SFUMATO: SFUMATO: A self-gravitational MHD AMR codeA self-gravitational MHD AMR code

Tomoaki Matsumoto

（ Hosei Univerisity ）Circumstellar disk

Outflow

Magnetic field

Protostar

Computational domain is1,000 times larger.

Matsumoto (2006) Submitted to PASJ, astro-ph/0609105

Introduction:Introduction:From a cloud to a protostarFrom a cloud to a protostar

H13CO+ 　core

Orion molecular cloud（ optical ＋ radio ）

Molecular cloud corein Taurus （ radio ）

Outflow and Protostar(radio)

Introduction:Introduction:From a cloud core to a protostarFrom a cloud core to a protostar

B

0.1 – 0.01 pc

Gravitationalcollapse B

Molecular cloud core Protostar, protoplanetary diskand 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

MULTI-SCALE SIMULATION

EXTREMELY HIGH-RESOLUTION

EXTREMELY HIGH-RESOLUTION

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

What is SfumatoWhat is Sfumato Sfumato originally denotes a

painting technique developed by Leonardo da Vinci (14521519).

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)

Several types of AMRSeveral 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, e

tc.

(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 ar

e fitted to boundaries/body

Level = 0 ～ 2

(a) Block-structured (b) Self-similar block-structured

(c) Unstructured rectilinear

(d) Unstructured triangle

AMR in astrophysicsAMR in astrophysicsMHD and Self-gravity areimplemented in many AMR codes

Code name Author(s) Main targets Grid type MHDSelf-gravity

Dark Matter

Radiative transfer

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 ) ( 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) Y Y N N

(a) Block-structured (b) Self-similar block-structured

(c) Unstructured rectilinear

(d) Unstructured triangle

Summary of implementation of SfumatoSummary 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 curv

e)

HD ・ MHD Based on the method of Berger & Colella (1989) . Numerical fluxes are conserved Scheme: TVD, Roe scheme, predictor-corrector method (2nd order accur

acy 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

Conservation of numerical fluxConservation of numerical flux

cyclesub surface

hhhHHH tSFtSF

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 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 hydr

o-dynamics) Double Mach reflection problem (Hydro-dynamics) MHD rotor problem (MHD)

Convergence test of self-gravty

Binary formation by AMR:Binary formation by AMR:Initial condition. Initial condition.

0.14 pc

Number of cells inside a block 　＝　 83

Initial condition Almost equilibrium Slowly rotation Non-magnetized Small velocity perturbation of m = 3. Isothermal gas

Isothermal gas

Same model as Matsumoto & Hanawa (2003)

Binary formation by AMR:Binary formation by AMR:The cloud collapses and a oblate first core formsThe cloud collapses and a oblate first core forms

30 AU

Number of cells inside a block 　＝　 83

Isothermal gas

Polytorpe gas

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

30 AU

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

30 AU

Binary formation by AMR: Binary formation by AMR: A binary system forms.A binary system forms.

30 AU

Spiral arm

Close binary

Binary formation by AMR: Binary formation by AMR: A spiral arm becomes a new companion.A spiral arm becomes a new companion.

30 AU

Companion

Close binary

Binary formation by AMR: Binary formation by AMR: A triplet system forms (last stage).A triplet system forms (last stage).

30 AUClose binary

Companion

Binary formation by AMR: Binary formation by AMR: Zooming-outZooming-out （（ 1/21/2 ））

500 AU

Binary formation by AMR: Binary formation by AMR: Zooming-outZooming-out （（ 2/22/2 ））

2000 AU

Cloud collapse and outflow formationCloud collapse and outflow formationSelf-gravitational MHDSelf-gravitational MHD

Density distribution

Magnetic field lines

Radial velocity

Level 11

Level 12

Level 13

Same model as Matsumoto & Tomisaka (2004)

Fragmentation of a rotating isothermal cloudFragmentation of a rotating isothermal cloud10% of bar perturbation, 10% of bar perturbation, = 0.26, = 0.26, = 0.16 = 0.16

ORION: Truelove et al. (1998) SFUMATO: Matsumoto (2006)

Level = 3 - 7

Double Mach reflection problemDouble Mach reflection problem

Wall

Wind

Shock wave

density

blocks

Level 0: h = 1/64Level 1: h = 1/128Level 2: h = 1/256Level 3: h = 1/512Level 4: h = 1/1024

MHD rotor problemMHD rotor problem

Toth (2000) Crockett et al. (2005) This work

B = 5P = 1= 10, 1 = 20

10.2

pressure

Estimation of error of gravity Estimation of error of gravity for binary spheresfor binary spheres

||/||log exex ggg

Convergence testchanging number of cells inside a block as23, 43, 83, 163,323 cells

Uniform spheres

Level 0 Level 3

Convergence test of multi-grid method:Convergence test of multi-grid method:22ndnd order accuracy order accuracy

Source: binary stars Maximum level = 4 Distribution of

blocks is fixed. Number of cells

inside a block is changed.

Error h∝ max2

◇ level = 0○ level = 1

◆ level = 2● level = 3■ level = 4

323/block

163

83

43

23

Cell width of the finest level

L2

no

rm o

f err

or o

f gra

vity

SummarySummary 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