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Numerical simulations of the magnetorotational instability (MRI)

Numerical simulations of the magnetorotational instability (MRI). 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. The magnetorotational instability.

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Numerical simulations of the magnetorotational instability (MRI)

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  1. Numerical simulations of the magnetorotational instability (MRI) 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. The magnetorotational instability (Balbus & Hawley, 1991) nonlinear evolution  numerical simulations

  3. I. Setup & numerical issues

  4.  The shearing box (1/2) r x y H H z y H x • Local approximations • Ideal MHD equations + EQS (isothermal) • vy=-1.5x • Shearing box boundary conditions (Hawley et al. 1995)

  5.  The shearing box (2/2) z x Transport diagnostics • Maxwell stress: TMax=<-BrB>/P0 • Reynolds stress: TRey=<vrv>/ P0 • =TMax+TRey  rate of angular momentum transport Magnetic field configuration Zero net flux: Bz=B0 sin(2x/H) Net flux: Bz=B0

  6. The 90’s and early 2000’s Local simulations (Hawley & Balbus 1992) • Breakdown into MHD turbulence (Hawley & Balbus 1992) • Dynamo process (Gammie et al. 1995) • Transport angular momentum outward: <>~10-3-10-1 • Subthermal B field, subsonic velocity fluctuations BUT: low resolutions used (323 or 643)

  7. 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) ZEUS code (Stone & Norman 1992), zero net flux The decrease of  with resolution is not a property of the MRI. It is a numerical artifact!

  8. Dissipation Small scales dissipation important  Explicit dissipation terms needed (viscosity & resistivity) • Reynolds number: Re =csH/ • Magnetic Reynolds number: ReM=csH/ Magnetic Prandtl number Pm=/

  9.  Case I Zero net flux

  10. 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. Pm=/=4, Re=6250 (Nx,Ny,Nz)=(256,400,256) Density Vertical velocity By component Movie: B field lines and density field (software SDvision, D.Polmarede, CEA)

  12. 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

  13.  The Pm effect Pm =/ <<1 Viscous length << Resistive length Schekochihin et al. (2004) Velocity Magnetic field Velocity Magnetic field Schekochihin et al. (2007) Schekochihin et al. (2007) Pm=/>>1 Viscous length >> Resistive length • No proposed mechanisms…but: • Dynamo in nature (Sun, Earth) • Dynamo in experiments (VKS) • Dynamo in simulations

  14. Parameter survey Pm ? MHD turbulence ? No turbulence Re • Small scales important in MRI turbulence • Transport increases with the Prandtl number • No transport when Pm≤1 For a given Pm, does α saturates at high Re?

  15. Pm=4, Transport Re=3125 Re=6250 Re=12500 (Nx,Ny,Nz)=(128,200,128) (Nx,Ny,Nz)=(256,400,256) (Nx,Ny,Nz)=(512,800,512) Total stress =9.2 ± 2.8  10-3 Total stress =7.6 ± 1.7  10-3 Total stress =2.0 ± 0.6  10-2 No systematic trend as Re increases…

  16.  Case II Vertical net flux

  17. Influence of Pm - Pseudo-spectral code, resolution: (64,128,64) - (Lx,Ly,Lz)=(H,4H,H) - =100 Lesur & Longaretti (2007)

  18. Conclusions & open questions Pm MHD turbulence ? 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 • Global simulations? What is the effect of large scales? • State of PP disks very uncertain (Pm<<1) • Dead zone location/structure very uncertain…

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