Numerical simulations of the magnetorotational instability (MRI) S.Fromang CEA Saclay, France...
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Transcript of Numerical simulations of the magnetorotational instability (MRI) S.Fromang CEA Saclay, France...
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 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=/
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
sont requis pour visionner cette image.
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…
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…