SFUMATO A selfgravitational MHD AMR code Tomoaki Matsumoto

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

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

Introduction: From a cloud core to a protostar N O I T U L O S E Protostar, R Molecular cloud core H G protoplanetary disk I H Y L and outflow E Gravitational M E R collapse T B X EB MUL TI-S CAL 0. 1 – 0. 01 pc E SI MUL ATIO 1 AU/0. 1 pc = 5× 10 N 1 -10 AU -5 First core ⇒ Second core ⇒ CTTS ⇒ WTTS ⇒ Main sequence Protostar 100 - 1000 AU

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

What is Sfumato originally denotes a painting technique developed by Leonardo da Vinci (14521519). o It was used by many painters in the Renaissance and Baroque. o The outline of an object becomes obscure and diffusive as it is located in dense gas. o Artists expressed AIR. o The code expresses GAS. o Sfumato = Smoky in Italian o NOT anagram of Matsumoto Mona Lisa, Leonardo da Vinci (1503– 1507)

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

AMR in astrophysics Grid type MHD and Self-gravity are implemented in many AMR codes MHD Selfgravity Dark Matter Radiative transfer Code name Author(s) Main targets ORION R. Klein Star formation (a) Y Y N Y Enzo M. Norman Cosmology (a) Y Y Y N FLASH ASC/U-Chicago Any (b) Y Y (Y) BAT-R-US K. G. Powell Space weather (b) Y Y N N NIRVANA U. Ziegler Any (b) Y Y N N RIEMANN D. Balsara ISM (a) Y Y N N RAMSES R. Teyssier Cosmology (c) Y Y Y N ? M. A. de Avillez ISM (b) Y N N N VPP-AMR H. Yahagi Cosmology (c) N Y Y N SFUMATO T. Matsumoto Star formation (b) (c) Unstructured rectilinear (a) Block-structured Y Y (b) Self-similar blockstructured (d) Unstructured triangle N N

Summary of implementation of Sfumato o Block structured AMR n n n o HD・MHD n n n o 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) 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 n n Multi-grid method (FMG-cycle, V-cycle) Numerical fluxes are conserved in FMG-cycle

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

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

Same model as Matsumoto & Hanawa (2003) Binary formation by AMR: Initial condition. Number of cells inside a block　 ＝　83 Isothermal gas 0. 14 pc Initial condition n Almost equilibrium n Slowly rotation n Non-magnetized n Small velocity perturbation of m = 3. n Isothermal gas

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

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

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

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

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

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

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

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

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

Fragmentation of a rotating isothermal cloud 10% of bar perturbation, a = 0. 26, b = 0. 16 ORION: Truelove et al. (1998) SFUMATO: Matsumoto (2006) Level = 3 - 7

Double Mach reflection problem Shock wave Wind Wall 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

MHD rotor problem 0. 2 Toth (2000) 1 B=5 P=1 r = 10, 1 W = 20 Crockett et al. (2005) This work pressure

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

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

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