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
QuickTime™ et undécompresseur codec YUV420
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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…
