1. Studies of the MRI with a New Godunov Scheme for MHD: Jim Stone & Tom Gardiner Princeton University Recent collaborators: John Hawley (UVa) Peter Teuben (UMd)

2. Outline of Talk A. Motivation B. Basic Elements of the Algorithm C. Some Tests & Applications D. Evolution of vortices in disks E. MRI, with comparison to ZEUS

3. Global simulation Local simulation Hawley, Gammie, & Balbus 1995; 1996; Stone et al. 1996; Armitage 1998; etc. ZEUS-like algorithms have been used successfully to study the MRI in both 3D global and local simulations

4. These methods have been extended to study the radiation dominated inner regions of disks around compact objects. Equations of radiation MHD: (Stone, Mihalas, & Norman 1992) Use ZEUS with flux- limited diffusion module (Turner & Stone 2001)

5. e.g. photon bubble instability in accretion disk atmospheres Gammie (1998) and Blaes & Socrates (2001) have shown magnetosonic waves are linearly unstable in radiation dominated atmospheres Turner et al. (2004) have shown they evolve into shocks in nonlinear regime: Limitation is FLD

6. 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) would be easier with nested grids. Nested (and adaptive) grids work best with single-step Eulerian methods based on the conservative form MHD 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 Constrained Transport (CT) So why develop a new MHD code?

7. A variety of authors have combined CT with Godunov schemes previously 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: 1. method by which EMFs are computed at corners. 2. calculation of PPM interface states in MHD 3. extension of unsplit integrator to MHD Gardiner & Stone 2005 B. Basic elements of the algorithm

8. The CT Algorithm Finite Volume / Godunov algorithm gives E-field at face centers. “CT Algorithm” defines E-field at grid cell corners. Arithmetic averaging: 2D plane-parallel flow does not reduce to equivalent 1D problem Algorithms which reconstruct E-field at corner are superior Gardiner & Stone 2005

9. Simple advection tests demonstrate differences Field Loop Advection (  10 6 ): MUSCL - Hancock Arithmetic averageGardiner & Stone 2005

10. CornerTransportUpwind [Colella 1991] (12 R-solves in 3D) Optimally Stable, CFL < 1 Complex & Expensive for MHD... CTU (6 R-solves) Stable for CFL < 1/2 Relatively Simple... MUSCL-Hancock Stable for CFL < 1/2 Very Simple, but diffusive... Directionally unsplit integration

11. C. Some Tests & Applications 1.Convergence Rate of Linear Waves 2.Nonlinear Circularly Polarized Alfven Wave 3.RJ Riemann problems rotated to grid 4.Advection of a Field Loop (Current Cylinder) 5.Current Sheet 6.Hydro and MHD Blast Waves 7.Hydro implosion 8.RT and KH instability See http://www.astro.princeton.edu/~jstone/tests for more tests.http://www.astro.princeton.edu/~jstone/tests

12. Linear Wave Convergence (2N x N x N) Grid

13. A test An application Multi-mode in hydrodynamics Multi-mode in with strong B 3D RT Instability (200x200x300 grid) Goal: measure growth rate of fingers, structure in 3D

14. Codes are publicly available Download a copy from www.astro.princeton.edu/~jstone/athena.html Current status: 1D version publicly available (C and F95) 2D version publicly available (C and F95) 3D version being tested on applications (C only) Expect to release parallelized 3D Cartesian grid code in ~1yr Code, documentation, and tests posted on web.

15. D. Evolution of vorticity in hydrodynamical shearing disk HBS 1996 showed nonlinear random motions decay in hydrodynamical shearing box using PPM Recent work has renewed interest in evolution of vortices in disks, especially transit amplification of leading->trailing waves 2D incompressible Umurhan & Regev 2004; Yecko 2004 2D compressible Johnson & Gammie 2005 3D anelastic in stratified disks Barranco & Marcus 2005 Our goal: Study evolution of vortices in 3D compressible disks at the highest resolutions possible, closely following JG2005

16. A test: shear amplification of a single, leading, incompressible vortex Initial vorticity (k x /k y ) 0 = 4

17. Dimensionless time:   = 1.5  t + k x0 /k y KE is amplified by ~ (k x /k y ) 2 : can code reproduce large amplification factors? Johnson & Gammie 2005

18. There is no aliasing at shearing-sheet boundaries using this single-step unsplit integrator Trailing wave never seeds new leading waves

19. Evolution of specific vorticity Initially W(k) ~ k -5/3 2D 512 2 simulation Box size: 4Hx4H Evolved to  =100 y x

20. T = 0.5  t 2D simulation W z initialized identically in 2D and 3D Random V z added in 3D 2D grid: 256 2 (4Hx4H) 3D grid: 256 2 x64 (4Hx4Hx1H) Comparison of evolution in 2D and 3D

21. T = 2  t

22. T = 5  t

23. T = 10  t

24. T = 20  t In 2D, long-lived vortices emerge In 3D, vorticity and turbulence decays much more rapidly

25. Evolution of KE: 2D versus 3D NB: Initial evolution identical

26. Evolution of Stress: 2D versus 3D In 2D, long-lived vortices emerge In 3D, vorticity (and KE and stress) decays much more rapidly Further 3D runs in both stratified and unstratified boxes warranted

27. E. MHD studies of the MRI Start from a vertical field with zero net flux: B z =B 0 sin(2 p x) Sustained turbulence not possible in 2D – dissipation rate after saturation is sensitive to numerical dissipation

28. 2D MRI Animation of angular velocity fluctuations: d V y =V y +1.5 W 0 x shows saturation of MRI and decay in 2D 3 rd order reconstruction, 256 2 Grid b min =4000, orbits 2-10

29. Magnetic Energy Evolution ZEUS vs. Athena Numerical dissipation is ~ 1.5 times smaller with CTU & 3 rd order reconstruction than ZEUS.

30. 3D MRI Animation of angular velocity fluctuations: d V y =V y +1.5 W 0 x Initial Field Geometry is Uniform B y CTU with 3 rd order reconstruction, 128 x 256 x 128 Grid b min = 100, orbits 4-20 Goal: Since Athena is strictly conservative, can measure spectrum of T fluctuations from dissipation of turbulence Requires adding optically- thin radiative cooling

31. Stress & Energy for  0 No qualitative difference with ZEUS results (Hawley, Gammie, & Balbus 1995)

32. Energy Conservation Saturation implies: Previously observed for Re M = 1 (Sano et al. 2001).

33. Dependence of saturated state on cooling Red line: no cooling; Green line:  cool = Q Cooling has almost no effect except on internal energy Internal energy Reynolds stress Magnetic energy Maxwell stress

34. Density Fluctuations Compressible fluctuations lead to spiral waves & M ~ 1.5 shocks Temperature fluctuations are dominated by compressive waves. Will this be a generic result in global disks?

35. Going beyond MHD If mean-free-path of particles is long compared to gyroradius, anisotropic transport coefficients can be important (Braginskii 1965) Generically, astrophysical plasmas are diffuse, and kinetic effects may be important in some circumstances These effects can produce qualitative changes to the dynamics: e.g. with anisotropic heat conduction, the convective instability criterion becomes dT/dz < 0 (Balbus 2000)

36. Evolution of field lines in atmosphere with dS/dz > 0 and dT/dz < 0 including anisotropic heat conduction, Details: 2D 128 2, initial  =100, B x proportional to sin(z), fixed T at vertical boundaries, isotropic conduction near boundaries Stable layer Unstable Stable layer z

37. Summary Developed a new Godunov scheme for MHD Code now being used for various 3D applications Evolution of vortices in disks Energy dissipation in MRI turbulence Further algorithm development is planned Curvilinear grids Nested and adaptive grids Future applications 3D global models of thin disks Fragmentation and collapse in self-gravitating MHD turbulence Beyond ideal MHD Kinetic effects in diffuse plasmas Particle acceleration (CRs are important!) Dust dynamics in protoplanetary disks

38. A. Astrophysical motivation Accretion disks Star formation Protoplanetary disks X-ray clusters Numerical challenges 3D MHD Non-ideal MHD Radiation hydrodynamics (P rad >> P gas ) Nested and adaptive grids Kinetic MHD effects