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SFUMATO: A self-gravitational MHD AMR code

SFUMATO: A self-gravitational MHD AMR code. Outflow. Circumstellar disk. Tomoaki Matsumoto （ Hosei Univerisity ）. Matsumoto (2006) Submitted to PASJ, astro-ph/0609105. Magnetic field. Protostar. Computational domain is 1,000 times larger. H 13 CO + core.

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SFUMATO: A self-gravitational MHD AMR code

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  1. SFUMATO: A self-gravitational MHD AMR code Outflow Circumstellar disk Tomoaki Matsumoto （Hosei Univerisity） Matsumoto (2006) Submitted to PASJ, astro-ph/0609105 Magnetic field Protostar Computational domain is 1,000 times larger.

  2. H13CO+core Introduction:From a cloud to a protostar Outflow and Protostar (radio) Molecular cloud core in Taurus （radio） Orion molecular cloud （optical＋radio）

  3. B B 0.1 – 0.01 pc Introduction:From a cloud core to a protostar EXTREMELY HIGH-RESOLUTION Protostar, protoplanetary disk and outflow Molecular cloud core Gravitational collapse MULTI-SCALE SIMULATION 1AU/0.1pc = 5×10-5 1-10 AU First core ⇒ Second core ⇒ CTTS ⇒ WTTS ⇒ Main sequence 100 - 1000 AU Protostar

  4. ★ ★ ★ ★ Self-gravitational Fluid-dynamics Utilizing Mesh Adaptive Technique with Oct-tree. AMR, dynamically allocated grids Nested Grid, static grids Developed in 2003 Matsumoto & Hanawa (2003) Cf., Talks of Mikmi, Tomisaka, Machida(male), Hanawa Matsumoto (2006) Submitted to PASJ, astro-ph/0609105

  5. Mona Lisa, Leonardo da Vinci (1503–1507) What is Sfumato • Sfumato originally denotes a painting technique developed by Leonardo da Vinci (1452-1519). • It was used by many painters in the Renaissance and Baroque. • The outline of an object becomes obscure and diffusive as it is located in dense gas. • Artists expressed AIR. • The code expresses GAS. • Sfumato=Smoky in Italian • NOT anagram of Matsumoto

  6. Block-structured grid Origin of AMR Most commonly used Enzo, ORION, RIEMANN, etc. Self-similar block-structured grid Commonly used FLASH, NIRVANA, SFUMATO, etc. Unstructured rectilinear grid (cell-by-cell grid) Also used in astrophysics Unstructured triangle grid Not used in astrophysics It takes advantage so that cells are fitted to boundaries/body (b) Self-similar block-structured (a) Block-structured (c) Unstructured rectilinear (d) Unstructured triangle Several types of AMR Level = 0 ～ 2

  7. (b) Self-similar block-structured (a) Block-structured (d) Unstructured triangle (c) Unstructured rectilinear AMR in astrophysics MHD and Self-gravity are implemented in many AMR codes

  8. Summary of implementation of Sfumato • Block structured AMR • Every block has same size in memory space. • Data is managed by the oct-tree structure. • Parallelized and vectorized (ordering via Peano-Hilbert space filling curve) • HD・MHD • Based on the method of Berger & Colella (1989) . • Numerical fluxes are conserved • Scheme: TVD, Roe scheme, predictor-corrector method (2nd order accuracy in time and space) • Cell-centered sheme • Hyperbolic cleaning of ∇・B(Dedner et al. 2002) • Self-gravity • Multi-grid method (FMG-cycle, V-cycle) • Numerical fluxes are conserved in FMG-cycle

  9. Conservation of numerical flux Flux conservation requires Flux on coarse cell surface = sum of four fluxes on fine cell surfaces Berger & Collela (1989) FH is modified for HD, MHD, and self-gravity Matsumoto & Hanawa (2003)

  10. Numerical results • Recalculation of our previous simulations • Binary formation (self-gravitational hydro-dynamics) Matsumoto & Hanawa (2003) • Outflow formation (self-gravitational MHD) Matsumoto & Tomisaka (2004) • Standard test problems • Fragmentation of an isothermal cloud (self-gravitational hydro-dynamics) • Double Mach reflection problem (Hydro-dynamics) • MHD rotor problem (MHD) • Convergence test of self-gravty

  11. Same model as Matsumoto & Hanawa (2003) Binary formation by AMR:Initial condition. Number of cells inside a block ＝　83 Isothermal gas Initial condition • Almost equilibrium • Slowly rotation • Non-magnetized • Small velocity perturbation of m = 3. • Isothermal gas 0.14 pc

  12. 30 AU Binary formation by AMR:The cloud collapses and a oblate first core forms Isothermal gas Number of cells inside a block ＝　83 Polytorpe gas

  13. 30 AU Binary formation by AMR:It deforms into a ring.

  14. 30 AU Binary formation by AMR:The ring begins to fragment.

  15. 30 AU Binary formation by AMR: A binary system forms. Spiral arm Close binary

  16. 30 AU Binary formation by AMR: A spiral arm becomes a new companion. Companion Close binary

  17. 30 AU Binary formation by AMR: A triplet system forms (last stage). Companion Close binary

  18. Binary formation by AMR: Zooming-out（1/2） 500 AU

  19. Binary formation by AMR: Zooming-out（2/2） 2000 AU

  20. Level 11 Level 12 Level 13 Same model as Matsumoto & Tomisaka (2004) Cloud collapse and outflow formationSelf-gravitational MHD Magnetic field lines Radial velocity Density distribution

  21. Fragmentation of a rotating isothermal cloud10% of bar perturbation, a = 0.26, b = 0.16 ORION: Truelove et al. (1998) SFUMATO: Matsumoto (2006) Level = 3 - 7

  22. Shock wave Wind Wall Double Mach reflection problem Level 0: h = 1/64 Level 1: h = 1/128 Level 2: h = 1/256 Level 3: h = 1/512 Level 4: h = 1/1024 density blocks

  23. MHD rotor problem B = 5 P = 1 r = 10, 1 W = 20 0.2 1 Crockett et al. (2005) Toth (2000) This work pressure

  24. Estimation of error of gravity for binary spheres Uniform spheres Level 3 Level 0 Convergence test changing number of cells inside a block as 23, 43, 83, 163,323 cells

  25. Convergence test of multi-grid method:2nd order accuracy • Source: binary stars • Maximum level = 4 • Distribution of blocks is fixed. • Number of cells inside a block is changed. ◇ level = 0 ○ level = 1 ◆ level = 2 ● level = 3 ■ level = 4 23 43 83 L2 norm of error of gravity 163 323/block Error ∝ hmax2 Cell width of the finest level

  26. Summary • A self-gravitational MHD AMR code was developed. • Block-structured grid with oct-tree data management • Vectorized and parallelized • Second order accuracy in time and space. • HD・MHD • Cell-centered, TVD, Roe’s scheme, predictor-corrector method • Hyperbolic cleaning of ∇・B • Conservation of numerical flux • Self-gravity • Multi-grid method • Conservation of numerical flux • Numerical results • Consistent with the previous simulations • Pass the standard test problems

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