MHD turbulence in protoplanetary disks S.Fromang CEA Saclay, France J.Papaloizou (DAMTP, Cambridge, UK) G.Lesur (DAMTP, Cambridge, UK), T.Heinemann (DAMTP, Cambridge, UK) R.Nelson (QMUL, London, UK) Background: ESO press release 36/06

The need for MHD turbulence Molecular transport Timescale:with l~10 11 m u~1 km/s ~1 cm  ~10 5 cm 2 /s et  diff ~10 13 yr >>  disk ~10 6 yrs Hydrodynamic transport - Linarly stable (Rayleigh criterion) - No proof yet of nonlinear instability Theoretical: Rincon et al Numerical: Lesur & Longaretti 2005 Experimental: Ji et al Need for more… Source for angular momenum transport

Over 17 years ago… See also Velikhov (1959) and Chandrasekhar (1960)

The magnetorotational instability (Balbus & Hawley, 1991) nonlinear evolution  numerical simulations

Numerical strategy Local simulations (high resolution, long integration times, influence of the small scales) Global simulations « Realistic models » PROPERTIES OF MHD TURBULENCE (dynamo mechanism, transport, velocity fluctuations)

Outline I. LOCAL SIMULATIONS Shearing box model, historical results Zero mean field simulations Mean field simulations Consequences for protoplanetary disks II. GLOBAL SIMULATIONS Disk models Dust settling III. CONCLUSIONS

I. Local simulations

The shearing box (1/2) H H HH x y z r y x Local approximations Ideal MHD equations + EQS (isothermal) v y =-1.5  x Shearing box boundary conditions (Hawley et al. 1995)

The shearing box (2/2) Magnetic field configuration Transport diagnostics Maxwell stress: T Max = /P 0 Reynolds stress: T Rey = / P 0  =T Max +T Rey  rate of angular momentum transport Zero net flux: B z =B 0 sin(2  x/H) Net flux: B z =B 0 x z

The 90’s and early 2000’s Local simulations (Hawley & Balbus 1992) Breakdown into MHD turbulence (Hawley & Balbus 1992) Dynamo process (Gammie et al. 1995) Transport angular momentum outward: ~ Subthermal B field, subsonic velocity fluctuations BUT: low resolutions used (32 3 or 64 3 )

The issue of convergence (Nx,Ny,Nz)=(128,200,128) Total stress:  =2.0  (Nx,Ny,Nz)=(256,400,256) Total stress:  =1.0  (Nx,Ny,Nz)=(128,200,128)(Nx,Ny,Nz)=(256,400,256)(Nx,Ny,Nz)=(64,100,64) Total stress:  =4.2  Fromang & Papaloizou (2007) ZEUS code (Stone & Norman 1992), zero net flux

Dissipation Reynolds number: Re =c s H/ Magnetic Reynolds number: Re M =c s H/  Small scales dissipation important  Explicit dissipation terms needed (viscosity & resistivity) Magnetic Prandtl number Pm= / 

Case I Zero net flux

Pm= /  =4, Re=3125 ZEUS :  =9.6  (resolution 128 cells/scaleheight) NIRVANA :  =9.5  (resolution 128 cells/scaleheight) SPECTRAL CODE:  =1.0  (resolution 64 cells/scaleheight) PENCIL CODE :  =1.0  (resolution 128 cells/scaleheight)  Good agreement between different numerical methods NIRVANA SPECTRAL CODE PENCIL CODE ZEUS Fromang et al. (2007)

Pm= /  =4, Re=6250 (Nx,Ny,Nz)=(256,400,256) DensityVertical velocityBy component Movie: B field lines and density field (software SDvision, D.Polmarede, CEA)

Effect of the Prandtl number Take Rem=12500 and vary the Prandtl number…. (Lx,Ly,Lz)=(H,  H,H) (Nx,Ny,Nz)=(128,200,128)   increases with the Prandtl number  No MHD turbulence for Pm<2 Pm= /  =4 Pm= /  = 8 Pm= /  = 16 Pm= /  = 2 Pm= /  = 1

The Pm effect Pm= /  >>1 Viscous length >> Resistive length Schekochihin et al. (2004) Schekochihin et al. (2007) VelocityMagnetic field Pm = /  <<1 Viscous length << Resistive length No proposed mechanisms…but: Dynamo in nature (Sun, Earth) Dynamo in experiments (VKS) Dynamo in simulations Schekochihin et al. (2007) VelocityMagnetic field

Parameter survey ? MHD turbulence No turbulence Re Pm Small scales important in MRI turbulence Transport increases with the Prandtl number No transport when Pm≤1 For a given Pm, does α saturates at high Re? ?

Pm=4, Transport (Nx,Ny,Nz)=(128,200,128) Re=3125 Total stress  =9.2 ± 2.8  Total stress  =7.6 ± 1.7  (Nx,Ny,Nz)=(256,400,256) Re=6250 Total stress  =1.5 ± 0.3  (Nx,Ny,Nz)=(512,800,512) Re=12500

Pm=4, flow structure Re=3125Re=6250Re=12500 By in the (x,z) plane Power spectra Kinetic energy Magnetic energy

Case II Vertical net flux

Influence of Pm Lesur & Longaretti (2007) - Pseudo-spectral code, resolution: (64,128,64) - (Lx,Ly,Lz)=(H,4H,H) -  =100

Relation to the MRI modes Growth rates of the largest MRI mode  No obvious relation between  and the MRI linear growth rates

Consequences for protoplanetary disks

Dead zones in protoplanetary disks Very low ionisation in protoplanetary disks (dense and cold)  Gas and magnetic field may decouple  Pm<<1 (Brandenburg 2005, Balbus & Henri 2008) Gammie (1996) Cosmic ray ionization

Critical Rem Fleming et al. (2000) Local simulations, zero net flux, resolution: (59,123,59) Finite resistivity, no viscosity Critical magnetic Reynolds number: Rem>10 4 (Rem>100 in case of a net flux) Fromang et al. (2007) Numerical artifact!

Dynamical/Chemical aspects Dynamical aspects: - Fleming & Stone (2003): local layered disk model - Fromang & Papaloizou (2006): dust settling in layered disk - Turner et al. (2007): turbulent mixing of electrons in the dead zone - Ilgner & Nelson (astroph 0802:4409): turbulent mixing of electrons in the dead zone - Turner & Sano (astroph 0804:2916): diffusion of B field in the dead zone Chemical aspects: - Sano et al. (2000) - Fromang, Terquem & Balbus (2002) - Semenov et al. (2004) - Ilgner & Nelson (2006abc) Location/properties/existence of the dead zone very uncertain!

II. Global simulations

The issue of resolution 1 AU 10 AU H/R~ H 6H over the disk vertical extent  Nz~ H to cover the radial range  Nr~N  ~6000 (non uniform grid spacing)  Nr~N  ~25000 (uniform grid spacing) Impossible with present and near (and far!) future resources! Global simulations are always under-resolved No explicit dissipation can be included with present day resources

Two types of models Torus Approach, Poloidal or toroidal field (Hawley et al. 2000, …) Thin disk approach, toroidal field (Fromang & Nelson 2006) z r z r Hot, thick disk orbiting around black hole Cold, thin disk orbiting YSO

Thin disk H/R=0.07 to 0.1,  r -1/2 EQS locally isothermal, 9 scaleheights in total Initial B field toroidal (but the toroidal flux escapes…) Resolution: (N r,N ,N  )=(360,120,213) to (455,150,213)  about 15 cells per scaleheight Fromang & Nelson (2006)

Disk properties Time history of the Maxwell stress Radial profile for alpha ~5   Vertical profile of the velocity perturbation Vertical profile of the magnetic pressure

Dust Settling Radial migration Dust MHD Turbulence Solid body dynamics -  s <1  strong coupling (Garaud et al, 2004) -  s ~1  weak coupling Coupling between gas and solids

Dust dynamic in global models MRI-TURBULENT LAMINAR Single dust size:  s =10 -2 Dust to gas ratio initially uniform Comparison hydro vs. MHD

Advection/diffusion equation (Dubrulle et al. 1995, Schrapler & Henning 2004, Dullemond & Dominik 2004) Settling toward the midplane Diffusion by turbulence Steady state solution: (assuming ) D dust =(  c s H)/S c with  =0.01 (measured from the simulations) Sc=1.5 (Johansen et al. 2005, Fromang & Papaloizou 2006)

The case  s =10 -2 Gas density Dust density (solution of the advection diffusion equation)

Dust sizes  s =10 -3  s =10 -2 (~1 mm)  s =10 -4 (~10  m)  Dust disk layer thicker than expected from simple modelling  Dust distribution in the upper disk probes turbulence properties (cf talk by Jean- Charles)

Conclusions & perspectives LOCAL SIMULATIONS Small scale dissipation : important (small Re) Value of  in real disks (Re->∞): unknown Effect of vertical stratification : unknown Relevance to global models : unknown GLOBAL SIMULATIONS Resolution : bad Effect of small scale dissipation : unknown Link with local simulations : unknown Constraint from dust observations : promising

Physical origin Gas disk thicker due to magnetic support Simulations results Isothermal profile Velocity fluctuations increase in the disk corona Vertical profile of the velocity perturbation

Origins & Applications Gas disk thicker due to magnetic support Observations Model Observations of GG Tau Pinte et al. (2007)  h(a)  a -1.5 h: dust disk thickness a: particle size Simulations results Isothermal profile Velocity fluctuations increase in the disk corona

Protoplanetary disks properties Size: R d ~ AU Mass: M d ~10 -2 M sol Lifetime:  d ~ yr Accretion rate: M acc ~ M sol.yr -1  need for a source of turbulence

Vertical settling r z Gas orbit Dust orbit z Weak coupling limit t Strong coupling limit (Garaud et al. 2004)

The effect of turbulence 1.5 orbits5 orbits10 orbits15 orbits gasdust Early evolution Fromang & Papaloizou (2006) /c s H=5.5  ~  /3 Turbulent diffusion Local simulation of stratified shearing box resolution (32,100,192)