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

2. Introduction: From a cloud to a protostar H 13 CO + core Orion molecular cloud （ optical ＋ radio ） Molecular cloud core in Taurus （ radio ） Outflow and Protostar (radio)

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

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

5. 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 Mona Lisa, Leonardo da Vinci (1503–1507)

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

7. AMR in astrophysics MHD and Self-gravity are implemented in many AMR codes Code nameAuthor(s)Main targetsGrid typeMHD Self- gravity Dark Matter Radiative transfer ORIONR. KleinStar formation (a)YYNY EnzoM. NormanCosmology (a)YYYN FLASHASC/U-ChicagoAny (b)YY( Y ) BAT-R-USK. G. PowellSpace weather (b)YYNN NIRVANAU. ZieglerAny (b)YYNN RIEMANND. BalsaraISM (a)YYNN RAMSESR. TeyssierCosmology (c)YYYN ?M.A. de AvillezISM (b)YNNN VPP-AMRH. YahagiCosmology (c)NYYN SFUMATOT. MatsumotoStar formation (b)YYNN (a) Block-structured (b) Self-similar block- structured (c) Unstructured rectilinear (d) Unstructured triangle

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 (2 nd 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 F H is modified for HD, MHD, and self-gravity Berger & Collela (1989) 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. Binary formation by AMR: Initial condition. 0.14 pc Number of cells inside a block ＝ 8 3 Initial condition Almost equilibrium Slowly rotation Non-magnetized Small velocity perturbation of m = 3. Isothermal gas Same model as Matsumoto & Hanawa (2003)

12. Binary formation by AMR: The cloud collapses and a oblate first core forms 30 AU Number of cells inside a block ＝ 8 3 Isothermal gas Polytorpe gas

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

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

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

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

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

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

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

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

21. Fragmentation of a rotating isothermal cloud 10% of bar perturbation,  = 0.26,  = 0.16 ORION: Truelove et al. (1998) SFUMATO: Matsumoto (2006) Level = 3 - 7

22. Double Mach reflection problem Wall Wind Shock wave density blocks 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

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

24. Estimation of error of gravity for binary spheres Convergence test changing number of cells inside a block as 2 3, 4 3, 8 3, 16 3,32 3 cells Uniform spheres Level 0 Level 3

25. Convergence test of multi-grid method: 2 nd order accuracy  Source: binary stars  Maximum level = 4  Distribution of blocks is fixed.  Number of cells inside a block is changed. Error ∝ h max 2 ◇ level = 0 ○ level = 1 ◆ level = 2 ● level = 3 ■ level = 4 32 3 /block 16 3 8383 4343 2323 Cell width of the finest level L2 norm of error of gravity

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