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)