Relativistic MHD Simulations of Relativistic Jets - LUTH

Relativistic MHD Simulations
of Relativistic Jets
Yosuke Mizuno
NASA Marshall Space Flight Center (MSFC)
National Space Science and Technology Center (NSSTC)
University of Nevada, Las Vegas (UNLV)
Co-authors and active collaborators:
M. Takahashi (Aichi Univ. of Edu.), K. Shibata (Kwasan Obs/Kyoto Univ.), S. Yamada (Waseda
Univ.), S. Koide (Kumamoto Univ.), S. Nagataki (Kyoto Univ./YITP), K.-I. Nishikawa
(NSSTC/UAH), P. Hardee (Univ. of Alabama), G.J. Fishiman (NSSTC/NASA-MSFC),
D. H. Hartmann (Clemson Univ.), B. Zhang (UNLV), D. Proga (UNLV), S.V. Fuerst (Stanford
Univ.), K. Wu (UCL), Y. Lyubarsky (Ben-Gurion Univ.), K. Ghosh (NSSTC), C. Fendt (MPIA),
P. Coppi (Yele Univ.), P. Biermann (MPIfR)

1. Introduction
Context
2. Development of 3D GRMHD code
3. 2D GRMHD simulations of Jet Formation
4. Stability of relativistic jets
5. MHD boost mechanism of relativistic jets
6. Summary and Future Research Plan

Astrophysical Jets
• Astrophysical jets: outflow of highly
collimated plasma
– Microquasars, Active Galactic Nuclei,
Gamma-Ray Bursts, Jet velocity ~c,
Relativistic Jets.
– Generic systems: Compact object
(White Dwarf, Neutron Star, Black
Hole)+ Accretion Disk
• Key Problems of Astrophysical Jets
– Acceleration mechanism and
radiation processes
– Collimation
– Long term stability
M87

Mirabel & Rodoriguez 1998
Relativistic Jets in Universe

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
– Magneto-centrifugal force and/or magnetic pressure
• Jet velocity ~ Keplerian velocity of accretion disk
– Can keep collimated structure by magnetic hoop-stress
• Direct extract of energy from a rotating black hole (Blandford &
Znajek 1977, force-free model)

Outflow (jet)
MHD model
• Acceleration
– Magneto-centrifugal force (Blandford-Payne
1982)
• Like a force worked a bead when swing a
wire with a bead
– Magnetic pressure force
• Like a force when stretch a spring
– Direct extract a energy from a rotating black hole
• Collimation
– Magnetic pinch (hoop stress)
• Like a force when the shrink a rubber
band
Magnetic
field line
Magnetic
field line
Magnetic
field line
outflow (jet)
accretion
Centrifugal
force

Requirment of Relativistic MHD
• Astrophysical jets seen AGNs show the relativistic
speed (~0.99c)
• The central object of AGNs is suppermassive black
hole (~10 5 -10 10 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 (MHD) simulations

Applicability of MHD Approximation
• MHD describe macroscopic behavior of plasmas
if
– Spatial scale >> ion Larmor radius
– Time scale >> ion Larmor period
• But MHD can not treat
– Particle acceleration
– Origin of resistivity
– Electromagnetic waves

Recent Work for Relativistic Jets
• Investigate the role of magnetic fields in relativistic jets against
three key problems
– Jet formation
– Jet acceleration and radiation process
• Acceleration of particles to very high energy
– Jet stability
• Recent research topics
– Development of 3D general relativistic MHD (GRMHD) code “RAISHIN”
– GRMHD simulations of jet formation and radiation from Black Hole
magnetosphere
– A relativistic MHD boost mechanism for relativistic jets
– Stability analysis of magnetized spine-sheath relativistic jets
– Particle-In-Cell (PIC) simulations of relativistic jets

1. Development of 3D GRMHD
Code “RAISHIN”
Mizuno et al. 2006a, Astro-ph/0609004
Mizuno et al. 2006, PoS, MQW6, 45

Numerical Approach to Relativistic MHD
• RHD: reviews Marti & Muller (2003) and Fonts (2003)
• SRMHD: many authors
• GRMHD
• Application: relativistic Riemann problems, relativistic jet propagation, jet
stability, pulsar wind nebule, etc.
– 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, Tchekhovskoy et al. 2008)
• 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, Cedra-Duran et al.
2008)

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 (γ>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:
∇ ν ( ρ U ν ) = 0 (conservation law of particle-number)
∇ ν T µν = 0 (conservation law of energy-momentum)
∂ µ F νλ + ∂ ν F λµ + ∂ λ F µν = 0
∇ µ F µν = - J ν
• Ideal MHD condition: F νµ U ν = 0
• metric: ds 2 =g µν dx µ dx ν
• Equation of state : p=(Γ-1) u
ρ : rest-mass density. p : proper gas pressure. u: internal energy. c: speed of light.
h : specific enthalpy, h =1 + u + p / ρ.
Γ: specific heat ratio.
U µυ : velocity four vector. J µυ : current density four vector.
∇ µν : covariant derivative. g µν : 4-metric,
(Maxwell equations)
T µν : energy momentum tensor, T µν = ρ h U µ U ν +pg µν +F µσ F ν
σ -g µνF λκ F λκ/4.
F µν : field-strength tensor,

Metric:
Conservative Form of GRMHD
Equations (3+1 Form)
U (conserved variables) F i (numerical flux) S (source term)
α: lapse function,
β i : shift vector,
γ ij: 3-metric
(Particle number conservation)
(Momentum conservation)
(Energy
conservation)
(Induction equation)
√-g : determinant of 4-metric
√γ : determinant of 3-metric
Detail of derivation of
GRMHD equations
Anton et al. (2005) etc.

New 3D GRMHD Code “RAISHIN”
Mizuno et al. (2006)
• RAISHIN utilizes conservative, high-resolution shock
capturing schemes (Godunov-type scheme) 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 problem
– Maintain divergence-free magnetic field by numerically
– Use constant Gamma-law or variable equation of states
– Parallelized by Open MP

Detailed Features of the Numerical
Schemes
Mizuno et al. 2006a, astro-ph/0609004
• RAISHIN utilizes conservative, high-resolution shock
capturing schemes (Godunov-type scheme) to solve the
3D GRMHD equations (metric is static)
* Reconstruction: PLM (Minmod & MC slope-limiter function),
convex ENO, PPM
* Riemann solver: HLL, HLLC approximate Riemann solver
* Constrained Transport: Flux interpolated constrained transport
scheme
* Time evolution: Multi-step Runge-Kutta method (2nd & 3rd-order)
* Recovery step: Koide 2 variable method, Noble 2 variable method,
Mignore-McKinney 1 variable method

Relativistic MHD Shock-Tube Tests
Exact solution: Giacomazzo & Rezzolla (2006)

Relativistic MHD Shock-Tube Tests
Balsara Test1 (Balsara 2001)
FR
CD
Black: exact solution, Blue: MC-limiter,
Light blue: minmod-limiter, Orange: CENO,
red: PPM
400 computational zones
SR
SS
FR
• 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.

Relativistic MHD Shock-Tube Tests
KO MC Min CENO PPM
• Komissarov: Shock Tube Test1 △ ○ ○ ○ ○ (large P)
• Komissarov: Collision Test × ○ ○ ○ ○ (large γ)
• Balsara Test1(Brio & Wu) ○ ○ ○ ○ ○
• Balsara Test2 × ○ ○ ○ ○ (large P & B)
• Balsara Test3 × ○ ○ ○ ○ (large γ)
• Balsara Test4 × ○ ○ ○ ○ (large P & B)
• Balsara Test5 ○ ○ ○ ○ ○
• Generic Alfven Test ○ ○ ○ ○ ○

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 (ρ d /ρ c =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) r S < r < 20 r S , 0.03<
θ < π/2, with 128*128 mesh
points
Schematic picture of the jet formation near
a black hole

Time evolution (Density)
non-rotating BH case (B 0=0.05,a=0.0)
Parameter
B 0=0.05
a=0.0
Color: density
White lines: magnetic
field lines (contour of
poloidal vector
potential)
Arrows: poloidal
velocity

Time evolution (Density)
rotating BH case (B 0=0.05,a=0.95)
Parameter
B 0=0.05
a=0.95
Color: density
White lines: magnetic
field lines (contour of
poloidal vector
potential)
Arrows: poloidal
velocity

Non-rotating BH Fast-rotating BH
ρρρρ
ββββ
v tot
Bφ
Results
• The matter in the disk loses its angular
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.
White curves: magnetic field lines (density), toroidal
magnetic field (plasma beta)
vector: poloidal velocity

Results (Jet Properties)
Non-rotating BH
Fast-rotating BH
W EM : Lorentz force
W gp : gas pressure gradient
• Outer jet: toroidal velocity
is dominant. The magnetic
field is twisted by rotation of
Keplerian disk. It is
accelerated mainly by the
gas pressure gradient (inner
part of it may be accelerated
by the Lorentz force).
• Inner jet: toroidal velocity
is dominant (larger than
outer jet). The magnetic field
is twisted by the framedragging
effect. It is
accelerated mainly by the
Lorentz force

Relativistic Radiation Transfer
• 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.).
α
u em
Wu et al., 2008, CJAA, submitted
Image of Emission, absorption &
scattering
Accretion Disk
Distribution of Absorbing Clouds
Black Hole
α
ucl α
uab u p α
Photon
Observer

Radiation images of black hole-disk system
• We have calculated the thermal freefree
emission and thermal synchrotron
emission from a relativistic flows in
black hole systems (2D GRMHD
simulation, rotating BH cases).
• We consider a GR radiation transfer
formulation and solve the transfer
equation using a ray-tracing algorithm.
• 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 GR effects.
• We can see the formation of twocomponent
jet based on synchrotron
emission and the strong thermal
radiation from hot dense gas near the
BHs.
Radiation image seen from
θ=85 (optically thin)
Radiation image seen from
θ=45 (optically thick)
Radiation image seen from
θ=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
•When jets propagate outward, there are possibility to grow of two
major instabilities
• Kelvin-Helmholtz (KH) instability
• Important at the shearing boundary flowing jet and external medium
• Current-Driven (CD) instability
• Important in 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
disruption etc
Limb brightening of M87 jets (observation)

Spine-Sheath Relativistic Jets
(observations)
M87 Jet: Spine-Sheath (two-component) 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
Radio ~ outside
optical emission
Sheath wind ?
• Observations of QSOs 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
• In many GRMHD simulation
of jet formation (e.g., Hawley &
Krolik 2006, McKinney 2006, Hardee
et al. 2007), 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.
(GRMHD Simulations)
Non-rotating BH Fast-rotating BH
Disk Jet/Wind
BH Jet Disk Jet/Wind
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 possibility to
grow of two instabilities
– Kelvin-Helmholtz (KH) instability
– Current-Driven (CD) instability
• How do jets remain sufficiently stable?
• What are the Effects & Structure of KH / CD
Instability in particular jet configuration (such as
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)
• Jet (spine): ujet = 0.916 c (γj =2.5), ρjet = 2 ρext (dense, cold jet)
• External medium (sheath): uext = 0 (static), 0.5c (sheath wind)
• Jet spine precessed to break the symmetry (frequency, ω=0.93)
• RHD: weakly magnetized (sound velocity > Alfven velocity)
• RMHD: strongly magnetized (sound velocity < Alfven velocity)
• Numerical box and computational zones
• -3 rj < x,y< 3rj , 0 rj < z < 60 rj (Cartesian coordinates) with 60*60*600 zones,
(1rj =10 zones)
Previous works: jet propagation simulation of Spine-Sheath jet model
(e.g., Sol et al. 1989; Hardee & Rosen 2002)

Simulation results: global structure
(nowind, weakly magnetized case)
3D isovolume of density with B-field lines show the jet is
disrupted by the growing KH instability
y
Longitudinal cross section
z
y
Transverse cross section show the strong
interaction between jet and external medium
x

Effect of magnetic field and sheath wind
vw =0.0 vw =0.5c vw =0.0
vw =0.5c
• Previous works: Study the effect of sheath wind on KH modes for non-rel HD
RMHD and RHD simulations (Hanasaz & Sol 1996, 1998; Hardee & Rosen 2002)
•The sheath flow reduces the growth rate of KH modes and slightly increases the
wave speed and wavelength as predicted from linear stability analysis.
•The magnetized sheath reduces growth rate relative to the weakly magnetized case
•The magnetized sheath flow damped growth of KH modes.
Criterion for damped KH modes:
(linear stability analysis)
1D radial velocity profile along jet

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
– If the jet is hotter and at much higher
pressure than a denser, colder external
medium, and moves with a large velocity
tangent to the interface, the relative motion
of the two fluids produces a hydrodynamical
structure in the direction perpendicular to the
flow.
– The rarefaction wave propagates into the jet
and the low pressure wave leads to strong
acceleration of the jet fluid into the
ultrarelativistic regime in a narrow region
near the contact discontinuity.
Schematic picture of simulations

Motivation
• This hydrodynamical boosting mechanism is
very simple and powerful.
• But it is likely to be modified by the effects of
magnetic fields present in the initial flow, or
generated within the shocked outflow.
• We investigate the effect of magnetic fields on
the boost mechanism by using Relativistic
MHD 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 γ~12 from an
initial Lorentz factor of γ~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
poloidal
toroidal
• 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.
• In both of cases, when the magnetic
field strength increases, maximum
Lorentz factor also increases.
• Toroidal magnetic field provides the
most efficient acceleration.

Multi-dimensional Simulations
(Initial Condition)
• 2D RMHD simulations of MHDA case (poloidal field).
• Pre-existing jet flow is established across the computational
domain.
• Simulation region, 0.5 < x,r < 1.5, 0 < z < 5.0 with (Nx * Nz)
= (2000 * 250)
• In order to investigate a possible influence of the chosen
coordinate system, we perform Cartesian and cylindrical
coordinates.

Multi-dimensional Simulation
(Results)
• A thin surface is accelerated by
the MHD boost mechanism to
reach a maximum Lorentz factor
g~15 from an initial Lorentz factor
g~7.
• The jet in cylindrical coordinates
is slightly more accelerated than
the jet in Cartesian coordinates,
which suggests that different
coordinate systems can affect
sideways expansion, shock profile,
and acceleration (slightly).
• The field geometry is an
important factor.

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
• Code Development
– Kerr-Schild Coordinates: long-term simulation in GRMHD
– Resistivity: extension to non-ideal MHD; (e.g., Watanabe &
Yokoyama 2007; Komissarov 2007)
– Couple with radiation transfer: link to observation
• Research of Jet Formation and Propagation
– Dependence on Magnetic field structure, BH spin parameter,
disk structure and perturbation etc.
• Research of Jet Stability
– Dependence on EoS
– Current-Driven instability
• Apply to astrophysical phenomena in which relativistic
outflows and/or GR essential (AGNs, microquasars,
neutron stars, and GRBs etc.)