Numerical simulations of Numerical simulations of the MRI: the effects of the MRI: the effects of dissipation coefficientsdissipation 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)

H

H H

x

yz

• Local approximation• Code ZEUS (Hawley & Stone 1995)• Ideal or non-ideal MHD equations• Isothermal equation of state• vy=-1.5x

• Shearing box boundary conditions• (Lx,Ly,Lz)=(H,H,H)

Magnetic field configurationZero net flux: Bz=B0 sin(2x/H) Net flux: Bz=B0

x

z

 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

(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)

The decrease of with resolution is not a property of the MRI. It is a numerical artifact!

Code ZEUSZero net flux

Numerical dissipation

Numerical resisitivity(Nx,Ny,Nz)=(128,200,128)

No explicit dissipation included

BUT: numerical dissipation depends on the flow itself in ZEUS…

Residual-k2B(k)2

Fourier Transform and dot product with the FT magnetic field:

=0 (steady state) Balanced by numerical dissipation (k2B(k)2)

ReM~30000 (~ Re)

Pm=/=4, Re=3125(Nx,Ny,Nz)=(128,200,128)

Maxwell stress: 7.4 10-3

Reynolds stress: 1.6 10-4

Total stress: =9.1 10-3

Residual-k2B(k)2

balanced by numerical dissipation

Explicit dissipation

Statistical issues at large scale

Varying the resolution(Nx,Ny,Nz)=(128,200,128)

Maxwell stress: 7.4 10-3

Reynolds stress: 1.6 10-3

Total stress: =9.1 10-3

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

Maxwell stress: 9.4 10-3

Reynolds stress: 2.1 10-3

Total stress: =1.1 10-2

(Nx,Ny,Nz)=(64,100,64)

Maxwell stress: 6.4 10-3

Reynolds stress: 1.6 10-3

Total stress: = 8.0 10-3

Good agreement

but…

Residual-k2B(k)2

Numerical & explicit dissipation comparable!

Code comparison: 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)

Code comparison: 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)

RAMSES

=1.4 10-2

(resolution 128 cells/scaleheight)

 Zero net flux: parameter survey

Flow structure: 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)

Schekochihin et al. (2007)Large Pm case

Velocity Magnetic field

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

Pm=/=4

(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

By in the (x,z) plane

Pm=4, Re=12500

Total stress=2.0 ± 0.6 10-2

(Nx,Ny,Nz)=(512,800,512)

BULL cluster at the CEA ~500 000 CPU hours (~60 years)

1024 CPUs (out of ~7000)2106 timesteps600 GB of data

No systematic trend as Re increases…

Power spectra

Re=3125 Re=6250

Re=12500

Kinetic energy

Magnetic energy

Summary: zero mean field case

• Transport increases with Pm• No transport when Pm≤1• Behavior at large Re, ReM?

Fromang et al. (2007)

Transition

Pm=3

Pm=4

Pm=2.5

~4.510-3

(Lx,Ly,Lz)=(H,H,H)(Nx,Ny,Nz)=(128,200,128)

Re=3125

 Vertical net flux

The mean field caseLesur & Longaretti (2007)

- Pseudo-spectral code, resolution: (64,128,64)- (Lx,Ly,Lz)=(H,4H,H)- =100

Pm

1

min

max

Critical Pm?Sensitivity on Re, ?

 Flow structure Pm=/>>1

Viscous length >> Resistive length

Schekochihin et al. (2007)

Velocity Magnetic field

Pm =/ <<1

Viscous length << Resistive length

Schekochihin et al. (2007)

Velocity Magnetic field

vz Bz vz BzRe=800 Re=3200

 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

?

MHD turbulence

No turbulenceRe

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?

Pm

1

min

max

Critical Pm?Sensitivity on Re, ?