General Relativistic MHD Simulations with Finite Conductivity

Shinji Koide (Kumamoto University)Kazunari Shibata (Kyoto University)

　　Takahiro Kudoh 　 (NA

OJ)

EANAM2006 @KASI,Daejeon, Korea, 2006.11.3(Fri)

Outline

• Numerical results of Ideal general relativistic MHD (ideal GRMHD) simulation: Jet formation by magnetic bridges between the ergosphere and disk around a rapidly rotating black hole. Anti-parallel magnetic field is formed along the jet Magnetic reconnection⇒

• Numerical method of GRMHD with finite conductivity (σGRMHD): Numerical algorithm and simple tests– Essential role of implicit method

Motivation: Relativistic Jets in the Universe

Mirabel, Rodriguez 1998

γ>100

AGN

γ>10 γ ～ 3~

~

Se

vera

l M ly

s

Se

vera

l lys

Gamma-ray burst

Forming SpinningBlack hole (?)

~ 1 km

~ 1AU

-rays

X-ray, optical,radio emission

~ Light years

Relativisticjet

Gravitationalcollapse

• Active galactic nuclei, Quasars: γ>10, Ljet~ several M pc

• Stellar mass black hole binaries (Microquasars): γ~ 3, Ljet ~ several pc

• Gamma-ray bursts: γ> 100, Ljet ~ 1AU-several pc

~

~

Acceleration of plasma/gasCollimation of plasma/gas outflow

1) Magnetic field

2) Radiation pressure

3) Gas pressure

Models: Relativistic Jets

The jet formation mechanism may be common. These relativistic jets are formed by drastic phenomena around black holes. However, the confirmed model has not yet shown.

Points ofmodels

Black Hole Magnetosphere

Black Hole Magnetosphere

（ Corona ）

Plasma Disk

BlackHole Ergosphere

Magnetic Field LinesMagnetic Field Lines

Plasma

Closed magnetic field linesbetween ergosphere and disk

Plasma

Magnetic Field induced by Current Loop around Black Hole

Plasma DiskBlackHole

Ergosphere

Magnetic Field LinesMagnetic Field Lines

Current loop

Magnetic bridgesMagnetic bridges

R0

Hayashi, Shibata, and Matsumoto (1996)

Nonrelativistic MHD Simulation with Dipole-Magnetic Field and Disk

Magneticbridge

Anomalous resistivity:

d 01.0 vJ/ρ d 0 vJ/ρ

Magnetic island(Plasmoid)

B

Frame-dragging

effect

Twist of magnetic bridge by ergosphere

Plasma

Ergosphere

Magnetic bridgeMagnetic bridge

Disk rotation

B

Twist by frame-dragging effect

RapidlyRotating

BlackHole

Current loop

Ideal General Relativistic Magnetohydrodynamics

To investigate dynamics of the magnetic bridge between the ergosphere and the disk, we have to consider the interaction of the plasma and magnetic field near the black hole. Simplest approximation for it is given by ideal GRMHD where electric conductivity σ is infinite (σ→∞).

(Ideal GRMHD)

general relativistic effect

Special relativistic total energy density

3+1 Formalism of Ideal GRMHD Equation 　　

σPfPTP

:)()]([ curv2

2

c

cDc

t

)]([ βv cDt

D

σTPβvP :)()]([ 222

cecDcct

)( BβEB

ct

Eβ

BE

βJctc

c 2e

1

0B E2e c

0BvE

where

(conservation of particle number)

(equation of motion)

(equation of energy)

(Maxwell equations)

(ideal MHD condition)

: (Lapse function),c

h iii : (shift vector)23

1

20

i

ii

c

hh

special relativistic effect

Special relativistic mass density,

Special relativistic total momentum density

No coupling with other Eqs.

～ similar to nonrelativistic ideal MHD(conservative form)

22 /ˆ cphch (equation of state)

Numerical Method• The ideal GRMHD equations are similar to those of

nonrelativistic ideal MHD. Therefore, we can use the numerical techniques developed for nonrelativistic MHD calculations. In this study, we use simplified TVD method.

• Simplified TVD method• This method is developed by Davis (1984) as a

simplest shock capturing scheme for hydrodynamics.• Merit: We don’t need eigen-vector of Jacobian matrix

of equations like primary TVD scheme. Just maximum of eigen-value of the Jacobian is used. It is easily applied for complex equations like GRMHD equations.

Results of Ideal GRMHD Simulations

Physical Review D 74, 044005 (Aug., 2006)

http://link.aps.org/abstract/PRD/v74/e044005

Solid white line: Magnetic field line

Color:

Initial condition of Ideal GRMHD simulation

t =0

Almost maximally rotating Black hole

logMagneticbridge

Disk: Keplerrotation

-4

-2

0

2

4

Corona: hydrostatic+background pressure

99995.0

maxJ

Ja

(Specific-heat ratio: )3/5

Ergosphere

-6

Condition of Ideal GRMHD simulation

-4

-2

0

2

4

-6

Axi

sym

met

ry

Mirror symmetry

HH 200006.1 rrr 2/01.0

( 210 × 70 mesh2 )

Solid white line: Magnetic field line

Color:

t =0

log

Calculation region:

Time evolution:Mass density, magnetic configuration

Solid white line: Magnetic field surface

Color:

Arrow: velocity

log

log

Solid line: Magnetic field surface

Color:

Arrow: Velocity

Solid line: Magnetic field line, Arrow: Velocity vmax : 0.4c - 0.6c

Mass density, velocity, magnetic pressure at S20t

cr /SS

log 2/log 2BMagnetic pressure,

-4

-2

0

2

4

-5

1

0

-1

-2

-3

-4

Solid line: Magnetic field line

Color: 　

Arrow: Velocity

vmax : 0.4c - 0.6c

Final stage of calculation ： Density, velocity, magnetic configuration

S110t

log

-4

-2

0

2

4

Solid line: Magnetic flux surfaceColor: 　Arrow: Velocity

Magnetic configuration of final stage:Numerical magnetic island

S110t

log

-4

-2

0

2

4Magnetic island(Plasmoid)

Ideal GRMHD:No magneticreconnection

• Magnetic Island:Numerical• Anti-parallel magnetic field

:Numerical

Schematic picture of phenomena caused by

the magnetic bridge near the black hole

Magnetic surface

Accretion disk

Ergosphere

Kerrblackhole

Kerrblackhole

Current loop

Magnetic surface

Accretion disk

Ergosphere

Magnetic bridge

Sub-relativisticjet

Initial

Ideal GRMHDresult

Summary of Results of Ideal GRMHD and Expected Phenomena beyond Ideal case

Kerrblackhole

Magnetic surface

Kerrblackhole

Flare of X-ray

Magneticreconnection

Accretion disk

Ergosphere Ergosphere

Accretion disk

heating

Intermittent Jet

Magnetic surface

Ideal GRMHDresult

GRMHDwith finite conductivity

Anti-parallel magnetic fieldis formed Mixture of hot and cool plasma:

Constant polytropic index EoSis not good approximation

Development of Numerical Method for GRMHD Simulation with Finite Conductivity

Fairly new topic. But no new results of physics.Only explanation of new required method and preliminary tests.

Previous GRMHD Simulations= ideal GRMHD with polytropic EoS

• Koide, Shibata, Kudoh 1999• Gammie 2003• DeVillier & Hawley 2003• Komissarov 2004• McKinney 2005

σ=∞ ，Γ=5/3, 4/3

This assumptionneglect astrophysicallyimportant effects

But no GRMHD simulation with finite conductivityand more appropriate EoS.

general relativistic effect

Special relativistic total energy density

GRMHD Equations with Finite Conductivity (σGRMHD)

σPfPTP

:)()]([ curv2

2

c

cDc

t

)]([ βv cDt

D

σTPβvP :)()]([ 222

cecDcct

)( BβEB

ct

Eβ

BE

βJctc

c 2e

1

0B E2c

(conservation of particle number)

(equation of motion)

(equation of energy)

(Maxwell equations)

(Ohm’s law with finite conductivity)

special relativistic effect

Special relativistic mass density,

Special relativistic total momentum density

vJvJBvE

2e2 11

c

conductivity

no correspondence to non-relativistic MHD

J

J

e

e

general relativistic effect

Special relativistic total energy density

GRMHD Equations with Finite Conductivity (σGRMHD)

σPfPTP

:)()]([ curv2

2

c

cDc

t

)]([ βv cDt

D

σTPβvP :)()]([ 222

cecDcct

)( BβEB

ct

Eβ

BE

βJctc

c 2e

1

0B E2c

(conservation of particle number)

(equation of motion)

(equation of energy)

(Maxwell equations)

(Ohm’s law with finite conductivity)

special relativistic effect

Special relativistic mass density,

Special relativistic total momentum density

J

J

e

e

vvEvBvEJ e2

1

c

βJEβ

BE

ccct e2

βJ ct ee

～ N. Watanabe & T. Yokoyama, ApJ 647, pp. L123-L126(astro-ph/0607285)

　 Electric conductivity → finite: Explicit(before improved EoS (Equation of State))

• Recalculation of dynamics of magnetic bridge with large conductivity (σ=100c2/τ)

　 Electric conductivity → finite: Implicit(before improved EoS (Equation of State))

• Recalculation of dynamics of magnetic bridge with large con

ductivity (σ=10,000c2/τ) Improved EoS (Electric conductivity: finite)• Recalculation of dynamics of magnetic bridge with large con

ductivity (σ=100c2/τ)

Numerical method ofσGRMHD: Tests

Solid line: Magnetic field line, Arrow: Velocity

Explicit method: Comparison of results of ideal and finite GRMHD simulations at (no anti-parallel magnetic field)

S18t

2/ cB

-4

-2

0

2

4

Color: S/100 cr

Ideal GRMHD: Finite :

Solid line: Magnetic field line, Arrow: Velocity

Explicit method: Comparison of results of ideal and finite GRMHD simulations at (no anti-parallel magnetic field)

S18t

-4

-2

0

2

4

S4 /10 cr

Ideal GRMHD: Finite :

Stop due to numericalinstability

01.0CFL

t

t

Solid line: Magnetic field line, Arrow: Velocity

Implicit method: Comparison of results of explicit and implicit methods with very large conductivity at

-4

-2

0

2

4

Color:σ=104/crS

01.0CFL

t

t

Explicit (ideal) Implicit (simplified) 2/ cB

S15t

― Comparison between different EoS’s ―

Ryu, Chattopadhyay, & Choi 2006

RP : Exact (Synge 1957)Γ=4/3, 5/3: Constant polytropic indexTM : Mignone et al (2005)RC : Ryu et al (2006)

p

ch 2

2/log cp

Equation of State (EoS)

3/4

3/5

Exact

improved

Solid line: Magnetic field line, Arrow: Velocity

Comparison of results of finite GRMHD simulations before/after improved EoS at (explicit)S20t

2/ cB

-4

-2

0

2

4

Color:

S/100 cr

Γ ＝５／３ (before improvement) Improved EoS (TM)

Summary• Ideal GRMHD:

– The magnetic bridges between the ergosphere and disk around rapidly rotating black hole can not be stationary and expand explosively to form a jet.

– The anti-parallel magnetic field is formed along the jet where the magnetic reconnection will take place, which may influence the jet propagation.

• GRMHD with finite conductivity (σGRMHD) is required to investigate the magnetic reconnection. We showed the new numerical method of σGRMHD and test calculations for it.– Implicit method is essential.

Near future plan

• Development of correct implicit σGRMHD code

• σGRMHD simulations of magnetic bridge between the ergosphere and disk around rapidly rotating black hole;Importance of magnetic reconnection in the mechanism of relativistic jet formation.