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

• 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 e
B
=E mag
/E int
=10 -3 -10 -1 .
• 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)

•
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 J1713.7-3946 (Uchiyama et al. 2007)

• 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

Mizuno et al. 2006a, 2011c, & progress
• 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 G
-law and variable Equation of State (Synge-type)
• Parallel computing (based on OpenMP, MPI )

• Relativistic shock propagates in an inhomogeneous medium
• Density: mean rest-mass density ( r
0
= 1.0) + small fluctuations ( following 2D
Kolmogorov-like power-law spectrum, P ( k ) ∝ 1/[1+( kL ) 8/3 ], < dr 2
> 1/2 =0.012
r
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 ( perpendicular ( B y
• Boundary:
) to shock direction b
=p gas
/ p mag
=10 3 ), parallel ( B x
) or
– periodic boundary in y direction
– a rigid reflecting boundary at x=x max direction) to create a shock wave. (shock propagates in
–x
– 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

v x
=0.4c, B x case

Parallel shock
Perpendicular 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)

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

Parallel shock Perpendicular shock
Spherically integrated spectra in post shock region
Solid: t=4, Dashed: t=6
Dotted: t=8, Dash-dotted: t=10
• 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)

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.

x
Mean magnetic field in
Peak total magnetic field strength in postshock region 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

• 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