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.