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Emerging symmetries and condensates
in turbulent inverse cascades
Gregory FalkovichWeizmann Institute of Science
Cambridge, September 29, 2008 כט אלול תשס''ח
Lack of scale-invariance in direct turbulent cascades
2d Navier-Stokes equations
E
1
2u
2d2x
Z
1
22d2x
Kraichnan 1967
lhs of (*) conserves
(*)
pumping
k
Family of transport-type equations
m=2 Navier-Stokes m=1 Surface quasi-geostrophic model,m=-2 Charney-Hasegawa-Mima model
Electrostatic analogy: Coulomb law in d=4-m dimensions
Small-scale forcing – inverse cascades
Strong fluctuations – many interacting degrees of freedom → scale invariance. Locality + scale invariance → conformal invariance
Polyakov 1993
_____________=
P Boundary Frontier Cut points
Boundary Frontier Cut points
Bernard, Boffetta, Celani &GF, Nature Physics 2006, PRL2007
Vorticity clusters
Schramm-Loewner Evolution (SLE)
C=ξ(t)
Different systems producing SLE
• Critical phenomena with local Hamiltonians • Random walks, non necessarily local • Inverse cascades in turbulence• Nodal lines of wave functions in chaotic systems • Spin glasses • Rocky coastlines
Bose-Einstein condensation and optical turbulenceGross-Pitaevsky equation
Condensation in two-dimensional turbulence
M. G. Shats, H. Xia, H. Punzmann & GF, Suppression of Turbulence by Self-Generated and Imposed Mean Flows, Phys Rev Let 99, 164502 (2007) ;
What drives mesoscale atmospheric turbulence? arXiv:0805.0390
Atmospheric spectrum Lab experiment, weak spectral condensate
Nastrom, Gage, J. Atmosph. Sci. 1985Nastrom, Gage, J. Atmosph. Sci. 1985
1E-10
1E-09
1E-08
1E-07
1E-06
10 100 1000
k -3
k -5/3
k -3
k (m )-1
E k( )
Shats et al, PRL2007
Mean shear flow (condensate)
changes all velocity moments:
0.02 0.04 0.06 0.08
turbulence condensate
S3 (10 m s )-7 3 -3
2
4
r (m)
6
0
-2
(b) 10-6
10 100 1000
k -3
k -5/3
k -3
k (m )-1
E k (m /s ) 3 2
10-7
10-8
10-9
10-10
ktkf
turbulencecondensate
(a)
VVV~
22 ~~2 VVVVV
32233 ~~3
~3 VVVVVVV
Inverse cascades lead to emerging symmetries but eventually to condensates which break symmetries in a different way for different moments
Mean subtraction recovers isotropic turbulence1.Compute time-average velocity field (N=400):
0.02 0.04
S3 ( )10 m s-9 3 -3
r (m) -2
0
4
6
2
10 100 1000
10 -6
10 -8
10 -9
10 -7
k (m ) -1
k -5/3E (k)
0
6
12
18
0 0.02 0.04-0.3
0.0
0.3
Flatness Skewness
r (m)
(a) (b) (c)
N
n ntyxVNyxV1
),,(1),(
2. Subtract from N=400 instantaneous velocity fields),( yxV
Recover ~ k-5/3 spectrum in the energy range
Kolmogorov law – linear S3 (r) dependence in the “turbulence range”;
Kolmogorov constant C≈7
Skewness Sk ≈ 0 , flatness slightly higher, F ≈ 6
Weak condensate Strong condensate
Conclusion
Inverse cascades seems to be scale invariant.
Within experimental accuracy, isolines of advected quantities are conformal invariant (SLE) in turbulent inverse cascades.
Condensation into a system-size coherent mode breaks symmetries of inverse cascades.
Condensates can enhance and suppress fluctuations in different systems
For Gross-Pitaevsky equation, condensate may make turbulence conformal invariant
Case of weak condensate
10 100 1000
k -3
E k ( )10 -5
10 -6
10 -7
10 -8
10 -9
k -5/3
k (m ) -1
(a) (b)
0.1
1
S 3 (10 )-7
0.01 0.1r (m)2
3
4
0.01 0.10
0.2
0.4Flatness
Skewness
r (m)
(c)
rrS L 2
3VVV)( 2
TL3L3 2
24 / SSF
2/323 / SSSk
Weak condensate case shows small differences with isotropic 2D turbulence
~ k-5/3 spectrum in the energy range
Kolmogorov law – linear S3 (r) dependence; Kolmogorov constant C≈5.6
Skewness and flatness are close to their Gaussian values (Sk=0, F=3)