Emerging symmetries and condensates in turbulent inverse cascades
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Transcript of Emerging symmetries and condensates in turbulent inverse cascades
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)