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 Dissipation Sunspot cell scale
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 H
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

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
Experimental results
Theoretical 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
Lewis and Swinney, 1999
Flattening of angular
momentum

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 Simulation SES (Hersant 2003)
HD A
Simulation SES (Hersant 2003)