# Presentation: 183783

## 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)

- Tucson, Math:  03/08/04

## Applications:

-  *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

- 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?

## Lagrangian Approach/View

- Burgulence

- MHD Turb.

- Collapse Turb.

- Kinematic Dynamo

- Passive Scalar Turb. *

- Wave Turb. *

- Rayleigh-Taylor Turb. *

- Elastic Turb.
- Polymer
- stretching

- Chem/Bio reactions
- in chaotic/turb flows

-  Spatially
- smooth flows
- (Kraichnan * model)

-  Spatially
- non-smooth flows
- (Batchelor * model)

- Intermittency

- Dissipative anomaly

- Cascade

- _[menu](slide2.xml)_

## Lagrangian

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

_**Lagrangian**_

_**Eulerian**_

- movie

- snapshot

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

- _[menu](slide2.xml)_

## Scalar Turbulence

_**Scalar Turbulence**_

_**Examples**_

- Given that statistics of _velocity field_ and _pumping field_ are known
- to 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 1996*
*LANL global circulation model.*

- _[menu](slide2.xml)_

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

## Navier-Stokes Turbulence

- cascade

-  integral
- (pumping)
-     scale

-    viscous
- (Kolmogorov)
-    scale

- Kolmogorov, Obukhov ‘41

- _[menu](slide2.xml)_

- _[inverse](slide17.xml)_

## Anomalous scaling. Intermittency.

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

- NS

- PS

- _[menu](slide2.xml)_

## From Eulerian to Lagrangian

_**From Eulerian to Lagrangian**_

- Average over “random”
- trajectories of 2n particles

- r

- L

- _[menu](slide2.xml)_

## Kraichnan model    ‘74

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

- _Lagrangian (path-integral)_
- MC’97

- _[menu](slide2.xml)_

## 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

- menu

## menu

- MC, G. Falkovich ‘96

## Batchelor model       ‘59

- CLT for
- matrix process

- -  concave

- smooth
- velocity

- IK ’86; MC, IK ’94;’96
- – quantum magnetism
- IK ’91 -1d localizaion
- MC, YF,IK ’94, ... – passive scalar statistics

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

- Kolokolov transformation

- Exponential
- stretching

- _[menu](slide2.xml)_

## Statistics of scalar increment   (Batchelor/smooth flow)

**convective range**

- _Statistics of scalar increment _  (Batchelor/smooth flow)

- MC,YF,IK ’94
- BS, ES ’94;’96
- MC,IK,VL,GF ‘95

- _Statistics of scalar dissipation _  (Batchelor-Kraichnan flow)

- MC,IK,MV ’97
- MC,GF,IK ‘98

- Major tool: _*separation of scales*_

- Green coresponds to naïve reduction
-                                 - - does not work

- Effective dissipative scale
- is strongly fluct. quantity

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

- _[menu](slide2.xml)_

## Lagrangian phenomenology of Turbulence

-    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*

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

- QM approx. to FT

- _[menu](slide2.xml)_

- Intermittency: structures         corr.functions

## Phenomenology of Rayleigh-Taylor Turbulence

- Idea:            Cascade +    Adiabaticity:

- - decreases with r

_*Results:*_

- M. Chertkov, PRL 2003

- 3d

- 2d

- “passive”

- “buoyant”

-     viscous and
-     diffusive scales

- _[menu](slide2.xml)_

## ...

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

- “The life and legacy of
- G.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

- _[menu](slide2.xml)_

## 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.*
* *
*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 for 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 ****inverse****** and dual cascades in wave turbulence****.*
* *
* *
* *
* *
- 8 August 2003

- _[menu](slide2.xml)_

- _[cascade](slide6.xml)_