Numerical simulations of the
MRI: the effects of
dissipation coefficients
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
Setup
The shearing box (1/2)
• Local approximation
• Code ZEUS (Hawley & Stone 1995)
• Ideal or non-ideal MHD equations
• Isothermal equation of state
• vy=-1.5x
• Shearing box boundary conditions
• (Lx,Ly,Lz)=(H,H,H)
z
H
y
x
H
Magnetic field configuration
Zero net flux: Bz=B0 sin(2x/H)
z
x
Net flux: Bz=B0
The shearing box (2/2)
Transport diagnostics
• Maxwell stress: TMax=<-BrB>/P0
• Reynolds stress: TRey=<vrv>/ P0
• =TMax+TRey
Small scale dissipation
• Reynolds number:
Re =csH/
• Magnetic Reynolds number: ReM=csH/
• Magnetic Prandtl number:
Pm=/
The issue of convergence
Fromang & Papaloizou (2007)
Code ZEUS
Zero net flux
(Nx,Ny,Nz)=(64,100,64)
Total stress: =4.2  10-3
(Nx,Ny,Nz)=(128,200,128)
Total stress: =2.0  10-3
(Nx,Ny,Nz)=(256,400,256)
Total stress: =1.0  10-3
The decrease of  with resolution is not a property of
the MRI. It is a numerical artifact!
Numerical dissipation
Numerical resisitivity
(Nx,Ny,Nz)=(128,200,128)
No explicit dissipation included
Fourier Transform and dot product
with the FT magnetic field:
=0 (steady state)
Balanced by numerical
dissipation (k2B(k)2)
 ReM~30000 (~ Re)
Residual
-k2B(k)2
BUT: numerical dissipation depends on the flow itself in ZEUS…
Pm=/=4, Re=3125
(Nx,Ny,Nz)=(128,200,128)
Explicit
dissipation
balanced by numerical dissipation
Statistical issues
at large scale
Maxwell stress: 7.4  10-3
Reynolds stress: 1.6  10-4
Total stress: =9.1  10-3
Residual
-k2B(k)2
Varying the resolution
(Nx,Ny,Nz)=(64,100,64)
(Nx,Ny,Nz)=(128,200,128)
(Nx,Ny,Nz)=(256,400,256)
Maxwell stress: 6.4  10-3
Reynolds stress: 1.6  10-3
Total stress: = 8.0  10-3
Maxwell stress: 7.4  10-3
Reynolds stress: 1.6  10-3
Total stress: =9.1  10-3
Maxwell stress: 9.4  10-3
Reynolds stress: 2.1 10-3
Total stress: =1.1  10-2
Residual
-k2B(k)2
Good agreement
but…
Numerical & explicit
dissipation comparable!
Code comparison: Pm=/=4, Re=3125
Fromang et al. (2007)
ZEUS
PENCIL CODE
SPECTRAL CODE
NIRVANA
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
Code comparison: Pm=/=4, Re=3125
Fromang et al. (2007)
ZEUS
NIRVANA
PENCIL CODE
RAMSES
SPECTRAL CODE
 =1.4 10-2
(resolution 128 cells/scaleheight)
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
Zero net flux: parameter
survey
Flow structure: Pm=/=4, Re=6250
(Nx,Ny,Nz)=(256,400,256)
Density
Vertical velocity
By component
QuickTime™ et un
décompresseur codec YUV420
sont requis pour visionner cette image.
Velocity
Movie: B field lines and density field
(software SDvision, D.Polmarede, CEA)
Magnetic field
Schekochihin et al. (2007)
Large Pm case
Effect of the Prandtl number
(Lx,Ly,Lz)=(H,H,H)
(Nx,Ny,Nz)=(128,200,128)
Take Rem=12500 and vary
the Prandtl number….
Pm=/=4
Pm=/= 8
Pm=/= 16
Pm=/= 2
Pm=/= 1
  increases with the Prandtl number
 No MHD turbulence for Pm<2
Pm=/=4
Re=3125
Re=6250
(Nx,Ny,Nz)=(128,200,128)
(Nx,Ny,Nz)=(256,400,256)
Total stress
=9.2 ± 2.8  10-3
Total stress
=7.6 ± 1.7  10-3
Pm=4, Re=12500
(Nx,Ny,Nz)=(512,800,512)
BULL cluster at the CEA
~500 000 CPU hours (~60 years)
1024 CPUs (out of ~7000)
2106 timesteps
600 GB of data
Total stress
=2.0 ± 0.6  10-2
No systematic trend as Re increases…
Power spectra
Re=3125
Re=6250
Kinetic energy
Magnetic energy
Re=12500
Summary: zero mean field case
Fromang et al. (2007)
• Transport increases with Pm
• No transport when Pm≤1
• Behavior at large Re, ReM?
Transition
Pm=4
~4.510-3
Pm=3
Pm=2.5
(Lx,Ly,Lz)=(H,H,H)
(Nx,Ny,Nz)=(128,200,128)
Re=3125
Vertical net flux
The mean field case
Lesur & Longaretti (2007)

Critical Pm?
Sensitivity on Re, ?
max
min
1
- Pseudo-spectral code, resolution: (64,128,64)
- (Lx,Ly,Lz)=(H,4H,H)
- =100
Pm
Flow structure
Pm=/>>1
Pm =/ <<1
Viscous length >> Resistive length
Viscous length << Resistive length
vz
Velocity
Re=800
Bz
Magnetic field
Schekochihin et al. (2007)
vz
Velocity
Re=3200
Bz
Magnetic field
Schekochihin et al. (2007)
Relation to the MRI modes
Growth rates of the largest MRI
mode
 No obvious relation between  and the MRI
linear growth rates
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
Pm

MHD turbulence
No turbulence
?Re
Critical Pm?
Sensitivity on Re, ?
max
min
1
Pm
• Vertical stratification? Compressibility (see poster by T.Heinemann)?
• Global simulations? What is the effect of large scales?
• Is brute force the way of the future? Numerical scheme?
• Large Eddy simulations?