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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
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Setup
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 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)

Magnetic field configuration
Zero net flux BzB0 sin(2?x/H)
Net flux BzB0
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 The shearing box (2/2)
Transport diagnostics

• Maxwell stress TMaxlt-BrB?gt/P0
• Reynolds stress TReylt?vrv?gt/ P0
• ?TMaxTRey

Small scale dissipation

• Reynolds number Re csH/?
• Magnetic Reynolds number ReMcsH/?
• Magnetic Prandtl number Pm?/?

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The issue of convergence
Fromang Papaloizou (2007)
Code ZEUS Zero net flux
The decrease of ? with resolution is not a
property of the MRI. It is a numerical artifact!
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Numerical dissipation
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Numerical resisitivity
(Nx,Ny,Nz)(128,200,128) No explicit dissipation
included
Fourier Transform and dot product with the FT
magnetic field
? ReM30000 ( Re)
BUT numerical dissipation depends on the flow
itself in ZEUS
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Pm?/?4, Re3125
(Nx,Ny,Nz)(128,200,128)
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Varying the resolution
(Nx,Ny,Nz)(128,200,128)
(Nx,Ny,Nz)(256,400,256)
(Nx,Ny,Nz)(64,100,64)
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
Maxwell stress 6.4 ? 10-3 Reynolds stress 1.6
? 10-3 Total stress ? 8.0 ? 10-3
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Code comparison Pm?/?4, Re3125
Fromang et al. (2007)
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
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Code comparison Pm?/?4, Re3125
Fromang et al. (2007)
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
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 Zero net flux parameter survey
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Flow structure Pm?/?4, Re6250
(Nx,Ny,Nz)(256,400,256)
Density
Vertical velocity
By component
Movie B field lines and density field (software
SDvision, D.Polmarede, CEA)
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Effect of the Prandtl number
Take Rem12500 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 Pmlt2

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Pm?/?4
By in the (x,z) plane
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Pm4, Re12500
(Nx,Ny,Nz)(512,800,512)
Total stress ?2.0 0.6 ? 10-2
No systematic trend as Re increases
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Power spectra
Re3125
Re6250
Re12500
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Summary zero mean field case
Fromang et al. (2007)

• Transport increases with Pm
• No transport when Pm1
• Behavior at large Re, ReM?

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Transition
(Lx,Ly,Lz)(H,?H,H) (Nx,Ny,Nz)(128,200,128) Re31
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 Vertical net flux
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The mean field case
Lesur Longaretti (2007)
- Pseudo-spectral code, resolution (64,128,64) -
(Lx,Ly,Lz)(H,4H,H) - ?100
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 Flow structure
Pm?/?gtgt1
Viscous length gtgt Resistive length
vz
Bz
vz
Bz
Re800
Re3200
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 Relation to the MRI modes
Growth rates of the largest MRI mode
? No obvious relation between ? and the MRI
linear growth rates
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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

• 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?