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Studies of the MRI with a New Godunov Scheme for MHD:. Jim Stone & Tom Gardiner Princeton University. Recent collaborators: John Hawley (UVa) Peter Teuben (UMd). Outline of Talk. Motivation Basic Elements of the Algorithm Some Tests & Applications Evolution of vortices in disks

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studies of the mri with a new godunov scheme for mhd

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

Jim Stone & Tom Gardiner

Princeton University

Recent collaborators:

John Hawley (UVa)

Peter Teuben (UMd)

outline of talk
Outline of Talk

Motivation

Basic Elements of the Algorithm

Some Tests & Applications

Evolution of vortices in disks

MRI, with comparison to ZEUS

slide3

Local simulation

Global simulation

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

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

slide4

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
e.g. photon bubble instability in accretion disk atmospheres

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:

Limitation is FLD

slide6

So why develop a new MHD code?

Global model of geometrically thin (H/R << 1) disk covering 10H in R, 10H in Z, and 2p in 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)

a variety of authors have combined ct with godunov schemes previously

B. Basic elements of the algorithm

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:
    • method by which EMFs are computed at corners.
    • calculation of PPM interface states in MHD
    • extension of unsplit integrator to MHD

Gardiner & Stone 2005

the ct 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
simple advection tests demonstrate differences
Simple advection tests demonstrate differences

Field Loop Advection (b = 106): MUSCL - Hancock

Arithmetic average Gardiner & Stone 2005

slide10

Directionally unsplit integration

CornerTransportUpwind [Colella 1991] (12 R-solves in 3D)

  • Optimally Stable, CFL < 1
  • Complex & Expensive forMHD...

CTU (6 R-solves)

  • Stable for CFL < 1/2
  • Relatively Simple...

MUSCL-Hancock

  • Stable for CFL < 1/2
  • Very Simple, but diffusive...
c some tests applications
C. Some Tests & Applications

Convergence Rate of Linear Waves

Nonlinear Circularly Polarized Alfven Wave

RJ Riemann problems rotated to grid

Advection of a Field Loop (Current Cylinder)

Current Sheet

Hydro and MHD Blast Waves

Hydro implosion

RT and KH instability

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

slide13

3D RT Instability (200x200x300 grid)

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

A test An application

Multi-mode in hydrodynamics Multi-mode in with strong B

codes are publicly available
Codes are publicly available

Code, documentation, and tests posted on web.

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
d evolution of vorticity in hydrodynamical shearing disk
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
slide16

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

Initial vorticity

(kx/ky)0 = 4

slide17

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

Johnson & Gammie 2005

Dimensionless time:

t = 1.5W t + kx0/ky

slide18

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

Trailing wave never seeds new leading waves

slide19

Evolution of specific vorticity

Initially W(k) ~ k-5/3

2D 5122 simulation

Box size: 4Hx4H

Evolved to t=100

y

x

slide20

Comparison of evolution in 2D and 3D

Wz initialized identically in 2D and 3D

Random Vz added in 3D

2D grid: 2562 (4Hx4H) 3D grid: 2562x64 (4Hx4Hx1H)

2D simulation

T = 0.5W t

slide24

T = 20W t

In 2D, long-lived vortices emerge

In 3D, vorticity and turbulence decays much more rapidly

evolution of ke 2d versus 3d
Evolution of KE: 2D versus 3D

NB: Initial evolution identical

evolution of stress 2d versus 3d
Evolution of Stress: 2D versus 3D

In 2D, long-lived vortices emerge

In 3D, vorticity (and KE and stress) decays much more rapidly

Further 3D runs in both stratified and unstratified boxes warranted

e mhd studies of the mri
E. MHD studies of the MRI

Start from a vertical field with zero net flux: Bz=B0sin(2px)

Sustained turbulence not possible in 2D – dissipation rate

after saturation is sensitive to numerical dissipation

2d mri
2D MRI

Animation of angular velocity fluctuations: dVy=Vy+1.5W0x

shows saturation of MRI and decay in 2D

3rd order reconstruction,

2562 Grid

bmin=4000, orbits 2-10

magnetic energy evolution zeus vs athena
Magnetic Energy EvolutionZEUS vs. Athena

Numerical dissipation is ~1.5 times smaller with

CTU & 3rd order reconstruction than ZEUS.

3d mri
3D MRI

Animation of angular velocity fluctuations: dVy=Vy+1.5W0x

Initial Field Geometry is Uniform By

CTU with 3rd order reconstruction,

128 x 256 x 128 Grid

bmin= 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

stress energy for b z 0
Stress & Energy for <Bz>  0

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

energy conservation
Energy Conservation

Saturation implies:

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

dependence of saturated state on cooling
Dependence of saturated state on cooling

Red line: no cooling; Green line:tcool = Q

Internal energy

Magnetic energy

Reynolds stress

Maxwell stress

Cooling has almost no effect except on internal energy

density fluctuations
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
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)

slide36

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

Stable layer

Unstable

Stable layer

z

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

summary
Summary
  • Developed 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
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