1. Review of MHD simulations of accretion disks MHD simulations of disk winds & protostellar jets Describe new Godunov+CT MHD Code Tests Application to MRI MHD Models of Accretion Disks and Outflows Jim Stone Princeton University

2. Goals Challenges Understand angular momentum transport mechanism Compute structure and evolution of accretion flows Understand how disks produce jets Must be MHD from start Multiple length and time scales (esp. for thin disks) Adding additional physics (radiation, microphysics, etc.) Curvilinear coordinates

3. Saturation of the MRI has been studying in small, local patches of the flow using the shearing box Hawley, Gammie, & Balbus 1995; 1996; Brandenburg et al. 1995; Stone et al. 1996; Matsumoto et al. 1996; Miller & Stone 1999

4. The outcome is always MHD turbulence.

5. Significant angular momentum transport is associated with MHD turbulence driven by the MRI Also note: Sustained amplification of B indicates dynamo action Time-evolution of volume-averaged quantities: = 0.3 = 0.07 Time in orbits

6. Current focus of studies using shearing box: adding more physics Protostellar disks Ionisation fraction is so low, non-ideal MHD effects (Ohmic dissipation, ambipolar diffusion, Hall effect) must be included Add dust Radiation dominated disks Inner regions of BH disks are so hot that P rad >> P gas. Does the saturation amplitude of the MRI depend on P rad, P gas, or some combination of the two?

7. EXAMPLE: Radiation dominated disks: Studying this regime requires solving the equations of radiation MHD: (Stone, Mihalas, & Norman 1992)

8. Linear growth rates of the MRI are changed by radiative diffusion (Blaes & Socrates 2001) (Turner, Stone, & Sano 2002) Linear growth rates make good code test

9. Density on faces of computational volume

10. (Stone & Pringle 2000; Hawley & Krolik 2001; Hawley, Balbus, & Stone 2001; Machida, Matsumoto, & Mineshige 2001) 3-D global models of geometrically thick (H/R ~ 1) black hole accretion disks demonstrate action of MRI (density over orbits 0 – 3): Latest models include full GR in Kerr metric

11. MHD models of outflows from disks Density Field lines I.Outflows from sub- Keplerian disks e.g. Uchida & Shibata Don’t get steady flows Studies can be classified based on initial/boundary conditions

12. II. Outflows from disks modeled as a boundary condition e.g. Ustyugova et al, Ouyed & Pudritz d Bp d Vz Disk is rotating plate at base of flow Internal dynamics of disk and feedback not included

13. III.Propagation of perfectly collimated jet (including cooling) Toroidal B helps to keep jet collimated images of log(d) Structure of jet is assumed

14. IV. Stability of perfectly collimated, uniform jets Temp Density

15. Future directions for disk & wind models Global models of thin disks (requires cooling) Global models with more physics non-ideal MHD for protostellar disks radiation dominated disks Synthetic spectra computed from dynamical models Outstanding issue: can we understand how jets are formed using global disk models?

16. These problems might benefit from improved methods Global model of geometrically thin (H/R << 1) disk covering 10H in R, 10H in Z, and 2  in azimuth with resolution of shearing box (128 grid points/H) will require nested grids. Nested (and adaptive) grids work best with single-step Eulerian methods based on the conservative form Algorithms in ZEUS are 15+ years old - a new code could take advantage of developments in numerical MHD since then. Our Choice: higher-order Godunov methods combined with CT

17. Constrained Transport is a conservative scheme for the magnetic flux. Difference using a staggered B and EMFs located at cell edges. Appropriately upwinded EMFs must computed from face-centered fluxes given by Riemann solver. Integrate the induction equation over cell face using Stoke’s Law to give Keeping div(B) = 0

18. A variety of previous authors have combined CT with Godunov schemes Ryu, Miniati, Jones, & Frank 1998 Dai & Woodward 1998 Balsara & Spicer 1999 Toth 2000 Londrillo & Del Zanna 2000 Pen, Arras, & Wong 2003 However, scheme developed here differs in: method by which EMFs are computed at corners. extension of unsplit integrator to MHD.

19. Test 1. Convergence Rate of Linear Waves Initialize pure eigenmode for each wave family Measure RMS error in U after propagating one wavelength quantitative test of accuracy of scheme C s = 1, V Ax = 1, V At = 3/2, L x = L y,  x =  y 1D 2D

20. Test 2. Circularly Polarized Alfven Wave  = 1, P = 0.1,  = 0.1, wave amplitude = 0.1 (Toth 2000) L x = 2L y,  x =  y, wave propagates at tan -1  Exact, nonlinear solution to MHD equations - quantitative test Subject to parametric instability (e.g. Del Zanna et al. 2001), but: Growth rate of perturbations must match dispersion relation Growing modes should not be at grid scale Animation of B z

21. Scatter plot showing all grid points - no parametric instability present

22. Test 3. RJ2a Riemann problem rotated to grid Initial discontinuity inclined to grid at tan -1  Magnetic field initialized from vector potential to ensure div(B)=0  x =  y, 512 x 256 grid Final result plotted along horizontal line at center of grid L x = 2 L y = 1 URUR ULUL Problem is Fig. 2a from Ryu & Jones 1995

23.  PE VxVx VyVy VzVz BxBx ByBy BzBz

24. Test 4. Hydrodynamical Implosion From Liska & Wendroff; 400 x 400 grid, P = 1  P = 0.125  Additional benefit of using unsplit integration scheme: Code maintains symmetry

25. Test 5. Spherical Blast Waves Not a very quantitative test, BUT check of whether blast waves remain spherical late term evolution interesting  x =  y, 400 x 600 grid, periodic boundary conditions P = 0.1  L X = 1 L Y = 1.5 P = 100 in r < 0.1 B at 45 degrees,  = 0.1 HYDRO MHD P = 0.1 

26. Hydrodynamic Blast Wave 400 x 600 grid MHD Blast Wave 400 x 600 grid

27. Test 6. Orszag-Tang vortex 512^2 grid, animation of d

28. Test 7. RT Instability Check linear growth rates One of tests in Liska & Wendroff (single mode in 2D) 200 x 600 grid d=2 d=1

29. Single mode in 3D 200x200x300 grid

30. Multi-mode in 3D Multi-mode in 3D with strong B Both 200x200x300 grids

31. A 2D Test/Application: MRI Start from vertical field with zero net flux B z = B 0 sin(2  x) Sustained turbulence not possible in 2D - rate of decay after saturation sensitive to numerical dissipation X Z

32. Animation of angular velocity fluctuations  V = V y -  0 x shows saturations of MRI and decay in 2D 3rd order Roe scheme, 256 2 grid,  min = 4000, orbits 2 - 10.

33. (Numerical) dissipation of field is slower with 3rd order Roe fluxes than with ZEUS, by a factor of about 1.5. Plot of B 2 - B 0 2 at various resolutions 256 2 64 2 128 2 ZEUS Athena

34.  Vy on faces of volume 3D shearing box with Athena is also similar to ZEUS results (32x64x32 grid -- best working resolution 10 yrs ago)

35. Code is publicly available Project is funded by NSF ITR; source code public. Code, documentation, and training material posted on web. 1D, 2D, and 3D versions are/will be available from www.astro.princeton.edu/~jstone/athena.html Future Extensions to Algorithm Curvilinear coordinates Nested grids and adaptive grids Testing and applications with fixed grid 3D version of code. Current Focus of Effort