! 10 kHz Re ! 106
velocity spectrum Mordant, Metz, Michel, Pinton, Phys. Rev. Lett., 87, 214501, (2001)
RL(!) ! exp(- ! / TL) EL(") = v2 TL / (1+ TL2 "2) Kolmogorov EL(") = C0# "$2
Lagrangian acceleration Mordant, Lévêque, Pinton, New J. Phys, 6, 34 (2004)
around a vortex
! 500 g
Lagrangian acceleration Mordant, Crawford, Bodenschatz, Physica D, 193, (2004)
R% = 690 = 85 m/s2
Heisenberg-Yaglom scaling :
Lagrangian acceleration Mordant, Delour, Leveque, Michel, Arnéodo, Pinton, J. Stat. Phys., 113, 701, (2002)
Short-time correlation for the acceleration direction Long-time correaltion for the acceleration magnitude
Correlation of velocity increments • !u!0(t) = v(t+!0)-v(t) ->
C(t)= < u!0(t’) u!0(t’+t) >t’
DNS R%=75
exp R%=740
Lagrangian intermittency Mordant, Metz, Michel, Pinton, Phys. Rev. Lett., 87, 214501, (2001)
CASCADE (1 time stat, exponents)
scale
CORRELATED WALK (2 times stat, dynamics)
" !v = & " !'v time
" !'v correlated ?
A Lagrangian Random Walk
$%2
< log| u!0(t’)| - < log| u!0| > )( log| u!0(t’+t)| - < log| u!0| >t’ ( $%2 log(t)
MRW model Bacry, Delour, Muzy, Phys. Rev. E, 64, (2001).
stochastic equation for the velocity increments
‘K41’ theory : )(t) is *-correlated noise, Model, from observations :
G(t) : gaussian , white in time, and :
MRW model Mordant, Metz, Michel, Pinton, Phys. Rev. Lett., 87, 214501, (2001)
Langevin models of Lagrangian acceleration Aringazin & Mazhitov, Phys. Rev. E, 68, 026305, (2004)
when L(t) is *-correlated Gaussian white noise F(a)=-a
QUESTION : statistics f(+) ?
Langevin models of Lagrangian acceleration Aringazin & Mazhitov, Phys. Rev. E, 68, 026305, (2004)
• f(+) : ,-square distribution • f(+) : log-normal distribution • unifying concept : + = +(u) • +(u) = u2 : ,2 • +(u) = exp(u) : log-normal • associated Langevin eqn.
• link with Laval-Dubrulle-Nazarenko turbulence model
Two-time scales stochastic models Sawford, Phys. Fluids 3, 1577 (1991) A.M. Reynolds, Phys. Fluids 15, 2773 (2003)
• Sawford’s second order model
• Reynold’s refinement with exponential correlation in the noise
Question: emergence of long time dynamics in turbulence ?
Exemple 1: motion of fluid particles
Exemple 2: dynamics of velocity gradients
Exemple 3: the dynamo instability
More questions …
Magnetohydrodynamics B-eqn:
U-eq:
Induction at low Rm • magnetic Reynolds number • B0 applied + low Rm : quasistatic approximation
• induction is a mirror of velocity gradients
diffusion scale:
Measurements VK-Gallium Velocity feed-back
3D Hall probe
Velocity feed-back
Power
Power
R
Motor 1
B0- B0// Motor 2
60
60
40
40
20
20
0
0
applied B0
mean induced bz
-20 0
10
20
30
time (s)
40
50
60
-20 1 10
10
2
60 40 20 0 30
30.5
31
31.5
time (s)
32
32.5
10
3
10
4
10
5
histogram
Bind,z (G)
Bz (G)
Time dynamics
33
Time dynamics
Slow dynamics T >> .-1
K41 scaling B2(k) ! k-11/3
Time dynamics
Time dynamics Distance of induction profile to mean profile
Time dynamics Distance of induction profile to mean profile
Dynamo : a “Bullard” cycle • Spontaneous bifurcation to B!" state
• Positive loop-back induction cycle : B
B1 !
i
U
B2
B3 // B1 …
Dynamo : a “Bullard” cycle
M. Bourgoin et al., NJP 8 (2006)
Dynamo : a “Bullard” cycle
M. Bourgoin et al., NJP 8 (2006)
VKB: “on-off” bifurcation
M. Bourgoin et al., NJP 8 (2006) Aumaitre et al., PRL 95 (2005)
Slow dynamics with reversals • Coupled modes:
• Bifurcation with additive noise ?
De la Tore, Burguete, PRL 99 (2007) Narteau et al., EPSL 262 (2007)
Question: emergence of long time dynamics in turbulence ?
Exemple 1: motion of fluid particles
Exemple 2: dynamics of velocity gradients
Exemple 3: the dynamo instability
More questions …
The VKS dynamo experiment
Re ! 106
Rm ! 30
a turbulent dynamo
Monchaux et al., PRL, 98 (2007)
a turbulent dynamo
Axisym Cowling’s thm
Monchaux et al., PRL, 98 (2007)
Magnetic field reversals
Berhanu et al. EPL, 77 (2007)
Question: emergence of long time dynamics in turbulence ?
Exemple 1: motion of fluid particles
Exemple 2: dynamics of velocity gradients
Exemple 3: the dynamo instability
More questions / observations …
• long time dynamics in convective flows in Karman flows Resagk et al. PoF 18 (2004)
• existence of turbulent « states » axisym stat. mech. Leprovost, Dubrulle, Chavanis,PRE (2005) Monchaux et al., PRL 96 (2006)
cascade models
Chilla et al. EPJB 40 (2004)
Benzi PRL 95 (2005)
Lohse & Sugiyama, preprint
• long time dynamics in convective flows in Karman flows • existence of turbulent « states »
De la Tore et al. PRL 99 (2007)
axisym stat. mech. Leprovost, Dubrulle, Chavanis,PRE (2005) Monchaux et al., PRL 96 (2006)
cascade models Benzi PRL 95 (2005)
Ravelet et al. PRL 93 (2004)
• low Pm MHD turbulence & dynamo
slow B , fast U
