1. Tomoaki Matsumoto (Hosei Univ. / NAOJ) Masahiro N. Machida (NAOJ)
Self-Gravitational Collapse of a Magnetized Cloud Core: High Resolution Simulation with Thee-dimensional MHD Nested Grid
Tomoaki Matsumoto (Hosei Univ. / NAOJ)
Masahiro N. Machida (NAOJ)
Koji Tomisaka (NAOJ)
Tomoyuki Hanawa (Chiba Univ.)
Introduction and Overview of Nested Grid
Application to Astronomical Problems
Outflow Formation in collapsing cloud
(single star formation)
Fragmentation of collapsing cloud
(binary star formation)
Methods
MHD
Self-gravity

2. Introduction: Scenario of Single Star Formation
H13CO+　core
Orion molecular cloud
Optical + radio image
Molecular cloud core
Radio image
Young star and outflow
Near-infrared image
Molecular Cloud Core
0.1pc
Mass Accretion,
and Outfow
Single Star
5×10-5
1AU
This work
High dynamic range!
Important physics
Self-gravity of gas cloud
Thermal pressure of gas
Interstellar magnetic field

3. Introduction: Scenario of Binary Star Formation
Molecular Cloud Core
Mass Accretion
and Outflow
Fragmentation
0.1pc
Binary Stars
5×10-5
This work
High dynamic range!
1AU

4. Introduction of Nested Grid
Nested grid is a cheap edition of AMR B-)
AMR is somewhat expensive for nearly spherically collapsing gas cloud.
Cell width Dxl = 1/2 Dxl-1
Subgrids are formed automatically according to physical conditions.
Location of the subgrid is fixed.
Schemes：
Roe method for HD and MHD
Multigrid method for selfgravity.
Numerical fluxes are conserved in grid-interfaces, surfaces between fine and coase grids.

5. Application 1 Outflow from a Rotating Collapsing Cloud (1) Model Construction
Matsumoto & Tomisaka (2003)
Rigid rotation
Gravitational collapse of magnetized rotating cloud
Initial condition
Spherical cloud
Uniform magnetic field
Rigid rotation
Isothermal gas (γ=1) is assumed for low density, and polytrope gas (γ=5/3) for high density. This mimics formation of a seed of a stellar core.
Ideal MHD
Ambipolar diffusion (skiped in this talk) q
Uniform Magnetic Field

6. Outflow from a Collapsing Cloud (2) Mechanism of Outflow
Magneto-centrifugal driven outflow (Blandford & Peyne 1982)
standard mechanism of outflow
Velocity of outflow
M=10
Outflow
Centrifugal Force
Fc ・ Bp/B
Gas disk rotates faster than ambient gas because central region spins up by collapse.
Magnetic field is twisted.
Centrifugal force launch gas

7. Application 2 Fragmentation of a Collapsing Cloud (1)
This is a very traditional but important problem for binary formation.
Almost stars (~70%) are formed as binary stars.
We show the first MHD numerical simulation in this problem.
In almost simulations, magnetic field has been neglected.
The initial condition is similar to the previous model of q=0, but includes the bar-mode perturbation.
Bar-mode perturbation promotes fragmentation of a cloud core.

8. Fragmentation of a Collapsing Cloud (2) non MHD model
Matsumoto & Hanawa(2003)
Collapse of the cloud
Fragmentation
Fragments rotate around each other.
Circum binary disk forms in late stage.
No outflow forms in non magnetized model.
Disk-bar model

9. Fragmentation of a Collapsing Cloud (3) MHD model
Each fragment ejects outflow.
Stream lines:
magnetic field
Arrows:
velocity

10. Implementation of Nested Grid Code MHD part
Numerical Flux is obtained by Roe’s method (Fukuda & Hanawa 2000).
Linearlized Reman solver.
2nd order accuracy in space by MUSCL
2nd order accuracy in time by predictor-corrector method.
Multi-timestep (c.f., standard AMR)
Numerical flux is conserved in grid interface by means of flux correction.
Two approach to div B:
div B free using staggered mesh (Balsara 2001)
div B cleaning (Dender et al. 2002)
With either methods, low b regions of b = are solved stably.

11. MHD part Multi timestep (1)dt
dt/2
dt/4 l
High resolution in time in sub-grid
Dtl = Min( Dtl-1 , CFL Dx / v)
The fine grid takes two or more steps while the coarse grid does one step.
This ensure that CFL condition is satisfied in every grid.

12. Self-gravity part Overview of multigrid method
Poisson equation is solved by multigrid iteration in nested grid (Matsumoto & Hanawa 2003 ApJ)
Scalability
Computation time ∝ Number of cell (=Nx×Ny×Nz×Nl)
Fast convergence
Residual is reduces by factor in each iteration.
Good solution
Numerical flux (gravity) is conserved at grid interfaces.
Solution satisfies Gauss’s and Stokes’ laws.
Important feature as mass conservation in astrophysics.
High accuracy
2nd order accuracy except for cells in grid interaces.
1st order accuracy for these cells.

13. Introduction of Multigrid method Multigrid method @ uniform grid
Fine
Ｓ =Gauss-Seidel
／= Interpolation
＼= Restriction
coarse
interpolation: Good initial guess
A few times of GS iterations converge the sol.
Number of cells
Real grid
Working grids
Number of operatons ∝ Number of cells in real grid

14. Multigrid method @ Nested Grid Straight forward extension
Real grid
Working grids
Number of operations ∝ Number of cell in a grid×Number of level

15. Multigrid method: Flux Conservation
This may be common sense in plasma physics, but not in astrophysics.
Poisson equation should be solved in the every grid level simultaneously and consistently.
Solve every grid simultaneously 
Solve each grid separately  + -
coarse grid
g≠0
fine grid + -
coarse grid
g=0
fine grid

16. Multigrid method: Flux Conservation (cont.)
Gauss-Seidel iteration
We pay special attention to Gauss-Seidel iteration (smoothing operator).
Flux = gravity, g = ∇y
Flux in coarse grid is obtained by sum of fluxes in fine grid at grid interface similar to HD method.
This ensures
Continuity of field lines of g
Gauss’s and Stokes’ laws
Important in astronomical simulation + =

17. Multigrid method: Fast convergence： Residual reduces by factor 10-2 – 10-3 in each FMG iteration.
Residuals are evaluated
in every level.
log (Maximum residual)
Round off error
Number of FMG

18. Multigrid method: Scalability: computation cost ∝ number of cell in comp domain
Computation time is in proportion to total number of cell in the computation domain, and independent of number of grid level.

19. Summary
High resolution, MHD, self-gravitational simulations are performed by using nested grid. The nested grid is a powerful method for astrophysical simulations.
The nested grid achieves high dynamic range.
MHD code solves even low beta region, b=10-6, stably.
Multigrid method for self-gravity shows high perfomance (fast convergence, scalably, and flux conservation). Especially, the solution satisfies Gauss’s and Stokes’ laws.