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)