Numerical simulations Numerical simulations of the of the

magnetorotational magnetorotational instability (MRI)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

x

yz

r

y

x

• Local approximations• Ideal MHD equations + EQS (isothermal)• vy=-1.5x

• Shearing box boundary conditions (Hawley et al. 1995)

 The shearing box (2/2)

Magnetic field configuration

Transport diagnostics

• Maxwell stress: TMax=<-BrB>/P0

• Reynolds stress: TRey=<vrv>/ P0

• =TMax+TRey

rate of angular momentum

transport

Zero net flux: Bz=B0 sin(2x/H) Net flux: Bz=B0

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: <>~10-3-10-1

• Subthermal B field, subsonic velocity fluctuations

BUT: low resolutions used (323 or 643)

The issue of convergence

(Nx,Ny,Nz)=(128,200,128)Total stress: =2.0 10-3

(Nx,Ny,Nz)=(256,400,256)Total stress: =1.0 10-3

(Nx,Ny,Nz)=(64,100,64)Total stress: =4.2 10-3

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 =csH/• Magnetic Reynolds number: ReM=csH/

Small scales dissipation important Explicit dissipation terms needed

(viscosity & resistivity)

Magnetic Prandtl numberPm=/

 Case I

Zero net flux

Pm=/=4, Re=3125

ZEUS : =9.6 10-3 (resolution 128 cells/scaleheight) NIRVANA : =9.5 10-3 (resolution 128 cells/scaleheight)SPECTRAL CODE: =1.0 10-2 (resolution 64 cells/scaleheight)PENCIL CODE : =1.0 10-2 (resolution 128 cells/scaleheight)

Good agreement between different numerical methods

NIRVANASPECTRAL CODE

PENCIL CODEZEUS

Fromang et al. (2007)

Pm=/=4, Re=6250(Nx,Ny,Nz)=(256,400,256)

Density Vertical velocity By component

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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=/=4Pm=/= 8Pm=/= 16

Pm=/= 2

Pm=/= 1

 The Pm effect Pm=/>>1

Viscous length >> Resistive length

Schekochihin et al. (2004)

Schekochihin et al. (2007)

Velocity Magnetic 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)

Velocity Magnetic 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 10-3

Total stress=7.6 ± 1.7 10-3

(Nx,Ny,Nz)=(256,400,256)

Re=6250

Total stress=2.0 ± 0.6 10-2

(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…