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)