G. Falkovich
description
Transcript of G. Falkovich
![Page 1: G. Falkovich](https://reader031.fdocuments.us/reader031/viewer/2022020721/56815311550346895dc135db/html5/thumbnails/1.jpg)
G. Falkovich
February 2006
Conformal invariance in 2d turbulence
![Page 2: G. Falkovich](https://reader031.fdocuments.us/reader031/viewer/2022020721/56815311550346895dc135db/html5/thumbnails/2.jpg)
Simplicity of fundamental physical laws manifests itself infundamental symmetries.
Strong fluctuations - infinitely many strongly interacting degrees of freedom → scale invariance.
Locality + scale invariance → conformal invariance
![Page 3: G. Falkovich](https://reader031.fdocuments.us/reader031/viewer/2022020721/56815311550346895dc135db/html5/thumbnails/3.jpg)
Conformal transformation rescale non-uniformly but preserve angles z
![Page 4: G. Falkovich](https://reader031.fdocuments.us/reader031/viewer/2022020721/56815311550346895dc135db/html5/thumbnails/4.jpg)
![Page 5: G. Falkovich](https://reader031.fdocuments.us/reader031/viewer/2022020721/56815311550346895dc135db/html5/thumbnails/5.jpg)
2d Navier-Stokes equations
E
1
2u
2d2x
Z
1
22d2x
In fully developed turbulence limit, Re=UL-> ∞ (i.e. ->0):
(because dZ/dt≤0 and Z(t) ≤Z(0))
u
t uu
p
2u u f
u0
u
t uu
p
2u u f
u0
![Page 6: G. Falkovich](https://reader031.fdocuments.us/reader031/viewer/2022020721/56815311550346895dc135db/html5/thumbnails/6.jpg)
The double cascade Kraichnan 1967
The double cascade scenario is typical of 2d flows, e.g. plasmas and geophysical flows.
kF
Two inertial range of scales:•energy inertial range 1/L<k<kF
(with constant )•enstrophy inertial range kF<k<kd
(with constant )
Two power-law self similar spectra in the inertial ranges.
![Page 7: G. Falkovich](https://reader031.fdocuments.us/reader031/viewer/2022020721/56815311550346895dc135db/html5/thumbnails/7.jpg)
![Page 8: G. Falkovich](https://reader031.fdocuments.us/reader031/viewer/2022020721/56815311550346895dc135db/html5/thumbnails/8.jpg)
![Page 9: G. Falkovich](https://reader031.fdocuments.us/reader031/viewer/2022020721/56815311550346895dc135db/html5/thumbnails/9.jpg)
_____________=
![Page 10: G. Falkovich](https://reader031.fdocuments.us/reader031/viewer/2022020721/56815311550346895dc135db/html5/thumbnails/10.jpg)
P Boundary Frontier Cut points
Boundary Frontier Cut points
![Page 11: G. Falkovich](https://reader031.fdocuments.us/reader031/viewer/2022020721/56815311550346895dc135db/html5/thumbnails/11.jpg)
Schramm-Loewner Evolution (SLE)
![Page 12: G. Falkovich](https://reader031.fdocuments.us/reader031/viewer/2022020721/56815311550346895dc135db/html5/thumbnails/12.jpg)
![Page 13: G. Falkovich](https://reader031.fdocuments.us/reader031/viewer/2022020721/56815311550346895dc135db/html5/thumbnails/13.jpg)
![Page 14: G. Falkovich](https://reader031.fdocuments.us/reader031/viewer/2022020721/56815311550346895dc135db/html5/thumbnails/14.jpg)
C=ξ(t)
![Page 15: G. Falkovich](https://reader031.fdocuments.us/reader031/viewer/2022020721/56815311550346895dc135db/html5/thumbnails/15.jpg)
![Page 16: G. Falkovich](https://reader031.fdocuments.us/reader031/viewer/2022020721/56815311550346895dc135db/html5/thumbnails/16.jpg)
Vorticity clusters
![Page 17: G. Falkovich](https://reader031.fdocuments.us/reader031/viewer/2022020721/56815311550346895dc135db/html5/thumbnails/17.jpg)
Phase randomized Original
![Page 18: G. Falkovich](https://reader031.fdocuments.us/reader031/viewer/2022020721/56815311550346895dc135db/html5/thumbnails/18.jpg)
![Page 19: G. Falkovich](https://reader031.fdocuments.us/reader031/viewer/2022020721/56815311550346895dc135db/html5/thumbnails/19.jpg)
![Page 20: G. Falkovich](https://reader031.fdocuments.us/reader031/viewer/2022020721/56815311550346895dc135db/html5/thumbnails/20.jpg)
Possible generalizations
Ultimate Norway
![Page 21: G. Falkovich](https://reader031.fdocuments.us/reader031/viewer/2022020721/56815311550346895dc135db/html5/thumbnails/21.jpg)
Conclusion
Within experimental acuracy, zero-vorticity lines in the 2d inverse cascade have conformally invariant statistics equivalent to that of critical percolation.
Isolines in other turbulent problems may be conformally invariant as well.