3D GRMHD code RAISHIN and Relativistic Jet Simulations

Yosuke MizunoCenter for Space Plasma and Aeronomic Research

University of Alabama in Huntsville

CFD-MHD seminar, ASIAA, Taiwan, 3/31/10

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

Requirement 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

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/0609004Mizuno 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 & Hawle

y 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 MethodLinear wave equation Hydrodynamic equations

are a set of wave equationsFinite difference in each term

df/dx as a function of fj+1,n fj,n fj-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: FU= 0

• metric ：　 ds2=-2 dt2+gij (dxi+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,ishift vector, gij : 3-metric

T : energy momentum tensor, T = pg +h UU+FF -gF

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) Fi (numerical flux) S (source term)

√-g : determinant of 4-metric√ : determinant of 3-metric

Detail of derivation of GRMHD equationsAnton et al. (2005) etc.

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-C

T* 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/0609004 and progress

ReconstructionCell-centered variables (Pi)→ right and left side of Cell-interface variables(PL

i+1/2, PRi+1/2)

PLi+1/2 PR

i+1/2

Piecewise linear interpolation

• Minmod and MC Slope-limited Piecewise linear Method

• 2nd order at smooth region• Convex CENO (Liu & Osher 1998)

• 3rd order at smooth region• Piecewise Parabolic Method (Marti & Muller 1996)

• 4th order at smooth region• Weighted ENO5 (Jiang & Shu 1996)

• 5th order at smooth region• Monotonicity Preserving (Suresh & Huynh 1997)

• 5th order at smooth region• MPWENO5 (Balsara & Shu 2000)

Pni

Pni+1

Pni-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

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 SolverMignore & 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)

HLL HLLC

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-1/2 j+1/2

k+1/2

k-1/2

２ D case Toth (2000)

Time evolution System of Conservation Equations

We use multistep TVD Runge-Kutta method for time advance of conservation equations (RK2: 2nd-order, RK3: 3rd-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 fro

m 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≡h2 and v2 by using 2-variable Newton-Raphson iteration method

W and v2 →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/K2, K3: 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

Ability of RAISHIN code (current status)

• Multi-dimension (1D, 2D, 3D)• Special and General relativity (static metric)• Different coordinates (RMHD: Cartesian, Cylindrical, Spherical a

nd 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 TestsBalsara 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

FR

SR

CD

SS

Mizuno et al. 2006

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/WindDisk 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)vj

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)

ue=0.0 ue=0.0ue=0.5c ue=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)

– Vj=0.2c, Rj=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=xmax to create the shock. (shock propagates in –x direction)

Time evolution

Mizuno et al. 2010 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.

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 w

orks with Dr. Wu) • Kerr-Schild coordinates (to avoid singularity at BH radius in GRM

HD 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.)