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Numerical simulations of the MRI: the effects of dissipation coefficients

Explore the impact of dissipation coefficients in MRI using numerical simulations in a shearing box model. Analyzing codes, stress measures, convergence issues, and resolution effects to highlight the role of numerical dissipation. Comparing various numerical methods for accuracy and agreement. Investigate the influence of Prandtl number on transport behavior and turbulence patterns in MHD systems.

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Numerical simulations of the MRI: the effects of dissipation coefficients

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  1. 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

  2. Setup

  3.  The shearing box (1/2) z x H H z y H x • 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: Bz=B0 sin(2x/H) Net flux: Bz=B0

  4.  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=/

  5. The issue of convergence (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 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!

  6. Numerical dissipation

  7. Numerical resisitivity =0 (steady state) Balanced by numerical dissipation (k2B(k)2) Residual -k2B(k)2 (Nx,Ny,Nz)=(128,200,128) No explicit dissipation included Fourier Transform and dot product with the FT magnetic field:  ReM~30000 (~ Re) BUT: numerical dissipation depends on the flow itself in ZEUS…

  8. Pm=/=4, Re=3125 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 (Nx,Ny,Nz)=(128,200,128)

  9. Varying the resolution Residual -k2B(k)2 Good agreement but… Numerical & explicit dissipation comparable! (Nx,Ny,Nz)=(64,100,64) (Nx,Ny,Nz)=(256,400,256) (Nx,Ny,Nz)=(128,200,128) 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

  10. Code comparison: Pm=/=4, Re=3125 ZEUS PENCIL CODE SPECTRAL CODE NIRVANA 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

  11. Code comparison: Pm=/=4, Re=3125 ZEUS NIRVANA 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

  12.  Zero net flux: parameter survey

  13. Flow structure: Pm=/=4, Re=6250 Velocity Magnetic field Schekochihin et al. (2007) Large Pm case (Nx,Ny,Nz)=(256,400,256) Density Vertical velocity By component Movie: B field lines and density field (software SDvision, D.Polmarede, CEA)

  14. Effect of the Prandtl number Pm=/=4 Pm=/= 8 Pm=/= 16 Pm=/= 2 Pm=/= 1 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

  15. 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 By in the (x,z) plane

  16. Pm=4, Re=12500 BULL cluster at the CEA ~500 000 CPU hours (~60 years) 1024 CPUs (out of ~7000) 2106 timesteps 600 GB of data (Nx,Ny,Nz)=(512,800,512) Total stress =2.0 ± 0.6  10-2 No systematic trend as Re increases…

  17. Power spectra Kinetic energy Magnetic energy Re=3125 Re=6250 Re=12500

  18. Summary: zero mean field case Fromang et al. (2007) • Transport increases with Pm • No transport when Pm≤1 • Behavior at large Re, ReM?

  19. 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

  20.  Vertical net flux

  21. The mean field case Critical Pm? Sensitivity on Re, ?  max min Pm 1 Lesur & Longaretti (2007) - Pseudo-spectral code, resolution: (64,128,64) - (Lx,Ly,Lz)=(H,4H,H) - =100

  22.  Flow structure Pm =/ <<1 Viscous length << Resistive length Velocity Magnetic field Velocity Magnetic field Schekochihin et al. (2007) Schekochihin et al. (2007) Pm=/>>1 Viscous length >> Resistive length vz Re=800 Bz vz Re=3200 Bz

  23.  Relation to the MRI modes Growth rates of the largest MRI mode  No obvious relation between  and the MRI linear growth rates

  24. Conclusions & open questions Critical Pm? Sensitivity on Re, ?  Pm max MHD turbulence min Pm ? 1 No turbulence Re • 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?

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