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Chad Meyer & Dinshaw Balsara
Core Collapse to Young Stellar Objects – UWO
May 19, 2010
15/19/2010 CC2YSO 2010
Outline Magnetorotational Instability (MRI) – regulates
accretion in disks
Previous large 3D simulations – effective α found to be resolution dependent
Imposing a physical dissipation limits the reachable Reynolds numbers in simulations
This work: Try to reach very high Reynolds numbers with higher order Godunov methodology
Quality of reconstruction plays a very important role
Riemann solver regulates the Magnetic Prandtl Number
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The physical problem depends on radiation flux, ionization fraction, vertical stratification, dust, etc.
Idealized problem presented here only depends on the Reynolds number and magnetic Prandtl number.
Astrophysical systems have high Reynolds numbers, and the plot below depicts the anticipation for large Reynolds numbers which motivates this work.
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Fromang, Papaloizou, Lesur, & Heinemann (2007)
Problems Fromang & Papaloizou (2007) report a very strong
dependence of α on numerical resolution.
For each doubling of resolution, α was halved.
Poses questions
Convergence?
Reliability of simulations?
Will a turbulent inertial range develop?
Is there dependence on numerical resolution?
On intuitive grounds, we expect that when the turbulence becomes scale free, α should be resolution-independent
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This work Simulate the MRI in 3D with a higher order Godunov
scheme using RIEMANN Cubic domain with 64 to 192 zones per scale height
Evaluate the effect of different algorithmic elements Slope limiters (MinMod, Monotone Centered, WENO)
Riemann solvers (HLLE, HLLD, Linearized)
Run for a sufficiently long time (>100 orbital periods: makes simulations difficult) Time averaged scalar quantities
Evaluate numerical dissipation of this scheme
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Algorithmic Elements
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Left state Right state
HLLE
HLLD
Linearized
Fast magnetosonic
Fast magnetosonic
Fast magnetosonic
Slow magnetosonic
Alfvén
Alfvén
t
t
t
x
x
x
Reconstruction Riemann Solver
MinMod, Monotone Centered and WENO
WENO will not clip extrema
MRI – Numerical Simulation Resolution study: 643 to 1923
(2883 in progress)
Prior studies at these resolutions only done with ZEUS; ATHENA at 128
Variety of reconstruction algorithms tried
Several different Riemann solvers used
Used RIEMANN code, which offers several algorithms and Riemann solvers
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θ
z
r
Local Shearing Box
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vr
Br
θ
z
r
vθ
B θ
vrvθ
Br Bθ
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Time history of α
Averaged from 40 orbits onward
Linearized RS, WENO reconstruction
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3.4 × 10-3 3.8 × 10-3
Balsara & Meyer 2010
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Balsara & Meyer 2010
Results – Numerical dissipation Spectral analysis of magnetic field equation
Take Fourier transform – build transfer functions
V
-V B shsh x t t t
d
dt y x
B By v B v B B v
2
div
1
2bb v bvA S T T T
t
B k
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Results – Numerical dissipation Spectral analysis of magnetic field equation
Take Fourier transform – build transfer functions
Should be zero (steady state turbulence)
V
-V B shsh x t t t
d
dt y x
B By v B v B B v
2
div
10
2bb v bvA S T T T
t
B k
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Results – Numerical dissipation Spectral analysis of magnetic field equation
Take Fourier transform – build transfer functions
Should be zero (steady state turbulence)
Numerical dissipation provided by the Riemann Solver
Can compare to a physical resistivity
V
-V B shsh x t t t
d
dt y x
B By v B v B B v
2
div
10
2bb v bv numA S T T T D
t
B k
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Comparison of Dissipation
Fromang & Papaloizou 2007 N=256Balsara & Meyer 2010, N=192
Linearized RS, WENO reconstruction
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Rem=3.4×104
Resolution trends of dissipation
16
Linearized RS, WENO reconstruction5/19/2010 CC2YSO 2010
Balsara & Meyer 2010
Energy Spectrum
Comparison of energy spectrum with expected Kolmogorov spectrum
A very small inertial range can be seen in the velocity spectrum at 1923
5/3k 5/3k
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Balsara & Meyer 2010
Magnetic Reynolds Number
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Balsara & Meyer 2010
Conclusions Higher order Godunov schemes have favorable properties
for simulating the MRI
Less dependence of α on resolution
More favorable dissipation characteristics
Can produce a high Reynolds number, needed to probe the asymptotic regime.
Higher resolutions will serve to shed more light on the issue.
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References Balsara and Meyer, 2010 (Submitted MNRAS; astro-ph/1003.0018 )
Fromang and Papaloizou, (2007) A&A 476, 1113-1122.
Fromang, Papaloizou, Lesur, & Heinemann, (2007) A&A 476, 1123-1132.
Simon, Hawley, and Beckwith, (2009) Ap.J. 690, 974-997
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