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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
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The magnetorotational instability (Balbus & Hawley, 1991) nonlinear evolution numerical simulations
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I. Setup & numerical issues
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The shearing box (1/2) H H HH x y z r y x Local approximations Ideal MHD equations + EQS (isothermal) v y =-1.5 x Shearing box boundary conditions (Hawley et al. 1995)
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The shearing box (2/2) Magnetic field configuration Transport diagnostics Maxwell stress: T Max = /P 0 Reynolds stress: T Rey = / P 0 =T Max +T Rey rate of angular momentum transport Zero net flux: B z =B 0 sin(2 x/H) Net flux: B z =B 0 x z
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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 (32 3 or 64 3 )
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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) 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!
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Dissipation Reynolds number: Re =c s H/ Magnetic Reynolds number: Re M =c s H/ Small scales dissipation important Explicit dissipation terms needed (viscosity & resistivity) Magnetic Prandtl number Pm= /
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Case I Zero net flux
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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 NIRVANA SPECTRAL CODE PENCIL CODE ZEUS Fromang et al. (2007)
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Pm= / =4, Re=6250 (Nx,Ny,Nz)=(256,400,256) DensityVertical velocityBy component Movie: B field lines and density field (software SDvision, D.Polmarede, CEA)
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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= / =4 Pm= / = 8 Pm= / = 16 Pm= / = 2 Pm= / = 1
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The Pm effect Pm= / >>1 Viscous length >> Resistive length Schekochihin et al. (2004) Schekochihin et al. (2007) VelocityMagnetic field Pm = / <<1 Viscous length << Resistive length No proposed mechanisms…but: Dynamo in nature (Sun, Earth) Dynamo in experiments (VKS) Dynamo in simulations Schekochihin et al. (2007) VelocityMagnetic field
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Parameter survey ? MHD turbulence No turbulence Re Pm 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? ?
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Pm=4, Transport (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 Total stress =2.0 ± 0.6 10 -2 (Nx,Ny,Nz)=(512,800,512) Re=12500 No systematic trend as Re increases…
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Case II Vertical net flux
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Influence of Pm Lesur & Longaretti (2007) - Pseudo-spectral code, resolution: (64,128,64) - (Lx,Ly,Lz)=(H,4H,H) - =100
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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 ? MHD turbulence No turbulence Re Pm 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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Pm=4, flow structure Re=3125Re=6250Re=12500 By in the (x,z) plane Power spectra Kinetic energy Magnetic energy
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Protoplanetary disks properties Size: R d ~100-500 AU Mass: M d ~10 -2 M sol Lifetime: d ~10 6-7 yr Accretion rate: M acc ~10 -7-8 M sol.yr -1 need for a source of turbulence
