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STABILITY AND TRANSPORT IN TAYLOR- COUETTE FLOW: APPLICATION TO PROTOPLANETARY DISKS B. DUBRULLE CNRS, Groupe instabilité et Turbulence SPEC/DRECAM/DSM,

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Presentation on theme: "STABILITY AND TRANSPORT IN TAYLOR- COUETTE FLOW: APPLICATION TO PROTOPLANETARY DISKS B. DUBRULLE CNRS, Groupe instabilité et Turbulence SPEC/DRECAM/DSM,"— Presentation transcript:

1 STABILITY AND TRANSPORT IN TAYLOR- COUETTE FLOW: APPLICATION TO PROTOPLANETARY DISKS B. DUBRULLE CNRS, Groupe instabilité et Turbulence SPEC/DRECAM/DSM, CEA Saclay F. HERSANT Obs. Meudon J-M HURE Obs. Meudon O. DAUCHOT CEA Saclay F. DAVIAUD CEA Saclay P-Y LONGARETTI Obs. Grenoble D. RICHARD Obs. Meudon J-P. ZAHN Obs Meudon

2 Astrophysical flows Disk/Galaxies Planetary Atmospheres Stars Navier-Stokes equations: Control parameter:

3 Turbulence Phenomenology Passive scalar Dispersion Passive vector stretching « Cascade » Création of finer and finer structures until dissipation scale

4 Turbulence Phenomenology Robust Result: Kolmogorov spectrum Interpretation (Kolmogorov 1941) Energy Cascade L Cascade constant dissipation rate Number of degrees of freedom

5 Example: the sun Too many degrees of freedom! Paramétrization of decimated degrees Giant Convection cell Sunspot Granule Dissipation scale 0.1 km Decimation of degrees (projection))

6 Influence of decimated degrees Typical time at scale l: Decimated degrees (small scales) vary rapidly They can be replaced by noise with short time corrélation Generalized Langevin equation

7 Influence of decimated degrees: transport Stochastic computation Effective viscosityAKA effect

8 Parametrization: Viscosity Not necessarily isotropic (cf shear flows) Isotropic case Dimensionnal Charactéristic Scale Characteristic Velocity Constant Kolmogorov theory RANS: Viscosité

9 Example: Mixing length RANS: Viscosité Radiative Core Hp Vc Convection BuyoancyInertia = Fc

10 MOTIVATION: PROTOPLANETRAY DISKS

11 DISK OBSERVATIONS Dust Sedimentation Boundary Layer Fu Ori

12 THIN DISK EQUATIONS R H H/R<<1 Vertical hydrostatic equilibrium Surface averaged quantities Negligible radial pressure gradients L

13 Parametrization: Viscosity Dimensionnal Charactéristic Scale Characteristic Velocity Constant Other possibility RANS: Viscosité

14 LABORATORY ANALOG Taylor-Couette experiment With porous boundaries Astrophysical disks

15 POROUS TAYLOR-COUETTE FLOW Stationary axisymmetric incompressible solutions K, A et B fixed by boundary conditions Non-porous material:

16 Control parameters Traditional choice Physical choice -4/3 0 Super- critical Sub- Critical Anti cyclonic Sub- Critical cyclonic Re Keplerian

17 Stability: supercritical case Experimental results Theoretical results Esser and Grossman Small gap (rotating PC):

18 Stability: subcritical Experimental data Theory None Taylor (1936), Wendt(1933), Richard (2001)

19 Stability: influence of body forces Experimental results Magnetic Stratification Theoretical results Whittaker and Chen (1974) Donnelly and Ozima (1962) Dubrulle et al, 2003 Chandrasekhar-Velikhov Necessary conditions for stability Anticyclonic flows: unstable!

20 Mean profile: supercritical Experimental results Lewis and Swinney, 1999 Theoretical results Busse, 1972 r Flattening of angular momentum Maximization of transport

21 Mean profile subcritical Busse Laminar Evolution vers Busse More rapid for cyclonic Cyclonic Anti-cyclonic

22 Transport: torque Supercritical: 2 regimes Subcritical: 1 regime Theoretical results Dubrulle and Hersant, 2002 Taylor, 1936, Wendt, 1933 Lewis and Swinney, 1999 Supercritical case Logarithmic corrections Analogy with thermal convection

23 ANALYTICAL PREDICTIONS Mean flow dominatesFluctuations dominates Low Re

24 TORQUE IN TAYLOR-COUETTE No adjustable parameter Dubrulle and Hersant, 2002

25 Transport: universality Relative torque does not depend on gap size, nor Re

26 Transport: influence of BC Experimental results Theoretical results Dubrulle, 2001 Rough boundaries destroy boundary layer No logarithmic correction Van den Berg et al, 2003 Increase of transport with Rough BC

27 Turbulent viscosity Dubrulle et al, 2005

28 Parametrization: Viscosity In disk: RANS: Viscosité

29 Disk structure: observations Interferpmetric obs. Inversion via 20 parameter minimization Keplerian model assumed Radial structure of disks (Dutrey et al) Classic thin disk Model with exces IR

30 Reynolds number in protoplanetary disks

31 Stability lines Protoplanetary disks are turbulent!

32 INSTABILITIES- THEORY-Summary Non-linearStrato Magneto Linar 1000 3000 Critical Reynolds number in protoplanetary disk Inviscid stability criterion

33 COMPARISON EXP/ASTRO fluctuations flickering Mean dissipationStatistics BPTau

34 ELARGISSEMENT DE RAIES Au laboratoire Dans un disque protoplanetaire Limite turb/lam

35 TURBULENCE ET FORMATION PLANETAIRE Turbulence+cisaillement+rotation=tourbillons Concentration locale de densité Freine la migration interne des poussières

36 IMPORTANCE DE LA CYCLONICITE BRACCO ET AL, 1999 Seuls les anti cyclones survivent dans un écoulement képlerien

37 ARGUMENTS GENERAUX u l Ro>1: la turbulence nest pas influencée par la rotation Ro<1: la turbulence est modifiée par la turbulence Naivement: la turbulence bi-dimensionalise => ralentit la cascade denergie vers les petites échelles => favorise lapparition de structures à longue durée de vie

38 TOURBILLONS Observation avec Hubble HD 141569A Simulation SES (Hersant 2003)


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