1. 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

2. 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?

3. Navier-Stocks Turb. ** 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)* IntermittencyDissipative anomalyCascade Lagrangian Approach/View menu

4. E. Bodenschatz (Cornell) Taylor based Reynolds number : 485 frame rate : 1000fps area in view : 4x4 cm particle size 46 microns LagrangianEulerian movie snapshot 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

5. Formulation of the (stationary) passive scalar problem 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 Convective penetration in stellar interrior (Bogdan, Cattaneo and Malagoli 1993, Apj, Vol. 407, pp. 316-329)

6. Navier-Stokes Turbulence cascade integral (pumping) scale viscous (Kolmogorov) scale Kolmogorov, Obukhov ‘41 Passive scalar turbulence cascade integral (pumping) scale dissipation scale Obukhov ’49; Corrsin ‘51 menu inverse

7. Anomalous scaling. Intermittency. More generically: Intermittency --- different correlation functions are formed/originated from different phase-space configurations NS PS menu

8. Field formulation (Eulerian) Particles Particles (“QM”) (Lagrangian) From Eulerian to Lagrangian Average over “random” trajectories of 2n particles r L menu

9. 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

10. 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 1 4 3 2 Perturbative (spectral) calculations Gaussian limit(s)Non-Gaussian perturbation Scale invariance ++ MC,GF,IK,VL ’95 MC,GF ‘96 KG, AK ’95 Bernard,GK,AK ‘96 menu

11. Perturbative calculations requires thus n/a for large moments dissipative anomaly MC, G. Falkovich ‘96 Non-perturbative evaluation of anomalous scaling Lagrangian instanton (saddle-point) method MC ’97 + Gaussian fluctuations 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 in statistical hydrodynamics/ turbulence menu

12. 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

13. “hat” “tail” 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

14. 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 scale anisotropy and intermittency) to non-stationary world, e.g. of Rayleigh-Taylor Turbulence Rayleigh-Taylor Turbulence QM approx. to FT menu Intermittency: structures corr.functions

15. Phenomenology of Rayleigh-Taylor Turbulence Idea: Cascade + Adiabaticity: - decreases with r L(t) ~ turbulent (mixing) zone width also energy-containing scale also energy-containing scale Sharp-Wheeler ’61 Input: Results: M. Chertkov, PRL 2003 3d2d “passive” “buoyant” viscous and diffusive scales decrease with time increase with time 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 Next ? Lagrangian! menu

16. 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 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

17. 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. the inverse cascade for two-dimensional turbulence inverse 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 cascade