Magnetic Field Amplification by Turbulence in A Relativistic Shock Propagating through An Inhomogeneous Medium Yosuke Mizuno Institute of Astronomy National Tsing-Hua University Collaborators M. Pohl (Univ Potsdam), J. Niemiec (INP, PAN), B. Zhang (UNLV), K.-I. Nishikawa (NSSTC/UAH), P. E. Hardee (UA) Mizuno et al., 2011, ApJ, 726, 62

Introduction In Gamma-Ray Bursts (GRBs), radiation is produced in a relativistic blastwave shell propagating weakly magnetized medium. Detail studies of GRB spectrum and light curves show  B =E mag /E int = But, simple compressional amplification of weak pre-existing magnetic field can not account for such high magnetization (e.g., Gruzinov 2001). ⇒ Need magnetic field amplification process Leading hypothesis for field amplification in GRBs Microscopic plasma process –Relativistic Weibel instability (e.g., Medvedov & Loeb 1999, Spitkovsky 2008, Nishikawa et al. 2009) –But it remains unclear whether magnetic fields will persist at sufficient strength in the entire emission region (e.g., Waxman’s talk) In MHD (Macroscopic plasma process), relativistic magnetic turbulence (e.g., Sironi & Goodman 2007) –If preshock medium is strongly inhomogeneous, significant vorticity is produced in shock transition –Vorticity stretches and deforms magnetic field lines leading to its amplification Direct observational motivation for relativistic turbulence in GRB outflows –Significant angular fluctuation is invoked to explain large variation of gamma-ray luminosity in prompt emission (Relativistic turbulent model) (e.g., Narayan & Kumar 2009, Kumar & Narayan 2009; Lazar et al. 2009, Zhang & Fan 2010)

Introduction (cont.) Direct observational motivation for relativistic turbulence in GRB outflows –Significant angular fluctuation is invoked to explain large variation of gamma-ray luminosity in prompt emission (e.g., Narayan & Kumar 2009, Lazar et al. 2009) –Relativistic turbulent model is proposed as alternative model of well-known internal shock model –Zhang & Yan (2010) developed a new GRB prompt emission model in highly magnetized regime, internal- collision-induced magnetic reconnecton and turbulence (ICMART) model

Introduction (cont.) Fast variable flares (X-ray/TeV gamma) observed in blazars may come from small regions ~ a few Schwarzschild radii Marscher et al. (1992) proposed relativistic shock passes through turbulent jet plasma in the jet flow Synchrotron emission from Supernova remnant (SNRs) (expanding nonrelativistic spherical blast wave) is generally consistent with compression of interstellar magnetic field (~ a few micro-Gauss) However, year-scale variability in synchrotron X-ray emission of SNRs suggests to magnetic field amplification up to milli-Gauss level (e.g., Uchiyama et al. 2007) Magnetic field amplification beyond simple shock compression is necessary to achieve this level in SNRs Chandra X-ray image of western shell of SNR RX J (Uchiyama et al. 2007)

Interaction of SNRs with a Turbulent Magnetized ISM Recently, Giacalone & Jokipii (2007) performed non-relativistic MHD shock simulation including preshock density fluctuation and observed a strong magnetic field amplification caused by turbulence in postshock region Inoue et al. (2009) obtained strong magnetic field amplification by turbulence associated with a strong thermal instability driven shock wave propagating through inhomogeneous two-phase ISM Balsara et al. (2004) investigated role of magnetic field amplification in SN-driven turbulence via 3D MHD simulations Giacalone & Jokipii (2007) Inoue et al. (2009)

Dynamo Mechanism Turbulent dynamo (small-scale dynamo) also known as a fast dynamo, different from the classical mean-field dynamo mechanism Mean-field dynamo theory (e.g, Steenbeck et al. 1966; Parker 1971) has been used to study the growth of large-scale magnetic field in our Galaxy. In general, mean-field dynamos are slow, involve almost incompressible motions, and growth rate decreases with decreasing resistivity The theory of fast dynamos has been developed to investigate the rapid growth of magnetic field on smaller scales in the limit of infinite magnetic Reynolds number, and growth rate does not depend on resistivity (e.g., Childress & Gilbert 1995)

Turbulent Dynamo Amplification of magnetic field arises from sequences of “stretch, twist and fold” regime While R em =∞ is unrealistic in astrophysical simulations, some incompressible and compressible MHD simulations are nevertheless in agreement with the existence of small-scale turbulent-driven dynamo (e.g., Schekochihin et al. 2004; Balsara et al. 2004) Schematic images of stretch, twist and fold regime

Propose Recently, Giacalone & Jokipii (2007) performed non- relativistic MHD shock simulation including preshock density fluctuation and observed a strong magnetic field amplification caused by turbulence in postshock region A relativistic blast wave as in GRBs should experience strong magnetic field amplification by turbulence. In order to investigate magnetic field amplification by relativistic turbulence we perform 2D RMHD simulations of a relativistic shock wave propagating through a inhomogeneous medium

Propose Non-relativistic MHD shock simulations including preshock density fluctuation are shown a strong magnetic field amplification caused by turbulence in postshock region (e.g., Giacalone & Jokipii 2007) A relativistic blast wave as in GRBs, AGN jets should experience strong magnetic field amplification by turbulence. In order to investigate magnetic field amplification by relativistic turbulence we perform 2D Relativistic MHD simulations of a relativistic shock wave propagating through a inhomogeneous medium

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

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 function), convex ENO, PPM, WENO5, MP5, MPWENO5 * Riemann solver: HLL, HLLC, HLLD 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

RAISHIN Code (3DGRMHD) RAISHIN utilizes conservative, high-resolution shock capturing schemes (Godunov-type scheme) to solve the 3D GRMHD equations (metric is static) Ability of RAISHIN code Multi-dimension (1D, 2D, 3D) Special & General relativity (static metric) Different coordinates (RMHD: Cartesian, Cylindrical, Spherical and GRMHD: Boyer-Lindquist of non-rotating or rotating BH) Different schemes of numerical accuracy for numerical model (spatial reconstruction, approximate Riemann solver, constrained transport schemes, time advance, & inversion) Using constant  -law and variable Equation of State (Synge-type) Parallel computing (based on OpenMP, MPI) Mizuno et al. 2006a, 2011c, & progress

Initial Condition Relativistic shock propagates in an inhomogeneous medium Density: mean rest-mass density (  0 =1.0) + small fluctuations ( following 2D Kolmogorov-like power-law spectrum, P(k) ∝ 1/[1+(kL) 8/3 ], 1/2 =0.012  0 ) established across the whole simulation region (e.g., Giacalone & Jokipii 2007). Relativistic flow: v x =0.4c in whole simulation region Magnetic field: weak ordered field (  =p gas /p mag =10 3 ), parallel (B x ) or perpendicular (B y ) to shock direction Boundary: – periodic boundary in y direction –a rigid reflecting boundary at x=x max to create a shock wave. (shock propagates in –x direction) –new fluid continuously flows in from the inner boundary (x=0) and density fluctuations are advected with the flow speed Computational Domain: (x, y)=(2L, L) in 2D Cartesian with N/L=256 grid resolution Simulation method: WENO5 in reconstruction, HLL Approximate Riemann solver in numerical flux, and CT scheme for divergence-free magnetic field

Time Evolution: parallel shock case v x =0.4c, B x case

Time Evolution: perpendicular shock case v x =0.4c, B z case

Postshock Structure Parallel shockPerpendicular shock v x =0.4c, t=10.0 Density fluctuation in preshock medium induces turbulent motion in postshock region through a process similar to Richtmyer-Meshkov instability Since preexisting magnetic field is much weaker than the post shock turbulence, turbulence motion can easily stretch and deform the frozen-in magnetic field resulting in its distortion and amplification Amplified magnetic field evolves into a filamentary structure. This is consistent with previous non-relativistic work (Giacalone & Jokipii 2007; Inoue et al. 2009)

Magnetic Field Components Parallel shock Perpendicular shock BxBx BzBz BzBz v x =0.4c, t=10.0 Turbulent motion makes counterpart components of magnetic field (B x => B z, B z => B x ) and amplify the magnetic field. Perpendicular shock case, Bz component has little negative component because compression of magnetic field by shock makes large mean Bz component in shocked region. BxBx

1D Cross Section Profile Parallel shockPerpendicular shock v x =0.4c, t=10.0 Plot on z=1.0 Shock propagation speed v sh ~ 0.17c, relativistic Mach number M s ~4.9 (shock strength) Density jumps by nearly a factor of 4 Transverse velocity is strongly fluctuating (v y_max ~0.04c, ~0.02c) and subsonic ( ~0.32c), mostly super-Alfvenic ( ~ 0.002c). Total magnetic field strength is also strongly fluctuated and amplified locally more than 10 times. Shock front up downstream

Energy Spectrum Kinetic energy spectra almost follow a Kolmogorov spectrum (initial density spectrum still exists in postshock region) Magnetic energy spectra are almost flat and strongly deviate from a Kolmogorov spectrum. Flat magnetic energy spectrum is generally seen in turbulent dynamo simulation (e.g., Brandenburg 2001; Schekochihin et al. 2004). Same properties are also observed in super-Alfvenic driven turbulence (e.g., Cho & Lazarian 2003) and in RMHD turbulence induced by a KH instability (Zhang et al. 2009) Parallel shockPerpendicular shock Spherically integrated spectra in post shock region Solid: t=4, Dashed: t=6 Dotted: t=8, Dash-dotted: t=10

Small-scale dynamo System size: L, forcing scale l 0, l 0 << L Energy cascade from l 0 to viscous dissipation scale l via pure hydrodynamic system (Kolmogorov inner scale) l  >1 If magnetic field are present: dissipation scale l  l  >1 If weak magnetic field amplified by Kolmogorov turbulence, small-scale kinetic dynamo spreads magnetic energy over subviscous range (k >> k ) and pile up at resistive scale (Kulsrud & Anderson 1992)

Small-scale dynamo Consider turbulent velocity field with Kolmogorov spectrum Smaller eddies rotate faster Magnetic field most efficiently amplified by the smallest eddies in which it is frozen The size of such eddies are defined by registivity Sketch of scale range and energy spectra in large-Pr m medium B

Magnetic Field Amplification Mean magnetic field in postshock region Peak total magnetic field strength in postshock region Mean postshock magnetic field is gradually increasing with time and not saturated yet. Mean postshock magnetic field is stronger for perpendicular shock (B y ) than parallel shock (B x ) The perpendicular magnetic field is compressed by a factor of 3 as the shock, and additional magnetic field amplification by turbulent motion is almost same as for a parallel magnetic field Peak field strength is much larger than mean magnetic field.

Magnetic Field Amplification (fast flow case, v x =0.9c) Mean magnetic field in postshock region Peak total magnetic field strength in postshock region Mean postshock magnetic field is gradually increasing with time and not saturated yet. Mean postshock magnetic field is stronger for perpendicular case (B y ) than parallel case (B x ) The perpendicular magnetic field is compressed by a factor of 3 as the shock, and additional magnetic field amplification by turbulent motion Peak field strength is much larger than mean magnetic field In comparison with slow flow case (v x =0.4c), growth time is faster and magnetic field strength (mean and peak) is larger

Discussion: 2D vs 3D It is known that turbulence in 2D usually leads to intermittent structures of velocity and magnetic fields compared to that of 3D case due to inverse cascade of enstrophy (Biskamp 2008) Therefore, the evolution of turbulence and resulting magnetic field amplification should be verified by 3D simulations. Recently Inoue et al. (2011) have performed 3D RMHD simulations of MHD turbulence behind relativistic shock wave and obtained similar results what we have shown.

3D RMHD Simulation of Shock Propagations in inhomogeneous Medium 2D slice of density and magnetic field strength flow shock Maximum (line) Averaged (points) averaged Time evolution of magnetic field strength and energy density Inoue et al. (2011)

Discussion: Saturation of Magnetic Field Amplification In our simulations, mean magnetic field strength in postshock region is still increasing when the simulations were terminated Therefore, longer simulations with a longer simulation box are needed to follow the magnetic field amplification to saturation Recently, Inoue et al. (2011) have performed long- term evolution of turbulence and magnetic field using 3D RMHD simulations and followed the saturation of magnetic field amplification.

3D RMHD Simulations of turbulence Inoue et al. (2011) Slice of magnetic field strength Time evolution of magnetic field strength and energy density

3D RMHD Simulations of turbulence (cont.) Power spectra of velocity and magnetic field from 3D turbulent simulations Power spectra of velocity and magnetic field from 3D shock propagation simulations

Discussion: Magnetization GRB C showed a featureless smoothly joint broken power- law spectra covering 6-7 decades in energy (Abdo et al 2009) Zhang & Pe’er (2009) argued that non-detection of a thermal component strongly suggests that outflow is Poynting-flux dominated with  >15 (Poynting to kinetic energy ratio) If we consider internal shock in GRBs, magnetization in preshock medium can cover a wide range of values In stronger magnetized regime, turbulence is anisotropic and a different scaling in the direction along and perpendicular to the homogeneous field (Goldreich & Sridhar 1995; Cho & Lazarian 2003) Eddies of smaller scales are more stretched and appear elongated along magnetic field RMHD turbulence in highly-magnetized regime is not well studied

Summery We have performed 2D RMHD simulations of propagation of a relativistic shock through an inhomogeneous medium The postshock region becomes turbulent owing to the preshock density inhomogeneity Magnetic field is strongly amplified by the turbulent motion in postshock region The magnetic energy spectrum is flatter than Kolmogorov spectrum, which is typical for a small-scale dynamo The total magnetic field amplification from preshock value depends on the direction of homogeneous magnetic field The time scale of magnetic field growth depends on the shock strength The mean magnetic field strength in postshock region is still increasing. So longer simulations with a larger simulation box are needed to follow the magnetic field amplification to saturation

Fluid treatment Strong collision radius for Coulomb collision –r col ~ e 2 /kT ~ /T Comoving collision mean free path of electrons –L e,col = (n e  r 2 col ) -1 ~ cm In order to plasma in the collisional regime, l e,col <  (shell thickness) ~ cm Therefore GRB shock must be collisionless But GRB fluid stays together through magnetic interactions More relevant mean free path would be magnetic gyro-radius and plasma skin depth

Fluid treatment (cont.) Gyro(cyclotron) radii in the comoving frame –r B =  e m e c 2 /eB~ 1 cm (for electron), 10 3 cm (for proton) Comoving relativistic plasma skin depths –  =c/  p =  e m e c 2 /4  n e e 2 ~ 10 2 cm (for electron), 10 3 cm (for proton) For magnetized ejecta, plasma skin depths are relevant in the direction parallel to magnetic field lines Gyro-radii are more relevant in the direction perpendicular to the magnetic field lines Both scales are << shell thickness of GRB flow Therefore this justifies fluid treatment of the GRB plasma

Turbulent Model in GRBs Relativistic turbulent eddies have a characteristic Lorentz factor The light curve shows a peak when an eddy enters into the line of sight. It can produce significant variability The size of an eddy in comoving frame, R e ~ f 1/a R/  ’, f  filling factor,  ’: random Lorentz factor of an energy bearing eddy Lazar et al. (2009), 2< a < 3 Narayan & Kumar (2009): R e ~R/  ’ turbulence sub-jets

Shock quantities Preshock region –Sound speed ~ 0.128c –Alfven velocity ~ c Postshock region Slow velocity case (v x =0.4c) –Turbulent velocity ~0.02c –Sound speed ~ 0.32c (B x & B y ) –Alfven velocity ~ 0.012c (B x ) & c (B y ) –Shock strength M s ~ 4.9 –Shock propagation speed v sh ~ 0.171c

Postshock Structure (v 0 =0.9c) Parallel shock Perpendicular shock v x =0.9c, t=4.4

1D Cross Section Profile (v 0 =0.9c)

Energy Spectrum

3D RMHD simulations of KH Instability Time evolution of ratio of kinetic and magnetic energy to total energy kinetic magnetic Zhang & MacFadyen (2009) Spectrum kinetic magnetic