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Numerical simulations of the MRI: the effects of dissipation coefficients

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). z. x.

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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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