g. falkovich february 2006 conformal invariance in 2d turbulence
Post on 20-Dec-2015
218 views
TRANSCRIPT
G. Falkovich
February 2006
Conformal invariance in 2d turbulence
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
Conformal transformation rescale non-uniformly but preserve angles z
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
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.
_____________=
P Boundary Frontier Cut points
Boundary Frontier Cut points
Schramm-Loewner Evolution (SLE)
C=ξ(t)
Vorticity clusters
Phase randomized Original
Possible generalizations
Ultimate Norway
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.