Relativistic MHD Simulations of
Relativistic Jets
Yosuke Mizuno
NASA Postdoctoral Program Fellow
NASA Marshall Space Flight Center (MSFC)
National Space Science and Technology Center (NSSTC)
Context
1.
2.
3.
4.
5.
6.
Introduction
Development of 3D GRMHD code
2D GRMHD simulations of Jet Formation
Stability of relativistic jets
MHD boost mechanism of relativistic jets
Summary
Astrophysical Jets
•
Astrophysical jets: outflow of
highly collimated plasma
– Active Galactic Nuclei, Microquasars,
Gamma-Ray Bursts, Jet velocity ～c.
– Generic systems: Compact object
（White Dwarf, Neutron Star, Black
Hole）+ Accretion Disk
•
Key Problem of Astrophysical Jets
– Acceleration mechanism
– Collimation
– Long term stability
•
Modeling of Astrophysical Jets
– Magnetohydrodynamics (MHD)+
Relativity
•
•
MHD centrifugal acceleration
Extraction of rotational energy from
rotating black hole
M87
Relativistic Jets in Universe
Mirabel & Rodoriguez 1998
Modeling of Astrophysical Jets
Energy conversion from accreting matter is the most efficient
mechanism
• Gas pressure model
– Jet velocity ~ sound speed (maximum is ~0.58c)
– Difficult to keep collimated structure
• Radiation pressure model
– Can collimate by the geometrical structure of accretion disk (torus)
– Difficult to make relativistic speed with keeping collimated structure
• Magnetohydrodynamic (MHD) model
– Jet velocity ~ Keplerian velocity of accretion disk
 make relativistic speed because the Keplerian velocity near the black hole is
nearly light speed
– Can keep collimated structure by magnetic hoop-stress
• Direct extract of energy from a rotating black hole (Blandford &
Znajek 1977, force-free model)
Requirment of Relativistic MHD
• Astrophysical jets seen AGNs show the relativistic
speed (~0.99c)
• The central object of AGNs is supper-massive black
hole (~105-1010 solar mass)
• The jet is formed near black hole
Require relativistic treatment (special or general)
• In order to understand the time evolution of jet
formation, propagation and other time dependent
phenomena, we need to perform relativistic
magnetohydrodynamic simulations
1. Development of 3D GRMHD
Code “RAISHIN”
Mizuno et al. 2006a, Astro-ph/0609004
Numerical Approach to Relativistic MHD
• RHD: reviews Marti & Muller (2003) and Fonts (2003)
• SRMHD: many authors
– Application: relativistic Riemann problems, relativistic jet propagation, jet
stability, pulsar wind nebule, etc.
• GRMHD
– Fixed spacetime (Koide, Shibata & Kudoh 1998; De Villiers &
Hawley 2003; Gammie, McKinney & Toth 2003; Komissarov 2004; Anton
et al. 2005; Annios, Fragile & Salmonson 2005; Del Zanna et al. 2007)
– Application: The structure of accretion flows onto black hole and/or
formation of jets, BZ process near rotating black hole, the formation of
GRB jets in collapsars etc.
– Dynamical spacetime (Duez et al. 2005; Shibata & Sekiguchi 2005;
Anderson et al. 2006; Giacomazzo & Rezzolla 2007 )
Propose to Make a New GRMHD Code
• The Koide’s GRMHD Code (Koide, Shibata & Kudoh 1999;
Koide 2003) has been applied to many high-energy
astrophysical phenomena and showed pioneering results.
• However, the code can not perform calculation in highly
relativistic (g>5) or highly magnetized regimes.
• The critical problem of the Koide’s GRMHD code is the
schemes can not guarantee to maintain divergence free
magnetic field.
• In order to improve these numerical difficulties, we have
developed a new 3D GRMHD code RAISHIN (RelAtIviStic
magnetoHydrodynamc sImulatioN, RAISHIN is the Japanese
ancient god of lightning).
4D General Relativistic MHD Equation
•
General relativistic equation of conservation laws and Maxwell equations:
∇n ( r U n ) = 0
∇n T mn = 0
(conservation law of particle-number)
(conservation law of energy-momentum)
∂mFnl  nFlm  lF mn = 0
∇mF
mn
=-J
n
(Maxwell equations)
FnmUn = 0
•
Ideal MHD condition:
•
metric： ds2=gmndxmdxn
•
Equation of state : p=(G-1) u
r : rest-mass density. p : proper gas pressure. u: internal energy. c: speed of light.
h : specific enthalpy, h =1 + u + p / r.
G: specific heat ratio.
mu
U : velocity four vector. Jmu : current density four vector.
mn
∇ : covariant derivative. gmn : 4-metric,
mn
mn
m n
mn
ms n
lk
T : energy momentum tensor, T = r h U U +pg +F F s -gmnF Flk/4.
Fmn : field-strength tensor,
Conservative Form of GRMHD
Equations (3+1 Form)
a: lapse function,
bi: shift vector,
gij: 3-metric
Metric:
(Particle number conservation)
(Momentum conservation)
(Energy
conservation)
(Induction equation)
U (conserved variables)
Fi (numerical flux) S (source term)
√-g : determinant of 4-metric
√g : determinant of 3-metric
Detail of derivation of
GRMHD equations
Anton et al. (2005) etc.
3D GRMHD Code “RAISHIN”
Mizuno et al. (2006)
• RAISHIN utilizes conservative, high-resolution shock
capturing schemes to solve the 3D general relativistic
MHD equations (metric is static)
• Ability of RAISHIN code
– Multi-dimension (1D, 2D, 3D)
– Special (Minkowski spcetime) and General relativity (static metric;
Schwarzschild or Kerr spacetime)
– Different coordinates (RMHD: Cartesian, Cylindrical,
Spherical and GRMHD: Boyer-Lindquist of non-rotating or
rotating BH)
– Use several numerical methods to solving each problems
– Use Gamma-law or variable equation of state
Relativistic MHD Shock-Tube Tests
Exact solution: Giacomazzo & Rezzolla (2006)
Relativistic MHD Shock-Tube Tests
Balsara Test1 (Balsara 2001)
FR
SR
SS
FR
CD
Black: exact solution, Blue: MC-limiter,
Light blue: minmod-limiter, Orange: CENO,
red: PPM
400 computational zones
• The results show good
agreement of the exact solution
calculated by Giacommazo &
Rezzolla (2006).
• Minmod slope-limiter and
CENO reconstructions are more
diffusive than the MC slopelimiter and PPM reconstructions.
• Although MC slope limiter and
PPM reconstructions can resolve
the discontinuities sharply, some
small oscillations are seen at the
discontinuities.
2. 2D GRMHD Simulation of Jet
Formation
Mizuno et al. 2006b, Astro-ph/0609344
Hardee, Mizuno, & Nishikawa 2007, ApSS, 311, 281
Wu et al. 2008, CJAA, submitted
2D GRMHD Simulation of Jet Formation
Initial condition
– Geometrically thin Keplerian
disk (rd/rc=100) rotates
around a black hole (a=0.0,
0.95)
– The back ground corona is
free-falling to a black hole
(Bondi solution)
– The global vertical magnetic
field (Wald solution)
Numerical Region and Mesh
points
– 1.1（0.75) rS < r < 20 rS, 0.03<
q < p/2, with 128*128 mesh
points
Schematic picture of the jet formation near
a black hole
Time evolution (Density)
non-rotating BH case (B0=0.05,a=0.0)
Parameter
B0=0.05
a=0.0
Color: density
White lines: magnetic
field lines (contour of
poloidal vector
potential)
Arrows: poloidal
velocity
Non-rotating BH
r
Fast-rotating BH
Results
• The matter in the disk loses its angular
b
Bf
momentum by magnetic field and falls to a black
hole.
• A centrifugal barrier decelerates the falling
matter and make a shock around r=2rS.
• The matter near the shock region is accelerated
by the J×B force and the gas pressure gradient
and forms jets.
• These results are similar to previous work
(Koide et al. 2000, Nishikawa et al. 2005).
• In the rotating black hole case, additional inner
jets form by the magnetic field twisted resulting
from frame-dragging effect.
vtot
White curves: magnetic field lines (density), toroidal
magnetic field (plasma beta)
vector: poloidal velocity
Relativistic Radiation Transfer
Wu et al., 2008, CJAA, submitted
• We have calculated the thermal free-free
emission and thermal synchrotron emission
from a relativistic flows in black hole
systems based on the results of our 2D
GRMHD simulations (rotating BH cases).
• We consider a general relativistic radiation
transfer formulation (Fuerst & Wu 2004,
A&A, 424, 733) and solve the transfer
equation using a ray-tracing algorithm.
• In this algorithm, we treat general
relativistic effect (light bending, gravitational
lensing, gravitational redshift, framedragging effect etc.).
Image of Emission, absorption &
scattering
Radiation images of black hole-disk
system
• The radiation image shows
the front side of the accretion
disk and the other side of the
disk at the top and bottom
regions because the general
relativistic effects.
• We can see the formation of
two-component jet based on
synchrotron emission and the
strong thermal radiation from
hot dense gas near the BHs.
• A wired synchrotron
emission (green-spark) is seen
the surface of the disk (timedependent). It becomes a
origin of QPOs?
Radiation image seen from
q=85 (optically thin)
Radiation image seen from
q=45 (optically thick)
Radiation image seen from
q=85 (optically thick)
3. Stability Analysis of Magnetized
Spine-Sheath Relativistic Jets
Mizuno, Hardee & Nishikawa, 2007, ApJ, 662, 835
Hardee, 2007, ApJ, 664, 26
Hardee, Mizuno & Nishikawa, 2007, ApSS, 311, 281
Instability of relativistic jets
• Kelvin-Helmholtz (KH) (this talk topics)
– Presence of velocity gradients in the flow
– Important at the shearing boundary flowing jet and external
medium
• Current-driven (CD) (e.g., kink instability)
– Presence of strong axial electric current
– Important in strongly twisted magnetic field
• Interaction of jets with external medium caused by
such instabilities leads to the formation of shocks,
turbulence, acceleration of charged particles etc.
• Used to interpret many jet phenomena
–
quasi-periodic wiggles and knots, filaments, limb
brightening, jet distuption etc
Spine-Sheath Relativistic Jets
(observations)
M87 Jet: Spine-Sheath Configuration?
HST Optical Image (Biretta, Sparks, & Macchetto 1999)
Typical Proper
Motions > c
Optical ~ inside
radio emission
Jet Spine ?
VLA Radio Image (Biretta, Zhou, & Owen 1995)
Typical Proper
Motions < c
Sheath wind ?
• Obsevations of OSOs show the evidence of high speed wind （~0.1-0.4c）(Pounds et
al. 2003):
•Related to Sheath wind
• Spine-sheath configuration proposed to explain
•limb brightening in M87, Mrk501jets （Perlman et al. 2001; Giroletti et al. 2004）
•TeV emission in M87 (Taveccio & Ghisellini 2008)
•broadband emission in PKS 1127-145 jet (Siemiginowska et al. 2007)
Spine-Sheath Relativistic Jets
(GRMHD Simulations)
• In recent general relativistic MHD
simulation of jet formation (e.g., Hawley &
Non-rotating BH
Fast-rotating BH
Krolik 2006, McKinney 2006, Hardee et al. 2007),
simulation results suggest that
• a jet spine driven by the magnetic
fields threading the ergosphere
• may be surrounded by a broad
sheath wind driven by the magnetic
fields anchored in the accretion disk.
• This configuration might additionally
be surrounded by a less highly
collimated accretion disk wind from the
hot corona.
Disk Jet
BH Jet
Disk Jet
Total velocity distribution of 2D
GRMHD Simulation of jet formation
(Hardee, Mizuno & Nishikawa 2007)
Key Questions of Jet Stability
• When jets propagate outward, there are several
possibility to grow of instabilities
• How do jets remain sufficiently stable?
• What are the Effects & Structure of KelvinHelmholtz (KH) / current driven (CD)
Instability in spine-sheath configuration?
• We investigate these topics by using 3D
relativistic MHD simulations
3D Simulations of Spine-Sheath Jet Stability
Initial condition
Mizuno, Hardee & Nishikawa, 2007
• Cylindrical super-Alfvenic
jet established across the
computational domain with a
parallel magnetic field (stable
against CD instabilities)
• Solving 3D RMHD equations in Cartesian coordinates
(using Minkowski spacetime)
•ujet = 0.916 c (γj=2.5), rjet = 2 rext (dense, cold spine jet)
• External medium, uext = 0 (static), 0.5c (sheath wind)
• Spine Jet precessed to break the symmetry (frequency, w=0.93)
• RHD: weakly magnetized (sound velocity > Alfven velocity)
• RMHD: strongly magnetized (sound velocity < Alfven velocity)
• Numerical Resion and mesh points
• -3 rj< x,y< 3rj, 0 rj< z < 60 rj (Cartesian coordinates) with
60*60*600 computational zones, (1rj=10 computational zone)
Simulation results (nowind, weakly
magnetized case)
3D isovolume of density with B-field lines show the jet is
disrupted by the growing KH instability
Longitudinal cross section
y
y
z
x
Transverse cross section show the strong
interaction between jet and external medium
Effect of magnetic field and sheath wind
vw=0.0
vw=0.5c
vw=0.0
vw=0.5c
• The sheath flow reduces the growth rate of KH modes and slightly
increases the wave speed and wavelength as predicted from linear
stability analysis.
• Substructure associated with the 1st helical body mode is eliminated
by sheath wind as predicted.
• The magnetized sheath reduces growth rate relative to the fluid case
and the magnetized sheath flow damped growth of KH modes.
4. MHD Boost mechanism of
Relativistic Jets
Mizuno, Hardee, Hartmann, Nishikawa & Zhang, 2008,
ApJ, 672, 72
A MHD boost for relativistic jets
• The acceleration mechanism boosting relativistic jets
to highly-relativistic speed is not fully known.
• Recently Aloy & Rezzolla (2006) have proposed a
powerful hydrodynamical acceleration mechanism of
relativistic jets by the motion of two fluid between
jets and external medium
• This hydrodynamical boosting mechanism is very
simple and powerful
• But is likely modified by the effect of magnetic fields
• We have investigated the effect of magnetic fields on
the boost mechanism by using RMHD simulations
Initial Condition (1D RMHD)
• Consider a Riemann problem consisting of two uniform initial states
• Right (external medium): colder fluid with larger rest-mass density and
essentially at rest.
• Left (jet): lower density, higher temperature and pressure, relativistic velocity
tangent to the discontinuity surface
• To investigate the effect of magnetic fields, put the poloidal (Bz: MHDA) or
toroidal (By: MHDB) components of magnetic field in the jet region (left state).
• For comparison, HDB case is a high gas pressure, pure-hydro case (gas
pressure = total pressure of MHD case)
Simulation region
-0.2 < x < 0.2 with 6400 grid
Schematic picture of simulations
Hydro Case
Solid line (exact solution), Dashed line (simulation)
In the left going rarefaction region,
the tangential velocity increases
due to the hydrodynamic boost
mechanism.
jet is accelerated to g~12 from an
initial Lorentz factor of g~7.
MHD Case
HDA case (pure hydro) :
dotted line
MHDA case (poloidal)
MHDB case (toroidal)
HDB case (hydro, high-p)
• When gas pressure becomes large, the normal velocity
increases and the jet is more efficiently accelerated.
• When a poloidal magnetic field is present, stronger
sideways expansion is produced, and the jet can achieve
higher speed due to the contribution from the normal
velocity.
• When a toroidal magnetic field is present, although the
shock profile is only changed slightly, the jet is more
strongly accelerated in the tangential direction due to the
Lorentz force.
• The geometry of the magnetic field is a very important
geometric parameter.
Dependence on Magnetic Field Strength
Solid line: exact solution, Crosses: simulation
Magnetic field strength is measured in fluid flame
• When the poloidal magnetic field
increases, the normal velocity
increases and the tangential velocity
decreases.
• When the toroidal magnetic field
increases, the normal velocity
decreases and the tangential velocity
increases.
• Toroidal magnetic field provides the
most efficient acceleration.
Summary
• We have developed a new 3D GRMHD code
``RAISHIN’’by using a conservative, high-resolution
shock-capturing scheme.
• We have performed simulations of jet formation from a
geometrically thin accretion disk near both non-rotating
and rotating black holes. Similar to previous results (Koide
et al. 2000, Nishikawa et al. 2005a) we find magnetically
driven jets.
• It appears that the rotating black hole creates a second,
faster, and more collimated inner outflow. Thus, kinematic
jet structure could be a sensitive function of the black hole
spin parameter.
Summary (cont.)
•We have investigated stability properties of magnetized
spine-sheath relativistic jets by the theoretical work and
3D RMHD simulations.
• The most important result is that destructive KH modes
can be stabilized even when the jet Lorentz factor
exceeds the Alfven Lorentz factor. Even in the absence of
stabilization, spatial growth of destructive KH modes can
be reduced by the presence of magnetically sheath flow
(~0.5c) around a relativistic jet spine (>0.9c)
Summary (cont.)
• We performed relativistic magnetohydrodynamic
simulations of the hydrodynamic boosting mechanism
for relativistic jets explored by Aloy & Rezzolla (2006)
using the RAISHIN code.
•We find that magnetic fields can lead to more efficient
acceleration of the jet, in comparison to the purehydrodynamic case.
• The presence and relative orientation of a magnetic
field in relativistic jets can significant modify the
hydrodynamic boost mechanism studied by Aloy &
Rezzolla (2006).
Future Work
• Resistivity (extension to non-ideal MHD; e.g., Watanabe &
Yokoyama 2007; Komissarov 2007)
• Couple with radiation transfer (link to observation:
collaborative works with Fuerst)
• Improve the realistic EOS
• Include Neutrino (cooling, heating)
• Include Nucleosysthesis post processing
• Couple with Einstein equation (dynamical spacetime)
• Adaptive mesh refinement
• Apply to astrophysical phenomena in which relativistic
outflows and/or GR essential (AGNs, microquasars,
neutron stars, and GRBs etc.)