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

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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, 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
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Astrophysical flows
Disk/Galaxies
Planetary Atmospheres
Stars
Navier-Stokes equations
Control parameter
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Turbulence Phenomenology
 Cascade 
Création of finer and finer structures until
dissipation scale
Passive scalar Dispersion Passive vector
stretching
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Turbulence Phenomenology
Robust Result Kolmogorov spectrum
Cascade constant dissipation rate
Interpretation (Kolmogorov 1941) Energy Cascade
L
Number of degrees of freedom
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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
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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
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Influence of decimated degrees transport
Stochastic computation
Effective viscosity
AKA effect
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ParametrizationViscosity
Not necessarily isotropic (cf shear flows)
Isotropic case
Charactéristic Scale
Dimensionnal
Characteristic Velocity
Constant
Kolmogorov theory
RANS Viscosité
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Example  Mixing length 
Convection
Fc
Radiative Core
Hp
Buyoancy
Inertia

Vc
RANS Viscosité
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MOTIVATION PROTOPLANETRAY DISKS
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DISK OBSERVATIONS
Fu Ori
Dust Sedimentation
Boundary Layer
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THIN DISK EQUATIONS
L
R
Vertical hydrostatic equilibrium Surface averaged
quantities Negligible radial pressure gradients
H
H/Rltlt1
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ParametrizationViscosity
Dimensionnal
Charactéristic Scale
Characteristic Velocity
Constant
Other possibility
RANS Viscosité
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LABORATORY ANALOG
Taylor-Couette experiment With porous boundaries
Astrophysical disks
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POROUS TAYLOR-COUETTE FLOW
Stationary axisymmetric incompressible solutions
K, A et B fixed by boundary conditions
Non-porous material
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Control parameters
Traditional choice
Physical choice
Re
Super- critical
Sub- Critical cyclonic
Sub- Critical Anti cyclonic
Keplerian
-4/3
-1
0
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Stability supercritical case
Theoretical results
Experimental results
Esser and Grossman
Small gap (rotating PC)
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Stability subcritical
Experimental data
Theory
None
Taylor (1936), Wendt(1933), Richard (2001)
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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!
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Mean profile supercritical
Experimental results
Theoretical results
Busse, 1972
Maximization of transport
r
Flattening of angular momentum
Lewis and Swinney, 1999
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Mean profile subcritical
Cyclonic
Busse
Busse
Laminar
Anti-cyclonic
Busse
Evolution vers Busse More rapid for cyclonic
Laminar
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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
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ANALYTICAL PREDICTIONS
Mean flow dominates
Fluctuations dominates
Low Re
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TORQUE IN TAYLOR-COUETTE
No adjustable parameter
Dubrulle and Hersant, 2002
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Transport universality
Relative torque does not depend on gap size, nor
Re
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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
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Turbulent viscosity
Dubrulle et al, 2005
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Parametrization Viscosity
In disk
RANS Viscosité
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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
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Reynolds number in protoplanetary disks
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Stability lines
Protoplanetary disks are turbulent!
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INSTABILITIES- THEORY-Summary
Inviscid stability criterion
Critical Reynolds number in protoplanetary disk
3000
1000
Magneto
Non-linear
Strato
Linar
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COMPARISON EXP/ASTRO
flickering
fluctuations
BPTau
Mean dissipation
Statistics
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ELARGISSEMENT DE RAIES
Dans un disque protoplanetaire
Au laboratoire
Limite turb/lam
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TURBULENCE ET FORMATION PLANETAIRE
Turbulencecisaillementrotationtourbillons
Concentration locale de densité Freine la
migration interne des poussières
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IMPORTANCE DE LA CYCLONICITE
BRACCO ET AL, 1999
Seuls les anti cyclones survivent dans un
écoulement képlerien
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ARGUMENTS GENERAUX
u
l
Rogt1 la turbulence nest pas influencée par la
rotation Rolt1 la turbulence est modifiée par la
turbulence
Naivement la turbulence
bi-dimensionalise gt ralentit la cascade
denergie vers les petites échelles gt
favorise lapparition de structures à longue
durée de vie
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TOURBILLONS
Observation avec Hubble HD 141569A
Simulation SES (Hersant 2003)