Chad Meyer & Dinshaw Balsara

Core Collapse to Young Stellar Objects – UWO

May 19, 2010

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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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