Numerical simulations of the magnetorotational instability (MRI) S.Fromang CEA Saclay, France J.Papaloizou (DAMTP, Cambridge, UK) G.Lesur (DAMTP, Cambridge, UK), T.Heinemann (DAMTP, Cambridge, UK) Background: ESO press release 36/06

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

I. Setup & numerical issues

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)=(64,100,64) Total stress:  =4.2  Fromang & Papaloizou (2007) ZEUS code (Stone & Norman 1992), zero net flux The decrease of  with resolution is not a property of the MRI. It is a numerical artifact!

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  =2.0 ± 0.6  (Nx,Ny,Nz)=(512,800,512) Re=12500 No systematic trend as Re increases…

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

Conclusions & open questions Include explicit dissipation in local simulations of the MRI: resistivity AND viscosity Zero net flux AND nonzero net flux  an increasing function of Pm Behavior at large Re is unclear ? MHD turbulence No turbulence Re Pm Global simulations? What is the effect of large scales? State of PP disks very uncertain (Pm<<1) Dead zone location/structure very uncertain…

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

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