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
Astrophysical flows
Disk/GalaxiesPlanetary Atmospheres
Stars
∂tu +u•∇ u=−∇ p+ν Δu+ fNavier-Stokes equations:
Control parameter: Re=LUν
Re≈1013
Re=108Re=109
Turbulence Phenomenology
Passive scalar DispersionPassive vector stretching
« Cascade »
Création of finer and finer structures until dissipationscale
Turbulence Phenomenology
Nl
Robust Result:Kolmogorov spectrum
Interpretation (Kolmogorov 1941)Energy Cascade
L
η = ν3
ε( )14
Cascade constant dissipationrate
du2
dt=ε ≅
u3
l=cte
⇒ u∝ (εl)13
Number of degrees of freedom
N =Lη
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
3
∝ Re94
Example: the sun
E(k)kGrandes EchellesPetites EchellesFlux d’Energie(Viscosité Turbulente)(Paramétrisées)Explosion90 % des ressources informatiques
Too many degrees of freedom!
Paramétrization of decimated degrees
GiantConvectioncell
SunspotGranuleDissipationscale
0.1 km 103km 3⋅104 km 2⋅105 km
N = 106( )
3=1018
Decimation of degrees (projection))
Influence of decimated degrees
Typical time at scale l: δt≈lu
∝ l2
3
Decimated degrees (small scales) vary rapidlyThey can be replaced by noise with short time corrélation
u=u +u'
Dtu' i =Aiju' j +ξj
ξi x,t( )ξj x',t'( ) =κ ij x,x'( )δ t−t'( )
Generalized Langevin equation
Influence of decimated degrees: transport
r x •
=r u +
r u '
r Ω =
r ∇ ×
r u
r Ω •
=(r Ω •
r ∇ )
r u +(
r Ω •
r ∇ )
r u '
∂tΩi +u k∇kΩi =Ωk∇kui +∇ k βkl∇l Ωi[ ]+2αkil∇kΩ l
βkl = uk' ul
'
αijk = ui'∂kuj
'Stochastic computation
Effective viscosity AKA effect
Parametrization:Viscosity
Not necessarily isotropic (cf shear flows)
Isotropic case βijkl =νTδjkδil
Dimensionnal νT =KVL
CharactéristicScale
CharacteristicVelocityConstant
Kolmogorov theory V = εL( )13
νT =K(εL)4
3
RANS: Viscosité
Example: Mixing length
RANS: Viscosité
RadiativeCoreHp
Vc
Convection
∂t
r u +
r u •∇
r u =−
1ρ
∇p+βθg+νTΔr u
BuyoancyInertia =
vc2
L≈βgθ
Fc =vcθ
Fc
vc = gβFcL( )13
νT = gβFc( )13L
43
L =α Hp 1<α <2
Hp =−dr
dlnp
MOTIVATION: PROTOPLANETRAY DISKS
DISK OBSERVATIONS
DustSedimentation
BoundaryLayer
0
20
40
60
80
100
5600 6400 7200 8000 8800 9600 1.04 10 4
Luminosity/L
0
JD-2.44E6
Fu Ori
THIN DISK EQUATIONS
10 -12
10 -11
10 -10
10 -9
10 -8
10 -7
10 -6
10 -5
10 -4
0.001 0.01 0.1 1 10 100 1000
Ω ( s-1 )
( . . )r A U
R
H
H/R<<1
€
ur = −3
2
M•
Σ
uθ =GM
r1/ 2+
ri
rΩ* − ΩK ri( )( )
Vertical hydrostatic equilibriumSurface averaged quantitiesNegligible radial pressure gradients
€
Ω ≈r−3 / 2 ≡ ΩK r( )
€
M• L
Parametrization:Viscosity
Dimensionnal
€
νT = α csH
CharactéristicScale
CharacteristicVelocityConstant
Other possibility
€
νT = β r2ΩRANS: Viscosité
€
cs = HΩ
LABORATORY ANALOG
Taylor-Couette experimentWith porous boundariesAstrophysical disks
POROUS TAYLOR-COUETTE FLOW
Stationary axisymmetric incompressible solutions
∇ •u=0 ur =Kr
ν Δ uϑ =1
r∂r ruruθ( ) uθ =
Ar
+B
r−(1+K /ν)
K, A et B fixed by boundary conditionsur =0 r =ri K =0
Non-porous material:
Control parametersTraditional choice Physical choice
-4/3 -1 0
Super-critical
Sub-CriticalAnticyclonic
Sub-Criticalcyclonic
Re
€
RΩ
Keplerian
€
Ri =riΩ id
ν
Ro =roΩod
ν
€
Re =Sd2
ν=
2
1+ ηηRo − Ri
RΩ =2Ω
S= 1−η( )
Ri + Ro
ηRo − Ri
Stability: supercritical case
100
1000
-1.5 -1 -0.5 0 0.5 1
Re
RΩ
€
η =1€
η =0.2
Experimental results Theoretical results
Esser and Grossman
€
Re2(RΩ +1)(RΩ − RΩc η( )) = −1708a(η )
€
RΩc (η ) ≈
1−η
η
€
a(η ) →1 η →1
Small gap (rotating PC):
€
Re =1708
−(RΩ +1)RΩ
Stability: subcritical
1000
104
105
-1.5 -1 -0.5 0 0.5
Re
RΩ
Experimental data Theory
None
Taylor (1936), Wendt(1933), Richard (2001)
Stability: influence of body forces
10
100
1000
10 4
10 5
10 6
10 7
-2 -1.5 -1 -0.5 0 0.5 1 1.5
Re
RΩ
Experimental results
Magnetic
Stratification
Theoretical results
Whittaker and Chen (1974)Donnelly and Ozima (1962)
Dubrulle et al, 2003
Chandrasekhar-Velikhov
€
RΩ > −4m2 I0
I1
Ω2
N 2
€
RΩ > −4m2 I0
I1
Ω2
ΩA2
Necessary conditions for stability
Anticyclonic flows: unstable!
Mean profile: supercritical
Experimental results
Lewis and Swinney, 1999
Theoretical results
Busse, 1972
€
S∞ r( ) =1
4S lam (r)
€
L = r2Ω
r
Flattening of angular momentum
Maximization of transport
Mean profile subcritical
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1 2 3 4 5 6
S / S
lam
Re / R g
Busse
Busse
Busse
Laminar
Laminar
Evolution vers BusseMore rapid for cyclonic
Cyclonic
Anti-cyclonic
Transport: torque
1
10
100
1000
10 100 1000 10 4 10 5 10 6 10 7
G / G
lam
Re
R T
R c
R g
Supercritical: 2 regimes
Subcritical: 1 regime
€
Re3 / 2
€
Re2
€
Re2
Theoretical results
Dubrulle and Hersant, 2002
Taylor, 1936, Wendt, 1933Lewis and Swinney, 1999
Supercritical caseLogarithmic correctionsAnalogy with thermal convection
ANALYTICAL PREDICTIONS
Mean flow dominates Fluctuations dominates
Low Re
G =1.46η2
1−η( )7/ 4 Re3/2
G =0.5η2
1−η( )3/2
Re2
ln(Re2)3/ 2
TORQUE IN TAYLOR-COUETTE
10 5
10 6
10 7
10 8
10 9
10 10
10 11
100 1000 10 4 10 5 10 6
G
Re
10 4
10 5
10 6
10 7
10 8
10 9
10 10
100 1000 10 4 10 5
G
R
η = 0.68
η = 0.935
η = 0.85No adjustable parameter
Dubrulle and Hersant, 2002
Transport: universality
0
0.2
0.4
0.6
0.8
1
1.2
-2 -1.5 -1 -0.5 0 0.5 1
G / G (
Ωo
=0 )
RΩ
Relative torque does not depend on gap size, nor Re
Transport: influence of BC
10 5
10 6
10 7
10 8
10 9
10 10
10 11
10 12
1000 10 4 10 5 10 6
G
Re
rough boundaries
smooth boundaries
Experimental results Theoretical results
Dubrulle, 2001
Rough boundaries destroy boundary layerNo logarithmic correction
Van den Berg et al, 2003Increase of transport withRough BC
Turbulent viscosity
10 -5
0.0001
0.001
0.01
1000 10 4 10 5 10 6
RC
4 G i
/ Re 2
Re
smooth boundaries
rough boundaries
Dubrulle et al, 2005
€
ν t = g(Re)h(RΩ ) Sr*2
Parametrization: Viscosity
In disk:
€
νT = α (Re,RΩ ) ΩH 2
RANS: Viscosité
€
S = −3
2Ω
r* = H
Disk structure: observations
105
107
109
1011
1013
1 10 100 1000
profils -GM Aur
n (cm^-3)
r(au)
MD=0.03 M
SOL
n = r -2.75
0,1
1
10
100
1 10 100 1000
profils -GM Aur
vrot
(km s
-1)
r(au)
M*=0.5 M
sol
v = r -0.5 +_0.1
M*=0.8 M
sol
1
10
100
1000
1 10 100 1000
profils -GM Aur
T (K)
r(au)
T = r -0.6 +-0.05
Interferpmetric obs.Inversion via 20 parameter minimization Keplerian model assumed
Radial structure of disks
(Dutrey et al)
Classic thin disk
Model with exces IR
Reynolds number in protoplanetary disks
r =100AU;Re=3×102
r =10 AU;Re=6×104
r =1 AU; Re=1.3×107
r =0.1AU;Re=2×109
Re=9×1014 v*
10kms−1
⎛ ⎝ ⎜
⎞ ⎠ ⎟
r*
1011cm⎛ ⎝ ⎜
⎞ ⎠ ⎟
104KT
n4×1019cm−3
⎛
⎝ ⎜
⎞
⎠ ⎟
Stability lines
1
100
10 4
10 6
10 8
10 10
10 12
10 14
10 16
0.001 0.01 0.1 1 10 100 1000
Reynolds
r (A. U.)
HD critical Reynolds number
MHD critical Reynolds number
Effective local Reynolds number
Protoplanetary disks are turbulent!
INSTABILITIES- THEORY-Summary
Non-linear Strato Magneto Linar
−1<2ΩS
<02ΩS
>−4m2 I0
I1
Ω2
N2
2ΩS
>−4m2 I0
I1
Ω2
Ω A2
105∞ 1000 3000
Critical Reynolds number in protoplanetary disk
Inviscid stability criterion
COMPARISON EXP/ASTRO
0
5
10
15
20
25
30
35
40
8840 8860 8880 8900 8920 8940 8960
Luminosity/L
0
JD-2.44E6
fluctuations flickering
Mean dissipation Statistics
BPTau
10 -13
10 -11
10 -9
10 -7
10 -5
0.001
0.1
10
10 12 10 13 10 14 10 15
Energy dissipation
Reynolds number
TTauri
FU Ori
laminar
turbulent min
turbulent max
ELARGISSEMENT DE RAIES
0.01
0.1
1
0.01 0.1 1 10 100 1000
Δ
( v km s
-1 )
( . . )r A U
Azimuthal velocity dispersion
0.01
0.1
1
10 4 10 5 10 6
Δ
( -1)v laboratory km s
*Re
Au laboratoire
Dans un disque protoplanetaire
Limite turb/lam
TURBULENCE ET FORMATION PLANETAIRE
Turbulence+cisaillement+rotation=tourbillons
Concentration locale de densitéFreine la migration interne des poussières
IMPORTANCE DE LA CYCLONICITE
BRACCO ET AL, 1999
Seuls les anti cyclones survivent dans un écoulement képlerien
ARGUMENTS GENERAUX
ul Ω
Ro=u
2lΩ≈l−2/3
Ro>1: la turbulence n’est pas influencée par la rotationRo<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
TOURBILLONS
Observation avec HubbleHD 141569A
Simulation SES (Hersant 2003)
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