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

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

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)

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.

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

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

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)

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

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　　　 (conservation law of particle-number) 　　　　　　　∇n T mn = 0 (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,

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

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

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

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

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

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　　　　　　 ○ ○　 ○　 ○ 　 ○

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

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

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

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

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

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)

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) 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 （~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 & 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)

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

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), 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)

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

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.

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.

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

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

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

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 Poloidal Field (Bz) Toroidal Field (By) Bz ↑ Vx ↑ Vz ↓ g ↑ By ↑ Vx ↓ Vz ↑ g ↑

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 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).

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.)

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

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)

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

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

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

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 >0 FHLL=FL SL < 0 < SR , FHLL=FM SR < 0 FHLL=FR

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

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))

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

Constrained Transport (Toth 2000)

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

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)

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)

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

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

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

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 = 0.916 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