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) H H HH x y z Local approximation Code ZEUS (Hawley & Stone 1995) Ideal or non-ideal MHD equations Isothermal equation of state v y =-1.5  x Shearing box boundary conditions (Lx,Ly,Lz)=(H,  H,H) Magnetic field configuration Zero net flux: B z =B 0 sin(2  x/H) Net flux: B z =B 0 x z

4. The shearing box (2/2) Transport diagnostics Maxwell stress: T Max = /P 0 Reynolds stress: T Rey = / P 0  =T Max +T Rey Small scale dissipation Reynolds number: Re =c s H/ Magnetic Reynolds number: Re M =c s H/  Magnetic Prandtl number: Pm= / 

5. 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) The decrease of  with resolution is not a property of the MRI. It is a numerical artifact! Code ZEUS Zero net flux

6. Numerical dissipation

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

8. Pm= /  =4, Re=3125 (Nx,Ny,Nz)=(128,200,128) Maxwell stress: 7.4  10 -3 Reynolds stress: 1.6  10 -4 Total stress:  =9.1  10 -3 Residual -  k 2 B(k) 2 balanced by numerical dissipation Explicit dissipation Statistical issues at large scale

9. Varying the resolution (Nx,Ny,Nz)=(128,200,128) Maxwell stress: 7.4  10 -3 Reynolds stress: 1.6  10 -3 Total stress:  =9.1  10 -3 (Nx,Ny,Nz)=(256,400,256) Maxwell stress: 9.4  10 -3 Reynolds stress: 2.1  10 -3 Total stress:  =1.1  10 -2 (Nx,Ny,Nz)=(64,100,64) Maxwell stress: 6.4  10 -3 Reynolds stress: 1.6  10 -3 Total stress:  = 8.0  10 -3 Good agreement but… Residual -  k 2 B(k) 2 Numerical & explicit dissipation comparable!

10. Code comparison: 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)

11. Code comparison: 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) RAMSES  =1.4  10 -2 (resolution 128 cells/scaleheight)

12. Zero net flux: parameter survey

13. Flow structure: 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) Schekochihin et al. (2007) Large Pm case VelocityMagnetic field

14. 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

15. Pm= /  =4 (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 By in the (x,z) plane

16. Pm=4, Re=12500 Total stress  =2.0 ± 0.6  10 -2 (Nx,Ny,Nz)=(512,800,512) BULL cluster at the CEA ~500 000 CPU hours (~60 years) 1024 CPUs (out of ~7000) 2  10 6 timesteps 600 GB of data No systematic trend as Re increases…

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

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

19. Transition Pm=3 Pm=4 Pm=2.5  ~4.5  10 -3 (Lx,Ly,Lz)=(H,  H,H) (Nx,Ny,Nz)=(128,200,128) Re=3125

20. Vertical net flux

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

22. Flow structure Pm= /  >>1 Viscous length >> Resistive length Schekochihin et al. (2007) VelocityMagnetic field Pm = /  <<1 Viscous length << Resistive length Schekochihin et al. (2007) VelocityMagnetic field vzvz BzBz vzvz BzBz Re=800Re=3200

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 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 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? Pm  1  min  max Critical Pm? Sensitivity on Re,  ?