Conformal invariancein two-dimensional turbulence

Guido BoffettaDipartimento di Fisica Generale

University of Torino

D.Bernard, G.Boffetta, A.Celani, G. Falkovich, Nature Physics, 2 124 (2006)

www.ph.unito.it/~boffetta

A physical motivation for two-dimensional turbulence

2D Navier-Stokes equation are a simplemodel for large scale motion of atmosphere and oceans: thin layers offluid in which stratification and rotation supress vertical motions.

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2d Navier-Stokes equations

Two inviscid quadratic invariants:

E =

1

2u2d2x∫

Z =

1

2ω2d2x∫

Energy/enstrophy balance in viscous flows:

dE

dt=−2νZ

dZ

dt=−2νP

(palinstrophy)

In fully developed turbulence limit, Re=ULν-> ∞ (i.e. ν->0):

limν→ 0

dE

dt=0 (because dZ/dt≤0 and Z(t) ≤Z(0))

no dissipative anomaly for energy in 2d: no energy cascade to small scales !

P =

1

2∇×ω

2

d2x ≥0∫

∂ω∂t

+u⋅∇ω =ν∇2ω

u=z×ψ−Δψ =ω

∂ω∂t

+u⋅∇ω =ν∇2ω

u=z×ψ−Δψ =ω

The double cascade

(Kraichnan 1967)

In the limit Re->∞, 2d turbulenceshows a direct enstrophy cascade tosmall scales at rate .Energy flows to large scales atrate generating the inversecascade.

The double cascade scenario is typical of 2d flows, e.g. plasmas and other geophysical models.

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.

kF1/L kd

Exact results

S3(r ) = δ

ru( )

3=3

2r

(δru ; 1/ 3r 1/ 3 )

Following the derivation obtainedby Kolmogorov for 3d turbulence(Kolmogorov 4/5 law) is it possibleto obtain for 2d cascades two exactresults:

inverse energy cascade:

direct enstrophy cascade:

δru δ

rω( )

2=−2r

(δru ; 1/ 3r )

Kolmogorov’s 4/5 law (1941)

Geophysical data

Mesoscale wind variability(radar and balloon): k-5/3

K.S. Gage, J.Atmos.Sciences 36 (1979)

GASP aircraft dataset: k-5/3 forwavelenghts 10-300 kmNastrom, Gage, Jasperson, Nature 310 (1984)

Lenght (km)

k-5/3

Early laboratory experiments

Thin layer of mercury with electricalforcing in a uniform magnetic fieldsuppressing vertical motions (linearfriction due to Hartmann layer).J.Sommeria, JFM 170, 139 (1986)

Energy spectrum

Laboratory experiments:soap films

(Y. Couder, W. Goldburg, H. Kellay,M.A. Rutgers, M. Rivera, R.E. Ecke)

interferometry, LDV, PIV

M.A. Rutgers, PRL 81, 2244 (1998)

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Laboratory experiments:electrolyte cell

J. Paret, P.Tabeling, PRL 79 4162 (1997)

(P. Tabeling, J. Gollub, A. Cenedese)

Direct numerical simulations of 2d turbulence

U.Frisch, P.L. Sulem,Phys. Fluids 27, 1921 (1984): 2562

G.Boffetta, A.Celani and M.Vergassola,Phys. Rev. E 61, R29 (2000): 20482

S5(r)

S7(r)

Kolmogorov scaling: no intermittency

Direct numerical simulations of 2d turbulence

Set of simulations athigh resolutions witha parallel pseudospectral code.

(G. Boffetta and A. Celani, 2005)

N L/rF rF/r I I

2048 2x10-5 0.015 100 26 0.54 0.97

4096 5x10-6 0.024 100 53 0.82 0.92

8192 2x10-6 0.025 100 81 0.92 0.90

16384 1x10-6 0.0 100 115 0.95 0.95

Energy/enstrophy fluxes in spectral space Energy spectra

k-5/3

k-3

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Simultaneous observation of direct and inverse cascadeψ ω

Conformal invariance in 2d statistical physics

Conformal invariance for the inverse cascade:geometrical properties (vorticity domains)

stochastic Loewner equation

Under broad conditions:homogeneity + isotropy + scale invariance = invariance under conformal transformations

(local combination of translation, rotation and dilatation, preserve angles)

There are counterexamples (e.g. elasticity in 2d, Riva and Cardy 2005)

Is there conformal invariance in two-dimensional turbulence?First attempt by Polyakov (enstrophy direct cascade, 1993)

Conformal mappingConformal mapping is a powerful tool for characterizingshapes in 2D by means of analytic functions.

Consider a curve tH starting from the origin (t parameterizes the curve)The complement of the hull K (the set ofpoints which cannot be reached from infinitytogether with ) is simply connected, thus analytic function

g : H\K H

g(z) maps the hull K on the real axis(and the growing tip on a point R)

This map is unique if we fix normalization, e.g. g(z)~z+O(1/z) as z

Example: a vertical segment 0 z i a=

t

0

g

t(z ) = z 2 +a2

Introducing the “time” t=a2/4 , gt(z)z+2t/z, for a verticalsegment starting from R:

g

t(z ) = + (z −)2 + 4t

Kt

t Htip

trace

hull

t

Loewner Equation

The growth of the curve t can be mapped on the evolution of theconformal mapping gt(z)

For a trace growing in the upper half plane H from 0 to ∞

dgt(z )

dt=

2

gt(z ) −

t

* The trace t is univocally (i.e. no branching) generated by the (continuous) driving t which is at any time the map of the tip g(t)= t

* Conversely, given t we can determine the hull Kt and thus the map gt(z) and the driving t=gt(t)

g0(z) = z

t

t+δt

t

trace driving

Loewner equation (1923)

g

t(z ) = + (z −)2 + 4t

E.g: the map for the vertical segmentis solution to LE with t==const

Loewner equation (Loewner, 1923)

A curve tH starting from the origin defines a analytic functionwhich maps the complement of the hull K to H: g : H\K H g(z) maps the hull K on the real axis (and the growingtip on a point R)

Example: a vertical segment oflength a starting from the origin:

ia

0 g(z ) = z 2 + a2

The growth of the curve t can be mapped on the evolution of the conformalmapping gt(z) (t parameterizes the curve):

dgt(z )

dt=

2

gt(z ) −

t

with g0(z) = zand driving t=gt(t)

Kt

t Htip

trace

hull

t

The trace t is univocally generated by the (continuous) driving t which is at anytime the map of the tip g(t)= t and conversely, t determines gt(z) and thus t

trace driving

g

t(z ) = + (z −)2 + 4t

Example: The solution to LE witht==const

i.e. a segment of length a=2√t

gt

An exampleof Loewnerevolution(from drivingto trace)

driving

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trace

for other examples see e.g. Kager, Nienhuis and Kadanoff, J. Stat. Phys. 115, 805 (2004)

dgt(z )

dt=

2

gt(z ) −

t

Stochastic Loewner Equation

For applications in statistical mechanics we are interestedin random curves t: Loewner equation with a random driving t

(O.Schramm, 2000)

dgt(z )

dt=

2

gt(z ) −

t

t= κB

t

diffusion coefficient parameterizes different universality classesof critical behavior. Problems in 2d critical systems reduced to problems in1d Brownian motion

(see Cardy, SLE for theoretical physicists, Ann.Phys. 318, 81 (2005)

then (assuming reflection symmetry and continuity)t is proportional to a random walk:

0

A

- - - - - + + + + +

---

-

+++

+

0

- - - - - + + + + +

---

-

+++

+ - - - - - + + + + +

0

conformalinvariance

Markovproperty

Phases of SLE (Rohde & Schramm, 2001):The shape of the trace depends on the value of κ:increasing κ the trace turns more frequently* 0 < κ < 4 simple curve* 4 < κ < 8 non-simple curve ( intersections)* κ > 8 space filling

Fractal dimension of SLE traces: DF=1+κ/8(Beffara, 2002)

tracefrontier

For κ > 4, the external frontier of the hull(i.e. the boundary of H\Kt) is a simple curvedescribed by SLEκ’ with k’=16/k(thus D’F=1+2/κ)Duplantier (2000); proven by Beffara (2002) for κ=6

SLE duality

Brownian motionOld conjecture by Mandelbrot (1982):the frontier of BM is a SAW with D=4/3

Lawler, Schramm & Werner, 2000 (via SLE):• pioneer points: D=7/4 (SLE6)• frontier: D=4/3 (SLE8/3)• cut points D=3/4

SLEκ and critical systems

κ=2 loop-erased random walk

κ=8/3 self avoiding random walk

κ=3 cluster boundaries in Ising

κ=6 cluster boundaries in percolation

κ=8 uniform spanning trees

Vorticityclusters in theinverse cascadeof 2d turbulence

Single vorticitycluster

Fractal dimensions of vorticity clusters Boundary Frontier Cut points

Boundary Frontier Cut points

L=side of squarecovering the cluster

κ=6, κ’=8/3as in critical percolation

H.Saleur and B.Duplantier, PRL 58, 2325 (1987)

Probability distribution of vorticity clusters

Size Boundary__ prediction SLE6

Size Boundary__ prediction SLE6

size s= # connected sites of same signboundary t= # connected sites adjacent to opposite sign

Vorticity isoline as SLE6 traces ?

see Cardy and Ziff,J.Stat. Phys. 110, 1 (2003)

Are vorticity isoline compatible with SLE traces ?From traces to driving functions

* Generate isolines from vorticity field

* Numerical inversion of SLE for obtaining associate driving functions

* Compute statistical properties of driving functions (Brownian ?, κ ?)

Deterministic exampleof slit maps inversion

gt+δt = gδt º gt with gδt solution to LE with constant from t and t+ t:

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t = t sin(t)

Inversion of SLE as composition of discrete slit maps over δt

t-1

z=t

gδt gδt(z ) =

t+ (z −

t)2 + 4δt

with t = Re( t) and t = Im2( t)/4

O(N2) algorithm

The problem of boundary conditions

Locality: For κ=6 the trace does not feelboundaries until it doesn’t hit them

(obvious for percolation)

Locality: For κ=6 the trace does not feelboundaries until it doesn’t hit them

(obvious for percolation)

t

SLE is defined for traces from two points on the boundary of a domain

How we can apply to NS simulation in a periodic domain without boundaries ?

Locality

For κ=6 the trace does not feel boundaries until it doesn’t hit them

tA

H\Agt

A’

t

H\A

ht

‘t

H

'

t= κB

t+ (κ −6)Ct

’t

H

h0

g’t

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Unrolling a vorticity isoline

Driving functions

Driving (t) is Brownian motion zero-vorticity lines are SLEκ

κ = 5.9 0.3

Vorticity clusters and percolation

* Independent percolation: short correlated

* Correlated percolation:

* For H>3/4 same universality class of percolation (Harris, 1974)

* For vorticity in inverse cascadei.e. H = 2/3 < 3/4

* In principle, different class from percolation (but maybe close)

* Independent percolation: short correlated

* Correlated percolation:

* For H>3/4 same universality class of percolation (Harris, 1974)

* For vorticity in inverse cascadei.e. H = 2/3 < 3/4

* In principle, different class from percolation (but maybe close)

θ(x + r )θ(x) ; r −2H

ω(x + r )ω(x) ; r −4 / 3

Comparison with a Gaussian field with same Fourier spectrum(phase randomization): check of the importance of dynamics

Comparison with a Gaussian field with same Fourier spectrum(phase randomization): check of the importance of dynamics

Is the inverse cascade just a complicate way to generate a percolation field ?

Phase randomized Original

Gaussian field is not SLE

z

θ

P =1

2+

Γ4

κ

⎛

⎝⎜

⎞

⎠⎟

πΓ 8 −κ2κ

⎛

⎝⎜

⎞

⎠⎟

cot(θ)2F1

1

2,4

κ;3

2;−cot2 (θ)

⎛

⎝⎜

⎞

⎠⎟

(Schramm, 2001)

Probability that the trace passesto the left of a point z (for κ=6)

Calculating with SLE: Schramm’s formula

Calculating with SLE: Crossing formulae

Probability that in a rectangle ofaspect ratio r=y/x:- a cluster crosses from top to bottom- four-legged cluster connects 4 sides

Cardy (1992), Watts (1996) CFTSmirnov (2001), Dubeat (2004) SLE

Statistical mechanics of two-dimensional turbulent inverse cascadeZero-vorticity isoline are conformally invariant random curvesThey are compatible with SLE6

What about other 2D turbulent systems ?Is conformal invariance a general property of inverse cascade ?Is it always κ=6 (percolation-like) ?... see next talk !