The Nature of Turbulence in Protoplanetary Disks

Jeremy Goodman

Princeton Universityjeremy@astro.princeton.edu

“Astrophysics of Planetary Systems”

Harvard

18 May 2004

Why do we care?

• Spectrum depends on accretion rate only:– from boundary-layer emission

• Viscosity determines surface density: – not obviously compatible with viscosity

• Agglomeration of solids (grains/planetesimals)• Gap formation & migration

– & planetary eccentricities?

• Unsteady behaviors– FU Orionis outbursts

– Waves and wakes

€

σTeff4 =

3Ω2

8π˙ M

€

Σ= ˙ M 3π ν

€

˙ M ≈10−8±1 M∗yr−1

€

Σ∝ r−3 / 2

Turbulence/Transport MechanismsCandidate Pro Con

Magnetorotational Instability (MRI)

Robust linear instability.

Well studied. ~10-2

Uncertain nonthermal ionization required

Finite-amplitude hydro instability

Independent of ionization. Demonstrated in lab (?)

Poorly understood. Not confirmed by simulation

Selfgravity Can be local. Reasonably well understood.

Q>>1 in T Tauri phase

Vertical convection Expected result of radiative cooling

Not driven by shear. Transports J inwards.

Radial convection / baroclinic instab.

Ditto. Seems to make

large vortices, ~10-3

Poorly understood. Linear instab. obscure

Planetary wakes Calculable. Inevitable at some level.

Requires planets.

Migration. <10-4

MRI in Resistive Disks

• MRI dynamo requires– ReM 1 with imposed field

• Ionization frac. crucial:– electron-neutral collisions

• Thermal xe negligible @ T<1000K

• Nonthermal xe uncertain– Ionization rate: CR, Xrays,…– Recombination: dust,

molecular ions, metal ions

• Other wrinkles:– Layered accretion (Gammie ‘96)

– Hall conductivity (Wardle ‘99)

Fleming, Stone, & Hawley 2000

Fleming & Stone 2003

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ReM ≡ΩH 2

η≥104

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ReM ≈1013T3001/ 2rAU

3 / 2xe

Resistive turbulence (Fleming et al. 2000)

Further remarks on layered MRI

• If CR=10-17s-1 & dissociative recomb. (after Gammie ‘96)

– & accretion rate is too small:

€

ReM (r,z) ≈ exp(z2 4H 2)rAU5 / 4

⇒ ReM =104 at z ≈ 6H

Σactive ≈10−9 Σ

then in MMSN,

€

˙ M = 3π α ΩH 2Σactive

≈ 3 ×10−14α rAU−1/ 2M∗yr−1

Finite-amplitude hydro instabilityRichard & Zahn (1999):

outer

in

n

er

€

Re∗ ≡ r3 ΔΩ

Δ r≥ 6 ×105

ν T ≈ −β r3 dΩ

dr

β ≈1.5 ± 0.5 ×10−5

˙ M ≈ 3×10−7 rAU−1 M∗yr−1

In MMSN:

Richard 2001

r-

3/2

“Keplerian” profile found turbulent (Richard 2001)

Objections to FAHI

• Nonlocal: r not H is the lengthscale– H > r >> r in experiments

– H << r ≈ r in accretion disks• Also compressible

• No local linear instability for– But e.g. pipe flow is also linearly stable

• Not found in local (shearing-sheet) simulations– But viscosity is explicitly nonlocal

– Resolution or numerical Re may be inadequate• E.g. Longaretti 2002

• Doesn’t explain outbursts (e.g. dwarf novae)

€

d

dr(r2Ω)2 ≥ 0

Princeton MRI Experiment (H. Ji et al.)

B= 0.7 TRe*~107

ReM ~ 1

Vortices & Baroclinic Instability

• Anticyclonic vortices hold together by Coriolis force– Local maximum in P & Σ– Local minimum in vorticity: & vortensity:

• Realistically,

• Wakes of persistent vortices transmit angular momentum

Godon & Livio 1999

Klahr & Bodenheimer 2003

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ω ≡∇×v

€

≡ω Σ

€

Dζ

Dt=∇P ×∇Σ

Σ3→ 0 if P → P(Σ)

€

P = P(Σ,T)

Baroclinic Instability, continued

disks are typically unstably stratified in radius:

– e.g. with dust opacity

• Growth is nonaxisymmetric– Axisym’ly stable since– Linear growth is only transient due to shear (swing

amplification)

• Self-consistent ~10-3 in 2D & 3D is claimed– Klahr & Bodenheimer 2003

• Confirmation is needed!

€

N 2 ≡ −1

γρ

∂p

∂r

∂

∂rln

p

ρ γ< 0

€

p ρ 7 / 5 = r−2 / 5

€

κ 2 + N 2 > 0 : κ ≈ Ω, N ~ Ω H r( )

A plug for planetary wakes

• A corotating obstacle---vortex or planet---has a wake– Wavelike angular-momentum transport

– Dissipation of gas orbits where wake shocks/damps

• One planet:– Goodman & Rafikov ‘01; Rafikov ‘02

• Many planets: assuming – all metals in planets of equal mass Mp

– planets distributed like gas

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˙ E ≈ 7.ΣΩ3H 4 M p M1( )8 / 5

, M1 ≡ 2Ω2H 3 3G ~ 8rAU3 / 4 M⊕

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˜ α ~ 10−4 rAU−1/ 5Z−2M p,⊕

3 / 5

Linearized wakein shearing sheet

Philosophical remarks

• Turbulent “viscosity” probably depends on frequency ωturb ~ , ωwake ~ (r/H) ωturb

• Angular momentum transport need not be turbulent– winds, wakes, …

• Disks need not be smooth, even on lengthscales H & timescales -1

– Surely not on smaller scales!

Nelson & Papaloizou ‘04

Peroration

• MRI is the leading candidate but depends on uncertain microphysics and HE irradiation– ISM theorists needed!

• Finite-amplitude instability should be taken seriously– Higher-resolution simulations

– Experiments with d(r2)/dr > 0

• Baroclinic instability needs to be confirmed– Simulations with independent codes

• Investigate T(ω)