3D GRMHD code RAISHIN and Relativistic Jet Simulations Yosuke Mizuno Center for Space Plasma and Aeronomic Research University of Alabama in Huntsville CFD-MHD seminar, ASIAA, Taiwan, 3/31/10

Context 1.Introduction 2.Development of 3D GRMHD code 3.2D GRMHD simulations of Jet Formation 4.Stability of relativistic jets (may be skip) 5.2D RMHD simulations of relativistic turbulence 6.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 10 4 K, the gas becomes fully ionized plasmas (4 th phase of matter). Plasmas are applied to many astrophysical phenomena. Plasmas are treated two way, –particles (microscopic) –magnetohydrodynamics, MHD (macroscopic)

Relativistic Jets Relativistic jets: outflow of highly collimated plasma –Microquasars, Active Galactic Nuclei, Gamma-Ray Bursts, Jet velocity ～ c –Generic systems: Compact object （ White Dwarf, Neutron Star, Black Hole ） + Accretion Disk Key Issues of Relativistic Jets –Acceleration & Collimation –Propagation & Stability Modeling for Jet Production –Magnetohydrodynamics (MHD) –Relativity (SR or GR) Modeling of Jet Emission –Particle Acceleration –Radiation mechanism Radio observation of M87 jet

Relativistic Jets in Universe Mirabel & Rodoriguez 1998

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) Modeling of Astrophysical Jets

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

Requirement 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

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

1. Development of 3D GRMHD Code “RAISHIN” Mizuno et al. 2006a, preprint, Astro-ph/ Mizuno et al. 2006, proceedings of science, MQW6, 045

Numerical Approach to Relativistic MHD RHD: reviews Marti & Muller (2003) and Fonts (2003) SRMHD: many groups developed their own code –Application: relativistic Riemann problems, relativistic jet propagation, jet stability, pulsar wind nebule, relativistic shock/blast wave etc. GRMHD –Fixed spacetime (Koide, Shibata & Kudoh 1998; De Villiers & Hawley 2003; Gammie, McKinney & Toth 2003; Komissarov 2004; Anton et al. 2005, 2010; Annios, Fragile & Salmonson 2005; Del Zanna et al. 2007, Nagataki 2009…) –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 (statement at 2006) 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 (  >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).

Finite Difference Method Linear wave equationHydrodynamic equations are a set of wave equations Finite difference in each term df/dx as a function of f j+1,n f j,n f j-1,n Forward derivative Backward derivative Central derivative

Finite Difference Method flux Conservative form of wave equation Finite difference FTCS scheme Upwind scheme Lax-Wendroff scheme

4D General Relativistic MHD Equation General relativistic equation of conservation laws and Maxwell equations: ∇  (  U  ) = 0 (conservation law of particle- number) ∇ T  = 0 (conservation law of energy-momentum) ∂  F  ∂ F   ∂ F  = 0 ∇  F  = - J Ideal MHD condition: F  U  = 0 metric ： ds 2 =-  2 dt 2 +g ij (dx i +  i dt)(dx j +  j dt) Equation of state : p=(  -1) u  : rest-mass density. p : proper gas pressure. u: internal energy. c: speed of light. h : specific enthalpy, h =1 + u + p / .  : specific heat ratio. U  : velocity four vector. J  : current density four vector. ∇  : covariant derivative. g  : 4-metric.  lapse function,  i   shift vector, g ij : 3-metric T  : energy momentum tensor, T  = p g  +  h U  U +F  F  - g  F  F  /4. F  : field-strength tensor, (Maxwell equations)

Conservative Form of GRMHD Equations (3+1 Form) (Particle number conservation) (Momentum conservation) (Energy conservation) (Induction equation) U (conserved variables)F i (numerical flux)S (source term) √-g : determinant of 4-metric √  : determinant of 3-metric Detail of derivation of GRMHD equations Anton et al. (2005) etc.

Basics of Numerical RMHD Code Non-conservative form (De Villier & Hawley (2003), Anninos et al.(2005)) 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 U=U(P) - conserved variables, P – primitive variables F- numerical flux of U

U=U(P) - conserved variables, P – primitive variables F- numerical flux of U, S - source of U System of Conservation Equations 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) 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)

Detailed Features of the Numerical Schemes 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), convex ENO, PPM, Weighted ENO5, Monotonicity Preserving5, MPWENO5 * Riemann solver: HLL, HLLC approximate Riemann solver * Constrained Transport: Flux CT, Fixed Flux-CT, Upwind Flux- CT * 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 * Equation of states: constant  -law EoS, variable EoS for ideal gas Mizuno et al. 2006a, astro-ph/ and progress

Reconstruction Cell-centered variables (P i ) → right and left side of Cell-interface variables(P L i+1/2, P R i+1/2 ) P L i+1/2 P R i+1/2 Piecewise linear interpolation Minmod and MC Slope-limited Piecewise linear Method 2 nd order at smooth region Convex CENO (Liu & Osher 1998) 3 rd order at smooth region Piecewise Parabolic Method (Marti & Muller 1996) 4 th order at smooth region Weighted ENO5 (Jiang & Shu 1996) 5 th order at smooth region Monotonicity Preserving (Suresh & Huynh 1997) 5 th order at smooth region MPWENO5 (Balsara & Shu 2000) PniPni P n i+1 P n i-1

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 magnetosonic mode) HLL flux F R =F(P R ), F L =F(P L ); U R =U(P R ), U L =U(P L ) S R =max(0,c +R, c +L ); S L =max(0,c -R,c -L ) If S L >0 F HLL =F L S L < 0 < S R, F HLL =F M S R < 0 F HLL =F R

HLLC Approximate Riemann Solver Mignore & Bodo (2006) Honkkila & Janhunen (2007) HLL Approximate Riemann solver: single state in Riemann fan HLLC Approximate Riemann solver: two-state in Riemann fan (HLLD Approximate Riemann solver: six-state in Riemann fan) (Mignone et al. 2009) HLLHLLC

Constrained Transport - 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 (flux-CT) (Gammie et al.(2003), Duez et al.(2005)) Fixed Flux-CT, Upwind Flux-CT (Gardiner & Stone 2005, 2007) Diffusive cleaning (Annios et al.(2005) etc) (better method for AMR or RRMHD) Differential Equations

Flux interpolated Constrained Transport 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. j-1/2j+1/2 k+1/2 k-1/2 ２ D case Toth (2000)

Constrained Transport (Toth 2000)

Time evolution System of Conservation Equations We use multistep TVD Runge-Kutta method for time advance of conservation equations (RK2: 2 nd -order, RK3: 3 rd -order in time) RK2, RK3: first step RK2: second step (  =2,  =1) RK3: second and third step (  =4,  =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) Mignone & McKinney’s method (Mignone & McKinney 2007)

Noble’s 2D method Conserved quantities(D,S, ,B) → primitive variables ( ,p,v,B) Solve two-algebraic equations for two independent variables W ≡ h  2 and v 2 by using 2-variable Newton-Raphson iteration method W and v 2 →primitive variables  p, and v Mignone & McKinney (2007): Implemented from Noble’s method for variable EoS

Variable EoS In the theory of relativistic perfect gases, specific enthalpy is a function of temperature alone (Synge 1957)  temperature p/  K 2, K 3 : the order 2 and 3 of modified Bessel functions Constant  -law EoS:  : constant specific heat ratio Taub’s fundamental inequality (Taub 1948)  → 0,  eq → 5/3,  → ∞,  eq → 4/3 TM EoS (Mathew 1971, Mignone et al. 2005) Mignone & McKinney 2007

Variable EoS Balsara Test2 exact solution with  =5/3 (solid line), variable EoS (dashed line)

Ability of RAISHIN code (current status) 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 (7) Different approximate Riemann solver (2) Different constrained transport schemes (3) Different time advance algorithms (2) Different recovery schemes (3) Using constant  -law and variable Equation of State (Synge-type) Parallelized by OpenMP

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

Relativistic MHD Shock-Tube Tests Balsara Test1 (Balsara 2001 ) Black: exact solution, Blue: MC-limiter, Light blue: minmod-limiter, Orange: CENO, red: PPM 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. 400 computational zones FR SR CD SS Mizuno et al. 2006

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

2. Highlights of Jet Simulations

Two component (Spine-Sheath) jet structure is seen in recent GRMHD simulations of jet formation in black hole-accretion disk system (e.g., Hawley & Krolik 2006, McKinney 2006, Hardee et al. 2007) jet spine: Formed by twisted magnetic field by frame- dragging effect of rotating black hole broad sheath wind: Formed by twisted magnetic field by rotation of accretion disk Non-rotating BH Fast-rotating BH BH Jet Disk Jet/Wind Disk Jet/Wind 2D GRMHD Simulation of jet formation Spine-Sheath Relativistic Jets (GRMHD Simulations) Color: density Color: total velocity Hardee, Mizuno & Nishikawa (2007)

Radiation Images of Black Hole- Disk System Calculation of thermal free-free emission and thermal synchrotron emission by ray-tracing method considered GR radiation transfer from a relativistic flows in black hole systems (2D GRMHD simulation, rotating BH cases). 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 two- component jet based on synchrotron emission and the strong thermal radiation from hot dense gas near the BHs. Radiation image seen from  =85 (optically thin) Radiation image seen from  =85 (optically thick) Wu, Fuerst, Mizuno et al. (2008) Schematic picture of Ray-tracing method

Stability of Magnetized Spine-Sheath Relativistic Jets We investigate the stability of magnetized two-component (spine-sheath) relativistic jets against Kelvin-Helmholtz (KH) instability by using 3D relativistic MHD simulations. Cylindrical super-Alfvenic jet established across the computational domain with a parallel magnetic field Put precession perturbation from jet inlet to break symmetry The jet is disrupted by the growing KH instability T=0 T=60 (Weakly magnetized, static external medium case) vjvj Mizuno, Hardee, & Nishikawa (2007)

Effect of magnetic field and sheath wind 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) u e =0.0 u e =0.5c 1D radial velocity profile along jet Mizuno, Hardee & Nishikawa (2007)

Current-Driven Instability: Static Plasma We studied the development of current-driven (CD) kink instability of a static force-free helical magnetic field configuration by using 3D RMHD simulations. We found the initial configuration is strongly distorted but not disrupted. The linear growth and non- linear evolution depends on the radial density profile and strongly depends on the magnetic pitch profile. Mizuno et al. (2009b) Increase pitch Decrease pitch Decrease density with Constant pitch case: CD kink instability leads to a helically twisted density and magnetic filament

CD kink instability of Sub-Alfvenic Jets: Spatial Properties Initial Condition Cylindrical sub-Alfvenic jet established across the computational domain with a helical force-free magnetic field (stable against KH instabilities) –V j =0.2c, R j =1.0 Radial profile: Decreasing density with constant magnetic pitch Jet spine precessed to break the symmetry Preliminary Result Precession perturbation from jet inlet produces the growth of CD kink instability with helical density distortion. Helical structure propagates along the jet with continuous growth of kink amplitude in non-linear phase. 3D density with magnetic field lines Mizuno et al. 2010, in preparation

Magnetic Field Amplification by Relativistic Shocks in Turbulent Medium Initial condition Density: mean + small inhomogenity with 2D Kolmogorov-like power-law spectrum Relativistic flow in whole region with constant magnetic field (parallel to shock propagation direction) a rigid reflecting boundary at x=x max to create the shock. (shock propagates in –x direction) Time evolution Mizuno et al in prep Preliminary result Density inhomogenity induces turbulent motion in shocked region Turbulence motion stretch and deform the magnetic field lines and create filamentary structure with strong field amplification.

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 Implementation of RAISHIN Resistivity (extension to non-ideal MHD; e.g., Watanabe & Yokoyama 2007; Komissarov 2007; etc ) 2 fluid MHD with resistivity (Zenitani et al. 2008) Couple with radiation transfer (link to observation: collaborative works with Dr. Wu) Kerr-Schild coordinates (to avoid singularity at BH radius in GRMHD simulations) Improve the realistic EOS Include Effect of radiation and neutrino emisson (cooling, heating) Include Nucleosysthesis (post processing) Couple with Einstein equation (dynamical spacetime) Adaptive mesh refinement (AMR) Parallerization by MPI for PC cluster type supercomputer Apply to astrophysical phenomena in which relativistic outflows/shocks and/or GR essential (AGNs, microquasars, neutron stars, and GRBs etc.)

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

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

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

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

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

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: u j = c Sound speeds: a j = a e = 0.4 c Surface mode: growth rates (dash-dotted lines) reduced as sheath speed increases from u e = 0 to 0.3 c. Resonance: disappears for sheath speed u e > 0.35 c Body mode: downwards arrows indicate damping peaks  R j /u j >> 1: damping for sheath speed u e > 0.5 c  R j /u j 0.5 c u e : sheath flow speed Black: u e = 0.0 c, Blue: u e = 0.2 c, Green: u e = 0.4 c, Red: u e = 0.6 c