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STABILITY AND TRANSPORT IN TAYLOR-COUETTE FLOW: APPLICATION TO PROTOPLANETARY DISKS

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

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STABILITY AND TRANSPORT IN TAYLOR-COUETTE FLOW: APPLICATION TO PROTOPLANETARY DISKS

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

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

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

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

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

  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 viscosity AKA effect

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

  9. Example:  Mixing length  Convection Fc Radiative Core Hp Inertia Buyoancy = Vc RANS: Viscosité

  10. MOTIVATION: PROTOPLANETRAY DISKS

  11. DISK OBSERVATIONS Fu Ori Dust Sedimentation Boundary Layer

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

  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 Re Super- critical Sub- Critical cyclonic Sub- Critical Anti cyclonic Keplerian -4/3 -1 0

  17. Stability: supercritical case Theoretical results Experimental 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 Theoretical results Experimental results Necessary conditions for stability Dubrulle et al, 2003 Stratification Chandrasekhar-Velikhov Magnetic Whittaker and Chen (1974) Donnelly and Ozima (1962) Anticyclonic flows: unstable!

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

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

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

  23. ANALYTICAL PREDICTIONS Mean flow dominates Fluctuations 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 Increase of transport with Rough BC Van den Berg et al, 2003

  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 Model with exces IR (Dutrey et al) Classic thin disk Radial structure of disks

  30. Reynolds number in protoplanetary disks

  31. Stability lines Protoplanetary disks are turbulent!

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

  33. COMPARISON EXP/ASTRO flickering fluctuations BPTau Mean dissipation Statistics

  34. ELARGISSEMENT DE RAIES Dans un disque protoplanetaire Au laboratoire 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 n’est pas influencée par la rotation Ro<1: la turbulence est modifiée par la turbulence Naivement: la turbulence bi-dimensionalise => ralentit la cascade d’energie vers les petites échelles => favorise l’apparition de structures à longue durée de vie

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

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