Download General Relativistic MHD Simulations with Finite Conductivity

General Relativistic MHD Simulations
with Finite Conductivity
Shinji Koide (Kumamoto University)
Kazunari Shibata (Kyoto University)
Takahiro Kudoh
(NAOJ)
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
Gamma-ray burst
AGN
X-ray,
optical,
radio
emission
Mirabel, Rodriguez 1998
g-rays
γ～ 3
γ>100
~
Several lys
Several M lys
γ>10
~
~ Light
years
~ 1AU
Relativistic
jet
~ 1 km
Gravitational
collapse
Forming Spinning
Black hole (?)
Models: Relativistic Jets
• 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
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 of Acceleration of plasma/gas
Collimation of plasma/gas outflow
models
1) Magnetic field
2) Radiation pressure
3) Gas pressure
Black Hole Magnetosphere
（Corona）
Closed magnetic field lines
between ergosphere and disk
Magnetic Field Lines
Plasma Disk
Black
Hole
Ergosphere
Plasma
Black Hole Magnetosphere
Magnetic Field induced by Current Loop
around Black Hole
Magnetic Field Lines
Magnetic bridges

Black
Hole
Plasma Disk
Current loop
R0
Ergosphere
Plasma
Nonrelativistic MHD Simulation
with Dipole-Magnetic Field and Disk
Magnetic
bridge
Hayashi, Shibata,
and Matsumoto
(1996)
Anomalous resistivity:
  0.01
 0
J/ρ  vd 
J/ρ  vd 
Magnetic island
(Plasmoid)
Twist of magnetic bridge by ergosphere
B
Twist by
frame-dragging
effect
Magnetic bridge
B
Rapidly
Rotating
Black
Current loop
Hole Frame-dragging
Ergosphere
effect
Plasma
Disk rotation
Ideal General Relativistic
Magnetohydrodynamics
(Ideal GRMHD)
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→∞).
3+1 Formalism of Ideal GRMHD Equation
～ similar to nonrelativistic ideal MHD
(conservative form)
(conservation of particle number)
Special relativistic mass density, g
D
   [D( v  cβ)]
t
general relativistic effect
Special relativistic total momentum density
P
 

   [ (T  cP)]   D  2 (c 2 )  f curv  P : σ
t
c 

Special relativistic total energy density
(equation of motion)
special relativistic effect

(equation of energy)
   [ (c 2 P  Dc 2 v  ecβ)]  ( )  c 2 P  T : σ
t
No coupling
 
1 E
β

B
J  e cβ  2
     B   E  with other Eqs.
    (E  cβ  B)
c t
c
t

 
e 
B  0
E  vB  0

h  c 2 hˆ p / c 2
c
2
2
0
E
(Maxwell equations)
(ideal MHD condition)
(equation of state)

 h 
  h   i i 
i 1  c 
3
where

2
: (Lapse function),
i 
hii
: (shift vector)
c
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
Initial condition of Ideal GRMHD simulation
4
Corona: hydrostatic
2
t =0
+background pressure
(Specific-heat ratio:
  5/3 )
Ergosphere
Magnetic
bridge
Disk: Kepler
rotation
0
Solid white line:
Magnetic field line
-2
Color:
log 
-4
-6


J
Almost maximally rotating Black hole  a  J  0.99995 
max


Condition of Ideal GRMHD simulation
Axisymmetry
4
Calculation region:
1.006rH  r  200rH
0.01     / 2
2
t =0
0
Solid white line:
Magnetic field line
-2
Color:
( 210 × 70 mesh2 )
-4
Mirror symmetry
-6
log 
Time evolution:
Mass density, magnetic configuration
Solid white line:
Magnetic field surface
Color:
log 
Arrow: velocity
Solid line:
Magnetic field surface
Color:
log 
Arrow: Velocity
Mass density, velocity,
magnetic pressure at t  20 S
log 
Magnetic pressure, log B / 2
2

1
4
0
2
-1
0
-2
-3
-2
-4
-4
-5
Solid line: Magnetic field line, Arrow: Velocity
vmax : 0.4c - 0.6c
 S  rS / c
Final stage of calculation：Density,
velocity, magnetic configuration
t  110 S
4
2
Solid line:
Magnetic field line
0
Color:
-2
Arrow: Velocity
-4
log 
vmax : 0.4c - 0.6c
Magnetic configuration of final stage:
Numerical magnetic island
t  110 S
Magnetic island
(Plasmoid) :
Numerical
4
2
0
-2
Ideal GRMHD:
No magnetic
reconnection
• Magnetic Island:
Numerical
• Anti-parallel
magnetic field
-4
Solid line:
Magnetic flux surface
Color: log 
Arrow: Velocity
Summary of Results of Ideal GRMHD and
Expected Phenomena beyond Ideal case
Initial
Magnetic bridge
Current
loop
Magnetic surface
Accretion disk
Kerr
black
hole
Ergosphere
Sub-relativistic
jet
Magnetic surface
Kerr
black
hole
Accretion disk
Ergosphere
Ideal GRMHD
result
Schematic picture of phenomena caused by
the magnetic bridge near the black hole
Ideal GRMHD
result
GRMHD
with finite conductivity
Intermittent Jet
Magnetic surface
Flare of X-ray
Magnetic surface
Magnetic
reconnection
Accretion disk
Kerr
black
hole
Kerr
black
hole
Ergosphere
Anti-parallel magnetic field
is formed
Accretion disk
heating
Ergosphere
Mixture of hot and cool plasma:
Constant polytropic index EoS
is 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 assumption
neglect
astrophysically
important effects
But no GRMHD simulation with finite conductivity
and more appropriate EoS.
GRMHD Equations with Finite Conductivity
(σGRMHD)
Special relativistic mass density, g
D
(conservation
   [D( v  cβ)]
t
general relativistic effect
Special relativistic total momentum density
P
 

   [ (T  cP)]   D  2 (c 2 )  f curv  P : σ
t
c 

Special relativistic total energy density
of particle number)
(equation of motion)
special relativistic effect

(equation
   [ (c 2 P  Dc 2 v  ecβ)]  ( )  c 2 P  T : σ
t
 
1 E
β

B



J

c
β





B


E



    (E  cβ  B)
e

2
c t
c
t

 
B  0
e 

2
E
of energy)
(Maxwell
equations)
no correspondence to non-relativistic MHD
1 
1
 
2


E  vB 
J

g


v

J
 e
v
(Ohm’s law with finite conductivity)

2
g 
c

 
conductivity
c
GRMHD Equations with Finite Conductivity
(σGRMHD)
Special relativistic mass density, g
D
(conservation
   [D( v  cβ)]
t
general relativistic effect
Special relativistic total momentum density
P
 

   [ (T  cP)]   D  2 (c 2 )  f curv  P : σ
t
c 

Special relativistic total energy density
of particle number)
(equation of motion)
special relativistic effect

(equation of energy)
   [ (c 2 P  Dc 2 v  ecβ)]  ( )  c 2 P  T : σ
t
 1 E β  
E
β  cβ 
B





B


E


J







J

c
β





B

 Ee 


    (E  cβ  B)
2

2
e


t
c
c




c
c t
t
 (Maxwell



e
 e 2 
equations)
B  0

 JE  e cβ 
c
t
1


J  g E  v J B  2 v  E v    e v
(Ohm’s law with finite conductivity)
c


～ N. Watanabe & T. Yokoyama, ApJ 647, pp. L123-L126
(astro-ph/0607285)
Numerical method ofσGRMHD: Tests
 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
conductivity (σ=10,000c2/τ)
 Improved EoS (Electric conductivity: finite)
• Recalculation of dynamics of magnetic bridge with large
conductivity (σ=100c2/τ)
Explicit method: Comparison of results of ideal and
finite  GRMHD simulations at t  18 S
(no anti-parallel magnetic field)
Finite :
Ideal GRMHD:
 
  100 / crS
4
2
0
-2
-4
Solid line: Magnetic field line, Arrow: Velocity
Color:
B / c 2
Explicit method: Comparison of results of ideal and
finite  GRMHD simulations at t  18 S
(no anti-parallel magnetic field)
Finite :
Ideal GRMHD:
 
  10 4 / crS
4
2
0
-2
Stop due to
numerical
instability
-4
Solid line: Magnetic field line, Arrow: Velocity
t
 0.01
tCFL
Implicit method: Comparison of results of explicit and
implicit methods with very large conductivity at t  15 S
σ=104/crS
Color:
Implicit (simplified) B / c 2
Explicit (ideal)
4
2
0
-2
-4
Solid line: Magnetic field line, Arrow: Velocity
t
 0.01
tCFL
Equation of State (EoS)
― Comparison between different EoS’s ―
  4/3
h  c 2
p
Exact
improved
  5/ 3

log p / c 2
RP
: Exact (Synge 1957)
Γ=4/3, 5/3: Constant polytropic index
TM
: Mignone et al (2005)
RC
: Ryu et al (2006)

Ryu, Chattopadhyay,
& Choi 2006
Comparison of results of finite  GRMHD simulations
before/after improved EoS at t  20 S (explicit)
Γ＝５／３ (before improvement)
Improved EoS (TM)
Color:
B / c 2
4
2
0
-2
-4

 100 / crS
Solid line: Magnetic field line, Arrow: Velocity
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.