emerging symmetries and condensates in turbulent inverse cascades
DESCRIPTION
Emerging symmetries and condensates in turbulent inverse cascades. Gregory Falkovich Weizmann Institute of Science. Cambridge, September 29, 2008 כט אלול תשס''ח. Lack of scale-invariance in direct turbulent cascades. 2d Navier-Stokes equations. Kraichnan 1967. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Emerging symmetries and condensates in turbulent inverse cascades](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814b87550346895db86c2a/html5/thumbnails/1.jpg)
Emerging symmetries and condensates
in turbulent inverse cascades
Gregory FalkovichWeizmann Institute of Science
Cambridge, September 29, 2008 כט אלול תשס''ח
![Page 2: Emerging symmetries and condensates in turbulent inverse cascades](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814b87550346895db86c2a/html5/thumbnails/2.jpg)
![Page 3: Emerging symmetries and condensates in turbulent inverse cascades](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814b87550346895db86c2a/html5/thumbnails/3.jpg)
Lack of scale-invariance in direct turbulent cascades
![Page 4: Emerging symmetries and condensates in turbulent inverse cascades](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814b87550346895db86c2a/html5/thumbnails/4.jpg)
2d Navier-Stokes equations
E
1
2u
2d2x
Z
1
22d2x
Kraichnan 1967
![Page 5: Emerging symmetries and condensates in turbulent inverse cascades](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814b87550346895db86c2a/html5/thumbnails/5.jpg)
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
![Page 6: Emerging symmetries and condensates in turbulent inverse cascades](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814b87550346895db86c2a/html5/thumbnails/6.jpg)
Small-scale forcing – inverse cascades
![Page 7: Emerging symmetries and condensates in turbulent inverse cascades](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814b87550346895db86c2a/html5/thumbnails/7.jpg)
Strong fluctuations – many interacting degrees of freedom → scale invariance. Locality + scale invariance → conformal invariance
Polyakov 1993
![Page 8: Emerging symmetries and condensates in turbulent inverse cascades](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814b87550346895db86c2a/html5/thumbnails/8.jpg)
![Page 9: Emerging symmetries and condensates in turbulent inverse cascades](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814b87550346895db86c2a/html5/thumbnails/9.jpg)
![Page 10: Emerging symmetries and condensates in turbulent inverse cascades](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814b87550346895db86c2a/html5/thumbnails/10.jpg)
_____________=
![Page 11: Emerging symmetries and condensates in turbulent inverse cascades](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814b87550346895db86c2a/html5/thumbnails/11.jpg)
P Boundary Frontier Cut points
Boundary Frontier Cut points
Bernard, Boffetta, Celani &GF, Nature Physics 2006, PRL2007
![Page 12: Emerging symmetries and condensates in turbulent inverse cascades](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814b87550346895db86c2a/html5/thumbnails/12.jpg)
Vorticity clusters
![Page 13: Emerging symmetries and condensates in turbulent inverse cascades](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814b87550346895db86c2a/html5/thumbnails/13.jpg)
Schramm-Loewner Evolution (SLE)
![Page 14: Emerging symmetries and condensates in turbulent inverse cascades](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814b87550346895db86c2a/html5/thumbnails/14.jpg)
C=ξ(t)
![Page 15: Emerging symmetries and condensates in turbulent inverse cascades](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814b87550346895db86c2a/html5/thumbnails/15.jpg)
![Page 16: Emerging symmetries and condensates in turbulent inverse cascades](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814b87550346895db86c2a/html5/thumbnails/16.jpg)
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
![Page 17: Emerging symmetries and condensates in turbulent inverse cascades](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814b87550346895db86c2a/html5/thumbnails/17.jpg)
Bose-Einstein condensation and optical turbulenceGross-Pitaevsky equation
![Page 18: Emerging symmetries and condensates in turbulent inverse cascades](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814b87550346895db86c2a/html5/thumbnails/18.jpg)
![Page 19: Emerging symmetries and condensates in turbulent inverse cascades](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814b87550346895db86c2a/html5/thumbnails/19.jpg)
![Page 20: Emerging symmetries and condensates in turbulent inverse cascades](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814b87550346895db86c2a/html5/thumbnails/20.jpg)
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
![Page 21: Emerging symmetries and condensates in turbulent inverse cascades](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814b87550346895db86c2a/html5/thumbnails/21.jpg)
![Page 22: Emerging symmetries and condensates in turbulent inverse cascades](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814b87550346895db86c2a/html5/thumbnails/22.jpg)
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
![Page 23: Emerging symmetries and condensates in turbulent inverse cascades](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814b87550346895db86c2a/html5/thumbnails/23.jpg)
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
![Page 24: Emerging symmetries and condensates in turbulent inverse cascades](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814b87550346895db86c2a/html5/thumbnails/24.jpg)
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
![Page 25: Emerging symmetries and condensates in turbulent inverse cascades](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814b87550346895db86c2a/html5/thumbnails/25.jpg)
Weak condensate Strong condensate
![Page 26: Emerging symmetries and condensates in turbulent inverse cascades](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814b87550346895db86c2a/html5/thumbnails/26.jpg)
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
![Page 27: Emerging symmetries and condensates in turbulent inverse cascades](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814b87550346895db86c2a/html5/thumbnails/27.jpg)
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)
![Page 28: Emerging symmetries and condensates in turbulent inverse cascades](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814b87550346895db86c2a/html5/thumbnails/28.jpg)
![Page 29: Emerging symmetries and condensates in turbulent inverse cascades](https://reader035.vdocuments.net/reader035/viewer/2022062409/56814b87550346895db86c2a/html5/thumbnails/29.jpg)