Lagrangian View of TurbulenceLagrangian View of Turbulence

Misha Chertkov (Los Alamos)

In collaboration with: E. Balkovsky (Rutgers) G. Falkovich (Weizmann) Y. Fyodorov (Brunel) A. Gamba (Milano) I. Kolokolov (Landau) V. Lebedev (Landau) A. Pumir (Nice) B. Shraiman (Rutgers) K. Turitsyn (Landau) M. Vergassola (Paris) V. Yakhot (Boston)

Tucson, Math: 03/08/04

Kraichnan model: • Anomalous scaling. Zero modes. Perturbative.’95;’96 • Non-perturbative - Instanton. ’97 Batchelor model: •Lyapunov exponent. Cramer/entropy function. • Statistics of scalar increment.’94;’95;’98 • Dissipative anomaly. Statistics of Dissipation. ’98 •Inverse vs Direct cascade in compressible flows. ’98•Slow down of decay. ‘03•Regular shear + random strain ‘04

Applications: •Kinematic dynamo ‘99•Chem/bio reactions in chaotic/turbulent flows. ’99;’03•Polymer stretching-tumbling. ’00;’04•Lagrangian Modeling of Navier-Stokes Turb. ’99;’00;’01 •Rayleigh-Taylor Turbulence. ’03 + in progress

Passive Scalar Turbulence:

Intro: •“Big picture” of statistical hydrodynamics •Lagrangian vs Eulerian •Scalar Turb.Examples. •Cascade •Intermittency. Anomalous Scaling.

Why Lagrangian?

Navier-Stocks Turb. *

Burgulence MHD Turb.

Collapse Turb.

Kinematic Dynamo

Passive Scalar Turb. *

Wave Turb. *

Rayleigh-Taylor Turb. *

Elastic Turb.

Polymer stretching

Chem/Bio reactionsin chaotic/turb flows

Spatially smooth flows

(Kraichnan * model)

Spatially non-smooth flows

(Batchelor * model)

Intermittency Dissipative anomaly Cascade

Lagrangian Approach/View menu

E. Bodenschatz (Cornell) Taylor based Reynolds number : 485frame rate : 1000fpsarea in view : 4x4 cmparticle size 46 microns

Lagrangian Eulerianmovie snapshot

);();( 2211 rtrt );();( 21 rtrt

Curvilinear channel in the regime of elastic turbulence (Groisman/UCSD, Steinberg/Weizmann)

Non-Equilibrium steady state (turbulence)

Equilibrium steady state vs

Gibbs Distribution exp(-H/T) ??????

Fluctuation Dissipation Theorem (local “energy” balance)

Cascade (“energy” transfer over scales)

Need to go for dynamics (Lagrangian description) any case !!!menu

Formulation of the (stationary) passive scalar problem

Scalar Turbulence Examples

ut

Given that statistics of velocity field and pumping field are knownto describe statistics of the passive scalar field

Flow visualization/die[A. Groisman and V. Steinberg, Nature 410, 905 (2001)]

Temperature field Pollutant (atmosphere, oceans)Pacific basin chlorophyll distribution

simulated.in bio-geochemical POP, Dec 1996LANL global circulation model.

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Convective penetration in stellar interrior (Bogdan, Cattaneo and Malagoli 1993, Apj, Vol. 407, pp. 316-329)

Navier-Stokes TurbulenceNavier-Stokes Turbulence

fupuuut

cascade

integral (pumping) scale

viscous(Kolmogorov) scale Lr

22 22 uuP tu 5/4)()( 123

2||1|| rPruru uKolmogorov, Obukhov ‘41

Passive scalar turbulencePassive scalar turbulence

ut

cascade integral (pumping) scale

dissipation scale

Lrrd

22/ 22 tP 3/4)()()()( 122

212||1|| rPrrruru

Obukhov ’49; Corrsin ‘51

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inverse

Anomalous scaling. Intermittency.

n

n

r

ru

nr

n

r

~

~||

More generically: Intermittency --- different correlation functions are formed/originated from different phase-space configurations

NS

PS

333 nnn

222 nnn

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21

LT 12

Field formulationField formulation (Eulerian)

ParticlesParticles (“QM”) (Lagrangian)

ut

)()(;)(

)(;);(

udd

drtt

rt

)(

2121 2)()( tttt ji

From Eulerian to Lagrangian

n21 Average over “random”trajectories of 2n particles

L

uTtd 12

0;1221 )0()(

0

0;0;)( rdr

rL

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Kraichnan model ‘74,~)(

)()(

)()();();(

2

0

12212211

rrK

rKVrV

rVttrturtu

jjiiii

R

r

n

i

t

iinn

pKpipS

tSdttpDtDRRrrtQii

ii

)(

)'('exp)()(),,;,,()0(

)0( 1 0

11

Eulerian (elliptic Fokker-Planck)Kraichnan ‘94MC,G.Falkovich, I.Kolokolov,V.Lebedev ’95B.Shraiman, E.Siggia ’95K.Gawedzki, A.Kupianen ’95

20

Lagrangian (path-integral) MC’97

)()()(ˆ

)(ˆ2

1

2

1,

RrtQRLQ

rKL

t

n

i

n

jijiij

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Anomalous scaling. Zero modes. Kraichnan model

n

i

n

jijiij

nnnn

nn

rKL

permutFFL

F

2

1

2

1,

2;2222121

22121

)(ˆ

.ˆ

ZFF oghomhomogeneous

termzero

modes+

(elliptic operator)

responsible foranomalous scaling !!

MC,GF,IK,VL ’95KG, AK ’95BS, ES ‘95

1

43

2

34123412 TTTT

0ˆ ZL

2~)( rrK

Perturbative (spectral) calculations

2

d

Gaussian limit(s)

nn rZ ~)2(0

1/ln rL

Non-Gaussian perturbationScale

invariance+ +

r

LZZrLZ 000 /ln

exp)2(),2/()2)(1(2

.exp/1,/)2)(1(22

dnn

ddnnn

MC,GF,IK,VL ’95MC,GF ‘96

KG, AK ’95Bernard,GK,AK ‘96

Zn ~221

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Perturbative calculationsrequires thus n/a for large moments

nn 2

2

const0dissipative anomaly n

n

n

nnn

d

n

rr

L

r

L

2

2

2

~

~

221

221

2

MC, G. Falkovich ‘96

Non-perturbative evaluation ofanomalous scaling

Lagrangian instanton (saddle-point) method

MC ’97

ii

ii

R

r

n

iii

n pQtpDtD

)0(

)0(

2

1

;exp)()(

0

Q

pQ

+ Gaussian fluctuations

0

1

2

210 ttt

constdn );(2 n

n2

n

1/d-expansion MC, GF, IK,VL ‘95 ``almost diffusive” limit KG, AK ‘95 ``almost smooth” limit BS, ES ’95 exponent saturation (large n) MC ’97; E.Balkovsky, VL ’98 Lagrangian numerics U.Frisch, A.Mazzino, M.Vergassola ’99

Fundamentally important!!!First analytical confirmation

of anomalous scaling instatistical hydrodynamics/

turbulence

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Batchelor model ‘59

)(ˆ);( ttu

')'(ˆexp)(ˆ

0

dttTtWt

WtW

tt

t

t

ˆ)(ˆˆ

)()(ˆ

t

tWtWtr T

2

)(ˆ)(ˆlog CLT for

matrix process- concave

smoothvelocity

]ˆ[)(ˆ)(ˆ WJtWDtDIK ’86; MC, IK ’94;’96 – quantum magnetism

IK ’91 -1d localizaion

MC, YF,IK ’94, … – passive scalar statistics

t

iii ttWrtW0

1 )'()'(ˆ)( (d-1)-dimensional “QM”for any (!!!) type of correlation functions

Kolokolov transformation

)(

))(exp(~)|(

G

tGtPExponentialstretching

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21 r

)( rP

)/ln(2exp~

2

rLr

rexp~

“hat”

“tail”

]/ln[~ rL

Lrrd

convective range

Statistics of scalar increment (Batchelor/smooth flow)

MC,YF,IK ’94BS, ES ’94;’96

MC,IK,VL,GF ‘95

)(P

ln~ BA

3/1exp~

C]exp[~ 1C

]exp[~ 2 C

Statistics of scalar dissipation (Batchelor-Kraichnan flow)

MC,IK,MV ’97MC,GF,IK ‘98

PerLtt ddifstr ln/ln~~ 111 Major tool: separation of scales

1~,, iCBA

Green coresponds to naïve reduction - - does not work

Effective dissipative scaleis strongly fluct. quantity 2~

drrr

1/3 is consistent with numerics (Holzer,ES ’94) ~0.3-0.36 and experiment (Ould-Ruis, et al ’95) ~0.37

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Lagrangian phenomenology of TurbulenceLagrangian phenomenology of Turbulence

velocity gradient tensorcoarse-grained over the blob

blob

blob

blob

blob

V

rrrdg

V

rturdtM

);(

)(

tensor of inertia of the blob

ˆˆˆˆˆˆ

ˆˆˆˆ

ˆˆˆ1ˆ 21

1

2

MggMgdt

d

Mtrtr

MMdt

d

Stochastic minimal modelStochastic minimal model verified againstverified against DNSDNS

Chertkov, Pumir, Shraiman Phys.Fluids. 99, Phys.Rev.Lett. 02

Steady, isotropic Navier-Stokes turbulenceSteady, isotropic Navier-Stokes turbulence

Challenge !!!Challenge !!! To extend the To extend the Lagrangian phenomenologyLagrangian phenomenology (capable of describing small (capable of describing small scale anisotropy and intermittency) to non-stationary world, e.g. ofscale anisotropy and intermittency) to non-stationary world, e.g. of Rayleigh-Taylor TurbulenceRayleigh-Taylor Turbulence

QM approx. to FT

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Intermittency: structures corr.functions

Phenomenology of Rayleigh-Taylor Turbulence

Idea: Cascade + Adiabaticity: - decreases with rr

r ur ~

L(t) ~ turbulent (mixing) zone widthL(t) ~ turbulent (mixing) zone width also energy-containing scalealso energy-containing scale

constT

constgt

tLt

tu

L

L

~

~)(~)(2

Sharp-Wheeler ’61

Input:

Results:

M. Chertkov, PRL 2003

3d

3/1

3/1

)(~

)()(~

tL

rT

tL

rtuu

r

Lr

5/1

5/25/35/3

5/1

~)()(

~

)(~

t

grtu

tL

ru

tL

rT

Lr

r

2d

“passive”

“buoyant”)()( trtL

viscous and diffusive scales

gtt

4/1

4/3

~)(

decrease with time

4/1

8/18/5

~)(g

tt

increase with time

0

0

u

TTuT

Tgup

uuu

t

t

Boussinesq

(extends to the generic misscible case)

Setting:

Schematic evolution of a heavy parcel: falling down towards the Mixing Zone (MZ) center + brake down in& breaking into smaller parcels

Time evolution

1T 2T 3T

TowardsMZ

center

MZ edge

Time evolution

1T 2T 3T 1T 2T 3T

TowardsMZ

center

MZ edge

Next ?Lagrangian!

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And after all … why “Lagrangian” is so hot?!

Soap-film 2d-turbulence:R. Ecke, M. Riviera, B. Daniel MST/CNLS – Los Alamos

“The life and legacy ofG.I. Taylor”, G. Batchelor

High-speed digital cameras,Promise of particle-image-velocimetry (PIV)

Powefull computers+PIV -> Lagr.Particle. Traj.

Now

Promise (idea) of hot wire anemometer(single-point meas.)

1930s

Taylor, von Karman-Howarth, Kolmogorov-Obukhov…

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2003 Dirac Medal  

On the occasion of the birthday of P.A.M. Diracthe Dirac Medal Selection Committee takes pleasure in announcing that

the 2003 Dirac Medal and Prizewill be awarded to:

Robert H. Kraichnan(Santa Fe, New Mexico) 

and Vladimir E. Zakharov

(University of Arizona, Tucson and Landau Institute for Theoretical Physics, Moscow) 

The 2003 Dirac Medal and Prize is awarded to Robert H. Kraichnan and Vladimir E. Zakharov for their distinct contributions to the theory of turbulence, particularly the exact results and the prediction of inverse cascades, and for identifying classes of turbulence problems for which in-depth understanding has been achieved.

 Kraichnan’s most profound contribution has been his pioneering work on field-theoretic approaches to

turbulence and other non-equilibrium systems; one of his profound physical ideas is that of the inverse cascade the inverse cascade for two-dimensional turbulencefor two-dimensional turbulence. Zakharov’s achievements have consisted of putting the theory of wave turbulence on a firm mathematical ground by finding turbulence spectra as exact solutions and solving the stability problem, and in introducing the notion of inverseinverse and dual cascades in wave turbulence.

   

 8 August 2003

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cascade