Lagrangian View of Turbulence Misha Chertkov (Los Alamos) In collaboration with: E. Balkovsky...
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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