1. 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)

2. Context Introduction Development of 3D GRMHD code
2D GRMHD simulations of Jet Formation
Stability of relativistic jets
MHD boost mechanism of relativistic jets
Summary

3. Magnetic field in the Universe
Magnetic and gravitational fields play a important role in determining the evolution of the matter in many astrophysical objects
Magnetic field can be amplified by the gas contraction or shear motion.
Even when the magnetic field is weak initially, the magnetic field glows in the short time scale and influences gas dynamics of the system

4. Plasmas in the Universe
The major constituents of the universe are made of plasmas.
When the temperature of gas is more than 104K, the gas becomes fully ionized plasmas (4th phase of matter).
Plasmas are applied to many astrophysical phenomena.
Plasmas are treated two way,
particles (microscopic)
magnetohydrodynamics, MHD (macroscopic)

5. Applicability of Hydrodynamic Approximation
To apply hydrodynamic approximation, we need the condition:
Spatial scale >> mean free path
Time scale >> collision time
These are not necessarily satisfied in many astrophysical plasmas
E.g., solar corona, galactic halo, cluster of galaxies etc.
But in plasmas with magnetic field, the effective mean free path is given by the ion Larmor radius.
Hence if the size of phenomenon is much larger than the ion Larmor radius, hydrodynamic approximation can be used.

6. 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

7. Astrophysical Jets
M87
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

8. Relativistic Jets in Universe
Mirabel & Rodoriguez 1998

9. 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)

10. MHD model outflow (jet) Outflow (jet) Magnetic field line 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
Centrifugal force
outflow (jet)
accretion
Outflow (jet)
Magnetic field line
Magnetic field line

11. Requirment of Relativistic MHD
Astrophysical jets seen AGNs show the relativistic speed (~0.99c)
The central object of AGNs is supper-massive black hole (~ 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

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

13. 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 )

14. 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).

15. 4D General Relativistic MHD Equation
General relativistic equation of conservation laws and Maxwell equations:　　　
　　　　　　　∇n ( r U n ) = 0　　　 (conservation law of particle-number)
　　　　　　　∇n T mn = (conservation law of energy-momentum)
　　　　　　　∂mFnl + ∂nFlm + ∂lF mn = 0
　　　　　　　∇mF mn = - J n
Ideal MHD condition: FnmUn = 0
metric：　ds2=gmndxmdxn
Equation of state : p=(G-1) u
(Maxwell equations)
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.
Umu : velocity four vector. Jmu : current density four vector.
∇mn : covariant derivative. gmn : 4-metric,
Tmn : energy momentum tensor, Tmn = r h Um Un+pgmn+FmsFns -gmnFlkFlk/4.
Fmn : field-strength tensor,

16. 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.

17. 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

18. Detailed Features of the Numerical Schemes
Mizuno et al. 2006a, astro-ph/
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

19. Ability of a New GRMHD code
Multi-dimension (1D, 2D, 3D)
Special and General relativity (static metric)
Different coordinates (RMHD: Cartesian, Cylindrical, Spherical and GRMHD: Boyer-Lindquist of non-rotating or rotating BH)
Different spatial reconstruction algorithms (4)
Different approximate Riemann solver (2)
Different time advance algorithms (2)
Different recovery schemes (3)
Using constant gamma-law and variable Equation of State
Parallelized by OpenMP

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

21. Relativistic MHD Shock-Tube Tests
Balsara Test1 (Balsara 2001)
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 slope-limiter and PPM reconstructions.
Although MC slope limiter and PPM reconstructions can resolve the discontinuities sharply, some small oscillations are seen at the discontinuities. FR SR SS FR CD
Black: exact solution, Blue: MC-limiter, Light blue: minmod-limiter, Orange: CENO, red: PPM
400 computational zones

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

23. 2. 2D GRMHD Simulation of Jet Formation
Mizuno et al. 2006b, Astro-ph/
Hardee, Mizuno, & Nishikawa 2007, ApSS, 311, 281
Wu et al. 2008, CJAA, submitted

24. 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

25. 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

26. Time evolution (Density) rotating BH case (B0=0.05,a=0.95)
Parameter
B0=0.05
a=0.95
Color: density
White lines: magnetic field lines (contour of poloidal vector potential)
Arrows: poloidal velocity

27. Results Non-rotating BH Fast-rotating BH r
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. b Bf
vtot
White curves: magnetic field lines (density), toroidal magnetic field (plasma beta)
vector: poloidal velocity

28. Results (Jet Properties)
WEM: Lorentz force
Wgp: gas pressure gradient
Non-rotating BH
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 frame-dragging effect. It is accelerated mainly by the Lorentz force
Fast-rotating BH

29. 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, frame-dragging effect etc.).
Image of Emission, absorption & scattering

30. 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 (time-dependent). It becomes a origin of QPOs?
Radiation image seen from q=85 (optically thin)
Radiation image seen from q=85 (optically thick)
Radiation image seen from q=45 (optically thick)

31. 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

32. 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

33. Spine-Sheath Relativistic Jets (observations)
M87 Jet: Spine-Sheath Configuration?
HST Optical Image (Biretta, Sparks, & Macchetto 1999)
VLA Radio Image (Biretta, Zhou, & Owen 1995)
Typical Proper Motions > c
Optical ~ inside radio emission
Jet Spine ?
Typical Proper Motions < c
Sheath wind ?
Obsevations of OSOs show the evidence of high speed wind （~ c）(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 jet (Siemiginowska et al. 2007)

34. Spine-Sheath Relativistic Jets (GRMHD Simulations)
In recent general relativistic MHD simulation of jet formation (e.g., Hawley & 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.
Non-rotating BH Fast-rotating BH
Disk Jet
BH Jet Disk Jet
Total velocity distribution of 2D GRMHD Simulation of jet formation
(Hardee, Mizuno & Nishikawa 2007)

35. 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 Kelvin-Helmholtz (KH) / current driven (CD) Instability in spine-sheath configuration?
We investigate these topics by using 3D relativistic MHD simulations

36. 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 = c (γj=2.5), jet = 2 ext (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)

37. 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

38. 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.

39. 4. MHD Boost mechanism of Relativistic Jets
Mizuno, Hardee, Hartmann, Nishikawa & Zhang, 2008, ApJ, 672, 72

40. Hydrodynamic Boost Mechanism
Aloy & Rezzolla (2006) investigated a potentially powerful acceleration mechanism in the context of purely hydrodynamical flow, posing a Riemann problem.
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.

41. 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 investigated the effect of magnetic fields on the boost mechanism by using Relativistic MHD simulations.

42. 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

43. 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

44. A MHD boost for relativistic jets (initial condition)
Consider a Riemann problem consisting of two uniform initial states (left: jet with vz=0.99c, right: external medium )
To investigate the effect of magnetic fields, put the poloidal (Bz) or toroidal (By) magnetic fields in the jet region (left state)
Computational domain -0.2 < x < 0.2 with 6400 zones
Schematic picture of simulations

45. Results Hydro case HDA (black) HDB (green) MHDA (blue) MHDB (red)
Left-going
rarefaction
Right-going shock
Boosting

46. 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.

47. 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.

48. Dependence on magnetic field
Poloidal
Field (Bz)
Toroidal
Field (By)
Bz ↑
Vx ↑
Vz ↓
g ↑
By ↑
Vx ↓
Vz ↑
g ↑

49. 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.

50. 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.

51. 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)

52. 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 pure-hydrodynamic 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).

53. 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.)

55. Basics of Numerical RMHD Code
Non-conservative form (De Villier & Hawley (2003), Anninos et al.(2005))
U=U(P) - conserved variables,
P – primitive variables
F- numerical flux of U
where
Merit:
they solve the internal energy equation rather than energy equation. → advantage in regions where the internal energy small compared to total energy (such as supersonic flow)
Recover of primitive variables are fairly straightforward
Demerit:
It can not applied high resolution shock-capturing method and artificial viscosity must be used for handling discontinuities

56. Basics of Numerical RMHD Code
Conservative form (Koide et al. (1999), Kommisarov (2001), Gammie et al (2003), Anton et al. (2004), Duez et al. (2005), Shibata & Sekiguchi (2005) etc)
System of Conservation Equations
U=U(P) - conserved variables,
P – primitive variables
F- numerical flux of U,
S - source of U
Merit:
High resolution shock-capturing method can be applied to GRMHD equations
Demerit:
These schemes must recover primitive variables P by numerically solving the system of equations after each step (because the schemes evolve conservative variables U)

57. Reconstruction Cell-centered variables (Pi)
→ right and left side of Cell-interface variables(PLi+1/2, PRi+1/2)
Piecewise linear interpolation
Minmod and MC Slope-limited Piecewise linear Method
2nd-order 1st -order at local extrema
Convex CENO (Liu & Osher 1998)
3rd -order, 1st -order at local extrema
Piecewise Parabolic Method (Marti & Muller 1996)
4th -order, 1st -order at local extrema
PLi+1/2
PRi+1/2
Pni-1
Pni
Pni+1

58. Piecewise Linear Method
Reconstructed cell-interface variables
Slope-limiter flunction
Minmod function
Monotonized Centeral (MC) function

59. Piecewise Parabolic Method
1st Step: interpolation
quartic polynomial interpolation determined by the five zone-averaged values.
2nd Step: contact steepening
slightly modified to produce narrower profiles in the vicinity of a contact discontinuity
3rd Step: Flattening
near strong shocks the order of the method is reduced locally to avoid spurious postshock oscillations
4th Step: Monotonization
monotonize by the interpolation parabola between smooth and shock region

60. HLL Approximate Riemann Solver
Calculate numerical flux at cell-inteface from reconstructed cell-interface variables based on Riemann problem
We use HLL approximate Riemann solver
Need only the maximum left- and right- going wave speeds (in MHD case, fast mode)
HLL flux
FR=F(PR), FL=F(PL); UR=U(PR), UL=U(PL)
SR=max(0,c+R, c+L); SL=max(0,c-R,c-L)
If SL > FHLL=FL
SL < 0 < SR , FHLL=FM
SR < FHLL=FR

61. HLLC Approximate Riemann Solver
Mignore & Bodo (2006)
HLL Approximate Riemann solver: single state in Riemann fan
HLLC Approximate Riemann solver: two-state in Riemann fan
HLL
HLLC

62. Constrained Transport
Differential Equations
The evolution equation can keep divergence free magnetic field
If treat the induction equation as all other conservation laws, it can not maintain divergence free magnetic field
→ We need spatial treatment for magnetic field evolution
Constrained transport method
Evans & Hawley’s Constrained Transport (Komissarov (1999,2002,2004), de Villiers & Hawley (2003), Del Zanna et al.(2003), Anton et al.(2005))
Toth’s constrained transport (Gammie et al.(2003), Duez et al.(2005))
Diffusive cleaning (Annios et al.(2005))

63. Constrained Transport (Toth 2000)
２D case
Use the “modified flux” f that is such a linear combination of normal fluxes at neighbouring interfaces that the “corner-centred” numerical representation of divB is kept invariant during integration.
k+1/2
k-1/2
j-1/2
j+1/2

64. Constrained Transport (Toth 2000)

65. Evans & Hawley’s Constrained Transport
Use staggered grid (with B defined at the cell-interfaces) and evolve magnetic fluxes through the cell interfaces using the
electric field evaluated at the cell-edges.
This keeps the following “cell-centred”
numerical representation of divB invariant

66. Time evolution
System of Conservation Equations
We use multistep Runge-Kutta method for time advance of conservation equations (RK2: 2nd-order, RK3: 3rd-order in time)
RK2, RK3: first step
RK2: second step (a=2, b=1)
RK3: second and third step (a=4, b=3)

67. Recovery step
The GRMHD code require a calculation of primitive variables from conservative variables.
The forward transformation (primitive → conserved) has a close-form solution, but the inverse transformation (conserved → primitive) requires the solution of a set of five nonlinear equations
Method
Koide’s 2D method (Koide, Shibata & Kudoh 1999)
Noble’s 2D method (Noble et al. 2005)

68. Recovery step (Koide’s 2D method)
Conserved quantities(D,P,e,B) → primitive variables (r,p,v,B)
2-variable Newton-Raphson iteration method

69. Noble’s 2D method
Conserved quantities(D,S,t,B) → primitive variables (r,p,v,B)
Solve two-algebraic equations for two independent variables W≡hg2 and v2 by using 2-variable Newton-Raphson iteration method
W and v2 →primitive variables r p, and v

70. Stability Properties of RMHD Relativistic Jet Spine & Sheath
Dispersion Relation:
Surface  << *
Body Mode Condition:
Resonance (*) :
Growth Rate Reduction:
Stability:

71. Dispersion Relation of KH modes: Effect of Sheath Flow on a RMHD Jet
Dispersion Relation of Kelvin-Helmholtz (KH) modes
Effect of Sheath Flow on a RMHD Jet
Jet speed: uj = c
Sound speeds: aj = ae = 0.4 c
Surface mode: growth rates (dash-dotted lines) reduced as sheath speed increases from ue = 0 to 0.3 c.
Resonance: disappears for sheath speed ue > 0.35 c
Body mode: downwards arrows indicate damping peaks
 Rj/uj >> 1: damping for sheath speed ue > 0.5 c
 Rj/uj << 1: growth for sheath speed ue > 0.5 c
ue: sheath flow speed
Black: ue = 0.0 c, Blue: ue = 0.2 c,
Green: ue = 0.4 c, Red: ue = 0.6 c