Studies of the MRI with a New Godunov Scheme for MHD:

Jim Stone & Tom GardinerPrinceton University

Recent collaborators:John Hawley (UVa)Peter Teuben (UMd)

Outline of Talk

A. Motivation

B. Basic Elements of the Algorithm

C. Some Tests & Applications

D. Evolution of vortices in disks

E. MRI, with comparison to ZEUS

Global simulation

Local simulation

Hawley, Gammie, & Balbus 1995; 1996; Stone et al. 1996; Armitage 1998; etc.

ZEUS-like algorithms have been used successfully to study the MRI in both 3D global and local simulations

These methods have been extended to study the radiation dominated inner regions of disks around compact objects. Equations of radiation MHD: (Stone, Mihalas, & Norman 1992)

Use ZEUS with flux-limited diffusion module (Turner & Stone 2001)

e.g. photon bubble instability in accretion disk atmospheres

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Gammie (1998) and Blaes & Socrates (2001) have shown magnetosonic waves are linearly unstable in radiation dominated atmospheres

Turner et al. (2004) have shown they evolve into shocks in nonlinear regime:

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Limitation is FLD

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) would be easier with nested grids.

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

MHD 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 Constrained Transport (CT)

So why develop a new MHD code?

A variety of authors have combined CT with Godunov schemes previously • 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:1. method by which EMFs are computed at corners.2. calculation of PPM interface states in MHD3. extension of unsplit integrator to MHD

Gardiner & Stone 2005

B. Basic elements of the algorithm

The CT Algorithm

• Finite Volume / Godunov algorithm gives E-field at face centers.

• “CT Algorithm” defines E-field at grid cell corners.

• Arithmetic averaging: 2D plane-parallel flow does not reduce to equivalent 1D problem

• Algorithms which reconstruct E-field at corner are superior Gardiner & Stone 2005

Ez,i 1 2,j 1

Ez,i 1 2,j

Ez,i , j 1 2 Ez,i 1, j 1 2

Ez,i 1 2,j 1 2

Simple advection tests demonstrate differences

Field Loop Advection (106): MUSCL - Hancock

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Arithmetic average Gardiner & Stone 2005

CornerTransportUpwind [Colella 1991] (12 R-solves in 3D)• Optimally Stable, CFL < 1

• Complex & Expensive for MHD...

CTU (6 R-solves)• Stable for CFL < 1/2

• Relatively Simple...

MUSCL-Hancock• Stable for CFL < 1/2

• Very Simple, but diffusive...

Directionally unsplit integration

C. Some Tests & Applications

1. Convergence Rate of Linear Waves2. Nonlinear Circularly Polarized Alfven Wave3. RJ Riemann problems rotated to grid4. Advection of a Field Loop (Current Cylinder) 5. Current Sheet 6. Hydro and MHD Blast Waves7. Hydro implosion8. RT and KH instability

See http://www.astro.princeton.edu/~jstone/tests for more tests.

Linear Wave Convergence(2N x N x N) Grid

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A test An applicationMulti-mode in hydrodynamics Multi-mode in with strong B

3D RT Instability (200x200x300 grid)

Goal: measure growth rate of fingers, structure in 3D

Codes are publicly available

Download a copy from

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

Current status:• 1D version publicly available (C and F95)• 2D version publicly available (C and F95)• 3D version being tested on applications (C only)

Expect to release parallelized 3D Cartesian grid code in ~1yr

Code, documentation, and tests posted on web.

D. Evolution of vorticity in hydrodynamical shearing disk

• HBS 1996 showed nonlinear random motions decay in hydrodynamical shearing box using PPM • Recent work has renewed interest in evolution of vortices in disks, especially transit amplification of leading->trailing waves 2D incompressible

Umurhan & Regev 2004; Yecko 2004 2D compressible

Johnson & Gammie 2005 3D anelastic in stratified disks

Barranco & Marcus 2005

Our goal: Study evolution of vortices in 3D compressible disks at the highest resolutions possible, closely following JG2005

A test: shear amplification of a single, leading, incompressible vortex

Initial vorticity(kx/ky)0 = 4

Dimensionless time: = 1.5t + kx0/ky

KE is amplified by ~ (kx/ky)2 : can code reproduce large amplification factors?

Johnson & Gammie 2005

There is no aliasing at shearing-sheet boundaries using this single-step unsplit integrator

Trailing wave never seeds new leading waves

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Evolution of specific vorticityInitially W(k) ~ k-5/3

2D 5122 simulationBox size: 4Hx4HEvolved to =100

y

x

T = 0.5t

2D simulation

Wz initialized identically in 2D and 3DRandom Vz added in 3D2D grid: 2562 (4Hx4H) 3D grid: 2562x64 (4Hx4Hx1H)

Comparison of evolution in 2D and 3D

T = 2t

T = 5t

T = 10t

T = 20t

In 2D, long-lived vortices emergeIn 3D, vorticity and turbulence decays much more rapidly

Evolution of KE: 2D versus 3D

NB: Initial evolution identical

Evolution of Stress: 2D versus 3D

In 2D, long-lived vortices emergeIn 3D, vorticity (and KE and stress) decays much more rapidlyFurther 3D runs in both stratified and unstratified boxes warranted

E. MHD studies of the MRI

Start from a vertical field with zero net flux: Bz=B

0sin(2x)

Sustained turbulence not possible in 2D – dissipation rateafter saturation is sensitive to numerical dissipation

2D MRIAnimation of angular velocity fluctuations: V

y=V

y+1.5

0x

shows saturation of MRI and decay in 2D

3rd order reconstruction,2562 Grid

min=4000, orbits 2-10

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Magnetic Energy EvolutionZEUS vs. Athena

Numerical dissipation is ~1.5 times smaller withCTU & 3rd order reconstruction than ZEUS.

3D MRIAnimation of angular velocity fluctuations: V

y=V

y+1.5

0x

Initial Field Geometry is Uniform By

CTU with 3rd order reconstruction, 128 x 256 x 128 Grid

min= 100, orbits 4-20

Goal: Since Athena is strictly conservative, can measure spectrum of T fluctuations from dissipation of turbulence

Requires adding optically-thin radiative cooling

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Stress & Energy for <Bz> 0

No qualitative difference with ZEUS results (Hawley, Gammie, & Balbus 1995)

Energy Conservation

Saturation implies:

Previously observed for ReM = 1 (Sano et al. 2001).

Dependence of saturated state on coolingRed line: no cooling; Green line: cool = Q

Cooling has almost no effect except on internal energy

Internal energy

Reynolds stress

Magnetic energy

Maxwell stress

Density Fluctuations

• Compressible fluctuations lead to spiral waves & M ~ 1.5 shocks• Temperature fluctuations are dominated by compressive waves.• Will this be a generic result in global disks?

Going beyond MHD

If mean-free-path of particles is long compared to gyroradius, anisotropic transport coefficients can be important (Braginskii 1965)

Generically, astrophysical plasmas are diffuse, and kinetic effects may be important in some circumstances

These effects can produce qualitative changes to the dynamics:e.g. with anisotropic heat conduction, the convective

instability criterion becomes dT/dz < 0 (Balbus 2000)

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Evolution of field lines in atmosphere with dS/dz > 0 and dT/dz < 0 including anisotropic heat conduction,

Details: 2D 1282 , initial =100, Bx proportional to sin(z), fixed T at vertical boundaries, isotropic conduction near boundaries

Stable layer

Unstable

Stable layer

z

SummaryDeveloped a new Godunov scheme for MHD

Code now being used for various 3D applications• Evolution of vortices in disks• Energy dissipation in MRI turbulence

Further algorithm development is planned• Curvilinear grids• Nested and adaptive grids

Future applications• 3D global models of thin disks• Fragmentation and collapse in self-gravitating MHD turbulence

Beyond ideal MHD• Kinetic effects in diffuse plasmas• Particle acceleration (CRs are important!)• Dust dynamics in protoplanetary disks

A. Astrophysical motivation• Accretion disks• Star formation• Protoplanetary disks• X-ray clusters

Numerical challenges• 3D MHD• Non-ideal MHD• Radiation hydrodynamics (Prad >> Pgas)• Nested and adaptive grids• Kinetic MHD effects