Lagrangian View of Turbulence

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Tucson, Math 03/08/04
Lagrangian 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)
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Intro

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

Passive Scalar Turbulence

• Kraichnan model
• Anomalous scaling. Zero modes.
  Perturbative.9596
• Non-perturbative - Instanton. 97
• Batchelor model
• Lyapunov exponent. Cramer/entropy function.
• Statistics of scalar increment.949598
• 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.
  9903
• Polymer stretching-tumbling. 0004
• Lagrangian Modeling of Navier-Stokes Turb.
  990001
• Rayleigh-Taylor Turbulence. 03 in progress

Why Lagrangian?
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Rayleigh-Taylor Turb.
Burgulence
MHD Turb.
Collapse Turb.
Navier-Stocks Turb.
Kinematic Dynamo
Wave Turb.
Chem/Bio reactions in chaotic/turb flows
Passive Scalar Turb.
Elastic Turb. Polymer stretching
Spatially smooth flows (Kraichnan model)
Spatially non-smooth flows (Batchelor model)
Intermittency
Dissipative anomaly
Cascade
Lagrangian Approach/View
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Lagrangian
Eulerian
snapshot
movie
E. Bodenschatz (Cornell) Taylor based Reynolds
number 485frame rate 1000fpsarea in view
4x4 cmparticle size 46 microns
Curvilinear channel in the regime of elastic
turbulence (Groisman/UCSD, Steinberg/Weizmann)
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Scalar Turbulence
Examples
Temperature field
Flow visualization/die A. Groisman and V.
Steinberg, Nature 410, 905 (2001)
Pollutant (atmosphere, oceans) Pacific basin
chlorophyll distribution simulated.in
bio-geochemical POP, Dec 1996 LANL global
circulation model.
Convective penetration in stellar interrior
(Bogdan, Cattaneo and Malagoli 1993, Apj, Vol.
407, pp. 316-329)
Formulation of the (stationary) passive scalar
problem
Given that statistics of velocity field and
pumping field are known to describe statistics of
the passive scalar field
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Navier-Stokes Turbulence
inverse
integral (pumping) scale
viscous (Kolmogorov) scale
cascade
Kolmogorov, Obukhov 41
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Anomalous scaling. Intermittency.
NS
PS
More generically Intermittency ---
different correlation functions are
formed/originated from different phase-space
configurations
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From Eulerian to Lagrangian
Average over random trajectories of 2n particles
r
L
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Kraichnan model 74
Lagrangian (path-integral) MC97
Eulerian (elliptic Fokker-Planck) Kraichnan
94 MC,G.Falkovich, I.Kolokolov,V.Lebedev
95 B.Shraiman, E.Siggia 95 K.Gawedzki,
A.Kupianen 95
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Anomalous scaling. Zero modes.
Kraichnan model
homogeneous term
zero modes

(elliptic operator)
responsible for anomalous scaling !!
MC,GF,IK,VL 95 KG, AK 95 BS, ES 95
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MC, G. Falkovich 96
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Batchelor model 59
smooth velocity
(d-1)-dimensional QM for any (!!!) type of
correlation functions
IK 86 MC, IK 9496 quantum magnetism IK
91 -1d localizaion MC, YF,IK 94, passive
scalar statistics
Kolokolov transformation
Exponential stretching
CLT for matrix process
- concave
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Statistics of scalar increment
(Batchelor/smooth flow)
convective range
MC,YF,IK 94 BS, ES 9496 MC,IK,VL,GF 95
Statistics of scalar dissipation
(Batchelor-Kraichnan flow)
Major tool separation of scales
1/3 is consistent with numerics (Holzer,ES
94) 0.3-0.36 and experiment
(Ould-Ruis, et al 95) 0.37
MC,IK,MV 97 MC,GF,IK 98
Green coresponds to naïve reduction
- - does not work
Effective dissipative scale is strongly fluct.
quantity
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Lagrangian phenomenology of Turbulence
QM approx. to FT
velocity gradient tensor coarse-grained over
the blob
tensor of inertia of the blob
Stochastic minimal model verified against DNS
Chertkov, Pumir, Shraiman Phys.Fluids. 99,
Phys.Rev.Lett. 02
Steady, isotropic Navier-Stokes turbulence
Intermittency structures corr.functions
Challenge !!! To extend the Lagrangian
phenomenology (capable of describing small
scale anisotropy
and intermittency) to non-stationary world, e.g.
of
Rayleigh-Taylor Turbulence
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Phenomenology of Rayleigh-Taylor Turbulence
M. Chertkov, PRL 2003
Idea Cascade Adiabaticity
- decreases with r
Results
3d
2d
buoyant
passive
viscous and diffusive scales
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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
Now
1930s
High-speed digital cameras, Promise of
particle-image-velocimetry (PIV) Powefull
computersPIV -gt Lagr.Particle. Traj.
Promise (idea) of hot wire anemometer (single-po
int meas.)

Taylor, von Karman-Howarth, Kolmogorov-Obukhov
The life and legacy of G.I. Taylor, G.
Batchelor
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2003 Dirac Medal    On the occasion of the
birthday of P.A.M. Dirac the Dirac Medal
Selection Committee takes pleasure in announcing
that the 2003 Dirac Medal and Prize will 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.   Kraichnans
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 for two-dimensional turbulence.
Zakharovs 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 inverse
and dual cascades in wave turbulence.         8
August 2003
cascade
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