review of mhd simulations of accretion disks mhd simulations of disk winds & protostellar jets...
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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
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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
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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
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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
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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?
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EXAMPLE: Radiation dominated disks:Studying this regime requires solving the equations of radiation MHD:
(Stone, Mihalas, & Norman 1992)
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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
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Density on faces of computational volume
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(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
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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
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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
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IV. Stability of perfectly collimated, uniform jets
Temp
Density
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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?
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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
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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
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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.
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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
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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
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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
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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
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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
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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.
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(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)
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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