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Lagrangian View of Turbulence

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)

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Lagrangian View of Turbulence

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  1. 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)

  2. 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.’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 * Why Lagrangian?

  3. 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 menu

  4. Non-Equilibrium steady state (turbulence) vs Equilibrium steady state Fluctuation Dissipation Theorem (local “energy” balance) Cascade (“energy” transfer over scales) Gibbs Distribution exp(-H/T) ?????? Need to go for dynamics (Lagrangian description) any case !!! 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) menu

  5. 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 ofthepassive scalar field menu

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

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

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

  9. Kraichnan model ‘74 Lagrangian (path-integral) MC’97 Eulerian (elliptic Fokker-Planck) Kraichnan ‘94 MC,G.Falkovich, I.Kolokolov,V.Lebedev ’95 B.Shraiman, E.Siggia ’95 K.Gawedzki, A.Kupianen ’95 menu

  10. 1 2 4 3 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 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

  11. Perturbative calculations requires thus n/a for large moments dissipative anomaly Lagrangian instanton (saddle-point) method Non-perturbative evaluation of anomalous scaling MC ’97 + Gaussian fluctuations Fundamentally important!!! First analytical confirmation of anomalous scaling in statistical hydrodynamics/ turbulence 1/d-expansion MC, GF, IK,VL ‘95 ``almost diffusive” limitKG, AK ‘95 ``almost smooth” limitBS, ES ’95 exponent saturation (large n) MC ’97; E.Balkovsky, VL ’98 Lagrangian numerics U.Frisch, A.Mazzino, M.Vergassola ’99 MC, G. Falkovich ‘96 menu

  12. Batchelor model ‘59 smooth velocity (d-1)-dimensional “QM” for any (!!!) type of correlation functions IK ’86; MC, IK ’94;’96 – quantum magnetism IK ’91 -1d localizaion MC, YF,IK ’94, … – passive scalar statistics Kolokolov transformation Exponential stretching CLT for matrix process - concave menu

  13. Statistics of scalar increment (Batchelor/smooth flow) “hat” “tail” convective range MC,YF,IK ’94 BS, ES ’94;’96 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 menu

  14. Lagrangian phenomenology of Turbulence QM approx. to FT velocity gradient tensor coarse-grained over the blob tensor of inertia of the blob Stochastic minimal modelverified againstDNS Chertkov, Pumir, Shraiman Phys.Fluids. 99, Phys.Rev.Lett. 02 Steady, isotropic Navier-Stokes turbulence Intermittency:structurescorr.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 menu

  15. Phenomenology of Rayleigh-Taylor Turbulence L(t) ~ turbulent (mixing) zone width also energy-containing scale Input: Setting: Boussinesq Sharp-Wheeler ’61 (extends to the generic misscible case) Schematic evolution of a heavy parcel: falling down towards the Mixing Zone (MZ) center + brake down in& breaking into smaller parcels Next ? Lagrangian! decrease with time increase with time M. Chertkov, PRL 2003 Idea: Cascade + Adiabaticity: - decreases withr Results: 3d 2d “buoyant” “passive” viscous and diffusive scales menu

  16. 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 computers+PIV -> Lagr.Particle. Traj. Promise (idea) of hot wire anemometer (single-point meas.) … Taylor, von Karman-Howarth, Kolmogorov-Obukhov “The life and legacy of G.I. Taylor”, G. Batchelor 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. 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 cascade menu

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