1. Recent advances in Astrophysical MHD Jim Stone Department of Astrophysical Sciences & PACM Princeton University, USA Recent collaborators: Tom Gardiner (Cray Research) John Hawley (UVa) Peter Teuben (UMd)

2. Eagle nebula (M16) Numerical methods for MHD are crucial for understanding the dynamics of astrophysical plasmas.

3. Modern schemes solve the equations of ideal MHD in conservative form Many schemes are possible: central schemes WENO schemes TVD schemes We have adopted finite-volume techniques using Godunov’s method. CC

4. Basic Algorithm: Discretization BxBx ByBy EVEV Scalars and velocity at cell centers Magnetic field at cell faces Cell-centered quantities volume-averaged Face centered quantities area-averaged Area averaging is the natural discretization for the magnetic field.

5. For pure hydrodynamics of ideal gases, exact/efficient nonlinear Riemann solvers are possible. In MHD, nonlinear Riemann solvers are complex because: 1.There are 3 wave families in MHD – 7 characteristics 2.In some circumstances, 2 of the 3 waves can be degenerate (e.g. V Alfven = V slow ) Equations of MHD are not strictly hyperbolic (Brio & Wu, Zachary & Colella) Thus, in practice, MHD Godunov schemes use approximate and/or linearized Riemann solvers. Key element of Godunov method is Riemann solver

6. Many possible approximations are possible: 1. Roe’s method – keeps all 7 characteristics, but treats each as a simple wave. 2. Harten-Lax-van Leer (HLLE) method – keeps only largest and smallest characteristics, averages intermediate states in- between. 3. HLLD method – Adds entropy and Alfven wave back into HLLE method, giving four intermediate states. Good resolution of all waves Requires characteristic decomposition in conserved variables Expensive and difficult to add new physics Does not preserve positivity Very simple and efficient Guarantees positivity Very diffusive for contact discontinuities Reasonably simple and efficient Guarantees positivity Better resolution of contact discontinuities

7. Keeping B = 0  1.Do nothing. Assume errors remain small and bound. 2.Evolve B using vector potential defined through B= 3.Remove solenoidal part of B using “flux-cleaning”. That is, set B  B – where + B = 0 4.Use Powell’s “8-wave solver” (adds div(B) source terms) 5.Evolve integral form of induction equation so as to conserve magnetic flux (constrained transport).  .  2 Requires taking second difference numerically to compute Lorentz force Requires solving elliptic PDE every timestep – expensive May smooth discontinuities in B Gives wrong jump conditions for some shock problems Requires staggered grid for B (although see Toth 2000)

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 (Balsara & Spicer 1999)

10. Which Multidimensional Algorithm? CornerTransportUpwind [Colella 1991] (12 R-solves) 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...

11. Verification: Linear Wave Convergence (2N x N x N) Grid

12. Validation: Hydro RT instability 256 2 x 512 grid Random perturbations Isosurface and slices of density

13. Dimonte et al, Phys. Fluids (2004) have used time evolution of rising “bubbles” and falling “spikes” from experiments to validate hydro codes: Asymptotic slope too small in ALL codes by about factor 2 Probably because of mixing at grid scale Comparison with expts. using miscible fluids much better experiment

14. Application: 3D MHD RT instability 256 2 x 512 grid Random perturbations Isosurface and slices of density B = (B x, 0, 0) crit = L x /2

15. Codes are publicly available Download a copy from www.astro.princeton.edu/~jstone/athena.html Current status: 1D version publicly available 2D version publicly available 3D version will be released in ~1 month Latest 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.

16. EXAMPLE APPLICATION: MHD of Accretion disks e.g., mass transfer in a close binary

17. If accreting plasma has any angular momentum, it will form a rotationally supported disk L  r 1/2  r -3/2 Thereafter, accretion can only occur if angular momentum is transported outwards. microscopic viscosity too small anomalous (turbulent?) viscosity required MHD turbulence driven by magnetorotational instability (MRI) dominates Profiles of specific angular momentum (L) and orbital frequency (  ) for Keplerian disk:

18. 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: Code Test Animation of angular velocity fluctuations: d V y =V y +1.5 W 0 x CTU with 3 rd order reconstruction, 256 2 grid, b min =4000, orbits 2-10

19. Magnetic Energy Evolution Numerical dissipation is ~ 1.5 times smaller with CTU & 3 rd order reconstruction than Zeus.

20. 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 In 3D, sustained turbulence

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

22. Vertical structure of stratified disks Now using static nested-grids to refine midplane of thin disks with cooling in shearing box Next step towards nested-grid global models Density Angular momentum fluctuations

23. Future Extensions to Algorithm Curvilinear coordinates Nested and adaptive grids (already implemented) full-transport radiation hydrodynamics (w. Sekora) non-ideal MHD (w. Lemaster) special relativistic MHD (w. MacFayden) Star formation Global models of accretion disks Future Applications