Review of• MHD simulations of accretion disks

• MHD simulations of disk winds & protostellar jets

Describe new Godunov+CT MHD Code

• Tests

• Application to MRI

MHD Models of Accretion Disks and Outflows

Jim StonePrinceton University

Goals

Challenges

• Understand angular momentum transport mechanism• Compute structure and evolution of accretion flows• Understand how disks produce jets

• Must be MHD from start• Multiple length and time scales (esp. for thin disks)• Adding additional physics (radiation, microphysics, etc.)• Curvilinear coordinates

Saturation of the MRI has been studying in small, local patches of the flow using the shearing box

Hawley, Gammie, & Balbus 1995; 1996; Brandenburg et al. 1995; Stone et al. 1996; Matsumoto et al. 1996; Miller & Stone 1999

The outcome is always MHD turbulence.

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Significant angular momentum transport is associated with MHD turbulence driven by the MRI

Also note: Sustained amplification of B indicates dynamo action

Time-evolution of volume-averaged quantities:

<> = 0.3<> = 0.07

Time in orbits

Current focus of studies using shearing box: adding more physics

•Protostellar disks• Ionisation fraction is so low, non-ideal MHD effects (Ohmic dissipation, ambipolar diffusion, Hall effect) must be included• Add dust

•Radiation dominated disks• Inner regions of BH disks are so hot that Prad >> Pgas. Does the saturation amplitude of the MRI depend on Prad , Pgas , or some combination of the two?

EXAMPLE: Radiation dominated disks:Studying this regime requires solving the equations of radiation MHD:

(Stone, Mihalas, & Norman 1992)

Linear growth rates of the MRI are changed by radiative diffusion (Blaes & Socrates 2001)

(Turner, Stone, & Sano 2002)

Linear growth rates make good code test

Density on faces of computational volume

(Stone & Pringle 2000; Hawley & Krolik 2001; Hawley, Balbus, & Stone 2001; Machida, Matsumoto, & Mineshige 2001)

3-D global models of geometrically thick (H/R ~ 1) black hole accretion disks demonstrate action of MRI (density over orbits 0 – 3):

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Latest models include full GR in Kerr metric

MHD models of outflows from disks

Density

Field lines

I. Outflows from sub-Keplerian disks

e.g. Uchida & Shibata

Don’t get steady flows

Studies can be classified based on initial/boundary conditions

II. Outflows from disks modeled as a boundary conditione.g. Ustyugova et al, Ouyed & Pudritz

d Bp d Vz

Disk is rotating plate at base of flow

Internal dynamics of disk and feedback not included

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III. Propagation of perfectly collimated jet (including cooling)

Toroidal B helps to keep jet collimated

images of log(d)

Structure of jet is assumed

IV. Stability of perfectly collimated, uniform jets

Temp

Density

Future directions for disk & wind models

• Global models of thin disks (requires cooling)

• Global models with more physics

• non-ideal MHD for protostellar disks

• radiation dominated disks

• Synthetic spectra computed from dynamical models

Outstanding issue: can we understand how jets are formed using global disk models?

These problems might benefit from improved methods

Global model of geometrically thin (H/R << 1) disk covering 10H in R, 10H in Z, and 2in azimuth with resolution of shearing box (128 grid points/H) will require nested grids.

Nested (and adaptive) grids work best with single-step Eulerian methods based on the conservative form

Algorithms in ZEUS are 15+ years old - a new code could take advantage of developments in numerical MHD since then.

Our Choice: higher-order Godunov methods combined with CT

Constrained Transport is a conservative scheme for the magnetic flux.

Difference using a staggered B and EMFs located at cell edges.

Appropriately upwinded EMFs must computed from face-centered fluxes given by Riemann solver.

Integrate the induction equation over cell face

using Stoke’s Law to give

Keeping div(B) = 0

A variety of previous authors have combined CT with Godunov schemes

• Ryu, Miniati, Jones, & Frank 1998

• Dai & Woodward 1998

• Balsara & Spicer 1999

• Toth 2000

• Londrillo & Del Zanna 2000

• Pen, Arras, & Wong 2003

However, scheme developed here differs in:• method by which EMFs are computed at corners.• extension of unsplit integrator to MHD.

Test 1. Convergence Rate of Linear WavesInitialize pure eigenmode for each wave family

Measure RMS error in U after propagating one wavelength quantitative test of accuracy of scheme

Cs = 1, VAx = 1, VAt = 3/2, Lx = Ly,x = y

1D2D

Test 2. Circularly Polarized Alfven Wave

= 1, P = 0.1, = 0.1, wave amplitude = 0.1 (Toth 2000)Lx = 2Ly, x = y , wave propagates at tan-1

Exact, nonlinear solution to MHD equations - quantitative test Subject to parametric instability (e.g. Del Zanna et al. 2001), but:

• Growth rate of perturbations must match dispersion relation• Growing modes should not be at grid scale

Animation of Bz

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Scatter plot showing all grid points - no parametric instability present

Test 3. RJ2a Riemann problem rotated to grid

Initial discontinuity inclined to grid at tan-1 Magnetic field initialized from vector potential to ensure div(B)=0

x = y, 512 x 256 grid

Final result plotted along horizontal line at center of grid

Lx = 2

Ly = 1UR

UL

Problem is Fig. 2a from Ryu & Jones 1995

P E

VxVy Vz

BxBy Bz

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Test 4. Hydrodynamical ImplosionFrom Liska & Wendroff; 400 x 400 grid,

P = 1

P = 0.125

Additional benefit of using unsplit integration scheme: Code maintains symmetry

Test 5. Spherical Blast Waves

Not a very quantitative test, BUT• check of whether blast waves remain spherical• late term evolution interesting

x = y, 400 x 600 grid, periodic boundary conditions

P = 0.1

LX = 1

LY = 1.5

P = 100 in r < 0.1

B at 45 degrees, = 0.1

HYDRO MHD

P = 0.1

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Hydrodynamic Blast Wave400 x 600 grid

MHD Blast Wave400 x 600 grid

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Test 6. Orszag-Tang vortex

512^2 grid, animation of d

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Test 7. RT Instability

• Check linear growth rates

• One of tests in Liska & Wendroff

(single mode in 2D)

• 200 x 600 grid d=2

d=1

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Single mode in 3D200x200x300 grid

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Multi-mode in 3D Multi-mode in 3D with strong B

Both 200x200x300 grids

A 2D Test/Application: MRIStart from vertical field with zero net flux Bz = B0 sin(2x)

Sustained turbulence not possible in 2D - rate of decay after saturation sensitive to numerical dissipation

X

Z

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Animation of angular velocity fluctuations V = Vy - 0 x shows saturations of MRI and decay in 2D

3rd order Roe scheme, 2562 grid, min = 4000, orbits 2 - 10.

(Numerical) dissipation of field is slower with 3rd order Roe fluxes than with ZEUS, by a factor of about 1.5.

Plot of B2 - B02 at various resolutions

2562

642

1282

ZEUS

Athena

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Vy on faces of volume

3D shearing box with Athena is also similar to ZEUS results(32x64x32 grid -- best working resolution 10 yrs ago)

Code is publicly available• Project is funded by NSF ITR; source code public.• Code, documentation, and training material posted on web.• 1D, 2D, and 3D versions are/will be available from

www.astro.princeton.edu/~jstone/athena.html

Future Extensions to Algorithm

• Curvilinear coordinates• Nested grids and adaptive grids

Testing and applications with fixed grid 3D version of code.

Current Focus of Effort